Dark Energy Spectroscopic Instrument DESI uses more than 2.4 million Emission Line Galaxies ELGs for 3D largescale structure LSS analyses in its Data Release 1 DR1. Such large statistics enable thorough research on systematic uncertainties. In this study we focus on spectroscopic systematics of ELGs. The redshift success rate frm goodz is the relative fraction of secure redshifts among all measurements. It depends on observing conditions thus introduces noncosmological variations to the LSS. We therefore develop the redshift failure weight wrm zfail and a perfibre correction etarm zfail to mitigate these dependences. They have minor influences on the galaxy clustering. For ELGs with a secure redshift there are two subtypes of systematics 1 catastrophics large that only occur in a few samples 2 redshift uncertainty small that exists for all samples. The catastrophics represent 0.26 of the total DR1 ELGs composed of the confusion between Otextscii and sky residuals double objects total catastrophics and others. We simulate the realistic 0.26 catastrophics of DR1 ELGs the hypothetical 1 catastrophics and the truncation of the contaminated 1.31ltzlt1.33 in the textscAbacusSummit ELG mocks. Their Pell show nonnegligible bias from the uncontaminated mocks. But their influences on the redshift space distortions RSD parameters are smaller than 0.2sigma. The redshift uncertainty of Yone ELGs is 8.5 kms with a Lorentzian profile. The code for implementing the catastrophics and redshift uncertainty on mocks can be found in httpsgithub.comJiaxiYumodellingspectrosys.
['Astrophysics - Cosmology and Nongalactic Astrophysics']
ELG Spectroscopic Systematics Analysis of the DESI Data Release 1
{'ELG Spectroscopic Systematics Analysis of the DESI Data Release 1': '- J. Yu , 1 A. J. Ross , 2 , 3 , 4 A. Rocher , 1 , 5 O. Alves, 6 A. de Mattia, 5\n- D. Forero-Sánchez , 1 J. Kneib, 1 , 7 A. Krolewski, 8 , 9 , 10 T.-W. Lan , 11\n- M. Rashkovetskyi , 12 J. Aguilar, 13 S. Ahlen , 14 S. Bailey , 13\n- D. Brooks, 15 E. Chaussidon , 13 T. Claybaugh, 13 A. de la Macorra , 16\n- Arjun Dey , 17 Biprateep Dey , 18 P. Doel, 15 K. Fanning , 19 , 20\n- J. E. Forero-Romero , 21 , 22 E. Gaztañaga, 23 , 24 , 25 S. Gontcho A\n- Gontcho , 13 K. Honscheid, 2 , 26 , 4 C. Howlett , 27 S. Juneau, 17\n- T. Kisner , 13 A. Kremin , 13 A. Lambert, 13 M. Landriau , 13\n- L. Le Guillou , 28 M. E. Levi , 13 M. Manera , 29 , 30 P. Martini , 2 , 3 , 4\n- A. Meisner , 17 R. Miquel, 31 , 30 J. Moustakas , 32 E. Mueller, 33\n- A. Muñoz-Gutiérrez, 16 A. D. Myers, 34 J. Nie , 35 G. Niz , 36 , 37\n- N. Palanque-Delabrouille , 5 , 13 W. J. Percival , 8 , 9 , 10 C. Poppett, 13 , 38 , 39\n- F. Prada , 40 M. Rezaie , 41 G. Rossi, 42 E. Sanchez , 43\n- E. F. Schlafly , 44 D. Schlegel, 13 M. Schubnell, 45 , 6 H. Seo , 46\n- D. Sprayberry, 17 G. Tarlé , 6 B. A. Weaver, 17 P. Zarrouk , 28\n- C. Zhao , 47 R. Zhou , 13 H. Zou 35  \nAffiliations are in Appendix C  \nE-mail: jiaxi.yu@epfl.ch  \nAbstract. Dark Energy Spectroscopic Instrument (DESI) uses more than 2.4 million Emission Line Galaxies (ELGs) for 3D large-scale structure (LSS) analyses in its Data Release 1 (DR1). Such large statistics enable thorough research on systematic uncertainties. In this study, we focus on spectroscopic systematics of ELGs. The redshift success rate ( 𝑓 goodz ) is the relative fraction of secure redshifts among all measurements. It depends on observing conditions, thus introduces non-cosmological variations to the LSS. We, therefore, develop the redshift failure weight ( 𝑤 zfail ) and a per-fibre correction ( 𝜂 zfail ) to mitigate these dependences. They have minor influences on the galaxy clustering. For ELGs with a secure redshift, there are two subtypes of systematics: 1) catastrophics (large) that only occur in a few samples; 2) redshift uncertainty (small) that exists for all samples. The catastrophics represent 0.26% of the total DR1 ELGs, composed of the confusion between [O /i.pc/i.pc] and sky residuals, double objects, total catastrophics and others. We simulate the realistic 0.26% catastrophics of DR1 ELGs, the hypothetical 1% catastrophics, and the truncation of the contaminated 1 . 31 < 𝑧 < 1 . 33 in the A/b.pc/a.pc/c.pc/u.pc/s.pcS/u.pc/m.pc/m.pc/i.pc/t.pc ELG mocks. Their 𝑃 ℓ show nonnegligible bias from the uncontaminated mocks. But their influences on the redshift space distortions (RSD) parameters are smaller than 0 . 2 𝜎 . The redshift uncertainty of DR1 ELGs is 8.5 km s -1 with a Lorentzian profile. The code for implementing the catastrophics and redshift uncertainty on mocks can be found in https://github.com/Jiaxi-Yu/modelling\\_spectro\\_sys .', 'Contents': '1  \nIntroduction  \nGalaxy redshift surveys probe the 3D LSS of the Universe by measuring the redshifts of millions of galaxies and quasars. Baryonic acoustic oscillations (BAO; [1]) and redshift-space distortions (RSD; [2]) encoded in the galaxy clustering reflect the properties of dark energy and dark matter. However, artefacts, or observational systematics, may introduce biases or reduce the precision of the cosmological measurements via BAO and RSD [e.g., 3-7]. Therefore, identifying, describing, and correcting any systematics that may exist in the cosmological measurements is essential for redshift surveys.  \nDark Energy Spectroscopic Instrument (DESI, 2021-2026) [8-10] is conducting the largest galaxy redshift survey to date with the 4-meter Mayall Telescope at Kitt Peak, Arizona, US. DESI  \nhas already observed thousands of square degrees of the sky using robotic-controlled fibres [11]. It is on track to obtain the spectra of ∼ 40 million galaxies and quasars over 14 , 000 deg 2 of the sky. Its early data release in June 2023 (EDR /one.sup , [12]) published the data collected during the survey validation phase [13] from December 2020 to May 2021, covering several hundred square degrees of sky. Its 5-year survey then started in May 2021, and the data collected during the first year of observations will be made available in the first data release (DR1 , [14]). DR1 uses 300,017 Bright Galaxies between redshift 0 . 1 < 𝑧 < 0 . 4 (BGS; [15]), 2,138,600 Luminous Red Galaxies between redshift 0 . 4 < 𝑧 < 1 . 1 (LRGs; [16]), 2,432,022 Emission Line Galaxies between redshift 0 . 8 < 𝑧 < 1 . 6 (ELG; [17]) and 1,223,170 quasars (QSO) between redshift 0 . 8 < 𝑧 < 3 . 5 [18] that are observed over more than 7400 deg 2 of the sky for cosmological analysis [19]. These dark matter tracers are grouped into LSS catalogues with different weights [20], aiming at providing unbiased clustering of galaxies, quasars, and the Ly 𝛼 forest [19, 21]. Accurate estimations of the full covariance matrices [22-26] and data blinding [27] are also crucial for the robustness of the cosmological results. The percent-level precision measurements of BAO are presented in [21, 28], where we show that the BAO precisions of the 1-year DESI observation outperform those from the two-decadal Sloan Sky Digital Survey (SDSS /two.sup ; [29]). The RSD measurements of DR1 will come very soon in [30]. The cosmological constraints on Ω 𝑚 , 𝐻 0 , Ω 𝐾 , 𝑀 𝜈 and dark energy equation of state informed by BAO are concluded in [31], presenting a 3 𝜎 detection of a 𝑤 0 𝑤 𝑎 CDMcosmology compared with the Λ CDM. The cosmological constraints from RSD and the measurements of primordial non-Gaussianity 𝑓 NL will be introduced in [32, 33].  \nFor DESI DR1, we consider three types of observational systematics that could influence the cosmological measurements: target selection [34], fibre assignment [35] and redshift measurements [36, 37]. DESI Legacy Imaging Survey [38] was built to provide galaxy and quasar candidates for spectroscopic observations. The Beĳing-Arizona Sky Survey (G, R bands), and the Mayall Z-band Legacy Survey (BASS/MzLS ; [39]) comprise the north galactic cap of the DESI footprint with declination larger than 32 . 375 · . The rest of the DESI footprint is covered by the Dark Energy Camera Legacy Survey (DECaLS; [40]) and the Dark Energy Survey (DES; [41]) observed with the Dark Energy Camera (DECam; [42]). The systematics analyses of the redshift measurements are thus split into BASS/MzLS and DECam parts due to the difference in their instruments. Imaging systematics are artefacts introduced from the target selection by, for example, the accuracy of the photometric information and the selection criteria. The modelling and corrections of imaging systematics are described in detail in [43-46]. Fibre assignment with proper designs ensures that the redshift survey will be completed on time but will also introduce unphysical variations in the observed galaxy clustering. [47-50] address the effects of it from different aspects. [51] and this paper characterize the impact of bad redshift measurements for all types of tracers. Specifically, this paper focuses on ELGs, the largest group of objects in the DESI survey, due to their unique role in probing the LSS in the star-forming epoch at 1 < 𝑧 < 2.  \nThe success rate of the redshift measurement ( 𝑓 goodz ) varies with observing conditions, reflecting the unphysical galaxy density variations brought by the instruments of DESI. Correcting these effects requires fair up- or down-weighting on galaxy samples with secure redshift measurements, thus avoiding potential impact on the LSS analysis. For those with secure redshift measurements, their redshift are not necessarily the true redshift. A small fraction of ELGs, e.g., ∼ 0.3% for eBOSS DR16 [5] and DESI preliminary studies [17, 52], have secure redshifts that are very different from its true redshift for various reasons. This may also introduce bias to the clustering measurements by significantly altering the LSS. In most cases, secure redshift measurements are close to its true  \nredshift, with statistical uncertainties. It is as small as 10 km s -1 for DESI ELGs [17, 53], resulting in negligible clustering impact on 𝑠 > 5 ℎ -1 Mpc. We will study all three aspects of the spectroscopic systematics for DR1 ELGs at 0 . 8 < 𝑧 < 1 . 6 and quantify their clustering and cosmological impacts.  \nThis paper is arranged as follows. We describe the DR1 data, especially the corrections on the redshift success rate, galaxy mocks, and covariance matrices involved in our systematics analysis in Section 2. The two-point clustering estimators and DESI cosmological pipeline will be introduced in Section 3. Section 4 presents the characterization and corrections of the redshift failures. The feature and impact of redshift catastrophics and uncertainty of ELGs with reliable redshift measurements will be introduced in Section 5. Finally, we conclude our findings in Section 6.  \nIn this work, we adopt a flat Λ CDM fiducial cosmology from the mean results of Planck [54] /b.pc/a.pc/s.pc/e.pc\\_/p.pc/l.pc/i.pc/k.pcHM\\_TTTEEE\\_/l.pc/o.pc/w.pc/l.pc\\_/l.pc/o.pc/w.pcE\\_/l.pc/e.pc/n.pc/s.pc/i.pc/n.pc/g.pc with 𝜔 b = 0 . 02237 , 𝜔 cdm = 0 . 12 , ℎ = 0 . 6736 , 𝐴 s = 2 . 083 · 10 -9 , 𝑛 s = 0 . 9649 , 𝜎 8 = 0 . 8079 , 𝑁 eff = 3 . 044 , " 𝑚 𝜈 = 0 . 06 eV (single massive neutrino). This is the fiducial DESI cosmology when converting the positions of objects to Cartesian coordinates in the two-point statistics computation, and also the cosmology of ELG mocks [55] built on A/b.pc/a.pc/c.pc/u.pc/s.pcS/u.pc/m.pc/m.pc/i.pc/t.pc simulations [56].', '2.1 DESI DR1 Data': 'The DR1 data of DESI is the first year of the DESI Main Survey (May 2021 to June 2022, [14]). Covering over 7,400 square degrees of the sky, DR1 includes more than 6 million spectra of galaxies and quasars with accurate redshift at 0 . 1 < 𝑧 < 3 . 5 for cosmological measurements with LSS. We focus on the spectroscopic systematics of ELG\\_LOPnotqso samples (ELG hereafter) in DESI DR1 [20]. This corresponds to ELG targets that have a high observational priority ( ELG\\_LOP with a surface number density ≈ 1940 deg -2 ) and are not part of QSO targets ( ≈ 5% of ELG\\_LOP ) [17]. We include three types of data products in our study: 1) the full\\_HPmapcut catalogue that includes all DR1 ELG targets with a good observing prerequisite and imaging properties [19]. It means that all observed ELG targets (not necessarily true ELGs) with good and failed redshift measurements at all redshifts are included. This catalogue is used to construct 𝑤 zfail (Section 4.1) and explore the possible improvement of it (Section 4.2). We refer to it as full catalogue afterwards; 2) the DR1 ELG LSS catalogues with ELGs at 0 . 8 < 𝑧 < 1 . 6 [20]. It includes the total observational systematics weight 𝑤 tot , assuming systematics are decomposable and all corrections are non-overlapping. 𝑤 tot is obtained as follows [19, 20]:  \n𝑤 tot = 𝑤 comp 𝑤 sys 𝑤 zfail , (2.1)  \nwhere 𝑤 comp is for the correction of the target completeness due to fibre assignment, 𝑤 sys is to correct the target density fluctuation existing in the BASS/MzLS and DECam imaging surveys (see [19] for more details), and 𝑤 zfail is for the failed-redshift systematics (see Section 4.1). The LSS catalogue of ELGs is used to quantify the clustering impact of the redshift failure weight 𝑤 zfail and its corrections; 3) spectroscopically confirmed ELGs from the One-Percent Survey of EDR. This dataset will be used to construct a catalogue of repeated observation (Section 2.1.2) for systematics studies in Section 5 and provide the visual inspection of catastrophics in Appendix A.', '2.1.1 Redshift Success Rate 𝑓 goodz': '𝑤 zfail are weights that account for missing objects in the observed LSS (i.e., not in LSS catalogues) due to failed redshift measurements. An ELG redshift measurement is classified as successful if it  \nmeets the requirement as follows [17]  \nlog 10 GLYPH<0> 𝑆 [ O/i.pc/i.pc ] GLYPH<1> + 0 . 2log 10 ( Δ 𝜒 2 ) > 0 . 9 , (2.2)  \nwhere ( 𝑆 [ O/i.pc/i.pc ] ) is the signal-to-noise ratio (SNR) of the [O /i.pc/i.pc] emission line fit, and Δ 𝜒 2 is the 𝜒 2 difference between the best-fit redshift and the second-best redshift in the R/e.pc/d.pc/r.pc/o.pc/c.pc/k.pc pipeline. This means that a reliable ELG redshift measurement should have either a high [O /i.pc/i.pc] SNR or a large Δ 𝜒 2 .  \nThe success rate of redshift measurement is the complement of the redshift failure rate. We always use the normalized redshift success rate 𝑓 goodz in this study, defined as  \n𝑓 goodz ( 𝑥 ) = 𝑁 goodz ( 𝑥 )/ 𝑁 obs ( 𝑥 ) " 𝑥 𝑁 goodz ( 𝑥 )/ " 𝑥 𝑁 obs ( 𝑥 ) , (2.3)  \nwhere 𝑥 represents different observing conditions (e.g., the effective observing time, the position on the focal plane). 𝑁 obs is the number of ELG targets observed in appropriate conditions with no instrumental issue, and 𝑁 goodz is a subsample of 𝑁 obs with good redshift measurements (Eq. (2.2)). For DR1 ELGs, " 𝑥 𝑁 goodz ( 𝑥 )/ " 𝑥 𝑁 obs ( 𝑥 ) = 72 . 6% [51].  \nWeremind our readers that we measure 𝑓 goodz for BASS/MzLS and DECam footprint separately. It means that all 𝑓 goodz studies and discussions are divided into these two areas (i.e., area selections are applied on both 𝑁 goodz and 𝑁 obs ). Additional selections (e.g., on redshift range as discussed in Figure 2) should be implemented on 𝑁 goodz ( 𝑥 ) only as they have reliable properties including redshift measurements.', '2.1.2 Repeated Observation Catalogue': 'The studies of redshift catastrophics and uncertainty are based on repeated observations of the same object. The DESI One-Percent Survey, part of the survey validation program [13], has a footprint that overlaps with the DR1 footprint. In DR1, the same object can be observed repeatedly on different nights. So, we select all repeated redshift measurements of ELG\\_LOPnotqso samples (DR1 ELGs hereafter) by cross-match these two data sets processed with the same versions of R/e.pc/d.pc/r.pc/o.pc/c.pc/k.pc [37], and we compute the redshift difference Δ 𝑣 = Δ 𝑧𝑐 /( 1 + 𝑧 ) iteratively among pairs. Studies based on these repeated observations represent the properties of DESI DR1 ELGs since they are fair subsamples by construction (see, e.g. Section 5.1)  \nThere are 115,160 pairs of repeated observation for DR1 ELGs, and 307 pairs (0.26%) of them have | Δ 𝑣 | > 1000 km s -1 , which are the catastrophics in our study. This is similar to the ELG catastrophics rate of the DESI survey validation data [17]. We will provide a detailed description of these samples in Section 5.1. Pairs with | Δ 𝑣 | < 1000 km s -1 will be used to study the redshift uncertainty (Section 5.4). Δ 𝑣 is roughly symmetric w.r.t. 0 as illustrated in Section 5.4.', '2.2 Galaxy Mocks and Covariance Matrices': "We employ galaxy mocks that mimic the DR1 ELG samples to assist our study of spectroscopic systematics. The model ELGs were generated by implementing a modified high-mass-quenched HOD model [55] on A/b.pc/a.pc/c.pc/u.pc/s.pcS/u.pc/m.pc/m.pc/i.pc/t.pc simulations [56] in a 2 ℎ -3 Gpc 3 box at 𝑧 = 1 . 100. Next, these model galaxies in simulation boxes were downsampled to the observed redshift distribution 𝑛 ( 𝑧 ) of DR1 ELG samples and truncated to a spherical shell that matches the footprint of the DESI DR1 survey. The survey-like model ELGs then went through the data reduction pipeline of real observations [48] (i.e., altmtl ) to select the 'observed' ELGs. The output mock ELG catalogue is similar to the full catalogue as mentioned in Section 2.1, and we will apply the modelled spectroscopic systematics (see Section 5) to this catalogue. Finally, mocks with and without systematics will be processed by the  \npipeline to generate LSS catalogues for data (data type 2 in Section 2.1) and to calculate the two-point statistics (see Section 3.1) for comparison. There are 25 realizations of A/b.pc/a.pc/c.pc/u.pc/s.pcS/u.pc/m.pc/m.pc/i.pc/t.pc simulations. Therefore, we will implement the systematics on all 25 realizations of the ELG mocks, and their averaged clustering will be used for cosmological tests to reduce the effect of cosmic variance.  \nThe cosmological measurements require accurate estimations of the overall cosmic variance, i.e. the full covariance matrices. We also need accurate covariance matrices to quantify the clustering and cosmological impact of spectroscopic systematics. In our study, we use analytical covariances created by /t.pc/h.pc/e.pc/c.pc/o.pc/v.pc /three.sup [24] in the Fourier space for the power spectrum multipoles, 𝑃 ℓ ( 𝑘 ) , which is based on C/o.pc/v.pc/a.pcPT /four.sup [57] and R/a.pc/s.pc/c.pc/a.pc/l.pcC /five.sup /six.sup in the configuration space for the two-point correlation function (2PCF) multipoles, 𝜉 ℓ ( 𝑠 ) [25, 58]. We direct the reader to [26] for a detailed study of the covariance matrices in DESI DR1.", '3.1 Two-point Statistics': 'Two-point statistics describe the probability of finding excess pairs of galaxies compared to a random distribution. The detailed description of the computation in DESI DR1 can be found in [19]. In this section, we provide a summary of their computational techniques.  \nThe distribution of galaxies from LSS catalogues has been corrected by 𝑤 tot to screen the impact of observational systematics. In addition, the evolution of the galaxy number density as redshift will introduce extra clustering variance at BAO scale [59]. We thus include the FKP weights 𝑤 FKP in the clustering measurement, defined as  \n𝑤 FKP = 1 1 + 𝑛 local ( 𝑧 ) 𝑃 0 , (3.1)  \nwhere 𝑛 local ( 𝑧 ) is the tracer average number density at redshift 𝑧 , and 𝑃 0 = 4000 ℎ -3 Mpc 3 for ELGs, is the amplitude of the observed power spectrum at 𝑘 0 ≈ 0 . 15 ℎ Mpc -1 . Therefore, we weight each galaxy with 𝑤 tot 𝑤 FKP for 2-point statistics measurements.  \nIn the configuration space, we measure the two-point correlation function 𝜉 (2PCF). It depends on the comoving distance 𝑠 between pairs of galaxies and 𝜇 , which is the cosine of the angle between the distance vector 𝒔 and the line-of-sight. For galaxies observed from redshift surveys, we use the Landy-Szalay estimator (LS) to calculate the 𝜉 [60]:  \n𝜉 LS = DD -2DR + RR RR , (3.2)  \nwhere DD, DR and RR represent the number of galaxy pairs identified in data-data catalogues, datarandom catalogues, and random-random catalogues at a given distance and angle ( 𝑠, 𝜇 ) normalized by their corresponding total number of pairs in these catalogues. 𝜉 ( 𝑠, 𝜇 ) can be decomposed by Legendre polynomials 𝐿 ℓ ( 𝜇 ) to obtain the multipoles of 𝜉 as  \n𝜉 ℓ ( 𝑠 ) = 2 ℓ + 1 2 ∫ 1 -1 𝜉 ( 𝑠, 𝜇 ) 𝐿 ℓ ( 𝜇 ) d 𝜇. (3.3)  \nWe compute the galaxy pairs for observations using the DESI package /p.pc/y.pc/c.pc/o.pc/r.pc/r.pc /seven.sup , a wrapper of the CORRFUNC package [61, 62], between 0-200 ℎ -1 Mpc in 200 linear bins and with 200 linear 𝜇 bins and regroup them to obtain the 𝜉 ℓ with Δ 𝑠 = 4 ℎ -1 Mpc for ℓ = 0 , 2 , 4. We use the comoving distance at 𝑠 ∈ [ 30 , 200 ] ℎ -1 Mpc for 𝜒 2 calculations (Section 3.2).  \nPower spectra 𝑃 ( 𝑘 ) are two-point statistics in Fourier space. Its estimator [63] is based on the weighted field [59]:  \n𝐹 ( 𝒓 ) = 𝑛 𝑑 ( 𝒓 ) -𝛼𝑛 𝑟 ( 𝒓 ) . (3.4)  \nwhere 𝒓 is the 3D Cartesian coordinate, 𝑛 𝑑 and 𝑛 𝑟 are the binned 𝑤 tot -weighted data and 𝑤 tot -weighted random catalogues on a grid of cell size 6 ℎ -1 Mpc, and 𝛼 = " 𝑁 𝑑 𝑖 = 1 𝑤 tot , i ( d ) / " 𝑁 𝑟 𝑖 = 1 𝑤 tot , i ( r ) rescales the mean density of the random catalogue to the mean data density. The power spectrum multipoles 𝑃 ℓ ( 𝑘 ) can be written as  \n𝑃 ℓ ( 𝑘 ) = 2 ℓ + 1 𝐴𝑁 𝑘 ∑︁ 𝒌 𝐹 0 ( 𝒌 ) 𝐹 ℓ (-𝒌 ) - SN ℓ (3.5)  \nwhere 𝒌 is the wavenumber between galaxy pairs and 𝑁 𝑘 the number of modes in a 𝒌 bin, and  \n𝐹 ℓ ( 𝒌 ) = ∑︁ 𝒓 𝐹 ( 𝒓 )L ℓ ( ˆ 𝒌 · ˆ 𝒓 ) 𝑒 𝑖 𝒌 · 𝒓 . (3.6)  \nwhere ˆ 𝒌 · ˆ 𝒓 is the cosine of the angle between the wavenumber vector 𝒌 and the line-of-sight ˆ 𝒓 . The shot-noise SN ℓ is non-zero for ℓ = 0 as  \nSN 0 = 1 𝐴 " 𝑁 𝑑 ∑︁ 𝑖 = 1 𝑤 2 tot , i ( d ) + 𝛼 2 𝑁 𝑟 ∑︁ 𝑖 = 1 𝑤 2 tot , i ( r ) # . (3.7)  \nThe normalization term 𝐴 = 𝛼 / 𝑑𝑉 " 𝑘 𝑛 𝑑,𝑘 𝑛 𝑟,𝑘 is summed over a cell with 𝑑𝑉 1 / 3 = 10 ℎ -1 Mpc in size. We compute our observed 𝑃 ℓ ( 𝑘 ) with /p.pc/y.pc/p.pc/o.pc/w.pc/e.pc/r.pc /eight.sup [64] from 0-0.4 ℎ Mpc -1 with Δ 𝑘 = 0 . 005 for ℓ = 0 , 2 , 4. The 𝜒 2 calculations include 𝑘 ∈ [ 0 . 02 , 0 . 35 ] ℎ Mpc -1 (Section 3.2).', '3.2 Quantifying the Clustering Differences': 'We use 𝜒 2 sys to describe the differences between the standard clustering and the clustering for mocks with spectroscopic systematics, defined as  \n𝜒 2 sys = ( 𝑭 ℓ, std -𝑭 ℓ, sys ) 𝑇 C -1 ( 𝑭 ℓ, std -𝑭 ℓ, sys ) , (3.8)  \nwhere 𝑭 denotes the vector composed of the two-point statistics multipoles. We calculate the 𝜒 2 sys values for 𝑭 ℓ = 0 , 2 , 4 = ( 𝜉 0 , 𝜉 2 , 𝜉 4 ) at 30 < 𝑠 < 200 ℎ -1 Mpc in the configuration space and 𝑭 ℓ = 0 , 2 , 4 = ( 𝑃 0 , 𝑃 2 , 𝑃 4 ) at 0 . 02 < 𝑧 < 0 . 35 ℎ Mpc -1 in Fourier space. 𝑭 ℓ, std represents the standard clustering measurement which is 1) ELGs from LSS catalogues with 𝑤 tot (Table 1 from Section 4) and 2) A/b.pc/a.pc/c.pc/u.pc/s.pcS/u.pc/m.pc/m.pc/i.pc/t.pc ELG mocks without catastrophics (Table 2 from Section 5). 𝑭 ℓ, sys , correspondingly, is the clustering of 1) ELGs with other redshift success corrections and 2) A/b.pc/a.pc/c.pc/u.pc/s.pcS/u.pc/m.pc/m.pc/i.pc/t.pc ELG mocks with different catastrophics. C -1 is the inverse of the analytical covariance matrices. √︃ 𝜒 2 sys estimates the maximum deviations the systematics can introduce to cosmological measurement rescaled by the cosmic variance. We think the effect of a type of systematics is negligible if √︃ 𝜒 2 sys < 1 (i.e., cosmological impacts < 1 𝜎 ). To avoid confusion, we use 𝜀 𝜉 and 𝜀 𝑃 for the 1 𝜎 error of the galaxy clustering in the configuration space and Fourier space, respectively in the following sections. They are the square root of the diagonal terms of the analytical covariance matrices 𝜀 𝑖 = √︁ 𝐶 𝑖,𝑖 .', '3.3 RSD Tests: ShapeFit and Full Modelling': 'We study the influence of ELG spectroscopic systematics on RSD measurements with full modelling methodandShapeFitcompression[65-68]. Thefull modelling of RSD is to generate model-dependent theoretical power spectra 𝑃 theory with linear power spectra 𝑃 lin provided directly by Boltzmann codes such as CLASS [69] and CAMB [70]. This algorithm is thus computationally expensive, since each sampling of cosmology requires the calculation of the accurate 𝑃 lin . ShapeFit compression, in contrast, is a more efficient, model-independent way of producing 𝑃 theory from 𝑃 \' lin . The 𝑃 \' lin of ShapeFit is parameterized as follows [71]  \n𝑃 \' lin ( 𝑘 ) = 𝑃 fid lin ( 𝑘 ) exp GLYPH<26> 𝑚 𝑎 tanh GLYPH<20> 𝑎 ln GLYPH<18> 𝑘 𝑘 𝑝 GLYPH<19> GLYPH<21> GLYPH<27> , (3.9)  \nwhere 𝑃 fid lin is the linear power spectra under the DESI fiducial cosmology, 𝑘 𝑝 = 𝜋 / 𝑟 𝑑 is the pivot scale and 𝑟 𝑑 is the sound horizon scale, 𝑎 = 0 . 6. 𝑚 is the free parameter to approximate the accurate linear power spectra. In our tests, we use /v.pc/e.pc/l.pc/o.pc/c.pc/i.pc/l.pc/e.pc/p.pc/t.pc/o.pc/r.pc/s.pc /nine.sup for both tests wrapped in /d.pc/e.pc/s.pc/i.pc/l.pc/i.pc/k.pc/e.pc /one.sup/zero.sup , the DESI cosmological pipeline. To speedup the fitting, we employ the emulator in /d.pc/e.pc/s.pc/i.pc/l.pc/i.pc/k.pc/e.pc based on the Taylor expansion . The cosmological parameters for full modelling are Θ = { ℎ, 𝜔 cdm , 𝜔 𝑏 , log ( 10 10 𝐴 𝑠 )} where 𝜔 ∗ = Ω ∗ ℎ 2 . For ShapeFit Θ = { 𝑞 iso , 𝑞 AP , 𝑑𝑓 , 𝑑𝑚 } , representing the isotropic and anisotropic BAO dilation (see [28] for detailed explanations), 𝑑𝑓 = 𝑓 / 𝑓 fid is the difference between the measured linear growth rate 𝑓 and the fiducial value, 𝑑𝑚 = 𝑚 -1.  \nTo take into account the observational geometry, we need to apply the window matrix 𝑊 ˜ ℓ,ℓ on 𝑃 ℓ, theory as ˜ 𝑃 ˜ ℓ, theory ( ˜ 𝑘 𝑖 ) = " 𝑗 𝑊 ˜ ℓ,ℓ ( ˜ 𝑘 𝑖 , 𝑘 𝑗 ) 𝑃 ℓ, theory ( 𝑘 𝑗 ) . Assuming a Gaussian likelihood 𝐿 ( Θ ) = exp (-𝜒 2 ( Θ )/ 2 ) , where 𝜒 2 is defined as  \n𝜒 2 = ( 𝑷 ℓ, mock -˜ 𝑷 ˜ ℓ, theory ) 𝑇 C -1 ( 𝑷 ℓ, mock -˜ 𝑷 ˜ ℓ, theory ) , (3.10)  \n𝑷 ℓ = 0 , 2 = ( 𝑃 0 , 𝑃 2 ) represent the data vector of systematics-uncontaminated and contaminated mocks, ˜ 𝑷 ℓ = 0 , 2 = ( ˜ 𝑃 0 , ˜ 𝑃 2 ) is the data vector of geometry-modulated theory. We consider 0 . 02 < 𝑘 < 0 . 2 ℎ Mpc -1 in the RSD fitting.', '4.1 The Redshift Failure Weight 𝑤 zfail': "Redshift failures are observed objects that do not have reliable redshift measurements. They are dropped from the LSS catalogue and thus may lead to an underestimation of galaxy density in the observed 3D map of the Universe. A straightforward way of making up for the absence of objects is to up-weigh the nearest object to the failed observation on the sky, i.e., count it as two objects when calculating the galaxy clustering. This simple weighting scheme was used in LRGs from the Baryon Oscillation Spectroscopic Survey (BOSS, 2008-2014; [72]) from SDSS-III [73]. It worked well as the failed redshift measurements were distributed randomly on the focal plane and only composed of less than 2% of the total observation [74].  \nHowever, when the failure rate rises to 10% or higher, as seen with LRGs from the extended Baryon Oscillation Spectroscopic Survey (eBOSS, 2014-2020; [75]) of SDSS-IV [76] and ELGs from eBOSS and DESI [5, 17], such a simple up-weighting approach becomes ineffective. This is because it assumes that the properties of galaxies with failed redshifts are similar to those of the successful  \nsamples, which is often too strong (see [17] for the example of ELGs). In such cases, the goal of correcting failed redshift measurements should be to restore the uniformity of the redshift success rate across varying observing conditions rather than striving for a 100% success rate. Similar to [77], we model the observed redshift success rate 𝑓 goodz as a function of observing conditions and use it to construct the 𝑤 zfail weight. To simplify the 𝑓 m goodz , we normalize the 𝑓 goodz as shown in Eq. (2.3).  \nThe primary observing condition influencing 𝑓 goodz is the observational squared SNR of ELG spectra TSNR2\\_ELG ( 𝑆 spec , obs hereafter), which is proportional to the effective observing time. 𝑆 spec , obs is defined as follows [36]  \n𝑆 spec , obs = ∑︁ 𝑖 𝑇 2 𝑖 ⟨( 𝛿𝐹 ) 2 ⟩ 𝑖 𝜎 -2 𝑖 , (4.1)  \nwhere 𝑇 is the calibration coefficient of photon-electron conversion measured from each observed spectrum, involving the product of throughput /one.sup/one.sup (the system's efficiency of collecting incoming photons) and the real exposure time. ⟨( 𝛿𝐹 ) 2 ⟩ is the squared spectral line flux averaged over all spectral templates. 𝜎 2 is the noise variance, including the instrumental noise and sky spectra. 𝑖 is the wavelength index of the whole spectrum. All else held constant, 𝑓 goodz should be monotonic with 𝑆 spec , obs .  \nIn addition to 𝑆 spec , obs , we find that 𝑓 goodz also varies as the redshift. This is because the sky emission lines appear at distinct wavelengths corresponding to distinct redshifts at which [O /i.pc/i.pc] emission can be observed (see Section 7.2 of [17] for example). The 𝑓 goodz at these redshifts are lower than the others because the noise level (the flux of sky-spectrum residuals) is high for [O /i.pc/i.pc] detections.  \nWe aim to obtain a redshift failure weight 𝑤 zfail that corrects the 𝑓 goodz dependences on 𝑆 spec , obs and redshift. This correction is part of the total weight 𝑤 tot in DR1 ELG LSS catalogue as introduced in Section 2.1. But 𝑤 zfail is constructed for all observed ELG targets, including those without reliable redshift measurements. Bearing this in mind, it is necessary to find quantities other than the 'redshift' itself to embody the challenges of [O /i.pc/i.pc] detection in the presence of sky emission lines.", '4.1.1 [O/i.pc/i.pc] Emission and Redshift Measurement': "First, we compute the median [O /i.pc/i.pc] flux ratio as a function of redshift, i.e., [ O/i.pc/i.pc ] 𝑅 ( 𝑧 ) . It is an empirical measure of how difficult it is to obtain a successful ELG redshift measurement as a function of redshift, thus critical in modelling 𝑓 goodz . We define it as  \n[ O/i.pc/i.pc ] 𝑅 ( 𝑧 ) = 𝐹 [ 𝑂 /i.pc/i.pc ] ( 𝑧 ) GLYPH<12> GLYPH<12> 𝑆 spec , obs <𝑇 ELG 𝐹 [ 𝑂 /i.pc/i.pc ] ( 𝑧 ) GLYPH<12> GLYPH<12> 𝑆 spec , obs >𝑇 ELG , (4.2)  \nwhere 𝐹 [ 𝑂 /i.pc/i.pc ] is the median [O /i.pc/i.pc] flux of ELGs with reliable redshift measurements (samples from 𝑁 goodz in Eq. (2.3)). 𝑇 ELG ≈ 123 is the median 𝑆 spec , obs of ELGs with appropriate observing conditions (samples from 𝑁 obs in Eq. (2.3)). [ O/i.pc/i.pc ] 𝑅 ( 𝑧 ) is relevant, as the dominant factor in the redshift success criteria Eq. (2.2) is the SNR of the observed [O /i.pc/i.pc] flux. The object must have a larger [O /i.pc/i.pc] flux to secure a good redshift measurement when the spectrum is noisier. We, therefore, observe a larger [ O/i.pc/i.pc ] 𝑅 ( 𝑧 ) at 𝑧 > 1 . 5 due to the interference of the sky emission lines to [O /i.pc/i.pc] measurements at that redshift range on the left panel of Figure 1. In DESI DR1 , we assume that [O /i.pc/i.pc] emitters with higher [O/i.pc/i.pc] flux are the same population as those with lower [O /i.pc/i.pc] flux, i.e., their clustering is consistent (e.g., see [78] for example). [ O/i.pc/i.pc ] 𝑅 ( 𝑧 ) is calculated at 𝑧 ∈ [ 0 . 8 , 1 . 6 ] , the clustering measurement range, with 𝑑𝑧 = 0 . 01 to have a balance between true features and noise.  \nFigure 1 . Left: [O/i.pc/i.pc] 𝑅 (the dashed line) and 𝜁 fac (the solid line) as a function of the redshift. 𝜁 fac ( 𝑧 ) (from spectra template) is a good representation of the observed [ O/i.pc/i.pc ] 𝑅 ( 𝑧 ) , describing the difficulty (the larger they are, the more difficult) of obtaining a good redshift measurement. Middle: The 20 slopes 𝑘 and their corresponding 𝜁 fac (the bottom x-label) and 𝜁 fac percentiles (the top x-label) for BASS/MzLS . We implement linear regression on the 𝑘 -𝜁 fac relation and the best-fit relation is shown in the dotted line and the text. Right: the same as the middle panel but for DECam data.  \n<!-- image -->  \nNext, we need to find a theoretical curve that embodies the characteristics of [ O/i.pc/i.pc ] 𝑅 ( 𝑧 ) at 𝑧 ∈ [ 0 . 8 , 1 . 6 ] to build 𝑓 m goodz . The theoretical squared SNR of ELG spectra 𝑆 spec , theo ( 𝑧 ) plays a role here, defined as  \n𝑆 spec , theo ( 𝑧 ) = ∑︁ 𝑖 𝐹 2 𝑒,𝑖 ( 𝑧 ) 𝜎 -2 𝐹,𝑖 , (4.3)  \nwhere 𝐹 𝑒 ( 𝑧 ) is the flux of emission lines /one.sup/two.sup at redshift 𝑧 averaged over 50 processed ELG templates. These template spectra get their continuum removed, and their emission lines are shifted to the wavelength corresponding to redshift 𝑧 for 𝐹 𝑒 ( 𝑧 ) calculation. 𝜎 -2 𝐹 is the inverse variance of the sky spectra obtained in a standard exposure and averaged over the 10 DESI spectrographs /one.sup/three.sup . 𝑖 is the wavelength index, and thus, the sum is performed over all DESI wavelength for 𝜆 ∈ [ 3600 , 9800 ] Å . Thus, this quantity estimates how an ELG target's cumulative squared SNR should change with its redshift 𝑧 . Numerically, we compute 𝑆 spec , theo ( 𝑧 ) with a redshift resolution of 𝑑𝑧 = 0 . 001 at redshift 𝑧 ∈ [ 0 , 1 . 6 ] to embody the influence of sky-residual spikes on the redshift measurement.  \nWe find that a simple transformation of 𝑆 spec , theo ( 𝑧 ) as follows:  \n𝜁 fac ( 𝑧 ) = ( 1 + 1500 / 𝑆 spec , theo ( 𝑧 ) -1 15 /( 𝑧 -1 ) , 0 < 𝑧 < 1 . 6 1 , otherwise , (4.4)  \nmatch both the general and fine features in [ O/i.pc/i.pc ] 𝑅 ( 𝑧 ) as shown in the left panel of Figure 1 despite small deviations. In this way, we assign a 𝜁 fac value to all ELGs with successful redshifts to represent the difficulty of obtaining such a good redshift measurement.", '4.1.2 The 𝑓 goodz Model and 𝑤 zfail Weight': 'Now we have 𝜁 fac ( 𝑧 ) and 𝑆 spec , obs for every ELG with proper observing conditions and we expect our 𝑓 m goodz to increase with these two quantities monotonically. In practice, we split these ELGs in 20 bins of 𝜁 fac , each with the same number of galaxies. In each 𝜁 fac bin 𝑖 , we fit a linear relationship  \n𝑘 𝑖 𝑆 spec , obs + 𝑏 𝑖 to 𝑓 goodz ,𝑖 in 10 evenly spaced bins of 80 < 𝑆 spec , obs < 200. All fits assume Poissonian errors for 𝑓 goodz ,𝑖 , i.e., 𝜖 𝑓 = √︁ 𝑁 goodz / 𝑁 obs .  \nWe then take the 20 slopes 𝑘 𝑖 and their corresponding median 𝜁 fac , i and perform a least-squares linear fit (errors are assumed to be the same given the number of galaxies was the same for each 𝑘 𝑖 calculation) to obtain  \n𝑘 = 𝐶 1 𝜁 fac + 𝐶 0 . (4.5)  \nThis process is repeated separately for the BASS/MzLS and DECam regions, and the results, including the best-fit lines and parameters, are shown in the middle and right panel of Figure 1. Note that the largest 90% 𝜁 fac , though having significantly larger 𝑘 𝑖 , agrees with the 𝑘 -𝜁 fac trend of the smaller 𝜁 fac . Therefore, we do not implement more complicated regression and stability checks of these two 𝑘 -𝜁 fac relations in this study. We will discuss the consequence of this choice in Section 4.3.  \nFinally, we convert the parameterized slope to a weight that can be attached to the galaxy catalogues to remove any trends between 𝑓 goodz and ( 𝑆 spec , obs , 𝑧 ). We expect the redshift success should only increase as a function of 𝑆 spec , obs . Thus, we clip 𝑘 such that it has a minimum value of 0:  \n𝑘 ( 𝜁 fac ) = ( 𝐶 1 𝜁 fac + 𝐶 0 , 𝑘 > 0 0 , otherwise . (4.6)  \nAssuming the intercept 𝑏 = 1 -𝑘𝑇 ELG , with 𝑇 ELG being the median 𝑆 spec , obs of all ELG targets with appropriate observing conditions, we now have a 𝑓 m goodz as a function of redshift 𝑧 and 𝑆 spec , obs  \n𝑓 m goodz ( 𝑆 spec , obs , 𝑧 ) = 𝑘 ( 𝜁 fac [ 𝑧 ])( 𝑆 spec , obs -𝑇 ELG ) + 1 . (4.7)  \nWe simply use the inverse of this for the weight to attach to the galaxy catalog  \n𝑤 zfail ( 𝑆 spec , obs , 𝑧 ) = 1 / 𝑓 m goodz ( 𝑆 spec , obs , 𝑧 ) . (4.8)  \nFigure 2 presents the 𝑓 goodz -𝑆 spec , obs relation before and after 𝑤 zfail correction for samples from BASS/MzLS and DECam surveys. ELGs with good redshifts are split into finer bins with 𝑑𝑧 = 0 . 1 at 0 . 8 < 𝑧 < 1 . 6 for 𝑓 goodz computation. The errors are binomial errors 𝜖 zfail  \n𝜖 zfail = √︁ 𝑁 obs ( 1 -𝑁 goodz / 𝑁 obs ) 𝑁 obs (4.9)  \nand their 𝜒 2 zfail = ( 𝑓 goodz -1 ) 2 / 𝜖 2 zfail represents the deviation from a uniform 𝑓 goodz -𝑆 spec , obs relation. The mean values of 𝑓 goodz fluctuate a bit, but all relations are consistent with monotonic tendencies. The 𝑓 goodz -𝑆 spec , obs dependency in both footprints is close to unity before the 𝑤 zfail correction, except for the ∼ 10%-level variations at 1 . 3 < 𝑧 < 1 . 4 and the ∼ 30%-trend at 1 . 5 < 𝑧 < 1 . 6. They correspond to the big spikes of [ O/i.pc/i.pc ] 𝑅 ( 𝑧 ) and 𝜁 fac in the left panel of Figure 1. Therefore, 𝑤 zfail based on 𝜁 fac suppress the dependences of 𝑓 goodz on 𝑆 spec , obs at 1 . 2 < 𝑧 < 1 . 6, especially on 1 . 5 < 𝑧 < 1 . 6 given the decreasing 𝜒 2 zfail . Note that the 𝜒 2 zfail / dof values can still be as large as 3. But we do not worry about it as they represent a sub-percent 𝑓 goodz difference, leading to a minor impact on the clustering as presented in Section 4.3.', '4.2 The Focal Plane Correction 𝜂 zfail': "The 𝑓 goodz not only depends on ( 𝑆 spec , obs , 𝑧 ) . It also varies across the focal plane (e.g., [5]). But 𝑆 spec , obs is defined for each exposure, without information on the focal plane. The definition of 𝜁 fac used  \nFigure 2 . The 𝑓 goodz𝑆 spec , obs relation before (shades) and after (error bars with dots) 𝑤 zfail correction in fine redshift bins at 0 . 8 < 𝑧 < 1 . 6 with 𝑑𝑧 = 0 . 1. BASS/MzLS and DECam results are presented in red/closed circles and blue/open circles, respectively. 𝑤 zfail removes 𝑓 goodz dependences on 𝑆 spec , obs at 𝑧 > 1 . 2, especially at 1 . 5 < 𝑧 < 1 . 6, where the sky emissions are big issues for ELGs by construction.  \n<!-- image -->  \nquantities that are averaged over spectrographs, eliminating the information as well. Consequently, 𝑤 zfail does not remove the 𝑓 goodz variations on the focal plane as shown in Figure 3. The left and middle panel shows a 10% 𝑓 goodz variation across the focal plane, resulting in 𝜒 2 zfail / dof = 4753 / 4262 for BASS/MzLS and 𝜒 2 zfail / dof = 4716 / 4262 for DECam where 4262 is the number of fibres. Note that the median number of good redshift measurements per fibre for BASS/MzLS survey is 101, and that for DECam is 575. The smaller statistics of BASS/MzLS lead to larger 𝑓 goodz variations and error bars, thus similar 𝜒 2 zfail to DECam . Marginalising over the angular direction, the 𝑓 goodz of fibres that are close to the focal centre shows 1% higher 𝑓 goodz than that of the distant fibres as shown in the right panel of Figure 3. This difference still exists with 𝑤 zfail corrections despite the fact that it is already close to the ideal case of a uniform distribution. Therefore, we need an extra correction factor on 𝑤 zfail as a function of its position on the focal plane (or a fibre-dependent correction).  \nThe correction of 𝑤 zfail is defined fiber-wise as:  \n𝜂 zfail , i = 𝑓 goodz ( 𝑖 )/ 𝑓 goodz ( 𝑖 ) , (4.10)  \nwhere 𝑖 is the fibre ID ranging from 0 to 4999, but only 4262 of them provide 𝑓 goodz measurements. The other 738 fibres are not used for scientific observations /one.sup/four.sup . 𝑓 goodz ( 𝑖 ) is the 𝑓 goodz of fibre 𝑖 and 𝑓 goodz ( 𝑖 ) is the mean 𝑓 goodz of all samples. 𝜂 zfail null all 𝑓 goodz on the focal plane by definition, and thus, the 𝑓 goodz dependence on the distance to the focal centre is removed in both the BASS/MzLS and DECam footprints. This fibre-wise correction is then implemented on each observed ELG target, and thus, the new total weight 𝑤 ' tot is written as  \n𝑤 ' tot = 𝑤 comp 𝑤 sys ( 𝑤 zfail 𝜂 zfail ) , (4.11)  \nwhere 𝑤 zfail 𝜂 zfail is the corrected redshift failure weight.  \nFigure 3 . Left: The 𝑓 goodz (with 𝑤 zfail) of each fibre distributed on the focal plane for BASS/MzLS survey. The 𝑓 goodz differences on the focal plane can be as large as 10%. Middle: the same as the left panel but for DECam survey. Its 𝑓 goodz variations are smaller than that of BASS/MzLS survey. Right: 𝑓 goodz as a function of the distance to the focal centre without (shades) and with (error bars) 𝑤 zfail corrections. Red/closed circles and blue/open circles represents results from BASS/MzLS and DECam respectively. The 𝑓 goodz variations on the focal plane are independent of ( 𝑆 spec , obs , 𝑧 ) , thus 𝑤 zfail does not eliminate this dependence.  \n<!-- image -->  \nFigure 4 . The 𝑓 goodz dependencies on 𝑆 spec , obs (the first column) and on averaged ratios of spectroscopic and imaging flux of standard stars in G (the second column), R (the third column) and Z (the fourth column) bands without 𝑤 zfail (lines with shades), with 𝑤 zfail (closed error bars) and with 𝑤 zfail 𝜂 zfail (open error bars) for BASS/MzLS regions (the upper panel) and DECam regions (the lower panel). The remaining 𝑓 goodz trends not corrected by 𝑤 zfail were removed by 𝑤 zfail 𝜂 zfail.  \n<!-- image -->", '4.3 Performances of 𝑤 zfail and 𝜂 zfail': "We have proved in Section 4.1 and Section 4.2 that 𝑤 zfail and 𝜂 zfail can respectively correct the 𝑓 goodz dependences on 𝑆 spec , obs in fine redshift bins and the focal plane by construction. In this section, we will check their performance of removing other 𝑓 goodz trends and their influences on galaxy clustering.  \nIn the first column of Figure 4, the dependence of 𝑓 goodz on 𝑆 spec , obs for ELGs targets without redshift selection is around 5% (shades) and 𝑤 zfail suppresses the trend down to 1% (closed error bars). The improvements are also significant, as shown in the decreasing 𝜒 2 zfail values in both BASS/MzLS and DECam regions. Therefore, 𝑤 zfail is included in the ELG LSS catalogue as part of the correction of observational systematics (Eq. (2.1)). The over-suppression of BASS/MzLS 𝑓 goodz for those with large 𝑆 spec , obs ( ∼ 𝑆 spec , theo ) probably come from the imperfect linear regression on small 𝜁 fac values as shown in the middle panel of Figure 1. The open circles in the same subplots illustrate the effects  \nFigure 5 . The comparison of 2PCF monopole (first column), quadrupole (second column) and hexadecapole (third column) for ELGs implementing the standard weighting scheme 𝑤 tot (' 𝑤 zfail', solid lines), the 𝑤 tot but excluding 𝑤 zfail ('no 𝑤 zfail', dashed lines) and the new total weight 𝑤 ' tot (' 𝑤 zfail 𝜂 zfail', dash-dotted lines) at 20 < 𝑠 < 200 ℎ -1 Mpc. The first row is for ELGs at 0 . 8 < 𝑧 < 1 . 1, and the second is for 1 . 1 < 𝑧 < 1 . 6. The clustering influences of redshift failure weights are minor, with a difference of < 0 . 05 𝜀 𝜉 .  \n<!-- image -->  \nof 𝜂 zfail 𝑤 zfail on the same 𝑓 goodz trends as in Figure 4. This extra correction brings back missing information on the focal plane by 𝑤 zfail and, therefore, improves 𝑓 goodz weighting in small 𝑆 spec , obs . This agrees with our conclusion that the source of the over-correction of 𝑓 goodz in BASS/MzLS area is from the 𝑤 zfail modelling, not from any uncorrected 𝑓 goodz trend on the focal plane. We also notice that 𝜁 fac is not a perfect representation of [ O/i.pc/i.pc ] 𝑅 ( 𝑧 ) . We will develop a better 𝑓 m goodz for future data release as these aspects a minor effect on the clustering.  \nThe second to fourth panels of Figure 4 illustrates the 𝑓 goodz dependence on the spectrographto-imaging flux ratio of standard stars at 𝜆 ∈ [ 4500 , 5500 ] Å (G band), 𝜆 ∈ [ 6000 , 7300 ] Å (R band) and 𝜆 ∈ [ 8500 , 9800 ] Å (Z band). They represent the fibre aperture loss in the central wavelength of these three bands [36]. As the definition of 𝑆 spec , obs includes the throughput of the fibre (Eq. (4.1)), we expect that 𝑓 goodz varies with throughputs, and 𝑤 zfail can also remove most of the 𝑓 goodz dependences on throughputs. The fibre aperture loss in R and Z bands present a more significant trend than in G band as the [O /i.pc/i.pc] emissions of DESI ELG targets are observed in R and Z bands. But the 𝑓 goodz dependences on the throughputs remain after applying 𝑤 zfail as shown in Figure 4. This is consistent with our finding in Figure 3 that 𝑤 zfail by construction does not include the modelling of 𝑓 goodz variations on the focal plane. The new weight, 𝜂 zfail 𝑤 zfail , nullifies the 𝑓 goodz dependences on the throughputs by construction. This is because 𝜂 zfail , defined per-fibre, eliminates the 2D 𝑓 goodz variations on the focal plane. Consequently, any quantities defined based on fibres will have a unity mean 𝑓 goodz after applying 𝜂 zfail . However, such an ad hoc correction might have the risk of overfitting the DR1 , so it is not part of the LSS weight budget. We may consider adding it to 𝑤 tot in the future data release.  \nWe then study the effects of 𝑤 zfail and 𝑤 zfail 𝜂 zfail on galaxy clustering. To do this, we calculate the 2PCF 𝜉 ℓ ( 𝑠 ) and power spectra 𝑃 ℓ ( 𝑘 ) of ELGs from the LSS catalogues at 0 . 8 < 𝑧 < 1 . 1 and 1 . 1 < 𝑧 < 1 . 6 (redshift bins for LSS analyses). The galaxy weight of ELGs is the product of 𝑤 FKP and the total observational weight in three formats: 𝑤 tot (Eq. (2.1)), 𝑤 tot / 𝑤 zfail (no corrections on the failed redshift measurements) and 𝑤 ' tot (Eq. (4.11), the complete LSS weights with the new redshift failure weight). They are denoted as ' 𝑤 zfail ', 'no 𝑤 zfail ' and ' 𝑤 zfail 𝜂 zfail ' respectively in Figure 5 and Figure 6. These two figures present the clustering effects of the redshift failure weights compared to the standard 𝑤 tot weight in the configuration space and Fourier space. Both weights have < 0 . 05 𝜀 influence on all two-point statistics multipoles at both redshift bins (see Section 3.2 for the definition of the 1𝜎 clustering error 𝜀 ). Such a minor effect of the redshift failure weights has been observed  \nFigure 6 . Similar to Figure 5 but for power spectra multipoles 𝑃 ℓ ( 𝑘 ) at 0 . 02 < 𝑘 < 0 . 35 ℎ Mpc -1 . The clustering influences of redshift failure weights in the Fourier space are as small as those in the configuration space.  \n<!-- image -->  \nTable 1 . The 𝜒 2 sys values for the ELG clustering calculated with different weights compared to the standard clustering with the 𝑤 tot weight in both configuration space and Fourier space for ℓ = 0 , 2 , 4. The third column shows ELGs implementing 𝑤 tot but excluding 𝑤 zfail, and the fourth column is for ELGs applying 𝑤 ' tot (Eq. (4.11)), which includes the corrected failure weight 𝑤 zfail 𝜂 zfail. In both configuration and Fourier spaces, the influences of the redshift failure weight and its correction are minor regardless of the redshift range.  \nin eBOSS ELGs as well [5]. 𝑤 zfail leads to a larger clustering impact at 1 . 1 < 𝑧 < 1 . 6 compared to the clustering at 0 . 8 < 𝑧 < 1 . 1. This is implied in Figure 2 as the significant corrections that 𝑤 zfail accomplished are all at 1 . 1 < 𝑧 < 1 . 6. The effect of 𝑤 zfail 𝜂 zfail on redshift ranges and multipoles are all comparable, showing the impact of the 2D 𝜂 zfail correction is evenly applied to all samples.  \nTable 1 provides the 𝜒 2 sys values of ELG clustering computed without 𝑤 zfail and with the new total weight 𝑤 ' tot (Eq. (4.11)) compared to the standard clustering computed with 𝑤 tot . These values are no larger than 0.32. So, we conclude that the cosmological impact of 𝑤 zfail and its correction 𝜂 zfail can be neglected. Nevertheless, it is important to note that 𝑓 goodz not only depends on the effective exposure time 𝑆 spec , obs and redshift 𝑧 . 𝑓 goodz of each fibre also varies on the focal plane, independent of the variations brought by ( 𝑆 spec , obs , 𝑧 ). Therefore, future surveys must check the dependence of 𝑓 goodz on different observing conditions and their correlation to develop a complete weight to correct the failed redshift measurements.", '5 Redshift Catastrophics and Redshift Uncertainty': "Catastrophics and redshift uncertainty are systematics hidden in ELG samples with secure redshift measurements (Eq. (2.2)). With Δ 𝑣 = Δ 𝑧𝑐 /( 1 + 𝑧 ) from repeated observations (Section 2.1.2), the catastrophics refer to pairs of redshift with a large | Δ 𝑣 | . There are several definitions of 'a large | Δ 𝑣 | ', which are | Δ 𝑣 | > 1000 km s -1 [17, 75] (our definition), Δ 𝑣 > 5 √︁ ⟨ Δ 𝑣 2 ⟩ [79], or those with clear evidence of misidentification of emission lines or between the emission line and the sky residuals  \n- [3]. All these definitions mean that the radial position of a small number of tracers is shifted to a large extent, leading to imprints on the clustering. ELGs from DESI EDR and SDSS-IV eBOSS have 𝑓 catas ∼ 0 . 3% [5, 17]. Given such a small fraction, its clustering and cosmological impacts are neglected in eBOSS analyses.  \nRedshift uncertainty comes from redshift measurement and thus exists in every galaxy in LSS catalogues. Repeated observations without the catastrophics samples provide a statistical estimation via the width of the Δ 𝑣 distribution (e.g. [17, 80, 81] from SDSS-IV eBOSS). Note that repeated observations cannot capture the velocity shift of QSOs, which is part of QSO redshift uncertainty. But we can study this type of mixed uncertainty via their clustering effects (see [53, 82]). For galaxy samples, we assume statistical uncertainty represents the redshift uncertainty. The redshift uncertainty of DESI ELGs from EDR is ∼ 8 km s -1 , leading to little impact on the clustering [17, 53].  \nGiven their properties, neither catastrophics nor redshift uncertainty can be corrected target-wise like 𝑤 zfail but they influence the clustering collectively. Catastrophics of line emitters (i.e., ELGs) are poorly understood due to the limited statistics and small fraction as mentioned above. Now, with millions of reliable ELG samples from DESI DR1 [14] and well-modelled, realistic ELG mocks [55], we have enough good-quality samples to study the effect of catastrophics carefully. DESI DR1 have fewer samples with long exposure compared to EDR [12] to guarantee better redshift measurements. Therefore, we will revisit these two aspects in DESI DR1 ELGs, understand the data, model them with galaxy mocks, and see how to prevent them from biasing the cosmological measurements.", '5.1 Characterising the Catastrophics': "Catastrophics are discrepancies in the redshift measurements for the same object. As reported in Section 2.1.2, 0.26% of the repeated ELG pairs have | Δ 𝑣 | > 1000 km s -1 . There is no preference for their locations in the focal plane, and their redshift distributions are shown in Figure 7. The upper panel of Figure 7 compares the true redshift ( 𝑧 true ) and the catastrophics redshift ( 𝑧 catas ) for the ELG catastrophics detected with the repeated observation catalogue. We determine the 𝑧 true and 𝑧 catas via visual inspection of their reduced spectra with the help of prospect /one.sup/five.sup [52, 83]. If a redshift measurement matches secure spectral features (e.g., the [O /i.pc/i.pc] doublets or multiple emission lines), this redshift is 𝑧 true and the other measurement on the same object is 𝑧 catas . There are five types of catastrophics for ELGs shown in Figure 7 (see Appendix A for examples of their spectra):  \n- · The misidentification of the residuals of sky emissions at around 8600-8700 Å to be the [O /i.pc/i.pc] emission (sky confusion hereafter). They form a prominent feature in Figure 7 at 𝑧 catas ≈ 1 . 32 as shown in crosses, comprising 26.7% of the total catastrophics samples. We observe a significant doublet-like spike in one of their spectra at 8600-8700Å, which is difficult to smooth out and, therefore, is identified as the [O /i.pc/i.pc] emission. The evolving DESI redshift pipeline might resolve this issue by improving the sky subtraction and line-identification process [36]. There is no such feature in the other spectrum of these repetitively observed objects, and thus, the redshift obtained by the other spectrum is correct ( 𝑧 true ).\n- · The misidentification of the sky residuals at other wavelengths as the [O /i.pc/i.pc] emission. The sky emissions/residuals appear at 𝜆 > 8000 Å , corresponding to 𝑧 catas > 1 . 1. As the [O /i.pc/i.pc] flux of ELGs at 𝑧 true < 1 . 1 is systematically lower than the sky residuals (see Figure 15 of [17]), this type of catastrophics tends to increase the redshift of ELGs, i.e., from 𝑧 true < 1 . 1 to 𝑧 catas > 1 . 1. They make up 27.6% of the total catastrophic pairs.  \nFigure 7 . Top: The comparison between the true redshift and the catastrophics redshift for catastrophics of DESI ELGs. The exceptions are spectra that contain two objects (triangles), which means that both redshifts are true redshifts, and spectra that have no emission lines for redshift determination (squares), which means that both redshifts are catastrophically failed redshifts. The cross, small dots represent the redshift misidentification of the sky residuals to be the [O /i.pc/i.pc] emission that concentrates on 𝑧 catas ≈ 1 . 32 (the horizontal dotted line), the same misidentification of sky residuals at other wavelengths. Stars are all other types of catastrophics. The dashed lines in 𝑧 = 0 . 8 and 𝑧 = 1 . 6 indicate the redshift range of ELGs for clustering measurements. Bottom: The (true) redshift distribution of the full catalogue (grey shades), the ELG A/b.pc/a.pc/c.pc/u.pc/s.pcS/u.pc/m.pc/m.pc/i.pc/t.pc mocks (solid line), the catastrophics without 𝑧 catas ≈ 1 . 32 (step histogram with error bars only), and the catastrophics with 𝑧 catas ≈ 1 . 32 (step histogram with dotted error bars). The error bars of both catastrophics samples are assumed to be binomial errors. The redshift distribution of two types of catastrophics roughly follows the full catalogue.  \n<!-- image -->  \n- · Two objects in the spectra. The fibre might capture two overlapping objects with different redshifts, or the ELG spectra were contaminated by a bright, nearby object ( 𝑧 catas or 𝑧 true around 0). As there are two sets of spectral features and thus true redshifts, the redshift pipeline can take either as its output redshift. These samples are mainly at 0 . 6 < 𝑧 < 0 . 9 and comprise 22.0% of the total catastrophics.\n- · Total catastrophics. There are 12.1% catastrophics that do not have a correct redshift measurement at all. This is because the [O /i.pc/i.pc] emission is too faint to be found (by the R/e.pc/d.pc/r.pc/o.pc/c.pc/k.pc pipeline and visual inspection). Therefore, R/e.pc/d.pc/r.pc/o.pc/c.pc/k.pc can identify any of sky residuals as [O /i.pc/i.pc] doublets. Their 𝑧 true and 𝑧 catas concentrate at 𝑧 > 1 . 1 because most of the sky residuals are at 𝜆 > 8000 Å as mentioned above.\n- · Other catastrophics. The remaining 11.6% catastrophics include line confusion between [O /i.pc/i.pc] doublets and other emission lines such as [O /i.pc/i.pc/i.pc] and H 𝛼 , bad spectra, and QSO spectra misclassified as ELGs.  \nAlthough there are five patterns of catastrophics, we only need to model the excess sky confusion and all the other patterns (random catastrophics hereafter). The random catastrophics alone cannot reproduce the distinct feature of 𝑧 catas ≈ 1 . 32. Random catastrophics with additional sky confusions should represent more general catastrophics while avoiding overfitting our specific sample.  \nThe 𝑧 catas distribution of the sky confusion (left) and the log 10 (| Δ 𝑣 |) ( ˆ Δ 𝑣 hereafter) distribution of random catastrophics (right) are presented in Figure 8. We fit the excess sky confusion with a Gaussian distribution N( 𝜇 sky , 𝜎 2 sky ) where 𝜇 sky = 1 . 32 and 𝜎 sky = 0 . 006, enclosing 24.9% of the total catastrophics. This Gaussian function is shifted up by 0.61, representing the remaining 1.9% sky confusion that has been included in random catastrophics. The ˆ Δ 𝑣 distribution of the random catastrophics can be fitted with a Gaussian profile, a Lorentzian profile and a log-normal profile with an extra free parameter LN( 𝜇 ran , 𝜎 2 ran , ˆ 𝑣 0 ; ˆ Δ 𝑣 ) which is defined as  \nLN( 𝜇 ran , 𝜎 2 ran , ˆ 𝑣 0 ; ˆ Δ 𝑣 ) = 𝐴 √ 2 𝜋𝜎 ran (-ˆ Δ 𝑣 + ˆ 𝑣 0 ) exp n -h ln GLYPH<16> -ˆ Δ 𝑣 + ˆ 𝑣 0 GLYPH<17> -𝜇 ran i 2 2 𝜎 2 ran o . (5.1)  \nAssuming a binomial error, the log-normal profile provides a much better fitting to the ˆ Δ 𝑣 distribution of random catastrophics. Its best-fitting parameters are 𝐴 = 0 . 99 , 𝜇 ran = 0 . 64 , 𝜎 ran = 0 . 25 , ˆ 𝑣 0 = 6 . 62.  \nFigure 8 . Left: The zoom-in histogram of 𝑧 catas of the sky confusion samples at 1 . 25 < 𝑧 < 1 . 35 (empty histogram). The solid line is the best-fitting Gaussian function N( 1 . 32 , 0 . 006 2 ) with an offset of 0.61. They compose of 24.9% of catastrophics. Right: The histogram of ˆ Δ 𝑣 of the random catastrophics (empty histogram with binomial errors). It is fitted by a Gaussian profile ('G', the dotted line), a Lorentzian profile ('L', the dashed line), and a log-normal profile ('lnG', the solid line). The 𝜒 2 improvement brought by the extra 1 parameter of the log-normal distribution is significant. Therefore, LN( 0 . 64 , 0 . 25 2 , 6 . 62 ) is the best description of the random catastrophics ˆ Δ 𝑣 .  \n<!-- image -->", '5.2 Modelling the Catastrophics': 'The 𝑧 true of sky confusion and random catastrophics both follow the 𝑛 ( 𝑧 ) of the ELG targets ( full catalogue) as illustrated in Figure 7. Therefore, we randomly take 0.26% galaxies from the A/b.pc/a.pc/c.pc/u.pc/s.pcS/u.pc/m.pc/m.pc/i.pc/t.pc mocks and implement the following pattern to reproduce the realistic catastrophics  \n𝑧 catas = ( 𝑧 true + Δ 𝑣 ran , random catastrophics, 76%; N( 1 . 32 , 0 . 006 2 ) , extra sky confusion, 24% , (5.2)  \nwhere 𝑧 true is the redshift of A/b.pc/a.pc/c.pc/u.pc/s.pcS/u.pc/m.pc/m.pc/i.pc/t.pc galaxy mocks. Half of Δ 𝑣 ran is positive and the other half is negative, and | Δ 𝑣 ran | follows LN( 0 . 64 , 0 . 25 2 , 6 . 62 ) . We also realise a 1% random catastrophics on A/b.pc/a.pc/c.pc/u.pc/s.pcS/u.pc/m.pc/m.pc/i.pc/t.pc mocks for demonstration purposes. 1% is an upper limit of the ELG catastrophics rate in eBOSS and DESI [17, 75]. In addition, we remove A/b.pc/a.pc/c.pc/u.pc/s.pcS/u.pc/m.pc/m.pc/i.pc/t.pc ELGs at 1 . 31 < 𝑧 < 1 . 33, which is the easiest implementation on data if we want to avoid the impact on the sky confusion at 𝑧 ≈ 1 . 32. Given a similar 𝑛 local ( 𝑧 ) after implementing the catastrophics, the 𝑤 FKP of samples with 𝑧 catas is obtained with the 𝑛 local ( 𝑧 catas ) . All the patterns are implemented on 25 realizations of A/b.pc/a.pc/c.pc/u.pc/s.pcS/u.pc/m.pc/m.pc/i.pc/t.pc galaxy mocks.  \nFigure 9 . The 2PCF monopole (first column), quadrupole (second column) and hexadecalpole (third column) of model galaxies from the standard A/b.pc/a.pc/c.pc/u.pc/s.pcS/u.pc/m.pc/m.pc/i.pc/t.pc mocks (solid lines), A/b.pc/a.pc/c.pc/u.pc/s.pcS/u.pc/m.pc/m.pc/i.pc/t.pc mocks with 0.26% realistic catastrophics (dashed lines), A/b.pc/a.pc/c.pc/u.pc/s.pcS/u.pc/m.pc/m.pc/i.pc/t.pc mocks with hypothetical 1% random catastrophics (dashdotted lines) and A/b.pc/a.pc/c.pc/u.pc/s.pcS/u.pc/m.pc/m.pc/i.pc/t.pc mocks without 1 . 31 < 𝑧 < 1 . 33 (dotted lines). The grey areas are the 1𝜀 𝜉 errors of clustering obtained via analytical covariances (Section 2.2). The first and the second rows are the 2PCF multipoles and the differences between the standard A/b.pc/a.pc/c.pc/u.pc/s.pcS/u.pc/m.pc/m.pc/i.pc/t.pc mocks and catastrophics/cuts rescaled by the errors provided by the analytical covariance matrices at 0 . 8 < 𝑧 < 1 . 1. The third and fourth rows are similar but model galaxies at 1 . 1 < 𝑧 < 1 . 6. Catastrophics lead to negligible clustering effects at 0 . 8 < 𝑧 < 1 . 1 except for the hypothetical 1% case. For galaxies at 1 . 1 < 𝑧 < 1 . 6, the realistic 0.26% catastrophics shift the monopole and quadrupole by > 0 . 1 𝜀 𝜉 at 𝑠 < 60 ℎ -1 Mpc, similar to the effects of the hypothetical catastrophics. Hexadecapoles are robust to the catastrophics effect.  \n<!-- image -->', '5.3 Impact of Catastrophics': 'We present in Figure 9 and Figure 10 the impact of realistic 0.26% catastrophics, hypothetical 1% random catastrophics and removing ELGs at 1 . 31 < 𝑧 < 1 . 33 (the sky-confusion-contaminated range) on multipoles of the 2PCF 𝜉 ℓ and power spectra 𝑃 ℓ . Removing ELGs at 1 . 31 < 𝑧 < 1 . 33 is a possible solution to avoid the impact of the hidden sky confusion at 𝑧 ≈ 1 . 32, corresponding to 𝑓 catas = 2 . 54%.  \nTheir influences are mainly embodied in galaxies at higher redshift 1 . 1 < 𝑧 < 1 . 6. Realistic and 1% random catastrophics result in up to 0 . 2 𝜀 𝜉 impact on 𝜉 0 and 𝜉 2 on 𝑠 < 60 ℎ -1 Mpc. The  \nFigure 10 . Similar to Figure 9 but for power spectra 𝑃 ℓ ( 𝑘 ) ℓ = 0 , 2 , 4.  \n<!-- image -->  \ncatastrophics can also suppress the power spectra monopole by up to 0 . 5 𝜀 𝑃 . Removing model ELGs at 1 . 31 < 𝑧 < 1 . 33 slightly perturbs the 2PCF as data and randoms both implemented the truncation. So there is no bias in its 2PCF and the difference from the standard clustering is due to the reduction of galaxies. Removing ELGs boosts the amplitude of ELG 𝑃 ℓ at 1 . 1 < 𝑧 < 1 . 6 by up to 0 . 3 𝜀 𝑃 while the other catastrophics suppress the clustering.  \nAt 0 . 8 < 𝑧 < 1 . 1, only 1% random catastrophics lead to up to 0 . 2 𝜀 𝜉 and 0 . 5 𝜀 𝑃 influence in the configuration space and Fourier space, respectively. This is because the number of galaxies that the realistic catastrophics interferes at 0 . 8 < 𝑧 < 1 . 1 is too small, and removing galaxies at 1 . 31 < 𝑧 < 1 . 33 does not affect the LSS at lower redshifts. Hexadecapoles are robust to all three implementations.  \nTheir 𝜒 2 sys are presented in Table 2. The DESI BAO analysis focusing on the configuration space will not [28] be influenced by any of the listed catastrophics. This is because the catastrophics do not move the position of the BAO peak. Their maximum 𝜒 2 sys (the left column of each 𝜒 2 sys ) is as small as 0.32. Both lead to negligible changes in the BAO cosmological parameters. However, their impacts become larger in Fourier space at 1 . 1 < 𝑧 < 1 . 6 due to the systematical shift they cause in the monopole and quadrupole. This could interfere with the RSD measurements. We should remind our reader that all catastrophics and redshift-removal implemented to A/b.pc/a.pc/c.pc/u.pc/s.pcS/u.pc/m.pc/m.pc/i.pc/t.pc ELG mocks result in the change of window function. The difference in 𝑃 ℓ and 𝜒 2 𝑠𝑦𝑠 values is not equivalent to their influence on cosmological measurements. Therefore, we perform cosmological fitting with their corresponding window function to determine how they will interfere with the parameters.  \nThe differences in the best-fitting values of cosmological parameters between the catastrophics clustering and the standard clustering in Fourier space are shown in Table 3 and Table 4. The maximum-likelihood values 𝜒 2 /dof (Eq. (3.10)) of all these samples are all smaller than 3 . 2 / 59 as the clustering is calculated from mocks based on accurate 𝑁 -body simulations and thus is fitted well by theories. Despite the large 𝜒 2 sys , the cosmological impacts of all types of systematics are no larger  \nTable 2 . Similar to Table 1 but for catastrophics models: 0.26% realistic failures, 1% random catastrophics, removing ELGs at 1 . 31 < 𝑧 < 1 . 33 compared to the A/b.pc/a.pc/c.pc/u.pc/s.pcS/u.pc/m.pc/m.pc/i.pc/t.pc standard clustering. The catastrophics all present larger impacts for ELGs at 1 . 1 < 𝑧 < 1 . 6. In configuration space, their influence is minor. However, in Fourier space, the catastrophics might bias the cosmological measurement.  \nthan 0.2 𝜎 . That is because the nuisance parameters and window functions absorb most of the effects. The nuisance parameters adjust the shape and amplitude of the theoretical curves directly, while the window function influences 𝑃 ( 𝑘 ) through geometric effects. As shown in Figure 11, the clustering difference between the 1% failures and the standard case remains largely unchanged with or without the window function. This suggests that the catastrophics parameters absorb the nuisance effects rather than the window function for the catastrophics-contaminated mocks. In contrast, the clustering difference between the truncated and standard samples varies when the geometric effects (window function) are ignored. This indicates that the clustering changes caused by truncating a slice of the redshift result from changes in geometry. To conclude, the influence of catastrophics effects on DESI ELG in DR1 is not a concern.  \nFigure 11 . The best-fit theoretical 𝑃 ( 𝑘 ) at 1 . 1 < 𝑧 < 1 . 6 for the standard clustering (solid lines or squares), the clustering with hypothetical 1% failures (dashed lines or flipped triangles) and the clustering with truncation at 1 . 31 < 𝑧 < 1 . 33 (dash-dotted lines or plus signs). The first column is the 𝑃 ( 𝑘 ) monopole, and the second is the quadrupole. The upper panel shows the best-fit theoretical curves with window functions applied (i.e., agree with mock 𝑃 ( 𝑘 ) ), while the lower panel shows the same curves without window functions. The 𝑃 ( 𝑘 ) differences for 1% failures did not shift much without the window function, while that of the cut 1 . 31 < 𝑧 < 1 . 33 changed its values. This implies that the nuisance parameters absorb the clustering effect of the 1% failures, whereas the truncation effect is primarily accounted for by its window function.  \n<!-- image -->  \nTable 3 . The cosmological impact of catastrophics in full modelling measurements: 0.26% realistic failures, 1% random catastrophics, removing ELGs at 1 . 31 < 𝑧 < 1 . 33 compared to the A/b.pc/a.pc/c.pc/u.pc/s.pcS/u.pc/m.pc/m.pc/i.pc/t.pc standard clustering. 𝜎 represent the 68% confidence level of DESI DR1 cosmological constraints. All influences are smaller than 0.2 𝜎 .  \nTable 4 . Similar to Table 3 but for ShapeFit results. The cosmological impacts are all smaller than 0.2 𝜎 .', '5.4 Redshift Uncertainty': "Figure 12 . The distribution of DR1 ELG redshift difference Δ 𝑣 (histogram with binomial error bars) that are smaller than 1000 km s -1 . There are the best-fit Gaussian profile N( 0 . 06 , 12 . 7 2 ) ('G', the dotted line) and Lorentzian profile L( 0 . 03 , 8 . 5 ) ('L', the dashed line). The Lorentzian profile is a better description of the ELG redshift uncertainty than a Gaussian profile.  \n<!-- image -->  \nRepeated observations at 0 . 8 < 𝑧 < 1 . 6 with | Δ 𝑣 | < 1000 km s -1 are used to measure the redshift uncertainty of ELGs as shown in Figure 12. Compared to a Gaussian profile, a Lorentzian  \nprofile described as follows is a better model of DR1 ELG Δ 𝑣 distribution  \nL( 𝑝, 𝑤 Δ 𝑣 ) = 𝐴 1 + (( 𝑥 -𝑝 )/ 𝑤 Δ 𝑣 ) 2 . (5.3)  \nwhere 𝐴 = 0 . 04, 𝑝 = 0 . 03 and 𝑤 Δ 𝑣 = 8 . 5 km s -1 . This 𝑤 Δ 𝑣 is similar to the values of the visual inspection and survey validation programs for DESI ELGs [17, 52] that are at the level of ∼ 10 km s -1 , thus its impact on the DR1 ELG clustering is negligible on 𝑠 > 5 ℎ -1 Mpc[53]. Its percentile velocities at 50, 95, and 99.5 % are 8.2, 53.5 and 139.0 km s -1 , respectively. They are less than half of the values from eBOSS ELGs, which are 20, 100, and 300 km s -1 [5] thanks to the improvement of spectrographs and redshift pipeline [11, 36, 37].", '6 Conclusion': 'In this paper, we investigate the spectroscopic systematics of ELGs in DESI DR1 and their clustering impact on 2PCF 𝜉 ( 𝑠 ) and the power spectrum 𝑃 ( 𝑘 ) . Spectroscopic systematics arise from the failed redshift measurements and cause non-cosmological variations in the galaxy distribution. One of the aspects is the redshift success rate 𝑓 goodz dependences on the observing conditions. The other ones are collective effects introduced by the catastrophics and the redshift uncertainty in the redshift measurement.  \nThe actual 𝑓 goodz depends on the squared signal-to-noise for ELG spectra TSNR2\\_ELG ( 𝑆 spec , obs ) and redshift 𝑧 , leading to larger 𝑓 goodz for ELGs with larger 𝑆 spec , obs at different redshift bins. This is equivalent to unfairly up-weighting ELGs with high 𝑆 spec , obs . We develop the redshift failure weight 𝑤 zfail as the inverse of a linear function of 𝑆 spec , obs as a function of ( 𝑆 spec , obs , 𝑧 ). With 𝑤 zfail , we successfully recover a close-to-unity 𝑓 goodz -𝑆 spec , obs relation for ELGs at fine redshift bins with 𝑑𝑧 = 0 . 1. However, this weight is imperfect since there is no information for 𝑓 goodz variations on the focal plane. Therefore, the 𝑓 goodz dependencies on the distance from the focal centre, and the fibre aperture loss at G, R, and Z bands, do not show improvement after applying 𝑤 zfail . We thus create fibre-based corrections 𝜂 zfail on the 𝑤 zfail . 𝜂 zfail is the inverse of the 𝑤 zfail -weighted 𝑓 goodz for each fibre. Implementing 𝜂 zfail leads to a more uniform 𝑓 goodz -𝑆 spec , obs relation and a unity relation between 𝑓 goodz , the distance to the focal plane, and all the throughput in different bands. Due to the small 𝑓 goodz difference before any corrections, 𝑤 zfail and 𝑤 zfail 𝜂 zfail result in smaller than 0 . 05 𝜀 𝜉 in the monopole, quadrupole, and hexadecapole of 𝜉 and 𝑃 ( 𝑘 ) , corresponding to less than 𝜒 2 sys = 0 . 25 differences. It means that their cosmological influences are negligible.  \nCatastrophics and redshift uncertainty are systematics that are difficult to subtract from individual targets. They can be explored through repeated observations with the redshift difference Δ 𝑣 , and their clustering impact can be modelled in cosmological theories. The catastrophics rate of DESI DR1 ELGs is 0.26%, similar to that of ELGs from DESI survey validation and eBOSS. There are five types of failure patterns: sky confusions that result in 𝑧 catas ≈ 1 . 32 (26.7%), other misidentification between sky-emission residuals and [O /i.pc/i.pc] doublets (27.6%), double objects (22%), double failures (12.1%), and other failures (11.6%).  \nDespite the variety of the patterns, we can model their 𝑧 catas -𝑧 true relation in the A/b.pc/a.pc/c.pc/u.pc/s.pcS/u.pc/m.pc/m.pc/i.pc/t.pc galaxy mocks with 75.1% of random catastrophics and 24.9% of the excess sky confusions besides the random catastrophics (realistic catastrophics). We also generate catastrophics-contaminated galaxy mocks with a hypothetical with 1% random catastrophics and mocks without the sky-confusioncontaminated redshift range 1 . 31 < 𝑧 < 1 . 33. Their impacts on the 2PCF are as small as 0 . 3 𝜀 𝜉 for all multipoles at 0 . 8 < 𝑧 < 1 . 6, corresponding to 𝜒 2 sys < 1. Therefore, we do not expect catastrophics to affect the BAO and RSD measurement in the configuration space. However, realistic failures and  \n1% random catastrophics systematically suppress the power spectra by up to 0 . 5 𝜀 𝑃 at both redshift bins. Removing the contaminated shell of 1 . 31 < 𝑧 < 1 . 33 from the data is a simple solution to avoid bias in cosmological measurements based on the configuration space. In the Fourier space, removing a small shell of the observed 3D map boosts the monopole by 0 . 3 𝜀 𝑃 at 1 . 1 < 𝑧 < 1 . 6. All these lead to 𝜒 2 sys values larger than 1 in Fourier space. We calculate the new window function of all three mocks and do RSD fitting with full-modelling and ShapeFit compression. We find that the cosmological influence of the realistic catastrophics, the hypothetical 1% catastrophics and removing the contaminated 1 . 31 < 𝑧 < 1 . 33 are smaller than 0.2 𝜎 . This means that the deviation from the standard 𝑃 ℓ is mainly from the change of geometry and nuisance parameters. This level of systematics is also within the systematics budget of [30].  \nIn conclusion, the spectroscopic systematics of ELGs have a minor impact on DESI cosmological measurements. Nevertheless, any unexpected patterns in this type of systematics are still worth examining. The modelling of 𝑓 goodz needs to take into account the information on the focal plane as well despite the small variations. The catastrophics, composed of 0.26% of the ELGs, have minor impacts on the cosmological parameters. This is not problematic for the current redshift surveys, but it can be a problem for photometric surveys from space. The redshift uncertainty of ELGs is small thanks to the characteristic [O /i.pc/i.pc] doublet. But that of the Lyman-Alpha Emitters (LAEs), as a type of emission-line galaxies, though, might be more complicated. With more data and improved telescopes, we will be able to resolve all these issues before they become a significant problem.', 'Data Availability': 'The data used in this analysis will be made public along the Data Release 1 (details in https: //data.desi.lbl.gov/doc/releases/ ). Zenodo includes all data to reproduce the figures in this paper: https://doi.org/10.5281/zenodo.11302697 .', 'Acknowledgments': 'JY, DFS, and JPK acknowledge the support from the SNF 200020\\_175751 and 200020\\_207379 \'Cosmology with 3D Maps of the Universe" research grant. We would like to thank Anand Raichoor, Allyson Brodzeller, Ruiyang Zhao and Julien Guy for their helpful discussions. We also would like to thank Andrei Variu for his support in visualising DESI spectra.  \nThis material is based upon work supported by the U.S. Department of Energy (DOE), Office of Science, Office of High-Energy Physics, under Contract No. DE-AC02-05CH11231, and by the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility under the same contract. Additional support for DESI was provided by the U.S. National Science Foundation (NSF), Division of Astronomical Sciences under Contract No. AST0950945 to the NSF\'s National Optical-Infrared Astronomy Research Laboratory; the Science and Technology Facilities Council of the United Kingdom; the Gordon and Betty Moore Foundation; the Heising-Simons Foundation; the French Alternative Energies and Atomic Energy Commission (CEA); the National Council of Humanities, Science and Technology of Mexico (CONAHCYT); the Ministry of Science and Innovation of Spain (MICINN), and by the DESI Member Institutions: https://www.desi.lbl.gov/collaborating-institutions . Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the U. S. National Science Foundation, the U. S. Department of Energy, or any of the listed funding agencies.  \nThe authors are honored to be permitted to conduct scientific research on Iolkam Du\'ag (Kitt Peak), a mountain with particular significance to the Tohono O\'odham Nation.', 'A Visualise Spectra of ELG Catastrophics': 'We present the ELG spectra of sky confusion at 𝑧 catas ≈ 1 . 32, sky confusion at other redshifts, double objects in the spectra, totally failed redshift measurements, and the misidentification of H 𝛼 to be an [O /i.pc/i.pc] doublet in Figure 13 as introduced in Section 5.1. The insert subplots in the first, second and fifth spectra are enlarged spectral lines. The subplot on the left shows the spectra around the true [O /i.pc/i.pc] doublet (black solid lines with blue dashed vertical lines for [O /i.pc/i.pc] ) and the one on the right is the false [O /i.pc/i.pc] line that results in catastrophics. The physical [O /i.pc/i.pc] emission lines for the left subplot in the first and the second spectra differ from the sky residuals in the right subplot. The H 𝛼 emission in the right subplot of the fifth spectra was also different from the [O /i.pc/i.pc] doublet. However, R/e.pc/d.pc/r.pc/o.pc/c.pc/k.pc was unable to distinguish them for some reason. Both redshift measurements are true in the third spectrum: one is determined by multiple emission lines from the 𝑧 0 = 0 object, and the other is from [O /i.pc/i.pc] emission. In the fourth spectra, both redshift measurements are wrong as they regard sky residuals at different redshifts as [O /i.pc/i.pc] doublets. All of them pass the good-redshift selection and most of them are within 0 . 8 < 𝑧 < 1 . 6 redshift range. This indicates that the 0.26% is probably a lower limit of the catastrophics rate because there can be consistently wrong redshift measurements that pass the good-redshift criteria hidden in the current catalogue.', 'B Catastrophics Impacts on Small-Scale 2PCF': 'In Section 5, we demonstrate that the impact of catastrophics on cosmological scales in the configuration space is negligible. In this section, we discuss the influence of catastrophics on small scales as it has not been corrected in the current clustering. As the analytical covariance is no longer applicable at 𝑠 < 20 ℎ -1 Mpc, we used the jackknife error from the EDR ELG samples calculated with 128 subsamples using /p.pc/y.pc/c.pc/o.pc/r.pc/r.pc in our discussion. At 1 . 1 < 𝑧 < 1 . 6, the catastrophics can suppress the monopole by up to ∼ 0 . 5 𝜀 𝜉 and change quadrupole systematically as shown in Figure 14. This indicates that the galaxy-halo connection studies of ELGs in galaxy surveys, with HOD or SubHalo Abundance Matching (SHAM), may need to consider this effect when they fit the observed ELG clustering at small scales.', 'C Author Affiliations': "1 Institute of Physics, Laboratory of Astrophysics, École Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, CH-1290 Versoix, Switzerland  \n- 8 Department of Physics and Astronomy, University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada\n- 9 Perimeter Institute for Theoretical Physics, 31 Caroline St. North, Waterloo, ON N2L 2Y5, Canada 10 Waterloo Centre for Astrophysics, University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada\n- 11 Graduate Institute of Astrophysics and Department of Physics, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd., Taipei 10617, Taiwan\n- 12 Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA 13 Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA\n- 14 Physics Dept., Boston University, 590 Commonwealth Avenue, Boston, MA 02215, USA\n- 15 Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK\n- Instituto de Física, Universidad Nacional Autónoma de México, Cd. de México C.P. 04510, México\n- 16 17 NSF NOIRLab, 950 N. Cherry Ave., Tucson, AZ 85719, USA\n- 18 Department of Physics & Astronomy and Pittsburgh Particle Physics, Astrophysics, and Cosmology Center (PITT PACC), University of Pittsburgh, 3941 O'Hara Street, Pittsburgh, PA 15260, USA 19\n- Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, Menlo Park, CA 94305, USA\n- 20 SLAC National Accelerator Laboratory, Menlo Park, CA 94305, USA\n- 21 Departamento de Física, Universidad de los Andes, Cra. 1 No. 18A-10, Edificio Ip, CP 111711, Bogotá, Colombia\n- 22 Observatorio Astronómico, Universidad de los Andes, Cra. 1 No. 18A-10, Edificio H, CP 111711 Bogotá, Colombia\n- 23 Institut d'Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain\n- 24 Institute of Cosmology and Gravitation, University of Portsmouth, Dennis Sciama Building, Portsmouth, PO1 3FX, UK\n- 25 Institute of Space Sciences, ICE-CSIC, Campus UAB, Carrer de Can Magrans s/n, 08913 Bellaterra, Barcelona, Spain\n- 26 Department of Physics, The Ohio State University, 191 West Woodruff Avenue, Columbus, OH 43210, USA\n- 27 School of Mathematics and Physics, University of Queensland, 4072, Australia\n- 28 Sorbonne Université, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de Hautes Energies (LPNHE), FR-75005 Paris, France\n- 29 DepartamentdeFísica, Serra Húnter, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain\n- 30 Institut de Física d'Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra Barcelona, Spain\n- 31 Institució Catalana de Recerca i Estudis Avançats, Passeig de Lluís Companys, 23, 08010 Barcelona, Spain\n- 32 Department of Physics and Astronomy, Siena College, 515 Loudon Road, Loudonville, NY 12211, USA\n- 33 Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, U.K  \n- 37 Instituto Avanzado de Cosmología A. C., San Marcos 11 - Atenas 202. Magdalena Contreras, 10720. Ciudad de México, México\n- 38 Space Sciences Laboratory, University of California, Berkeley, 7 Gauss Way, Berkeley, CA 94720, USA\n- 39 University of California, Berkeley, 110 Sproul Hall #5800 Berkeley, CA 94720, USA\n- 40 Instituto de Astrofísica de Andalucía (CSIC), Glorieta de la Astronomía, s/n, E-18008 Granada, Spain\n- 41 Department of Physics, Kansas State University, 116 Cardwell Hall, Manhattan, KS 66506, USA\n- 42 Department of Physics and Astronomy, Sejong University, Seoul, 143-747, Korea\n- 43 CIEMAT, Avenida Complutense 40, E-28040 Madrid, Spain\n- 44 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA\n- 45 Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA\n- 46 Department of Physics & Astronomy, Ohio University, Athens, OH 45701, USA\n- 47 Department of Astronomy, Tsinghua University, 30 Shuangqing Road, Haidian District, Beĳing, China, 100190", 'References': '- [1] D.J. Eisenstein and W. 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Bautista, M.A. Bershady et al., Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: Cosmological implications from two decades of spectroscopic surveys at the Apache Point Observatory , Phys. Rev. D 103 (2021) 083533 [ 2007.08991 ].  \nFigure 13 . Five ELG spectra that suffer from catastrophics in redshift measurements due to the sky confusion at 𝑧 catas ≈ 1 . 32, other sky confusion patterns, double objects in the spectrum, total failures in the redshift measurement, and the other failures like the misidentification of H 𝛼 to be [O /i.pc/i.pc] . The calibrated and skysubtracted spectra are in solid lines. The orange shades in the foreground indicate the sky spectra. The blue vertical lines with 𝑧 true and [O /i.pc/i.pc] labels indicate the true [O /i.pc/i.pc] emissions positions except for the fourth spectra, in which the blue line indicates the sky residuals misidentified as [O /i.pc/i.pc] . The zoom-in spectra in their vicinity are presented in the inserted subplot on the left-hand side. The red vertical lines with 𝑧 catas point out the position of features misidentified as [O /i.pc/i.pc] line, except for the third spectra. Their enlarged features are the right-hand side inserted subplots. In the third spectra, there are H 𝛽 , H 𝛼 , [S /i.pc/i.pc] to determine the redshift of the second object to be 𝑧 true , 2 = 0.  \n<!-- image -->  \nÅ  \nFigure 14 . Similar to Figure 5 but for the comparison between the standard A/b.pc/a.pc/c.pc/u.pc/s.pcS/u.pc/m.pc/m.pc/i.pc/t.pc galaxy mocks and the catastrophics-contaminated mocks with 0.26% realistic catastrophics (dashed lines), 1% random catastrophics (dash-dotted lines) and no ELGs at 1 . 31 < 𝑧 < 1 . 33 (dotted lines) at 𝑠 < 30 ℎ -1 Mpc. The grey regions are the analytical 1𝜎 error of the ELG clustering. The impact of catastrophics at 1 . 1 < 𝑧 < 1 . 6 can be as large as ∼ 0 . 5 𝜀 𝜉 , but is much smaller at 0 . 8 < 𝑧 < 1 . 1.  \n<!-- image -->'}