Paper Title: SPECtrophotometer for TRansmission spectroscopy of exoplanets (SPECTR).







0009-0005-5145-5165]Yeon-Ho Choi
Department of Astronomy and Space Science, University of Science and Technology, Korea, 217 Gajeong-ro, Yuseong-gu, Daejeon 34113, Republic of Korea
Korea Astronomy and Space Science Institute,
776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea

0000-0003-1544-8556]Myeong-Gu Park
Department of Astronomy and Atmospheric Sciences, Kyungpook National University, 80 Daehak-ro, Buk-gu, Daegu 41566, Republic of Korea


Korea Astronomy and Space Science Institute,
776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea

0000-0001-8969-0009]Jae-Rim Koo
Earth Environment Research Center, Kongju National University, 56 Gongjudaehak-ro, Gongju-si, Chungcheongnam-do 32588, Republic of Korea

0009-0008-2592-9437]Tae-Yang Bang
Department of Astronomy and Atmospheric Sciences, Kyungpook National University, 80 Daehak-ro, Buk-gu, Daegu 41566, Republic of Korea


Korea Astronomy and Space Science Institute,
776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea


Korea Astronomy and Space Science Institute,
776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea


Korea Astronomy and Space Science Institute,
776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea


Korea Astronomy and Space Science Institute,
776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea


Division of Energy and Optical Technology Convergence, Cheongju University, 298 Daeseong-ro, Cheongwon-gu, Cheongju 28371, Republic of Korea


Korea Astronomy and Space Science Institute,
776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea




Department of Astronomy and Space Science, University of Science and Technology, 217 Gajeong-ro, Yuseong-gu, Daejeon 34113, Republic of Korea
Korea Astronomy and Space Science Institute,
776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea


Byeong-Cheol Lee
bclee@kasi.re.kr


§ ABSTRACT 

The SPECtrophotometer for TRansmission spectroscopy of exoplanets (SPECTR) is a new low-resolution optical (3800 Å - 6850 Å) spectrophotometer installed at the Bohyunsan Optical Astronomy Observatory (BOAO) 1.8 m telescope. SPECTR is designed for observing the transmission spectra of transiting exoplanets. Unique features of SPECTR are its long slit length of 10 arcminutes which facilitates observing the target and the comparison star simultaneously, and its wide slit width to minimize slit losses. SPECTR will be used to survey exoplanets, such as those identified by the Transiting Exoplanet Survey Satellite (TESS), providing information about their radii across the wavelength range. In this paper, we present the design of SPECTR and the observational results of the partial transit of HD 189733 b and a full transit of Qatar8 b. Analyses show the SPECTR's capability on the white light curves with an accuracy of one ppt. The transmission spectrum of HD 189733 b shows general agreement with previous studies. 




































§ INTRODUCTION
 


The Kepler/K2 and the TESS space missions so far have confirmed about 3,800 exoplanet candidates, 67% of which are confirmed exoplanets[https://exoplanetarchive.ipac.caltech.edu/]. 
There are about 10,000 candidates yet to be confirmed. While these successful missions have discovered a substantial number of exoplanets, further follow-up observations, such as precise radial velocity measurements, are still necessary to determine their masses. Furthermore, there has been a growing importance placed on studying the atmospheric characteristics of exoplanets, such as atmospheric composition and the presence of clouds using transmission and thermal emission spectroscopy methods.

In particular, transmission spectroscopy, employed for transiting exoplanets, stands out as an exceptional tool for investigating the upper atmospheres of these planets. This method has been widely used with both space- and ground-based telescopes over a range of spectral resolutions. Space telescopes such as the Hubble Space Telescope <cit.> and the JWST <cit.>, along with ground-based telescopes such as the Very Large Telescope <cit.>, the Gemini Telescope <cit.>, and the Keck Telescope <cit.>, and programs such as the ACCESS program with the Magellan Telescope <cit.> and the LRG-BEASTS program with the William Herschel Telescope <cit.> have successfully studied the atmospheric characteristics of exoplanets. Until recently, such studies were limited to space- and ground-based large telescopes due to the faintness of the stars.

However, TESS has been particularly effective in discovering exoplanets orbiting bright stars. The atmospheres of these exoplanets can be studied using smaller telescopes like the BOAO 1.8 m telescope. This is because the 1.8 m telescope at BOAO, with its limited collecting area, is best suited for observing bright stars, which are also the primary targets of TESS due to its own small telescope size.

With these opportunities, we aimed to develop a spectrophotometer capable of studying the atmospheric characteristics of the TESS targets with the Bohyunsan Optical Astronomy Observatory (BOAO) 1.8 m telescope located in South Korea.

Korea Astronomy and Space Science Institute (KASI) and Kyungpook National University (KNU) initiated the development of the SPECtrophotometer for TRansmission spectroscopy of exoplanets (SPECTR). 
Considering the given situation and budget, we concluded that a long-slit spectrophotometer type is most suitable for studying the atmospheric characteristics of exoplanets at BOAO. While a long-slit spectrograph was operating at BOAO, it was not suitable for performing transmission spectroscopy due to its small field of view (FOV, 3.3× 3.3) and the absence of an imaging mode. The FOV determines the likelihood of finding a proper comparison star, necessary to perform precision differential spectrophotometry from the ground, due to the variability of Earth's atmosphere. Also, an imaging mode is required to confirm the position angle between the target and the comparison star to ensure that both stars are located within the same slit. 

In this paper, we describe the optical design of SPECTR and its first observational results.



§ SPECTR DESIGN AND SYSTEM


    

 §.§ Optical design

    SPECTR is designed to fit into the BOao Echelle Spectrograph (BOES) Cassegrain Interface Module <cit.>. 
    The CIM, attached to the Cassegrain focus of the BOAO 1.8 m telescope, includes a slit assembly, a slit-monitoring system, and a calibration lamp system. 
    The CIM is designed to facilitate high-resolution observation with BOES and medium-resolution observations with the Long Slit Spectrograph (LSS). BOES is a fiber-fed echelle spectrograph capable of providing precise radial velocity measurements <cit.>. BOES covers a spectral range from 3,500 Å to 10,100 Å. Its spectral resolution varies from 30,000 to 90,000, depending on the fiber diameter, which ranges from 80 microns to 300 microns.
    
    The original design of SPECTR had a straight optical path to enable the imaging mode utilizing the volume phase holographic grating (VPHG). 
    
    However, due to the limited space and compartments formed by walls inside the CIM as shown in Figure <ref>, it was changed to an L-shaped design using a flat mirror. Figure <ref> shows the optical layout of SPECTR. SPECTR consists of a slit assembly, a collimator assembly, a grism assembly, and an imaging assembly.

    The optical design of an L-shape can cause an imbalance at the center of mass in the CIM. To address this, we conducted simulations to test the amount of deflection due to changing gravity. We confirmed that the maximum deflection of the outer part of SPECTR at 1 g was 22 microns, which corresponded to a 2.6 Å shift in the spectral axis. Given that we use 400 Å bins for the spectroscopic light curves, this 2.6 Å shift is minor and will not severely affect the results.
    
    
    
    
    
    
    

 §.§ Slit assembly

    SPECTR has a wavelength coverage from 3,800 Å to 6,850 Å and a spectral resolution (R) ranging from 200 to 1,300 depending on the slit width. Details of the spectral resolutions for each slit are listed in Table <ref>.
    SPECTR contains eight slits of different widths, in addition to one pinhole for testing and one blank space for imaging to check the positions of the target and the comparison stars. Slit #1 is mainly used for taking the spectrum from the FeAr lamp for wavelength calibration and slits from #2 to #4 for general long-slit spectroscopy. Since transit observations require minimizing the light loss which may arise due to a finite slit width, we use slits from #5 to #8, much wider than the typical seeing at BOAO. 
    
    The slit plates are made of stainless steel (STS), measuring 60 mm × 22 mm, and they are coated with SiO_2 after metal polishing. The slits are manufactured by wire electric discharge machining and have the same physical length of 42 mm, which corresponds to 10 arcminutes in the FOV. 
    The blank space has the physical dimensions of 46 mm × 20 mm.
    
    
    
    
    In ground-based transmission spectroscopy, it is essential to observe a comparison star simultaneously with the target to correct for the effects of the changing of the Earth's atmosphere. A longer slit length will increase the probability of finding a suitable comparison star for the target. The LSS is already installed for low-resolution spectroscopy, but its FOV covers only one-third of that of SPECTR, limiting its capability to capture comparison stars simultaneously.
    In addition, the slit of SPECTR can be rotated to match the position angle of the two stars.
    To arrive at the optimal length of the slit within the optical and physical limitations of the telescope and the CIM, we estimated the number of suitable comparison stars available within a given FOV using the TESS Input Catalog <cit.>. During test observations, we found that a proper comparison star should have a difference in photometric magnitude of at most 1 magnitude in the V band and have colors or spectral types similar to the target. The TIC provides TESS magnitudes for about 471 million bright stars. Moreover, TESS provides light curves of stars, that can be used to check the photometric variability of potential comparison stars.
    Figure <ref> shows the number of stars with magnitudes within the interval of 1 magnitude for slit FOVs of 3, 5, and 10 arcminutes.
    
        
    
    
    
    

 §.§ Collimator assembly

    A collimator with an effective focal length of 280 mm was designed to change the f/8 beam from the 1.8 m telescope to a 35 mm diameter at the SPECTR grism. The collimator consists of two symmetric achromatic doublet lenses, which collimate the incident light towards the grism. The doublet lenses are fixed in place using two spacers, axial springs, radial springs, and retainers.
    

 §.§ Grism assembly

    The collimated beam is reflected by the folded mirror toward the grism and dispersed toward the imaging lens. The folded mirror has a physical dimension of 94 mm × 66 mm and is coated with enhanced aluminum for durability and reflectance with SPECTR's wavelength coverage.

    The grism assembly houses the grism. The grism consists of a VPHG and a wedge prism. The detailed parameters of the VPHG are given in Table <ref>. 
    
    It is manufactured by Wasatch Photonics, USA[https://wasatchphotonics.com/]. Two N-BK7 wedge prisms with an apex angle of 25.3 are glued to either side of the VPHG which together produces a straight optical path at 5261 Å. The grism is installed on the X-stage with a step motor and therefore can be moved the grsim away from the light path for the imaging mode.
       
    
    
    
    
    
    

 §.§ Imaging assembly

    We selected a Canon RF 85 mm f/1.2 lens as the imaging lens. The lens has excellent imaging capability despite its small form factor and large FOV. The imaging lens is backed by Andor iKon-L 936 CCD, which serves as the CCD for SPECTR. 
    This Peltier-cooled CCD has a pixel array of 2048 × 2048 and a pixel size of 13.5  × 13.5 . The designed FOV in imaging mode is 10× 4.6 with a plate scale of 0.62/pixel. The gain and readout noise are 2.3 e^-/ADU and 9 e^-, respectively. 
    
    

    
    

 §.§ Slit monitoring and lamp systems of CIM

    SPECTR installed on the CIM of the BOAO 1.8 m telescope is shown in Figure <ref>.
    We designed SPECTR in a cartridge form factor, which can be installed and removed easily, and has compatibility with existing systems. Such a design has three advantages. First, it allows us to utilize the observing time before and after the transit event. 
    Transit observations typically require only a fraction of a night. Hence, a simple procedure to exchange SPECTR with BOES allows efficient usage of the allocated time.
    
    Second, SPECTR can utilize the slit-monitoring and auto-guiding systems within the CIM. The FOVs of each system are 3 × 3 and 40 × 40, respectively. Precise guiding is crucial for transit observations lasting several hours to ensure that the whole point spread functions of both the target and the comparison star remain within the slit to avoid slit losses in the presence of changing seeing conditions.
    Although the slit-monitoring system has a limited FOV, it provides more precise guidance compared to the auto-guiding system. We utilize the auto-guiding system if no adequate star for guiding is available within this FOV.
    We checked the performance of each guiding system during test observations. The observations with the slit-monitoring system showed gradual positional drifts of approximately 1.8 over three hours, while the auto-guiding system showed drifts of approximately 3.6 shifts over four hours. 
    Third, it allows us to utilize the FeAr wavelength calibration lamp system in the CIM. Although SPECTR is designed to be integrated with the CIM, the existing calibration lamp system was not fully optimized for the SPECTR. When the calibration lamp is set to the optical axis, it only covers 50 which is 8% of the FOV of the SPECTR (red lines in Figure <ref>). We modified the lamp system so that the lamp along the spatial axis covers the whole FOV of SPECTR. Figure <ref> shows the resulting. The narrowest #1 slit width of 1.4 (R ∼ 1300) was selected to identify the wavelength solution and the rms value of the residuals after fitting was about 0.1 Å.
    
    
    





§ SYSTEM PERFORMANCE

We tested the performance of SPECTR by observing transiting close-in exoplanets. The first test target was the well-known exoplanet HD 189733 b <cit.>. 
The other was Qatar8 b, discovered by the Qatar Exoplanet Survey <cit.>. Our main purpose of the test was to validate the spectrophotometric stability and accuracy of SPECTR by comparing our results with existing spectroscopic light curves.


    

 §.§ Observations

    
    We observed with SPECTR one partial transit of HD 189733 b, starting at ingress, and one full transit of Qatar8 b. A summary of the observations is provided in Table <ref>. Observations were carried out with the #7 slit, which has a width of 15, to minimize slit losses.
    Although the spectral resolution R corresponding to this slit width is approximately 150, the effective spectral resolution is determined by the seeing when the seeing is smaller than the slit width. The mean values of seeing for each observation were about 1.5, resulting in a mean spectral resolution of approximately 1200.
    The auto-guiding system was used during the HD 189733 b observations, while the slit-monitoring system was used during the Qatar8 b observations. All observations were carried out with the target and a comparison star simultaneously within the slit. We collected calibration data such as bias, dark, wavelength calibration, and sky flat in both observations.
    
    The partial transit of HD 189733 b was observed on October 14, 2021. The airmass changed during the observations from 1.03 to 1.61. We selected the comparison star HD 345459 from <cit.>. This comparison star had been chosen for photometry and spectrophotometry in <cit.> and <cit.>. The target and the comparison stars have similar spectral types and magnitudes, K2V with V = 7.64 and K0V with V = 8.09, respectively <cit.>. A timeseries of 248 spectra of HD 189733 and the comparison star were obtained during in and out of transit with an exposure time of 15 seconds per frame. The cadence of the observations was 20 seconds, with five seconds of readout time, which was fixed throughout the entire test observation.

    One full transit of Qatar8 b was observed on April 8, 2022. The airmass changed during the observations from 1.22 to 1.81. Qatar8 is known to be a G0 V star <cit.>. Qatar8 is much fainter than HD 189733 and was chosen to test the spectrophotometric stability of SPECTR for a fainter system. We chose the comparison star TYC 4387-01409-1 (also known as TIC 103753216) from TIC v8 <cit.>. Although the spectral type of the comparison star is not known, we selected it based on its (B-V) color and checked its photometric variability from TESS light curves. The  () light curves of TYC 4387-01409-1 in TESS Sectors 40, 41, and 47 appear stable at the rms = 0.8 ppt across all sectors, as shown in Figure <ref>. 
    The target and the comparison star have similar magnitudes and colors, V = 11.71 with B-V = 0.534 and V = 11.65 with B-V = 0.573. A timeseries of 78 spectra of Qatar8 and the comparison star were obtained in and out of transit with each exposure time of 300 seconds without readout time.


    

    
    
 
    

 §.§ Data reduction
 

    Data reduction was performed with standard IRAF <cit.> procedures for bias, dark subtraction, illumination correction, aperture tracing, and wavelength calibration. From flat images of SPECTR, we found that uneven illumination does not significantly affect the results if the angular separation between the target and the comparison star is smaller than 7 arcminutes. For the HD 189733 b observations, the angular separation between the target and the comparison star was 8.72 arcminutes. Although we applied illumination correction to both targets, only the results for the HD 189733 b showed a significant difference.

    Aperture tracing and extraction were performed using the onedspec/apextract package. To confirm the aperture radii for both targets, we incrementally increased the radii from 3 to 15 pixels and extracted the aperture at each step. In this procedure, we fixed the background region as the pixel ranges from -30 to -20 and from 20 to 30 from the center to maintain consistency. Then we constructed white light curves of each aperture radius and compared the standard deviations of the out-of-transit light curves. The lowest scatters were found when we adopted aperture radii of 5.5 pixels for the HD 189733 and 11 pixels for the Qatar8 observations. These aperture radii were fixed for the subsequent analysis. The difference in aperture radii between the two observations seems to arise from different seeing conditions.

    Despite guiding, there were shifts along the spectral axis during the observations. We aligned the spectral axis of all spectra by comparing each spectrum with the reference spectrum using  function. Figure <ref> shows the pixel shifts over time in both spatial and spectral axes. Given that the integration time for Qatar8 b observations was approximately twice that of HD 189733 b observations, guiding with the slit-monitoring system appears significantly more effective. However, it is important to note that various confounding factors, such as atmospheric turbulence, the gravitational shifts depending on the position angle, and the position of the target in the sky, can impact guiding precision.
     
    Figure <ref> shows the extracted sample spectra of HD 189733 and Qatar8, along with those of the comparison stars. Qatar8 and its comparison star show quite similar spectral shapes, albeit with slightly different reported temperatures. The passbands of 400 Å width, used to construct seven spectroscopic light curves, are indicated by grayed vertical regions.

    


    
    










    

 §.§ White and spectroscopic light curves

    We integrated the flux across 4000 to 6800 Å to derive the white light curves for both targets, excluding wavelengths below 4000 Å  due to low SNRs.
    The white light curve of each target was obtained by dividing the integrated flux of the target by that of the comparison star. This removes the variations caused by the changing Earth's atmosphere as much as possible. Figure <ref> shows the resulting white light curves, illustrating the observed flux variations during the transit events of HD 189733 b and Qatar8 b, with model fits overlaying the data.
    
    We constructed the spectroscopic light curves by dividing the time-series spectra using uniform bins of 400 Å width.  
    Seven channels were extracted from 4,000 Å to 6,800 Å for each target and the light curve fits with the data from each channel are shown in Figure <ref>. 
    Spectroscopic light curves of both HD 189733 b and Qatar8 b show similar shapes to the white light curves of each target before detrending. Since the systematics appear consistent across wavelength bins, we detrended the white light curves and the spectroscopic light curves in a similar manner.


    

 §.§ Light curve fitting


    We fitted the transit light curves using a quadratic limb-darkening transit model <cit.> implemented in  <cit.>. 
    Given that only a single transit event was observed for each target, we adopted the orbital parameters such as the period, semi-major axis, and inclination from previous studies <cit.>.
    Mid-transit time, planet-to-star radius ratio R_ p/R_ *, limb-darkening coefficients, and baseline trend were modeled using .
    
    As shown in the middle panels of Figure <ref>, the relative fluxes show both downward and upward parabolic trends. 
    To account for these trends over time, we applied the Gaussian Process (GP) with a Matérn 3/2 kernel, implemented in  <cit.>.
    The transit parameters and baseline trends were determined using a Bayesian framework.
    The baseline trends were derived by dividing the fluxes by the transit models and subsequently fitting them with the GP.
    
    We set a normal prior for all parameters, except for R_ p/R_ *, which had a uniform prior range between 0.01 and 0.2.
    The mid-transit time prior was set using a normal distribution with a standard deviation of 0.01 days. For HD 189733 b, the center value was 1.96368 days, and for Qatar8 b, the center value was 178.125434 days, both referenced to BJD - 2,459,500 days, and the limb-darkening coefficients were calculated using the Python Limb Darkening Tool Kit <cit.> based on literature values for stellar parameters (, , and ) listed in Table <ref>. These values were used as means and standard deviations for the normal priors. 
    
    Optimal parameters were initially found using the differential evolution optimization method <cit.> implemented in . This was followed by MCMC sampling with  <cit.> to estimate the final parameters while fixing the baseline trends.
    The mid-transit times for each object were fixed for the spectroscopic light curves based on the white light curve estimates.
    The limb-darkening coefficients remained consistent with the  calculations after parameter estimation. 


    The best-fit models for the white light curves of HD 189733 b and Qatar8 b are shown in Figure <ref>. 
    The summary of the best-fit and fixed parameters is listed in Table <ref>. 
    The rms values of the residuals after the light curve model fitting are 0.969 ppt for HD 189733 b and 0.158 ppt for Qatar8 b. 

    We tabulate the spectral radius ratio from the spectroscopic light curve fits in Table <ref>. Our transmission spectra along with those of <cit.> and <cit.> are shown in Figure <ref>. The R_ p/R_ * values for HD 189733 b determined in this work show a general agreement with those reported in previous studies, although two points deviate significantly from the general trend. This discrepancy may have been caused by the effects of the absence of data before ingress. 
    For Qatar8 b, the R_ p/R_ * values show a relatively smooth trend across the wavelengths with small increases in the 5800 Å and 6200 Å bins. 
    By additional monitoring of these targets, we aim to confirm whether these anomalies are indeed real. The mean rms values of the spectroscopic light curve fit residuals are 1.060 ppt for HD 189733 b and 0.843 ppt for Qatar8 b. 


    To illustrate how sensitive our transmission spectra are to potential atmospheric features in these two planets, we calculated the atmospheric scale heights of our targets using the equation from <cit.>, under the assumption of hydrogen-dominated atmospheres with a mean molecular weight (μ_m=2.37). The atmospheric temperatures were taken to be equal to the equilibrium temperatures of the planets, and we used the known radii and masses of the planets. For HD 189733 b (R_ * = 0.776 R_ Sun from <cit.>), the calculated scale height is 192 km, which corresponds to 0.0004 R_ p/R_ *. For Qatar8 b (R_ * = 1.315 R_ Sun from <cit.>), the calculated scale height is 911 km, which corresponds to 0.0010 R_ p/R_ *. Figure <ref> shows the amplitudes that would correspond to ± 5 scale heights with respect to the median of the transmission spectrum for each planet. This illustrates that SPECTR is sensitive to that range of atmospheric heights
    
    
    
     



     



    
    


    







§ SUMMARY



We present the instrumental design and spectrophotometric performance of the SPECtrophotometer for TRansmission spectroscopy of exoplanets (SPECTR) to be used with the BOAO 1.8 m telescope.
With its 10 long-slit capability and spectrophotometrical stability, SPECTR will provide observational opportunities for 1 – 2 m class telescopes to contribute to the understanding of exoplanetary atmospheres, particularly those transiting bright stars. While many of these targets will be discovered by TESS, others will come from various surveys and instruments. 
The test observations showed that SPECTR can obtain precise spectroscopic light curves with a mean rms precision of 0.951 ppt. Although the analysis of the HD 189733 b and Qatar8 b can be further improved, our preliminary results highlight the potential of SPECTR to perform atmospheric characterization of transiting exoplanets.
As the importance of exoplanet atmospheric research is increasing along with the increasing number of bright exoplanet systems, we hope to make a meaningful contribution to the characterization of exoplanets at visible wavelengths with the SPECTR and the BOAO 1.8 m telescope.








§ ACKNOWLEDGMENTS


We would like to express our sincere gratitude to the referee for their extensive valuable and insightful comments, which have greatly improved the quality and clarity of this paper. This paper is based on observations collected with the BOAO 1.8 m telescope. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program.
B.-C. Lee acknowledges partial support by the KASI (Korea Astronomy and Space Science Institute) grant 2024-1-832-03 and acknowledges support by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No-2021R1A2C1009501).
M.-G. Park was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No-2019R1I1A3A02062242 and No-2018R1A6A1A06024970) and KASI under the R&D program supervised by the Ministry of Science, ICT and Future Planning.
J.-R. Koo was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT; grant No-2022R1C1C2004102)


Software:  <cit.>,   <cit.>,  <cit.>,  <cit.>



aasjournal




 Paper Title: FCN4Flare: Fully Convolution Neural Networks for Flare Detection.



A-Li Luo
* lal@nao.cas.cn 
0009-0001-0257-3377]Ming-Hui JIa
CAS Key Laboratory of Optical Astronomy, National Astronomical Observatories, Beijing 100101, China
University of Chinese Academy of Sciences, Beijing 100049, China
0000-0001-7865-2648]A-Li Luo^*
CAS Key Laboratory of Optical Astronomy, National Astronomical Observatories, Beijing 100101, China
University of Chinese Academy of Sciences, Beijing 100049, China
Nanjing Institute of Astronomy and Optical Technology, University of Chinese Academy of Sciences, Nanjing 211135, China

University of Science and Technology Beijing,Beijing, 100083, People's Republic of China


§ ABSTRACT  
Stellar flares originate from the sudden reconnection of magnetic flux lines within the star's atmosphere, particularly in the chromosphere. Automated flare detection enables exploiting vast photometric datasets from missions like Kepler. Prior methods rely on outlier detection, facing challenges of complexity, detection accuracy, and scalability. This paper presents FCN4Flare, a deep learning approach using fully convolutional networks for precise point-to-point flare prediction regardless of light curve length. Key innovations include the NaN Mask to handle missing data, dilated convolutions to preserve local information, and the MaskDice loss to mitigate severe class imbalance. Experiments demonstrate significantly improved detection performance over previous models, with a 0.64 Dice coefficient on Kepler data. Applying FCN4Flare to Kepler and LAMOST, we compile a catalog of 30,285 high-confidence flares across 1426 stars. Flare energies are estimated and stellar/exoplanet properties analyzed, identifying pronounced activity for an M-dwarf hosting a habitable zone planet. This work overcomes limitations of prior flare detection methods via deep learning, enabling new scientific discoveries through analysis of photometric time-series data.






§ INTRODUCTION

Stellar flares may arise from the magnetic reconnection <cit.>. They offer profound revelations into the enigmatic magnetic dynamo processes transpiring deep within stellar cores, as well as the intricate mechanisms orchestrating stellar activity cycles <cit.>. By meticulously scrutinizing the distinctive properties of these flares, we gain invaluable insights into the intricate nature of stars themselves. Moreover, beyond their significance in comprehending stellar phenomena, flares emerge as pivotal agents molding the genesis and evolution of exoplanets <cit.>. The discernment of stellar flare occurrences thus assumes a critical role in our endeavors to assess the potential habitability of exoplanets <cit.>.

In the contemporary astronomical landscape, the advent of photometric space missions, exemplified by the likes of Kepler <cit.> and TESS <cit.>, has ushered in unprecedented opportunities for astronomers to delve into the study of stellar flares. These missions have facilitated the acquisition of vast troves of light curves, presenting us with an exhilarating prospect. However, the sheer magnitude of this data also presents us with a novel challenge: the development of efficient methods for automated flare detection.

To address this challenge, various methodologies, predominantly anchored in outlier detection, have been meticulously crafted <cit.>. These methods adroitly follow a systematic three-step procedure:

    
  (1) In the initial stage, a meticulous fitting of the quiescent flux or baseline is accomplished through the skillful deployment of a prewhitening procedure, complemented by an iteratively smooth filter. The combined effect of these techniques is to deftly eliminate extraneous noise and spurious data points, thus laying the foundation for a pristine analysis.
    
  (2) With the quiescent flux duly established, stringent criteria, encompassing amplitude, profile, and duration, are artfully applied to the candidate outliers, artfully winnowing out the most promising flare events. These selection criteria typically involve setting a threshold on the amplitude, mandating an impulsive surge followed by an exponential decay in the flare profile, and considering the temporal extent of the event.
    
  (3) The results are validated using mathematical tests or simulations to establish selection criteria and assess errors. 


To provide concrete instances, <cit.> adroitly employed an iterative light curve detrending approach to effectively segregate intrinsic variability from potential flares. Flare candidates were duly identified as instances where consecutive points surpassed 2.5 σ above the local mean flux for a sustained period. Subsequently, manual confirmation of flares ensued to ensure precision. In a similar vein, <cit.> meticulously employed thresholds governing the intensity increase and duration of the flares, stipulating a minimum of 3 consecutive points exhibiting an intensity above 3σ, accompanied by a rapid flux augmentation. Flares, once again, were judiciously validated through manual inspection. <cit.> adeptly resorted to iterative light curve detrending to model the quiescent brightness. Subsequently, flare candidates were discerningly extracted as outliers from the quiescent model, courtesy of a matched filter approach featuring an analytical flare template. <cit.>, meanwhile, adroitly detrended light curves via the proficient application of Fourier series fitting, mandating that flare candidates exceed a 3σ threshold for a minimum duration of 3 data points.

Undoubtedly, the method's utilization of traditional outlier detection comes with two primary limitations, impeding its broad application in large-scale astronomical data analysis. Firstly, the detection process's inherent complexity necessitates continuous iteration, leading to protracted computational times. Such extended processing periods can hinder timely analyses of extensive datasets, making it challenging to keep up with the ever-expanding volume of astronomical information <cit.>. Secondly, to effectively suppress the false positive sample rate, the methods rely on the development of highly detailed and stringent criteria. As observed in the study conducted by <cit.>, the earlier flare catalogs suffered from contamination by various false-positive signals and artifacts. Consequently, more rigorous criteria were introduced by <cit.> to discern flares on Kepler light curves. While stricter evaluation criteria enhance the accuracy of flare detection, they introduce further intricacy to an already involved detection process, exacerbating the challenges faced in analyzing large datasets.

The recent application of deep neural networks in detecting stellar flares has shown considerable promise. Notably, Stella <cit.> and Flatwrm2 <cit.> have revealed the effectiveness of using convolutional and recurrent networks to determine whether the center of a fixed-length segment of light curves indicates a flare event. By utilizing a sliding window method, these networks can accurately predict flare occurrences throughout a light curve.

The key advantage of employing deep neural networks lies in their inherent capacity for highly parallelizable computations. This attribute allows for expedient processing of substantial datasets, enabling the rapid and efficient detection of flares on a large scale <cit.>. The ability to analyze copious data points in parallel not only accelerates the identification process but also contributes to the overall scalability of the method. Consequently, these deep learning-based approaches hold great promise in tackling the challenges posed by the vast amounts of astronomical data acquired through modern photometric space missions like Kepler and TESS <cit.>.

However, the precision of these methods still need to be improved. Additionally, their use of a fixed sliding window to detect flares in an entire light curve is not efficient enough for large-scale datasets.

Drawing inspiration from the groundbreaking work of <cit.>, our endeavor is to accomplish point-to-point dense prediction, eschewing the sliding window approach. In this paper, we present a fully convolutional network designed for the purpose of detecting stellar flares—henceforth referred to as FCN4Flare. This approach holds the potential to revolutionize the domain of flare detection, propelling it towards unprecedented levels of accuracy and efficiency.

In the following article, Section <ref> will introduce the details of FCN4Flare, Section <ref> will discuss the model performance, Section <ref> will explore the practical applications of our approach, and Section <ref> is the conclusion.



§ FCN4FLARE





The crux of our task lies in predicting the probability that each flux value in a given light curve belongs to a flare event. To accomplish this, we employ neural networks denoted by f(X|Θ), where X={x_i ∈ℝ | i=1, 2, ⋯, T} represents the input light curve with a length of T. Our objective is to generate a probability vector P={ p_i ∈ [0, 1] | i=1, 2, ⋯, T }, where p_i signifies the likelihood that the flux value x_i is part of a flare event.

A notable feature of our approach is its flexibility regarding input length. Unlike recent methods that utilize equal-length segments of the light curve, generated by sliding a window point by point along the curve, our method can handle light curves of varying lengths. This adaptability allows us to analyze and predict flare probabilities effectively for light curves of different durations, which is particularly advantageous given the inherent variability in astronomical data.

By leveraging the power of neural networks and their capability to discern intricate patterns and relationships within data, our method aims to yield accurate and reliable predictions regarding the likelihood of flare events at individual flux values. With the ability to process light curves of diverse lengths, we anticipate that our approach will contribute significantly to the efficient analysis of large-scale astronomical datasets, unlocking new possibilities for the exploration and understanding of stellar phenomena.



 §.§ Architecture Overview


As depicted in Figure <ref>, FCN4Flare is composed of five essential components: NaN Mask, Mapping, Backbone, SegHead, and MaskDice. Each of these components plays a crucial role in the process of flare detection and optimization of the neural network:

    
  * NaN Mask This component serves to guide the model in learning where the flux values are NaN (Not-a-Number) and how to handle these NaN values during the flare detection process. Proper handling of NaN values is crucial for accurate predictions in light curves with missing data. See section <ref>.
    
  * Mapping The Mapping component utilizes a convolution layer with a 1 × 1 filter size to map the extracted features to a high-dimensional space. This step enhances the representation capabilities of the model. See Section <ref>.
    
  * Backbone The Backbone is the primary component responsible for feature extraction. It is formed by stacking ten Conv Blocks, each contributing to the extraction of relevant features from the input normalized flux sequences. See Section <ref>.
    
  * SegHead The SegHead is responsible for point-wise flare detection of the light curves, regardless of their variable lengths. This component is essential for generating predictions at each point in the light curve, enabling detailed and precise flare detection. See Section <ref>.
    
  * MaskDice MaskDice serves as the optimization objective function for the neural network. Its design is aimed at addressing the challenge of extremely imbalanced samples, which commonly occurs in astronomical datasets. See Section <ref>.


The input to FCN4Flare is a batch of entire normalized flux sequences of light curves. The normalization process involves dividing the flux values by the median flux, ensuring consistent and comparable representations.

The model's output consists of point-wise flare probabilities, which have the same shape as the input batch. This design allows FCN4Flare to achieve end-to-end training and point-to-point prediction. This advantage not only facilitates efficient model training but also enables accurate and granular predictions, making FCN4Flare a powerful tool for the comprehensive analysis of large-scale astronomical datasets and the identification of stellar flares with precision.



 §.§ NaN Mask


NaN values, which can cause abnormal calculation results and disrupt the normal training of neural networks are common in light curves. Previous studies <cit.> addressed this by simply removing NaN data points before feeding light curves into the model. Post-prediction, any flares distributed on both sides of the NaN timestemps were manually excluded. In contrast, our approach automates this process within the model itself.

We introduce the NaN Mask module to guide the model in learning to distinguish NaN values on its own. This module first replaces all NaN flux values with placeholder values (e.g., zeros). It then creates a concatenated matrix by combining the flux vector with an indicator vector, where `1' denotes NaN and `0' denotes non-NaN values. For instance, a light curve with a flux vector x=[0.1, NaN, 0.3, NaN, NaN] results in a matrix x̂=[[0.1, 0, 0.3, 0, 0], [0, 1, 0, 1, 1]]. This method allows the model to discern between valid data points and missing data, preventing NaN values from adversely affecting the training and prediction processes.



 §.§ SegHead and Mapping


The variable feature shape inherent in light curves poses a significant obstacle when attempting point-to-point prediction using neural networks. Previous approaches, such as those seen in <cit.> and <cit.>, were compelled to resort to a sliding window methodology with a fixed width due to the constraints imposed by fully connected (FC) layers. These FC layers demand input data with fixed shapes, rendering them incompatible with variable-length input curves.

In our quest to overcome this limitation, we introduce SegHead module, which enables us to generate variable-length outputs, thereby triumphing over the constraint of fixed input shapes.

Consider a flux sequence with an input shape of 1 × L, where L denotes the length of the sequence. Upon feature extraction, the resulting feature shape manifests as F × L^', where the dimension F represents the feature channels, and L^' denotes the spatial dimension. It is important to note that L^' may differ from the original length L due to the utilization of down-sampling layers, which are strategically designed to expand the receptive field of subsequent convolutional filters.

To achieve precise point-to-point prediction, it is crucial that the shape of the output matches that of the input. In other words, we need to convert the feature channels F to 1 and the spatial dimension L^' to L. To facilitate this process and for the sake of ease during prediction, we set L^' equal to L during the feature extraction phase (see section <ref> for further elucidation on this aspect). To achieve the conversion of feature dimensions, we resort to employing a convolutional layer, which adeptly modifies the feature channels without altering the spatial dimension. Specifically, we employ convolution filters of size 1 × 1 and set the output channels to 1. As a result, this convolutional layer adeptly accomplishes weighted summation of different channels for each data point, ultimately yielding probabilities through the application of the sigmoid function.

It is noteworthy that the use of a 1 × 1 convolutional layer is tantamount to employing a fully connected (FC) layer for each individual point's feature in mathematical terms. However, this convolutional approach imparts greater flexibility to both the input and output, facilitating the accommodation of variable-length input sequences. 

Additionally, in the context of neural networks, as the depth of features increases, the network gains the ability to capture more global and high-level patterns in the data. However, in the case of flare detection, it is crucial to focus on local and fine-grained information within the light curves. To address this challenge effectively, we introduce an additional convolutional layer with a small filter size before the 1 × 1 convolution layer as depicted in Figure <ref>. This strategic inclusion of a small filter size convolutional layer serves the purpose of capturing and preserving important local information within the data.

Similarly, the application of a 1 × 1 convolution layer (Mapping) before the Backbone serves to rapidly increase feature channels while preserving the spatial dimension, as depicted in Figure <ref>. This enables the model to effectively capture and leverage higher-level features without compromising the crucial spatial details necessary for accurate flare detection.




 §.§ Backbone


The Backbone, serving as the primary component for feature extraction, plays a pivotal role in the efficacy of the model. As mentioned in section <ref>, we need to keep feature spacial size the same during feature extracting. So unlike <cit.>, we use dilated convolution layers <cit.> instead of combination of traditional convolution and pooling layers.

Dilated convolution is a powerful convolutional operation widely employed in deep learning models for image processing tasks. In this technique, the filter is dilated by inserting zeros between its values, effectively creating a larger filter with gaps or holes between its elements. This dilation strategy augments the receptive field of the filter without necessitating pooling layers, thereby preserving the feature map size.

Inspired by the insightful work of <cit.>, our backbone architecture consists of ten residual convolutional blocks. Each of these blocks comprises two 1-D convolutional layers, with a dilation parameter, as illustrated in Figure <ref>.




 §.§ MaskDice


We have achieved point-to-point prediction on light curves of variable lengths using the aforementioned four components. However, this solution has given rise to a novel challenge, namely, an exceedingly pronounced sample imbalance. As is well-known, the number of flare points within a light curve is significantly smaller in comparison to the number of no-flare points. This profound disparity in the distribution of data may compel our neural networks to favor predictions of the no-flare label for all data points, as such an approach would conveniently yield high accuracy (for instance, in the case of a light curve where the flare point rate is merely 1%, a model that merely predicts the no-flare label would attain a seemingly impressive accuracy of 99%, although such a model would bear no meaningful significance).

It is important to emphasize that such an issue was not prevalent in previous methods <cit.>, as their input was confined to individual curve segments rather than the entire curve. Consequently, they were able to control the distribution of different segment types within the training set, effectively circumventing this predicament. However, in our particular scenario, we confront the daunting task of addressing the challenge of extremely imbalanced samples, necessitating a more sophisticated and nuanced approach.

We use the MaskDice component shown in Figure <ref> to solve the problem. It is a loss function based on Dice loss <cit.>.

The Dice loss, specifically tailored to address the significant imbalance between foreground and background pixels, is a widely employed loss function in the domain of image semantic segmentation. This loss is determined through the Dice coefficient, a metric employed to gauge the similarity between distinct sets. A higher Dice coefficient signifies a greater degree of similarity between the sets. The Dice coefficient is computed using the formula shown in Equation <ref>, where DiceCoef ∈ [0, 1] represents the Dice coefficient, and X and Y represent the respective sets involved.


    DiceCoef = 2|X ∩ Y|/|X| + |Y|


To ensure efficient computation and prevent potential division by zero, the loss is often calculated using the formulation shown in Equation <ref>, which is equivalent to Equation <ref>. In this context, N represents the length of a light curve, p_i and r_i correspond to the model's prediction and the ground truth for the i-th point in the light curve respectively, and ϵ is a small constant introduced to prevent the denominator of the Dice coefficient from becoming zero.


    DiceLoss = 1 - DiceCoef = 1 - 2|X ∩ Y|/|X| + |Y|



    DiceLoss = 1 - 1/N2 ∑^N_i=1 p_i · r_i + ϵ/∑^N_i=1 p_i + ∑^N_i=1 r_i + ϵ


Although Dice Loss has been effective in handling sample imbalance <cit.>, it may not perform optimally in the context of flare detection when dealing with a very small percentage of flare points (approximately 0.5% in our dataset). To better understand the issue, let's consider a prediction vector P={ p_i ∈ [0, 1] | i=1, 2, ⋯, T } and a label vector R = { r_i ∈{ 0, 1 } | i=1, 2, ⋯, T }, where r_i = 1 denotes that the timestamp is a flare point, and r_i = 0 denotes that it is not. In the ideal scenario, p_i → 0 (e.g., p_i = 1e-9) when r_i = 0, and it is similar when r_i = 1, ensuring that ∑_i=1^N p_i →∑_i=1^N r_i, and consequently, DiceLoss → 0.

However, during the initial stages of training, the values of p_i may not be ideal (e.g., p_i = 0.1 when r_i = 0). In this situation, ∑_i=1^N p_i might be much larger than ∑_i=1^N r_i due to the overwhelming proportion of non-flare points. Consequently, the DiceLoss might not decrease significantly, regardless of the nature of the label vector R for a given curve, making it challenging to optimize the neural network parameters effectively.

In essence, the main reason why Dice Loss does not perform well in this context is the exceedingly large proportion of non-flare timestamps. To address this issue, the paper presents a solution in the form of the Mask Dice Loss, which aims to tackle the sample imbalance problem more effectively and improve the performance of flare detection in light curves.

The Mask Dice Loss function is expressed in Equation <ref>, where u(x) represents a step function, as defined in Equation <ref>, and m is a hyperparameter. The objective of this function is to address the issue of sample imbalance in the context of flare detection. It achieves this by utilizing a simple step function to mask the timestamps for which the prediction probabilities are less than the threshold m. By doing so, the loss function penalizes the model for misclassifying non-flare timestamps, thereby mitigating the numerical advantages associated with the significant sample imbalance.



    Mask   Dice   Loss = 1 - 1/N2 ∑^N_i=1 u(p_i - m) p_i · r_i + ϵ/∑^N_i=1 u(p_i - m) p_i + ∑^N_i=1 r_i + ϵ




    u(x) = {[ 1, x > 0.5; 0, x ≤ 0.5 ].


The step function u(x, m) acts as a masking mechanism, identifying the timestamps for which the prediction probabilities (p_i) are less than the threshold m. These masked timestamps, for which the model can easily make correct predictions, have less impact on improving the overall performance compared to the probabilities near the classification threshold boundary. By selectively masking such timestamps, the model is encouraged to pay more attention to the flare samples, enabling the optimizer to guide the optimization of neural network parameters in the right direction from the initial stages of training, even in the presence of the significant sample imbalance problem. As a result, the Mask Dice Loss function helps improve the performance of the model for flare detection in light curves by mitigating the challenges posed by imbalanced data.




§ RESULTS AND DISCUSSION





 §.§ Data and Training

The data used in this study is sourced from the catalog presented by <cit.>, which identifies flares in Kepler's long-cadence light curves. The catalog records a total of 162,262 flare events across 3,420 distinct stars on 33214 quarters of light curves. The method of detecting flares used in <cit.> involves a rigorous and detailed approach to ensure the accuracy and credibility of the catalog. It is a multi-step process that includes the following key steps:

    
  (1) Each observation quarter is divided into blocks at points where there are discontinuities longer than 6 hours. Each block is independently adjusted by removing long-term instrumental trends using a third-order polynomial. A Lomb–Scargle periodogram determines the most significant frequency, and a spline filter fits the baseline. An iterative σ-clipping method removes outliers to clean the data further.
    
  (2) Flare candidates are identified under strict conditions:
    
        
  (i) At least three consecutive points must exceed a 3σ threshold without interruption within 3 hours, with the flare ending at the last point above 1σ.
        
  (ii) The decay phase of a flare must be longer than its rise phase.
        
  (iii) Flares must be observed in at least two different quarters if multiple quarters of data are available.
        
  (iv) Flares energy must be greater than 10^34 ergs and last least 2 hours if only single quarter is available.
    
    
  (3) A contamination check is performed to ensure the purity of the flare signal. This involves spatial analyses to rule out photometric contamination from nearby sources, correlation checks with other instruments to confirm the uniqueness of the events, and comparison with known artifacts to identify false positives. Critical cases undergo a manual review by experts to further validate the findings.


The stringent process can significantly reduce the false positive rate of flare detection. While label errors can not be completely cleared up in the catalog, it still provides a robust basis for effectively training and testing deep learning models. This issue is common in machine learning benchmarks, as highlighted by <cit.>, which shows that even widely used datasets like ImageNet comprises label errors at least 6% of the validation set.

The 33,214 light curves described in the catalog were divided into training, validation, and test sets using a 6:2:2 ratio. We train all the models including ours and previous methods on the training set from scratch. Model selection was based on performance in the validation set, and the chosen models were then evaluated on the test set. Comparative performance results are discussed in Section <ref>.



Our model is trained using two NVIDIA V100 GPUs with a batch size of 150, utilizing the AdamW optimizer at an initial learning rate of 2 × 10^-5. The training is set to run for a maximum of 50 epochs, incorporating an early stopping mechanism based on validation dice scores. The entire training process takes approximately one hour. Figure <ref> illustrates the loss and dice scores during training and evaluation.



 §.§ Detection Performance





To rigorously evaluate the efficacy of the FCN4Flare method for stellar flare detection, we employ a suite of statistical metrics that provide a multi-dimensional view of performance. Here are the metrics:


    
  * Recall measures the proportion of actual flare events that are correctly identified by the model. A higher recall rate indicates a lower likelihood of missing real flare events. It is calculated as:
    
    Recall = TP/TP + FN

    where TP and FN correspond true positives and false negatives, respectively.
    
    
  * Precision assesses the accuracy of the positive predictions by the model. A higher precision rate indicates that the flares detected are indeed true flares, thereby minimizing the time and resources spent on following up false leads. It is calculated as:
    
    Precision = TP/TP + FP

    where FP corresponds false positives.
    
    
  * F1 is the harmonic mean of recall and precision, providing a single measure to balance both metrics. It is calculated as:
    
    F1 = 2 ×Recall × Precision/Recall + Precision

    
    
  * AP is the average precision scores at each threshold. Each precision score is weighted by the increase in recall compared to the previous threshold. AP provides a comprehensive measure that captures the performance across all thresholds which we decide the probability as the flare or not. It is calculate as:
    
    AP = ∑ (R_n - R_n-1) × P_n

    where R_n and P_n represent precision and recall at n-th threshold, respectively.
    
    
  * AUC-ROC is the area under the curve plotted with true positive rate (TPR) against false positive rate (FPR) at various threshold settings. It tells us how likely the model is to correctly identify a positive case as being more likely than a negative one when picking randomly. It is calculated as:
    
    AUC-ROC = ∫_0^1 TPR(x)   dx

    where x corresponds to varying FPR from 0 to 1.
    
    
  * IoU also known as the Jaccard index, is the intersection of union. It provides a straightforward metric that quantifies how much the predicted set covers actual set while penalizing false positives and false negatives. It calculated by:
    
    IoU = |A ∩ B|/|A ∪ B|

    where A and B are the predicted and actual labels set.
    
    
  * Dice coefficient is similar with IoU, but it calculates the size of the overlap times two divided by the sum of the sizes of both sets. It is more sensitive to the size of the intersection relative to the total size of each individual set. It is calculated as:
    
    Dice = 2 × |A ∩ B|/|A| + |B|





Table <ref> presents a comparative analysis of FCN4Flare with established flare detection methods using selected performance metrics. FCN4Flare's superiority is evident as it outperforms other models in all metrics, particularly showing significant improvements in precision, AP, IoU, and Dice coefficient. These enhancements suggest greater specificity in flare detection, as demonstrated by fewer false positives. To further highlight the model's predictive accuracy, the confusion matrix is shown in Figure <ref>. This matrix clearly visualizes the reduction in false positives, providing solid evidence of FCN4Flare's enhanced detection capabilities.



 §.§ Ablation Study

We present an ablation study to systematically evaluate the contribution of individual components within the FCN4Flare architecture. By sequentially removing key components, we assess their impact overall performance of the model focusing on precision and IoU metrics.

The ablation study involves training variants of the FCN4Flare model, each with a specific component removed. Each modified version of the model is trained under identical conditions to the full FCN4Flare model to ensure comparability of the results. Performance metrics focusing on precision and IoU are recorded in Table <ref>. The table displays the metrics scores for each model alongside the percentage change from the full FCN4Flare model, which serves as the baseline. 



The ablation study provides clear evidence of the importance of NaN Mask and Mask Dice loss components in the FCN4Flare architecture. Each component significantly contributes to model's overall performance, and their removal leads to pronounced decreases in both precision and IoU.



§ APPLICATION



To advance our scientific investigation, we deploy FCN4Flare to detect flares on the Kepler-LAMOST dataset, which is generated by cross-matching the Kepler <cit.> DR25 long cadence catalog and LAMOST <cit.> DR9 low-resolution catalog with a tolerance of 3^''. This dataset encompasses a total of 1,282,142 light curves, representing 87,549 distinct stars.



It is imperative to acknowledge that the data partitioning approach we adopt, utilizing the complete Kepler flare catalog <cit.> for training and a subset of Kepler data for prediction, may be deemed unconventional in machine learning due to concerns of potential data leakage. Nevertheless, in the context of our study, this concern is mitigated. The primary objective of our research is the development of a novel flare detection methodology and the identification of candidates for flare events, rather than a direct comparison of various methods based on specific performance indicators.

In light of these considerations, we have included the outcomes of our model's predictions, which can be invaluable for further scientific research and the exploration of flare phenomena.



 §.§ Flare Catalog




Following the application of our model to the set of light curves, we obtained probabilities assigned to each time step, signifying the likelihood of it belonging to a flare event. To ensure a higher level of reliability and to minimize the risk of false positive events, we implemented specific screening rules on the prediction results. The rigorous screening process involved the following criteria:


    
  (1) A minimum of three consecutive points with probabilities above 0.5 was required to qualify as a flare event.
    
  (2) For these points, the duration from start to peak time must be shorter than from peak to end time, adhering to flares' typical “rapid rise and slow fall” pattern.
    
  (3) Flux levels before and after the event must be close to each other, indicating a clear deviation from background levels.
    
  (4) In the case of a target exhibiting multiple quarters, flares had to occur in at least two quarters to be considered valid.


These screening rules were systematically applied during a post-processing step and significantly contributed to enhancing the overall accuracy of our predictions. By carefully filtering the candidate flare events based on these criteria, we could mitigate false detections and ensure the reliability of our flare detection methodology.

In our investigation, we successfully identified a total of 30,285 distinct flare events occurring in 1426 stars. For each of these events, Table <ref> furnishes crucial details, including the start time, end time, and energy estimation.

To estimate the energy associated with each flare event, we adopted a well-established approach based on integrating normalized flux, as described in <cit.> and <cit.>. Initially, we normalized the PDC flux by dividing it by the median value. Next, we employed linear interpolation to determine the start and end times of the flare, using the stellar background flux during the flare as reference. These measures helped create a suitable baseline for the estimation process. Subsequently, the flare energy is computed by this equation:

    E_flare = 4 π R_*^2 σ_sb T^4 ∫ F_flare dt

where R_* and T represent the stellar radius and effective temperature, respectively, both of which were sourced from the Kepler Catalog. The symbol σ_sb denotes the Stefan-Boltzmann constant, while F_flare corresponds to the difference between the normalized flux and the interpolated background flux. The energy estimation is expressed in erg units.



Figure <ref> illustrates the energy estimation and the frequency distribution of flares, which intriguingly conform to a power-law relation, expressed as dN/dE ∝ E^α. We apply an ordinary least-square to fit the solid line, whose slope is the α index. Drawing from <cit.>, it is noteworthy that the exponent α was found to be approximately 2.

Except for the parameters of the flare events, the parameters of the 1426 flare stars are also listed in Table <ref>, in which the spectral type is taken from the LAMOST catalog, Teff, log g, [Fe/H] and radius are all taken from the Kepler catalog, and the rotation period is estimated by the Lomb–Scargle periodogram <cit.> algorithm, T_obs represents the observation time of Kepler, T_fl/T_obs represents the proportion of the flare duration, E_max represents the maximum detected flare energy, N_flare represents the number of detected flare events, and L_fl/L_bol represents the flare activity, which is calculated as the ratio of the total flare energy to the bolometric luminosity (i.e., ∑ E_flare / ∫ L_bol dt = L_flare / L_bol) <cit.>.



Subsequently, we computed the incidence of flare stars stratified by their spectral types, and the findings are thoughtfully presented in Table <ref>. To further encompass the intrinsic properties of stars, we deliberately differentiated between M-type dwarfs (dM) and M-type giants (gM). Intriguingly, our analysis revealed a pronouncedly elevated incidence of flare stars among dM, while the occurrence of flare events in gM stars was found to be nearly negligible. This intriguing observation seamlessly aligns with our current comprehension of the underlying flare mechanism inherent to M-type stars.



 §.§ Flare activity distribution




In the quest to elucidate flare activity across varied stellar spectral types, two quintessential diagrams have been proffered, delineated in Figure <ref>. The first panel presents a histogram detailing the flare activity distribution of different stellar types. Subsequently, the secondary illustration, rendered as a density plot, provides a normalized perspective, artfully portraying the probabilistic density of flare activity within each stellar type. This normalization ensures an equivalency in the integral area beneath each curve, thereby facilitating an unambiguous juxtaposition amongst the subclasses. It is noteworthy that the abscissa of both depictions adopts a logarithmic scale.

One striking observation emerges: the zenith of flare activity recedes progressively from M-type stars to F-type counterparts. Intriguingly, the flare activity linked with A-type stars surpasses that of both F and G spectral categories. Several postulations arise from this conundrum. It could be postulated that the cohort of A-type stars, wherein flares are detected, is minuscule, thereby deeming it statistically inconsequential, or perhaps an observational predilection is at play. Alternatively, the underpinnings governing the flare processes in Type A stars might diverge from other types, a phenomenon that remains shrouded in mystery.




 §.§ Chromospheric Dynamics and Stellar Eruptions




The intricate interplay between chromospheric dynamics and stellar flares has been a subject of sustained academic enquiry. Among the myriad indicators of such dynamism, the H_α equivalent width (EW) and the S Index emerge as particularly salient. Leveraging the LAMOST spectral data, we have discerned the H_α EW and the S Index for our stellar catalog, subsequently elucidating their correlation with stellar flare proclivities. These findings are delineated in Figure <ref>. Gleanings from the figure encompass:


    
  * Interplay of H_α and Flare: An overwhelming majority of the stellar types, with an especial emphasis on M-type stars, exhibit a pronounced trend wherein an uptick in the H_α EW directly corresponds to an escalation in flare activity. This observation is congruent with antecedent investigations underscoring the nexus between elevated chromospheric dynamism and augmented flare manifestations, particularly within this stellar classification.
    
  * Correlation between S Index and Flare: Analogous to the patterns discerned in H_α EW, the S Index also evinces a discernible positive association with flare activities. Notably, the symbiotic relationship for G-type and F-type stars is more palpable in the S Index than in the H_α EW.
    
  * Nuances between H_α and Max Flare Energies: The relationship between H_α EW and the max flare energy presents a nuanced scenario, challenging extant paradigms. It appears that elevated H_α EW metrics do not unerringly predicate the magnitude of flares, hinting at the presence of latent variables modulating flare energies. 
    
  * Interrelation of S Index and Peak Flare Energies: Mirroring the dynamics between S Index and stellar eruptions, the correlation between the S Index and the apogee of flare energies appears elusive, suggesting a potentially multifaceted interrelation between chromospheric dynamism and peak flare intensities.


It is imperative to underscore that the paucity of flare-star specimens in our compilation, especially within the Type A stars, could engender observational biases in discerning correlations.






 §.§ Habitable Zone Exoplanets

Planetary bodies represent the sole confirmed repositories for the phenomenon of life as we know it. The irradiative influence from a celestial body's central luminary plays a pivotal role in determining the habitability of any given planetary environment. Consequently, the scientific examination of stellar flares assumes paramount importance in the discrimination and selection of potentially life-bearing planets.



The NASA Kepler mission has facilitated the discovery of an extensive array of planetary candidates, many of which have been corroborated by subsequent observational analysis <cit.>. We have executed a cross-match between our flare star catalog and the cumulative table of Kepler Objects of Interest (KOI) <cit.>. The resulting data is delineated in Table <ref>. Within the table, `kepid' refers to the identification, `kepoi_name' designates a numerical value utilized to identify and monitor a KOI, `kepler_name' is the nomenclature assigned to the specific planet, and `subclass' denotes the spectral classification of the star as obtained from LAMOST. Additionally, `radius', `teff', `feh', and `logg' symbolize stellar parameters extracted from the Kepler catalog. The table is methodically organized in accordance with flare activity.

Numerous investigations have focused on the intricate subject of habitable zone exoplanets, contributing to a comprehensive understanding of these celestial bodies <cit.>. In this work, we execute a meticulous cross-reference between the previous table and the Catalog of Habitable Zone Exoplanets, as assembled by <cit.>. Remarkably, one stellar object situated in the inner habitable zone has emerged from this analysis, and it is distinctively highlighted in Table <ref>. This stellar entity, identified by the Kepler ID 5938970, manifests indications of pronounced magnetic activity.

In the present manuscript, we delineate an exhaustive investigation of this particular star, detailing the salient characteristics such as flare events, flare energy, flare activity indicators, and other pertinent parameters. We harbor the aspiration that our scholarly contribution will significantly augment and influence future research endeavors directed towards the understanding of the habitability potential of this specific stellar object.





§ CONCLUSION


In this paper, we have presented FCN4Flare, a novel deep learning approach for detecting stellar flares in light curves. By leveraging fully convolutional networks, our model achieves end-to-end training and precise point-to-point flare prediction regardless of input length. This overcomes limitations of prior methods relying on fixed-length input segments. 

Through components like NaN Mask, dilated convolutions, and MaskDice loss, FCN4Flare handles challenges unique to flare detection, including missing data, feature resolution, and class imbalance. Our experimental results demonstrate a significant improvement in flare detection performance compared to previous models, with a Dice coefficient of 0.64 versus 0.21 for the state-of-the-art.

Applying FCN4Flare to Kepler-LAMOST data, we have compiled a catalog of 30,285 flare events across 1426 stars. By estimating flare energies and cross-matching with exoplanets, our analysis provides new insights into stellar magnetic activity and planetary habitability. One noteworthy finding is the identification of pronounced flaring behavior on an M-dwarf hosting a planet within the optimistic habitable zone. 

Overall, this work makes multiple contributions - an innovative deep learning architecture for flare detection, a sizable catalog of high-quality flare predictions, and new perspectives on connections between stellar flares and exoplanetary environments. FCN4Flare demonstrates the power of deep learning to unlock new scientific discoveries in astronomical time-series data. 

Key directions for future work include enhancing the model, expanding the stellar sample analyzed, and combining our flare predictions with other astronomical data to answer open questions about stellar physics and planetary system evolution. By overcoming long-standing challenges in flare detection, this research opens new avenues for mining the wealth of photometric data from missions like TESS. The transformative potential of deep learning in astronomy is only beginning to be realized.




§ ACKNOWLEDGEMENTS

This work is supported by the National Science Foundation of China (Nos. U1931209, 12261141689) and the National Key R&D Program of China (No. 2019YFA0405502).  Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope, LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences.  
Funding for the project has been provided by the National Development and Reform Commission.  
LAMOST is operated and managed by the National Astronomical Observatories, the Chinese Academy of Sciences.


 



 Paper Title: Spurious Solar-Wind Effects on Acceleration Noise in LISA Pathfinder.





Spurious Solar-Wind Effects on Acceleration Noise in LISA Pathfinder]Spurious Solar-Wind Effects on Acceleration Noise in LISA Pathfinder



^1Institute for Computing in Research, 145 Washington Ave, Santa Fe, NM 87501
^2California Institute of Technology, 1200 East California Boulevard Pasadena, California 91125, United States
^3Los Alamos National Laboratory, Los Alamos National Laboratory P.O. Box 1663 Los Alamos, NM 87545
gdmeadors@lanl.gov




  August 5, 2024

§ ABSTRACT 
Spurious solar-wind effects are a potential noise source in the measurements of the future Laser Interferometer Space Antenna (LISA). Comparative models are used to predict the possible impact of this noise factor and estimate spurious solar-wind effects on acceleration noise in LISA Pathfinder (LPF). Data from NASA’s Advanced Composition Explorer (ACE), situated at the L1 Lagrange point, served as a reliable source of solar-wind data. The data sets were compared over the 114-day time period from March 1, 2016 to June 23, 2016. To evaluate these effects, the data from both satellites were formatted, gap-filled, and adapted for comparison, and a coherence plot comparing the results of the Fast Fourier Transformations. The coherence plot suggested that solar-wind had a minuscule effect on the LPF, and higher frequency coherence (LISA’s main observing band) can be attributed to random chance correlation. This result indicates that measurable correlation due to solar-wind noise over 3-month timescales can be ruled out as a noise source. This is encouraging, although another source of noise from the sun, solar irradiance pressure, is estimated to have a more significant effect and has yet to be analyzed.








Keywords: Space weather, solar wind, gravitational waves, LISA















§ INTRODUCTION

The future Laser Interferometer Space Antenna (LISA) will be the first space-based gravitational wave detector on a heliocentric orbit <cit.>. The LISA project expands upon the work of the ground-based Laser Interferometer Gravitational-wave Observatory (LIGO), which has made significant contributions to our understanding of the universe <cit.>. LISA will measure frequencies ranging from approximately 0.1 millihertz to 100 millihertz <cit.>.  LISA will consist of three spacecraft in an equilateral triangle with arms 2.5 million km long. The antenna will follow 20 degrees behind the Earth. Each spacecraft will contain two centered, free-falling test masses. Deviations from the geodesic motion of the test masses are detected with time-delay interferometry (TDI) <cit.>, using six laser links that connect each of the three spacecraft bidirectionally. TDI constructs a set of time series data, sampled every few seconds, from the combination of these laser links that represents that stretching of space, known as (dimensionless) strain. LISA's TDI output is, in effect, three virtual, correlated space-based interferometer channels sensitive to gravitational waves in the millihertz band.



To confirm the viability of the LISA concept, the European Space Agency (ESA) developed and launched the LISA Pathfinder (LPF). LPF consists of a test version of a LISA module - two free acceleration test masses held in the module's center. The LPF was launched on December 3, 2015, and traveled 1.5 million km from Earth to orbit the L1 Lagrange point; see Figure 1 for positioning in relation to the Sun and Earth. The goal of LPF is to show that a drag-free control measurement technique is viable for a long baseline gravitational wave observatory. LPF was sent into space with 2 test masses, the second one located 38 cm from the first, to avoid common noise sources: solar radiation pressure, solar-wind, magnetic environment, and particle impacts. In addition, spacecraft acceleration was measured to observe external disturbances on a single-test mass signal. One of these sources, solar-wind, has been generally understood to be a very modest noise source <cit.>, below other factors - actuation fluctuations, Brownian noise, micrometeoroids, and more.

Theoretical extrapolation from solar wind and solar irradiance data to LISA <cit.> has set the stage for the analysis in this manuscript, which considers whether correlations are measurable between solar wind and LPF acceleration data. Analyzing LPF's data can be proof that solar-wind noise impact is minimal or demonstrate that in the future, LISA might act as a serendipitous solar-wind sensor at an unusual orbit, similar to STEREO <cit.>. Constant solar wind, sunlight radiation pressure, or other static forces are not expected to have an impact on LISA. However, a time-varying force could accelerate the LISA spacecraft, particularly if the three spacecraft experience the differential forces due to orbital position and orientation. Studies of solar wind variability <cit.> demonstrate that the solar wind can vary from 350 km/s to 700 km/s in the span of a week, doubling the solar-wind force on a timescale of less than 1 Mega-second, motivating an investigation of the micro-Hertz to milli-Hertz frequencies even though they are below the typical LISA science band.

The idea was tested by comparing the LPF spacecraft's Z and X axis data with the corresponding ACE solar-wind data from March 1, 2016, to June 23, 2016 <cit.>. The methods used can find direct, linear couplings at the measured timescales, but nonlinear “upconversion" of solar-wind noise to higher frequencies cannot be ruled out. The Advanced Composition Explorer (ACE), orbiting the L1 point as the LPF did, is used to measure solar-wind data, such as proton speed, proton density, proton to alpha particle ratio, x, y, z components of proton velocity, and Coordinated Universal Time (UTC) <cit.>. This data can be modeled onto LPF and compared with the data from LPF to analyze the effects of solar-wind.



 §.§ Related Works

Space weather has been identified as a key noise factor in possible future LISA measurements. Past work that assessed the impact of solar wind involved using data from the Solar and Heliospheric Observatory’s (SOHO) Virgo Solar irradiance and the ACE Solar-wind to model possible impacts on the future LISA mission <cit.>. Another major source of noise identified for LPF is micrometeoroids. Through various models, 54 impact candidates were identified across a 4348-hour time period. They were shown to be similar to those resulting from Jupiter Family Comets (JFC), Oort cloud comets, Halley-type comets, and asteroids <cit.>. Additionally, the development of LISA is renewing the interest in the environment of space <cit.><cit.><cit.>. This environment includes solar energetic particles, galactic cosmic ray fluxes, and the effects of solar neutrons and interplanetary electrons <cit.>.




§ METHODS

 Figure 2 shows the steps to plot and compare the ACE and LPF data sets.
 We first formatted both data sets to include only data present is within the time frame and for the axes being studied. Next, we filter and gap-fill the ACE data set because the data set is in time-series format. An Inverse Fast Fourier Transform (IFFT) is applied to the LPF data set because the set is in the frequency domain <cit.>. Once the IFFT is performed, the LPF data is filtered and gap-filled. The ACE data set must be modeled onto LPF. Lastly, both data sets are plotted and compared.




 §.§ Process for Creating Plottable ACE Data

Solar-wind data was obtained from the Advanced Composition Explorer (ACE). ACE data included information on solar-wind velocity components (x, y, z), alpha particle to proton ratios, proton densities, and proton speeds at 64-second intervals measured in Coordinated Universal Time (UTC) <cit.>. The area of the solar array was modeled as a 2.9 meter diameter flat circular array, based off of the concept S/C design <cit.>. ACE's UTC measurements were converted to GPS time because the LPF data is in GPS time. Bad data values from the ACE data were also removed (the ACE data header specifies that bad data is indicated by the value -9999.9) <cit.>. The gaps created by removing the bad data were split into three categories based on the process described in Modeling Spurious Forces on the LISA Spacecraft Across a Full Solar Cycle <cit.>. Type A gaps are missing one data point and are filled using linear interpolation. Type B gaps are missing two to twenty-four data points and are filled using linear interpolation with Gaussian Noise. Eq. 1 derives Gaussian Noise (σ) where σ_1 and σ_2 are the standard deviations found by sampling data of equal length to the gap on either side of the ACE data gap. In the equation, for the ith entry in a gap of length l such that 1 ≤ i ≤ l, a point was generated based on σ_1 and σ_2 <cit.>. 



    σ=l-(i-1)/l+1σ_1+i/l+1σ_2


Type C gaps are missing twenty-five or more data points; they are filled using a Hann window with a slow fall-off to zero, as defined by Eq. 2 <cit.>. Twenty-five data points on either side of the gap are used to calculate simulated interpolation data ω(n) where N=51 and 0<n<N-1, creating the data to fill a type C gap.


    ω(n)=0.5[1-cos(2π n/N-1)]




Following previous methods <cit.>, the force exerted on the satellite by solar-wind particles was calculated using ACE data.

Eq. 3 calculates the number of protons (N_p) and alpha particles (N_α) hitting the satellite per unit of time,


    N=n v Acos(ϕ)

with n being particle density and v being wind speed. The surface area of the LPF solar array is represented by A. R is the fraction of particles reflected. ϕ represents the angle between the normal of the array and the orbital plane <cit.>. LPF's eta sensor data of the second test mass showed ϕ was less than ±1 × 10^-7 for all values, making it reasonable to assume ϕ in that case to be 0. The eta sensor data was stated to be within a 2-degree margin of error. Therefore, both a 0-degree and 2-degree value for the angle were tested. Equations 4,5,6 were used to calculate the final forces for the x and z axis. The y-axis equation was disregarded as the amount of force occurring on that axis was negligible. <cit.>.

The following equations are reproduced verbatim from previous LISA analysis <cit.>.

    F_x    = (N_pm_p+N_αm_α)[(1+Rcos(2ϕ))v_x+Rsin(2ϕ)v_z], 
    
    F_y    = (N_pm_p+N_αm_α)[(1-R)v_y], 
    
    F_z    = (N_pm_p+N_αm_α)[(1+Rcos(2ϕ))v_z+Rsin(2ϕ)v_x]


N_p, the proton collision rate, and N_α, the alpha particle collision rate, were found using Eq. 3. The proton and alpha-particle masses from the ACE solar-wind data are represented by m_p and m_α. The x, y, z components of solar-wind velocity are represented by v_x,v_y, and v_z. Assuming the worst-case scenario (that all the particles reflect), R will be defined as 1. Then Equations 4,5,6 can be simplified to the following <cit.>:

    F_x    = (N_pm_p+N_αm_α)[(1+cos(2ϕ))v_x+sin(2ϕ)v_z], 
    
    F_y    = 0, 
    
    F_z    = (N_pm_p+N_αm_α)[(1+cos(2ϕ))v_z+sin(2ϕ)v_x].



The ACE solar-wind force values derived using Equations 7,8,9 are now able to be compared to the processed LPF data by being plotted and transformed as described in Section III.



 §.§ Process for Creating Plottable LPF Data

Data from the LPF was used to predict the effect of solar-wind on the future LISA data. The LPF data obtained from the NASA Goddard Space Flight Center combines information from various sensors to estimate the free-body motion of LPF. The data spans 114 days and is organized into 370 files, each 16384 seconds long. The data was taken at 2.5-second intervals from the second sensor on the x-axis and the second sensor on the z-axis. The data for LPF was in the frequency domain; thus, an IFFT was applied. The IFFT added the complex conjugates of negative values and zeros to the positive, non-zero values, returning the data to the original time series format. To calculate force, the acceleration from the IFFT was multiplied by the mass 422 kg <cit.>. The 31 gaps in both the x and z axes were filled using the Hann Window (Type C) method (Table. 1) <cit.>.




 §.§ Plotting Methods

A simple time domain comparison and a Fourier Transform comparison were used in Figures 3 and 4, respectively, to see if there were any apparent overlaps when the data were graphed over each other. The time domain data was zero-meaned to highlight the differences in the force data collected between LPF and ACE. The  function is used to determine coherence between LPF and ACE data and the averages of different lengths of correlation values <cit.>. As described in the legends of Figures 5, 6, 7, and 8, the coherence comparison used increasing segment lengths multiplied by factors of 4 from 1 to 4^5.




§ RESULTS


Figure 3 depicts a time series that presents a direct comparison between the LPF time domain and the ACE force time domain, with both datasets having been zero-meaned. The ACE force data was calculated using the force equations and a 2-degree estimate for angle ϕ. The time series do not show any direct correlation between ACE and LISA data. Long constant segments in LPF indicate data gaps. 



Figure 4 illustrates some overlap between the ACE and LPF data but nothing significant enough to be seen as a solid correlation between the two data sets. FFT was done with a 1/N normalization coefficient and a sampling time interval of dt = 2.5 seconds and dt = 64 seconds for LPF and ACE respectively.


From the coherence plot (Figure 5), it was observed that the coherence between the two data sets was minimal, and any correlation is probably due to random chance. In particular, spectral lines in LPF, due to instrumental artifacts, the line above 10 millihertz in ACE, do not show coherence on closer inspection.


Figure 5 shows six different lengths of coherence value averages. Each decrease in the values averaged increases the accuracy of the line. As the accuracy increases, the coherence decreases. The low coherence indicates that solar-wind will have little to no direct effects on the future LISA. Although with an estimated normal angle of 0, there should be no effect on the x-axis, the 2-degree margin of error of the angle was tested for coherence in the x-axis in Figure 6.




§ TEST VERIFICATION

To ensure the accuracy of the previous tests and to create a visual of what would happen if there was a finding, the correlation methods were tested with simulated data. A measured known excitation of the sinusoidal form was added to both the LPF and ACE data sets after they had been filtered, gap-filled, and post-processed.


    F(t) = A sin(2π f t+ϕ)


In Eq. 10, F(t) represents the Force excitation added to the data where A is amplitude, f is frequency, t is time, and ϕ is phase. The phase was set to 0 because it does not affect the coherence measurement. Amplitude and frequency were varied as follows. Adding this excitation in both sets of data simulates a specific frequency of the wave that should then be picked up by the coherence function, thus showing an excitation in that frequency on the coherence plot.

In each test, a single sinusoidal frequency was excited. Multiple frequencies were excited in independent tests. The test frequencies 1 × 10^-2 Hz, 1 × 10^-3 Hz, 2 × 10^-3 Hz, 5 × 10^-3 Hz, 1 × 10^-4 Hz, 5 × 10^-4 Hz were measured. All frequencies displayed distinct excitation at the injected frequency in the coherence spectrum. They were also tested for the least excitation needed for at least 50% correlation when 1024 points were used per segment.
To illustrate this, Figure 7 shows the frequency 2 × 10^-3 Hz at 99.5% coherence for 1024 segments, and Figure 8 shows the frequency 5 × 10^-3 Hz at 68.36% coherence for 1024 segments per average with only 0.2 nanoNewtons of force. Figure 9 shows the frequency 5 × 10^-3 Hz again, but with the same 5 nanoNewtons of force as Figure 7 and displays a similar 99.9% correlation. Thus showing that the coherence function works accurately and can properly compare the 2 datasets.







The excitation frequencies of 2 and 5 millihertz in this paper represent the low-to-mid range of LISA, relevant to both the galactic white-dwarf binary population and black-hole mergers around a million solar masses. At these frequencies, prior work <cit.> has estimated the force coupling of solar wind pressure into LISA: Figure 3 of that paper shows forces in the year 2000, near solar maximum, resulting in a predicted force at 2 millihertz of roughly 20 picoNewtons, and less at 5 millHertz. While a comprehensive statistical sampling from that work’s force spectrum is reserved for future research, this paper shows that near 200 picoNewtons, magnitude-squared coherence levels do reach above 0.68, or about two-thirds, at 5 millihertz. Comparing the two frequencies, the coherence would be expected to be lower at 2 millihertz, consistent with the noise spectrum.




§ CONCLUSION

The effects of spurious solar-wind on acceleration noise of LISA Pathfinder (LPF) data from the x and z axes were thoroughly analyzed. As shown in Figures 5 and 6, the coherence between the Advanced Composition Explorer (ACE) and LPF data was sporadic for the given frequency range, and the average correlation for each data section was minuscule. The coherence test has verified that there is a measurable correlation for a simulated excitation of known amplitude, as seen in Figures 7, 8 and 9. The analysis shows that the correlation seen in the real, unexcited data is indistinguishable from chance given the data available and no relationship was found between ACE solar-wind data and LPF data. The models and plots of the x and z axes found a negligible relation between the ACE solar-wind data set and the LPF.

This result suggests that there is a reasonable safety margin between solar wind forces and measurable acceleration noise in LISA. More detailed analysis is warranted, as solar variability could reach levels even higher than in 2000, but comparison with solar irradiance <cit.> suggests that those forces may pose a larger concern. To the benefit of LISA, albeit a loss to the space weather community, it currently appears difficult, with such low coherence at expected force levels, to solve the inverse problem of finding solar wind force from LISA acceleration, as proposed previously <cit.>.

These findings are encouraging for the future of the LISA as less noise will allow LISA to see much more of the universe. However, solar irradiance data, another noise factor with likely greater effects on the LPF, has yet to be tested <cit.>, and the limited 3-month time frame of data that were analyzed means that correlations over longer timescales are possible as well. Future analyses should also focus on working with individual subsets of the data and plotting the relationships between those to eliminate the possible confounding effects created when using gap-fill methods. 




§ ACKNOWLEDGMENTS

Thank you to the Institute for Computing in Research: Mark Galassi and Rhonda Crespo. Thank you I. Thorpe and J. Slutsky at the NASA Goddard Space Flight Center for the data and helpful comments. Additionally, thank you to B.M. Frank, B. Piotrzkowsky, B. Bolen, M. Cavaglià, and S.L. Larson for their methods and helpful comments. Indie Desiderio-Sloane and Arnold Yang were supported by the Institute for Computing in Research. GDM was supported by multiple Los Alamos National Laboratory internal funding sources, including the Intelligence and Space Research Early Career (ISR-EC) 2022 Pitch Day award XB4F00 and the Center for Space and Earth Science project development award W88600. This document has Los Alamos Unlimited Release (LA-UR) number LA-UR-23-30801. This work was supported by the U.S. Department of Energy through the Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of U.S. Department of Energy (Contract No. 89233218CNA000001).




§ SUPPLEMENTARY DATA

The source code and procedures to run it are saved at codeberg.org/Poarthan/solarWind- DataAnalysisCode. The ACE and LPF data used can be found respectively at zenodo.org/records/6955182 and zenodo.org/records/6954044.



§ REFERENCES


iopart-num







 Paper Title: Catalog-based pseudo-$C_\ell$s.


Quantum Simulation via Stochastic Combination of Unitaries
    Prineha Narang
    August 5, 2024
==========================================================





§ INTRODUCTION

  In cosmology and astrophysics, a plethora of observable quantities can be described by projected fields on the sky. These fields may be observed and mapped in a continuous manner, as is the case of Cosmic Microwave Background (CMB) observations, or may be observed only at the discrete positions of a finite number of sources. Depending on the physical nature of the quantity at hand, these quantities will transform differently under rotations of our angular coordinates, leading to spin-0 fields (i.e., scalars, such as the CMB temperature anisotropies, the lensing convergence, or the galaxy overdensity), spin-1 fields (i.e., vectors such as transverse peculiar motions or lensing displacements), spin-2 fields (i.e., tensors such as the CMB polarization anisotropies or weak lensing shear), and higher-spin fields (e.g. the spin-4 polarization of gravitational waves). The angular power spectrum, corresponding to the covariance between angular fluctuations in two different fields at a given multipole scale ℓ, is one of the most used summary statistics in cosmology and astrophysics and, in particular, encodes all available information in the case of Gaussian fields. For this reason, being able to accurately estimate the angular power spectrum is a key requirement for many cosmological probes. In the case of galaxy weak lensing, for instance, a precise and accurate reconstruction of the cosmic shear power spectrum on sub-arcminute scales is of vital importance to obtain meaningful constraints on the gravity-dominated, nonlinear regime of cosmic structure formation, scrutinizing the Universe’s matter content and nature of Dark Energy <cit.>. This is a key goal for current and next-generation galaxy imaging surveys, such as the Kilo Degree Survey (KiDS)[<http://kids.strw.leidenuniv.nl>], the Dark Energy Survey (DES)[<https://www.darkenergysurvey.org>], the Hyper Suprime-Cam[<https://www.naoj.org/Projects/HSC/>], Euclid[<https://www.euclid-ec.org>], and the Rubin Observatory Legacy Survey of Space and Time (LSST)[<https://www.lsst.org>]. Valuable complementary insight may be gained from the angular power spectrum of other discretely sampled tracers of the matter density, such as the overdensity of galaxies <cit.>, fluctuations in their sizes <cit.>, fluctuations in the luminosity distance of standard sirens <cit.>, transverse peculiar motions <cit.>, or the dispersion measure of extragalactic Fast Radio Bursts (FRBs) <cit.>.

  There is a large literature of unbiased power spectrum estimators that are optimal, have well-understood statistical properties, and robustly meet common observational challenges such as partial sky coverage, linear sky contamination, and inhomogeneous noise <cit.>. For small angular scales, optimal quadratic estimators are usually computationally prohibitive, and the most common alternative is their simpler incarnation, the so-called “pseudo-C_ℓ” estimator <cit.>. The pseudo-C_ℓ estimator effectively assumes a diagonal covariance matrix for the data, and is therefore close to optimal for spatially uncorrelated or white-noise-dominated data. As a consequence of this assumption, the mode-coupling induced by the spatial variation of the resulting inverse-variance weights (i.e., the “mask”) can be estimated using fast, analytical methods. Variations of the method allow it to account for the impact of linear contaminant deprojection, and for the effects of E- and B-mode purification in non-scalar fields <cit.>.

  In its standard implementation, the pseudo-C_ℓ method takes, as input, pixelized maps of the fields in question. When the original observations are in the form of discrete measurements at arbitrary sky positions, as is the case for any catalog-based observable, this results in two fundamental problems. First, generating a pixelized map from the discrete sources induces a bias at scales comparable to, and smaller than, the pixel size. The form of this bias depends on a combination of pixel smoothing and aliasing of small-scale power, and is non-trivial to characterize in general <cit.>. Second, when the source number density is low, the mask is dominated by Poisson noise, also known as shot noise, which, as we will see, leads to statistical couplings between distant angular multipoles. If not treated carefully, this can lead to strong numerical instabilities and catastrophic biases in the estimator. The issue of small-scale power in the mask has been traditionally handled via apodization and smoothing of the mask edges <cit.>, but this approach is highly impractical in the context of catalog-based observations for sparse samples.

  This work addresses the issue of how to define an unbiased power spectrum estimator for fields of arbitrary spin discretely sampled on the sphere. Building on the recent work on this topic by <cit.>, we forgo the pixelization step completely and directly estimate spherical harmonic transforms of the field and its measurement weights at the discrete source positions. This allows us to build a pixel-independent version of the pseudo-C_ℓ estimator, addressing the first problem described above. We address the second problem (shot noise instabilities) by introducing a simple modification to the estimator that accounts for uncorrelated noise, both in the mask and the field, and simultaneously eliminates both the associated noise bias and the numerical instabilities of the mask-induced mode coupling. More specifically, the contribution to mode coupling from the mask shot noise is estimated analytically and subtracted from the measured pseudo-C_ℓ, together with a naïve, data-based estimate of the noise bias. As we will show, the resulting estimator is then immune to any bias from uncorrelated noise, and benefits from a numerically stable mode-coupling matrix, independently of the number density of sources in the catalog. We implement this methodology in the publicly available package[<https://github.com/LSSTDESC/NaMaster/>]. As we were preparing this manuscript, we learned of related work presented in Tessore et al. 2024 <cit.>. The results there agree with those presented here.

  This paper is structured as follows. In Section <ref>, we lay out the formalism that defines our unbiased estimator. In Section <ref>, we validate it on simulated cosmic shear catalogs with a realistic source distribution, based on real data from the Kilo Degree Survey, also demonstrating its validity on significantly lower number densities. In Section <ref>, we present several example applications of the methodology, including state-of-the-art cosmic shear data from the Dark Energy Survey, transverse proper motion surveys, and FRB-based dispersion measure maps. We conclude in Section <ref>.




§ METHODS

  This section presents the formalism to estimate the angular power spectrum from sparse observations of a given field at a discrete set of points on the sphere using the pseudo-C_ℓ formalism. Section <ref> reviews the main concepts behind the standard pseudo-C_ℓ method, and the application to discrete tracers is described in Section <ref>. We first present the formalism in the simpler case of scalar fields, and then, in Section <ref> present the generalization to spin-s quantities.

  

 §.§ Pseudo-C_ℓs

    Let f() be a scalar field on the sphere, and let w() be a sky mask (often also called weights maps, or sky window function). The mask may be as simple as a binary map (0 for unobserved regions, 1 for observed regions) but, more usefully, it should represent the precision with which f has been measured at different points (e.g. an inverse-variance map will be optimal for noise-dominated observations). Because of this, w() is often referred to as a “window function” or “weights map”, but we will stick to the simpler “mask” in what follows.

    The field is related to its harmonic coefficients f_ℓ m through a spherical harmonic transform (and its inverse):
    
    f()=∑_ℓ m Y_ℓ m() f_ℓ m↔f_ℓ m=∫ d f() Y^*_ℓ m(),

    where Y_ℓ m() are the spherical harmonic functions. The spherical harmonic coefficients of the mask, or any other sky map, are defined in the same way.

    Let f^w()≡ f()w() be the masked field. The “pseudo-C_ℓ” of f is simply the naïve estimate of the masked field's power spectrum ignoring the fact that it's masked:
    
    C̃^f_ℓ≡1/2ℓ+1∑_m=-ℓ^ℓ f^w_ℓ mf^w*_ℓ m.

    We commonly assume the field to receive signal and noise contributions: f=s+n, where the signal s is a statistically isotropic field, such that
    
    ⟨ s_ℓ ms^*_ℓ' m'⟩=δ_ℓℓ'δ_mm'S_ℓ,

    here S_ℓ is the signal power spectrum, and is the quantity we are ultimately after.

    The key result of the pseudo-C_ℓ estimator is the relation between the ensemble average of C̃^f_ℓ and S_ℓ:
    
    ⟨C̃^f_ℓ⟩=∑_ℓ'M^w_ℓℓ'S_ℓ'+Ñ_ℓ,

    where M^w_ℓℓ' is the mode-coupling matrix:
    
    M^w_ℓℓ'=2ℓ'+1/4π∑_ℓ”C̃^w_ℓ” (2ℓ”+1)ℓℓ'ℓ”000^2,

    here C̃^w_ℓ≡∑_m |w_ℓ m|^2/(2ℓ+1) is the pseudo-C_ℓ of the mask. Ñ_ℓ is the “noise bias”, given by the ensemble average of the pseudo-C_ℓ of the masked noise:
    
    Ñ_ℓ≡⟨1/2ℓ+1∑_mn^w_ℓ mn^w*_ℓ m⟩.

    Depending on the type of observations, and our understanding of the noise properties, Ñ_ℓ may be estimated analytically, via Monte-Carlo simulations, or avoided by using cross-correlations between observations with the same signal but different noise realizations (e.g. CMB maps constructed from disjoint sets of timestreams). As we will see, in the case of discretely-sampled fields, the noise bias can be entirely avoided by construction under the assumption that the noise contributions from different points are uncorrelated. The expressions above may be generalized to the case of cross-correlations between different fields, by simply replacing C̃^w_ℓ with the cross-spectrum between the masks of both fields.

  

 §.§ The case of discretely-sampled fields

    Consider now observations of the field f only at a discrete set of N points on the sphere. Most commonly, these will be the sky positions of sources in a catalog. Let w_i be the weight associated to the i-th point, with sky position _i, and f_i≡ f(_i). To ease notation, in what follows, the subscript _i applied to any function defined on the sphere will be shorthand for the value of that function at _i. The mask and masked field can, in this case, be written as a sum over delta functions:
    
    w()=∑_i w_i δ^D(,_i),f^w()≡ w() f()=∑_i w_if_i δ^D(,_i),

    where δ^D(_1,_2) is the Dirac delta function on the sphere[That is, δ^D(_1,_2)≡δ(φ_1-φ_2)δ(cosθ_1-cosθ_2), where (θ_n,φ_n) are the spherical coordinates of the unit vector _n.]. The harmonic coefficients of w and f^w are then sums over the harmonic functions sampled at the positions of the discrete sources
    
    w_ℓ m=∑_i w_i Y_ℓ m,i^*,f^w_ℓ m=∑_i w_if_i Y_ℓ m,i^*.

    An efficient evaluation of such discrete harmonic transform over a non-regular set of points is, in general, computationally more expensive than standard approaches making use of regular pixelization schemes <cit.>. Nevertheless, approaches for dealing with this problem have been recently developed <cit.>, and we will use the implementation of such “general” spherical harmonic transforms built in the ducc library[<https://mtr.pages.mpcdf.de/ducc>], as described in <cit.>. This paper will not discuss the details of the numerical implementation of these methods and instead will deal with the implementational details of the pseudo-C_ℓ algorithm itself, once these discrete transforms are available. We refer readers to <cit.> further details.

    The pseudo-C_ℓ of both mask and masked field is given by:
    
    C̃^w_ℓ=∑_ijw_iw_j 1/2ℓ+1∑_mY_ℓ m,i^*Y_ℓ m,j,
       C̃^f_ℓ=∑_ijw_iw_jf_if_j 1/2ℓ+1∑_mY_ℓ m,i^*Y_ℓ m,j.

    These equations may be further simplified making use of the addition theorem for spherical harmonics (Eq. <ref>). However, it will be more useful to leave them as is for now.

    

  §.§.§ Noise bias from uncorrelated measurements

      Let us now consider our measurements of the field f=s+n to contain some common signal s and noise component n, which we assume to be uncorrelated among different sources. Under this assumption, the noise contribution to C̃^f_ℓ reads
      
    ⟨ n_i n_j⟩ = δ_ijσ_N,i^2,

      where σ_N,i^2 is the noise variance in the i-th point. In this case, the noise bias only receives contributions from the variance at the same source (i.e., the zero-lag correlation function):
      
    Ñ_ℓ=∑_ijw_iw_j⟨ n_in_j⟩1/2ℓ+1∑_mY^*_ℓ m,iY_ℓ m,j=1/4π∑_i w_i^2σ_N,i^2,

      where, in the last equality, we have made use of the addition theorem for spherical harmonics (Eq. <ref>) for the special case i=j. The noise bias is therefore “white” (i.e., scale-independent), and non-zero at all ℓs.

      Since the standard pseudo-C_ℓ estimator requires us to estimate Ñ_ℓ and subtract it from C̃^f_ℓ, in order to obtain an unbiased estimator of the signal power spectrum S_ℓ, it is important for us to consider practical ways to estimate Ñ_ℓ. Assuming our measurements of f to be dominated by noise at any single point, one potentially simple way to do this would be to estimate Ñ_ℓ directly from the variance of the data itself. This methodology has been commonly applied, in the past, when estimating cosmic shear power spectra, where measurements are indeed dominated by noise <cit.>. 
      In this situation, and looking at Eq. <ref>, we can thus propose the following estimator for Ñ_ℓ:
      
    Ñ^f≡1/4π∑_i w_i^2f_i^2.

      Expanding the field measurements into signal and noise components, we can thus see that Ñ^f is a biased estimate of Ñ_ℓ due to the presence of signal in our measurements:
      
    ⟨Ñ^f⟩=Ñ_ℓ+1/4π∑_i w_i^2⟨ s_i^2⟩.

      For a statistically isotropic field, such as s, the field variance is related to its power spectrum via
      
    σ_S^2≡⟨ s_i^2⟩=∑_ℓ2ℓ+1/4πS_ℓ,

      and thus we can write the bias of Ñ^f as:
      
    ⟨Ñ^f⟩=Ñ_ℓ+Ñ^w σ_S^2,

      where we have defined the combination
      
    Ñ^w≡1/4π∑_iw_i^2.


    

  §.§.§ Mask noise

      Drawing intuition from our discussion of uncorrelated noise in the field, we can see that the pseudo-C_ℓ of the mask C̃^w_ℓ will also receive a white-noise-like contribution from the terms i=j in Eq. <ref>. By inspection, we can see that this is none other than the quantity Ñ^w we just calculated. We will thus refer to Ñ^w as the “mask shot noise”. In terms of it, we can thus write the mask power spectrum as
      
    C̃^w_ℓ=S̃^w_ℓ+Ñ^w,

      where the “signal” component is, mathematically:
      
    S̃^w_ℓ=1/4π∑_i≠ jw_iw_j P_ℓ(μ_ij),

      with μ_ij≡_i·_j, and P_ℓ(μ) the Legendre polynomials. In practice, the fastest way to estimate S̃^w_ℓ is not to compute the double sum above, but to compute C̃^w_ℓ from the harmonic transform of w() (Eq. <ref>), and then subtract Ñ^w from it, calculated using Eq. <ref>.

      The presence of Ñ^w is a feature exclusive of the discrete origin of our observations. It sets a floor below which C̃^w_ℓ cannot deviate significantly and, for most common point distributions of interest in astrophysics, C̃^w_ℓ will, in fact, tend to Ñ^w at high ℓ. This is contrary to the usual scenario for power spectrum estimation over incomplete skies. In more common cases, in which the mask is a smooth function, and has support over a significant fraction of the sky (compared to the angular scales of interest), C̃^w_ℓ decays with ℓ. This causes the sum over ℓ” in Eq. <ref> to converge quickly, and the resulting mode-coupling matrix to be highly concentrated along the diagonal. It is therefore important to understand what the contribution from Ñ^w would be to M^w_ℓℓ' and, in general to the ensemble average of C̃^f_ℓ. Calling these contributions M^w,N_ℓℓ', and (ΔC̃^f_ℓ)_N, we can calculate them by replacing C̃^w_ℓ with Ñ^w in Eq. <ref>, and then substituting the result into Eq. <ref>. The resulting contribution to the mode-coupling matrix is:
      
    M^w,N_ℓℓ'=2ℓ'+1/4πÑ^w,

      where we have made use of one of the orthogonality relations of the Wigner-3j coefficients (Eq. <ref>). We can thus see that the mask shot noise leads to a contribution that couples all multipole ℓs with all other multipoles ℓ' (in principle up to ℓ'=∞). Although this does not imply that the mode-coupling matrix is necessarily singular, it does mean that an accurate implementation of the pseudo-C_ℓ estimator might involve (depending on the size of Ñ^w), estimating M^w_ℓℓ' up to very large values of both ℓ and ℓ' in order to account for mode coupling. This is problematic, since estimating M^w_ℓℓ' is the most expensive step of the pseudo-C_ℓ estimator.

      An alternative to avoid this, is to estimate the contribution to ⟨C̃^f_ℓ⟩ from the mask shot noise term, and to bring it to the left hand side of Eq. <ref>. From our result above, this contribution is, simply:
      
    (ΔC̃^f_ℓ)_N=∑_ℓ'M^w,N_ℓℓ'S_ℓ'=Ñ^wσ_S^2.

      Thus, we find that the contribution to the masked field's pseudo-C_ℓ coming from the mask shot noise is precisely equivalent to the unwanted contribution to our naive estimate of the noise bias in Eq. <ref>. This result allows us to construct a fully unbiased estimator for the power spectrum of a discretely-sampled field with uncorrelated noise, which we describe below.

    

  §.§.§ An unbiased estimator

      Gathering our results from the last two sections, the algorithm presented in this paper is as follows:
      
        
  * Estimate the spherical harmonic coefficients of the masked field, and of the mask (Eq. <ref>).
        
  * From these, compute the pseudo-C_ℓ of the mask (C̃^w_ℓ) and of the masked field (C̃^f_ℓ).
        
  * Estimate the “noise” components of both power spectra, Ñ^w, Ñ^f directly from the measurements, using Eqs. <ref> and <ref>, respectively.
        
  * Subtract both noise contributions to form the “signal” pseudo-C_ℓs of the mask and the masked field[As we discussed, S̃^f_ℓ is actually a biased estimator of the signal component of C̃^f_ℓ, since we are subtracting a contribution sourced by the signal variance. Nevertheless, having corrected C̃^w_ℓ in the same way, the resulting estimator of S_ℓ will be unbiased.] S̃^w_ℓ≡C̃^w_ℓ-Ñ^w, S̃^f_ℓ≡C̃^f_ℓ-Ñ^f.
        
  * Apply the standard pseudo-C_ℓ estimator to these “signal-only” spectra. This means, compute the mode-coupling matrix using S̃^w_ℓ instead of C̃^w_ℓ, and apply its inverse[The usual caveats about power spectrum binning and invertible mode-coupling matrices should be borne in mind, however. See <cit.>.] to S̃^f_ℓ to obtain an unbiased estimate of the signal power spectrum.
      
      To reiterate, subtracting the noise-like contribution from the mask pseudo-C_ℓ has two desirable consequences: first, it automatically allows us to account for the bias induced by correcting for a noise power spectrum computed from our own data. Secondly, the resulting mode-coupling matrix is now computed from the noise-debiased mask spectrum which, by construction, will decay with ℓ (at least for the usual point distributions that make up most cosmological observations). Hence, the sum over ℓ” in <ref> will have better convergence properties, and the mode-coupling matrix will be less numerically unstable.

    

  §.§.§ Noise hardening

      The result presented in Section <ref> may seem slightly disconcerting: by correcting for the noise-like component of the mask pseudo-C_ℓ, which we can estimate entirely from the data, we are able to exactly correct for the bias induced by the contribution to our estimate of the noise power spectrum coming from the signal variance. However, this contribution in principle depends on the true signal power spectrum, which we do not know a priori. For this to be possible, it must be that our correction of the mask power spectrum automatically makes the estimator immune to any noise-like (i.e., constant additive) contribution to the power spectrum. In other words, the corrected mode-coupling matrix M^w,S_ℓℓ'≡ M^w_ℓℓ'-M^w,N_ℓℓ' must satisfy that ∑_ℓ' M^w,S_ℓℓ'S_ℓ'=0 if S_ℓ= constant for all ℓ. In linear algebra terms, M^w,S_ℓℓ' has a zero eigenvalue with eigenvector v_ℓ=1 ∀ℓ. We can show this explicitly:
      
    ∑_ℓ'M^w,S_ℓℓ'   =1/4π∑_ℓ'ℓ”(2ℓ'+1)(2ℓ”+1)ℓℓ'ℓ”000^2[C̃^w_ℓ”-Ñ^w],
       =∑_ℓ”2ℓ”+1/4π[C̃^w_ℓ”-Ñ^w],
       =∑_i≠ jw_iw_j/4π∑_ℓ” m”Y^*_ℓ” m”,iY_ℓ” m”,j,
       =∑_i≠ jw_iw_j/4πδ^D(_i,_j)=0,

      where, in the second line, we have used the orthogonality relation Eq. <ref>, in the third line we have expanded C̃^w_ℓ and Ñ^w using Eqs. <ref> and <ref>, and in the last line we have used the completeness of the spherical harmonic functions (Eq. <ref>).

      In practice, what this means is that this power spectrum estimator is not able to recover any completely flat component of the true power spectrum or, in other words, that it has infinite variance for these flat modes. This is not a strong shortcoming, since the amplitude of such modes can be reconstructed from the zero-lag correlation function (i.e., the variance of the field across all sources), assuming a model for the noise. This also means that, to avoid numerical instabilities, it will be useful to employ singular value decomposition methods before inverting the noise-corrected mode-coupling matrix. The crucial advantage is the fact that we do not need to know which fraction of the total field variance is attributed to uncorrelated noise and which fraction to the signal autocorrelation: changes in either of them will only affect, and are exactly captured by, Ñ^f. This fundamentally establishes the unbiasedness of our estimator.

    

  §.§.§ Cross-correlations

      Cross-correlations between two different fields measured on two disjoint sets of points, or cross-correlations between standard maps and discretely-sampled fields, can be estimated by making use of the standard pseudo-C_ℓ algorithm, taking advantage of discrete harmonic transform methods in the case of the sampled fields. No additional care must be taken in this case about subtracting noise components in the mask or field power spectra, since these are zero by construction. Care must be taken, however, if the two different fields to be correlated have been sampled at the same discrete positions. A good example is the cross-correlation between the cosmic shear signal of a set of sources and the PSF ellipticity at their position, as a diagnostic for potential systematics. In the most general case, different weights may be applied to the sources when mapping either field. In this case, the mask and masked-field pseudo-C_ℓs must be corrected for noise bias.

      Let e and f be the two fields being correlated, and let v_i and w_i, respectively, be the weights applied to each source when mapping either field. The noise-like contributions to the mask pseudo-C_ℓ and to the masked-field pseudo-C_ℓ are simple generalizations of those presented in the case of auto-correlations:
      
    Ñ^vw≡1/4π∑_i v_i w_i,Ñ^ef≡1/4π∑_i v_i w_i e_i f_i.


  

 §.§ Spin-s fields

    Let f^a(), with a∈{1,2} be a spin-s_f field, and let f^α_ℓ m be its spherical harmonic coefficients (with α∈{E,B}, corresponding to E- and B-modes). The two are related via a spin-s_f spherical harmonic transform (and its inverse):
    
    f^a()=∑_ℓ m _s_fY^aα_ℓ m()f^α_ℓ m,↔ f_ℓ m^α=∫ d f^a() _s_fY^aα*_ℓ m(),

    where the generalized spin-s spherical harmonic functions _sY^aα_ℓ m() are defined in Appendix <ref>, and we implicitly sum over all repeated indices (e.g. a and α above).

    The pseudo-C_ℓ of the masked field is defined in the same way as in the scalar case, except now we have to keep track of E/B-mode indices:
    
    C̃^f,αβ_ℓ≡1/2ℓ+1∑_m f_ℓ m^α f_ℓ m^β*.

    The generalization of the scalar pseudo-C_ℓ estimator from Eq. <ref> to spin-s fields is:
    
    ⟨C̃^f,αβ_ℓ⟩=∑_ℓ'( M^w_ℓℓ')^αβ_α'β' S^α'β'_ℓ'+Ñ^αβ_ℓ,

    where the mode-coupling matrix is <cit.>:
    
    M_ℓℓ'^w=2ℓ'+1/4π∑_ℓ”C̃^w_ℓ” (2ℓ”+1)ℓℓ'ℓ”s_f-s_f0^2  D_ℓ+ℓ'+ℓ”,

    and the matrix D_n is:
    
    ( D_n)^EE_EE=( D_n)^EB_EB=( D_n)^BE_BE=( D_n)^BB_BB=1+(-1)^n/2,
       ( D_n)^EE_BB=-( D_n)^EB_BE=-( D_n)^BE_EB=( D_n)^BB_EE=1-(-1)^n/2.


    Following the same logic used in Section <ref>, in the case of discretely-sampled fields, the spherical harmonic transform of a sampled and masked spin-s_f field is
    
    f^w,α_ℓ m=∑_iw_if_i^a _sY^aα*_ℓ m,i,

    and its pseudo-C_ℓ is
    
    C̃^f,αβ_ℓ=∑_ijw_iw_jf_i^af_j^b1/2ℓ+1∑_m _sY^aα*_ℓ m,i _sY^bβ_ℓ m, j.


    Now, let us separate the field into signal and noise components f^a=s^a+n^a, where the signal is statistically isotropic (⟨ s^α_ℓ ms^β*_ℓ'm'⟩=δ_ℓℓ'δ_mm'S^αβ_ℓ), and the noise is uncorrelated between different sampling points, which in this case means ⟨ n_i^an_j^b⟩=δ^abδ_ijσ^2_N,i. Inserting this in the previous equation, and using Eq. <ref>, we obtain the noise bias:
    
    Ñ^αβ_ℓ=∑_iw_i^2σ_N,i^21/2ℓ+1∑_m _sY^aα*_ℓ m,i _sY^aβ_ℓ m,i=δ^αβ/4π∑_iw_i^2σ_N,i^2.

    Based on this result, a potentially useful estimate of the noise bias, using only our measurements of f, would be:
    
    Ñ^f,αβ≡δ^αβ/4π∑_i w_i^2 1/2f_i^af_i^bδ_ab.

    As in the spin-0 case, this estimator will be biased due to the contribution from the signal component of f. To calculate this bias, we take the expectation value of the previous equation, and find:
    
    ⟨Ñ^f,αβ⟩   =Ñ_ℓ^αβ+δ^αβ/4π∑_iw_i^2δ_ab/2⟨ s^a_is^b_i⟩
       =Ñ_ℓ^αβ+δ^αβ/4π∑_iw_i^21/2∑_ℓS^α'β'_ℓ∑_m _sY^aα'*_ℓ m, i _sY^aβ'_ℓ m, i
       =Ñ_ℓ^αβ+δ^αβÑ^w∑_ℓ2ℓ+1/4πS_ℓ^EE+S_ℓ^BB/2
       ≡Ñ_ℓ^αβ+δ^αβÑ^wσ_S^2,

    in analogy with Eq. <ref>, after generalising the expression for the field variance σ_S^2 in Eq. <ref>. In the second line above, we have simply expanded s_i^a in its spherical harmonic coefficients, and used their statistical isotropy. We have then made use of Eq. <ref> in the third line.

    The final step in our derivation shows that the contribution from Ñ^w to Eq. <ref> cancels out the term ∝σ_S^2 in the last equation, thus yielding and unbiased power spectrum estimator. To begin with, let us replace C̃^w_ℓ with N̅^w in Eq. <ref> to calculate the mask noise contribution to the mode-coupling matrix, M^w,N_ℓℓ':
    
    M^w,N_ℓℓ'=Ñ^w 2ℓ'+1/4π∑_ℓ”(2ℓ”+1)ℓℓ'ℓ”s_f-s_f0^2  D_ℓ+ℓ'+ℓ”.

    To simplify this further, note that all elements of D_ℓ+ℓ'+ℓ” either are zero, or take the form [1±(-1)^ℓ+ℓ'+ℓ”]/2 (see Eq. <ref>). We can use the term ∝(-1)^ℓ+ℓ'+ℓ” to swap the sign of the lower spin indices of one of the two Wigner 3j symbols, using Eq. <ref>. The resulting sum over ℓ” for this term is then zero by virtue of Eq. <ref>, and we are left only with the contribution from the first term in the D matrix elements, which is always 1/2 (or zero). Again, using Eq. <ref> on this term, we obtain the simplified result:
    
    M^w,N_ℓℓ'=Ñ^w2ℓ'+1/4π  d/2,

    where the matrix d has elements:
    
    ( d)^EE_EE=( d)^EB_EB=( d)^BE_BE=( d)^BB_BB=( d)^EE_BB=( d)^BB_EE=1,  ( d)^EB_BE=( d)^BE_EB=-1,

    and zero otherwise. Substituting this in Eq. <ref>, and after a bit of algebra[This can be done by writing out d as a 4×4 matrix, and multiplying it by the vector (S_ℓ^EE,S_ℓ^EB,S_ℓ^BE,S_ℓ^BB), with S_ℓ^BE=S_ℓ^EB.], we find that the resulting contribution to ⟨C̃^f,αβ_ℓ⟩ is
    
    (ΔC̃^f,αβ_ℓ)_N=δ^αβÑ^wσ_S^2.


    Thus, the recipe for an unbiased estimator described in Section <ref> can be directly applied to spin-s fields, simply replacing the scalar noise bias Ñ^f by its spin generalization Ñ^f,αβ in Eq. <ref>.

    

  §.§.§ Different spins

      One important generalization of the formalism we just described is the calculation of cross-power spectra between two fields of different spin sampled at the same source positions. A good example of this would be measurements of both the shape (spin-2) and size (spin-0) of galaxies, as proxies for weak lensing shear and convergence. The formalism we just described, in which the mode-coupling matrix is estimated from the noise-debiased mask pseudo-C_ℓ, can be applied directly to these situations without any modifications. In fact, the orthogonality relation Eq. <ref> guarantees that, for fields of different spin, the contribution from Ñ^w to the mode-coupling matrix vanishes exactly. Likewise, by virtue of the addition theorem for spin-weighted spherical harmonics (Eq. <ref>), the noise bias to the pseudo-C_ℓ of the masked fields also vanishes in the limit of uncorrelated noise. Thus, in these cases, one may even forego the calculation of Ñ^w and Ñ^f altogether.

  

 §.§ Clustering

    A conceptually different type of discretely sampled field is the case of galaxy clustering, in which the overdensity of points in the discrete source catalog is itself the field. This case has been discussed in detail in <cit.>, but we repeat part of the discussion here for completeness. In this case, the number density of objects is
    
    n_d()=∑_i=1^N w^d_i δ^D(,_i),

    where w^d_i is the weight of the i-th galaxy (which may be used to account for observational conditions, instrumental effects, such as fiber collisions, and other systematics <cit.>).

    Critical in the study of galaxy clustering is our ability to account for spatial variations in the expected mean number density of sources n̅(), caused by depth variations across the footprint, contamination by galactic or instrumental systematics, and the complex geometry of the observed footprint itself. This has been commonly done through the use of “random catalogs”: synthetic catalogs containing discrete points that trace the underlying expected density of sources, but that are otherwise not clustered. The mean number density of sources in this case is then
    
    n̅()=α∑_i=1^N_r w_i^r δ^D(,_i),

    where w_i^r is the weight of the i-th random point, N_r is the total number of random points, and the normalization factor α accounts for the potentially different number of objects in the data and random catalogs:
    
    α≡∑_i=1^Nw_i^d/∑_i=1^N_rw_i^r.

    Ideally, the field whose power spectrum we wish to estimate is the galaxy overdensity f()≡δ_g()≡ n_d()/n̅()-1, although, obviously, the discretized nature of n̅() complicates the construction of such a map. We can proceed, however, under the following consideration: a close to the optimal choice of mask for the standard pseudo-C_ℓ method is one that is proportional to the inverse noise variance of the field under study. Assuming n_d to be a Poisson realization of n̅(1+δ), a reasonable choice of mask is therefore w()=n̅(). With this choice, the masked field and the associated mask are simply
    
    f^w()=n_d()-n̅()=∑_i=1^Nw_i^d δ^D(,_i)-α∑_i=1^N_rw_i^r δ^D(,_i),
       w()=α∑_i=1^N_r w_i^rδ^D(,_i).


    The methodology described in Section <ref> can therefore be extended directly to the case of galaxy clustering with the following changes:
    
      
  * The SHTs of the masked field and the mask are
      
    f^w_ℓ m=∑_i=1^Nw_i^dY^*_ℓ m,i-w_ℓ m, w_ℓ m=α∑_i=1^N_rw_i^rY^*_ℓ m,i,

      
  * The shot noise components to field and mask are:
      
    Ñ^f=1/4π∑_i=1^N(w_i^d)^2+Ñ^w,
            Ñ^w=1/4π∑_i=1^N_r(α w_i^r)^2.

    

    Although the use of randoms to characterize survey homogeneity has been common, particularly in the analysis of spectroscopic galaxy surveys, it is also possible to map spatial variations in survey homogeneity at every point in the celestial sphere. It is therefore important to also consider the case in which a “completeness map” W is provided, such that W()∝n̅(). In this case, the method must be adapted as follows:
    
      
  * The sky mask is generated by normalizing the completeness map, to make it correspond to the expected number density of galaxies:
      
    w()=α W(),α=∑_i=1^Nw_i^d/∫ d W().

      
  * The SHT of the masked field is still given by the first equality in Eq. <ref>, with the mask SHT w_ℓ m estimated from the continuous w() as usual.
      
  * Since the mask is now a map, rather than a collection of sources, its shot noise is Ñ^w=0. The shot-noise component of the field's pseudo-C_ℓ, in turn, is Ñ^f=∑_i=1^N (w_i^d)^2/(4π).
    



§ VALIDATION
 
  Having presented the mathematical formalism of an unbiased power spectrum estimator for discretely sampled fields, we now proceed to validate it against simulations. We focus on what is likely the most relevant probe for which this formalism is useful: cosmic shear catalogs. The results presented here will therefore correspond to spin-2 fields, but other spin cases do not change the problem conceptually[Except perhaps in the case s=0, where the separation between  E- and B-modes is meaningless.]. We showcase examples of the spin-1 and spin-0 cases in Sections <ref> and <ref>, respectively.

  Cosmic shear measures the apparent distortion of a galaxy's shape due to the gravitational lensing effect caused by the foreground matter distribution integrated along the line of sight. This effect, while weak and indistinguishable from the galaxy's intrinsic ellipticity when considering single objects, represents a powerful probe of the cosmic large-scale structure when studied in a statistical way on large catalogs of galaxies. In the weak lensing regime, the shear γ_a,i that the i-th galaxy's image has undergone, provides a linear contribution to its measured ellipticity e_a,i (estimated from the traceless component of the projected inertia tensor, see e.g. <cit.>):
  
    e_a,i = γ_a,i + n_a,i,

  where a=1,2 labels the two components of ellipticity and shear, and the noise component n_a,i includes both the intrinsic (i.e., unlensed) galaxy ellipticity and any additional measurement noise. Crucially, it is common to work under the assumption that all contributions to n_a,i are uncorrelated between different sources.

  Section <ref> below describes the setup used to validate the catalog-based methodology described in the previous section, comparing it against the results of the standard map-based approach. The results of this validation, as a function of the number density of catalog sources, are then presented in Section <ref>.

  

 §.§ Validation setup

    To validate our estimator, we generate 500 synthetic cosmic shear catalogs. Each catalog is created by first generating a sky realization of the spin-2 cosmic shear field as a Gaussian random field from a theoretical E-mode power spectrum. This realization is generated using the pixelization scheme with a resolution parameter N^ hires_ side=4096, corresponding to a pixel size of ∼0.86'. The native resolution is chosen to be significantly higher than the angular scales we will try to extract using either the map-based or catalog-based methods, to ensure that the finite-resolution effects inherent to these ground-truth maps are negligible compared to the typical statistical error. From this map, we then produce a mock observed shear catalog, by sampling it at the positions of a discrete set of sources. Specifically, to each source we assign the value of the shear field corresponding to the pixel it falls in[Unfortunately, more sophisticated interpolation schemes are highly computationally intensive for .]. We do not add any shape noise to the catalog, in order to reduce the statistical uncertainties of the recovered power spectra, thereby maximizing our sensitivity to any potential bias.

    Once a mock shear catalog has been generated, we estimate its angular power spectrum using the catalog-based method described above, and the standard map-based method. For the map-based case, we follow the procedure outlined in <cit.>: a map of the average shear in each pixel is constructed at a resolution N_ side=1024. In each pixel, the local shear field is calculated as the weighted average of the source ellipticities in it. As our simulated catalogs do not contain any additional shape noise, we do not subtract any data-based noise bias in the standard map-based approach (although we do subtract Ñ^f and Ñ^w in the catalog-based method, as described in Section <ref>). The pixelization suppresses the small-scale information in the shear map, reducing the map RMS by a pixel window function, which needs to be accounted for when computing pseudo-C_ℓs <cit.>. Assuming that the pixelization is coarse enough so that every pixel contains at least one galaxy, we can approximate the pixel window function in real space by a Gaussian smoothing kernel with an FWHM of ∼ 41.7 deg/N_ side size <cit.>. We do not correct for this bias in the simulated data, but account for it when comparing with the bandpower-convolved theory power spectrum.

    In order to produce mock catalogs with realistic source distributions, we use the angular positions and shape weights of real data from the public KiDS-1000 shear catalog <cit.>. In particular, we use all sources from the fourth photometric bin, which contains 4,544,395 sources, observed in two different fields, covering an area of approximately 800  deg^2 after masking. The ground-truth maps were generated using the theoretical prediction for this redshift bin based on the best-fit ΛCDM parameters from <cit.>, assuming no intrinsic alignments. We note that, by choosing to use real KiDS-1000 sources, we ensure that the method is validated on catalogs with not only a realistic number density, but also a spatial distribution that reproduces both the expected clustering of sources, and their inhomogeneity caused by depth variations across the observed footprint. In order to quantify the performance of the catalog- and map-based methods as a function of source number density, we repeat our analysis on down-sampled versions of these catalogs, in which we select a fraction of the KiDS-1000 sources. Specifically, we will study the results of selecting only 1% and 0.01% of the sources. We choose a multipole binning of constant width Δℓ=30 up to a maximum of ℓ_ max=2N_ side=2048.

  

 §.§ Validation results


    

    Figure <ref> shows the main results of our validation exercise. The upper right panel shows the results in the fiducial setup, in which all the KiDS-1000 sources are used to measure the power spectrum. The power spectrum averaged over all simulations is shown as orange and blue circles for the catalog- and map-based approaches, respectively. The theoretical predictions, estimated by convolving the input power spectrum with the bandpower window functions corresponding to the mode-coupling matrix derived with either method, are shown as dashed red and solid grey lines (the latter barely visible behind the red dashed lines). We multipliy the theory spectra by the squared pixel window function (N_ side^ hires=4096 for the catalog-based estimator, N_ side=1024 for the map-based one). The narrow subpanel below shows the residual of each method with respect to its corresponding prediction normalized by the error on the mean of the 500 simulations (i.e., the scatter across simulations divided by √(500)). The horizontal grey bands show 3-σ confidence intervals.
    
    The catalog-based method is able to produce an unbiased estimate of the power spectrum, while at ℓ≳ 500, the map-based approach deviates significantly due to pixelization effects at the map level and shot-noise-related bias in the estimator itself. Note that, in a realistic scenario, the statistical uncertainties would be at least √(500)∼22 times larger, not accounting for the impact of shape noise, and hence these deviations would likely be subdominant. We discuss the various sources of this bias in Appendix <ref>. For the catalog-based estimator, we must account for the pixelization of our ground truth shear map, and can do so accurately by means of the window function.[The simulated catalog contains field values that are, by design, constant within every pixel of the ground truth map. This makes the catalog-based field effectively band-limited, with no power at scales below the pixel scale, and we can account for this by means of the analytic pixel window function.] Ideally, we could choose an even higher N_ side^ hires to avoid this effect, but given the higher numerical cost, we prefer to correct the theory accordingly. With the map-based approach, the estimate is biased both for the uncorrected case (not shown in the figure) and the case where the theory is convolved with the pixel window function (shown). As explained in Appendix <ref>, the two main reasons are the impact of aliasing, and the numerical instability of the pixel-based mode-coupling matrix due to mask shot noise. Note that this coupling matrix is truncated to multipoles ℓ<3N_ side, and thus fails to capture the full correlation between scales induced by the mask for the sparsest catalogs.

    The bottom panels of Fig. <ref> show the results of repeating this analysis on catalogs containing 100 times and 10,000 times fewer sources (left and right panels, respectively). We find that, in spite of the substantial drop in number density, the catalog-based method is able to recover unbiased power spectrum measurements in both cases within the range of scales of interest. The map-based approach, however, recovers significantly biased measurements, which cannot be corrected for by our analytical pixel window function. This bias becomes gradually larger on smaller scales for the 1% catalog, reaching very significant levels after ℓ∼100. For the 0.01% catalog, in turn, the map-based method fails catastrophically, due to significant shot noise in the field pseudo-C_ℓ estimator, as well as numerical instabilities in the estimated mode-coupling matrix. These lead to the method misestimating the measured power spectrum at all scales (bottom right panel). In contrast, the catalog-based approach remains unbiased even for this extremely low number density (the shear field is sampled at the positions of just 454 sources). For the thinned catalogs, the catalog-based estimator is again fully unbiased even at scales ℓ>1024 after accounting for pixelization-related bias mentioned above, which becomes less important for smaller source densities due to higher estimator noise (∼2σ and ≲1 σ at ℓ=1024 for 1% and 0.01% thinning, respectively).
    
    
    The upper left panel of Fig. <ref> shows the mask pseudo-C_ℓ in the catalog-based case, C̃^w_ℓ, for the fiducial catalog, and the 1% and 0.01% catalogs (from top to bottom). In each case, the mask shot noise level Ñ^w is marked by horizontal black dashed lines, and the raw mask pseudo-C_ℓs are shown as the solid lines. We see that, for the original KiDS source density, the mask shot noise is low enough to not affect the mode-coupling matrix significantly, except above ℓ∼2000. In turn, the 1% catalog becomes shot-noise dominated at ℓ∼200 (compared to ℓ∼100 where the decoupled power spectrum estimate starts deviating significantly from the input). Finally, the 0.01% catalog is shot-noise dominated at almost all scales, leading to severe numerical instabilities in the standard pixel-based algorithm, which does not account for the impact of mask shot noise.

    

    Figure <ref> shows the mode coupling matrices computed from mock catalogs, and compares the result to the map-based analog. We show this for the full KiDS-1000 catalog (blue) and the two thinned versions (orange, green) but focus on the former for now. The left panels show specific columns of the ^EE_EE component of the catalog-based corrected mode coupling matrix M_ℓℓ'^w,S (solid lines, see Sect. <ref>) and the standard pixel-based mode coupling (dashed lines), normalized to one at ℓ=ℓ'. The right panels show, in the same units, the relative difference between the columns of the map-based mode coupling matrix (<ref>) and the catalog-based one[For a direct comparison between the catalog-based and the map-based mode coupling matrix elements, we need to divide the latter by the squared pixel area, (4π/N_ pix)^2.]. From top to bottom, we display columns with ℓ=(30,  100,  800,   2000). We find that, in the case of the full catalog, both matrices differ by a factor of less than 10^-4, indicating that the mask shot noise correction is subdominant. The matrix columns peak at ℓ=ℓ' as expected, rapidly decline to percent level at |ℓ-ℓ'|∼ 200, and converge to a nearly constant value at the 0.1 percent level. 

    For the 1%-thinned catalog, both methods start differing at the percent level. The elements of the catalog-based matrix match on average those for the full catalog, but are much noisier for couplings between distant multipoles, while the pixel-based coupling matrix includes a now very significant contribution due to mask shot noise, with the expected scaling ∝ (2ℓ+1), see Eq. (<ref>). For the highly shot-noise dominated case with 454 sources, the catalog-based mode coupling becomes more noisy, with fluctuations reaching percent level, but its mean stays compatible with the one for the full catalog. On the other hand, the pixel-based coupling matrix is almost completely dominated by the shot noise component, and differs from the catalog-based version by more than 100% at ℓ≳500. This shows that even with an extremely low source number density, the catalog-based approach recovers a stable mode coupling matrix that allows us to estimate unbiased power spectra from sparsely sampled sources on the sky. 
    
    

    To understand what causes the bias in the pixel-based estimator, we compare both approaches in terms of the coupled pseudo-C_ℓ estimators. Figure <ref> shows, in the same color scheme as in Fig. <ref>, the coupled shear EE pseudo-C_ℓs (see Eq. <ref>) computed using the catalog-based and pixel-based approach, and compared to the input theory, multiplied by the square of the respective pixel window, and coupled with the respective mode coupling matrix. The field noise level N_f is shown as a horizontal black dashed line for comparison. The left and the right panel show the cases of 1% and 0.01% thinning, respectively, and the lower panels show the bias with respect to the mode-coupled theory prediction. The pixel- and catalog-based coupled estimators differ at similar scales as their decoupled versions shown in Fig. <ref>, but without being affected by the numerical instability of the inverse binnned mode-coupling matrix.
    
    In contrast, the coupled theory bandpowers do depend on the mode coupling matrix. We see that the catalog-based theory prediction becomes noisy for small scales, which is inherited from the noisy mode-coupling matrix seen in Fig. <ref>. For both thinned catalogs, the catalog-based estimator is unbiased, while the map-based estimator receives a bias at intermediate and small scales. In summary, Fig. <ref> illustrates that for low source number density, the pixel-based coupled pseudo power spectrum is shot-noise dominated. Analogously, the mask shot noise dominates the pixel-based mode coupling at small scales and leads to a severe misestimation of the mode-coupled theory theory prediction. The decoupled pixel-based estimator inherits this field noise bias, and suffers from numerical instabilities from applying the inverse mode coupling matrix. In contrast, the catalog-based estimator does not deviate significantly from theory predictions, even when considering the statistical power of 500 noiseless simulations combined.




§ EXAMPLE APPLICATIONS
 
  

 §.§ Spin-2 example: cosmic shear from DES-Y3
 
  As a first real-data example, we apply our estimator to the highest redshift bin of the galaxy shear catalog from DES-Y3 <cit.>. This catalog contains 𝒪(10) more galaxies in a four times larger survey footprint than the fourth redshift bin of KiDS-1000 used in Sect. <ref>. Cosmic shear is a common example of a spin-2 field sampled at discrete positions, and one of the most powerful probes of the cosmic LSS. It measures the lensing-induced distortion of galaxy shapes due to the foreground matter distribution along the line of sight. In the Limber approximation <cit.>, the cosmic shear angular power spectrum is related to the 3D matter power spectrum P(k,   z(χ)), and can be modeled as
  
    C_ℓ = G_ℓ^2 ∫χ/χ^2  q^2(χ)   P( k=ℓ + 1/2/χ,   z(χ) )   ,

  where k is the modulus of the Fourier wave vector, z is the redshift, and χ is the comoving radial distance. The lensing kernel q(χ) quantifies the radial contribution of foreground matter to the angular galaxy lensing power spectrum, and depends on the redshift distribution p(z) of galaxies that may act as lensed images. In the case of a flat universe in which general relativity is valid and c=1, the lensing kernel is given by
  
    q(χ) = 3/2 H_0^2 Ω_m χ/a(χ)∫_z(χ)^∞ z’   p(z’)χ(z’) - χ/χ(z’)  ,

  where H_0 is the Hubble constant, Ω_m is the matter density parameter, and a=1/(1+z) is the scale factor. Transitioning from the 3D Laplacian of the gravitational potential to the angular Hessian of the associated lensing potential requires the correction factor <cit.>
  
    G_ℓ≡√((ℓ+2)!/(ℓ-2)!)1/(ℓ + 1/2)^2.

  Galaxy shear data are notoriously biased by galaxy shape noise, which has to be corrected for explicitly in the pixel-based approach <cit.>, but is removed at the catalog level using our estimator.

  We use the fourth redshift bin of the DES-Y3 data, containing 25,091,297 catalog galaxies between redshifts ∼0.4 and ∼1.5. We correct the catalog galaxy ellipticities for linear and multiplicative bias as described in <cit.>. The survey footprint covers an effective sky area of 4143 deg^2 as shown in the left panel of Figure <ref>. The weighted source number density of DES-Y3 is n_ eff = 5.59  gal/ arcmin^2, compared to 6.17  gal/ arcmin^2 for KiDS-1000, yielding a similar mask shot noise level.

  Figure <ref> in the right panel shows the results in terms of the decoupled power spectrum, comparing our catalog-based approach with recent pixel-based measurements using the pixel-based approach published in <cit.>. Both estimators use the same set of multipole bins up to ℓ=8192, corresponding to 2 N_ side of the map used with the pixel-based estimator, or an angular scale of ∼26 arcsec. We corrected the map-based estimate by the analytical pixel window function. Both results use 1-sigma error bars from <cit.>, derived from an empirical power spectrum covariance that includes a Gaussian and a non-Gaussian super-sample contribution. The lower panel shows the relative difference between both methods in units of sigma. We find that both methods agree within 1 sigma within the shown multipole range.

  


  

 §.§ Spin-1 example: proper motion surveys

    
    Although most quantities of cosmological interest are either scalars or spin-2 objects (e.g. cosmic shear, CMB polarization), there are examples of particularly useful spin-1 fields, especially in terms of catalog-based observations. Examples of these are lensing displacements, galaxy angular momenta, and transverse peculiar velocities <cit.>. Astrometric surveys, such as the satellite, are able to track the angular position of sources over time with high precision. Although the main use of such observations is reconstructing the dynamics of stars in the Milky Way, it has been argued that they may be useful for cosmology when applied to extra-galactic sources, placing constraints on the local rate of expansion <cit.>, detecting the Solar System's proper motion dipole <cit.>, and even constraining structure growth <cit.>.

    As an example application, we generate simulated observations of quasar proper motions for a -like sample. In particular, we use the positions of quasars in the public quasar sample <cit.>. We select 342,971 quasars with spectro-photometric redshifts in the range 1<z_ sp<1.5. The spatial and redshift distribution of this sample is shown in Fig. <ref>. Next, we generate the theoretical prediction for the angular power spectrum of the expected cosmological proper motion signal. Following the formalism in <cit.>, the projected proper motion map Ȧ is a spin-1 field given by
    
    Ȧ()=∫dz/(1+z)χp(z) (1-^T)· v(χ),

    where p(z) is the redshift distribution of the sample, v is the peculiar velocity, projected on the plane transverse to the line of sight via the projector 1-^T. At linear order in perturbation theory, Ȧ is a pure E-mode field, since it can be written as the covariant derivative of a projected velocity potential <cit.>. Thus, in what follows, we study only the power spectrum of its E-mode component.

    We compute a theoretical prediction for the signal power spectrum using the redshift distribution of the sample and assuming standard ΛCDM cosmological parameters. We then generate Gaussian realizations of this field at resolution N_ side=1024, and sample them at the positions of the quasars. Since the peculiar motion signal is most relevant at large scales, and given the relative sparsity of the sample, we will only attempt to reconstruct the power spectrum up to scales ℓ=360, and thus this resolution is sufficient. We generate 100 such simulated catalogs, estimate their power spectrum using the catalog-based method presented here and the standard pixel-based approach, using maps of resolution N_ side=128 in the latter case.
    

    The result of this exercise is shown in Fig. <ref>. The upper panel shows the E-mode power spectrum bandpowers measured and averaged over 100 simulations, as well as the associated theoretical prediction, given by the input power spectrum convolved with the bandpower window functions. The lower panel shows the relative residuals between simulations and theory prediction in both cases. We find that, while the catalog-based approach is able to obtain an unbiased estimate of the power spectrum at all scales, the pixel-based method under-predicts it severely on small scales (ℓ≳70).

  

 §.§ Spin-0 example: FRB-based dispersion measure maps
 
    As an example of a scalar (spin-0) quantity sampled at discrete positions, we study the case of maps of the projected electron density distribution constructed from the dispersion measure of fast radio bursts (FRBs). FRBs are powerful millisecond-long radio pulses emitted by (most likely) extra-galactic sources of unknown nature. As they propagate through the interstellar and intergalactic plasma, the group velocity of these pulses changes with frequency. This leads to a time delay between pulses at different frequencies ν that scales as Δ t=t_0  DM/ν^2, where t_0≡ 4.15×10^-3   s  GHz^2  cm^3  pc^-1, and the dispersion measure DM is
    
    DM()≡∫_0^χ_sdχ/an_e(χ),

    with χ_s the comoving radial distance to the source, a the scale factor, and n_e( x) the comoving electron density. As in the case of cosmic shear and peculiar transverse velocities, one can thus make use of FRB catalogs to construct maps of the projected electron density. For a catalog with a redshift distribution p(z), this is given by <cit.>
    
    DM()=∫dz/H(z)(1+z) P(>z) n_e(χ(z))  , P(>z)≡∫_z^∞ dz' p(z').

    where P(>z) is the cumulative redshift distribution.
    

    Current FRB catalogs are significantly sparser than the transverse velocity and cosmic shear examples from previous sections, with existing FRB samples containing a few hundred sources (although this number will grow significantly in the future <cit.>). Another distinctive feature of FRB catalogs is the fact that DM measurements are highly accurate. Typically, the DM can be measured with a precision of ∼0.1  cm^-3 pc, while extragalactic DM values are orders of magnitude higher (e.g. DM_z∼0.8∼1000  cm^-3  pc), and the contributions from the Milky Way are also comparatively small (DM_ MW∼ 50  cm^-3 pc). Typical host contributions are likely of the order DM_ host∼100  cm^-3  pc (although the distribution of host DM may have significant tails <cit.>). Thus, in this case, the per-source measurements of the field are signal-dominated, unlike the previous examples.

    To showcase the ability of the method presented here to recover accurate power spectrum measurements in this regime, we use the latest public catalog of FRBs made available by the CHIME collaboration <cit.>. After removing duplicates, the catalog contains 492 unique sources, with DM values in the range DM≲2000. The left panel of Fig. <ref> shows the spatial distribution of this sample and their measured dispersion measures. We assign a nominal redshift to each source from their measured DM using the background relation between DM and redshift:
    
    DM(z)- DM_0=∫_0^zdz'/H(z') (1+z') n̅_e,n̅_e=3H_0^2Ω_b/8π G m_px_e(1+x_H)/2,

    where m_p is the proton mass, H_0 is the expansion rate today, Ω_b is the fractional density of baryons (which we set to Ω_b=0.05), x_e is the ionization fraction (which we set to x_e=1), and x_H is the hydrogen mass fraction (which we set to x_H=0.75). We fix the offset DM_0=100  cm^-3  pc to account for the typical host / Milky Way contribution (although our results do not depend on this choice). The corresponding redshift distribution of the sample is shown in the right panel of Fig. <ref>, and can be approximated as p(z)∝ z^2exp(-z/0.15) (solid line in the same figure).
    

    We use this redshift distribution to generate a theoretical prediction for the power spectrum of the DM field in Eq. <ref>, assuming a linear bias for the electron overdensity of b_e=0.8 <cit.>. We generate a suite of 1000 simulated Gaussian realizations of DM() at a resolution of N_ side=512, which we then sample at the positions of the CHIME catalog sources. Given the sparsity of the sample, we only attempt to reconstruct the power spectrum up to scales ℓ=180, and hence this reduced resolution is sufficient. The left panel of Fig. <ref> shows, in orange error bars, the reconstructed power spectrum averaged over all simulations, together with its theoretical prediction (i.e., the bandpower-convolved power spectrum used to generate the simulated maps) in solid black lines. In blue error bars, we show also the result of using the standard, pixelized version of the estimator, using a resolution N_ side=64. The numerical instability of the mode-coupling matrix, and aliasing due to limited pixel resolution, causes the standard estimator to fail catastrophically in this case.
    
    The right panel of the same figure shows rows of the normalized mode-coupling matrix M_ℓℓ' for four different values of ℓ. Results are shown for the catalog-based and pixel-based approaches as solid and dotted lines, respectively. In the catalog-based case, M_ℓℓ' displays significant coupling between multipoles separated by up to Δℓ∼20, with a more non-trivial structure at larger multipole separations. In turn, the pixel-based approach, which does not correct for the effect of mask shot noise, shows significant correlation extending across the whole range of multipoles, growing like (2ℓ'+1) (see Eq. <ref>). The mode-coupling matrix in this case is thus significantly more numerically unstable. Together with the impact of aliasing (see Appendix <ref>), this leads to the failure observed in the right panel.

  

 §.§ Clustering example

    We end this section showcasing the applicability of the discrete pseudo-C_ℓ approach to the case of source clustering, in which the density distribution of sources is itself the field under study. The formalism for this case was first presented in <cit.>, and we described it within the general framework presented here in Section <ref>. To validate our implementation of the formalism in , we generate fast simulations of a galaxy clustering sample as follows:
    
      
  * We calculate an initial angular power spectrum corresponding to the projected ovedensity of sources with a linear galaxy bias b=1 and a normal redshift distribution centered at z=0.6 with a standard deviation σ_z=0.1.
      
  * We create a completeness map c() to simulate the observational survey inhomogeneities that the random catalogs described in Section <ref> aim to trace. To do this, we draw 200 random unit vectors and select a spherical cap with a 10^∘ radius centered at each of them. For each cap, we then draw a random number between 0.97 and 1 to represent the sky completeness in that patch (i.e., up to 3% incompleteness). We multiply the resulting map by a sky mask that removes regions of high dust extinction and star contamination, constructed as described in <cit.>.
      
  * We generate a Gaussian overdensity field, δ_G() from this power spectrum, and perform a lognormal transformation to turn it into a positive-definite angular density field:
      
    n^ true_g()=n̅_g exp(δ_G()-σ_G^2/2),

      where σ_G is the standard deviation of δ_G, and n̅_g is the desired average number of sources per steradian[The -σ_G^2/2 counterterm ensures that the mean of the resulting field is indeed n̅_g, assuming δ_G is Gaussian.]. The observed mean source density is then constructed by multiplying this field by the completeness map described above: n^ obs_g()=c() n^ true_g().
      
  * We Poisson-sample this field, assigning each source a random position within its pixel, to generate the “data” catalog. The “random” catalog is generated by drawing random positions on the sphere, and discarding them with a probability given by the local value of the completeness map. The random catalog thus constructed has 50 times more sources than the data.
    
    

    Keeping the completeness map and associated random catalog fixed, we generate 100 data simulations at N_ side=1024 with a number density n̅_g=0.01  arcmin^-2, and estimate the power spectrum of each simulation using the procedure described in Section <ref>. The use of a lognormal transformation to generate the galaxy density field prevents us from using the initial angular power spectrum as a theoretical prediction to compare against. Although the effect of a lognormal transformation can be propagated to the power spectrum analytically (see <cit.>) we resort to a more direct approach that avoids potential numerical instabilities of Hankel transforms: for each simulation, we calculate the power spectrum of the lognormal density field (δ_ LN=exp(δ_G-σ_G^2/2)-1) on the full sky (i.e., in the absence of completeness, masking, and discrete-source effects). The “true” power spectrum against which we compare the result of our simulations is then given by the average of this power spectrum over simulations corrected for the impact of the pixel window function used in these simulations.

    The left panel of Fig. <ref> shows the completeness map used in this exercise. The right panel of the same figure shows the mean power spectrum recovered from our simulation (blue points with error bars), the true C_ℓ (orange line), and the relative residual between them (bottom subpanel). The catalog method implemented here is able to recover the true underlying power spectrum at very high accuracy (the error bars shown represent the statistical uncertainties in the mean, corresponding to 1/10 of the error of a single realization).



§ CONCLUSION
 
  In this work, we present a simple extension to the standard pseudo-C_ℓ estimator to tackle the problem of fields sampled at the positions of a discrete and finite catalog of sources. By treating the problem directly at the catalog level, without resorting to an intermediate pixelization of the sources onto a regular grid, we demonstrate that the method is able to tackle three prevalent sources of systematic bias in the standard pixel-based approach: resolution and aliasing effects due to the finite pixel area, shot noise in the field estimate due to the discrete nature of the field tracers, and numerically unstable mode-coupling matrices due to shot noise in the pixelized mask constructed from the sources. We interpret the catalog data as representing a masked field, with a mask given by a set of delta functions at the position of each source, and take advantage of new methods developed for the calculation of spherical harmonic transforms on irregular grids. We show that a simple correction of the shot-noise contributions to the power spectra of the masked field and of the mask itself, using fast and model-independent analytical expressions, is able to significantly improve the numerical stability of the mode-coupling matrix, while simultaneously rendering the estimator immune to any uncorrelated noise bias, irrespective of the size of the signal contribution. The method is a generalization of the technique presented in <cit.> in the context of galaxy clustering. The most evident use of this method is in the analysis of cosmic shear data, although it is broadly applicable to any datasets described by observed properties of discrete sources.
  
  We validate our implementation of this method on a set 500 simulated KiDS-1000-like cosmic shear catalogs, with sources sampled at their real sky positions. We show that the resulting estimator is unbiased even for unrealistically small samples (100 and 10,000 times less dense than the real KiDS-1000 sample), where mode-coupling instabilities due to shot noise render standard pixel-based methods unusable. Even at realistic sample sizes, including the larger densities achieved by deeper samples, we show that pixel-based methods encounter difficulties to accurately characterize the effects of pixelization on the small-scale power spectrum. While these can be circumvented through the use of higher pixel resolutions, this is at a significantly higher cost in computational time and complexity, which the catalog-based method presented here avoids altogether.

  To demonstrate the practical use of our estimator on other suitable astrophysical datasets, with varying sample sizes and signal-to-noise ratios, we apply it in four additional distinct scenarios. First, we demonstrate its feasibility in the analysis of significantly larger catalogs, reproducing the analysis of the 3-year DES cosmic shear sample, comparing the results to those presented in <cit.>. We then study the case of a spin-1 field: the peculiar transverse motion of extragalactic objects as traced by an astrometric survey, using the quasar sample. Besides the different spin, this type of dataset features a significantly lower source density and signal-to-noise ratio than cosmic shear. We also consider measurements of the spin-0 projected electron density in the form of Fast Radio Burst catalogs. These correspond to samples with extremely low number densities (10^2-10^3 objects) but a very high signal-to-noise ratio. Finally, we showcase the applicability of our formalism to the analysis of galaxy clustering, in which the density of catalog sources is itself the field of interest.  Our results from all four examples confirm our previous findings that our estimator is unbiased and can overcome the numerical challenge of source sparseness when characterising mode coupling.

  We expect the method presented to be useful in the analysis of upcoming large-scale structure datasets, particularly in the context of cosmic shear, as well as novel probes such as FRBs, radial and transverse peculiar motion surveys, weak lensing of gravitational waves, and any other catalog-based observations. Our implementation is made available with the latest release of the public power spectrum package [<https://github.com/LSSTDESC/NaMaster/>]. 



§ ACKNOWLEDGMENTS

  When the work presented here was at an advanced stage, we learnt of similar work being developed by Tessore et al. 2024 <cit.>. We thank the authors of this work, particularly Nicolas Tessore and Benjamin Joachimi, for sharing their draft paper and for useful discussions. We also thank Anton Baleato, Elyas Farah, Robert Reischke, and Martin White, for useful comments and feedback. KW acknowledges support from the Science and Technology Facilities Council (STFC) under grant ST/X006344/1. DA acknowledges support from the Beecroft trust. We made extensive use of computational resources at the University of Oxford Department of Physics, funded by the John Fell Oxford University Press Research Fund.



§ GENERALITIES OF SPHERICAL HARMONIC FUNCTIONS

  We use the following definition for the generalized spin-s spherical harmonic functions[The -1 sign preceding these definitions is a mostly inconvenient convention that is dropped for the case of spin-0 fields, so that _0Y^1E_ℓ m= _0Y^2B_ℓ m=Y_ℓ m, and _0Y^2E_ℓ m= _0Y^1B_ℓ m=0.]:
  
    _sY^1E_ℓ m=-1/2[ _sY_ℓ m+(-1)^s _-sY_ℓ m],
        _sY^1B_ℓ m=-i/2[ _sY_ℓ m-(-1)^s _-sY_ℓ m],
    
        _sY^2E_ℓ m=i/2[ _sY_ℓ m+(-1)^s _-sY_ℓ m],
        _sY^1B_ℓ m=-1/2[ _sY_ℓ m-(-1)^s _-sY_ℓ m],

  where _sY_ℓ m() are the spin-weighted spherical harmonic functions of spin s.
  These functions satisfy the following properties:
  
    ∫ d _sY_ℓ m^aα*()_sY_ℓ' m'^aβ()=δ_αβδ_ℓℓ'δ_mm',
       ∫ d _sY_ℓ m^aα*()_sY_ℓ' m'^aβ() Y_ℓ”m”()=(-1)^s+m√((2ℓ+1)(2ℓ'+1)(2ℓ”+1)/4π)
       ℓℓ'ℓ”-mm'm”ℓℓ'ℓ”s-s0 d^αβ_ℓ+ℓ'+ℓ”,
       d^EE_n=d^BB_n=[1+(-1)^n]/2,d^EB_n=-d^BE_n=-i[1-(-1)^n]/2.
       ∑_m _sY_ℓ m^aα*() _s'Y_ℓ m^aβ()=2ℓ+1/4πδ_αβδ_ss'.

  The last relation can be derived from the addition theorem for spin-weighted spherical harmonics <cit.>. For scalar harmonics, it reads simply
  
    ∑_m Y^*_ℓ m(_1)Y_ℓ m(_2)=2ℓ+1/4π P_ℓ(_1·_2)   ,

  which, for _1 = _2, simply evaluates to (2ℓ+1)/(4π). Furthermore, the spherical harmonic functions are a complete basis on the sphere, so that:
  
    ∑_ℓ mY^*_ℓ m(_1)Y_ℓ m(_2)=δ^D(_1,_2).

  We will also employ the following properties of the Wigner 3j symbols:
  
    ∑_mm'ℓℓ'ℓ”mm'm”ℓℓ'ℓ”'mm'm”'=δ_ℓ”ℓ”'δ_m”m”'/2ℓ”+1,
       ∑_ℓ”(2ℓ”+1)ℓℓ'ℓ”s-s0ℓℓ'ℓ”s'-s'0=δ_ss',
       ℓℓ'ℓ”-m-m'-m”=(-1)^ℓ+ℓ'+ℓ”ℓℓ'ℓ”mm'm”.




§ PIXELIZATION EFFECTS IN CATALOG-BASED OBSERVABLES

  The main text has emphasized the impact of the numerical instability of the mode-coupling matrix due to the presence of shot noise in the mask power spectrum for catalog-based observables that are first pixelized before estimating their angular power spectrum. This appendix explores the two other sources of potential bias that affect pixelized catalog-based data: the impact of pixel window functions, and aliasing.
  

  For this exercise we generate two types of simulated realizations of a  scalar Gaussian random field using a base pixelization of N_ side=1024:
  
      
  * Band-limited realizations with a power spectrum of the form
      
    C_ℓ=exp[-(ℓΘ)^2]/ℓ+50.

      The exponential cutoff, with Θ=0.3^∘ sharply reduces any structure on scales ℓ≳200.
      
  * Non-band-limited realizations with a power spectrum of the form
      
    C_ℓ=1/ℓ+50,

      which is rather flat, decaying significantly more slowly at high ℓ than the previous case.
  

  First, to isolate the effects of pixel averaging and aliasing from the practicalities of fields sampled at arbitrary discrete positions, we construct maps at lower resolution, N_ side=256 and N_ side=64 by simply averaging the N_ side=1024 maps onto larger pixels. We calculate the angular power spectrum of each of these maps, and estimate the effective pixel window function of the measured power spectrum as:
  
    b^ eff_ℓ≡√(Ĉ_ℓ/C_ℓ^ in),

  where Ĉ_ℓ is the estimated power spectrum, and C_ℓ^ in is the input power spectrum (given by Eq. <ref> or Eq. <ref>). We repeat this for 100 simulations and compute the mean and scatter of the effective pixel window function.

  The top left and top right panels of Fig. <ref> shows the result of this exercise for the band-limited and non-band-limited realizations, respectively. We find that, for the band-limited case, the impact of pixel averaging is captured by the pixel window function with relatively good accuracy[The slight difference for the N_ side=256 maps is due to the fact that averaging over only 4×4 sub-pixels is less representative of a true area averaging than the 16×16 pixels for N_ side=64.]. In the case of non-band-limited simulations, however, we see very significant differences in the effective pixel window function. These differences are caused by aliasing of the small-scale power in the base maps onto the larger pixels via averaging. Interestingly, the resulting effective pixel window function has a similar shape both in the N_ side=256 and the N_ side=64 case.

  We now repeat the exercise with a key modification: we sample the base maps at the discrete positions of 10^6 sources drawn at random on the sphere, and we then construct maps of the resulting field using the standard pixel-based approach at N_ side=256 and N_ side=64 (corresponding to ∼30 and ∼1.3 sources per pixel on average, respectively). When calculating the power spectra of theses maps we use the density of said sources as their corresponding masks. Finally, to factor out the effects caused by shot noise in the mask, we subtract the analytical shot noise contribution to the mask and field pseudo-C_ℓ (Ñ^w and Ñ^f), mimicking the procedure described in the main text to avoid these effects. The bottom left and right panels of Fig. <ref> show the results of this exercise for the band-limited and non-band-limited maps, respectively. The conclusions are analogous to those found in the previous exercise: regardless of the average number of sources per pixel, the most relevant effect is aliasing in the case of non-band-limited data. The effects of pixelization on band-limited data are well captured by the pixel window function.





 Paper Title: Blind and robust estimation of adaptive optics point spread function and diffuse halo with sharp-edged objects.

 








Application to asteroid deconvolution and moon enhancementBased on observations made with ESO Telescopes at the Para­nal Observatory under programmes ID 60.A-9362(A) (Yang et al. 2014), ID 296.C-5038(A) (Yang et al. 2016), ID 199.C-0074 (Vernazza et al. 2021) and reduced data publicly available at https://observations.lam.fr/astero/https://observa-tions.lam.fr/astero/ and observations obtained at Keck Observatory under programme ID U58N2 (de Pater et al. 2005).




	LESIA, Observatoire de Paris, Université PSL, Sorbonne Université, Université Paris Cité, CNRS, 5 place Jules Janssen, 92195 Meudon, France
	National Astronomical Research Institute of Thailand, Center for Optics and Photonics, 260 Moo 4, T. Donkaew, A. Maerim, Chiang Mai 50180, Thailand
	

	anthony.berdeu@obspm.fr






 



Initially designed to detect and characterise exoplanets, extreme adaptive optics (AO) systems open a new window onto the Solar System by resolving its small bodies. Nonetheless, their study remains limited by the accuracy of the knowledge of the AO-corrected point spread function (AO-PSF) that degrades their image and produces a bright halo, potentially hiding faint moons in their close vicinity.



To overcome the random nature of AO-PSFs, I aim to develop a method that blindly recovers the PSF and its faint structured extensions directly into the data of interest, without any prior on the instrument or the object's shape. The objectives are both to deconvolve the object and to properly estimate and remove its surrounding halo to highlight potential faint companions.



My method first estimated the PSF core via a parametric model fit, under the assumption of a sharp-edged flat object. Then, the resolved object and the PSF extensions were alternatively deconvolved with a robust method, insensitive to model outliers, such as cosmic rays or unresolved moons. Finally, the complex halo produced by the AO system was modelled and removed from the data.



The method is validated on realistic simulations with an on-sky AO-PSF from the SPHERE/ZIMPOL instrument. On real data, the proposed blind deconvolution algorithm strongly improves the image sharpness and retrieves details on the surface of asteroids. In addition, their moons are visible in all tested epochs despite important variability in turbulence conditions.



My method shows the feasibility of retrieving the complex features of AO-PSFs directly from the data of interest. It paves the way towards more precise studies of asteroid surfaces and the discovery and characterisation of Solar System moons in archival data or with future instruments on extremely large telescopes with ever more complex AO-PSFs.




Blind and robust estimation of adaptive optics point spread function and diffuse halo with sharp-edged objects
    
	Anthony Berdeu1,2
	
    Received 2 August 2023 / Accepted 7 May 2024
==============================================================================================================
















§ INTRODUCTION



The performance of ground-based instruments is limited by the atmospheric turbulence that corrugates the incident wavefront of the observed target <cit.>. For short exposures, the diffraction-limited point spread function (PSF) of the telescope breaks down to a random speckle field. For long exposures, their average is equivalent to the PSF of a 10 to 20 telescope <cit.>, strongly degrading the sensitivity and the resolution.


Introduced in the 1990s, adaptive optics (AO) systems are now commonly deployed in instruments of ground-based observatories to mitigate the effects of the turbulence <cit.>. The coupling of (i) PSF prediction to optimise the instrument design and performance <cit.>, with (ii) PSF modelling <cit.> to improve the data post-processing via model-fitting or deconvolution <cit.>, has resulted in a leap forward in numerous fields such as photometry and astrometry of stellar populations <cit.>, spectroscopy and kinematics of distant galaxies <cit.>, or the study of extended object surface topology <cit.>.

In the last decade, the arrival of extreme AO systems pushed the performance in high-contrast and high-resolution imaging even further  <cit.>. In this context, PSF prediction and modelling reach their limits, unable to faithfully reproduce the complexity of observed AO-corrected PSFs (AO-PSFs). Point spread function estimation or reconstruction techniques must then be considered <cit.>. But direct estimation of the PSF parameters from the AO telemetry <cit.> or reference PSFs obtained on calibration sources (internal or natural stars) before or after the observation <cit.> are not always sufficient, and can for example lead to strong deconvolution artefacts <cit.>.

Among others, the challenge arises from the evolving nature of AO-PSFs. By essence, the turbulence is random and the AO loops are not perfect, suffering from measurement noise, or temporal and aliasing errors <cit.>. Finally, flexions or thermal dilations in the instruments induce quasi-static speckles that slowly drift with time, changing the PSF's small and faint structures <cit.>. As a consequence, a given observation represents one realisation of an AO-PSF that is barely reproducible, or even impossible to reproduce.

To overcome this random nature of the AO-PSF, the only solution is to extract and reconstruct it directly from the data of interest, a problem known as blind deconvolution <cit.>. The challenge is to overcome the degeneracy between the estimation of the PSF and the object <cit.> and to solve an ill-posed problem with potentially twice more unknowns than measurements if both the images of the PSF and the object are to be retrieved from a single data image. As a consequence, no method is strictly `blind', and there is a need to make some assumptions about the PSF or the deconvolved object (or both) to avoid local minima of the problem and converge on the best `PSF and object' pair.

One solution is to implement marginal approaches <cit.>. The problem is rewritten to split the PSF and the object contributions, in order to integrate over the space of the object parameters to marginalise them out of the problem. The deconvolution of the object is only performed as a second step. Marginalisation has proved to be efficient when combined with parametric PSF models that strongly limit the number of unknowns to fit <cit.>. These methods nonetheless imply a need to know some prior on the object, such as the structure of its power spectrum density  <cit.> or its overall shape <cit.>. In addition, such PSF models, relying on a limited number of parameters, cannot fully grasp the complexity of real and potentially broadband AO-PSFs of high-contrast and high-resolution imagers. The challenge will be even greater with the incoming extremely large telescopes (ELTs). Their segmented primary mirror and large spiders holding the secondary mirror will produce intricate and structured PSFs that are very sensitive to AO residuals <cit.>. More versatile methods than parametric approaches are consequently needed.

Finally, in the context of asteroid study, beyond the deconvolution of the images of asteroids' surfaces <cit.>, the approximate knowledge of the PSF also limits the study of asteroids' close vicinity and the detection of faint moons. Similar constraints apply when observing the moon systems around the giant planets of our Solar System <cit.>. To get rid of the bright halo induced by the PSF extensions around the main object, techniques were adapted from exoplanet detection algorithms, based on local averaging or median filters <cit.>. But the problem to solve is quite different, the halo being produced not by a coherent source (an unresolved star) but by an extended object (the resolved asteroid or planet). These techniques consequently do not account for the physics at the origin of the halo that is poorly estimated. They thus inherently suffer from self-subtraction problems and can bias the moon photometry or even prevent its detection <cit.>.


It is thus essential to properly recover the AO-PSF extensions. But compared to the PSF core, their intensity is several orders of magnitude fainter. This echoes high-contrast imaging challenges where outliers must be carefully handled to avoid any corruption of the model. Some outliers, such as dead pixels, can be identified through proper calibration <cit.>, handled, and removed from the problem. But others may be random or specific to a given acquisition, such as hot pixels, cosmic rays, or model errors (signals not predicted by the forward model of the problem). This second kind of outlier must be identified on the fly, directly in the data of interest.

To reduce the incidence of such outliers, robust penalisation approaches have been developed <cit.>. They consist of replacing the conventional quadratic penalisation of the problem by a robust estimator that is approximately quadratic around zero but that grows sub-quadratically for large deviations to reduce their impact <cit.>.

One solution is to directly change the cost function of the problem <cit.>. Another solution, so-called iterative reweighted least squares <cit.>, is to stay in the least squares framework but to solve a sequence of least squares problems whose weights are iteratively updated with a robust estimator. The mathematical formalism associated with such robust penalisations will be further detailed in sec:wing.


This paper is a continuation of a previous pragmatic method that was initially developed to detect and characterise the third moon of  <cit.>. In this previous work, the performance of an AO system being better in the infrared, a simple parametric model was sufficient to describe the PSF, to deconvolve the object, and to properly model and subtract its bright halo. In addition, the PSF was directly estimated on a bright moon in the data of interest, acting as a reference point source.

In this work, I intend to provide a more general and blind approach for more complex AO-PSFs with limited priors on the object and the PSF, while removing the need for a point source to be present in the field of view (FoV). There are three main objectives: (i) to reconstruct the AO-PSF in its complexity, reproducing its faint structured extensions, so-called wings, directly from the data; (ii) to deconvolve the object image with this blindly estimated AO-PSF; and (iii) to carefully model the AO residual halo in order to remove it from the data and enhance the signal of potential moons.

The proposed method, which alternates between object and PSF deconvolution, is described in sec:method. It is then validated on realistic simulated data in sec:res_sim and applied to real data in sec:res_real. Finally, in 
sec:conclu, I discuss several ways of further improving the proposed method. For readers interested in knowing more about the algorithms and their implementation, the pseudo-codes of the different steps are detailed in app:algo.





§ PROPOSED METHOD






In the isoplanetic domain (the PSF does not depend on the position in the FoV), the forward model of the problem  canonically states that the image data,  (discretised onto the sensor pixel grid), is the (discrete) convolution  of the extended object, , with the long exposure PSF, , that combines the telescope and instrument response and the AO residuals <cit.>,

    =  + n ,

where n is a nuisance term. This equation is pictured in fig:deconv_model. As is seen in fig:deconv_modeld, the nuisance term encompasses the classical detector readout noise and photon shot noise, but also outliers such as defective pixels (`salt & pepper' noise) and cosmic rays that may hit the sensor during the acquisition (orange arrows). I emphasise here that the way the problem is posed – the convolution of an extended object with a PSF – implies that the unresolved moons (point sources), are also considered to be outliers, and thus belong to the nuisance term (coloured circles). As was stated above, the objective is to split  and  only from the knowledge of  without being corrupted by n.



The PSF presented in fig:deconv_modelc was obtained on a star, as part of the European Southern Observatory (ESO) Large Programme ID 199.C-0074 <cit.> and using the imaging mode of the Zurich IMaging POLarimeter instrument <cit.>. This figure emphasises all the complexity of an AO-PSF. The AO cut-off frequency (orange annulus) delimits the classical two regimes of AO-PSFs: (i) the outer region (red), dominated by the atmospheric turbulence halo and left uncorrected by the AO system, and (ii) the inner region dominated by the AO residuals. The roughly symmetric core of the PSF (white and grey) is surrounded by a wind-driven halo (green) elongated in the wind direction (temporal error of the AO system, so called servo-lag) and a speckle field produced by the non-common path aberrations <cit.> between the science part of the instrument and the AO system (yellow and orange). The diffraction spikes produced by the spiders holding the secondary mirror are also visible (yellow cross along the diagonals). All these features are responsible for the extended and structured bright halo visible in fig:deconv_modela that hides the neighbouring moons.

I propose to solve eq:forward_model step by step, using minimal assumptions on the object and the PSF, summarised in tab:deconv_assumtions. (I) Firstly, as is pictured in fig:deconv_core, the problem can be approximated, `from a distance', as a sharp-edged flat object, fig:deconv_coreb, convolved with a simple PSF core, fig:deconv_corec. In the following, such an object will qualify as `binary' in the sense of morphological operations <cit.>. This strongly simplifies the problem in order to get a first estimated separation between the object, , and the PSF, . (II) Secondly, as is pictured in fig:deconv_obj, this PSF core was used to deconvolve the main extended object, fig:deconv_objb. (III) Thirdly, as is pictured in fig:deconv_wing, the deconvolution paradigm is reversed and the object was used to deconvolve the faint PSF extensions, so-called wings, fig:deconv_wingc. Finally, after the removal of the halo model, fig:deconv_winga, the moon can be seen in the residuals, fig:deconv_wingd. A detection algorithm could then be applied to find the signal of potential faint moons further hiding in the noise. This is nonetheless beyond the scope of this paper. All of these steps are further detailed in the following.







 §.§ Step 1. Estimation of the PSF core



The main objective of this step is to estimate a first separation between the object, , and the PSF, , as is shown in fig:deconv_core. As a consequence, it must depend on a very limited number of parameters to be robustly constrained by the data.




Concerning the PSF, the method assumes that its core can be approximated by a parametric description. Such a simple model does not include the physics of an AO system and cannot properly model the turbulent halo. It thus dilutes the energy and is consequently not quantitative <cit.>. Parametric model-based profiles can partially solve this issue <cit.> but they imply some knowledge of the instrument (such as, for example, its pupil shape, the number of actuators of its AO system, or the wavelength). As a consequence, they are harder to tune. But such precision is not needed for this given step. Indeed, <cit.> showed that a Moffat profile <cit.> is a general but sufficient approximation to qualitatively retrieve the morphology of the observed objects, while depending only on a limited number of parameters. A two-dimensional (2D) Moffat pattern is defined as

    , α, β, θ = 
        	1 + 
            r_1^2/α_1^2
            + r_2^2/α_2^2^-β ,

where β is the power parameter of the pattern, θ is its orientation, r_1 = _1cosθ + _2sinθ and r_2 = -_1sinθ + _2cosθ are the 2D coordinates in its rotated frame, and α = α_1, α_2 are its elongations along its two axes.


As we shall see later in sec:res_simsec:res_real, this Moffat approximation worked well with all the tested datasets. Nonetheless, in the hypothetical case that this model is not sufficient for a given dataset, eq:Moffat can be replaced by another more adapted parametric model if needed. Doing so does not change the presented methodology as long as the number of parameters remains limited.


The PSF core is thus simply proportional to this 2D pattern:

    _0, α, β, θ, γ≜γ-_0, α, β, θ .

Its parameters mainly control how the object is blurred by the PSF. Most of the signal is then given by the object's edges rather than its surface texture. If the shape of the object is overall smooth with sharp edges, applying a threshold, d̅, on the data, , defined as

    thr, d̅ = 
        
    		1  if ≥d̅
    
            0  otherwise ,

is a sufficient approximation of its support to extract the information needed to estimate the parameters of eq:PSF_core. Indeed, in a parametric approach with a limited number of arguments, namely the parameters of the core and the threshold on the data, a crude description of the object is sufficient because there are far more measurements than unknowns to recover.

The values of the different parameters of eq:PSF_coreeq:thres are obtained by minimising the following cost function,

    core, wd̅, γ, _0, α, β, θ≜   
       wls,
    			_0, α, β, θ, γthr, d̅,
    			 ,

which is a pure data fidelity term between the data, , and the model, _0, α, β, θ, γthr, d̅, weighted by . 𝒟^wls is defined as the weighted least square difference (wls)

    wlsφ_1,φ_2,≜   
       1/2∑_, ^', ^'φ_1 -φ_2φ_1^' -φ_2^' ,

where  is the inverse of the data covariance matrix. Its role is to weigh the measurements in the data fidelity term in terms of confidence.

The readout noise is assumed to be of a Gaussian statistic of variance, . On the other hand, for fluxes higher than a few photons, the Poisson statistic of photon shot noise can be approximated by a Gaussian statistic whose variance is proportional to the incident flux (and thus approximately to the data after proper calibration and pre-reduction) by a factor, η, that accounts for the conversion between photons and analogue to digital unit (ADU) and for the quantum efficiency of the pixels <cit.>. In the general case, these noise terms are independent from one pixel of the sensor to another. As a consequence, the inverse of the data covariance is diagonal,

    , ^' = 
         if  = ^'
    
            0  otherwise ,

and is given by <cit.>

    = 1/η+ .

The sum in eq:WLS_full is thus a sum on  only

    wlsφ_1,φ_2,≜1/2∑_φ_1 -φ_2^2 .

I emphasise that η and   are technically vectors as their values may depend on the position in the FoV, for example because of inhomogeneities in the pixel response or detector gains. Thus, contrary to other methods that approximate the noise level as being homogeneous across the pixels, to allow the marginalisation of the object and the PSF in the Fourier domain <cit.>, the proposed method includes a physical model of the noise statistics that depends on the pixels and that is a function of the data intensities (measured, see eq:weight_diag, or expected by the model, see eq:weight_mod below). This subtlety is critical: for the other methods, the unknown is the deconvolved image of the bright asteroid for which a homogeneous photon noise is a good enough approximation. But the objectives of my method are to additionally retrieve the PSF wings and remove the diffuse halo, while discarding faint outliers, which implies having a fine model of the noise according the intensity level.

As was discussed above, most of the signal is given by the blurring of the object's edges, and is condensed in the bright core of the halo. To avoid any corruption by the faint extensions of the halo that are not modelled under the parametric PSF core assumption, a threshold is applied to the data to reweight, , as is detailed in the alg:core of app:algo_core. Only the pixels not shaded in red in fig:deconv_corea are considered in the fit.

Finally, I mention that this first step can be replaced by more classical approaches, for example using reference PSFs obtained on a star or myopic approaches <cit.>. Indeed, its main objective is to get a good enough PSF core to initialise the object deconvolution of sec:deconv. The global PSF will be further refined later, in sec:wing.




 §.§ Step 2. Deconvolution of the main object





As is shown in fig:deconv_obj, the objective of this step is to deconvolve the image of the object, , assuming that the PSF, , is known, for example from the fit of its core done in the previous section sec:core. It is a classical deconvolution problem solved by minimising the following cost function <cit.>:

    obj,,≜wls, , 
    	+ 
    	^objobj ,

under the physical constraint that ≥0. This hard constraint also limits oscillating artefacts close to sharp edges, commonly seen in deconvolution problems <cit.>. ^obj is a hyperparameter to balance the regularisation, obj, compared to the data fidelity term. obj was chosen to favour smooth objects with sharp edges, by encouraging the sparsity of spatial gradients <cit.>

    objφ≜   
       ∑_√(∇_1φ^2+∇_2φ^2+ϵ^obj^2)-ϵ^obj ,

where ∇_1 and ∇_2 correspond to finite difference operators along the first and second spatial dimensions of the image. ϵ^obj>0 is a threshold controlling the transition between a ℓ^1-norm (edge-preserving) and a ℓ^2-norm (smoothness). ϵ^obj also ensures that eq:reg_obj is differentiable at zero. Such an edge-preserving regularisation or equivalents are common in asteroid deconvolution <cit.>. In practice, these hyperparameters, ^obj and ϵ^obj, were manually tuned. ^obj mainly depends on the object's characteristics. Since all asteroids share similar properties, it is not expected that this parameter changes a lot for resolved asteroids. ϵ^obj varies with the awaited contrast on the asteroid surface: (i) for a well-resolved asteroid, values of a few percent of the object's dynamics provide good results, (ii) for poorly resolved targets where surface details are hardly expected, values of around a thousandth or less better favour sharp-edged binary reconstructions.



At this stage, there is still no constraint on potential moons. They are thus in the deconvolved model and disappear from the residuals, as is shown in fig:deconv_objd. Once the object is deconvolved, it can be segmented with a threshold to remove the moons or other artefacts from the image model, as is described in app:algo_deconv, alg:obj_wing.



 §.§ Step 3. Deconvolution of the extended PSF wings









Now that a de-blurred image of the main object, , has been correctly estimated, it is possible to revert the problem and use this image as the convolution kernel to deconvolve the AO-PSF wings, as is shown in fig:deconv_wing. In this approach, the moons, which are a priori unknown, are not explainable by a model of an extended object convolved with a PSF, and should thus appear in the residuals; so should defective or hot pixels or random cosmic rays impacting the sensor, which cannot be calibrated in advance. These outliers should not corrupt the reconstruction of the PSF wings. As was introduced in sec:intro, the mean square error of eq:WLS is consequently not adapted. I propose to tackle this problem with robust penalisation <cit.>. This consists of replacing the quadratic norm of the residuals in eq:WLS with a function, , that limits the influence of large (and aberrant) values in the minimisation:

    φ_1,φ_2,w≜∑_√(w)φ_1 -φ_2 .


Of the different robust estimators, , available in the literature <cit.>, I chose the Cauchy function:

    r≜γ^2/2ln1+ r^2 / γ^2 .

If the confidence model,  in eq:robdata, is properly scaled and calibrated so that the argument

    r = √(w)φ_1 -φ_2
 follows a normalised Gaussian noise statistics (centred on zero and of a standard deviation equal to one) in the absence of outliers, setting γ = 2.385 ensures that minimising eq:WLS or eq:robdata gives similar results. In doing so, the Cauchy function has the advantages of having no tuning parameter and of being differentiable.

In more detail, 

    r=1/r∂r/∂ r=1+r^2 / γ^2^-1 ,

the so-called robust weight in the following, is the correction factor to apply to the weights in the least square error of eq:WLS to make it equivalent to the robust penalisation of eq:robdata. This is typically used in the context of IRLS, as was introduced in sec:intro:

    ^irls = r .


fig:weighta gives an example of a robust weight map, obtained by applying  of eq:Cauchy_weight to a weighted halo residual map when fitting the full PSF, . The outliers can directly be identified in black, with equivalent weights tending towards zero. It is then possible to discard them on the fly, using a conservative threshold,  (typically 50, as is discussed in app:algo_deconv), and as is shown in fig:weightb. This is done by updating eq:weight, with the additional remark that this equation is in fact only an approximation. Indeed, the noisy data, , is not the real flux. A better estimate of the weighting confidence term, , can be obtained by replacing  with the forward model

    ^mod =

in eq:weight. In total, as is detailed in alg:obj_wing of app:algo_deconv, the updated weight map now reads as

    =
    	
    		0  if ≤
    
    		1/η^mod+ otherwise .

It is iteratively updated in an approach similar to IRLS, eq:weight_IRLS, but with (i) a binary update when it comes to the robust estimator, and (ii) a refinement of the expected noise level with the intensity predicted by the forward model. Doing so prevents any further corruption of the deconvolved PSF wings by strong outliers.




The cost function that is iteratively minimised to reconstruct the PSF, , is

    psf, , ≜wls,,
    	+ ^psfpsf ,

under the physical constraint that ≥ 0. ^psf is a hyperparameter to balance psf. I define this regularisation term as

    psfφ≜∑_∇_1lnφ^2+∇_2lnφ^2 ,

which consists of the classical ℓ^2-norm on the gradient but applied to the logarithm of the PSF. The former ensures a smooth reconstruction, while the latter acts as a `dynamic whitening' term. Indeed, the challenge in regularising the PSF wings is that  spans multiple orders of magnitude between the PSF core vicinity and the external turbulence halo, as is seen in fig:deconv_modelc. psf must act through this high dynamic range. Similar to  in eq:WLSeq:robdata, which scales the residuals according to the admissible awaited noise level (noise whitening), the regularisation must scale the PSF wings. Since

    ∇_ilnφ = ∇_iφ / φ ,

I let the opportunity for the gradients in eq:reg_psf to evolve proportionally to the local intensity of the total PSF, despite the large dynamics of the unknowns, both refining the PSF core and recovering its wings. As for ^obj in sec:deconv, the hyperparameter ^psf is a function of the noise level of the data, , and was manually tuned in this study.

Technically, one could stop here if the main objective were to retrieve the faint moon hidden in the bright halo. Nonetheless, now that a better model of the AO-PSF is known, it is possible to loop back on the object deconvolution step and iterate the steps of sec:deconvsec:wing to further refine the reconstruction while robustly updating the noise model, , via eq:weight_mod in eq:cost_objeq:cost_wing. This implementation is further detailed in alg:obj_wing of app:algo_deconv. This proved to be efficient for complex PSF cores, badly described by a 2D Moffat pattern, for example in the presence of motion blur or for bad seeing conditions. On top of a better object deconvolution, this also improves the residual quality close to the object edges, potentially unveiling moons very close to the main body.





§ VALIDATION ON SIMULATIONS



The method was tested on simulated data representative of real conditions. The simulated object, shown in fig:deconv_modelb, is an image of 67P/Churyumov–Gerasimenko obtained by Rosetta (European Space Agency), downscaled to the resolution and size (between 100 and 200 milliarcseconds) that are typical of the targets observed within the ESO Large Programme described in <ref>. As was mentioned in sec:method, the PSF used to blur the object image is a real AO-PSF obtained with ZIMPOL on a star. Three moons with different contrasts and distances were added to the main body (coloured circles in fig:deconv_modeld). The photon shot noise and the readout noise parameters, η and , are assumed to be identical for all the pixels in the FoV:

    ∀∈FOV, η = η and  =  .

On top of these noises, random pixels are modelled as hot or dead (`salt & pepper' noise) and five cosmic rays are randomly simulated (orange arrows).



The results of the different steps of the method are shown in Figs. <ref>, <ref>, and <ref>. fig:sim_analysis focuses on the main body deconvolution, providing zooms on the deconvolved images, ^mod, the model residuals, - ^mod, and the object residuals, - κ^mod.  stands for the shifted true object, . Indeed, due to the PSF core deformation, the PSF in fig:deconv_modelc may not be properly centred. By construction (see app:algo), my method reconstructs a centred PSF. As a consequence, the deconvolved object may be translated compared to the truth. This shift is estimated by fitting a perfect 2D Moffat model in the true PSF.  is the cubic interpolation of  translated by the position of this fitted 2D Moffat pattern. fig:sim_analysisi shows that this shift is vertical, producing the bright arcs at the top and the bottom of the object residuals. To emphasise the surface texture retrieval rather than a bias in the global intensity, the deconvolved ^mod is further normalised with

    κ = κ̃∑_^mod-κ̃ ,

using the absolute value in the sum to be robust to outlier regions, badly deconvolved or smoothed by the deconvolution such as edges or very bright or dark areas.

fig:deconv_core shows that the oversimplified model of the first step efficiently fits the bright inner part of the halo that is correctly removed from the residuals of fig:deconv_cored and fig:sim_analysisf. By design, the outer part of the structured halo, mainly driven by the AO cut-off frequency, is still present in the residuals, as it cannot be explained by this simple model. As for the object, the threshold applied to the data produces a binary object (see fig:deconv_coreb and fig:sim_analysisb, whose edges are consistent with the blurred object, as is shown by the orange contour in fig:deconv_corea). This is further confirmed by the slices of fig:sim_analysism, the edges of the binary approximation (dark grey) being within a few pixels of the true slice of the object (light grey). As was expected, the overall object intensity is underestimated by more than 10 because the energy diluted by the PSF wings is not integrated and of course none of the surface structure is retrieved, fig:sim_analysisj. As for the PSF core, the radially averaged profile of the blindly estimated Moffat (blue curve in fig:sim_analysisn) shows a very good agreement with the theoretical radial profile of the 2D Moffat (red curve) fitted directly in the true PSF.

fig:deconv_obj shows that using the fitted PSF core is indeed sufficient to retrieve the morphology of the object, as is discussed in sec:core. Structures at the surface of the deconvolved object, fig:deconv_objb and fig:sim_analysisc, are consistent with the simulated truth, fig:deconv_modelb and fig:sim_analysisa, as is shown by the good residuals of fig:sim_analysisk. Nonetheless, the fact that the Moffat model cannot properly grasp the deformations of the PSF core produces two kinds of artefacts: (i) a bright edge on the left side of the object in fig:sim_analysisc,k that suggests that this edge is `over-deconvolved', and (ii) non-homogeneous structures in the residuals at the edge's location, shown in fig:sim_analysisg. This can also be seen in the blue slice in fig:sim_analysism, which shows unnatural deconvolution peaks at the edges of the reconstructed object. And as was already discussed, the fact that the PSF core does not account for the PSF wings induces an underestimation of the object's intensity, here by more than  6. I also emphasise that, by construction, the halo is found in this deconvolved image of fig:deconv_objb, as well as the moons and other noise artefacts. They consequently appear in the modelled data, fig:deconv_obja, and the moons are not visible in the residuals, fig:deconv_objd.

All of these problems are solved after a few iterations of the alternate algorithm, once the PSF wings are properly deconvolved, as is shown by fig:deconv_wing. Concerning the object, the zooms of fig:sim_analysisd,l and the green slice of fig:sim_analysism show that, despite being slightly smoothed by the regularisation of eq:cost_obj, the details are quantitatively retrieved with the correct intensity within 100.2-97.3 ≤ 3. The sharp edges are also well defined at the correct location (see the green and light grey curves). Compared with fig:sim_analysisc,k, the `over-deconvolution' problem on the left side of the object is greatly improved. Nonetheless, the combination of (i) the threshold to segment the main object, (ii) the regularisation in eq:cost_obj, and (iii) the approximate knowledge of the PSF core (or equivalently the cut-off of the optical transfer function of the system) within a blind approach, limits the precision of the object's edge recovery, as is visible in the data residuals of fig:sim_analysish. In addition, the spatial regularisation erases the finest details on the surface of the object, as is seen in the model residuals of fig:sim_analysisl. These are known limits of the edge-preserving regularisation <cit.> but are a necessary trade-off with the PSF reconstruction to control the alternate blind deconvolution convergence. Otherwise, the residuals reach the noise level, below the percent of the data dynamic (see colour bars of fig:deconv_wingd for fig:sim_analysish). There is no noticeable feature at the location of the object's surface that would imply an over-regularisation in the deconvolution, indicating that the details smoothed by the regularisation are indeed lost in the noise level.

Concerning the PSF, all the features mentioned in sec:method are quantitatively retrieved in fig:deconv_wingc. The wind-driven halo is well defined. The brightest structures on the AO cut-off frequency ring and the diffraction spikes are clearly visible and deconvolved. Only the resolution on the speckle pattern is smoothed by the regularisation but its global structure has the correct dynamics. In fig:sim_analysisn, the radial profile of the deconvolved PSF perfectly matches the radial profile obtained on the true PSF. As is explained in sec:wing, the moons and other outliers are robustly penalised in the equivalent weights in fig:weighta and the brightest ones are successfully rejected from the fit, fig:weightb. Thus, they do not impact the model of the halo, fig:deconv_winga, and are consequently nicely visible in the residuals, fig:deconv_wingd. Compared with the injected signals in fig:deconv_modeld, the moons are slightly dimmed and erased from the data, showing that the method suffers from slight self-subtraction. This is nonetheless a cost worth paying to efficiently retrieve the halo and constitutes a trade-off in the regularisation of eq:cost_wing between the reconstruction of the PSF structures and the moon rejection from their deconvolution.






§ EXAMPLES IN REAL DATA





 §.§ The multiple asteroids and 



The ESO Large Programme `Asteroids as tracers of Solar System formation: Probing the interior of primordial main belt asteroids' <cit.> targeted several asteroids of the Solar System's main belt in order to characterise the albedo, surface, shape, and mass of the ones accompanied by moons. The data were obtained with ZIMPOL, mounted on the Spectro Polarimetric High-contrast Exoplanet REsearcher <cit.> of the Very  Large Telescope (VLT) observatory, equipped with the SPHERE Adaptive eXtreme Optics system <cit.>. Mainly designed for polarimetry acquisitions, ZIMPOL has an imaging mode. Working in visible wavelengths, it offers an unprecedented resolution of small Solar System bodies at the cost of a more complex AO-PSF. I applied my method to two targets of this Large Programme: and . 

Among the 100 class asteroids, is quite original. It has a dumbbell shape and is orbited by two known moons, which constrain its gravitational field, and thus its mass and density <cit.>. Its elongated shape and the availability of state-of-the-art deconvolutions and 3D reconstructions make it a good case with which to test the deconvolution performance of the proposed method.


is also an asteroid of the main belt, surrounded by three moons <cit.>. Its faintest and closest companion was missed until recently due to the lack of proper data reduction and data analysis tools <cit.>. It was detected in archival data obtained in 2014 with the Integral Field Spectrograph of SPHERE <cit.>. The complexity of ZIMPOL AO-PSFs and the variability of the brightness and distance of the different moons relative to the main body represent a challenging case with which to test the proposed method. was observed at different epochs in 2018 and 2019 by <cit.>, each epoch gathering multiple frames. In order to assess the consistency of the method results, both in terms of deconvolution and moon enhancement, two different frames of four epochs were selected. To show the versatility of the method and the large acceptability of its assumptions, I also selected additional datasets from other instruments: the IFS (ID 60.A-9362(A), PI: Yang et al. 2014, YJH filter, λ from 0.95 to 1.65), the InfraRed Dual-band Imager and Spectrograph <cit.> of SPHERE, and the Near Infrared Camera 2 (NIRC2, ID U58N2, PI: de Pater et al. 2005, Kp filter, λ=1.248, Δλ=163) of the Keck II telescope.




 §.§ Main body deconvolution and surface detail recovery




fig:Elektra_profilesfig:Kleopatra gather the results obtained on and for different epochs with different seeing conditions. The noise model parameters, η,, of eq:weight, critical to scale the robust penalisation, are empirically adjusted in the data, as is described in app:noise_model.









fig:Elektra_profiles focuses on the final deconvolutions of , obtained after the alternate algorithm. The reconstructed edges are sharp without any `corona artefact' (see below discussion on fig:Kleopatra) or motion blur residual, even for 2019-07-30 (2), which is a challenging case, as is highlighted by the reference deconvolution or the important deformation of the PSF core in fig:comp_instru. Nonetheless, for the 2019-08-05 frames, the overshoot on the left side of the asteroid also indicates a slight over-deconvolution, as was already seen in the simulations, sec:res_sim. Features at the asteroid's surface are resolved, as is seen in the zooms of fig:Elektra_profiles. The evolution of the projected shadows on the surface is visible between the frames of the best epochs, 2019-08-05 and 2019-08-06, and the corresponding slices of fig:Elektra_profiles. This supports the facts that the method is consistent and that these are not deconvolution artefacts. For 2019-08-06, the slices show that the contrast in the structures decreases between the two frames, as the land forms face the Sun more. Except for the main shadow of 2019-08-05, none of these details can be seen in the reference deconvolution obtained by <cit.>.





fig:Kleopatra details the different steps of the blind deconvolution algorithm on . For each epoch, the first columns show a zoom of the data, the fitted binary model, and the deconvolved images based on the knowledge of the PSF core or the PSF wings (and the corresponding residuals) as well as intensity slices along the coloured lines. The deconvolutions obtained by <cit.> are also given to serve as references, b. c show that the binary threshold of the data provides a good approximation of the object support, both in terms of size and shape. Then, it clearly appears that the mere knowledge of the PSF core is sufficient to retrieve the object body, but the deconvolved images of f suffer from what is sometimes called `corona artefact', a stepwise artefact surrounding the object edges that originates from a coupling between the edge-preserving regularisation and an incorrect description of the PSF core <cit.>. These images are similar to the reference reconstructions. For some epochs, such as 2017-07-27, 2017-08-10, and 2017-08-22, the artefacts surrounding the object strongly suggest hiccups of the AO loop or even motion blur. This could be explained by a shift in the telescope pointing direction in between the four Detector Integration Times (DITs), which are averaged in one ZIMPOL-reduced frame by the data reduction pipeline <cit.>. This is particularly visible in the blue slices of 2017-07-27 and 2017-08-10 that cut along this motion blur.


Deconvolving the full PSF solves these issues. In the final step of the proposed method, the edges of are de-blurred and sharp, and the duplicated images are collapsed into a single image in the cases of strong motion blur. This is also emphasised by the green slices in fig:Kleopatra. These deconvolved images obtained via a blind deconvolution algorithm with limited priors on the object and the PSF compare well with state-of-the-art marginal deconvolutions and are consistent with the previously derived 3D model of the asteroid <cit.>. The orange slices of fig:Kleopatra show that the reference method underestimates the object's intensity and can only retrieve smooth and large-scale structures. Due to these limitations, only a difference in albedo between the two lobes of the asteroid was studied by <cit.>. The small-scale structures in the green slices of fig:Kleopatra suggest that finer details from 's surface can be retrieved with my method.



Interestingly, the deconvolution residuals in fig:Kleopatra show a line-by-line horizontal stripe pattern. This can be directly linked to the ZIMPOL imaging mode where only one out of two lines of the sensor is exposed. To produce `high-resolution' images, the reduction pipeline interpolates the missing lines <cit.>. This implies that the proposed deconvolution algorithm reaches the level of the artefacts introduced by the data reduction. To limit the impact of this `zebra' in the deconvolved image, I had to increase the regularisation parameter, ^obj, compared to sec:res_sim, limiting the performance of the method of edge and detail recovery. This explains why spatial features can still be seen in the residuals. These reduction features that correlate neighbouring pixels also mean that the assumption of independent noise previously mentioned in sec:core for eq:weight_diag is not strictly correct and that the diagonal weighting term, , in eq:cost_wing should actually be the inverse of a covariance matrix, as in eq:WLS_full. This effect was nonetheless neglected in this study. 






 §.§ Point spread function reconstruction and halo removal


A quick look at the halo brightness in the data column of fig:Kleopatra shows that the turbulence conditions change between the different epochs. This directly impacts the deconvolved AO-PSF wings and their contrast. For the two epochs 2017-07-14 and 2017-08-10, the PSF is even mainly dominated by the wind-driven halo. Nonetheless, the blind PSF deconvolution successfully retrieves the AO cut-off limit at the correct location as well as its brightest speckles, despite the absence of an instrumental prior in the PSF model. For good seeing conditions, in spite of the elongated shape of , the reconstructed speckles appear nicely resolved and split. In addition, the orientation and elongation of the wind-driven halo do not appear to correlate with 's orientation (see especially 2017-07-14 and 2017-08-22), as one could have feared from a potential cross-talk during the blind deconvolution. This overall consistency favours the robustness of the method in connection with the shape of the object and with the turbulence conditions.

Looking at fig:comp_instrufig:Elektra_data, the conclusions are similar with , with the additional supporting fact that for each epoch the two reconstructed PSFs are extremely consistent in terms of turbulence features (contrast and wind-driven halo). Furthermore, the rotation of the spider diffraction spikes is clearly visible (see 2019-07-30 and 2019-08-06 in particular).




After halo removal, the two moons of are clearly visible at all epochs in the halo residuals, as is shown in the last column of fig:Kleopatra (coloured circles). The transition between a readout noise-limited regime to a photon shot noise-limited regime is also nicely visible: the noise level increases in the object's vicinity where the halo is very bright in the data, especially for bad seeing conditions (2017-07-14, 2017-08-10, and 2017-08-22). Looking at the halo residuals in fig:comp_instru, this transition is also visible, especially for the bad seeing cases of 2019-07-30 and 2019-08-04 with an increased noise level in the object's vicinity. The three moons of are visible in all the presented ZIMPOL frames (coloured circles), even when the deconvolution is not perfect (see 2019-07-30). Indeed, by construction, even if the object and the PSF are not perfectly split, moons (point sources) remain outliers of the model and should appear in the residuals. This is an encouraging result that shows the robustness of the approach. The counterparts of the moons in the robust weight maps are also nicely visible. The orange arrows point towards discarded outliers, such as defective pixels or random cosmic rays as well as camera defects impacting some columns of the detector.

The last lines of fig:comp_instru present results on data obtained with other instruments and reference PSFs measured on a star. They show that the method is robust for a large variety of AO-PSFs.

Looking at the IFS line, my method is able to reproduce the results of my previous pragmatic approach <cit.>, extracting the three moons from the diffuse halo. The poor AO performance of this dataset explains why the diffraction rings of the reconstructed PSF differ from the IFS reference. Nonetheless, the diffraction spikes are visible and the reconstructed multi-lobed core nicely matches the PSF produced by the brightest moon (red circles): it presents a strong secondary lobe, at the top left of the main core. A similar complex PSF core is retrieved with ZIMPOL for the 2019-07-30 (2) frame (secondary lobe on the right). This shows that the method is, to some extent, robust for multi-lobed PSF cores diverging from a simple Moffat pattern, slightly relaxing this potential limitation of the method. The robust weights obtained with the IFS appear quite low. This comes from the wrong noise model fitted in the reduced data via the method presented in app:noise_model. Indeed, the reduction performed by PIC of IFS data <cit.> is based on a spatial regularisation that biases and correlates the noise in the reconstruction. Such an incorrect noise model can be handled by relaxing the constraint on  to lower values for automatic outlier rejection.

The IRDIS data are very noisy, with a dense pollution with defective pixels (common with infrared detectors), while only spans a few pixels. Despite this unfavourable situation, the first Airy rings of the PSF are reconstructed with the correct dynamics and with negligible signal in the residuals. The first ring is fragmented due to poor seeing conditions. Unfortunately, only the brightest moon is visible because of the high noise level and high density of unusable pixels.

Finally, with NIRC2, is barely resolved, with an angular size similar to that of the first diffraction ring of the instrument PSF. Despite this challenging situation, the characteristic hexagonal shape of this ring, due to the fragmentation of the Keck II mirror with hexagonal segments, is nicely recovered. So are the secondary diffraction peaks (two in green at the top right of the core for the first order and a few in yellow for the second order) that produce replicated images of in the saturated data. Concerning the moons, only the brightest is visible in the residuals, highlighting the need for more frames and epochs and dedicated detection algorithms to increase the signal-to-noise ratio.




§ CONCLUSIONS AND PERSPECTIVES



In this paper, I have presented a novel blind deconvolution technique that can retrieve both the object and the faint wings of AO-PSFs. The only main priors on the object are that it be contrasted with a smooth shape and sharp edges. Contrary to most of the existing blind deconvolution algorithms, it is not based on a parametric model of the PSF, allowing one to reconstruct a large variety of shapes and structures, as featured by PSFs after an AO correction. The proposed inverse problem approach is an alternate deconvolution with an edge-preserving regularisation for the object and a regularisation on the smoothness of the gradients for the PSF with intensity whitening. Tested on simulated and real data, my method outperforms state-of-the-art techniques <cit.>, by both efficiently reconstructing details at the surface of the objects and faithfully retrieving the features of the AO-PSFs. It produces a physical and realistic model of the bright halo surrounding the object, which can then be carefully removed. This enhances faint moons in its close vicinity with a better contrast and at closer distances than previous halo-subtraction methods <cit.>. 

Implemented in a general framework and with a limited number of assumptions, my method can be applied to different instruments mounted on different telescopes equipped with different AO systems. It paves the way for the study of resolved asteroids or moon systems around the giant planets of our Solar System via archival data of different instruments for which previous methods could not reveal the companions. It also prepares the arrival of the future generation of ELTs, whose complex and sensitive PSFs and AO systems will require robust and versatile processing techniques. Nonetheless, as is discussed below, there are several ways to further improve this method.

Solving the dual deconvolution problem implies carefully tuning the method. This is mainly done through the hyperparameters of the regularisations on the object and the PSF, namely ^obj, ϵ^obj, ^psf, and . In particular, one could decide to first keep a strong regularisation on the object to prioritise the PSF reconstruction, and to relax it the second time, once the PSF has been correctly estimated, to better retrieve the object's details. Providing an automatic tuning of such parameters to get a fully unsupervised algorithm is an active field of research <cit.> and implementing such methods was beyond the scope of this paper. In this work, these parameters were manually tuned and fixed once and for all for each instrument. The fact that they worked for all the tested datasets is encouraging and shows their good admissible range in terms of noise level, object shape, and seeing conditions. This thus limits the need for further manual tuning by the user.

It was emphasised in sec:res_real that lots of datasets suffer from poor AO-PSF quality or even motion blur. Robustness is obtained via a higher regularisation level, but this can degrade the image quality or produce over-deconvolution. To increase the resolution and the deconvolution quality, and since the targets are very bright, one should work on the individual ZIMPOL DITs rather than the stacked image. The results also show that the method reaches the limits of the ZIMPOL reduction pipeline, whose artefacts dominate the residuals in some datasets. Implementing such changes implies modifying the reduction pipeline and was beyond the scope of this paper.

As is shown in sec:res_real, most of the outliers, such as defective pixels or cosmic rays, are successfully and robustly identified and discarded on the fly by the proposed algorithm. Nonetheless, some of them are missed, and so are the faintest moons. As is discussed in sec:res_sim, the proposed method consequently suffers from slight self-subtraction, which could dim the signal of potential moons. Even if it is less pronounced than with other methods <cit.>, some precautions must be taken when fitting the photometry of a moon that can be underestimated, or when announcing the absence of moons below a given noise-limited contrast. It is possible to assess these biases by injecting false moons as is done in the standard exoplanet detection algorithm to quantitatively estimate the contrast performance <cit.>. This was beyond the scope of the paper, but should be kept in mind for further studies.

Finally, the natural next step of this method is to use to the reconstructed PSF within matching filter algorithms to detect even fainter moons in the residuals. Combined with the refined noise model and with the robust weights to limit false positive detections by discarding outliers, this would push the detection limits even further. Echoing the challenges of exoplanet detection, it comes nonetheless with the additional difficulty of the rapid motion of the moons around the main object <cit.> and is an open research field.











	The author would like to warmly thank the Referee Laurent Mugnier for his pertinent feedback and numerous remarks and suggestions that greatly contributed to improving the quality and clarity of the article.
	

	The author would like to thank Éric Thiébaut and Ferréol Soulez for the fruitful discussions on robust detection algorithms and adequate prior for high dynamic unknowns as well as Benoît Carry for his careful proofreading.
	

	All the ZIMPOL reductions are based on public data acquired for the ESO Large Programme ID 199.C-0074 (Vernazza et al. 2021, `Asteroids as tracers of Solar System formation: Probing the interior of primordial main belt asteroids') and available at https://observations.lam.fr/astero/https://observations.lam.fr/astero/
.
	

	The IFS and IRDIS reductions are based on public data provided by the ESO Science Archive Facility. The author would like to thank Antoine Kaszczyc and Maud Langlois who provided the PSFs and reduced the IRDIS raw data.
	

	The NIRC2 reductions are based data provided by the Keck Observatory Archive (KOA), which is operated by the W. M. Keck Observatory and the NASA Exoplanet Science Institute (NExScI), under contract with the National Aeronautics and Space Administration. The author would like to thank Kate Minker who reduced the raw data and provided the PSF.
	

	This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 101004719.






aa













§ PSEUDO-CODES OF THE METHOD



This appendix gathers the pseudo-codes of the different steps of the proposed method, described in sec:method. In practice, all the code is implemented in MATLAB within the open-source  framework <cit.>. I also detail in this appendix how the different cost functions are minimised.



 §.§ Estimation of the point spread function core



The different parameters of the first step of sec:core were alternately fitted, as was described in alg:core. Indeed, applying a threshold to the data is not a differentiable operation, complicating the minimisation with iterative gradient descent methods. A first set of parameters of the PSF core was guessed, corefirst_guess, on a coarse grid of threshold values, corerough_set. These values were then refined by alternating five fits on (i) the threshold on the data, along with the amplitude parameter, coreobj_thr, and (ii) the PSF core parameters, corePSF_fit. The optimisation problems were solved with the simplex search method of <cit.>. For each problem, 200 iterations were performed. The parameters of the Moffat profiles were initialised with α = 3 and β=1.6, which are standard values <cit.>. To reweight, , corereweight, a general threshold of 2.5 was applied to the data. From the perspective of the dynamic of the PSF, this value roughly corresponds to the transition between its wind-driven core and its wings (see fig:deconv_modelc). Only the meaningful pixels in the brightest part of the halo, not shaded in red,  fig:deconv_corea, are thus kept in the data fidelity term.






 §.§ Alternate object and point spread function deconvolution




alg:obj_wing details the implementation of the dual deconvolution algorithm of the object and the PSF wings of sec:deconvsec:wing. The role of obj_wingth_obj is to define the main object support. I empirically chose a threshold of d̅^sup between 15 and 25 of the maximal value of the median-filtered deconvolved object, med (kernel of 5×5 pixels, as in alg:noise). Such a threshold is low enough to account for partially illuminated pixels on the object's edge, while being conservative enough to correctly constrain and tackle the `corona artefacts' mentioned in sec:res_real. In practice, this threshold was applied only in the vicinity of the main body. To further  avoid crenellated edges on the segmented image, this support was dilated by one pixel. The rest of the FoV was automatically excluded to avoid bright moons or artefacts, such as cosmic rays in the deconvolved image, exceeding this threshold.

Conversely, for obj_wingw_rob_obj and obj_wingw_rob_wing, the  threshold was applied only in the halo to exclude the moons or artefacts, such as cosmic rays, but not in the main body vicinity, to keep this area (see fig:weightb). Indeed, this area is critical to fit the PSF core features but hard to fit due to the artificial threshold on the deconvolved image and the edge-preserving regularisation, as is seen in the robust weights of fig:comp_instru. I typically used the conservative value of 50, as it is important to remove any moon suspicion while the deconvolution of the smoothly varying PSF is robust for a high rate of missing data. In the presence of strong outliers (dead pixels) on the object support or its vicinity, as in fig:weightb, it is still possible to apply a less conservative threshold to this region, typically below 10.

The two minimisation problems of obj_wingth_obj and obj_wingwing were solved with 1000 iterations of the variable metric with limited memory-bounded algorithm <cit.>, a limited-memory quasi-Newton method with Broyden-Fletcher-Goldfarb-Shanno updates <cit.> that handles bound constraints. In the case of strong motion blur, typically alt=30 alternate iterations are looped.

In obj_wingw_rob_obj and obj_wingw_rob_wing, the weights were robustly updated only after a few alternate loops,  wgt=5. I would like to remark here that if a proper calibration of the reduced data has been made, the robust weights, obj_wingw_rob_0, can be initialised with a map of already known aberrant pixels. This was done for the IRDIS dataset.







§ EMPIRICAL NOISE MODEL



The attentive reader may notice that the proposed method implies knowledge of the noise model parameters, η,, in eq:weight. If they are known in the simulated case of sec:res_sim, they are a priori unknown for the real data of sec:res_real. In principle, the noise model can be obtained through proper calibration and analysis of both calibration and science data <cit.>. Nonetheless, most of the data used in this paper are ZIMPOL data from the ESO Large Programme described in sec:res_data that were already reduced <cit.>. It was beyond the scope of this paper to fully recharacterise the ZIMPOL sensor noise and rerun a reduction procedure. As a consequence, I chose to implement a pragmatic approach that could empirically fit these parameters directly in the data of interest and that could be easily applied to reduced data from other instruments. The principle of the method is pictured in fig:noise_model.



The noise model of eq:weight is basically an affine law that links the data intensity and its associated variance. With the assumption that the noise parameters are constant over the FoV, eq:weight_const, the two noise parameters, η,, can be fitted on the full data image using the diversity of information, as is implemented in alg:noise.


First, as is shown in fig:noise_modelb, the noise map was approximated by removing from the data, a smoothed image obtained after applying a median filter of 5×5 pixels, noisemed. The pixels were then split among arc arcs (with a typical width of 5 pixels and length of 20 pixels for the ZIMPOL data) centred on the main body, shown in fig:noise_modela. The intensities (resp. variances) of the model were empirically estimated by averaging the data (resp. by computing the variance of the noise map) on each arc, noiseavg (resp. noisevar). To be robust with regard to the presence of outliers among this point cloud, fig:noise_modelc, the affine law was robustly fitted via an approach based on the median absolute deviation of the model from the empirical points, noisemad, a robust estimator of the standard deviation <cit.>. 



This method was tested on the simulated data of sec:res_sim, for which the true noise parameters are known. The results of the fit, given in fig:noise_modelc, show that the noise model was successfully estimated.

I point out here that these noise parameters could be iteratively refined during the iterations of the alternate algorithm once the model of the convolution between the object and the PSF, namely the theoretical intensity map, and the model of the halo residuals, namely the theoretical noise map, are better constrained.[See for example the documentation of the Epifluorescence DEconvolution MICroscopy plug-in <cit.> at https://icy.bioimageanalysis.org/plugin/epidemic/https://icy.bioimageanalysis.org/plugin/epidemic/.] This was nonetheless beyond the scope of this paper and is kept for a future work. Indeed, the homogeneity of the weight maps in fig:comp_instrufig:Elektra_data shows that the proposed method is a good enough approximation, correctly whitening both the readout noise and the photon shot noise.








§ ADDITIONAL FRAMES OF 



fig:Elektra_data comes in addition to fig:comp_instru and presents the second frame of the couples introduced in fig:Elektra_profiles. This figure supports the consistency of my method that produces similar results within each epoch, in terms of PSF structures and dynamics as well as moon enhancement.







 Paper Title: FNet II: Spectral Classification of Quasars, Galaxies, Stars, and broad absorption line (BAL) Quasars.



firstpage–lastpage
Dirac eigenvalues and the hyperspherical radius
    Christian Bär
    August 5, 2024
===============================================




§ ABSTRACT 
In this work, we enhance the FNet, a 1-dimensional convolutional neural network (CNN) with a residual neural network (ResNet) architecture, to perform spectral classification of quasars, galaxies, stars, and broad absorption line (BAL)-quasars in the SDSS-IV catalog from DR17 of eBOSS. Leveraging its convolutional layers and the ResNet structure with different kernel sizes, FNet autonomously identifies various patterns within the entire sample of spectra.  Since FNet does not require the intermediate step of identifying specific lines, a simple modification  enabled our current network to classify all SDSS spectra. This modification involves changing the final output layer from a single value (redshift) to multiple values (probabilities of all classes), and accordingly adjusting the loss function from mean squared error (MSE) to cross-entropy. FNet achieves a completeness of 99.00% ± 0.20 for galaxies, 98.50% ± 0.30 for quasars, 99.00% ± 0.18 for BAL-quasars, and 98.80% ± 0.20 for stars. These results are comparable to those obtained using QuasarNET, a standard CNN employed in the SDSS routine, comprises convolutional layers without the ResNet structure with equal kernel sizes, and is utilized for redshift measurement and classification by identifying seven emission lines. QuasarNET, in order to overcome the problem of finding a CIV emission line with broad absorption which is slightly more challenging than that of detecting emission lines requires to add BAL CIV line to the list of lines that the network learns to identify. However, this procedure is not necessary in FNet as it learns the features through a self-learning procedure.






stars—galaxies—quasar: individuals— software: simulations — techniques: spectroscopic  — surveys










§ INTRODUCTION


Classifying astrophysical objects—such as stars, galaxies, and quasars—based on their spectral and photometric properties is crucial for advancing research and understanding the Universe's origin and evolution <cit.>. 

Paradoxically, the substantial quantity of data generated in astrophysical surveys renders the manual inspection of each spectrum unfeasible. For instance, the seventeenth data release (DR17) from the Sloan Digital Sky Survey (SDSS) encompasses over 623 TB of data on the Science Archive Server (SAS), containing approximately 1.5 million unique spectra and objects <cit.>. The SDSS catalog from DR17 of the main survey extended Baryon Oscillation Spectroscopic Survey (DR17-eBOSS) <cit.>, from which we derive data for our current research, contains 1.4 million spectra. Therefore, automated methods must be implemented to achieve classification with the same precision as human experts.

As of 2023, approximately ten distinct and intermediate methods have been recognized for the classification and identification of stars, galaxies, supernovae, dwarfs, and quasars <cit.>:

1. Template Matching (TM): Constructs a template database for each class and associates new data with the closest matching template <cit.>. This method is used in various applications, such as the redrock pipeline classifier in the Dark Energy Spectroscopic Instrument (DESI) quasar survey <cit.>. See also <cit.>.

2. K-Nearest Neighbor (KNN): Determines the label for an object based on the majority labels of its K closest objects in the feature space <cit.>.

3. Support Vector Machine (SVM): Determines decision boundaries to distinguish between various data classes <cit.>.

4. Decision Trees (DT): Utilize leaves to represent classification results and nodes to serve as criteria for distinguishing objects <cit.>.

5. Ensemble Learning: Connects numerous classifiers to form a robust classifier <cit.>.

6. Artificial Neural Networks (ANN): Mimic the functionality of the human brain to perform machine learning tasks <cit.>.

7. Gaussian Naive Bayes: Manages features characterized by a Gaussian distribution, with the outcome determined by the maximum posterior probability <cit.>.

8. Bayesian Logistic Regression (LR): Derives posterior probability distributions from linear regression <cit.>.

9. Partial Least Squares Discriminant Analysis (PLS-DA): Addresses data within the same cluster, as opposed to data in different clusters <cit.>.

10. Ranking-based (RB) Classification: Utilizes graph-based ranking methods. An example is the locally linear embedding classifier, which transforms data from its high-dimensional space to a lower-dimensional space based on the manifold <cit.>.



 A comprehensive study by <cit.> on spectra in their observed cosmological frame has shown that convolutional neural networks (CNNs) outperform other methods in classifying observed spectra. Since DR16 <cit.>, SDSS has officially adopted QuasarNET <cit.>, a CNN used to infer redshifts and identify candidates that are likely to be misclassified by the standard pipeline. However, CNNs can face challenges such as overfitting and prolonged computation times due to their strong feature selection capabilities. For optimal classification of stars, galaxies, and quasars, conducting the analysis in the observed cosmological frame is advisable to minimize redshift impact and gain a comprehensive understanding.

Other methods usually perform worse than human-expert classification <cit.>. In contrast, deep learning (DL) methods, with their ability to recognize patterns and identify spectral features such as emission/absorption lines and spectral breaks, achieve accuracy comparable to visual inspection <cit.>. For example, <cit.> showed that principal component analysis (PCA) <cit.> produces extreme redshift estimations for quasars, which are not consistent with visual inspection or DL-based networks like FNet <cit.> and QuasarNet <cit.>.

Efficient optimization procedures in DL and CNN-based networks have enabled the solution of previously unseen problems <cit.>. The successful application of CNNs in large surveys has made them increasingly applicable to astronomical tasks <cit.>.

In this paper, we enhance the previously implemented network FNet <cit.> to classify quasars, galaxies, stars, and BAL quasars in the DR17 eBOSS dataset. FNet is a CNN with a residual neural network (ResNet) structure, designed to be trained using the observed optical spectra/flux of quasars. Initially, it was created to estimate the redshift of quasars in the SDSS-IV quasar catalog from DR16 eBOSS <cit.>. FNet achieves an accuracy of 97.0% for the velocity difference in redshift, |Δν| < 6000  km/s, and 98.0% for |Δν| < 12000  km/s.

FNet provides similar accuracy to QuasarNET but is applicable to a wider range of SDSS spectra, especially those lacking the clear emission lines exploited by QuasarNET <cit.>. In DR16Q, around 8600 spectra were not recognized by QuasarNET due to the absence of at least two emission lines necessary for its line finder units. After visual inspection, approximately 5000 of these spectra were flagged as quasars, and their visual inspection (VI) redshifts were subsequently reported in the DR16Q catalog. FNet predicts the redshift of these VI quasars with an accuracy of 87.4% for |Δν| < 6000  km/s (see Fig. 4 in ).

Unlike QuasarNET, FNet does not require identifying specific lines, allowing straightforward modification to classify all SDSS spectra. This involved changing the final output layer of FNet from a single value (redshift) to multiple values (class probabilities) and adapting the loss function from mean squared error (MSE) to cross-entropy. Further details can be found in Sec. <ref>.

Both FNet and QuasarNET are CNN-based networks but have different designs. QuasarNET consists of 4 convolutional layers with kernel sizes of 10, following the traditional procedure to identify emission lines as local patterns hidden in spectra using its "line finder" units. FNet, on the other hand, consists of 24 convolutional layers with a ResNet structure and kernel sizes of 500, 200, and 15, enabling it to find both “local” and “global” patterns in spectra through a self-learning procedure. This makes FNet applicable to ambiguous spectra and a larger range of redshifts; further details can be found in Sec. <ref>.

In this work, we utilize the SDSS catalog from DR17 of the main survey eBOSS <cit.>, which comprises 1.4 million observations.

In Sec. <ref>, we briefly review the basics of quasar spectroscopy, the structure of their optical spectra, and the description of SDSS and eBOSS surveys.

In Sec. <ref>, we describe the CNN structure developed for this work.

In Sec. <ref>, we present the results of our training and compare them with similar works in the literature.

Finally, in Sec. <ref>, we present the conclusions of the paper.





§ DATA



The dataset used in this study is derived from the Sloan Digital Sky Survey IV (SDSS-IV) quasar catalog from Data Release 17 (DR17) of the extended Baryon Oscillation Spectroscopic Survey (eBOSS) <cit.>. Notably, there have been no changes in the primary eBOSS survey data since DR16; consequently, both DR16 eBOSS and DR17 eBOSS contain the same number of spectra <cit.>. Data is gathered using 500 fibers on a 2k CCD associated with each spectrograph, with wavelength coverage spanning approximately 361 to 1014 nm <cit.>.

The data in this sample is annotated with class identifiers for each spectrum: Star (CLASS_PERSON = 1), Galaxy (CLASS_PERSON = 4), Quasar (CLASS_PERSON = 3), or BAL-Quasar (CLASS_PERSON = 30). In total, there are 1,367,288 spectra classified with these identifiers. The distribution is as follows: Stars = 233,859, Galaxies = 39,054, Quasars = 396,843, and BAL-Quasars = 39,501. Additionally, 658,020 objects were designated as CLASS_PERSON = 0, indicating they were not inspected. Eleven spectra with high flux and continua matching an archetypal quasar continuum shape, but without recognizable absorption or emission features, have been marked as possible blazars (CLASS_PERSON = 50) <cit.>. See Fig. <ref>.


A vital aspect of data selection for training neural networks is the preparation of a training/test data sample that maximizes confidence, ensuring the neural network is trained with the most accurate training set possible. The (DR16Q/DR17Q)-eBOSS dataset includes automated classifications determined by version v5_13_0 of the SDSS spectroscopic pipeline <cit.>. The quasar-only sample is estimated to be contaminated by non-quasar objects at a rate of approximately 0.3%–1.3% <cit.>. Furthermore, it is affected by low-confidence quasars, stars, and galaxies labeled by a confidence rating recorded in the column Z_CONF, with 0 representing the lowest confidence and 3 indicating the highest confidence (a value of -1 indicates the object was not visually inspected). Consequently, due to incorrect pipeline classification and redshift estimation, the DR17Q-eBOSS dataset contains stars and galaxies that have been misclassified as quasars and vice versa <cit.>. To address this, we apply the following criteria to ensure the dataset is as confident as possible:


    

  * High Confidence Level: Each class is selected based on Z_CONF = 3, indicating the highest confidence level.
   

  * BAL-Quasar Probability: To ensure the inclusion of nearly all confident BAL-Quasars, we use BAL_PROB ≥ 0.9. The BAL_PROB represents the probability assigned to each quasar based on the statistical significance of the troughs associated with the C IV line and the quality of the χ^2 fit to the quasar continuum <cit.>.
   

  * Visual Inspection: To include only visually inspected sources for Galaxies, Quasars, and BAL-Quasars, we use Z_VI > 0.
   

  * Quasar Classification: For quasars, we use IS_QSO_FINAL = 1, while for Galaxies and Stars, we use IS_QSO_FINAL ≠ 1. The IS_QSO_FINAL parameter, introduced in DR16, takes integer values from -2 to 2. Quasar spectra have a value of 1, and questionable quasar spectra have a value of 2. All values of 0 or less denote non-quasars <cit.>.
    


Moreover, in addition to visually inspected quasars from DR7Q <cit.> and DR12Q <cit.>, some spectra in the dataset were classified by the automated classifier in DR14Q <cit.>. Some of these entries were flagged for visual inspection when automated methods failed to recognize their classifications, indicated by the flag AUTOCLASS_DR14Q = VI. This subset comprises 35,268 spectra. Many of these spectra exhibit low SNRs or display strong/weak absorption lines, which can confound the pipeline (see Sec. 3 of <cit.> and Sec. 3 of <cit.>). A small DELTA_CHI2, which is the case for many spectra flagged as AUTOCLASS_DR14Q = VI, indicates that they are unlikely to be useful for scientific study.

Consequently, these spectra of flag AUTOCLASS_DR14Q = VI have been excluded from the  dataset since they cannot provide the ground true for training and testing. The final dataset in this study comprises 639,906 spectra, consisting of 229,647 stars, 28,688 galaxies, 359,609 quasars, and 21,962 BAL-quasars (see Fig. <ref> (A).

It is important to highlight that the application of the aforementioned criteria does not impose any constraints on the signal-to-noise ratio (SNR) of our test and training sets. The distribution and range of SNR in the confident data used for the training and test sets are representative of the broader dataset, as illustrated in Fig.<ref>(B). 



 






 §.§ Prepossessing the data



To enable deep learning (DL) networks to process data effectively, data preprocessing plays a crucial role. We follow the same procedure presented by <cit.>. The observed spectrum, representing wavelength versus flux, is akin to a one-dimensional vector. In the DR16Q/DR17Q dataset, each observed spectrum contains nearly 4500 flux data points in the logarithmic space of wavelength. To create trainable data, we fit and extrapolate each spectrum to a one-dimensional row of 4618 data points uniformly distributed in the logarithmic space of wavelength in the range of 360  nm to 1032.5  nm. 

Due to significant variability in the observed flux of the objects, and to accelerate the network and enhance processing accuracy, we normalize the data of each spectrum using zero-mean normalization and unit-norm normalization <cit.>. The reduced, normalized data is then stored in an N × M matrix, where N represents the total number of objects in different classes and M = 4618 is the number of wavelength resolutions. Specifically, after applying the criteria to ensure a confident dataset, N = 639,906.






§ CONVOLUTIONAL NEURAL NETWORK (CNN)








 

 §.§ Model Selection



Convolutional Neural Networks (CNNs) have been widely utilized in machine learning <cit.>, demonstrating substantial improvements in computer vision <cit.>, and have become the most widely adopted architecture <cit.>. The architecture of FNet was initially presented in <cit.>. In this work, we provide an overview of FNet and highlight the significant modifications that enable it to excel in classification tasks.

As Residual Networks (ResNets) were employed in the architecture of FNet, we first examine whether this structure can be enhanced using novel and more complex methods. Introduced by <cit.>, ResNets are an advanced form of CNNs that incorporate "skip connections." These improvements allow inputs to bypass certain layers and be added directly to the outputs of later layers, mitigating the vanishing gradient problem and enabling the training of much deeper networks. Currently, ResNet serves as a baseline in many neural network applications and comparisons <cit.>.

 

Compared to more complicated or newly introduced architectures, such as Transformers <cit.>, ResNets often maintain competitive performance for simpler tasks <cit.>. In our tests of the transformer model with different numbers of headers (2, 4, and 8) and layers (2, 4, and 6), after training with the same data as FNet, the overall accuracy achieved was always below 95%, falling short of the performance achieved by the simpler ResNet-based FNet architecture. It is observed that beyond a certain model complexity, the improvements in accuracy tend to be marginal, especially in scenarios where the data characteristics do not demand such complexity, as is the case for some 1D data <cit.>. Additionally, complexities can sometimes lead to overfitting, particularly when not properly regularized or when the dataset is not sufficiently large <cit.>. The choice to move to a more complex architecture should be driven by specific needs that cannot be met by simpler models and should be justified by empirical evidence showing clear benefits.

Moreover, the accuracy of any machine learning model is highly dependent on the quality of the input data <cit.>. High-quality data are well-curated, balanced, and reflective of real-world scenarios. This includes comprehensive representation of all classes in classification tasks, minimal noise, and high precision. Often, efforts and resources are better spent on improving data quality through enhanced collection methods and rigorous cleaning processes. As shown in Sec. <ref>, we have carefully preprocessed and selected the data.

Building upon our previous work, where our earlier model achieved high accuracy, as demonstrated by our comparison with the Transformer and supported by the reasoning above, we continue to utilize FNet based on ResNet in this article, with some minor modifications for classification instead of regression. Below, we outline the architecture of FNet.



 

 §.§ FNet



FNet differs from the officially adopted SDSS QuasarNET <cit.>, a CNN-based model that mimics the traditional human method by identifying prominent spectral lines, classifying the spectrum, and calculating redshifts from these lines. In contrast, FNet is designed to autonomously uncover hidden patterns in the data without relying on predefined information. These patterns could include prominent and weak lines, global patterns, and specific patterns of different classes. This distinct design results in different outcomes and application scopes.

The architecture of FNet is demonstrated in Fig. <ref>. FNet employs a one-dimensional ResNet architecture designed for classifying spectral data. This architecture features 8 residual blocks, each containing 3 convolutional layers with small (15), medium (200), and large (500) kernel sizes to capture both local and global patterns in a spectrum. This design aligns with our tests, where we evaluated training efficiency and prediction accuracy using randomly selected kernel sizes (10 to 500 pixels) and model depths (8 to 72 layers). Our designed architecture represents an optimized balance of efficiency and accuracy.


For each residual block, the input spectrum x is processed through three convolutional layers initialized by the He Normal initializer <cit.>. Batch normalization is applied after each convolutional layer to maintain network stability. The ReLU activation function <cit.> is used following the first two batch normalization steps to introduce non-linearity, enabling the model to capture complex patterns. The output of each residual block H(x) is then added to the input x to form H(x) + x, ensuring gradients remain within an appropriate range during training. After the residual blocks, the network flattens the output and passes it through fully connected layers, culminating in the classification of the spectrum.



 

 §.§ Training



A crucial aspect of training a neural network for a particular task is choosing the right loss function and optimization method. For classification tasks, "Cross Entropy" is commonly used, whereas "Mean Squared Error" is often preferred for regression tasks <cit.>. In the earlier version of the FNet model, we employed mean squared error since the task involved predicting the redshift of quasars <cit.>. In this work, "Cross Entropy" <cit.> is used as the loss function for this classification problem, along with the AdamW optimizer to optimize the loss function.

AdamW, or Adam with weight decay <cit.>, is a modification of Adam, which combines the best properties of the AdaGrad <cit.> and RMSProp <cit.> algorithms to handle sparse gradients on noisy problems. AdamW retains the benefits of Adam’s adaptive learning rate and momentum while incorporating weight decay separately rather than adding it to the gradient as in Adam. This small but critical change helps in better regularization and often leads to better training outcomes. AdamW typically converges quickly and is effective in cases with large datasets and/or high-dimensional spaces. Experimental results demonstrate that AdamW leads to improved training loss and enhanced generalization compared to models trained with Adam <cit.>.

We also performed a comparison by testing FNet on 40,000 spectra from the DR17Q catalog over 100 epochs (see Fig. <ref>). The results clearly exhibit the advantage of using AdamW, which converges faster than other optimizers.





Optimizing a neural network model with the AdamW optimizer requires the right selection of hyperparameters. Experimentation with hyperparameter configurations is pivotal in identifying the optimal settings for achieving superior optimization and generalization outcomes. This includes fine-tuning the learning rate (lr), adjusting beta coefficients for gradient averaging, setting weight decay for regularization, and contemplating the utilization of AMSGrad <cit.>. 

In this work, specific parameter values were employed to obtain highly optimized results, including an initial learning rate of lr_1=0.00024, beta coefficients of (0.9, 0.999), weight decay of 1 e-6, and amsgrad=False to use the standard Adam update rule. Notably, the implementation of learning rate scheduling strategies, such as reducing the learning rate after designated epochs, can enhance convergence <cit.>. Techniques like L2 penalty weight decay play a crucial role in mitigating overfitting, particularly in datasets characterized by a profusion of features <cit.>.


 
We utilized the PyTorch framework for training and testing our neural network model <cit.>. The source code for our model is available at: https://github.com/AGNNet/FNet.githttps://github.com/AGNNet/FNet.git, and the dataset used can be accessed at: https://www.kaggle.com/ywangscience/sdss-iii-ivhttps://www.kaggle.com/ywangscience/sdss-iii-iv. Training of the FNet model was performed on an Nvidia Tesla V100 graphics card, with each training epoch taking approximately 90 minutes. The inference time for predicting the redshift of an SDSS spectrum is around 14 milliseconds.



 



§ RESULTS



As detailed in Sec. <ref>, we used confident quasars, galaxies, stars, and BAL-quasars from the DR17Q catalog to train and test FNet.

We quantify the precision of the predictions using the metric of completeness <cit.>, defined as the ratio of correctly predicted spectra to the total number of spectra in each class. This metric is assessed on the test sample.




 §.§ Imbalance in Data


Highly imbalanced data presents a significant challenge, as most networks tend to prioritize the majority class. In extreme cases where the minority class constitutes less than 1%, it may be completely disregarded. Addressing class imbalance in deep learning is an active area of research, and finding effective approaches for specific tasks and imbalance scenarios is challenging <cit.>. However, traditional methods such as data sampling and cost-sensitive learning have shown applicability in deep learning contexts in recent years <cit.>.


In our sample, most spectra belong to quasars (approximately 390,000) and stars (approximately 230,000), while fewer spectra are attributed to BAL-quasars (approximately 26,000) and galaxies (approximately 30,000). This creates a moderate imbalance, with only 9% of the data representing the less common classes. Testing the FNet model on this imbalanced data yielded expected results: galaxy completeness was 86.9%, BAL-quasar completeness was 78.8%, star completeness was 98.8%, and quasar completeness was 98.3%. As anticipated, training the network with this imbalanced sample prioritized quasars and stars while neglecting BAL-quasars and galaxies. We attempted oversampling the minority classes and using weighted cross-entropy to address the imbalance, but these methods did not significantly improve the results.


Fortunately, the number of minority class spectra (approximately 30,000 for each class) in our sample is sufficient to establish a confident training set. To mitigate data imbalance, we equalized the number of spectra for each class to N = 26,667, the total number of BAL quasars, which is the smallest class.




 §.§ Problem of BAL-quasar identification



It is widely acknowledged that detecting quasars with broad absorption line (BAL) features <cit.> is more complex than detecting emission lines at the human-expert level. Quasars with BAL features display a distinct absorption trough blueward of the emission lines, which must be sufficiently separated from the line center and possess adequate width. The balnicity index (BI), an automated measurement of the strength of the BAL feature in the CIV line, annotated in the SDSS survey as BI_CIV, quantifies this characteristic. The primary challenge in determining whether a spectrum exhibits a BAL feature, whether through visual inspection or automated methods, lies in accurately assessing the level of the unabsorbed continuum.


For example, in this context, QuasarNET <cit.> addresses the problem of identifying a CIV emission line with broad absorptions by treating it as equivalent to detecting any other emission line. To achieve this, they incorporate a BAL_CIV line into the list of lines that the network is trained to recognize. QuasarNET, which consists of a network with four convolutional layers followed by a final fully-connected layer, includes "six line finders" to detect Lyα (121.6 nm), CIV (154.9 nm), CIII (190.9 nm), MgII (279.6 nm), Hβ (486.2 nm), and Hα (656.3 nm) emission lines. Additionally, it is enhanced with a seventh “line finder” unit specifically designed to identify the CIV line with a broad absorption feature.


On the other hand, FNet utilizes a ResNet structure with 24 convolutional layers and kernel sizes of 500, 200, and 15. This design enables FNet to recognize both “global” and “local” patterns in spectra through a self-learning procedure, independently discovering hidden features and training on all spectra uniformly. The presence of the larger kernels, particularly the size 500, effectively bypasses complexities in detecting BAL features and eliminates the need to incorporate external information, such as the BAL_CIV threshold, for spectral classification. 

We tested FNet using only smaller kernel sizes of 100 and 10, and 200 and 17, instead of 500, 200, and 15. The completeness decreased with the smaller kernels, especially for BAL-quasars, suggesting that some broad features were overlooked by the network.



 §.§ training results


Eighty percent of the spectra were allocated to the training set, while 10% were reserved for validation using k-fold cross-validation <cit.>. The remaining spectra were designated for the test set, ensuring they were not used to train the CNN.





After exclusively using the training/test set primarily sourced from our criteria introduced in Sec. <ref> and excluding those labeled with AUTOCLASS_DR14Q = VI, consistent with the approach employed in QuasarNET <cit.>, the classification completeness achieved by FNet improved to 99.00% ± 0.20 for galaxies, 98.50% ± 0.30 for quasars, 99.00% ± 0.18 for BAL-quasars, and 98.80% ± 0.20 for stars. The uncertainty in completeness percentages reflects tests conducted over multiple data batches. This method of evaluating a model is particularly useful for ensuring robustness and understanding the consistency of the model's performance, with validity ensured by choosing large batches that include randomly selected test samples of all classes. The confusion matrix evaluating FNet's classification is depicted in Fig. <ref>. Figure <ref> and Figure <ref> show examples of both successes and failures of the final test. In order to reduce noise in the data and make underlying spectral features, such as emission or absorption lines, more pronounced, we used Gaussian smoothing on all spectra in the above figures only for presenting purpose.


Indeed, when training on imbalanced data (as illustrated in Sec. <ref>), the high number of misclassified rare sources indicates that the network tends to confuse rare sources with dominant ones, reducing the completeness for dominant sources. Conversely, using balanced data yields more accurate results for rare sources. This improvement in completeness for rare sources subsequently increases the overall number of correctly predicted sources, as there are fewer instances of rare sources being confused with dominant ones.












 §.§ Training result for less confident spectra and potential application to other datasets
 


In Sec. <ref>, we used only data with the highest confidence redshifts (Z_CONF = 3), the most secure BAL probabilities (BAL_PROB ≥ 0.9), and no ambiguity about quasar classification (IS_QSO_FINAL = 1). Relaxing these conditions to include sources with low redshift confidence (Z_CONF ≠ 3), less secure BAL probabilities (0.5 ≤ BAL_PROB < 0.9), and ambiguously classified quasars (IS_QSO_FINAL ≠ 1) adds an extra 2% of spectra to the main data for the training-test procedure. These spectra are mostly related to sources with too low SNRs to be confidently classified by human experts <cit.>.


We performed training and testing on the dataset with ambiguous data included and observed a slight decrease in completeness compared to using only clean data. Specifically, FNet achieves a completeness of 98.30% ± 0.40 for galaxies, 98.00% ± 0.40 for quasars, 98.50% ± 0.40 for BAL-quasars, and 98.20% ± 0.30 for stars. These results remain acceptable for scientific use.


For example, when testing QuasarNet on the sample of spectra without confident annotations (Z_CONF < 3), which represents about 2% of the spectra <cit.>, it classifies 20% of the spectra as quasars with reliable redshifts. In comparison, FNet achieves a completeness of 45% for quasars, 72% for BAL-quasars, and 62% for stars. For galaxies, the number of sources with Z_CONF ≠ 3 is negligible, so we do not consider them for this test. Fig. <ref> shows the confusion matrix for this 2% of ambiguous spectra.



In general, FNet learns to classify spectra not only by recognizing prominent emission and absorption lines but also by identifying subtle patterns that may not be immediately apparent <cit.>. Consequently, even after incorporating the 2% of train/test sets with less confident classifications, FNet still achieves high classification completeness. This robustness suggests that FNet has significant potential for application to other datasets. However, careful data preprocessing is essential when applying FNet to datasets other than SDSS-IV.






§ CONCLUSIONS
 


Accurate spectral classification in astrophysics requires reliable methods. With the increasing volume of data from current and forthcoming surveys, relying solely on visual inspection is impractical. Research into statistical approaches for analyzing spectra within their cosmological context <cit.> has shown that convolutional neural networks (CNNs) outperform other methods in classifying observed spectra. Therefore, developing automated methods that achieve human-expert accuracy has become essential.


In this study, we adapted and modified the previously introduced network, FNet, a 1-dimensional CNN with a ResNet architecture, to classify the spectra of quasars, galaxies, stars, and BAL-quasars in the SDSS-IV catalog from the DR17 superset of eBOSS. This network was previously successful in predicting the redshift of quasars in DR16 of eBOSS, achieving a velocity difference accuracy of 97.0% for |Δν| < 6000   km/s, 98.0% for |Δν| < 12000   km/s, and 98.9% for |Δν| < 30000   km/s when compared to visually inspected redshifts <cit.>.


As FNet does not rely on identifying specific lines, a simple modification enabled our current network to classify all SDSS spectra without external information about emission/absorption lines. This modification involves changing the last output layer of FNet from a single value (redshift) to multiple values (probabilities of all classes) and adjusting the loss function from mean squared error (MSE) to cross-entropy.

From a total of 1,367,288 spectra of quasars, stars, and galaxies in the SDSS-IV quasar catalog from DR17 of eBOSS, we selected 639,906 spectra (Stars: 229,647; Galaxies: 28,688; Quasars: 359,609; BAL Quasars: 21,962) for the classification training set. This subset represents the highest confidence level of source selection in this catalog.


Utilizing its convolutional layers and the ResNet structure, FNet autonomously discerns both “local” and “global” patterns within the spectra. The CNN employs convolutional layers with kernel sizes of 500, 200, and 15 to detect various patterns in the flux of quasars. The flux is then processed through 24 residual blocks: the first 21 blocks have a channel size of 32, followed by three blocks with channel sizes of 64, 32, and 16, respectively. This self-learning capability is crucial for classifying spectra from diverse astrophysical objects.


It is worth noting that detecting quasars with broad absorption lines (BALs), as in the case of BAL quasars, is slightly more challenging than detecting emission lines. Unlike QuasarNET, which addresses this by adding the BAL CIV line to the list of identifiable lines to find a CIV emission line with broad absorptions, this step is unnecessary in FNet due to its self-learning feature acquisition process.

FNet achieves a completeness of 99.00% ± 0.20 for galaxies, 98.50% ± 0.30 for quasars, 99.00% ± 0.18 for BAL quasars, and 98.80% ± 0.20 for stars. These results are comparable to those obtained using QuasarNET <cit.>, a standard CNN used in the SDSS catalog for redshift measurement and classification <cit.>.





§ ACKNOWLEDGMENT


We acknowledge the support from Dr. Jie Jiang and Prof. Yifu Cai from USTC for providing the Nvidia DGX Station. We would like to thank the anonymous referee for their comments, which have enhanced the scientific credibility of the paper, leading to more comprehensive discussions and clearer explanations. R. Moradi acknowledges support from the Institute of High Energy Physics, Chinese Academy of Sciences (E32984U810).




§ DATA AVAILABILITY


The catalog underlying this article is available in The Sloan Digital Sky Survey Quasar catalog: sixteenth data release (DR16Q), at https://www.sdss.org/dr16/algorithms/qso_catalog/https://www.sdss.org/dr16/algorithms/qsocatalog/. The code can be found in the public domain: https://github.com/AGNNet/FNet.githttps://github.com/AGNNet/FNet.git and the prepared data set can be found in https://www.kaggle.com/ywangscience/sdss-iii-ivhttps://www.kaggle.com/ywangscience/sdss-iii-iv.

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@doi@urlcharsother ifnextchar [ @doi@ @doi@[]
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[Abdurro'uf et al.,Abdurro'uf et al.2022]2022ApJS..259...35A
Abdurro'uf et al., 2022, @doi [] 10.3847/1538-4365/ac4414, https://ui.adsabs.harvard.edu/abs/2022ApJS..259...35A 259, 35

[Allen et al.,Allen et al.2019]allen2019deep
Allen G.,  et al., 2019, arXiv preprint arXiv:1902.00522

[Ball & BrunnerBall & Brunner2010]ball2010data
Ball N. M.,  Brunner R. J.,  2010, International Journal of Modern Physics D, 19, 1049

[Baqui et al.,Baqui et al.2021]baqui2021minijpas
Baqui P.,  et al., 2021, , 645, A87

[Barrientos, Solar  & MendozaBarrientos et al.2020]barrientos2020machine
Barrientos A.,  Solar M.,   Mendoza M.,  2020, Astronomical Data Analysis Software and Systems XXVII, 522, 385

[Brice & AndonieBrice & Andonie2019a]brice2019classification
Brice M.,  Andonie R.,  2019a, in Wang D.,  Doya K.,  eds, 2019 International Joint Conference on Neural Networks (IJCNN). IEEE, Budapest, Hungary, pp 1–8

[Brice & AndonieBrice & Andonie2019b]2019AJ....158..188B
Brice M. J.,  Andonie R.,  2019b, @doi [] 10.3847/1538-3881/ab40d0, https://ui.adsabs.harvard.edu/abs/2019AJ....158..188B 158, 188

[Busca & BallandBusca & Balland2018]2018arXiv180809955B
Busca N.,  Balland C.,  2018, arXiv e-prints, https://ui.adsabs.harvard.edu/abs/2018arXiv180809955B p. arXiv:1808.09955

[Carleo, Cirac, Cranmer, Daudet, Schuld, Tishby, Vogt-Maranto  & ZdeborováCarleo et al.2019]RevModPhys.91.045002
Carleo G.,  Cirac I.,  Cranmer K.,  Daudet L.,  Schuld M.,  Tishby N.,  Vogt-Maranto L.,   Zdeborová L.,  2019, @doi [Rev. Mod. Phys.] 10.1103/RevModPhys.91.045002, 91, 045002

[Cowie, Barger, Bauer  & González-LópezCowie et al.2020]2020ApJ...891...69C
Cowie L. L.,  Barger A. J.,  Bauer F. E.,   González-López J.,  2020, @doi [] 10.3847/1538-4357/ab6aaa, https://ui.adsabs.harvard.edu/abs/2020ApJ...891...69C 891, 69

[Daniel, Connolly, Schneider, Vanderplas  & XiongDaniel et al.2011]2011AJ....142..203D
Daniel S. F.,  Connolly A.,  Schneider J.,  Vanderplas J.,   Xiong L.,  2011, @doi [] 10.1088/0004-6256/142/6/203, https://ui.adsabs.harvard.edu/abs/2011AJ....142..203D 142, 203

[Dawson et al.,Dawson et al.2013]2013AJ....145...10D
Dawson K. S.,  et al., 2013, @doi [] 10.1088/0004-6256/145/1/10, https://ui.adsabs.harvard.edu/abs/2013AJ....145...10D 145, 10

[Deng, Zhu, Cheng, Zong  & ZhangDeng et al.2016]DENG2016143
Deng Z.,  Zhu X.,  Cheng D.,  Zong M.,   Zhang S.,  2016, @doi [Neurocomputing] https://doi.org/10.1016/j.neucom.2015.08.112, 195, 143

[Du, Luo, Yang, Hou  & GuoDu et al.2016]2016PASP..128c4502D
Du C.,  Luo A.,  Yang H.,  Hou W.,   Guo Y.,  2016, @doi [] 10.1088/1538-3873/128/961/034502, https://ui.adsabs.harvard.edu/abs/2016PASP..128c4502D 128, 034502

[Duchi, Hazan  & SingerDuchi et al.2011]JMLR:v12:duchi11a
Duchi J.,  Hazan E.,   Singer Y.,  2011, Journal of Machine Learning Research, 12, 2121

[Farr, Font-Ribera  & PontzenFarr et al.2020]2020JCAP...11..015F
Farr J.,  Font-Ribera A.,   Pontzen A.,  2020, @doi [] 10.1088/1475-7516/2020/11/015, https://ui.adsabs.harvard.edu/abs/2020JCAP...11..015F 2020, 015

[FleschFlesch2021]2021MNRAS.tmp..804F
Flesch E. W.,  2021, @doi [] 10.1093/mnras/stab812, https://ui.adsabs.harvard.edu/abs/2021MNRAS.tmp..804F 

[Freund & MasonFreund & Mason1999]10.5555/645528.657623
Freund Y.,  Mason L.,  1999, in Ivan Bratko S. D.,  ed., Proceedings of the Sixteenth International Conference on Machine Learning. ICML '99.
Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, p. 124–133

[Fuqiang, Yan, Yude  & GuodongFuqiang et al.2014]2014PASA...31....1F
Fuqiang C.,  Yan W.,  Yude B.,   Guodong Z.,  2014, @doi [] 10.1017/pasa.2013.38, https://ui.adsabs.harvard.edu/abs/2014PASA...31....1F 31, e001

[FushikiFushiki2011]10.1007/s11222-009-9153-8
Fushiki T.,  2011, @doi [Statistics and Computing] 10.1007/s11222-009-9153-8, 21, 137–146

[Glazebrook, Offer  & DeeleyGlazebrook et al.1998]glazebrook1998automatic
Glazebrook K.,  Offer A. R.,   Deeley K.,  1998, The Astrophysical Journal, 492, 98

[Golik, Doetsch  & NeyGolik et al.2013]golik13_interspeech
Golik P.,  Doetsch P.,   Ney H.,  2013, in Proc. Interspeech 2013. pp 1756–1760, @doi10.21437/Interspeech.2013-436

[Goodfellow, Bengio  & CourvilleGoodfellow et al.2016]Goodfellow-et-al-2016
Goodfellow I.,  Bengio Y.,   Courville A.,  2016, Deep Learning.
MIT Press

[Gorishniy, Rubachev, Khrulkov  & BabenkoGorishniy et al.2021]Gorishniy2021RevisitingDL
Gorishniy Y. V.,  Rubachev I.,  Khrulkov V.,   Babenko A.,  2021, in Neural Information Processing Systems. <https://api.semanticscholar.org/CorpusID:235593213>

[Guo et al.,Guo et al.2019]2019MNRAS.485.2167G
Guo Y. X.,  et al., 2019, @doi [] 10.1093/mnras/stz458, https://ui.adsabs.harvard.edu/abs/2019MNRAS.485.2167G 485, 2167

[Guy et al.,Guy et al.2023]2023AJ....165..144G
Guy J.,  et al., 2023, @doi [] 10.3847/1538-3881/acb212, https://ui.adsabs.harvard.edu/abs/2023AJ....165..144G 165, 144

[Guzmán, Esquivel, Hernández  & RoblesGuzmán et al.2018]guzman2018stellar
Guzmán A. P. A.,  Esquivel A. E. O.,  Hernández R. D.,   Robles L. A.,  2018, in Batyrshin I.,  ed., 2018 Seventeenth Mexican International Conference on Artificial Intelligence (MICAI). IEEE, Guadalajara, Mexico, pp 81–87

[Haehnelt & ReesHaehnelt & Rees1993]1993MNRAS.263..168H
Haehnelt M. G.,  Rees M. J.,  1993, @doi [] 10.1093/mnras/263.1.168, https://ui.adsabs.harvard.edu/abs/1993MNRAS.263..168H 263, 168

[He, Zhang, Ren  & SunHe et al.2015]he2015delving
He K.,  Zhang X.,  Ren S.,   Sun J.,  2015, in Proceedings of the IEEE international conference on computer vision. pp 1026–1034

[He, Zhang, Ren  & SunHe et al.2016]he2016deep
He K.,  Zhang X.,  Ren S.,   Sun J.,  2016, in Proceedings of the IEEE conference on computer vision and pattern recognition. pp 770–778

[Henning, Beluch, Fraser  & FriedrichHenning et al.2022]2022arXiv221004675H
Henning S.,  Beluch W.,  Fraser A.,   Friedrich A.,  2022, @doi [arXiv e-prints] 10.48550/arXiv.2210.04675, https://ui.adsabs.harvard.edu/abs/2022arXiv221004675H p. arXiv:2210.04675

[Hoyle, Rau, Paech, Bonnett, Seitz  & WellerHoyle et al.2015]2015MNRAS.452.4183H
Hoyle B.,  Rau M. M.,  Paech K.,  Bonnett C.,  Seitz S.,   Weller J.,  2015, @doi [] 10.1093/mnras/stv1551, https://ui.adsabs.harvard.edu/abs/2015MNRAS.452.4183H 452, 4183

[Hutchinson et al.,Hutchinson et al.2016]2016AJ....152..205H
Hutchinson T. A.,  et al., 2016, @doi [] 10.3847/0004-6256/152/6/205, https://ui.adsabs.harvard.edu/abs/2016AJ....152..205H 152, 205

[Inayoshi, Visbal  & HaimanInayoshi et al.2020]2020ARA   A..58...27I
Inayoshi K.,  Visbal E.,   Haiman Z.,  2020, @doi [] 10.1146/annurev-astro-120419-014455, https://ui.adsabs.harvard.edu/abs/2020ARA   A..58...27I 58, 27

[Jayalakshmi & SanthakumaranJayalakshmi & Santhakumaran2011]jayalakshmi2011statistical
Jayalakshmi T.,  Santhakumaran A.,  2011, International Journal of Computer Theory and Engineering, 3, 1793

[Jing-Min, Chen-Ye, Lu, Li-Ting, Ting-Ting, Lin-Pin  & Wei-HongJing-Min et al.2020]jing2020new
Jing-Min Z.,  Chen-Ye M.,  Lu W.,  Li-Ting D.,  Ting-Ting X.,  Lin-Pin A.,   Wei-Hong Z.,  2020, Chinese Astronomy and Astrophysics, 44, 334

[Johnson & KhoshgoftaarJohnson & Khoshgoftaar2019]Johnson2019SurveyOD
Johnson J. M.,  Khoshgoftaar T. M.,  2019, Journal of Big Data, 6, 1

[Khramtsov, Vavilova, Dobrycheva, Vasylenko, Melnyk, Elyiv, Akhmetov  & DmytrenkoKhramtsov et al.2022]2022KosNT..28e..27K
Khramtsov V.,  Vavilova I. B.,  Dobrycheva D. V.,  Vasylenko M. Y.,  Melnyk O. V.,  Elyiv A. A.,  Akhmetov V. S.,   Dmytrenko A. M.,  2022, @doi [Kosmichna Nauka i Tekhnologiya] 10.15407

[Kingma & BaKingma & Ba2015]kingma2014adam
Kingma D. P.,  Ba J.,  2015, in Bengio Y.,  LeCun Y.,  eds, 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings. <http://arxiv.org/abs/1412.6980>

[Kiranyaz, Avci, Abdeljaber, Ince, Gabbouj  & InmanKiranyaz et al.2021]kiranyaz20211d
Kiranyaz S.,  Avci O.,  Abdeljaber O.,  Ince T.,  Gabbouj M.,   Inman D. J.,  2021, Mechanical systems and signal processing, 151, 107398

[Koziarski & CyganekKoziarski & Cyganek2017]koziarski2017image
Koziarski M.,  Cyganek B.,  2017, Integrated Computer-Aided Engineering, 24, 337

[LeCun, Bottou, Bengio  & HaffnerLeCun et al.1998]lecun1998gradient
LeCun Y.,  Bottou L.,  Bengio Y.,   Haffner P.,  1998, Proceedings of the IEEE, 86, 2278

[Li et al.,Li et al.2018]2018ApJS..234...31L
Li Y.-B.,  et al., 2018, @doi [] 10.3847/1538-4365/aaa415, https://ui.adsabs.harvard.edu/abs/2018ApJS..234...31L 234, 31

[Li, Li, Guan, Liang, Lai  & LuoLi et al.2019a]li2019research
Li H.,  Li J.,  Guan X.,  Liang B.,  Lai Y.,   Luo X.,  2019a, in 2019 15th international conference on computational intelligence and security (CIS). pp 78–81

[Li, Lin  & QiuLi et al.2019b]2019RAA....19..111L
Li X.-R.,  Lin Y.-T.,   Qiu K.-B.,  2019b, @doi [Research in Astronomy and Astrophysics] 10.1088/1674-4527/19/8/111, https://ui.adsabs.harvard.edu/abs/2019RAA....19..111L 19, 111

[LiuLiu2021]liu2021stellar
Liu Z.,  2021, New Astronomy, 88, 101613

[Liu et al.,Liu et al.2015]2015RAA....15.1137L
Liu C.,  et al., 2015, @doi [Research in Astronomy and Astrophysics] 10.1088/1674-4527/15/8/004, https://ui.adsabs.harvard.edu/abs/2015RAA....15.1137L 15, 1137

[Liu, Hsaio  & TuLiu et al.2018]liu2018time
Liu C.-L.,  Hsaio W.-H.,   Tu Y.-C.,  2018, IEEE Transactions on Industrial Electronics, 66, 4788

[Liu et al.,Liu et al.2019]2019MNRAS.483.4774L
Liu W.,  et al., 2019, @doi [] 10.1093/mnras/sty3020, https://ui.adsabs.harvard.edu/abs/2019MNRAS.483.4774L 483, 4774

[Lloyd EvansLloyd Evans2010]2010JApA...31..177L
Lloyd Evans T.,  2010, @doi [Journal of Astrophysics and Astronomy] 10.1007/s12036-010-0017-6, https://ui.adsabs.harvard.edu/abs/2010JApA...31..177L 31, 177

[Loshchilov & HutterLoshchilov & Hutter2017]2017arXiv171105101L
Loshchilov I.,  Hutter F.,  2017, @doi [arXiv e-prints] 10.48550/arXiv.1711.05101, https://ui.adsabs.harvard.edu/abs/2017arXiv171105101L p. arXiv:1711.05101

[Luo, Wu, Zhao  & ZhaoLuo et al.2008]luo2008automated
Luo A.-L.,  Wu Y.,  Zhao J.,   Zhao G.,  2008, in Bridger A.,  Radziwill N. M.,  eds,   Vol. 7019, Advanced Software and Control for Astronomy II. SPIE, Marseille, France, pp 1055–1065

[Lupton, Ivezic, Gunn, Knapp, Strauss  & YasudaLupton et al.2002]sdssimagepipeline
Lupton R. H.,  Ivezic Z.,  Gunn J. E.,  Knapp G.,  Strauss M. A.,   Yasuda N.,  2002, in Tyson J. A.,  Wolff S.,  eds,   Vol. 4836, Survey and Other Telescope Technologies and Discoveries. SPIE, Waikoloa, Hawai'i, United States, pp 350 – 356

[Lyke et al.,Lyke et al.2020]2020ApJS..250....8L
Lyke B. W.,  et al., 2020, @doi [] 10.3847/1538-4365/aba623, https://ui.adsabs.harvard.edu/abs/2020ApJS..250....8L 250, 8

[Madau & ReesMadau & Rees2001]2001ApJ...551L..27M
Madau P.,  Rees M. J.,  2001, @doi [] 10.1086/319848, https://ui.adsabs.harvard.edu/abs/2001ApJ...551L..27M 551, L27

[Mao, Mohri  & ZhongMao et al.2023]2023arXiv230407288M
Mao A.,  Mohri M.,   Zhong Y.,  2023, @doi [arXiv e-prints] 10.48550/arXiv.2304.07288, https://ui.adsabs.harvard.edu/abs/2023arXiv230407288M p. arXiv:2304.07288

[Moradi, Rueda, Ruffini  & WangMoradi et al.2021]2021A   A...649A..75M
Moradi R.,  Rueda J. A.,  Ruffini R.,   Wang Y.,  2021, @doi [] 10.1051/0004-6361/201937135, https://ui.adsabs.harvard.edu/abs/2021A   A...649A..75M 649, A75

[Morice-Atkinson, Hoyle  & BaconMorice-Atkinson et al.2018]2018MNRAS.481.4194M
Morice-Atkinson X.,  Hoyle B.,   Bacon D.,  2018, @doi [] 10.1093/mnras/sty2575, https://ui.adsabs.harvard.edu/abs/2018MNRAS.481.4194M 481, 4194

[Pâris et al.,Pâris et al.2017a]paris2017sloan
Pâris I.,  et al., 2017a, Astronomy & Astrophysics, 597, A79

[Pâris et al.,Pâris et al.2017b]2017A   A...597A..79P
Pâris I.,  et al., 2017b, @doi [] 10.1051/0004-6361/201527999, https://ui.adsabs.harvard.edu/abs/2017A   A...597A..79P 597, A79

[Pâris et al.,Pâris et al.2018]2018A   A...613A..51P
Pâris I.,  et al., 2018, @doi [] 10.1051/0004-6361/201732445, https://ui.adsabs.harvard.edu/abs/2018A   A...613A..51P 613, A51

[Paszke et al.,Paszke et al.2019]2019arXiv191201703P
Paszke A.,  et al., 2019, arXiv e-prints, https://ui.adsabs.harvard.edu/abs/2019arXiv191201703P p. arXiv:1912.01703

[Pérez-Galarce, Pichara, Huijse, Catelan  & MeryPérez-Galarce et al.2021]2021MNRAS.503..484P
Pérez-Galarce F.,  Pichara K.,  Huijse P.,  Catelan M.,   Mery D.,  2021, @doi [] 10.1093/mnras/stab320, https://ui.adsabs.harvard.edu/abs/2021MNRAS.503..484P 503, 484

[Praveen, Srinivasu, Shafi, Wozniak  & IjazPraveen et al.2022]praveen2022resnet
Praveen S. P.,  Srinivasu P. N.,  Shafi J.,  Wozniak M.,   Ijaz M. F.,  2022, Scientific Reports, 12, 20804

[Pruzhinskaya, Malanchev, Kornilov, Ishida, Mondon, Volnova  & KorolevPruzhinskaya et al.2019]pruzhinskaya2019anomaly
Pruzhinskaya M. V.,  Malanchev K. L.,  Kornilov M. V.,  Ishida E. E.,  Mondon F.,  Volnova A. A.,   Korolev V. S.,  2019, , 489, 3591

[QuinlanQuinlan1996]10.1145/234313.234346
Quinlan J. R.,  1996, @doi [ACM Comput. Surv.] 10.1145/234313.234346, 28, 71–72

[Rastegarnia, Mirtorabi, Moradi, Sadr  & WangRastegarnia et al.2022]Rastegarnia_2022
Rastegarnia F.,  Mirtorabi M. T.,  Moradi R.,  Sadr A. V.,   Wang Y.,  2022, @doi [] 10.1093/mnras/stac076, 511, 4490

[Reddi, Kale  & KumarReddi et al.2019]2019arXiv190409237R
Reddi S. J.,  Kale S.,   Kumar S.,  2019, @doi [arXiv e-prints] 10.48550/arXiv.1904.09237, https://ui.adsabs.harvard.edu/abs/2019arXiv190409237R p. arXiv:1904.09237

[Refaeilzadeh, Tang  & LiuRefaeilzadeh et al.2009]refaeilzadeh2009cross
Refaeilzadeh P.,  Tang L.,   Liu H.,  2009, Encyclopedia of database systems, 5, 532

[Rosenfeld & VanderbrugRosenfeld & Vanderbrug1977]template_matching
Rosenfeld A.,  Vanderbrug G.,  1977, @doi [IEEE Transactions on Computers] 10.1109/TC.1977.1674847, 26, 384

[Schneider et al.,Schneider et al.2010]2010AJ....139.2360S
Schneider D. P.,  et al., 2010, @doi [] 10.1088/0004-6256/139/6/2360, https://ui.adsabs.harvard.edu/abs/2010AJ....139.2360S 139, 2360

[Smee et al.,Smee et al.2013]2013AJ....146...32S
Smee S. A.,  et al., 2013, @doi [] 10.1088/0004-6256/146/2/32, https://ui.adsabs.harvard.edu/abs/2013AJ....146...32S 146, 32

[Song, Wang, Maguire  & NiboucheSong et al.2018]SONG201879
Song W.,  Wang H.,  Maguire P.,   Nibouche O.,  2018, @doi [Chemometrics and Intelligent Laboratory Systems] https://doi.org/10.1016/j.chemolab.2018.08.011, 182, 79

[Sookmee, Suwannajak, Techa-Angkoon, Panyangam  & TanakulSookmee et al.2020]sookmee2020globular
Sookmee P.,  Suwannajak C.,  Techa-Angkoon P.,  Panyangam B.,   Tanakul N.,  2020, in Lursinsap C.,  ed., 2020 17th International Joint Conference on Computer Science and Software Engineering (JCSSE). IEEE, Bangkok, Thailand, pp 98–103

[SubbaRao, Frieman, Bernardi, Loveday, Nichol, Castander  & MeiksinSubbaRao et al.2002]subbarao2002sloan
SubbaRao M.,  Frieman J.,  Bernardi M.,  Loveday J.,  Nichol B.,  Castander F.,   Meiksin A.,  2002, in Starck J.-L.,  Murtagh F. D.,  eds,   Vol. 4847, Astronomical Data Analysis II. SPIE, Waikoloa, Hawaii, United States, pp 452–460

[Tabasi, Salmani, Khaliliyan  & FirouzjaeeTabasi et al.2023]2023ApJ...954..164T
Tabasi S. S.,  Salmani R. V.,  Khaliliyan P.,   Firouzjaee J. T.,  2023, @doi [] 10.3847/1538-4357/ace03f, https://ui.adsabs.harvard.edu/abs/2023ApJ...954..164T 954, 164

[Tao, Zhang, Cui  & ZhangTao et al.2018]tao2018automated
Tao Y.,  Zhang Y.,  Cui C.,   Zhang G.,  2018, @doi [arXiv:1801.04839] https://doi.org/10.48550/arXiv.1801.04839

[Tieleman & HintonTieleman & Hinton2012]Tieleman
Tieleman T.,  Hinton 2012, Lecture 6.5 - RMSProp, COURSERA

[Vaswani, Shazeer, Parmar, Uszkoreit, Jones, Gomez, Kaiser  & PolosukhinVaswani et al.2017]vaswani2017attention
Vaswani A.,  Shazeer N.,  Parmar N.,  Uszkoreit J.,  Jones L.,  Gomez A. N.,  Kaiser Ł.,   Polosukhin I.,  2017, Advances in neural information processing systems, 30

[Wallerstein & KnappWallerstein & Knapp1998]doi:10.1146/annurev.astro.36.1.369
Wallerstein G.,  Knapp G. R.,  1998, @doi [Annual Review of Astronomy and Astrophysics] 10.1146/annurev.astro.36.1.369, 36, 369

[Wang, Guo  & LuoWang et al.2017]2017MNRAS.465.4311W
Wang K.,  Guo P.,   Luo A. L.,  2017, @doi [] 10.1093/mnras/stw2894, https://ui.adsabs.harvard.edu/abs/2017MNRAS.465.4311W 465, 4311

[Wang et al.,Wang et al.2021]2021ApJ...907L...1W
Wang F.,  et al., 2021, @doi [] 10.3847/2041-8213/abd8c6, https://ui.adsabs.harvard.edu/abs/2021ApJ...907L...1W 907, L1

[Westfall et al.,Westfall et al.2019]Westfall_2019
Westfall K. B.,  et al., 2019, @doi [] 10.3847/1538-3881/ab44a2, 158, 231

[Weymann, Carswell  & SmithWeymann et al.1981]1981ARA   A..19...41W
Weymann R. J.,  Carswell R. F.,   Smith M. G.,  1981, @doi [] 10.1146/annurev.aa.19.090181.000353, https://ui.adsabs.harvard.edu/abs/1981ARA   A..19...41W 19, 41

[Weymann, Morris, Foltz  & HewettWeymann et al.1991]1991ApJ...373...23W
Weymann R. J.,  Morris S. L.,  Foltz C. B.,   Hewett P. C.,  1991, @doi [] 10.1086/170020, https://ui.adsabs.harvard.edu/abs/1991ApJ...373...23W 373, 23

[Whang & LeeWhang & Lee2020]whang2020data
Whang S. E.,  Lee J.-G.,  2020, Proceedings of the VLDB Endowment, 13, 3429

[Willott et al.,Willott et al.2010]2010AJ....140..546W
Willott C. J.,  et al., 2010, @doi [] 10.1088/0004-6256/140/2/546, https://ui.adsabs.harvard.edu/abs/2010AJ....140..546W 140, 546

[Wu, Zhang  & XuWu et al.2017]wu2017tiny
Wu J.,  Zhang Q.,   Xu G.,  2017, Technical report

[Wu, Tao, Fan, Cui  & ZhangWu et al.2023]10.1093/mnras/stad2913
Wu Y.,  Tao Y.,  Fan D.,  Cui C.,   Zhang Y.,  2023, @doi [Monthly Notices of the Royal Astronomical Society] 10.1093/mnras/stad2913, 527, 1163

[Xu, Ren, Liu  & JiaXu et al.2014]xu2014deep
Xu L.,  Ren J. S.,  Liu C.,   Jia J.,  2014, Advances in neural information processing systems, 27, 1790

[Xu, Wang, Chen  & LiXu et al.2015]xu2015empirical
Xu B.,  Wang N.,  Chen T.,   Li M.,  2015, arXiv preprint arXiv:1505.00853

[Yamashita, Nishio, Do  & TogashiYamashita et al.2018]yamashita2018convolutional
Yamashita R.,  Nishio M.,  Do R. K. G.,   Togashi K.,  2018, Insights into imaging, 9, 611

[Yang, Nguyen, San, Li  & KrishnaswamyYang et al.2015]yang2015deep
Yang J.,  Nguyen M. N.,  San P. P.,  Li X.,   Krishnaswamy S.,  2015, in Ijcai. pp 3995–4001

[Yang et al.,Yang et al.2020]2020ApJ...897L..14Y
Yang J.,  et al., 2020, @doi [] 10.3847/2041-8213/ab9c26, https://ui.adsabs.harvard.edu/abs/2020ApJ...897L..14Y 897, L14

[Yang, Zhou, Cai, Shi, Yang, Zhao, Duan  & YinYang et al.2023]2023MNRAS.518.5904Y
Yang H.,  Zhou L.,  Cai J.,  Shi C.,  Yang Y.,  Zhao X.,  Duan J.,   Yin X.,  2023, @doi [] 10.1093/mnras/stac3292, https://ui.adsabs.harvard.edu/abs/2023MNRAS.518.5904Y 518, 5904

[Zhang & ZhouZhang & Zhou2007]ZHANG20072038
Zhang M.-L.,  Zhou Z.-H.,  2007, @doi [Pattern Recognition] https://doi.org/10.1016/j.patcog.2006.12.019, 40, 2038

[Zhao & ZhangZhao & Zhang2008]zhao2008comparison
Zhao Y.,  Zhang Y.,  2008, Advances in Space Research, 41, 1955

[Zhao, Zhao, Chu, Jing  & DengZhao et al.2012]Zhao_2012
Zhao G.,  Zhao Y.-H.,  Chu Y.-Q.,  Jing Y.-P.,   Deng L.-C.,  2012, @doi [Research in Astronomy and Astrophysics] 10.1088/1674-4527/12/7/002, 12, 723

[Zhong et al.,Zhong et al.2023]2023arXiv231104146Z
Zhong F.,  et al., 2023, @doi [arXiv e-prints] 10.48550/arXiv.2311.04146, https://ui.adsabs.harvard.edu/abs/2023arXiv231104146Z p. arXiv:2311.04146

[Zou & el al.Zou & el al.2020]2020PASP..132d4503Z
Zou Z.,  el al. 2020, @doi [] 10.1088/1538-3873/ab7548, https://ui.adsabs.harvard.edu/abs/2020PASP..132d4503Z 132, 044503








 Paper Title: Holographic Beam Measurements of the Canadian Hydrogen Intensity Mapping Experiment (CHIME).




























































CHIME Collaboration
100The CHIME Collaboration:

0000-0001-6523-9029]Mandana Amiri


0000-0002-7758-9859]Arnab Chakraborty



0000-0002-0190-2271]Simon Foreman


0000-0002-1760-0868]Mark Halpern


0000-0001-7301-5666]Alex S Hill



0000-0002-4241-8320]Gary Hinshaw





0000-0001-8064-6116]Joshua MacEachern


0000-0002-4279-6946]Kiyoshi W. Masui



0000-0002-0772-9326]Juan Mena-Parra



0000-0001-8292-0051]Nikola  Milutinovic


0000-0002-7333-5552]Laura Newburgh


0000-0002-2465-8937]Anna Ordog



0000-0003-2155-9578 ]Ue-Li Pen






0000-0002-9516-3245]Tristan Pinsonneault-Marotte


0000-0001-6967-7253]Alex Reda


0000-0003-2631-6217]Seth R. Siegel




0000-0001-7755-902X]Saurabh Singh


0000-0002-1491-3738]Haochen Wang



0000-0001-7314-9496]Dallas Wulf



T. Pinsonneault-Marotte, A. Reda
tristpm@stanford.edu, alex.reda@yale.edu


§ ABSTRACT 
We present the first results of the holographic beam mapping program for the Canadian Hydrogen Intensity Mapping Experiment (CHIME). We describe the implementation of the holographic technique as adapted for CHIME, and introduce the processing pipeline which prepares the raw holographic timestreams for analysis of beam features. We use data from six bright sources across the full 400-800 MHz observing band of CHIME to provide measurements of the co-polar and cross-polar beam response of CHIME in both amplitude and phase for the 1024 dual-polarized feeds instrumented on CHIME. In addition, we present comparisons with independent probes of the CHIME beam which indicate the presence of polarized beam leakage in CHIME. Holographic measurements of the CHIME beam have already been applied in science with CHIME, e.g. in estimating detection significance of far sidelobe FRBs, and in validating the beam models used for CHIME's first detections of emission (in cross-correlation with measurements of large-scale structure from galaxy surveys and the Lyman-α forest). Measurements presented in this paper, and future holographic results, will provide a unique data set to characterize the CHIME beam and improve the experiment's prospects for a detection of BAO.





§ INTRODUCTION



The Canadian Hydrogen Intensity Mapping Experiment (CHIME) is a drift scan radio interferometer operating between 400-800 MHz in 1024 radio frequency bins. It consists of four parabolic cylindrical reflectors, each 20 m wide and 100 m long, oriented such that the long axis is aligned North-South. The cylinders are essentially reflective along their length and parabolic (focusing, with f/D = 0.25) along their width; the resulting primary beam has an East-West profile formed by diffraction (width ≳ 1^∘, from 800 MHz) and a North-South profile which is essentially the broad (∼ 120^∘ wide across the band) illumination pattern of the CHIME dipoles reflected to the sky. As a result, the telescope's instantaneous field-of-view (FoV) is a narrow strip on meridian extending nearly from horizon-to-horizon; over the course of a day, the rotation of the Earth passes the entire sky through this strip, allowing CHIME to map the full sky visible from its latitude (49.3215 deg north) each day. Each cylinder is populated with 256 dual-polarization feeds along the central 80 m of the focal line. It is located at the radio-quiet Dominion Radio Astrophysical Observatory (DRAO) site near Penticton, British Columbia, Canada. Its primary science goals are a measurement of large-scale cosmological structure using the redshifted 21 cm line of neutral hydrogen <cit.> and as a radio transient detector for Fast Radio Bursts <cit.> and pulsars <cit.>.  

The CHIME experiment design is described in more detail in <cit.>, and its analog and digital signal chain closely follows the design prototyped in the CHIME pathfinder experiment <cit.>. Amplified signals from all 2048 analog inputs of the CHIME radio feeds <cit.> are sent along 55 m of coaxial cable, where they are filtered and further amplified prior to being digitized, channelized, and sent by corner-turn network <cit.> to the cosmology backend to be correlated and integrated in a GPU-based correlator <cit.>, or sent to the transient backend for transient searching.

CHIME will map the redshifted 21 cm emission of neutral hydrogen between redshifts z = 0.8-2.5 across the northern sky to probe the evolution of the baryon acoustic oscillation (BAO) scale imprinted on the large scale structure of matter (of which the neutral hydrogen is a biased tracer). Cosmological measurements with CHIME are complicated by the presence of bright astrophysical foregrounds (primarily synchrotron emission from our own Galaxy and extragalactic point sources) which dwarf the faint signal by up to several orders of magnitude. Discriminating between the foregrounds and signal should be possible by leveraging their spectral differences <cit.>. However, any spectral structure intrinsic to the instrument (e.g. antenna gains, the beam response) will be imprinted on the foregrounds as well, so separating out the foregrounds requires a high-fidelity characterization of the instrument. Previous work which simulated a measurement of the power spectrum, corrupted by per-feed beam-width perturbations, suggests that the primary beam per feed as a function of angle and frequency must be known to much better than 1% <cit.>. To-date, measurements of with CHIME have relied on an aggressive foreground filtering scheme, filtering out low-delay (spectrally smooth) modes of the data dominated by foregrounds. With this scheme, CHIME has achieved detections of in cross-correlation with eBOSS ELG/LRG/QSO samples at redshifts 0.78-1.43 <cit.>, and with the Lyman-α forest at redshifts 1.8-2.5 <cit.>. However, as this filtering discards large scale modes along the line-of-sight, these detections were limited to smaller scales insensitive to BAO.

Recovering sensitivity to the large scale modes relevant to the study of BAO will require an improved characterization of the instrument systematics including the beam response. Achieving the aforementioned level of sub-1% calibration fidelity is challenging for the CHIME instrument for a variety of reasons: the primary and synthesized beams are broad and as a result confusion noise is high; the telescope is stationary and so scanning sources to build up a beam map is not possible; and its far-field is inaccessible to drones or other artificial sources that might be able to transmit a signal for beam mapping purposes. In addition, the CHIME reflectors have proven to be complicated optical systems; ab-initio analytical beam models and electromagnetic simulations must take into account both multi-path internal reflections and direct feed-to-feed coupling to achieve a realistic reconstruction of CHIME beam features, and these effects do not admit straightforward parametrizations. 
As a result, a novel program of CHIME beam measurements has evolved, as described in <cit.>: (i) solar measurements <cit.>, which can measure the common mode primary beam (i.e., the beam profile effectively averaged over the per-feed variations in the beam) with good signal to noise over the sun's semi-annual declination range -23.5^∘ < δ < +23.5^∘ DEC, and α∼± 20 degrees in Hour Angle (HA); (ii) source catalog measurements, which can be used to measure the North-South beam shape from transit data; and (iii) the holographic measurements described in this paper. Of these measurements, holography is unique in that it can measure the beam shape along East-West transit trajectories to high signal-to-noise, anywhere there is a bright enough source to target. 



In this paper we describe our implementation of the holographic technique for CHIME and present results from 529 observations of six sources, representing only a subset of the holographic measurements taken to date (1800 observations of 24 sources).  With this amount of data, these results provide measurements of the CHIME beam with typical radiometric noise at or below the few percent-level relative to the peak main beam amplitude. We identify and present a variety of features in the beams, including the per-feed pointing, beam-widths, feed non-redundancy, overall spectral structure of the sidelobes, etc. 
Section <ref> presents an overview of the holographic technique as applied to CHIME and a steerable telescope (the John A. Galt 26 m telescope at DRAO), including the formalism describing the measurement, a brief overview of the relevant configuration of the Galt telescope and CHIME correlator, and a procedure to estimate our noise floor due to confusion. Section <ref> describes the holographic observations, Section <ref> describes the data processing pipeline, Section <ref> describes the resulting co-added measurements of the CHIME beams, and Section <ref> describes consistency checks performed on the results. We conclude in Section <ref>.




§ DESCRIPTION OF THE CHIME HOLOGRAPHIC TECHNIQUE

 
Holographic measurements of a radio dish have a long history in radio astronomy, primarily as a means of measuring the surface figure of a telescope dish to search for deviations from specifications. For a theoretical overview of the technique, see e.g. <cit.>; for some examples of applications of the technique to more traditional radio telescopes, see e.g. <cit.> and the historical account from <cit.>.  Using holography to map the far-field beam of a cylindrical telescope was first demonstrated on the CHIME pathfinder array <cit.>.

Holographic measurements consist of two radio dishes: a `dish under test' whose beam is being characterized and a reference dish. For the holographic measurements of CHIME, the CHIME inputs are under test, and the co-located equatorially-mounted steerable 26 m John A. Galt telescope is the reference dish (see  for details on the Galt telescope, as instrumented for holography with CHIME). The Galt telescope tracks a source as it transits through the CHIME beam, and the signals from CHIME and the Galt telescope are correlated in the CHIME correlator. This correlation measures only what is common between the Galt beam and the larger CHIME primary beam, reducing the confusion noise on a given point source. The resulting interferometric data set provides a measurement of the amplitude and phase of the CHIME far-field beam pattern in both co-polarization and cross-polarization, for each of the 1024 dual-polarized CHIME feeds at all 1024 frequencies. This far-field pattern can be transformed to provide a map of the aperture (as would be done for most holography applications). Because the confusion noise is reduced, this technique provides high signal-to-noise measurements of the CHIME primary beam along the East-West trajectory at the declination of the transiting source. This provides an assessment of sidelobe power which is broadly useful for CHIME science; for example, this sidelobe information was used in <cit.> to obtain signal-to-noise ratios for detections of far-sidelobe FRBs.  After applying an overall calibration per observation (see Section <ref>), the holographic system is sufficiently stable that multiple observations of a single source can be co-added to improve signal-to-noise. Even with co-adding, only about ∼20 sources are bright enough to warrant targeting via the holographic technique, thus it cannot be used to fill the North-South beam. In spite of its sparse declination coverage, holography has excellent synergy with independent probes of the CHIME beam, as the polarization, phase, and especially per-feed variation of the beam are hard to measure without holography. 


 §.§ General formalism


Holography is an interferometric measurement of the complex (i.e. amplitude and phase) far-field response of a radio telescope, obtained by correlating the signals measured by the antenna under-test and the reference dish while observing a common calibrator. In general, the polarised response of the i-th CHIME feed (under-test), in polarization a as a function of sidereal time ϕ and frequency ν, to the sky can be written as follows:


    F^a_i(ϕ, ν) = ∫ d^2𝐧̂ J^ac_i(𝐧̂; ν)ϵ_c(𝐧̂; ϕ, ν)e^2π i 𝐧̂·𝐮_i


where J^ac_i is the (direction-dependent) Jones matrix of the CHIME feed, including the beam response, ϵ_c is the vector electromagnetic field density defined on the celestial sphere and incident on the feed,  𝐧̂ denotes a position on the sky in the coordinate frame of the telescope, and 𝐮_i is the position vector (in units of wavelength) of the feed measured from an arbitrary origin point (when we take the cross-correlation with the Galt response, the origin point will be irrelevant). One can write an analogous expression for the response of the (reference) Galt telescope, 

    F^b_26(ν) = ∫ d^2𝐧̂ J^bd_26(𝐧̂ - 𝐧̂_s(ϕ); ν)ϵ_d(𝐧̂; ν)e^2π i 𝐧̂·𝐮_26

where J^bd_26 is the Jones matrix of the Galt telescope and 𝐧̂_s(ϕ) denotes the direction of source s at time ϕ, which the Galt telescope tracks. 𝐮_26 is the position vector (in units of wavelength) of the Galt telescope from the same (arbitrary) origin point.


 For CHIME, we use a linear polarization basis a, b ∈{X, Y}, with X polarization orthogonal to the CHIME focal axis and Y parallel to it. For the Galt telescope, the polarization basis indexed by b is not fixed relative to CHIME due to the rotation of the dish over the course of an observation, however the feed is aligned such that its polarization axes align with that of CHIME when the telescope is pointed to the meridian. The indices c, d belong to an on-sky polarization basis, and the Jones matrices J_i, 26 then define a mapping between on-sky polarization and the polarization measured by the instruments. 

In CHIME holography, we form the cross-correlation of the i-th CHIME feed and the Galt feed. In general this cross-correlation takes the form:

    V^ab_i, 26(ϕ, ν)    ≡<F^a_iF^† b_26> 
       = ∫ d^2𝐧̂∫ d^2𝐧̂'J_i^ac<ϵ_cϵ^*_d>J_26^*dbe^2π i (𝐧̂'·𝐮_i- 𝐧̂·𝐮_26)

where the product <ϵ_cϵ^*_d> is the coherency matrix of the sky signal. 

In practice, the experimental setup allows us to simplify this expression considerably. First, we assume that the electric fields from the astronomical source are spatially incoherent; i.e., the coherency term will introduce a factor δ(𝐧̂ - 𝐧̂') which eliminates one of the integrals. Second, we note that we perform holographic measurements of point-source calibrators transiting over CHIME in tracks of constant declination. To CHIME, the source has a time-dependent position 𝐧̂_s(ϕ), while the Galt telescope tracks the source throughout the observation. This is expressed as a δ-function spatial dependence in the sky coherency term, i.e. we will write <ϵ_cϵ^*_d> ≡ T^s_cdδ(𝐧̂ - 𝐧̂_s), where T^s_cd are the elements of a matrix describing the polarized emission of the point source.  The δ-function collapses the remaining integral to the direction of the calibrator, leaving us with 

    V^ab_i, 26(ϕ, ν) = 
     J^ac_i(𝐧̂_s(ϕ);  ν)T^s_cd(ν)J^*db_26(ν) 
    ×
     e^2π i𝐧̂_s(ϕ)·𝐮_i, 26

where we have written 𝐮_i, 26 = 𝐮_i - 𝐮_26. We have dropped the direction-dependence of the 26 m's Jones matrix because when we collapse the integrals to the location of the source, only the constant boresight response of the 26 m remains. 
Combining the phase factor with the sky brightness factor T^s_cd, we can write the holographic response for all polarization products compactly in matrix form as: 

    𝐕_i, 26(ϕ, ν) = 𝐉_i(𝐧̂_s(ϕ); ν)𝐓_s(ν)𝐉^†_26(ν)  .

Given the Jones matrix of the Galt telescope and a suitable, well-known calibrator (i.e. a well-constrained 𝐓_s), one could invert this expression to solve for the Jones matrix of CHIME feed i. However, as we  have ignored instrumental noise, this represents only a schematic illustration of the technique rather than a formal estimator for CHIME's Jones matrix. In practice, there are additional confounding factors beyond instrumental noise: among the most pressing is that the Jones matrix of the Galt telescope is difficult to calibrate independently, as its equatorial mount precludes typical polarimetric calibration through measurements of polarised sources over a range of parallactic angles. In the past (for a different receiver system and observing band), electromagnetic simulations have been used to compute the response of the Galt telescope to polarized emission (see ). For our purposes, however, a holography-based full-Jones matrix calibration of CHIME, which would include the effects of parallactic angle rotation and polarisation leakage / differential gain in the signal chain, is beyond the scope of this paper. Instead, we directly measure and present the products 𝐕_i, 26(ϕ, ν); we discuss our ability to reconstruct independent measurements of the CHIME beam from these products in Section <ref>. 




 §.§ John A. Galt Telescope and Receiver


The John A. Galt telescope <cit.> is a 25.6 m f/D = 0.3 (focal length 7.6 m) equatorial-mount telescope located at the DRAO. It is (254, 22, 19) m east, north, and above the center of the CHIME array. A custom 400-800 MHz receiver for the Galt telescope was installed at the focus for holographic measurements with CHIME. The receiver system is described in <cit.>, below we highlight some salient features. 

The first element of the receiver chain consists of a dual polarization CHIME cloverleaf antenna <cit.> attached to a sleeve-and-choke structure to circularize the beam. The resulting beamwidths range from 1.2^∘-2^∘ across the band for both polarizations. One polarization is aligned north-south and the other aligned east-west when the equatorial mount is at Hour Angle = 0 (transit) such that the Galt telescope's polarization frame should align with that of CHIME at transit. Because the equatorial mount rotates with the sky, the polarization frame of the Galt telescope will otherwise rotate with parallactic angle relative to CHIME's fixed polarization. 

The feed is designed with an additional port for calibration signal injection from a local noise source. The signal is split and then attenuated with a 15 dB Pi-type attenuator that is matched in parallel to 50 Ω to keep the return loss at the signal injection port <-30 dB. The attenuator is followed by a 2 mm-wide transmission line which terminates, leaving a 0.7 mm gap between the transmission line and the microstrip carrying the antenna signals. This gap capacitively couples the injected noise signal to the antenna polarization ports at ∼-40 dB. This low coupling ensures the noise temperature from the attenuator at room temperature is small (0.03 K) when the noise source is off. 

The calibration source is a 51 dB ENR NC512/12 Noisecom noise module[<https://noisecom.com/Portals/0/Datasheets/NC500REV3_datasheet_WEB.pdf>], powered by a 12 V precision power supply and thermally regulated to be 50 C with a variation of 0.1 C. A gating signal from the CHIME FPGA is sent along a dedicated coaxial cable to the Galt telescope focus and is used to switch the noise source on and off at a cadence appropriate for gating in the CHIME correlator. The gating scheme switches the noise source on for one 5s integration per minute (i.e., the source is on for one out of every twelve integrations). Measurements of Cygnus A transits by the Galt telescope indicate the signal from the calibration source is ≈ 20 K for both polarizations. This noise source was functional during some of the observations presented in this paper, however the calibration estimation remains a work in progress. As a result, the calibration has not been applied to the data presented here, and all data used comes from the 'gated off' position. 




 §.§ Holography Acquisition by the CHIME Correlator System
 


The CHIME correlator has 2048 inputs, distributed over 128 ICE FPGA boards
<cit.> which digitize and channelize the input signals. These are
alias-sampled at 800 and Fourier-transformed into 1024
frequency channels streaming at a 2.56μs cadence. The digital
streams are transmitted over a high-speed network to an array of 256 GPU nodes where their spatial correlation (N^2) is computed and
accumulated in time to a cadence of ∼30. Following the GPU kernel, an
estimate of the radiometric noise is calculated by differencing even and odd
frames at the 30 cadence and recorded alongside the visibilities as a weighting dataset. The data is further accumulated
to 5 or 10 second frames (see below) and transferred to a dedicated receiver node where
additional processing, including calibration and compression, is performed
before saving to disk. 

Signals from the Galt telescope are input to the CHIME correlator and undergo
the same channelization and spatial correlation process alongside every CHIME
input. Following the N^2 step, the products between the Galt and CHIME inputs
are split into a separate stream and saved at 5s cadence, faster than the 10s CHIME cadence, to avoid smearing the
faster fringe rate of the longer CHIME-Galt baseline. No additional processing
or calibration is performed on the Galt products before they are saved to disk. 

Although we only report measurements of standard radio sources in this paper, the CHIME correlator is capable of gating on the 30 frames,
making it possible to perform holographic measurements of pulsars with periods ≳300 (a period larger than the observing cadence is required to accommodate the case where the number of observed pulses varies from frame-to-frame). Pulsars are attractive targets for holography because we can remove background emission by differencing frames based on the on/off chopped pulsar signal.

Pulsar timing is computed in
real-time to determine the on- and off-gates using timing model parameters
generated for every observation by the  package <cit.>. The
gated stream can be switched on or off and the gating parameters altered at any
time without interfering with the rest of the acquisition system. 





 §.§ Confusion noise
 



One of the limitations of CHIME-only techniques to measure its primary beam is the confusion noise associated with the instrument's large instantaneous field of view. Direct observations of calibrator sources quickly become confusion limited off-meridian. The Sun is bright enough to mitigate this problem, but these measurements are only available in the southern sky due to the limited declination range of the Sun. In holography, the effect of confusion is significantly reduced because the correlation between CHIME and the Galt telescope removes all confusing sources which are outside the Galt telescope's much smaller (compared to CHIME) beam. Here we outline a procedure to estimate the remaining confusion noise limit of the holography data, which sets a fundamental signal-to-noise limit for the holographic technique for non-pulsar sources. 

We proceed from Equation <ref>, where we will drop the polarization indices, considering scalar quantities (i.e. in only Stokes I)  instead of the matrix quantities appearing in the fully polarized treatment (i.e. following the discussion at the end of Section <ref>, we will replace Jones matrices J_i, J_26 with primary beam factors A_i, A_26). Then, a holographic visibility at a single frequency and sidereal time can be written schematically in the following way: 

    V_i, 26 = A_i(𝐧̂_s)A_26T_se^2π i𝐮_i, 26·𝐧̂_s

where A_i is the primary beam response of CHIME feed i, A_26 is the primary beam of the Galt telescope and the other symbols are as defined in Section <ref>.  

To account for the confusing background of unresolved sources within the Galt beam, we model the brightness temperature as a sum over sources (indexed by s' in the following equations) within the Galt beam (“FoV"). The visibility then reads 

    V_i, 26 = ∑_s'∈FoV A_i(𝐧̂_s')A_26(𝐧̂_s')T(𝐧̂_s')e^2π i (𝐮_i - 𝐮_26 )·𝐧̂_s' + src

where + src represents the contribution of a like term from the actual calibrator source; i.e., the proper holographic visibility we are attempting to measure. The 𝐧̂_s' are the sky positions of confusing sources which are near the telescope boresight, i.e., 𝐧̂_s' = 𝐧̂_s + δ𝐧̂_s'. 

Because the CHIME-Galt baseline fringes rapidly, the geometric phases of the confusing sources are expected to decohere so that the average confusion bias over all sources is negligible. However, the variance of the first term will become a noise term, computed as:

    Var[V_i, 26]_conf = |A_i|^2 ∑_s'∑_s”A^*_26(𝐧̂_s')A_26(𝐧̂_s”)T(𝐧̂_s')T(𝐧̂_s”) 
    ×<e^2π i (𝐮_i - 𝐮_26)·(𝐧̂_s' - 𝐧̂_s”)>

where the A_i(𝐧̂) factor from the CHIME beam has been factored out under the assumption that the CHIME beam varies slowly on scales comparable to the beam width of the Galt telescope.
 As the interferometric phase will decohere for different sources, the ensemble average over the confusing phase term will be non-zero only for 𝐧̂_s' = 𝐧̂_s”, collapsing one of the sums, so that the noise variance reduces to

    Var[V_i, 26]_conf = |A_i|^2 ∑_s'∈FoV|A_26(𝐧̂_s')T(𝐧̂_s')|^2

Thus, the confusion noise can be taken as the sum of the squares of the beam-weighted fluxes of the confusing sources within the Galt telescope's field of view.

To estimate this quantity, 1000 realizations of a point source sky were simulated in a HEALpix map (N_side = 256) using the open-source radio sky simulation package cora [<https://github.com/radiocosmology/cora/tree/master>]. The simulated point source sky has three ingredients: a Gaussian distributed background of dim sources below 0.1 Jy, a synthetic population drawn using the source count model of <cit.> between 0.1 Jy and 10 Jy, and a catalog of real sources drawn from NVSS and VLSS above 10 Jy <cit.>. These nominal flux thresholds are defined at 151 MHz and the latter 10 Jy threshold for real sources is scaled to the target frequency of the simulation. For each holography source, Equation <ref> was then evaluated by summing over the squared beam-weighted fluxes of the sources taken within a ring centered on the location of the holography source and of diameter 2 times the full-width at half-max (FWHM) of the Galt beam, using the beam model described in <cit.>. We take the square root to obtain a quantity dimensionally compatible with the source flux, and we ignore the prefactor of the CHIME beam as there will be a compensating factor multiplying the source flux.  

The results of this procedure are summarized in Table <ref> for 600 MHz. For most holography sources, with the exception of 3C84 and 3C10C which are near other real catalog sources, the results are dominated by the random, synthetic background of sources, which are effectively spatially isotropic. For these holographic sources, over the ensemble of 1000 simulations, the confusion noise from Eq <ref> has a median value under 1 Jy. As both the source flux and the confusion noise level are modulated by the CHIME beam in an actual measurement, in Table <ref> we present the average confusion noise as a percentage of the source flux at 600 MHz. The reciprocal of this ratio represents the maximum ratio of signal to noise that is achievable with holography; e.g. in our worst case scenario using the very dim source 3C138 (S_600∼10Jy), the confusion noise level is about 10% of the source flux, such that the peak achievable SNR we may expect is ∼10. 
 
c c c c
    The 24 sources used for holography, their declinations, and their fluxes at 600 MHz, along with estimates for the holographic confusion noise (also at 600 MHz), expressed as a percentage of the source flux. These estimates are defined as in Eq <ref>.  
    
    4
    
        Source     
        DEC [deg]    
        S_600 [Jy]    
        Conf. Noise [%]
    
    Cygnus A     40.7     3613     0.02 
 
    Cassiopeia A     58.8     2375     0.03 

    Taurus A     22.0     1142     0.06 

    Virgo A     12.4     436     .2 

    Hercules A     5.0     120     .6 

    3C353     -1.0     106     .7 

    Hydra A     -12.1     97     .7 

    Perseus B     29.7     87     .8 
 
    3C10C     64.2     71     7 

    3C273     2.1     55     1 

    3C84     41.5     38     10 

    3C295     52.2     37     2 

    3C58     64.8     31     2 

    3C161     -5.9     30     2 

    3C147     49.9     29     2 

    3C111     38.0     29     2 

    3C196     48.2     28     2 

    3C409     23.6     27     3 

    3C358     -21.5     26     3 

    3C48     33.2     25     3 

    3C286     30.5     18     4 

    3C454.3     16.1     12     6 

    3C78     4.1     11     6 

    3C138     16.6     10     7 

    




§ OBSERVATIONS



In a holographic observation, the Galt telescope tracks a particular source transiting overhead, typically within ±30^∘ of meridian. During this time, CHIME continues collecting data normally; the signals from the Galt telescope are treated as another input by the CHIME correlator, so that holography observations do not interfere with normal operation of the telescope.  

The Galt telescope is programmed to track sources ahead of time. Pointing during observations is corrected in real-time using a model of the Galt telescope derived in 1998 based on 1.42 observations for the Canadian Galactic Plane Survey <cit.>. A pointing model specifically for the holographic receiver has not yet been developed and is left to future work, however measurements presented in <cit.> found pointing offsets of ∼3', which we do not expect to affect the measurements presented here.

In assembling a selection of sources and a schedule for observations, several factors are taken into account. Brighter sources are preferred; for the faintest sources (fluxes on the order of 10 Jy), in ≈ -20 dB sidelobes and with thermal noise at the level of T_sys≈ 70 K, measuring the beam sidelobes with signal-to-noise of 10 requires in excess of 100 observations. At the same time, observations at a wide variety of source declinations are needed to fill out the 2D beam shape as much as possible. Low declination sources enable a direct comparison with CHIME's solar data-derived beam measurements <cit.>, while high declination sources are uniquely observable with holography. Figure <ref> gives the hitmap of all holography tracks taken to date since October of 2017, illustrating the maximum sky coverage of the current dataset. We use an orthographic projection of the sky onto a frame centered on CHIME with the y-axis pointing to celestial North, x pointing East and z pointing to zenith <cit.>, and refer to these coordinates as “Telescope X/Y". This dataset consists of over 1800 individual observations of 24 sources spanning the sky from -21 to +65 degrees in declination. The wide tracks across a large range of declinations offer a unique probe of CHIME's far sidelobes.

 Figure <ref> shows the time and extent of the holographic measurements presented in this paper, taken between 2018-2020, of six of the brightest sources available from DRAO: Cygnus A (which may be abbreviated in this work as Cyg A), Cassiopeia A (Cas A), Taurus A (Tau A), Virgo A (Vir A), Hercules A (Her A), and Hydra A. In each panel, the vertical lines denote a single observation of the source as a function of observing date. The vertical extent of the line indicates the span in hour angle of that observation. Typically, the Galt telescope will track a source within ± 30^∘ of transit over CHIME, but the exact span of the data will vary considerably from transit to transit.







§ DATA PROCESSING



In this section we describe the data processing pipeline for holography which interpolates the data onto a common grid in hour angle, corrects for known systematic effects, and averages over repeated observations to improve the signal-to-noise of the beam measurement. 
CHIME data are stored in multiple redundant archives at Compute Canada centres.
Data processing and analysis is mainly conducted on the
Cedar[<https://docs.alliancecan.ca/wiki/Cedar/en>] cluster using a
modular Python-based pipeline and analysis tools developed by members of the
CHIME collaboration[<https://github.com/radiocosmology>, <https://github.com/chime-experiment>]. 




 §.§ Fringestopping, regridding, decorrelation correction
 


To facilitate the comparison of beam measurements from multiple observations and
sources, the time-ordered holography data are resampled onto a common grid in
hour angle (HA) relative to the transit time of the source on the telescope's
meridian. This regridding is implemented as an inverse Lanczos interpolation, identical to the implementation in the CHIME offline pipeline used in . The chosen grid spans ±30 of hour angle in 720 samples, which is broad
enough to cover most holography observations and sets the resolution of our beam measurement to approximately 0.1^∘ in hour angle, taking the data from a 5 cadence to approximately
20. For
comparison, the minimum full width at half maximum of the CHIME primary (voltage-)beam is 2^∘ at 800 MHz, taking at least 480 for a source to transit, and therefore the
20 sampling is sufficient.

Prior to downsampling in HA, a phase correction is applied to remove the fringing
in the data produced by the changing delay between the CHIME and Galt feeds
as the source moves overhead. Without this correction, averaging adjacent time
frames when downsampling would lead to signal loss. The phase correction is 

    2 π𝐧̂·𝐮_i, 26,

the projection of the direction to the source 𝐧̂ onto the baseline
between the Galt and the i-th CHIME feed in units of wavelength λ.  For
a source at 0 declination on the meridian, fringestopping corresponds to removing
>4.5 of phase over a period of 20 at 600 MHz. 


The amplitude, but not phase, of the raw holographic measurement is affected by an hour-angle dependent multiplicative bias principally generated by cable delays. CHIME contains 55 m of coaxial cable, while the Galt telescope signal traverses 405 m of coaxial cable en-route to the correlator. The result is that the signal from the Galt telescope is delayed relative to CHIME's signal by at least 1.49 μs, which is a significant fraction of the 2.56 μs integration frame. This leads to a significant decorrelation as the signals from CHIME and the Galt telesope no longer overlap within the expected geometric delay. The amplitude of the decorrelation depends on the exact delay between the two signals, which changes over the course of an observation as the source transits, so that the result is not a constant overall suppression of the signal but instead a modulation of the amplitude of the holography measurement as a function of hour angle, distorting the inferred beam shape. 
Fundamentally, the decorrelation is a result of the finite window of time used when channelizing the data in the CHIME correlator's polyphase filter bank (PFB). By simulating the response of the PFB, we can determine an estimate for the amplitude of the decorrelation effect for arbitrary delays and correct the holography amplitude accordingly (see Appendix <ref>). Broadly, we find that for the range of hour angles used in the re-gridding step, ±30^∘, and for the sources considered in this work, this effect modulates the beam amplitude by as much as ±∼30% and the magnitude of the decorrelation increases approximately linearly through the transit (we use the full result, not an approximation, when correcting the data). Referencing to the first feed on each cylinder, there is also a linear dependence on the amplitude with feed, with the sign of the slope dependent on whether the source is north or south of CHIME's zenith (in this paper, the only source north of zenith is Cas A). See Figure <ref> in the Appendix.  

As the fringestopping and decorrelation correction steps require knowledge of the geometric delay between a CHIME input and the Galt telescope we must have accurate knowledge of their (relative) positions. Any error in the baseline distances 𝐮_i, 26 will introduce a phase error in the beam
measurement by biasing our estimate of the geometric delay associated with a source location, but this can be identified and corrected in later stages of
processing if necessary. For instance, an earlier version of the pipeline omitted the vertical component (w-term) of the holographic baselines when fringestopping. <cit.> details an analysis of that version of the data to fit for the w-term, including the contribution of an additional declination-dependent term originating from the geometry of the Galt telescope mount. This model tracks the remaining delay term to within .1% for 7 sources spanning >60^∘ of declination. We use the best-fit vertical displacement from this analysis for fringestopping in the present version of the pipeline, and include the Galt-specific term as an additional phase correction for the holography prior to averaging over repeated transits (see Section <ref>). We also determine a best-fit value for the cable delays and use these when applying the decorrelation correction.

In <cit.>, we reported evidence of an error in the North-South positions of the CHIME antennas, based on the observation of a phase that linearly scales with East-West position when beamforming to bright point sources on meridian. At the time, we remarked that this could be caused by either a rotation of the telescope from astronomic North, or by a small, linearly staggered North-South offset of the cylinders from one to the next (i.e., an observer looking down at the telescope from zenith would see a parallelogram with the top and bottom sides sloped, rather than a rectangle). In the interim, in all of CHIME's analyses which required knowledge of the baseline distances,  we assumed the rotation scenario; however, a recent professional survey of the telescope confirms the parallelogram scenario. As a result, the baseline distances used here contain an erroneous East-West component which linearly scales with North-South position along the cylinders, which was intended to correct for the rotation scenario. From end-to-end of a cylinder, this makes a difference of about 10 cm. This error sources a phase gradient in the fringestopped data proportional to sinHA; for Cygnus A at declination of ≈41^∘, at the extent of our grid (±30^∘)  this is as much as 23^∘ of phase error at 600 MHz. The delay error sourced by a 10 cm offset, however, produces no appreciable change in the strength of the decorrelation effect. This will be corrected in future iterations of the pipeline; for now we retain it as it makes no difference to the amplitude and the same phase error is present in the independent CHIME datasets we compare to in Section <ref>. Also note that the previously described phase analysis in <cit.> is unaffected by this issue because it used data taken on meridian, where the fringe phase due to the East-West component of the baseline is identically zero. 



 §.§ Data flagging
 



  §.§.§ Manual data cuts

Prior to any further processing, we visually inspect all observations used in this paper for overall quality. We completely omit any observations with significant outstanding features from the stacks presented here. Among the features we found which disqualified observations are: 

    
  * Significant RFI.
    
  * Significant disparities between, or anomalous features in, one or more cylinders.
    
  * Large chunks of missing data.
    
  * Presence of the Sun at any point in the transit. This is more typical among earlier observations, and is specifically an issue for the southern sources (Tau A, Virgo A, Hercules A, Hydra A). This is a conservative cut as in principle the Sun can be masked when it overlaps the source's track. However, as the overall quality of the data near the Sun is uncertain, we simply omit these data.


In total, these by-hand cuts remove 112 of the 529 transits (about 21% of the data) used for this work. Over half of the discarded transits are of Hercules A and Hydra A, primarily due to solar contamination. 


  §.§.§ RFI flagging

Although CHIME is located in a radio-quiet zone at DRAO, there is still contamination observed in the data over significant portions of the band due to radio interference from a variety of artificial sources, including satellites, airplanes,  wireless communications, and TV bands <cit.>. The Galt telescope sees the same RFI environment, and so holography data is also compromised by this interference. To account for this, prior to fitting and stacking over observations, we apply an RFI mask to the holography data. When available, we use the RFI masks produced by the CHIME offline pipeline, as described in <cit.>. 

The CHIME masks are produced for the entire sidereal day; to mask the holography, we select the daily data flags in a window around the transit of the holography source, corresponding to the time span of the observation. Individual time samples in the holography data are then flagged based on a nearest-neighbors interpolation along the time axis of the CHIME masks; i.e. the holography is flagged based on the nearest time sample in the daily mask. We note that some of the holography data presented in this paper was taken at a time before the offline analysis pipeline had been fully developed and so daily RFI masks may not be available for all holography observations. When this is the case, we opt instead to use a static RFI mask which flags known persistent bands of RFI but is not otherwise resolved in time.  

The daily masks are intended to flag anomalous, RFI-contaminated samples in the CHIME data and are not specifically tailored to holography. However, in general we do not expect the RFI in the DRAO environment to decorrelate between CHIME and the Galt telescope. Therefore, to be conservative, we adopt the daily masks to omit data wherever the masks indicate that CHIME was compromised by RFI. 





 §.§ Stacking
 

To integrate down the noise in our observations, we co-add all available holographic observations of a particular source together to form a lowest-noise estimate of the measured response along the source track, at all frequencies, all feeds, and in all polarization products. We compute the stacks as a weighted average over observations, where the weights are taken as the inverse of the fast-cadence estimate of the noise variance computed by the real-time pipeline as described in Section <ref> (i.e., we perform an inverse-variance weighted average).



  §.§.§ Transit normalization


Prior to averaging, we must account for variations in the instrument gains between transits. To do this, for every transit, we fit a simple model for the fully complex holographic response in the main lobe, for all frequencies and feeds, in both co-polarization (YY, XX) products only; we do not attempt an independent fit of the cross-polarized (XY, YX) response. The amplitude of the response is modeled as a Gaussian parameterized by amplitude A, centroid μ and width σ. The phase is modeled as a 5th degree polynomial in hour angle ϕ, ∑_i=0^5c_iϕ^i. Prior to fitting, we apply the phase corrections discussed in Section <ref> and RFI masks as discussed in Section <ref>. When averaging over observations, we then normalize each transit by its measured complex gain, which we take as the best-fit model evaluated on transit (i.e., at HA = 0); in terms of the best-fit parameters, this is given by Ae^ic_0. This normalization is applied in both co- and cross-polarization, so that the amplitude and phase of the cross-polarized response are now referenced to the on-meridian co-polarized response. All best-fit parameters are saved to disk for offline analysis; see Section <ref>.

As this procedure independently normalizes the data for all feeds, frequencies, polarizations, and sources (declinations) to 1 + 0j at transit, we are losing information about the modulation of the on-meridian beam response as a function of these dimensions of the data. Although we do not attempt this for the purposes of this work, the dependence of the beam on declination can be re-inserted using independent measurements of (ratios of) the source fluxes, referenced to a specific calibrator source which would provide the overall normalization of the beam. 



  §.§.§ Accounting for noise-source installation


The procedure described above applies the best-fit gains to both the co- and cross-polarized responses; in the cross-polarized response, this normalization leaves behind a factor of the ratio of the gains of the X- and Y- polarized inputs of the Galt telescope. We noticed that the cross-polarized holography normalized in this way exhibits a phase jump for all observations following March 8, 2019. This corresponds to the date on which the noise source described in Section <ref> was installed on the Galt receiver. This implies that the phase of this ratio, while otherwise stable in time, was impacted by the noise-source installation modifying the receiver signal chain. This phase jump will cause the cross-polarized response to decorrelate in a simple average of observations before/after the noise source installation, so we must first align the phases by applying a correction to the cross-polarized data of all of the observations taken before (or after, as the absolute reference of the phase does not matter) March 8, 2019. To do this, we split the data into two sets, one taken before the noise source was installed, and one after. Using the noise weights, we compute the stack over each of these two sets independently, then fit for the on-transit phase of the cross-polarized data in each stack. We apply the difference of the two phases as a correction to the data taken before the noise source was installed; the result is that all observations are aligned to the current instrumental phase, post-noise source installation. This procedure is carried out for each of our 6 sources, with the exception of Hydra A, for which there are no observations, within the range included in this work, that were taken after March 8, 2019.







  §.§.§ Final stacking

Finally, the split-stacks are combined into a single, final noise-weighted stack; the noise-weights are propagated to give an estimate of the radiometer noise in the final stack. The stack is saved with 
a record of the number of observations included in each frequency/feed/polarization/hour-angle voxel (this number is not trivially equal to the number of input transits owing both to gaps in the data from masking and the varying lengths of each observation), as well as the empirical variance of the ensemble. 







§ BEAM MEASUREMENTS



In this section we present and summarize some of the noteworthy features of the fully processed holography dataset. Each observation presented here is an estimate, as a function of frequency, hour angle, and polarization product, of the per-feed beam response, normalized as described in Section <ref>; moving forward we denote these quantities as V_i, 26^ab, with i indexing the CHIME feeds, a, b ∈{X, Y}, and where we choose the letter V as a reminder that we are measuring a holographic visibility as a proxy for the underlying beam shape. We note that as an interferometer, CHIME's fundamental data products are the correlations between feeds; i.e. the scale of the power-beam which modulates the sky as seen by a particular baseline is set approximately (due to feed non-redundancies) by the square of the profiles we show here. When referring to specific polarization products, i.e. XX, YY, XY, YX, we use the convention that the first letter indicates the polarization of the CHIME feed, and the second letter indicates that of the Galt telescope.

Some one-dimensional profiles, median-averaged over cylinders, are shown in Figure <ref>, featuring, for Cygnus A, the co- (XX) and cross-polarized (XY) amplitudes and phases. The profiles are consistent between cylinders, with one of the most notable discrepancies a systematic overall shift of the profile of Cylinder B in the left-middle panel. This suggests that the focal line of Cylinder B is slightly mis-aligned; we return to this point below when discussing the dependence of the beam centroids with feed. The co-polarized response in the main beam is approximately Gaussian, with a first sidelobe around 20% (4% in power), and far-sidelobes around ≲3-4% (<.1% in power) referenced to the peak. In the main beam, the amplitude of the cross-polarized response is ≲5% (∼0.2%) compared to the co-polarized response (although varies widely with frequency as will be discussed later), and is characterized by a decrement approximately on meridian where the polarization frame of the 26 m is nominally aligned with that of CHIME. The cross-polarized response in the far-sidelobe region, outside of the blue dashed region in the top row of Figure <ref>, is ∼2% (<.1% in power), and its shape is qualitatively similar to the co-polarized beam response. The phases of the co- and cross-polarized response vary smoothly within the main lobe and are again consistent between cylinders, except for points corresponding to zero-crossings in the complex beam pattern where otherwise small noise fluctuations may cause the phase to diverge on a different branch (as the ratio of real to imaginary parts is highly sensitive to perturbations near a zero-crossing).




Figure <ref> shows two-dimensional slices of the amplitude of the stacked observation of Cyg A, in frequency vs. sky degrees, for all four cylinders and all polarization products. Here, we take a median over all feeds of the selected polarization on the respective cylinders. Figures <ref>, <ref>, <ref>, <ref>, and <ref> in the Appendix show the same visualization for the other sources. Superimposed on the expected ν^-1 scaling of the main-beam FWHM is an additional widening of the FWHM every 30 MHz in frequency, generating a characteristic“vertebrae" pattern in the main-lobe. This 30 MHz ripple is associated with a standing wave in the cavity between the vertex of the cylinder and the focal line, which are separated by ∼ 5 meters as discussed in <cit.>. The X-polarized response is wider in the East-West direction than the Y response, due to differences in the illumination pattern of the X and Y dipoles of the CHIME cloverleaf feeds; see Figure 9 of <cit.>. The side-lobe pattern also differs between the two polarizations; in YY the sidelobes take on a “checkerboard" pattern while in XX the sidelobes appear relatively smoothed out. The cross-polarized signal is larger in XY than YX by about a factor of 2 in the main-lobe. In all cases, the average beam profile follows the same overall behavior on all of the cylinders.



The 30 MHz modulation and the polarization dependence in the widths is also apparent in Figure <ref>, which shows the beam FWHMs – for all six sources presented in this work – recovered from the fits prior to stacking as a function of frequency. The beam-widths are consistent between all declinations measured here, although the 30 MHz ripple patterns are offset. 











Figure <ref> shows two-dimensional slices of the amplitude of the stacked observation of Cyg A, in cylinder feed-number vs sky degrees, again for all four cylinders and all polarization products. Figures <ref>, <ref>, <ref>, <ref>, and <ref> in the Appendix show the equivalent plots for the other sources. Here, the East-West beam profile in co-polarization is consistent between feeds on a cylinder. Beyond the main beam, the sidelobes plateau at a level around -10 dB, with the next notable feature an abrupt drop, usually of about 5 dB across the cylinders and polarizations, at ≈±10^∘. The cross-polarization response is shown in the bottom two rows; as before, there is a central null on-transit when the polarization frame of the Galt feed is aligned with that of CHIME, so any signal is the result of genuine polarization either on-sky or inherent to the instrument. We again observe a secondary null at ≈±10^∘. We also find that feeds near the legs exhibit an enhanced response off-axis which appears as near-horizontal bands coinciding with the feed legs. The inset-panels at the top of each row show the median profile over the cylinder.  


In Figure <ref> we show an example of the variation in the beam across all feeds on Cylinder B in YY polarization for Cyg A. Within the main beam, deviations are ∼10% of the peak, driven by the feed-dependent centroid wander. In the sidelobes, essentially all (non-outlier) feeds lie within 5% of the median peak; this corresponds to order unity variation in the sidelobe level from feed to feed.


This centroid wander is one of the dominant modes of variation along the feed axis, as was described in detail in <cit.>. In that paper, using holographic data from only CygA, we noted that the centroid wander was suggestive of ≲1.7 cm physical offsets between the E-W feed position and the symmetry plane of the cylinders. Here we present the centroids from all six sources along each cylinder, for both co-polarization products, taken as a median across frequencies (although the amplitude of the centroid offset depends on frequency and is also subject to a 30 MHz ripple, the pattern across the feeds is correlated across all frequencies). The results are shown in Figure <ref>, where we plot the centroids of both co-polarizations along with measurements of the physical displacements of the feeds taken through photogrammetry of the focal lines. We find that the patterns in the centroids as a function of feed are correlated with the physical displacements (up to an overall misalignment which our photogrammetry is insensitive to), and are consistent between all of the sources, with the exception of those on Cylinder B. It is clear that the overall bias on Cylinder B is declination dependent; the average centroid offset on Cylinder B worsens with increasing declination, changing from an average of -0.05 degrees at δ=-12^∘ to -0.25 degrees at δ=59^∘, which is about 16% of the (voltage) primary beam width in YY at the top of the band (∼1.6^∘, see Figure <ref>); e.g. beamforming the North-South baselines on Cylinder B to meridian at high elevations would suffer about ≲5% (10% in power) signal loss due to primary beam attenuation. In <cit.>, we measured the overall centroid offset of Cylinder B with holography of only Cygnus A; from that offset we inferred an overall misalignment of the focal line of Cylinder B. While the measurements of Cygnus A here are consistent with that work, the declination dependence that becomes apparent with this expanded dataset is inconsistent with the interpretation of the overall offset as originating purely from a focal line misalignment. This is not a systematic inherent to the holography measurement; because we do not see the same trend in the other cylinders, and an overall linear trend with elevation in the centroids of Cylinder B is also observed in the solar measurement of the beam. The exact cause of this feature remains an area of active investigation. 


We also find a dependence of the holography amplitude with feed both along and between cylinders. In Figure <ref> we show the best-fit amplitudes for Cas A, Cyg A, and Tau A (these amplitudes, we reiterate, are removed from the data prior to stacking) for a few transits of each source. We note two features: a strongly linear dependence of the amplitude with feed within each cylinder, and an overall jump in the average amplitude between cylinders. This latter effect can be understood as the result of the holographic baselines resolving out a portion of the extended structure of the source; as we move from longer to shorter baselines (with cylinder A being the farthest cylinder from the Galt telescope and cylinder D the closest) the characteristic fringe pattern on the sky takes on longer wavelengths and thus resolves less of the source, increasing the apparent power. The overall slope within each cylinder, noting that the sign of the slope changes across zenith, is not yet understood. The 26 m is aligned ∼20 m north of the center of the CHIME array while the portion of the focal lines outfitted with feeds extends ±40 m from the center; i.e., the baseline lengths between CHIME and the 26 m do not increase monotonically along a given cylinder. As a result, we would not expect signal loss from resolving the sources to behave linearly along the cylinders. The linear behavior, along with the sign-flip over zenith, is qualitatively similar to the behavior of the decorrelation correction (see Figure <ref>), however, the decorrelation correction has the opposite overall sign. This remains an area of active investigation, but we do not expect this to impact the measurements of the beam shape we show here due to our normalization scheme.


Finally, we also find variation in the first-sidelobe levels. We note that for many feeds these sidelobes appear to be asymmetric. In Figure <ref> this can be seen both at the per-feed level and in the medians over the cylinders. There are multiple potential sources of asymmetry in the holography data, including an error in the applied decorrelation correction and comatic aberration. Coma (see for instance ) arises from a lateral misalignment of the feed of a radio telescope from the focal point of the reflector and can generate asymmetric sidelobes. As first shown in <cit.> and also described above, we infer deviations of the CHIME feeds from the symmetry axis of the cylinders from measurements of the beam centroids. In principle, this displacement can be used to model a phase error in the aperture illumination and predict the expected sidelobe asymmetry; we leave this to future work. 



 §.§ Aperture transforms


In the usual application of the holographic technique, one uses the phase information of the interferometric measurement to uniquely invert the Fourier transform mapping between the beam farfield and the electric fields in the aperture of the telescope.  

Typically one imagines the aperture plane as the finite two-dimensional surface through which all radiation incident on the telescope must pass; the electric fields across this surface are then Fourier transformed giving the two-dimensional (in terms of sky coordinates) beam response in the farfield regime. For example, for a typical parabolic reflector, the aperture plane is a disk of diameter equal to that of the reflector; if this plane is uniformly illuminated then the Fourier transform of the disk yields the familiar Airy diffraction pattern. 

The case of CHIME is more complicated to treat exactly as there is no finite two-dimensional surface to take for the aperture of the reflector, due to the open ends of the cylinders. Instead, for the purposes of the strictly qualitative overview of aperture features we give here, we take the following approach. We treat the CHIME cylinders as diffractive systems only along the parabolic axis; along the focal axes of the cylinders they merely reflect rather than focus incident light. We then treat the aperture-to-farfield mapping as a one-dimensional Fourier transform; i.e., a transform along the East-West (diffraction) axis. If we consider emission from a source at a fixed incidence angle θ measured with respect to the local zenith, then the result of our aperture transform can be interpreted as a measurement of the aperture illumination as a function of the aperture coordinate x (centered at 0 on the focal line and extending out in either direction with the cylinder edges at x = ±10 m), at a fixed position y = 2ftanθ along the cylinder axis.

When analyzing holography data in this context, this is only an approximate mapping; as a source moves overhead, it follows a curved track on the sky so that the incidence angle of its emission on the cylinders is not fixed. Thus for holography the aperture measurement will be affected by variations in the beam over the North-South extent of the track in telescope coordinates, and will also mix power at neighboring positions along the focal line. 

Figure <ref> shows the aperture transform of Cyg A as a function of feed along the cylinder and the aperture x coordinate. We find there is little variation in the x-profile of the aperture illumination as a function of feed. As a function of x, the dominant features are (i) a decrement in the central region corresponding to the ground-plane; and (ii) evidence for non-zero response beyond the edges of the cylinder, which would indicate cross-coupling between feeds of adjacent cylinders. This is most apparent in the Y-pol case; the inner cylinders, B and C, see excess power beyond the physical aperture on either side due to being flanked on both sides by other cylinders, while cylinders A and D have a negligible response on the side opposite of the inner cylinders. Finally, we note that as this is essentially a Fourier transform of the data in hour angle, the erroneous phase gradient in hour angle introduced to correct for the rotated-telescope scenario discussed in Section <ref> implies that from end-to-end of a cylinder, the center of the shape of the aperture transform is shifted from its true position in x-space by ≲10 cm.







§ VALIDATION AND SYSTEMATICS





 §.§ Noise-performance and jack-knives
 





An example of the final output of the processing pipeline described in Section <ref> is shown in Figure <ref>, where the stack of Hercules A, for a single feed and frequency in YY polarization, is shown in blue in the top panel with all individual input transits of Hercules A overplotted in gray. We see the intended effect of the averaging procedure reducing the noise in the beam measurement; outside of the main lobe, the individual transits hit a noise floor at ≈-10 dB (there is a noise bias as we are plotting the positive definite amplitude of the complex data) while beam features in the sidelobes are more clearly discerned in the stack. The middle panel shows all residuals between the stack and the individual transits along with the 2σ contours. 


The bottom panel compares the observed 1σ standard error on the mean (i.e. the standard deviation visualized in the middle panel, scaled by a factor 1 / √(N), with N ∼ 100 in this case) with the fast-cadence estimate of the thermal noise derived from the CHIME real-time pipeline and propagated through the full holography offline pipeline. As we have normalized all transits to be identically 1 + 0j on transit, the units of the estimates may be interpreted as a percentage of the on-transit response; in this example, the noise level is 1-2% of the peak. We see that with the exception of some larger excursions in certain pixels, the observed noise follows the trend set by the expected radiometer noise, which sets a lower bound for the observed variation as long as we are not dominated by confusion or other systematics.

For the four dimmer sources presented here, the real-time radiometric noise estimate is generally consistent with what is presented in the bottom panel of Figure <ref>, i.e. on the order of a percent of the peak or lower. For the two brightest sources, Cygnus A and Cassiopeia A, the radiometric noise is about an order of magnitude lower, at the <.1% level, and structured variations above the thermal noise limit by a factor of ∼ 2-3 are discernible. To characterize these features in the stacks we take two different jack-knives of the data, here using Cygnus A. We first consider a seasonal jack-knife; we assign all observations taken between October and March (in any year) to one set, and all other observations taken between April and September to the other set. We will label these as the “Winter" and “Summer" partitions, respectively; with this scheme, we assign 56 transits to the summer partition and 49 to the winter partition. We compute the stack over both sets independently, then take their difference. The difference is normalized such that its statistical noise level is equivalent to that of the weighted average of the partitions <cit.>. We also consider an alternative but equal (in the size of the sets) partitioning of the data in which transits are assigned randomly to one of two sets. 

Figure <ref> shows a corner plot from the two different jack-knives described above, with the difference between seasonal partitions labeled “Summer - Winter" and the difference between random partitions labeled “R0 - R1."  The lower corner plots show noise realizations of the jack-knives; we take the observed variance within each partition and add these in quadrature to estimate the noise level of the jack-knife σ_jack, then draw a normal random deviate with mean 0 (i.e. expecting the stacks in each partition to be identical) and standard deviation σ_jack. 

It is clear that the jack-knives are not unstructured, zero-mean noise. Along hour angle, the residuals are smallest in the main lobe in both the data-jackknives and noise realizations, indicating that with the current size of the dataset the variability of the beam in this region is comparable to the noise level. However, as we move away from transit to smaller values of the beam response, the fractional residuals dwarf the noise level. Within the frequency-feed subspace, the data-jackknives show vertical striping which is absent from the noise realizations, suggesting systematic variations in frequency which are correlated across the cylinder. Likewise within the hour angle-frequency subspace, there is a clear signature of residual 30 MHz structure in both data jackknives.

These residuals indicate that for the brightest, highest signal-to-noise sources, we have reached the point of becoming sensitive to and limited by systematic variation in the beam between transits. However, from the comparison of seasonal and random jack-knives, it is not clear that there is systematic variation between observations that is specifically correlated with the ambient temperature. This may have been expected, for example, if daily variation were dominated by expansion / contraction of the focal line with temperature, or if the per-transit normalization scheme were insufficient to account for temperature-dependent gain drifts over the course of the observation. We attempted a further split of the data into four seasonal bins (December - February, March - May, June - August, September - November), yielding an additional six jack-knives. We did not find that the difference of the (coldest) December - February and (warmest) June - August seasons was systematically larger than the other jack-knives, and the amplitudes of the jack-knives (using the variance across voxels as a metric) otherwise do not appear to follow a clear trend with the expected ambient temperature differences of the seasons. The jack-knives with the largest amplitudes tend to involve the spring season, which also has the largest observed variance among its constituent transits. Overall though, the differences in the jackknives saturate at the level of 1% of the beam value, even in the sidelobes. This is sub-dominant to the 𝒪(1) variation in the sidelobes observed over feed which we noted in Figure <ref>, and which is implicated in analyses that average over feeds. Some differences could be alleviated by improved cleaning of the data (i.e. masking) on a per-feed and frequency basis for each individual transit. 




 §.§ Comparison to solar data
 


<cit.> presented measurements of the CHIME primary beam obtained using the Sun as a calibrator source as it passes through a declination range ± 23^∘ throughout the year. This allows comparisons between holography sources and solar data where they overlap in the southern sky. In this work we present comparisons of solar data to the holography sources Tau A, Virgo A, Hercules A, and Hydra A. 

As described in <cit.>, estimates of the primary beam using the Sun were computed by beamforming visibilities corresponding to intra-cylinder baselines under 10 m long (to avoid resolving out the solar flux) to the location of the Sun. This technique is essentially a coherent average of the visibilities collapsing over the baseline span of the data, such that the resultant measurement is an estimate of the common-mode beam response of CHIME per cylinder. To form a comparable quantity from the holography dataset, we form the holographic estimate of the average co-polarized power beam of the baselines used in the solar analysis. In equation form, 


    B̅^co, co(ν, ϕ)    = 
       1/N_solar∑_ij ∈solar baselinesV_i, 26^co(ν, ϕ)V_j, 26^*co(ν, ϕ)


where the superscript “co" indicates either the XX or YY product and N_solar is the number of solar baselines.


The comparison between solar data and holography data is shown in Figure <ref>. The left hand panel shows a beam map of the solar data and holography data together, indicating regions where the sources in this paper overlap. In the holography tracks, the amplitude is taken from the holography data, scaled to the solar data amplitude on meridian, and shows general alignment in features between the data sets. This is also clear in the right hand panels, which shows profiles for a single frequency in Y polarization. 


The profiles in Figure <ref> are generally similar for the solar and holographic response, but a few discrepancies are noteworthy. Far off meridian and particularly at lower declinations, the holography prefers a lower response, as low as -50 dB in certain pixels. Around meridian, the solar measurement tends to prefer a systematically wider beam-width at all declinations, a feature which persists even if the finite size of the Sun on the sky is accounted for by smoothing the holography profiles with a boxcar of about half a degree width. As was discussed in Section <ref>, the beam response of the CHIME X feeds has a systematically wider main lobe than the Y feeds; the discrepant beam-widths in Figure <ref> are thus suggestive that the solar measurement is probing X signal that has leaked into Y. In addition, as shown in the top two panels of Figure <ref>, at certain frequencies and in XX polarization only, the main lobe of the solar beam appears significantly distorted from the expected Gaussian shape with an excess in power on either side of meridian. 

These discrepancies between CHIME and co-polarized holography are evidence of the presence of polarization leakage in CHIME; they do not appear in co-polarized holography because contaminating leakage does not correlate between like polarization inputs in CHIME and the 26 m. Schematically, if we write the voltage of an X polarized CHIME input as the sum of X sky signal and signal that has leaked in from Y due to the optics of CHIME:

    F_i^X∝ E^X + E^Y→ X

then when this voltage is correlated with the voltage of another CHIME input, there will be an overall contribution from the correlation of the leakage terms. However, if the 26 m receiver does not leak signal in this way, so that

    F_26^X∝ E^X ,

then the leaked Y signal above will decorrelate when forming a co-polarized visibility with the 26 m's X (Y) input. Instead, we should expect to see the effect of the leakage in the cross-polarized holography, i.e. in V^X, Y_i, 26 = <F_i^XF_26^Y>.

Thus, to reconstruct the solar measurement from the holography, we must form a combination of the co- and cross-polarized holography. Generalizing from Equation <ref>, 

    B̅^pq(ν, ϕ) = 1/N_solar∑_ij ∈solar baselinesV_i, 26^p(ν, ϕ)V_j, 26^*q(ν, ϕ)

where the indices p, q ∈{co, cross}. So, for each polarization, we now have four templates {B̅^co, co, B̅^cross, cross, B̅^co, cross, B̅^cross, co, }. We smooth each of these templates with a boxcar approximating the angular size of the Sun. We then model the solar measurement as some linear combination of these and solve for the coefficients with a linear least squares fit.

As seen in the bottom panel of Figure <ref>, this procedure resolves the distortion in the main lobe at certain frequencies where the polarization leakage is particularly severe. We summarize the results of this comparison procedure for the 3 other southern holographic sources in Figure <ref>, which show medians (over frequency) of the ratio of the solar response and the holographic reconstruction as described above, with the naive 1-template fit (i.e. fitting an overall amplitude) in black, a 2-template fit in cyan (which does not include the products of co- and cross-polarized holography, only co- with co- and cross- with cross-) and the full 4-template fit in blue. The inclusion of cross-polarized holography removes the large excursions on either side of transit for all sources. For Hydra A, which is biased low compared to the solar measurement across the entire East-West range seen in Figure <ref>, the inclusion of cross-polarized information brings the holography into close agreement with the solar measurement on average through the band. For these sources, though, there remain polarization and declination-dependent deviations as large as 45% (Tau A) in the far sidelobes. 

There is little improvement in the reconstruction when including all 4 templates compared to only 2, suggesting that the terms B̅^co, cr, B̅^cr, co are relatively unimportant and do not explain the remaining residuals with the solar measurement. These terms are expected to contribute only if the 26 m has its own polarization leakage. As a result, we conclude that any leakage in the 26 m system is subdominant to the source of the discrepancies observed in Figure <ref>. A possible source of this discrepancy is differential gain of the 26 m receiver; i.e. even if the 26 m Jones matrix is diagonal, the two polarization channels may not have equivalent gain. In this case, the 2-template holographic reconstruction will differ from the CHIME data by an additional term which depends both on parallactic angle and the spatial-dependence of the CHIME Jones matrix. 









 §.§ Point-source subtraction


Section 4.1 of <cit.> describes a stage of the CHIME data processing pipeline that performs a targeted removal of the four brightest point sources (Cygnus A, Cassiopeia A, Taurus A, and Virgo A) from the CHIME data by fitting and subtracting the following “simple model” from the visibilities:


    𝒱_ psrc(u⃗, ν, t) = ∑_s=1^4 a_s(ν, t)  e^i 2 πu⃗·n⃗̂⃗_s(t) .


Here u⃗ is the baseline vector, ν is the frequency, t is the time,  n⃗̂⃗_s(t) the direction of the source on the sky at time t. The source amplitude a_s(ν, t) is meant to encode the spectral flux density of source s modulated by the primary beam pattern of the instrument at the source location.  The amplitudes are estimated for each frequency and time by performing a weighted linear regression over the inter-cylinder baselines.  After performing a 2D smoothing of the best-fit amplitudes in frequency and time, the resulting model is subtracted from the visibilities.

The model given by Equation <ref> assumes that the source amplitudes are constant as a function of baseline, which is true if the angular extent of the source is much less than 1 / u⃗_ max≈ 10, if residual complex gain variations are negligible or common to all feeds, and if the primary beam pattern is the same for all feeds.  However, the holography observations indicate significant feed-to-feed variations in the primary beam pattern, which violate the last assumption.  The feed-to-feed variation was explored in Section <ref>.  In this section, we attempt to account for the feed-to-feed variation in the point source subtraction algorithm by incorporating the holographic measurements into our model for the signal from the four brightest sources.

The new “holography-based model” for the visibilities is given by


    𝒱_ psrc^ hol(u⃗, ν, t) = ∑_s=1^4∑_p∑_q a_s^pq(ν) B_s^pq(u⃗, ν, t) 
    × e^i 2 πu⃗·n⃗̂⃗_s(t) .


Here again p, q ∈{, } and B_s^pq is the now baseline-dependent (note the absence of the overhead bar) set of templates constructed from the holographic observations of source s as follows


    B_s^pq(u⃗, ν, t) = 1/N_b∑_ij ∈u⃗ V^s, p_i, 26(ν, ϕ) V^*s, q_j, 26(ν, ϕ)


where the summation runs over the N_b redundant pairs of feeds separated by baseline vector u⃗.  The 16 coefficients a_s^pq(ν) (corresponding to the 4 sources and 4 polarisation pairs) are estimated for each frequency by performing a weighted linear regression over the inter-cylinder baselines and all time samples spanning a full sidereal day.  Since the holographic measurements describe the time dependence of the source amplitudes, the total number of free parameters that must be solved for is greatly reduced compared to the simple method.



Figure <ref> compares the best-fit, simple model and the best-fit, holography-based model to the measured visibilities as a function of baseline for a single frequency channel and time integration.  The dominant contribution to the visibilities at this time is the signal from Cygnus A, which is at an hour angle of only 1^∘.  The visibilities vary by approximately 10% as a function of baseline.  The simple point source model is unable to describe this variation by definition.  However, the holographic observations of Cygnus A accurately predict this variation, indicating that it is caused by a change in the effective primary beam pattern with baseline.  At this specific hour angle of 1^∘, the variation is mostly driven by wander in the centroid of the primary beam with the position of the feed along the cylinder, which is illustrated in Figure <ref> and whose effect is also highlighted in Figure <ref>. Larger baseline separations exhibit larger deviations from the mean behavior, which is described by the simple model, because fewer redundant baselines are being averaged.  In addition, those baselines are sampling the centroid wander on different cylinders and/or disparate positions along the cylinder.  After subtracting the holography-based model, the residuals are in general less than 1% of the flux of the source.



Figure <ref> provides an alternative qualification of the effectiveness of the source subtraction by constructing a map from the visibilities focused on the region around Taurus A before and after applying the different algorithms.  The map making procedure is described in <cit.>.  The holography-based subtraction results in a significant improvement in the quality of the residual map compared to the simple method.  This is because it accounts for feed-to-feed variations in the primary beam, but also because it fully constrains the time dependence of the source amplitudes, greatly reducing the number of degrees of freedom and preventing the algorithm from subtracting the unmodelled, background sky.  Residual flux from the source is observed to be ≲ 1% of the peak flux.  However, the holography-based method clearly oversubtracts the signal when the source is in the far side lobes at positive hour angle (most obvious between 90^∘≲≲ 100^∘).  This is consistent with the large fractional discrepancy between the holography and solar-based reconstruction of the far side lobes at positive hour angles at the declination of Taurus A that is displayed in the top row of Figure <ref>.  This is suggestive of a slow drift in the normalization of the holography observations with respect to the CHIME data.  The origin of this drift is still under investigation. 

Future improvements to the source subtraction algorithm will involve fitting for an overall normalization of the holgraphy-based model that is slowly varying with time in order to capture the unexplained drift in the holographic measurements that are highlighted in Figure <ref>.  In addition, models for the extended emission of each source will also be incorporated into the algorithm.




§ CONCLUSION


In this work we presented an overview of the holographic beam mapping technique adapted for and used to measure the beam pattern of CHIME. This measurement provides us with a rich dataset, allowing us to measure the beam response of all CHIME feeds in amplitude and phase; and providing constraints on the beam-phase and variation between feed elements that are uniquely accessible to the holographic technique among CHIME's beam-mapping methods. The relatively small confusion noise limit of the measurement enables highly repeatable measurements of the beam sidelobes across a wide range of declinations, albeit with sparse sampling. 

We have detailed an offline analysis pipeline accounting for known systematic effects of the measurement and enabling combination of repeated observations of each source to mitigate thermal noise. This pipeline has been demonstrated on a small portion of the available data, for a subset of six of the brightest sources available for holography. 

We summarized these beam measurements as a function of frequency and feed vs position on the sky. These full 2D visualizations illustrate the frequency and feed dependence of structure in the sidelobes while most of the observed variation in the region of the main lobe is more concisely described by the best-fit centroids and beam-widths. We use the phase information provided by holography to sketch out some basic features of the CHIME aperture, and the possible interaction between the cylinders. 

We have cross-checked the resulting beam measurements against independent CHIME data, including the solar beam measurement and visibility timestreams. Initial discrepancies in the main lobe region, particularly the absence, in co-polarized holography, of the “double-lobe" distortion of Figure <ref> provided an early indication of the presence of polarization leakage in CHIME. When this leakage is accounted for by including cross-polarized holography in attempts to reconstruct the independent CHIME measurements, we find much-improved agreement in the main lobe region. However, as illustrated in Figure <ref> and the asymmetric subtraction of Tau A in Figure <ref>, there remain discrepancies between holography and independent CHIME data in the far sidelobes which are declination-dependent and as large as 40% in some directions. 

We have demonstrated a few key respects in which even the limited sample of observations presented here has provided invaluable insights on, and paths to further investigation of, the properties of CHIME. As we proceed with the full data set, we anticipate that holography's uniquely detailed window into the East-West sidelobes for these sources will provide important input to our beam modeling efforts in accurately capturing the behavior of the instrument in these regions on the sky. This should in turn facilitate an improved characterization of the signatures of foregrounds in the sidelobes as CHIME continues to pursue the power spectrum.



§ ACKNOWLEDGEMENTS


We thank Meiling Deng for her substantial contributions to the
instrument design and analysis of CHIME.

We thank the Dominion Radio Astrophysical Observatory, operated by the National
Research Council Canada, for gracious hospitality and expertise. The DRAO is
situated on the traditional, ancestral, and unceded territory of the Syilx
Okanagan people. We are fortunate to live and work on these lands.

We acknowledge that the holographic measurements presented in this work depended on DRAO's John A. Galt 26 m telescope. We thank the DRAO for their generosity in allowing CHIME to make extensive use of the Galt telescope. We extend special thanks to Tim Robishaw and Andrew Gray for coordinating observing time on the Galt telescope, to Benoit Robert, Rob Messing, and Mike Smith for supporting the operation of the Galt telescope, and to Andre Johnson for designing the customized CHIME receiver for the Galt telescope.

CHIME is funded by grants from the Canada Foundation for Innovation (CFI) 2012
Leading Edge Fund (Project 31170), the CFI 2015 Innovation Fund (Project 33213),
and by contributions from the provinces of British Columbia, Québec, and
Ontario. Long-term data storage and computational support for analysis is
provided by WestGrid, SciNet and the Digital Research Alliance of Canada, and we
thank their staff for flexibility and technical expertise that has been
essential to this work.

Additional support was provided by the University of British Columbia, McGill
University, and the University of Toronto. CHIME also benefits from NSERC
Discovery Grants to several researchers, funding from the Canadian Institute for
Advanced Research (CIFAR), from Canada Research Chairs, from the FRQNT Centre de
Recherche en Astrophysique du Québec (CRAQ) and from the Dunlap Institute for
Astronomy and Astrophysics at the University of Toronto, which is funded through
an endowment established by the David Dunlap family. This material is partly
based on work supported by the NSF through grants (2008031) (2006911) and
(2006548) and (2018490) and by the Perimeter Institute for Theoretical Physics, which in turn
is supported by the Government of Canada through Industry Canada and by the
Province of Ontario through the Ministry of Research and Innovation.

K.W. Masui holds and acknowledges the support of the Adam J. Burgasser Chair in Astrophysics. 


bitshuffle <cit.>,
caput <cit.>,
ch_pipeline <cit.>,

cora <cit.>,
Cython <cit.>,
draco <cit.>,
driftscan <cit.>,


hankl <cit.>,
h5py <cit.>,
HDF5 <cit.>,
HEALPix <cit.>,
healpy <cit.>,
Matplotlib <cit.>,
mpi4py <cit.>,

NumPy <cit.>,
OpenMPI <cit.>,
pandas <cit.>,
peewee <cit.>,



SciPy <cit.>,
Skyfield <cit.>,






§ CHIME PFB AND DECORRELATION CORRECTION


As described in <cit.>, the CHIME correlator follows an FX design where the analog signals from the CHIME antennas are first fed into an FPGA-based system (the “F-Engine") which samples, digitizes and channelizes the timestreams into a 400-800 MHz band subdivided into 1024 channels, each of width 390.625 kHz. 

The channelization step is implemented in FPGA firmware, as a polyphase filter bank (PFB) <cit.>, which aims to ensure that the frequency response within a given channel is flat and dies rapidly outside the channel. The PFB performs the following steps on the input digitized signal: 

    
  * Accumulate a finite number of digitized samples in time, x[t_i], organized in M chunks each of N samples. 
    
  * Apply a window function w[t_i] of length MN-samples to the accumulated data.
    
  * Perform a Fast Fourier Transform on the windowed data.
    
  * Select only every M-th output channel. 


For the PFB implemented in CHIME, M = 4, N = 2048 (and the underlying timestreams are sampled at 800 MSPS) and the window function w is a sinc-Hamming window

    w[t_i] = (t_i/N - M/2)(0.54 - 0.46cos2π t_i/NM -1)


The decorrelation which affects holography is a consequence of the fact that we are channelizing and correlating frames of finite length, so that when one timestream is delayed relative to another, some of the correlated information is not captured within the same frame. When correlating CHIME-only signals, the relevant delays between signals consist of only the expected geometric delays central to interferometry and are too small to lead to any appreciable signal loss. In the case of holography, however, the overall delay, due to the extra lengths of cables connecting the 26 m to the CHIME correlator, becomes a significant fraction of the 2.56μs integration window and the strength of the decorrelation becomes significant. 

Given the implementation of the PFB, it is possible to calculate (and thus correct for) the amount of signal loss due to decorrelation when the input signals are delayed by time τ. Note that this approach is not “recovering signal," but rather compensating for the fact that as the decorrelation varies continuously with τ (and thus with time during an observation as the geometric delay associated with the transiting source changes), the primary effect we are sensitive to in the holography is a distortion of the beam shape, which we attempt to estimate and correct for here. The output of the PFB can be written as 


    x̃[q, f] = ∫ dν e^2π i ν qNΔ tw̃(ν -ν_f)x̃(ν)

where q indexes the 2.56μs output integration frames, f is a channelized frequency with ν the underlying continuous frequency of the signal x, the overhead tildes denote Fourier transforms, ν_f = f / (NΔ t) and Δ t is the cadence in time of the digitized samples. 

The decorrelation results from taking a correlation of Eq. <ref> with an identical copy of itself but delayed by time τ. Taking this correlation and assuming that the underlying signal x has stationary statistics with power spectrum V(ν) (i.e. the visibility spectrum we are trying to measure) we find


    <x̃x̃_τ> = V(ν_f + ν_n)e^2π i(ν_f + ν_n)τ∫_-ν_n /2^ν_n /2dν e^2π iντ|w̃(ν)|^2


where ν_n is the Nyquist frequency associated with the 2.56μs integration cadence (i.e., the width of the output channels, 390 kHz). The term outside the integral is essentially the ideal visibility we wish to measure, but it is modulated by the value, at delay τ, of the Fourier transform of the square of the PFB window function. When τ is large, this term suppresses the measured visibility. Hence we can define the decorrelation ratio as


    D(τ) = ∫_-ν_n / 2^ν_n / 2dν e^2π iντ|w̃(ν)|^2/∫_-ν_n/2^ν_n/2|w̃(ν)|^2


For a given τ, this expression can be calculated numerically. In Figure <ref>, we show the result as a function of hour angle and feed index for the six sources presented here. Noting that we have an unconstrained overall amplitude scale in the holography, we reference the result of the calculation to unity at transit (HA = 0) in the hour angle plots and to feed 0 on the cylinders in the feed index plots.






§ FIGURES FOR OTHER SOURCES

In this Appendix we present versions of Figures <ref>, <ref> for each of the 5 other sources included in this paper: Cas A, Tau A, Virgo A, Hercules A, and Hydra A. Figure <ref> shows the calculated effect of the decorrelation discussed in Section <ref>.




























aasjournal



 Paper Title: Toward Spectrum Coexistence: First Demonstration of the Effectiveness of Boresight Avoidance between the NRAO Green Bank Telescope and Starlink Satellites.
























































0000-0001-5122-9997]Bang D. Nhan
National Radio Astronomy Observatory (NRAO),
520 Edgemont Road,
Charlottesville, VA 22903 USA

0000-0002-0786-7307]Christopher G. De Pree
National Radio Astronomy Observatory (NRAO),
520 Edgemont Road,
Charlottesville, VA 22903 USA

0009-0005-2746-9145]Matt Iverson
Space Exploration Technologies Corporation (SpaceX),
Hawthorne, CA USA

0000-0002-0320-6181]Brenne Gregory
Green Bank Observatory (GBO),
Green Bank, WV USA


Space Exploration Technologies Corporation (SpaceX),
Hawthorne, CA USA

0000-0001-5844-8359]Anthony Beasley
National Radio Astronomy Observatory (NRAO),
520 Edgemont Road,
Charlottesville, VA 22903 USA






0009-0008-5489-2215]Brian Schepis
Space Exploration Technologies Corporation (SpaceX),
Hawthorne, CA USA














§ ABSTRACT 
NRAO and SpaceX have been engaged in coordinated testing efforts since
Fall 2021, including conducting experiments on different interference
avoidance schemes for the Karl G. Jansky Very Large Array (VLA) in New
Mexico, and the Green Bank Telescope (GBT) inside the National Radio
Quiet Zone (NRQZ) in West Virginia. The Starlink system is capable of
avoiding direct illumination of telescope sites with their adaptive
tasking to place downlink beams far away. Nevertheless, even satellites
operating in this mode can potentially present strong signals into the
telescope's receiver system if they pass close to the telescope's main
beam at the boresight. For additional protection, Starlink
satellites can either momentarily redirect or completely disable their
downlink channels while they pass within some minimum angular
separation threshold from the telescope's boresight, methods that are referred
to as “telescope boresight avoidance”. In two separate experiments
conducted since Fall 2023, NRAO and SpaceX arranged to have the
GBT observe a fixed RA/Dec position in the sky, chosen to have a large
number of close-to-boresight Starlink passages. Preliminary analysis from these two experiments illustrate the feasibility of these avoidance methods to significantly reduce, if not eliminate, the negative impact of close-to-boresight
satellite passages. Importantly, these experiments demonstrate the value of continuing cooperative efforts between NRAO and SpaceX, and expanding cooperation between the radio astronomy and satellite communities more generally.



























§ INTRODUCTION


For decades, typical radio-frequency interference (RFI) issues faced by radio astronomy observatories either came from on-site or from emitters near the horizon. These spurious signals arose from nearby electronics and later from cellular telephone towers or other transmitters placed in the environment of observatories. For these reasons, observatories have typically been situated in remote locations, with some sites being protected by specially defined zones. The most well-known zone in the United States is the National Radio Quiet Zone (NRQZ), which was established in 1958 by FCC Docket 11745 and encompasses parts of West Virginia, Virginia, and Maryland. Meanwhile, some locations, such as the Karl G. Jansky Very Large Array (VLA) in New Mexico, have established coordination agreements with known local transmitters (e.g. White Sands Missile Range) without being surrounded by a defined protected zone. For a time, these arrangements provided adequate protection. In recent years, however, the growth of large satellite constellations has challenged this framework since no location is too remote for global satellite networks <cit.>. Satellite-borne signals have had significant impacts on radio astronomy at least since the advent of the Russian Global Navigation Satellite System <cit.>, which severely impacted observations the 1612 MHz Hydroxyl line.

The advent of satellite constellations like Starlink, operated by SpaceX, have added a significant complication to what had been the operating environment of remote radio astronomy observatories. Starlink satellites operate in Low Earth Orbit (LEO, ∼ 160-1,600 km) and transmit downward from their orbits in order to provide broad-band internet to ground-based receivers and to communicate with gateway stations. This new environment will require additional safeguards along with, in particular, coordinated and cooperative efforts for both passive (e.g., radio astronomy observatories) and active (e.g., satellite operators) spectrum users to meet their differing goals. The work described in these experiments is a response to the call from NSF's Spectrum Innovation Initiative (SII) to engage in creative methods for spectrum coexistence.



 §.§ Starlink Operational Parameters


SpaceX began launching production Starlink satellites in late 2019 and shipping Starlink User Terminals (UTs) at scale in 2021. Starlink uses ∼ 500 MHz bandwidth (14.0-14.5 GHz) for their UT uplinks, and ∼ 2 GHz bandwidth (10.7-12.75 GHz) for their UT downlinks. Uplink UT signals use one of eight 62.5-MHz wide channels in the uplink band, with all UTs in a given geological service region (or cell) sharing the same uplink channel. During normal operations, all Starlink UTs in a given cell will dynamically switch uplink and downlink channels. Uplink transmissions use 60 MHz of each 62.5 MHz channel. The UT emissions towards the horizon are limited to -72.76 dBW/Hz in 60 MHz, which corresponds to a maximum equivalent isotropically radiated power (EIRP) of ∼ 3.2 W <cit.>.

Downlinks are transmitted in one of eight 250-MHz wide channels. The highest EIRP density (-15.0 dBW/4 kHz for satellites in 570 km orbits) occurs at the maximum slant (40.5^∘ from nadir). Starlink satellites adjust transmitting power using phased-array beams, accounting for spreading loss and transmitting antenna gain to achieve a target power flux density (PFD) at each UT of -146 dBW/m^2/4 kHz. A UT can transmit with a maximum short-term duty cycle of 33%, although the long-term duty cycle is limited to 11% over 30 minutes (this is a radiation hazard requirement). In practice, the duty cycle is typically of the order of 1% or less, depending on network traffic. For reference, Starlink’s downlink 250-MHz wide channels, denoted as Channel 1-8, start at the following frequencies: 10.7, 10.95, 11.2, 11.45, 11.7, 11.95, 12.2, and 12.45 GHz.



 §.§ Precursor Experiments and Current Developments


Since Fall 2021, NRAO has conducted two types of tests with SpaceX: coordinated and uncoordinated. The former requires formal prearrangement at a given time (typically 48 hours ahead of the scheduled observation) and a pointing direction for our telescopes to intersect with a specific group of satellite passages, as informed by SpaceX. The latter does not involve input from SpaceX; NRAO schedules periodic observations on our telescopes at the same pointing position to assess the evolving impact of satellite's downlink signals during normal operation. 

The first coordinated experiment took place in Fall 2021 with the testing of a Starlink UT at several locations near the VLA in Socorro, NM <cit.>. This experiment showed clearly that the major RFI issue posed by the UTs were not their uplink signals (14.0-14.5 GHz), but the downlink signals from satellite to ground. In an effort to monitor and evaluate the impact of these space to ground transmissions (10.7-12.7 GHz), NRAO worked to install ∼ 60 Starlink UTs on the Alamo Navajo Indian Reservation (T'iistsoh) located about 25 miles northeast of the VLA and began monthly uncoordinated tests (still ongoing). Early results of this experiment indicate minor impacts of the Starlink transmissions <cit.>.


Additionally, the impacts of direct site illumination was conducted through a series of coordinated tests in April and July of 2023. The Starlink's adaptive beam tasking placed their downlink beams to fully illuminate of the cells in the NRQZ. In the April test, the GBT detected a large signal while the telescope was pointed at an elevation of 25^∘, and pointing to the North <cit.>. Subsequent analysis of the Starlink satellites ephemeris, computed from publicly available Two-Line Elements (TLE), in the vicinity the GBT's boresight (Figure <ref>) verified that this large signal was the result of a satellite passing very close to the GBT's boresight at that moment. Since the Starlink satellites are at LEO altitudes, the downlink signal only intersects with the GBT's narrow main beam[Simulation from <cit.> provides GBT's FWMH(11.6 GHz) ∼ 0.86 arcmin = 0.014^∘.] for a short amount of time, on the order of a few tenths of a second. In fact, this detection verified that a critical element to any coexistence scenario would be to account for and mitigate these close-to-boresight encounters by using any tasking capability available onboard, such as redirecting the downlink beams from the telescopes and possibly disabling downlink, when a satellite is within a certain angular separation from the telescope’s boresight during observation.


Recently, NRAO has been developing an autonomous telescope self-reporting system, the Operational Data Sharing <cit.> system, to provide near real-time telescope operational information queryable by satellite operators to be incorporated into their satellite tasking algorithms, including some of the telescope avoidance schemes. The ODS data will be frequently updated, ideally refresh every minute, to allow satellite systems, such as Starlink, to have awareness of the current pointing coordinates and observing frequencies of the VLA and GBT. A pilot testing program with the Starlink engineers to ingest the ODS data into their network algorithm is underway, but a detailed description of the program is outside the scope of this study. The boresight-avoidance experiments described in this study will play a key role in providing preliminary design requirements for the ODS system.



§ EXPERIMENTAL SETUP


The coordinated experiments described in this study were carried out on October 25, 2023 (Experiment #1) and February 19, 2024 (Experiment #2). The observational parameters for the two experiments were identical, covering the downlink frequencies in four frequency bands and excluding the uplink bands. For both experiments,  the GBT was outfitted with an upgraded X-band receiver <cit.> to make spectral measurements with the VEGAS[Versatile GBT Astronomical Spectrometer, <https://www.gb.nrao.edu/vegas/>] in four overlapping 1.5-GHz wide spectral windows centered at: 9.75, 10.5, 11.25, and 12 GHz, with a resolution bandwidth (RBW) of ∼91.6 kHz. These bands are respectively denoted as Spectral Bank (SpB) A, B, C, and D. As part of the coordination, the SpaceX engineering team provided a specified telescope pointing to track in equatorial coordinates (RA/Dec) over a given UTC time window to schedule an observation on the telescope beforehand. The telescope pointing was chosen to be a position that would result in a large number of close-to-boresight encounters using SpaceX's predicted satellite ephemeris (Figure <ref>, center panel). The SpaceX team also provided a list of satellites with the estimated time when their angular separation from boresight (bs) would be the smallest, based on their ephemeris. Satellite positions provided by SpaceX were consistent with those independently derived from the public TLE. 



In each experiment, the telescope observing script was executed by the authors manually in real time using the GBT's AstrID[Astronomer's Integrated Desktop, <https://www.gb.nrao.edu/softdev/>] and CLEO[Control Library for Engineers and Operators,<https://www.greenbankobservatory.org/ rmaddale/GBT/CLEOManual/index.html>] user interface software. Each observation consisted of two main steps. The first step was to conduct an auto-peak focus observing scan at a nearby source from the given pointing to correct for the pointing offset and the receiver feed's focus distance relative to GBT's secondary reflector. This procedure was followed by the main observation sequence with recorded data grouped into multiple scans, each accumulated ten-minutes worth of one-second long integrated spectra, for a total of 600 spectra per scan. 

One of the primary goals of this study is to assess potential impacts of Starlink satellites' downlink and their boresight avoidance capability for a representative GBT observation session. Although the VEGAS can sample spectra with much shorter time interval for high time resolution observation, most observations done on the GBT are commonly observed at one-second integration or longer for better signal-to-noise ratio (SNR). On rare occasions, such as when doing pulsar observation, that the observers would store spectra at sub-millisecond scale. We have planned to conduct follow-up observations at such a fine time scale to more accurately track the downlink signal from the fast-moving LEO satellites. Additionally, since obtaining the absolute flux calibration was not the priority in these tests, also due to the fast transition of the satellite passages over the telescope, we did not allocate time to observe a stronger astronomical flux density calibrator. 



 §.§ Experiment #1 - Boresight avoidance deactivated


This observation started at around 22:09 UTC. The telescope took the first 17 minutes to slew to the prearranged pointing position at (RA, Dec) = (14.257^∘, 45.745^∘) = (00h57m1.6s, +45d44m43.00s). The auto-peak scan used the calibrator source 1824+5651 (1.15 Jy at 9.0 GHz) at 5.3^∘ away from the pre-arranged pointing, to determine the focus correction of -9.59 mm and (azimuth, elevation) offset of (0.1660^∘, 0.1885^∘). This telescope pointing was provided by SpaceX to be a position that would result in 52 close-to-boresight encounters, ranging from 0.06^∘ to 2.96^∘ from the telescope's boresight. The main observation started at 22:25 UTC and concluded at 23:29 UTC.

During this experiment, the Starlink system (as it usually does) excluded the three cells located closest to the Green Bank Observatory (GBO), in order to avoid direct illumination of the GBT, but was available to serve mobile units (UTs for RVers and campers) using Channels 7 and 8 only within cells outside the three excluded ones. Also, the Starlink system excluded the illumination of any registered Starlink UTs in neighboring cells within the NRQZ listed as “not served” on the Starlink website[<https://www.starlink.com/map>] (Figure <ref>, right panel). Starlink Generation 2 (Gen2) satellites equipped with downlink capability in Channels 1 and 2 are designated with a five-digit identification numbers starting with a “3”. The Gen2 satellites were enabled to downlink in all eight Channels during this test, while all the Gen1 satellites were only capable of using Channels 3-8. The only major modification intended for the Experiment #1 to normal Starlink operations was the use of Channels 1 and 2 to serve locations greater than 180 km from the GBO site (approximately the distance between GBO and Washinton, DC). No boresight-avoidance scheme was implemented in this experiment.



 §.§ Experiment #2 - Boresight avoidance activated


This observation started at around 20:08 UTC. The telescope took the first 17 minutes to slew to the pointing at (RA, Dec) = (90.1^∘, 55.4^∘) = (06h00m23.9s, +55d24m00.00s). The auto-peak scan used the calibrator source 0541+5312 (0.77 Jy at 9.0 GHz) located 3.5^∘ away from the pre-arranged pointing, to determine the focus correction of -3.726 mm and (azimuth, elevation) offsets of (0.2911^∘, -0.0081^∘). This sky position was chosen so that during the test, there would be 49 close-to-boresight encounters, ranging from 0.17^∘ to 2.97^∘ from boresight. The main observation started at 20:25 UTC and concluded at 21:59 UTC.

Experiment #2 had the same parameters as described for Experiment #1, however, this time the Starlink network disabled downlink beams from satellites that were scheduled to pass within 0.5^∘ of the GBT's boresight. This boresight angular separation threshold is experimental, and would be subject to change from satellite operator to operator due to their distinct onboard capabilities and service requirements. Before and after the period of this experiment, the Starlink network operated normally without any beam avoidance activated nor downlinking with Channels 1 and 2. 



§ RESULTS
 




 §.§ Differential spectra
 

The spectral plots shown below were made by computing the difference between the raw spectra recorded at each one-second long integration and the median value of all the measured spectra. The raw (uncorrected) antenna temperature, T_ant(t,ν), is first estimated as <cit.>,  

    T_ant(t,ν) = 0.5T_calC_cal-ON + C_cal-OFF/C_cal-ON - C_cal-OFF,

where T_cal is the internal calibrator temperature, C_cal-ON/OFF are measured raw counts with internal noise diode calibration signal with a switching cycle of 1 Hz injected into the main RF signal path. Subsequently, the raw flux density, S(t,ν), in Jansky (Jy) were obtained by,

    S(t,ν) = T_ant/G,

where the antenna gain G = 2.84η_ap [Kelvin/Jy] is computed by assuming the GBT's aperture efficiency, η_ap = 0.71 across the entire band, and assuming no atmospheric correction is needed for the X-band. Since Equation (<ref>) assumes maximum gain of the GBT, this calculation of S(t,ν) could have underestimated the exact flux density of the satellite's downlink transmission. To remove any potential offset in individual spectra, we instead evaluated the differential flux density, which is computed as the difference between all the recorded spectra during the observation and their minimum value of the spectra over the observing time,

    Δ S(t,ν) = S(t,ν) - min[S(t,ν)],

where the minimum value of the spectra was chosen to ensure the least possible RFI is present in the reference spectrum.




For illustration purposes, Figure <ref> shows a subset of spectra recorded in the SpB-C centered at 11.25 GHz, covering the downlink Channels 1-5 and the protected radio astronomy (RA) band between 10.68-10.70 GHz. Examples of the GBT spectra recorded at instances when close-to-boresight Starlink passages with bs≤ 0.5^∘ (left panels) and bs > 0.5^∘ (right panels) are shown. Each panel shows eight spectra of one-second integration, centered at the time of the closest encounter in a given passage to show the varying signal strength as a satellite orbiting passes through the GBT's main beam. Note that strong narrow-band emissions are present at the transitions between channel pairs (Channels 1-2, 3-4 and 5-6) in some of the passages. The nature and a potential fix for this narrow-band emissions are discussed below.



In contrast to Figure <ref>, Figure <ref> shows the lack of apparent strong broad-band emission in Channels 1-5 for the spectra measured for close-to-boresight Starlink passages with bs≤ 0.5^∘. However, as in Experiment #1, strong narrow-band emissions present at the transitions between downlink channels pairs remains even when the boresight avoidance was activated. These narrow-band emissions so far do not seem to pose a problem in our data since we do not observe any induced inter-modulation harmonic components or direct power leakage into the RA-band at  10.68-10.70 GHz. As part of the collaborative effort, after our reporting, the SpaceX team identified the source of these emission features and devised a software patch to suppress them, which will be validated by our follow-up observations. This example demonstrates the importance of coordination and collaboration between passive and active spectrum users, so both parties can resolve such issues in the early development and deployment phases.



 §.§ Signal-to-Noise Ratio Comparison
 

To quantify the effectiveness of the boresight avoidance, beyond visual inspection of the spectra, two metrics are compared between the two experiments. Firstly, at an expected time for a close Starlink passage, we compute the relative SNR between the observed signal level in one of the eight downlink channels and the signal level of a clean channel far from the downlink channels. The clean channel is chosen to be between 9.60-9.80 GHz, observed in the SpB-A of the data. This clean channel is denoted as the reference (Ref) signal. Secondly, we assess the potential spectral leakage from the out-of-band emission (OOBE) into the 10.68-10.70 GHz RA-band when there is a nearby Starlink downlink in Channels 1-2 (10.70-11.20 GHz) by computing the SNR at the RA-band using the same clean reference signal at 9.6-9.8 GHz. 



There are instances that downlink transmission from satellite systems other than Starlink present in our data. Hence, only spectra measured at the timestamps when the Starlink satellites were expected to be the closest to the GBT's pointing are considered. Since the downlink channel has a bandwidth of 250 MHz, the median level of the signal across a given downlink channel is used to compute the SNR. Namely, for a given downlink Channel j, SNR_j is defined as 

    SNR_j = med[Δ S_j(ν)]/med[Δ S_Ref(ν)],

where S_Ref is the flux density in the clean reference channels at 9.60-9.80 GHz.The SNR for the RA-band is computed in the same manner using the median value of the flux density across the 10.68-10.70 GHz band.

If boresight avoidance is activated by the Starlink system and working as expected, the observed SNR in the downlink channels, along with one in the RA-band, are close to unity (zero in logarithmic scale) since no transmission is made during the close-to-boresight passages. Our analysis shows that the boresight avoidance is functional and helps to reduce the RFI significantly. In particular, as shown in Figure <ref>, for passages with bs≤ 0.5^∘, the SNR for the eight downlink channels in Experiment #1 (left panels) are at least two orders of magnitude higher than the ones in Experiment #2 (right panels) with boresight avoidance activated. 



§ DISCUSSION AND SUMMARY
 

Besides avoiding direct site illumination, the primary method to protect a telescope from Starlink satellite transmissions is through adaptive beam tasking that places a satellite's downlink beams far away from the telescope site when the satellite is within a certain angular separation from the telescope's boresight during observation. For example, a satellite that passes within 2^∘ of boresight could be directed to steer its beams no closer than 180 km from a radio telescope. An additional protection level can be achieved by completely disabling downlink beams from satellites that pass within an even tighter cone of a telescope's boresight during observation. This operational mode would further reduce the chance of a telescope main beam being illuminated by any satellite's downlink beam, including its inner sidelobes. At the moment, these two mitigation methods are referred to, both separately and collectively, as the “telescope boresight-avoidance” method by the NRAO and SpaceX collaborators. A separate designation for each of these two methods will be adopted in the follow-up studies.

This study has, for the first time, demonstrated the feasibility of implementing the telescope boresight-avoidance method between a radio astronomy telescope and a LEO satellite operator through collaborative experiments between NRAO's Green Bank Telescope and SpaceX's Starlink constellation. This experiment was made possible by sharing the radio telescope's pointing position and frequency of observation with the LEO satellite operator, who was then able to use this data to mitigate interference into the telescope. The two experiments conducted at the GBT in October 2023 and February 2024 demonstrated:

  
  * When informed about a telescope's pointing direction and the frequency band being observed, the Starlink system is capable of disabling downlink beams for satellite passages close to telescope boresight. While this action is planned for the closest of boresight passages, we expect that refraining from placing beams near the radio telescope will suffice for most near-boresight passages of consequence.
  
  * Briefly disabling satellite downlinks as a satellite passes close to boresight can significantly reduce the observed satellite emission in our data, indicated by statistically significant reductions in SNR by two orders of magnitude inside the 0.5^∘ radius. 
  
  * For Starlink Gen2 passages using Channels 1 and 2, although the SNR levels of the RA-band between 10.68-10.7 GHz in both experiments are approximately unity, a closer inspection suggests a slight increase (about a factor of three) in signal level in Experiment #1 for passages with bs≤ 0.5^∘ (Figure <ref>, top left panel). This potential leakage is no longer an issue when boresight avoidance is in use for close passages (Figure <ref>, top right panel). 
 


Although the boresight cutoff value of bs = 0.5^∘ was adopted in this study, it is subject to change as NRAO and SpaceX iterate on a final value for a future operational system. It is also worth noting that boresight avoidance is only in use for a small number of satellite passages that intersect telescope boresight (11 of 49 passages or 22.45% in Experiment #2). Adaptive beam tasking will be the primary avoidance mechanism for majority of the Starlink passages falling outside a given boresight cutoff, especially if a  more stringent limit (<0.5^∘) is required. However, follow-up experiments with higher time resolution will be required to fully assess the effects of beam interactions between the telescope and the fast-moving satellites on the measured signal.

This work has provided critical data for both the NRAO and SpaceX teams to refine the design parameters of the telescope boresight-avoidance method, such as a boresight degree separation threshold and a satellite's downlink disabling timescale. In particular, feedback for NRAO's development of an autonomous self-reporting system, the Operational Data Sharing (ODS)[ODS data format <https://obs.vla.nrao.edu/ods/>] system, which will provide near real-time telescope operational data (e.g., pointing coordinate, observing band, observation mode and duration) to satellite operators to implement similar telescope avoidance schemes in their network algorithms. NRAO and SpaceX are planning real-time tests of an early version of this system in Summer 2024. Preliminary tests for the ODS system will be based on similar procedures for the coordinated experiments reported here. Importantly, NRAO's cooperative experimentation with SpaceX demonstrates the value of, and a pathway for, continuing cooperative efforts between passive (e.g., radio astronomy observatories) and active (e.g., satellite operators) spectrum users in order to achieve their respective goals.

The telescope boresight-avoidance method being developed by NRAO and SpaceX is a novel way to ensure the coexistence of radio astronomy and commercial satellite operators in a way that mutually benefits the mission of both groups. The initial results from this work suggest that these avoidance methods, when properly implemented and tested, can simultaneously increase the range of communication services of a satellite operator while expanding the frequency bands on which a radio astronomy telescope can observe without harmful interference from the satellite constellation. As boresight avoidance can in theory be implemented across the radio spectrum, there is good reason to think that a telescope that implements the method could observe on a much larger set of frequency bands than may be protected by existing regulations, while the satellite operator could simultaneously provide communication services in and around the telescope using other frequency bands that are not being observed by the telescope. This would mark a fundamental shift in thinking about the coexistence of radio astronomy and satellite services, where radio frequency regulations have traditionally imposed a strict separation of the two services through the hard allocation of certain frequency bands to one service at the exclusion of the other. The authors of the paper invite radio astronomers, satellite operators, and regulatory agencies to consider the implication of the telescope boresight avoidance method on radio frequency regulations and ongoing efforts to coordinate passive space research services and active satellite services.


The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. This work is supported by the National Science Foundation SII-NRDZ (AST-2232159) and SWIFT-SAT (AST-2332422) grants. The authors acknowledge the contributions of many individuals at both NRAO and SpaceX who have made these experiments possible. At NRAO and GBO: James Robnett, Fred Schwab, Sheldon Wasik, Daniel Bautista; and at SpaceX: Doug Knox, Michael Nicolls, David Goldman, David Partridge, Joe McMichael, Tony Liang, and Mihai Albulet.



aasjournal












 Paper Title: Advanced pure tilt actuator for testing tilt-to-length coupling in space-based gravitational wave detection.







MOE Key Laboratory of Fundamental Physical Quantities Measurement & Hubei Key Laboratory of Gravitation and Quantum Physics, PGMF and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China


*yanhao2022@hust.edu.cn 



 
Tilt-to-length (TTL) coupling, caused by the jitter of test masses or satellites, is a significant noise source in space-based gravitational wave detection. Calibrating and suppressing TTL coupling noise at the sub-nanometer level is essential. One main challenge in current ground-based TTL coupling testing is the residual translational movement of the tilt actuator. This paper introduces the development of an advanced pure tilt actuator (APTA) specifically designed for testing TTL coupling. The APTA provides precise tilt motion and is monitored by a four-beam interferometer, which measures the displacement of attached array pyramids. We present a detailed theoretical model and experimental setup. Experimental results demonstrate that this optical test bed, equipped with the APTA, can achieve subnanometer-level TTL coupling calibration. In addition, a typical heterodyne interferometer was tested using the APTA test bed. Comparative testing demonstrated that the imaging system is capable of effectively suppressing TTL coupling errors. The TTL coupling coefficients were reduced from over ± 30  μm/rad to within ± 5  μm/rad across a range of ± 200  μrad, meeting the preliminary requirements for the TianQin mission. This APTA test platform has the potential to be widely utilized for ground-based TTL coupling inspection.

©2024 Optica Publishing Group under the terms of the https://www.osapublishing.org/submit/research-articles/authors/open-access-publishing-agreementOptica Publishing Group Open Access Publishing Agreement






§ INTRODUCTION

In space gravitational wave detection missions such as TianQin<cit.>, the Laser Interferometer Space Antenna (LISA) <cit.>, and Taiji <cit.>, the demand for displacement detection sensitivity is extremely high. Measurement optical path changes caused by tilt jitter, known as TTL coupling, have become a significant potential noise source, severely affecting the accuracy of displacement measurements along sensitive axes. TTL coupling occurs in both test mass interferometers and long-arm interferometers. The TianQin mission, a space-based gravitational wave detection project in geocentric orbit, constructs a large-scale laser interferometer with an arm length of approximately 1.7 × 10^5 km . It aims to measure inter-satellite displacement changes with a sensitivity limit of 1 pm/Hz^1/2 at 6 mHz [1]. This requirement imposes stringent demands on the suppression of various noise sources, particularly TTL coupling noise. To achieve this goal, the TTL coupling must be minimized to less than 10 μ m/rad for tilts of 10 nrad/Hz^1/2 at 6mHz.

Reaching this goal directly is almost impossible, but we can achieve it through several steps. First, we should minimize misalignment error as much as possible during design and installation. However, it is not feasible to perfectly align all components in the system using this method alone, so we must meet the required benchmarks as closely as possible at this phase. Next, we need to calibrate the misalignment. The beam alignment mechanism (BAM), composed of a pair of parallel glass windows, can adjust the transversal position of the beam <cit.>. The fast steering mirror can correct the beam's angular direction. These two devices are integrated into the system for compensatory adjustments to minimize alignment errors after installation. Finally, misalignments can occur during launch, transfer to orbit, and over time. The remaining TTL noise needs to be subtracted in ground-based post-processing by measuring angular jitter <cit.>, ensuring continuous compliance with gravitational wave detection requirements. By following the three steps, the TTL coupling can be suppressed to the level below our required threshold. However, before achieving this, a clear understanding of the TTL coupling formation mechanism is essential, along with the design and construction of ground-based experimental testing setups.

TTL coupling can arise from various mechanisms, broadly categorized into geometric and non-geometric types <cit.>. In the test mass (TM) interferometer, geometric TTL coupling occurs when the tilt of the TM increases the optical path length from the origin of the measuring light to the detector surface, a phenomenon known as the light lever effect. Misalignment between the TM's center of mass and the reflection point of the measurement laser can also result in a piston effect. Additionally, changes in the angle of the incident measurement light on a plane transmission mirror can alter the optical path along the sensitive axis. Non-geometric TTL coupling involves factors such as the sizes and positions of the beam waists of the two interfering beams, which can affect the readout of the interference signal. Wavefront errors, including higher-order modes and wavefront distortions, modify the wavefront curvature of the interfering beams, potentially impacting the signal readout and inducing TTL coupling errors.  To date, research in this area has been limited. Beyond the interference light itself, the characteristics and geometry of the optoelectronic detector—a critical component in detecting interference signals—directly influence signal readout. Factors such as the detector’s area, slit dimensions, and electrical non-uniformity still need to be accurately characterized.

Ground-based experimental testing on TTL coupling is crucial for the development of space laser interferometers. However, a challenge in experimental testing setups is how to generate pure rotation without introducing parasitic displacement along the sensitive axis, as such displacement would affect the ultimate precise calibration of TTL coupling. Various methods have been proposed in numerous studies, such as using the characteristic of identical optical parameters of two interference beams in a zero-difference interferometer, monitoring and deducting the parasitic displacement of the actuator with a large single element photo diode (SEPD), successfully calibrating the TTL coupling of the device, and verifying that a double lens imaging system can suppress TTL coupling <cit.>. To realistically simulate the LISA optical platform, the LISA team constructed an optical testing platform using a silicate bonding technique, successfully deducing the parasitic displacement of the actuator with technologies such as phase locking between multiple beams <cit.>. This device can simultaneously simulate TM interferometers and interstellar interferometers <cit.>. However, this device has a complex optical path structure, and since most optical components are glued on the substrate, it is not easily modifiable or adjustable to test different settings, such as different distances or beam characteristics. To make the TTL coupling calibration device more flexible, the LISA team developed the advanced tilt actuator (ATA), taking advantage of the characteristic that a conical prism is insensitive to changes in the angle of incident light, effectively suppressing the parasitic displacement produced by the actuator <cit.>. The ATA was designed and built as an independent experimental platform; hence it can be used not only for a specific TTL coupling experiment but also for other experiments requiring three-degree-of-freedom (DOF) motion of the ATA.

Based on our research on experimental testing setups for TTL coupling, we noticed that the pyramid produces a lateral displacement in the incoming light during rotation, thus affecting the final readout of the interference signal and thus influencing the calibration of the TTL coupling. To address this issue, we designed a self-aligning four-beam interferometer <cit.>. The feature of this device is that the conical prism does not cause lateral displacement of the incoming light when tilted. The device is constructed using a UV adhesive <cit.>, providing compactness and significantly improved displacement measurement sensitivity. This makes it potentially applicable for high-precision TTL coupling experiments in the future.

The paper is organized as follows. Section 2 provides an overview of the mechanisms of TTL coupling formation, which helps us to gain a clearer understanding of TTL coupling. Section 3 details the working principles, optical design, and error analysis of the ATPA, and how this device is applied to ground-based experimental calibration of TTL coupling. Section 4 presents the ground-based experimental testing setup for TTL coupling, demonstrates the calibration of system TTL coupling, and shows the suppression of TTL coupling using a confocal imaging system. Section 5 contains the discussion.



§ TTL COUPLING AND IMAGING SYSTEM



 §.§ Mechanism of TTL coupling

TTL coupling in laser interferometers is a complex effect. Geometric TTL coupling originates from the situation whereby one of the beams, usually the one carrying the main measurement signal, tilts and deviates from the initial interferometer light path, leading to a propagation distance greater than that of the completely aligned reference beam. The optical path difference between beams results in a phase difference, for example, the power fluctuation of the photodiode in a zero-difference interferometer, or the phase shift of the heterodyne signal in a heterodyne interferometer. 


Taking the TM interferometer as an example, when the TM tilts, geometric TTL coupling errors can occur due to the presence of a lever arm between the TM and the detector, a phenomenon known as the light lever effect. As shown in Fig. 1, assuming the laser is a plane wave, the expression for this effect can be calculated through simple geometric relationships as:

    OPD_lever ≈d_lever /2α^2.

Furthermore, an offset exists between the reflection point of the measuring light on the TM surface and the TM rotation point (center of mass), leading to an extra optical path, known as the piston effect. Its expression can be written as:

    OPD_piston ≈- d_lat α+ d_long α^2.

where d_lever is the lever arm length, d_lat is the lateral offset of the reflection point, d_long is the longitudinal offset, and α is the beam tilt angle. All the above results are retained to the second order of the tilt angle α.

However, in a real heterodyne interferometer, lasers have their own wavefront curvatures and power distributions. For instance, the TM interferometer uses Gaussian light while the interstellar interferometer uses both Gaussian light and flat-top light. If the two interference lasers are not perfectly matched and aligned on the detector surface (for example, due to laser axis offset, wavefront tilt, and distortion), they would interfere with the interference light field distribution on the detector surface and subsequently impact the final sensitive-axis optical path signal readout. This type of effect is known as a non-geometric TTL coupling. The sensitive axis optical path signal LPS readout can be written as the phase of the interference light field at the active area of the detector, integrated over that part,

    LPS=1/k(∬ E_m E_r^* d S_PD).

Where E_m represents the electric field of the measurement light, and E_r refers to the electric field of the reference light, any factors affecting this integral will cause TTL coupling, such as the previously mentioned mismatch of light parameters, properties of the photodetector <cit.>, and even different definitions of the sensitive axis optical path signal can result in various TTL couplings, as mentioned in literature <cit.>. In interstellar interferometers, the interfering beams are Gaussian light and flattop light, and the sensitive axis optical path signal needs to replace the electric field expression in the integral equation. The TTL coupling generation mechanism is roughly the same in both kinds of interferometers. However, due to the intercept and magnification of the far-end light by the interstellar telescope, the beam tilt angle is amplified, thereby affecting the level of TTL coupling.





 §.§ Imaging system

Imaging systems are an effective method for suppressing the TTL coupling <cit.>, as shown in Fig. 2. A strict optical design is employed to place the TM reflection surface in the entrance pupil of the imaging system and the quadrant photodiode (QPD) in the exit pupil. The Fermat principle is utilized to suppress the change in the geometric optical path of the measurement beam during tilting, as well as to restrain the offset of the light spot on the surface of the photodetector. However, imaging systems can alter the optical parameters of the interfering beams and the tilt angle of the measurement beam, potentially increasing non-geometric TTL coupling. Consequently, imaging systems can suppress geometric effects while leaving residual non-geometric effects. To reduce the system's TTL coupling below the required level, modifying the design of the imaging system is the only method to deliberately introduce optical path changes to offset non-geometric TTL coupling.

In this paper, we design a set of classic confocal imaging systems using existing lab lenses. Through fine-tuning the installation positions of the imaging system and QPD, we attempt to combine and test their suppression effects on TTL coupling within the system. The imaging system is installed on a three-axis displacement stage, allowing for precise multiaxis adjustments of the lens position. The detector installation is based on thermally stable single-blade aluminum flexors with ultrafine precision screws, achieving micrometer-level positional accuracy and milliradian angular accuracy.



§ APTA: ADVANCED PURE TILT ACTUATOR



 §.§ Measurement principle


In TTL coupling calibration, it is necessary to subtract the residual longitudinal displacement of the tilting actuator at the measurement light reflection point. This requires monitoring the three degrees of freedom (longitudinal displacement, yaw and pitch) at the reflection point to effectively cancel out the actuator's residual noise. As I understand it, two non-collinear lines define a plane, and similarly, three non-collinear points determine a plane. During plane movement, we can infer the three degrees of freedom at any point on the plane from the displacement information at these three points. By leveraging this property, we can monitor and subtract the residual longitudinal displacement at the measurement light reflection point while also reading the tilt angle of the measurement light. To achieve these objectives, two critical conditions must be met. Firstly, the precise measurement of the longitudinal displacement information at three points on the plane; Secendly, Accurate positioning of these three points.


Once these conditions are satisfied, we can theoretically decouple the three degrees of freedom at the measurement light reflection point. As shown in Figure 4, assume three points on the same plane in space have coordinates P_1(x_1, y_1, z_1), P_2(x_2, y_2, z_2)  and P_3(x_3, y_3, z_4), where the Z-axis is the sensitive direction. Given that the plane's rotation angle is very small (of the order of a hundred microradians), we assume the coordinates remain unchanged, treating x_1,2,3 and y_1,2,3 as constants during the tilt, while the longitudinal displacements z_1,2,3 change over time. The normal vector of the plane containing the mirror can be expressed through the cross-product of vectors between the three points:

    n⃗=P_1 P_2×P_2 P_3=(n_x, n_y, n_z).

Substituting the coordinate information, the elements of the normal vector can be expressed as:

    n_x=z_1 ·(y_3-y_2)+z_2 ·(y_1-y_3)+z_3 ·(y_2-y_1), 
        n_y=z_1 ·(x_2-x_3)+z_2 ·(x_3-x_1)+z_3 ·(x_1-x_2), 
        n_z=x_1 ·(y_2-y_3)+z_2 ·(y_3-y_1)+z_3 ·(y_1-y_2).

where n_z is a constant. Any point P(x,y,z) on the plane containing P_1, P_2 and P_3 can be expressed as:

    z=-n_x ·(x-x_1)+n_y ·(y-y_1)/n_z+z_1.


As can be seen from the equation above, knowing the longitudinal displacement and position information of three points on the plane enables accurate measurement of the longitudinal displacement z at any other point on the plane. By utilizing this property, the longitudinal displacement of the measurement light reflection point can be measured and the actuator's residual noise can be subtracted, thus calibrating the TTL coupling error in the interferometer more accurately.



 §.§ Optical design

The APTA (Advanced Pure Tilt Actuator) consists of two parts: a compact four-beam interferometer and a conical rotary actuator, as shown in Figure 5. The compact four-beam interferometer is composed of a collimator, a lateral displacement prism, a beam-splitting prism, and a punctured plane mirror, with all optical elements adhered to an ultra-low expansion microcrystalline glass substrate using UV glue <cit.>, ensuring a structure with robust thermal stability.


The optical path design is shown in Figure 6. The laser emitted from the collimator is split into two layers, upper and lower, with four beams in total, by the lateral displacement prism. The two layers of beams are completely parallel. Each beam is then split again via the beam-splitting prism, with three of the beams exiting after being transmitted through the punctured reflecting mirror. The position of the hole in the mirror is precisely measured to ensure that there is no truncation of the light spot. The remaining transmitted laser beam reflects off the mirror and returns along the original optical path, interfering with another beam reflected from the beam-splitting prism. Since the reflecting mirror does not move, this interference signal serves as a reference to remove noise from the upstream fiber link and environmental noise. The conical rotary actuator consists of a piezoelectric ceramic mirror flipper and three hollow conicals. Piezoelectric ceramic acts as an actuator by providing three degrees of freedom (pitch rotation, yaw rotation, and longitudinal translation) controlled by voltage. The hollow conicals serve as target mirrors, reflecting the three transmitted beams back to the punctured reflecting mirror. Thanks to the property of conical prisms, which keeps the incident and outgoing beams parallel, adjusting the attitude of the perforated reflector will ensure each beam reflected by the cone exits the same way it entered. Then these interfere with their corresponding reference light, forming three independent measurement signals.


APTA's unique structure provides several advantages. First, when the target cone rotates, the contrast of the interference signals hardly changes, maintaining a high level. This is because the measurement beam always returns along the same path, thus ensuring consistently well-aligned interfering light <cit.>. Second, the optical path within the hollow cones is a constant value, solely dependent on the height of the cones and not the angle of incidence. This makes the cone insensitive to the angle of incidence - TTL coupling can thus be considered insensitive to an interferometer using a cone as the target mirror. Third, its versatile use. As an independently designed and built experimental platform, the overall structure of APTA is compact and easy to transport. Therefore, it can be used not only for specific TTL coupling experiments but also for other experiments requiring multi-degree of freedom motion. Fourth, APTA has three independent measurement interferometers that can detect the three degrees of freedom motion information of the target mirror. Moreover, multi-beam measurement has a larger linear range and lower non-linearity [20], allowing for future comparison and calibration of the DWS signal in the test platform.



 §.§ Error analysis

APTA will be used for TTL coupling calibration, so it is necessary to analyze its own errors to avoid introducing them into the calibration results. The first concern is the dihedral angle error <cit.>. Due to the machining process, it is impossible to ensure that the three faces of the corner cube prism are perfectly perpendicular to each other, which affects the property of the constant optical path within the corner cube, thereby introducing calibration errors into the APTA. In this context, we need to account for the dihedral angle errors in the internal optical path calculations of the corner cube. Let's consider the dihedral angle errors as δ_12 and δ_13, where δ_12 is the dihedral angle error between the first and second reflective surfaces, and δ_13 is the dihedral angle error between the first and third reflective surfaces.


At this point, we only consider the change in the geometric optical path of the measurement light within the corner cube. Assuming the measurement light is a ray, we first assume that the vertex of the corner cube is positioned at the origin "(0,0,0)^T"  of the coordinate system. The installation position of the corner cube is as shown in Figure 7(a). The normal vector of the exit plane is n_0=(0,0,1)^T, and this plane is 1 meter away from the origin. The exit point of the light ray is P_0(x_0, 0,0), with the propagation direction l⃗=(0,0,-1)^T. Using ray-tracing methods for analysis and calculation, we can derive the analytical solution for the direction of the exit ray after reflection by the corner cube with errors (accurate to the second order of the dihedral angle error),

    l^'≈([ 0; 0; 1 ])+1/2([   -2 δ_12-2 √(2)δ_13-4 δ_12δ_13+√(2)δ_12^2; -2 √(2)δ_12+2 √(2)δ_12δ_13-δ_12^2+2 δ_13^2;          -3 δ_12^2-2 √(2)δ_12δ_13-2 δ_13^2 ])+([ -δ_12+√(2)δ_13; -δ_12+√(2)δ_13;              0 ])([ η; ϕ; 0 ]).

Here, η and ϕ represent the pitch and yaw angles of the corner cube prism, respectively. The first term in the equation can be understood as the direction of the reflected ray after passing through an ideal corner cube reflector, which is completely parallel and opposite to the direction of the incident ray. The second term represents the deviation in the direction of the exit ray when the corner cube reflector with dihedral angle errors is in a static state. The third term represents the coupling deviation in the direction of the exit ray due to the rotation of the corner cube reflector with dihedral angle errors. During the rotation of the corner cube reflector, the introduced geometric optical path error is:

    OPD_error≈   ( x_0 δ_12 + √(2) x_0 δ_13 + √(2)δ_12δ_13 + ( 32 + x_0√(2)) δ_12^2 + δ_13^2 )_static term
        + [ x_0 δ_12 - √(2) x_0 δ_13] η_first-order term+ {( 1 - x_0/√(2)) δ_12^2 - 2 δ_13^2 }η_first-order term
       + [ ( √(2) - x_0/2) δ_12^2 - ( 2 + √(2) x_0 ) δ_12δ_13 + x_0 δ_13^2 ] ϕ_first-order term
        + [ -x_0/2δ_12 - x_0/√(2)δ_13] η^2_second-order term+ [ -√(2) x_0 δ_12] ηϕ_second-order term

The first term in the equation (static term) represents the additional optical path introduced when the corner cube reflector is static, accounting for the exit point position and its dihedral angle error. The second and third terms represent the linear coupling of the pitch angle η. However, the third term involves second-order polynomials of the dihedral angle error, indicating that if the dihedral angle deviation is small, its contribution to the optical path length change is negligible. The fourth term denotes the change in optical path length corresponding to the second-order polynomial of the pitch angle η. Similarly to the third term, its impact is negligible for small rotation angles and dihedral angle errors. The fifth term shows the linear coupling of the yaw angle ϕ. Given that the second-order polynomial of the dihedral angle deviation is very small, the influence of the fifth term on the optical path length change is not significant. Similarly, the contribution from the sixth term may be negligible.


In practice, the dihedral angle error can be controlled in 3 arcseconds (1.45 μrad), the deflection of the corner cube reflector is approximately -300 ∼300 μrad, and the accuracy of displacement measurement is within the picometer range, so the influence of the third-order term can be ignored. To verify the correctness of the analytical solution, we numerically solved the ray-tracing process using Matlab, and compared the results of the two methods. Assuming the exit point is s⃗(0.005  m, 0,0) and the deflection of the corner cube reflector is -300 ∼ 300  μrad, and there is only one dihedral angle error δ_12=0, δ_13=0∼40 μrad, only the coupling term (excluding the static term) is calculated.When the corner cube undergoes pitch deflection η, the additional optical path introduced by the coupling of different dihedral angle errors is shown in Figure 8(a), where the solid line represents the numerical calculation results and the dashed line represents the analytical calculation results. It can be seen that the maximum TTL coupling coefficient is ∼0.28 μm/rad@δ_13=0∼40 μrad, and the analytical results are in good agreement with the numerical results.

Next, we consider the incident angle error of the measurement light on the corner cube prism. Ideally, the three measurement laser beams should be parallel to each other in pairs and perpendicular to the plane of the corner cube; otherwise, the measurement interferometer will not be able to accurately measure the longitudinal displacement of the corner cube vertex. In the experiment, we first use a flat mirror to reflect the three measurement beams back along their original paths, while the other end of the mirror is adjusted using an independent interferometer to reflect its measurement beam back along its original path. During the experiment, the power of the light at the return port of the circulator can be used to determine whether the measurement beam is returning along its original path. When the return power of all measurement beams is maximized, the mirror is replaced with a corner cube combined with a mirror, which forms the corner cube rotation actuator. Then, by adjusting its alignment to restore the return power of all measurement beams, it can be assumed that the three measurement beams in the APTA are aligned: parallel to each other in pairs and perpendicularly incident onto the plane containing the corner cube vertex. Despite efforts to align the beam angles, some residual misalignment may still exist. As shown in Figure 7(b), assuming the incidence angle of the beam is i=δ, the optical path S_pyramid of the laser inside the corner cube is calculated using the ray tracing method as follows,


    S_pyramid =2 n H/√(1-sin i^2/n^2).

where n is the refractive index of the corner cube material. For a hollow corner cube n, and H is the height of the corner cube (distance from the vertex to the incident plane). Since a hollow corner cube is used in this experiment, when the laser is incident non-orthogonally (i ≠ 0^∘) on the surface of the corner cube, the optical path error approximated to the second order is,

    OPD_error ≈ 2 H·(1+i^2/2)-2 H=H i^2.


In this experiment, the height of the corner cube H is 0.01 m. From the above equation, the optical path difference under various incidence angles can be analyzed and compared with the numerical calculation results, as shown in Figure 8(b). The simulation results indicate that the optical path error increases with the incidence angle; therefore, the incidence angle should be kept as small as possible. Based on the current experimental testing level, the incidence angle should be controlled below 100 μrad.

Finally, considering the installation errors related to the position of the corner cube, as derived in Section 3.1, it is necessary to combine the position and displacement information of the corner cube vertex to obtain the displacement information at the measurement light reflection point in the TTL coupling experiment. Therefore, the position of the pymarid vertex is crucial for the final calibration of the TTL coupling. In this experiment, a perforated metal base is used to install the hollow corner cube prism, which is then fixed by metal-set screws. This means that the installation accuracy of the corner cube vertex position depends on the machining accuracy. Ideally, the coordinates of the vertices of three corner cubes in the XY plane are P_1(-7  mm, 7  mm), P_2(7  mm, 7  mm) and P_3(7  mm,-7  mm). At present, the machining accuracy of the metal base is ±0.05 mm. Since the machining errors are random, in order to realistically reflect the actual situation, random errors within the range of ±0.05 mm  are assigned to the coordinates of the vertices of the three corner cubes in the XY plane, as shown in Table 1.





Assuming only pitch rotation around the X-axis occurs, substituting the above errors into formulas (6, 7), the errors in the three degrees of freedom at the origin O are given by:

    Δϕ_error ≈ 5.831 × 10^-3·η
       Δη_error ≈-4.157 × 10^-3·η
       Δ z_error ≈ 1.317 × 10^-3·η

where ϕ is the yaw angle, η is the pitch angle, and Δϕ_error, Δη_error and Δ z_error are the yaw angle error, pitch angle error, and longitudinal displacement error at the origin, respectively. From the above equations, it can be seen that even if the plane of the corner cube only undergoes one degree of freedom motion, errors will still be generated in all three degrees of freedom. This means that the lateral displacement of the vertex position will lead to cross-coupling between each readout signal of the APTA. The longitudinal displacement error Δ z_arb at any point on the vertex plane of the pymarid is expressed as:

    Δ z_arb≈ (-x_arb·Δϕ_error) + (-y_arb·Δη_error) + Δ z_error.

where x_arb and y_arb are the lateral position offsets of any point on the corner cube vertex plane with respect to the initial origin in the X and Y axes. Calculating the longitudinal displacement errors at three different positions, with coordinate information x = 0 mm,  y = 0, 2, 4, 6 mm, the results are shown in Figure 9. It can be seen that the longitudinal displacement increases with the increase in the offset, and exhibits a significant first-order linear relationship, which cannot be ignored.

To suppress the above errors, intentional introduction of errors can be used to offset the known errors. The specific method is to adjust the position of the measurement light-reflection point during the decoupling process to offset the existing linear errors. The feasibility of this method has been preliminarily verified in previous studies <cit.>.



§ EXPERIMENTAL SETUP AND RESULTS



 §.§ Experimental setup

A ground-based experimental setup was established to test the TTL coupling in the quality inspection interferometer, where APTA and the imaging system are used respectively for calibration and suppression of the TTL coupling. As shown in Figure 10, the entire setup consists of two parts: the TM interferometer and APTA. The heterodyne interferometer is used to simulate the quality inspection interferometer, while the APTA is used to simulate the tilt of the TM attitude and to compensate for the errors of longitudinal displacement during motion. The following sections will explain each part in detail.


In this experiment, a low-noise narrow-linewidth NPRO laser (Coherent, Mephisto) with a central wavelength of 1064 nm was used as the light source. A high resolution dual channel waveform generator (Keysight, 33600A series) was utilized to generate two sinusoidal signals with frequencies f_1 = 40 MHz and f_2 = 40.01 MHz, respectively, achieving a heterodyne frequency of 10 kHz for the heterodyne interferometer. The orthogonal interference signals were received by a variable-gain InGaAs detector (Thorlabs, PDA20CS2) and a quadrant photodetector (Kyphotonics). Furthermore, a data acquisition (DAQ) system with low quantization loss (NI-PXIe 4464) was used for phase acquisition and demodulation. The sampling rate was set to 1 MHz, with a phase readout rate of 50 samples/s.

A single-frequency laser is transmitted to the optical workstation through a single-mode polarization-maintaining fiber. In the optical workstation, the laser undergoes frequency shifting via a pair of acousto-optic modulators. The two laser beams, which have a frequency difference of 10 kHz and nearly equal optical path length and power, are used as the measurement beam f_1 and the reference beam f_1, respectively, and are injected into the space interferometry portion. To transmit the two heterodyne beams into the APTA and the heterodyne interferometer, each beam is split using a 50:50 beam splitter, resulting in a total of four laser beams coupled into the fiber collimators.

In the TM interferometer, the measurement beam and the reference beam pass through nonpolarizing beam splitters, respectively, where the two reflected beams interfere. The interference beam is detected by a large-area single-quadrant photodetector through a mirror, forming a reference signal used to subtract fiber noise and environmental noise at the front end of the TM interferometer. The measurement beam, after passing through another nonpolarizing beam splitter and reflecting off the APTA mirror, generates beam tilt and reflects again through the. At this point, the tilted measurement beam and the reference beam are combined in NPBS, forming two separate interference beams. One of these interference beams is detected by a quadrant photodetector without passing through the imaging system, using differential wavefront sensing to obtain the tilt angle between the two beams and measure the TTL coupling magnitude within the TM. The other interference beam, passing through a confocal dual-lens imaging system, is detected by another quadrant photodetector. This setup is used to verify the imaging system's ability to suppress TTL coupling. The parameters for the imaging system and the TM interferometer are listed in Table 2.


In high-precision experiments, interferometers are typically built on ultra-low expansion glass substrates and placed in a vacuum environment to suppress environmental disturbances such as temperature fluctuations and air currents. However, in this experiment, the test platform was constructed on a stainless steel honeycomb breadboard and placed in ambient air. The entire system was enclosed with an acrylic cover for passive temperature control and acoustic isolation. To further suppress noise that is not related to the tilt modulation signal and to more accurately calibrate the TTL coupling in the interferometer, we employed coherent filtering techniques <cit.>. This filtering technique requires applying a modulation signal to the tilt actuator and collecting multiple cycles of the interferometer signal. By performing a fast Fourier transform on the interferometer signal, we can consider any signal frequencies in the frequency domain that do not equal the modulation frequency or its harmonics as noise. By only computing the inverse transform corresponding to the modulation frequency and its harmonics, we retain the displacement signal related to the beam tilt and reconstruct the TTL coupling curve. To avoid frequency leakage, we applied a 0.5 Hz sinusoidal modulation signal to the tilt actuator, with the post-downsampled sampling rate of the phase meter set to 50 Hz, an integer multiple of the modulation frequency. Clock synchronization was performed between the modulation signal source and the phase meter, and the sampling time was approximately 10 minutes.


 §.§ Test results

After completing this setup, we compared the interferometer signals on the reference QPD and the measurement QPD, which respectively represent the original TTL coupling in the TM interferometer and the suppressed TTL coupling after the imaging system. By comparing these two signals, we can assess the effectiveness of the imaging system in suppressing TTL coupling. In the experiment, the longitudinal displacement signal LPS was calculated using the arithmetic mean phase of the four quadrants of the QPD,

    LPS^AP = ϕ_A + ϕ_B + ϕ_C + ϕ_D/4k.

where ϕ_A represents the phase measured by quadrant A of the QPD, and similarly for the other quadrants. k = 2π/λ is the wavenumber, with the laser wavelength being 1064 nm. The beam tilt angle information was obtained through the DWS signal.

The calibration results of TTL coupling in the TM interferometer before and after suppression by the imaging system are shown in Figure 11. After careful alignment and match of the reflection points <cit.>, the imaging system was aligned and the QPD measurement was adjusted to minimize TTL coupling. Without the imaging system, the longitudinal displacement readout of the system could change up to 3 nm within a tilt angle range of ± 200 μrad. After incorporating the imaging system, the longitudinal displacement readout variation was reduced to below 0.6 nm. It is evident that the TTL coupling curve exhibits a significant quadratic effect with respect to angle, although higher-order terms still remain. To calculate the variation of the angle-to-twist ratio coefficient, we fitted the TTL coupling curve with a fourth-order polynomial and plotted the first derivative against the angle. It can be seen that within a beam tilt range of ± 100 μrad, the TTL coupling coefficient, after suppression by the imaging system, decreased from ± 15 μm/rad to below ± 5 μm/rad.




§ CONCLUSION

TTL coupling is ubiquitous and any misalignment can lead to its occurrence. It is one of the main noise sources in space-based gravitational wave detection missions such as TianQin, LISA, and Taiji. Therefore, studying the formation mechanism of TTL coupling and the methods to suppress it are crucial for the development of ultra-high-precision space laser interferometers. One challenge in the experiments involving TTL coupling is that the longitudinal displacement error of the tilt actuator directly couples into the TTL calibration and interferes with the final results. To address this, we propose a novel tilt brake (APTA) to eliminate the longitudinal displacement error of the actuator, providing the interferometer system under test with a pure tilt drive. The APTA comprises two parts: a compact four-beam interferometer and a retroreflector tilt actuator. The APTA reflects the measurement beam in the interferometer under test and measures the three degrees of freedom information at the reflection point, thus enabling accurate TTL coupling calibration. The APTA features a compact and portable structure, making it applicable for TTL coupling experiments for future interferometers or any experiment requiring three degrees of freedom motion.

Additionally, we employ a classical confocal dual-lens imaging system to suppress TTL coupling. Experimental results show that after aligning the lenses and fine-tuning the detector position, the imaging system significantly suppresses TTL coupling. However, since off-the-shelf lenses were used and the actual layout of the optical platform in a space interferometer was not considered, future optimizations of the imaging system design based on the actual optical platform configuration are necessary.

During TTL coupling calibration, APTA still experiences high-order noise effects. We believe that misalignments during assembly and manufacturing errors in components such as the retroreflector are likely causes. Despite efforts to minimize errors during assembly, residual errors still interfere with the final calibration results. This study analyzes three error mechanisms within the APTA and employs data processing to minimize their impact on calibration. More research is required to investigate the error mechanisms within the APTA and optimize its structure and assembly process.

TTL coupling experiments show that without the imaging system and within a tilt range of ± 200 μrad, the measured TTL coupling reaches up to 3 nm, with a coupling coefficient of ± 15 μm/rad. Using the imaging system, the TTL coupling can be reduced to below 0.6 nm throughout the tilt range, with the coupling coefficient decreasing to within ± 5 μm/rad.

Furthermore, within a small tilt range of ± 100 μrad, the slope can be reduced to below 3 μm/rad, meeting the preliminary requirements of the TianQin mission.


Funding
National Natural Science Foundation of China (12105375).National Key Research and Development Program of China (2022YFC2203901).

Acknowledgments
We thank members of Prof. xxx for useful discussions. The authors acknowledge the experimental facility support from the PGMF (National Precision Gravity Measurement Facility).

Disclosures
The authors declare no conflict of interest.

Data Availability Statement
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request. 

Supplemental document
See Supplement 1 for supporting content. 







