# Measuring the global 21-cm signal with the MWA-II: improved characterisation
of lunar-reflected radio frequency interference

Himanshu Tiwari  International Centre for Radio Astronomy Research (ICRAR),
Curtin University, Kent Street, Bentley, Perth, Western Australia, 6102.  [
Benjamin McKinley  International Centre for Radio Astronomy Research (ICRAR),
Curtin University, Kent Street, Bentley, Perth, Western Australia, 6102.
Cathryn M. Trott  International Centre for Radio Astronomy Research (ICRAR),
Curtin University, Kent Street, Bentley, Perth, Western Australia, 6102.
Nithyanandan Thyagarajan  Commonwealth Scientific and Industrial Research
Organisation (CSIRO), Space & Astronomy, P. O. Box 1130, Bentley, WA 6102,
Australia.

###### Abstract

Radio interferometers can potentially detect the sky-averaged signal from the
Cosmic Dawn (CD) and the Epoch of Reionisation (EoR) by studying the Moon as a
thermal block to the foreground sky. The first step is to mitigate the Earth-
based RFI reflections (Earthshine) from the Moon, which significantly
contaminate the FM band $\approx 88-110$ MHz, crucial to CD-EoR science. We
analysed MWA phase-I data from $72-180$ MHz at $40$ kHz resolution to
understand the nature of Earthshine over three observing nights. We took two
approaches to correct the Earthshine component from the Moon. In the first
method, we mitigated the Earthshine using the flux density of the two
components from the data, while in the second method, we used simulated flux
density based on an FM catalogue to mitigate the Earthshine. Using these
methods, we were able to recover the expected Galactic foreground temperature
of the patch of sky obscured by the Moon. We performed a joint analysis of the
Galactic foregrounds and the Moon’s intrinsic temperature $(T_{\rm Moon})$
while assuming that the Moon has a constant thermal temperature throughout
three epochs. We found $T_{\rm Moon}$ to be at $184.4\pm{2.6}\rm~{}K$ and
$173.8\pm{2.5}\rm~{}K$ using the first and the second methods, respectively,
and the best-fit values of the Galactic spectral index $(\alpha)$ to be within
the $5\%$ uncertainty level when compared with the global sky models. Compared
with our previous work, these results improved constraints on the Galactic
spectral index and the Moon’s intrinsic temperature. We also simulated the
Earthshine at MWA between November and December 2023 to find suitable
observing times less affected by the Earthshine. Such observing windows act as
Earthshine avoidance and can be used to perform future global CD-EoR
experiments using the Moon with the MWA.

###### keywords:

Early Universe, Cosmic Dawn and Reionization, first stars, observations, MWA

Commonwealth Scientific and Industrial Research Organisation (CSIRO), Space &
Astronomy, P. O. Box 1130, Bentley, WA 6102, Australia. Himanshu
<EMAIL_ADDRESS>ARC Centre of Excellence for All
Sky Astrophysics in Three Dimensions (ASTRO-3D), Australia. ARC Centre of
Excellence for All Sky Astrophysics in Three Dimensions (ASTRO-3D), Australia.

##  1 Introduction

The Cosmic Dawn (CD) marked the end of the Dark Ages with the emergence of the
first luminous objects in the early Universe. These objects were the first
generation of stars, black holes and other compact objects. The X-ray and UV
radiation produced by these objects started heating and ionising their
surrounding matter. This ionising process gradually changed the state of the
IGM from neutral to fully ionised during the Epoch of Reionisation (EoR) until
the redshift $z\sim 5.3$ (Furlanetto_2004, 19, 5, 43, 33, see). The
observational evidence from high redshift quasar spectra fan03 (17, 3),
Ly-$\alpha$ emitters mcquinn06 (30) and Cosmic Microwave Background (CMB)
scattering from the ionised IGM at low redshifts implies that the cosmic
reionisation lasted between redshifts $z\approx 18-5.3$ komatsu11 (23, 2).
However, these studies have only placed weak constraints on the astrophysical
properties of the first stars, black holes and galaxies and the evolution of
the early Universe due to the difficulty of direct observations. The neutral
medium of atomic Hydrogen (HI ) is largely opaque to UV radiation, whereas the
X-ray tends to heat the IGM; therefore, it becomes challenging to observe the
CD-EoR directly at these frequencies due to absorption by the IGM. An indirect
approach of using the 21-cm emission line from the leftover HI from this era
is the most likely candidate to probe the CD-EoR. Mesinger_2016 (33)

The 21-cm signal arises from the spontaneous transition of Hydrogen from the
ground state triplet to the singlet state, causing the emission of a radio
photon of $\approx 1420$ MHz, which can be detectable by ground-based radio
antennae. Therefore, several radio instruments, e.g. Murchison Widefield Array
(MWA) Tingay_13 (52), Precision Array for Probing the Epoch of Reionization
(PAPER) Pober_2011 (42), Hydrogen Epoch of Reionization Array (HERA) deboer17
(12), LOw-Frequency ARray (LOFAR) Haarlem_2013 (56), New Extension in Nancay
upgrading LOFAR (NenuFAR) Mertens_2021 (32), Giant Meter wave Radio Telescope
(GMRT) Paciga13 (37), Long Wavelength Array (LWA) Eastwood_2019 (14) etc., aim
to detect the 21-cm signal from the CD-EoR. These instruments target either
the statistical features of the 21-cm signal (e.g. power spectrum) using radio
interferometers, e.g. MWA trott20 (54), GMRT GMRT_upper_limits (38), LOFAR
Mertens_2020 (31), HERA HERA_2021 (1), PAPER Parsons_2014 (39) or the volume
averaged 21-cm signal (global 21-cm signal) using a single antenna, e.g.
Experiment to Detect the Global EoR Signature (EDGES) Bowman_2008 (7, 6, 8),
Broad-band Instrument for Global HydrOgen ReioNization Signal (BIGHORNS)
Sokolowski_2015 (50), Shaped Antenna measurement of the background RAdio
Spectrum (SARAS) patra_2012 (40), SARAS2 singh_2018 (49), SARAS3
saras3_singh2022 (47), Large Aperture Experiment to Detect the Dark Age (LEDA)
Bernardi_2016 (4), Dark Ages Radio Explorer (DARE) Burns_2012 (46) now known
as Dark Ages Polarimeter Pathfinder (DAPPER) Burns_2019 (9), Sonda Cosmológica
de las Islas para la Detección de Hidrógeno Neutro (SCI-HI) Voytek_2014 (58),
Probing Radio Intensity at High-Z from Marion (PRIZM) prizm_17 (41). However,
the radiation received by these instruments is dominated by the foreground
emission from the Galactic and extra-galactic radio sources. Alongside these
foregrounds, instrument-based systematics, thermal noise, and radio frequency
interference (RFI) present enormous challenges in 21-cm signal detection.

The global EoR experiments measure the sky-averaged strength of the 21-cm
signal by estimating the total power of 21-cm brightness as a function of
frequency, which represents the redshift evolution of the early Universe. The
first claimed detection of the global 21-cm signal came from EDGES Bowman_18
(8), which found an absorption trough profile around 78 MHz, which aligned
with the theoretical predictions of early cooling and reheating
Furlanetto_2006 (18, 43). However, recent cross-validations from the SARAS3
experiment found that EDGES detection was mainly due to systematic errors
saras3_singh2022 (47). This signifies that, despite the high signal-to-noise
ratio (SNR) of these experiments, the non-thermal components of the receiver
noise can mimic the cosmic signal of interest, leading to false detection
scenarios. Therefore, minimisation of the instrumental systematics is the
major goal of these experiments Monsalve_2017a (34, 48, 50).

An alternative approach, known as the lunar occultation method, has the
potential to detect the global 21-cm signal. Instead of a single antenna or
dipole, this approach utilises radio interferometers to detect the sky-
averaged 21-cm signal. Interferometers, in general, are not sensitive to the
global sky as the response rapidly falls off for baselines $>1\lambda$.
However, the presence of the Moon introduces a mask in the sky, which helps
sustain the coherence in the global sky response at longer baselines
$(>50\lambda)$ (see fig. 1 from Vedantham_2015 (57) for reference). The idea
of lunar occultation was first proposed by Shavar_1999 (45) and has been
implemented by McKinley_2012 (29, 28) on MWA, and Vedantham_2015 (57) on
LOFAR. In lunar occultation, the Moon is treated as a thermal block in the
sky, and the interferometer measures the difference between the Moon’s
temperature and the global sky temperature. The benefits of using
interferometers are that they contain independent antenna elements, and the
voltage correlations at their receivers are devoid of frequency-dependent
receiver-noise bias. Also, unlike single antenna or dipole-based experiments,
which measure and mitigate the entire foreground sky to detect the global
21-cm signal, the lunar occultation method only measures the patch of sky
occulted by the Moon and hence is only required to mitigate the foregrounds
occulted by the Moon. This significantly reduces the Galactic emission and
spectral index anomalies, which otherwise are very difficult to deal with.
Therefore, lunar occultation can be used to cross-validate the findings from
the EDGES and SARAS-3 experiments, and since more sensitive instruments, such
as MWA phase II & III, are currently functional, and the upcoming SKA
operations are also on the horizon, it is worth further investigating the
lunar occultation approach for CD-EoR science.

In this work, we extended the approach of Ben_2018 (28) by incorporating
multiple nights of higher (time and frequency) resolution MWA-phase I data. We
also used a new simulation-based approach to model the reflected RFI
(Earthshine) in the FM band (88-110 MHz) to mitigate the reflected FM response
from the Moon. Using these two approaches, we obtained improved constraints on
the Moon’s intrinsic temperature, the Galactic foreground temperature and
spectral index. This paper is organised as follows: In §2, we briefly
summarise the lunar occultation technique. In §3 and §4, we provide
information about the MWA phase-I observations and data processing techniques
used in this work. In §5, we describe our approach to modelling the reflected
RFI from the Moon (Earthshine). In §6, we provide the main results and discuss
them in §7 along with the limitations and potential aspects of our project,
and finally, we provide the conclusion in §8.

##  2 Background

Lunar occultation provides a unique way to detect the EoR 21-cm global signal
using radio interferometers. In the radio frequencies corresponding to the CD-
EoR, the Moon is treated as a thermal source at a constant blackbody
temperature. The interferometer measures the difference between the Moon’s
temperature and the background sky temperature of the patch of sky occulted by
the Moon. This can be expressed as,

$\Delta T(\nu)=T_{\rm Lunar}(\nu)-T_{\rm
sky}(\nu)=\frac{c^{2}S_{m}(\nu)}{2k_{\rm B}\nu^{2}\Omega}\rm~{}~{}K$ (1)

where $S_{m}$ is the observed flux density of the Moon, $\Omega$ is the Moon’s
solid angle, $k_{\rm B}$ is Boltzmann’s constant, $c$ is the speed of light,
and ${\nu}$ is the frequency. The Moon reflects a part of the Galactic
foreground and terrestrial radio emission back to the observer on the Earth
Evans_1969 (15). Therefore, the Moon’s temperature includes two additional
factors; the reflected Galactic foreground and reflected RFI. Thus, we can
express eq.1 as,

$\Delta T(\nu)=T_{\rm Lunar}(\nu)-T_{\rm sky}(\nu)\\\ =[T_{\rm Moon}+T_{\rm
refl-Earth}(\nu)+T_{\rm refl-Gal}(\nu)]\\\ -[T_{\rm Gal}(\nu)+T_{\rm
CMB}+T_{\rm EoR}(\nu)],$ (2)

where $T_{\rm Moon}$ is the intrinsic temperature of the Moon,
$T_{\rm refl-Earth}(\nu)$, is the reflected RFI (Earthshine) temperature,
$T_{\rm refl-Gal}(\nu)$ is the reflected Galactic temperature. The sky
temperature $T_{\rm sky}(\nu)$ includes the contribution from the Galactic
temperature $T_{\rm Gal}(\nu)$ (temperature of the occulted patch of the sky),
Cosmic Microwave Background temperature $T_{\rm CMB}$ and EoR 21-cm global sky
temperature $T_{\rm EoR}$. We used $T_{\rm CMB}\approx 2.725\rm~{}K$
Mather_1994 (27) in our analysis.

In most cosmic reionisation scenarios predicted by the reionisation models,
the global 21-cm signal goes through major phase transitions between 88-110
MHz; however, in this band, the Moon temperature is hugely contaminated by the
Earthshine. Therefore, the first step in the detection of the global 21-cm
signal would be to mitigate FM reflections from the Moon. In a similar
fashion, the Galactic foregrounds would be required to be mitigated in the
subsequent steps, and once all components from the Moon are successfully
evaluated, the $T_{\rm EoR}$ can be isolated from the Moon.

##  3 Observations

In this work, we used six nights of MWA-phase I data from 2015. Our
observational strategy was to observe the same patch of sky on two different
nights at the same (Local Sidereal Time) LST, where the Moon was present on
one night (ON-Moon) and absent on the other (OFF-Moon). These observations
were typically separated by two to three days and observed at identical LST
(LST-locked). The benefit of using these LST-locked observations is to
eliminate artefacts and sidelobes from the sources in the observed field by
taking the difference between the ON and OFF-Moon images. The six nights of
observations were carried out on $30^{\rm th}$ August, $26^{\rm th}$ September
and $21^{\rm st}$ December for the ON-Moon and on $2^{\rm nd}$ September,
$29^{\rm th}$ September and $24^{\rm th}$ December $2015$ for the OFF-Moon,
respectively (see observation table 1). Throughout this paper, we used the
nomenclature of Epochs 1,2 and 3 to represent the ON-OFF Moon (paired)
datasets from August, September and December, respectively.

The effect of ionospheric activity can cause a significant difference in the
flux and positional offsets of sources over the period of two nights. However,
studies of ionospheric activity have shown that the majority of these effects
are minimal at the MRO Jordan_2017 (22, 55). We did not see a significantly
bad ionospheric shift, flux anomalies or positional offsets in our data.
Therefore, we ignored the effects of the ionosphere on the data. The MWA uses
discrete analog beamformer settings to track the Moon and takes the
observation in $30.72$ MHz contiguous bands, resetting the beamformer settings
close to the Moon’s location after every observation. This is known as the
drift and shift tracking method Trott_2014 (53). A full-band $72-230$ MHz
observation includes $5$ successive observations of such $30.72$ MHz
contiguous bands. In this work, we used a total of $340$ observations which
comprise a total of $68$ full-band observations for both ON and OFF Moon from
all six nights. Since each ON-Moon observation has an LST-locked OFF-Moon pair
to perform the differencing, this results in a total of $34$ full-band ON-OFF
Moon observations. The individual observations were carried out for $\approx
230$ seconds each. We processed these observations using
COTTER111https://github.com/MWATelescope/cotter Offringa_2015 (36) for a
frequency and time resolution of $40$ kHz and $4$ seconds, respectively. The
observational details are given in table 1. At each observation, the central
fine channel and two adjacent fine channels at the edge of each $1.28$ MHz
course channel were flagged. We also flagged the combined bad dipoles/tiles
from each pair of the ON-OFF Moon observations (i.e. we isolated the common
working dipoles from both ON and OFF Moon observations and flagged the rest),
which otherwise would have created a difference in the UV coverage on the ON
and OFF-Moon observations, which could cast some additional artefacts in the
final difference (ON-OFF) images. Additionally, the operating frequencies of
the ORBCOMM satellites from 121-136 MHz were avoided in our analysis.

$\rm ON-Moon$
---
$\rm Date$ | $\rm N_{obs}$ | $\rm N_{obs}^{full-band}$ | $\rm Bandwidth~{}(MHz)$ | $\rm Freq.~{}res.~{}(kHz)$ | $\rm Time~{}res.~{}(sec)$ | $\rm Total~{}Int.~{}time~{}(sec)$ | $\rm Obs.~{}duration~{}(hrs)$
$\rm 30^{th}Aug.2015$ | $60$ | $12$ | $30.76$ | $40$ | $4$ | $236$ | $3.93$
$\rm 26^{th}Sept.2015$ | $55$ | $11$ | $30.76$ | $40$ | $4$ | $236$ | $3.60$
$\rm 21^{th}Dec.2015$ | $55$ | $11$ | $30.76$ | $40$ | $4$ | $236$ | $3.60$
$\rm OFF-Moon$
$\rm 2^{nd}Sept.2015$ | $60$ | $12$ | $30.76$ | $40$ | $4$ | $236$ | $3.93$
$\rm 29^{th}Sept.2015$ | $55$ | $11$ | $30.76$ | $40$ | $4$ | $236$ |  $3.60$
$\rm 24^{th}Dec.2015$ | $55$ | $11$ | $30.76$ | $40$ | $4$ | $236$ | $3.60$
| $\rm N_{total}=340$ | $\rm N_{total}^{full-band}=68$ |  |  |  |  | $\rm Time_{total}^{ON-Moon}=11.13$
Table 1: ON-Moon and OFF-Moon observation details.

##  4 Data processing and modelling

The processed observations were arranged according to the ON-OFF Moon pairs
and phase-shifted to the precise location of the Moon to produce difference
images. We used astropy to locate the sky position of the Moon and shifted the
phase-centre of the measurements to the location of the Moon. In order to
calibrate the sky visibilities, we produced the sky model using the Positional
Update and Matching Algorithm (PUMA) catalogue line_2017 (25) with the $800$
brightest sources in the field around the Moon. The sky model was passed to
MWA-Hyperdrive222https://github.com/MWATelescope/mwa_hyperdrive to perform
Direction Independent (DI) calibration. The calibrated measurement sets were
then used to produce primary-beam-corrected images. We used
WSCLEAN333https://gitlab.com/aroffringa/wsclean offringa-wsclean-2014 (35) to
produce the Stokes-I beam-corrected images from the calibrated measurement
sets. Each measurement set produced $768$ images corresponding to $\approx
30.72~{}\rm MHz$ of bandwidth, with each image produced at $40~{}\rm kHz$
frequency resolution. The images were produced with $2048\times 2048$ pixels,
with each pixel covering approximately $0.0085^{\circ}$, and the image
spanning $\approx 17^{\circ}$ of the sky on each side.

The MWA’s observational strategy uses integer delays across the MWA’s $4\times
4$-dipole tiles (in multiple units of $\mu$ sec.) to set the beamformers. As a
result, there are a limited number of ”sweet-spot” pointing locations in the
sky. Thus, in the majority of cases, the pointing centre of the beam was not
at the precise sky position of the Moon during the observation. Since the
primary beam size reduces with increasing frequency, most of the high-band
observations were affected by the Moon being located beyond half power point
of the beam (resulting in a low SNR). Therefore, we limited our analysis to
the beam’s full-width half maximum (FWHM), discarding observations where the
Moon was located beyond the FWHM. As a result, we had a full-band frequency
range of our processed images from $\approx 72-180$ MHz.

Finally, we took the difference between the ON and the OFF Moon images and
proceeded to the flux-density estimation and Earthshine mitigation processes.
The detailed radar studies of the Moon by Evans1966StudyOR (16, 15, 21).
Evans_1969 (15) showed that the scattering of the reflected radio power can be
assumed to be coming from two distinct features of the Moon’s surface. The
specular reflections from the smooth Moon and the diffuse reflections from the
rough Moon. These studies showed that the radar cross-section area and the
reflected power of the specular component decrease with increasing frequency
while the increase with increasing frequency for the diffuse component. We
followed the same two-component model (previously used by Ben_2018 (28)) to
estimate the reflected RFI from the Moon. The first component was a quasi-
specular Earthshine component which was a point-like source in the middle of
the Moon’s disk and corresponded to the reflection from the smooth
surface444Moon’s disk is the $2$D projection of Moon’s area, and the second
component was the diffuse reflection from the rough disk of the Moon. (Note
that the quasi-specular Earthshine can have a typical angular size of $\approx
16~{}\rm arcsec$ at the centre of the Moon’s disk (Vedantham_2015 (57)),
whereas the diffuse is roughly equal to the Moon’s angular size). The two-
component Earthshine model can be described as:

$s_{\rm disk},s_{\rm spec}={\rm(H^{T}H)^{-1}H^{T}\textbf{D}}$ (3)

where the $s_{\rm disk}$ is the flux density (in Jy/pixels units) of the
Moon’s disk, $s_{\rm spec}$ is the flux density of the quasi-specular
Earthshine at the centre of the Moon, and [D] is the beam-corrected difference
image. The vector operator $\rm H$ is defined using the PSF [P] convolved over
unity masks [M, B],

$\rm{H=\left[\textbf{M}\ast\textbf{P}~{}~{}~{}~{}\textbf{B}\ast\textbf{P}\right]},$

the mask M represents the disk, B represents the quasi-specular mask, P is the
PSF of the image and $\ast$ represents the convolution operation. The disk
mask has the same angular size as the Moon, whereas the quasi-specular mask
occupies $8\times 8$ pixels at the centre of the Moon’s image. Fig.1 shows the
cropped $(252\times 252)$ central pixels of the difference image, the
reconstructed quasi-specular component, the disk component and the residual at
the middle of the FM-band $\approx 100$ MHz ($\nu_{0}$, hereafter). Finally,
the estimated values of $s_{\rm disk}$ and $s_{\rm spec}$ are summed over the
corresponding masks to get the total flux density of the total disk and quasi-
specular earthshine components.

$S_{\rm disk}=\sum s_{\rm disk}.{\rm\textbf{M}};~{}~{}S_{\rm spec}=\sum s_{\rm
spec}.{\rm\textbf{B}}$ (4)

(Note that the flux density of the disk component $S_{\rm disk}$ includes the
contribution of the flux density $(S_{m})$of the Moon and the diffuse
Earthshine $(S_{\rm diffuse})$). In this work, we used two different
approaches to separate the Earthshine from the Moon’s flux density. In the
first approach, we followed a similar method as Ben_2018 (28), whereas in the
second, we mitigated the Earthshine by simulating the FM flux density.
Finally, we estimated the flux density of the Moon, $S_{m}(\nu)$ from $S_{\rm
disk}$ as described later in §6.1.

Figure 1: {Top panel} Left: Difference image of the ON-OFF observation, Right:
The reconstructed model of the quasi-specular Earthshine component obtained by
multiplying $s_{\rm spec}$ with quasi-specular mask [B]. {Bottom panel}, Left:
The reconstructed disk of the Moon is obtained by operating the PSF [P] to the
disk mask [M], Right: The residual image, obtained by subtracting the
reconstructed disk and specular Earthshine components from the difference
image. The shown images are the average of all components from the second
observational Epoch (Sept. 2015) at the middle of the FM band
$(\approx\nu_{0}=100~{}\rm MHz)$

.

##  5 Earthshine Simulation

###  5.1 Simulation Motivation

At radio frequencies, the Moon reflects about $7\%$ of the incident radiation
falling on its surface Evans_1969 (15). As a result, the Moon reflects back a
significant amount of the residual radiation which escapes the Earth. We can
observe strong RFI reflections from the Moon in the FM band (88-110 MHz) and
Digital TV (180-220 MHz). The exact behaviour of these RFI reflections is
unknown, as it depends on the time and the location of the Moon and the
observatory during the observation. Therefore, simulating the reflected
radiation from the Moon in the context of MWA’s Moon observations can be
useful in understanding the nature of the reflected RFI from the Moon. In
addition, the simulations can help investigate the RFI reflections at
different LSTs, which can be used to identify the minimal RFI imprint during
the observation window and help in scheduling future observations.

###  5.2 Simulation Method

In this analysis, we used a catalogue of FM stations555https://fmlist.org to
estimate the reflected Earthshine from the Moon. The catalogue contains
information on the location of $\approx 171,000$ FM stations across the Earth,
including their operating frequencies and transmitting powers. In modelling
the simulated Earshshine, we made a few assumptions based on inadequate
information on the beam pattern and operating hours of the FM stations in the
catalogue. We assumed that all FM stations transmit isotropically throughout
the full day. Due to the isotropic beam assumption, the reflected RFI includes
the contribution from all of the FM stations where the Moon was above the
horizon at their location at the time of ON-Moon observation. Fig.2 shows the
location of all the stations appearing above the horizon as viewed from the
Moon at the time of observation on the world map.

Figure 2: The locations of FM stations considered for the reflected RFI
estimation during the ON-Moon observation. The figure represents a single
snapshot in the middle of the second observation epoch (Sept. 2015). The
colourbar represents the transmitting power of the stations in kW. The figure
represents all the stations from where the Moon is above the horizon at the
time of ON-Moon MWA observation.

FM broadcasting follows different operating standards in different countries;
therefore, the bandwidth of FM stations varies worldwide. We followed the
conservative approach of Carson’s rule carson (10) to estimate the bandwidth
of FM stations and assumed that all FM stations have the same frequency
deviation of $75\rm~{}kHz$ with a modulation frequency of $15\rm~{}kHz$. It
provided us with a bandwidth $\Delta B$ of $180~{}\rm kHz$ (see Chapter 4 from
Haykin2007 (20) for reference). We assumed all stations transmit a constant
power across $\Delta B$. Finally, we estimated the reflected FM flux density
from all the contributing FM stations at the MWA location using the following
equation:

$S_{{\rm FM}(\nu)}=\frac{P_{\rm rec}(\nu)}{\Delta BA_{\rm eff}},$ (5)

where $P_{\rm rec}(\nu)$ is the received power defined as radar equation (eq.
24 Evans_1969 (15))

$P_{\rm rec}(\nu)=\frac{P_{\rm emit}(\nu)\sigma_{\rm cross}A_{\rm
eff}}{{(4\pi)}^{2}D_{1}^{2}D_{2}^{2}},$

where $P_{\rm emit}(\nu)$ is the transmitted power, $D_{1}$ is the distance
between the FM station to the Moon, $D_{2}$ is the distance from the Moon to
the MWA, $\Delta B$ is the transmission bandwidth, and $A_{\rm eff}$ is the
effective area of the MWA telescope. $\sigma_{\rm cross}=0.081\pi R_{\rm
Moon}^{2}$ is the radar cross-section area; it includes the contribution of
$7\%$ of Moon’s albedo Evans_1969 (15). We sampled our simulation at every
$40$ kHz to match the frequency resolution of the data.

Figure 3: The variation of FM station count during the ON-Moon observations
made on all three Epochs ($30^{\rm th}$ August, $26^{\rm th}$ September and
$21^{\rm st}$ December 2015). It can be seen that the station counts change
significantly on all three epochs during the observation. The data points are
generated every $\approx 230$ seconds to match the observation time.

##  6 Results

###  6.1 Estimating $S_{m}(\nu)$

In the previous steps of modelling (§4), we separated out the specular
component from the Moon. The removal of the remaining diffuse component
$(S_{\rm diffuse})$ will provide an estimate of the flux density of the Moon
$(S_{m}(\nu))$,

$S_{m}(\nu)=S_{\rm disk}(\nu)-S_{\rm diffuse}(\nu)$ (6)

The reflected power of the specular and the diffuse components have a
frequency dependence (see eq. 31 and 32 from Evans_1969 (15)). Applying these
equations to the radar equation (eq. 24 from Evans_1969 (15), or a similar
equation described herein 5.2), one can describe a relation between specular
and diffuse components as,

$S_{\rm diffuse}(\nu)=R_{e}(\nu)S_{\rm spec}(\nu)$ (7)

Due to the usefulness of the FM band in our analysis, we defined $R_{e}(\nu)$
for a scaling frequency $(\nu_{0}=100$ MHz) in the middle of the FM band, with
the power-law index 0.58 arriving from the frequency dependence (see eq. 31
and 32 from Evans_1969 (15)).

$R_{e}(\nu)=\frac{S_{\rm diffuse}(\nu_{0})}{S_{\rm
spec}(\nu_{0})}\left(\frac{\nu}{\nu_{0}}\right)^{0.58},\nu_{0}=100\rm~{}MHz$
(8)

Once the $S_{\rm diffuse}(\nu)$ is estimated $S_{m}(\nu)$ can evaluated using
eq. 6 . Next describes the two methods we used to estimate $S_{\rm
diffuse}(\nu)$ as the final Earthshine mitigation step before estimating the
foreground sky temperatures.

####  6.1.1 From DATA

In the first method, the flux-density $S_{\rm diffuse}(\nu_{0})$ is obtained
by fitting a line to the $S_{\rm disk}(\nu)$ component (see fig.4, yellow
fitted line on the top panel). The fitted value at $\nu_{0}$ represents
$S_{m}(\nu_{0})$, and rearranging eq.6 can provide $S_{\rm diffuse}(\nu_{0})$.
Once we obtained the $R_{e}(\nu)$, we plugged it into eq.7 and determined
$S_{\rm diffuse}(\nu)$ for the entire frequency band, and finally we remove
the Earthshine component $S_{\rm diffuse}(\nu)$ from the $S_{\rm disk}(\nu)$
and estimated $S_{m}(\nu)$ using eq. 6.

Figure 4: The observed flux density of the disk component in the top panel and
the quasi-specular Earthshine component in the middle panel. The bottom panel
shows the flux density of the Moon after performing the Earthshine mitigation
using two methods. The blue errorbars correspond to the Earthshine mitigation
of the first kind, where we used the fitted value of $S_{\rm diffuse}(\nu)$ at
$\nu_{0}$ to determine $S_{m}(\nu)$. The line fitted to the disk component is
shown with yellow dashes in the top panel. The second method is shown by the
pink errorbars, where the simulated value of $S_{\rm diffuse}(\nu)$ at
$\nu_{0}$ is used to estimate $S_{m}(\nu)$. The data points are at 40 kHz fine
channels and averaged over 12 full-band MWA observations from the first epoch.
The corresponding errors are obtained using the inverse variance weighted
scheme. The black vertical dashed line is at 150 MHz, which corresponds to the
frequency where the Moon first appears in emission.

####  6.1.2 From Simulation

In the Earthshine simulations, we first calculated the number of FM stations
from where the Moon was above the horizon. This was used to estimate the total
RFI reflection from the disk of the Moon. It can be seen (see fig. 13) that
the reflected flux density received at the MWA increases with increasing
station count. For an entire ON-Moon observation epoch, we simulated the
Earthshine for every ON-Moon observation (see the variation of FM station
counts as it changes over the entire observation duration of $\approx 3$ hours
in fig. 3). However, in our analysis, we used the simulated Earthshine only at
those timestamps when our ON-Moon observations were in the FM band (i.e. when
the 30.72 MHz contiguous band included the FM frequencies). The simulated
Earthshine flux density can be used to replace the flux density from the data
in the FM band (here, we refer to $S_{\rm diffuse}$ between $88-110$ MHz as
the data which were evaluated using eq. 7).

To statistically quantify the simulations and the data we performed a simple
T-test on the data and the simulation. Our null hypothesis was based on the
argument that the two discrete samples are drawn from the same distribution.
We used the T-test with a confidence of $95\%$, which means we rejected the
null hypothesis if the $p$-values were less than 0.05 and accepted otherwise.
Our sample set comprised the LST variation (across a given observing epoch) of
the flux density of the data and simulation with the frequency resolution of
40 kHz and matching LST cadence to the observing epochs (note that while
estimating the simulated flux density we did not account for the integration
time but rather estimated the instantaneous flux density at the middle of each
observation). The $S_{\rm diffuse}$ from the data is obtained by performing a
similar fitting (mentioned as the first approach) to $S_{\rm disk}$ for all 34
full-band individually which also provides the LST variation of $S_{\rm
diffuse}$. Once the flux density $S_{\rm diffuse},S_{\rm FM}$ for every
individual epoch is evaluated we performed the T-test between them at every
$40$ kHz fine channel independently. The estimated $p$-values on the dataset
from the first epoch are shown in fig. 14. As we are limited by our FM
catalogue and simulations, our null hypothesis was rejected in nearly half of
the fine frequency channels. We note that the simulations were generated at
the frequency resolution of the data (i.e. 40 kHz) with the assumption that
the FM stations transmit constant power throughout the 180 kHz bandwidth, so a
constant flux density was estimated by the simulations, which is not true in
case of the data (see fig. 5 showing the flux-density of the data $(S_{\rm
diffuse})$ and simulation $(S_{\rm FM})$). Therefore, instead of directly
subtracting the reflected diffuse FM RFI component from the data, we
restricted our analysis to the single frequency channel, in particular, the
central FM band $(\approx\nu_{0})$ and evaluated the $R_{e}(\nu)$ (from eq.8)
to perform the Earthshine mitigation. The estimated $p$-values at the middle
of the FM band $(\approx\nu_{0})$ are presented in Table 2.

${\rm Epochs}$ |  $p$-value at $(\approx\nu_{0})$
---|---
${\rm Aug.}$ | 0.75
${\rm Sept.}$ | 0.69
${\rm Nov.}$ | 0.78
Table 2: $p$-values from the T-test at the middle of the FM band
$\approx\nu_{0}$. Figure 5: The mean flux density of the data and simulation
in the FM band at the first observing epoch.

Since the $p$-values at $\nu_{0}$ satisfy our null hypothesis, in the second
method of Earthshine mitigation, we used the simulated value of $S_{\rm
FM}(\nu_{0})$ as the $S_{\rm diffuse}(\nu_{0})$ and estimated $R_{e}(\nu)$ and
used it in eq.7 and eq.6 to determine $S_{m}(\nu)$. The flux density of the
Moon obtained via both methods is shown in fig.4.

In both methods of Earthshine mitigation, we obtained the values of
$S_{m}(\nu)$ for every 34 full-band observations separately, and to obtain the
uncertainties in $\Delta S_{m}(\nu)$. We used the RMS noise of the disk and
quasi-specular components, the analysis of which is presented in the appendix.
Finally, we used the inverse-variance weights to obtain the mean and variance
in $S_{m}(\nu)$. We averaged $S_{m}(\nu)$ according to their corresponding
epochs. As a result, we had 3 realisations of $S_{m}(\nu)$ for the
corresponding 3 ON-Moon observing nights which were used in eq.1 to find the
temperature difference $\Delta T(\nu)$.

###  6.2 Estimating $T_{\rm refl-Gal}(\nu)$

We used a similar vector ray-tracing algorithm to Ben_2018 (28) to estimate
the reflected Galactic emission from the Moon. The algorithm assumes that the
observatory is located on the Moon and generates the sky map as observed from
the Moon. The part of the sky map that reflects back to MWA’s location is
determined by estimating the angle of incidence and reflection of the sky map
at the Moon’s surface based on the radar cross-section criteria. We used
Python-based PYGDSM to produce the Low-Frequency Survey Model (LFSM)
LWA1_LFSM_Dowell (13), Haslam Haslam_Remazeilles (44), Global Sky Model 2008
GSM2008_de_Costa (11), Global Sky Model 2016 Zheng_2016 (60) models for all
three ON-Moon observation epochs. The sky models were then injected into our
algorithm to produce the reflected sky map from the Moon. The reflected sky
does not change significantly during our observational epochs (spanning approx
$2-3~{}$hours), as the Moon sweeps only about $(\approx
1^{\circ}-1.5^{\circ})$ on the sky during each epoch. Therefore, we used the
middle point of time at each MWA’s full band observations to estimate the
reflected Galactic temperature. We produced the sky maps at every $5$ MHz
frequency channel from $70-180$ MHz. Fig. 6 shows the reflected Galactic
temperature $T_{\rm refl-Gal}$ at $150$ MHz in the middle of the first ON-Moon
observation epoch.

Figure 6: Galactic temperature from the GSM2016 model at the middle of the
second epoch. The left panel shows the occulted sky temperature $T_{\rm
Gal150}$ as observed from the MWA. The single dot represents the location of
the Moon during the ON-Moon observation. The right panel shows the sky
temperature reflected by the Moon $T_{\rm refl-Gal}$. To make the pixels
visible on the left figure, we doubled the pixel counts of the Moon and
saturated the colourbar by using the same colour scale as the right figure.

$T_{\rm refl-Gal}(\nu)$ follows a power law, and it can be shown that the
reflected Galactic temperature fits well with the power-law equation:

$T_{\rm refl-Gal}(\nu)=T_{\rm refl-Gal150}\left(\frac{\nu}{150\rm
MHz}\right)^{\beta}$ (9)

Figure 7: Mean reflected Galactic temperature estimated using GSM2016
Zheng_2016 (60) at all three epochs. The reflected Galactic temperature (shown
by point, triangle, and diamond markers) is fitted with the power law eq.9. It
can be seen that the reflected Galactic temperature at $150$ MHz $(T_{\rm
refl-Gal150})$ does not change significantly between the epochs. The error
bars corresponding to $T_{\rm refl-Gal150}$ represent the $5\%$ model
estimation error of GSM2016Zheng_2016 (60), and the uncertainty in the $\beta$
represents the fitting error.

The spectral index $\beta$ is obtained by fitting the mean and $5\%$ model
error Zheng_2016 (60) of the reflected Galactic temperature to the power-law
equation. Fig.7 shows the reflected Galactic temperature for all of the three
epochs. The fitted values of $T_{\rm refl-Gal150}$ and $\beta$, are provided
in Table 3.

${\rm Epochs}$ | $T_{\\{\rm refl-Gal150;~{}GSM2016\\}}$ | $\beta_{\rm GSM2016}$
---|---|---
${\rm Aug.}$ | $23.83\pm{1.19}\rm K$ | $-2.541\pm{0.003}$
${\rm Sept.}$ | $19.64\pm{0.98}\rm K$ | $-2.562\pm{0.003}$
${\rm Nov.}$ | $27.41\pm{1.37}\rm K$ |  $-2.511\pm{0.003}$
Table 3: Showing the fitted values of the $T_{\rm refl-Gal150}$ and reflected
Galactic spectral index $\beta$.

###  6.3 Estimating $T_{\rm Moon}$ and $T_{\rm Gal}(\nu)$

So far, we have evaluated the quantities $\Delta T$, $T_{\rm refl-Earth}$ and
$T_{\rm refl-Gal}$ of eq.2. Note that the $T_{\rm refl-Earth}$ has already
been removed from the data during the Earthshine mitigation process (described
in the previous section §6.1). Therefore, we are left with the $T_{\rm
Gal}(\nu)$, $T_{\rm Moon}(\nu)$, $T_{\rm EoR}(\nu)$ and $T_{\rm CMB}$
variables. Our present analysis focuses on combining the observations from
different epochs and checking whether we can produce better constraints on the
Moon’s intrinsic temperature from our previous work. Detecting the EoR would
require proper foreground modelling, more observations to increase the SNR of
the occulted sky patch and improved Earthshine models, which we aim to address
in future works. Therefore, we ignore the contribution of EoR in the sky
temperature. The remaining variables $T_{\rm Gal}(\nu)$ and $T_{\rm Moon}$,
are the temperature of the occulted patch of sky and the intrinsic temperature
of the Moon, respectively. The variables in eq.1 can be rearranged to:

$T_{\rm Gal}(\nu)-T_{\rm Moon}=T_{\rm refl-Gal}(\nu)-\Delta T(\nu)-T_{\rm
CMB}$ (10)

Here we have information on the RHS variables of the equation, and as we
considered the Moon to be at a constant temperature, it will act as a
temperature offset to the LHS. We fit the temperature difference between the
Galactic emission and Moon with a similar power law from eq. 9, but with a
constant temperature offset $(T_{\rm offset})$.

$T_{\rm Gal}(\nu)-T_{\rm Moon}=T_{\rm Gal150}\left(\frac{\nu}{150\rm
MHz}\right)^{\alpha}-T_{\rm offset}$ (11)

Figure 8: Occulted sky temperature from the dataset from the first epoch. The
data points shown by light blue and light pink represent the occulted sky
temperature obtained using the two Earthshine mitigation methods. The data
points fitted with $T_{\rm Gal}(\nu)$, are shown by the dashed lines (black
and pink) for the respective methods. The quoted values of the $T_{\rm
Gal150;~{}Fitting}$, $\alpha_{\rm Fitting}$ and $T_{\rm Moon}$ are obtained by
taking the inverse-variance weighted mean and variance. For comparison,
different GSM sky models are plotted over the data points. The model
uncertainty in the Galactic temperature and spectral index of GSM2016 are
shown in the blue text in the figure.

####  6.3.1 Fitting for $T_{\rm Moon}$ (Individual Epochs)

In order to obtain $T_{\rm Moon}$, first, we estimated the model values of the
$T_{\rm Gal}(\nu)$, and we followed the same procedure as for $T_{\rm ref-
Gal}(\nu)$. We generated the GSM, GSM2016, LFSM and Haslam maps at the
frequency resolution of 5 MHz for all ON-Moon observations and estimated the
$T_{\rm Gal}(\nu)$ of the patch of sky occulted by the Moon.
We fit the model Galactic temperature with a similar power-law equation to
$T_{\rm refl-Gal}$ (see §A.2 and eq.15, and obtained the values of $T_{\rm
Gal150;~{}sky-model}$, and $\alpha_{\rm Gal150;~{}sky-model}$. In fig.8, we
show the data from the first epoch. The $T_{\rm Gal}(\nu)$ is estimated using
both methods of Earthshine mitigation, and the fitted values of $T_{\rm
Gal}(\nu)$ at 150 MHz and the Galactic spectral index $(\alpha)$ are presented
for the data and GSM2016 model. A comprehensive table showing the fitted
values of $T_{\rm Gal}(\nu=150\rm~{}MHz)$ and $T_{\rm Moon}$ estimates from
all sky models is given in the appendix (see 7). Please note that, in order to
get the fitting for $T_{\rm Moon}$ and $T_{\rm Gal150}$ for a given sky model,
we first estimated the reflected sky temperature from the same sky model and
repropagated it through eq. 10 and 11.

####  6.3.2 Joint Fitting for $T_{\rm Moon}$

We combined the measured values of $\Delta T(\nu)$ from all three epochs and
performed a joint fit to eq.11. We placed restrictions on the $T_{\rm Moon}$
and considered the Moon to have a constant temperature. In the joint fit of
the data to eq.11, there is a total of six independent and one dependent
parameter. The best-fit values of $T_{\rm Gal150}$, spectral index $\alpha$
and $T_{\rm Moon}$, along with the estimations of the sky models, are
presented in Table 4. The results presented in Table 4 are for both the
Earthshine mitigation approaches. The table also shows the expected Galactic
foreground temperature and the spectral index (obtained from the GSM2016 model
Zheng_2016 (60)) of the patch of sky occulted by the Moon during these
observations. Please note that the larger error bars in fig.(8, 9, 10) at the
lower frequencies could arise due to several factors, the first being the
observations itself. Since the Moon moves $\approx 6.5^{\circ}$ with respect
to the background sky during a single night, the mean occulted galactic
temperature obtained over multiple individual spectra shows a larger
deviation. Second, the thermal noise and the noise from the sidelobe confusion
are more at the lower frequencies. In general, the sky noise is dominated by
the lower frequencies.

$\rm Method~{}1$
---
$\rm Epochs$ | $T_{\\{\rm Gal150;~{}GSM2016\\}}\rm~{}(K)$ | $\alpha_{\rm GSM2016}$ | $T_{\\{\rm Gal150;~{}fitting\\}}\rm(K)$ | $\alpha_{\rm fitting}$ | $T_{\\{\rm Moon\\}}\rm(K)$
$\rm Aug.$ | $242.7\pm{12.1}$ | $-2.621\pm{0.003}$ | $192.4\pm{3.1}$ | $-2.745\pm{0.031}$ |
$\rm Sept.$ | $241.0\pm{12.0}$ | $-2.585\pm{0.002}$ | $171.3\pm{2.8}$ | $-2.598\pm{0.033}$ | $184.4\pm{2.6}$
$\rm Dec.$ | $380.4\pm{19.0}$ | $-2.497\pm{0.002}$ | $243.5\pm{2.9}$ | $-2.612\pm{0.022}$ |
$\rm Method~{}2^{\rm~{}(FM~{}sim.)}$
$\rm Aug.$ | $242.7\pm{12.1}$ | $-2.621\pm{0.003}$ | $179.1\pm{2.9}$ | $-2.798\pm{0.033}$ |
$\rm Sept.$ | $241.0\pm{12.0}$ | $-2.585\pm{0.002}$ | $159.1\pm{2.7}$ | $-2.640\pm{0.034}$ | $173.8\pm{2.5}$
$\rm Dec.$ | $380.4\pm{19.0}$ | $-2.497\pm{0.002}$ | $232.0\pm{2.7}$ | $-2.661\pm{0.021}$ |
Table 4: Table showing the best-fit parameters from the joint-fitting of the
combined epochs. Figure 9: The data points from the three observational epochs
are shown with different colour schemes (light blue, light grey, light pink).
The black dashed line corresponds to the best fit to the occulted sky
temperature from the first epoch (August 2015) and is plotted along with the
blue errorbars of the GSM2016. The data points and fitted values are the
estimates obtained using the first method of Earthshine mitigation.  Figure
10: Occulted sky temperature measured using the second method of Earthshine
mitigation (FM simulation). Data points from all three epochs are presented
with coloured errorbars. The best fit to the first epoch (August 2015) dataset
is plotted with the black dashed line, and the corresponding values of the
GSM2016 model are plotted with the blue errorbars.

##  7 Discussion

As the $T_{\rm Gal}(\nu)$, $T_{\rm Moon}$ and the spectral index $\alpha$ are
deduced from the same fitting function, these parameters are highly correlated
with each other. The correlation amongst the parameters from the joint
analysis for the first method of Earthshine mitigation is shown in Table 6.

Figure 11: Measurement of the $T_{\rm Moon}$ from our analysis along with the
Ben_2018 (28) result and Table 2 from Krotikov_1964 (24). The values of
$T_{\rm Moon}$ from this work are presented as points at $150$ MHz.

We aim to deal with these degeneracies in future work when we include the
observational datasets from the Engineering Development Array (EDA) wayth_2017
(59) and measure the Moon temperature independently, which can provide better
constraints on the Galactic foregrounds. Our finding of the Moon’s
temperature, along with Ben_2018 (28) and results from Table $2$ of
Krotikov_1964 (24), are shown in fig.11. In our joint analysis, we obtained
the Moon temperature from the first method to be $184.4\pm{2.6}\rm K$ and
$173.8\pm{2.5}\rm K$ from the second method, respectively. These estimates are
inconsistent with each other. The deviation in $T_{\rm moon}$ between the two
methods is $\approx 10\rm K$, which could be due to imperfect simulations;
however, these results put tighter constraints on the Moon temperature and
agree with the results of Ben_2018 (28), which predicted $T_{\rm
moon}=180\pm{12}\rm K$. On the other hand, similar to Ben_2018 (28), our
estimates of the occulted sky temperature at 150 MHz ($T_{\rm Gal150}$)
underestimated the model predictions. Our results agree within $2\sigma$
uncertainty levels when compared with the GSM2016 sky model Zheng_2016 (60)
(note that here we used the $\sigma$ levels by considering the uncertainties
as Gaussian around the mean predicted values of $T_{\rm Gal150}$).

When we compared our best-case scenario for a single epoch with the GSM2016
model, our estimates of $T_{\rm Gal150}$ had $\approx 19\%$ error with the
model predictions. In comparison, the same was $\approx 12\%$ in Ben_2018 (28)
results. It can be argued that our frequency resolution of 40 kHz, which is
higher compared to Ben_2018 (28) (1.28 MHz), could have provided excess noise
in the flux density estimation. The Galactic spectral index $(\alpha)$, when
compared to the corresponding GSM2016 predicted values, agrees within $5\%$
uncertainty levels in all three observation scenarios in the first method and
$7\%$ uncertainty in the second method. The spectral index measured at all
three epochs is consistent with the findings of Ben_2018 (28). When comparing
the spectral index from other global sky models (see. fig.8), our results
agreed, except for the LFSM, which predicted a higher steepness. We note that
the spectral index changes from one part of the sky to another.

We compared our joint-fit results with the individual fits to the epochs.
Since the three epochs are at different LSTs, one can argue how the joint fit
will affect the $T_{\rm Gal150}$ estimates of individual epochs. We see that
the deviation in the mean temperature estimates from the individual epoch
$T_{\rm Gal150}$ to the jointly fitted values is only about $2-3$K (about 1-2
per cent) and agrees with the estimated uncertainties. A comprehensive table 7
in the appendix section shows the individual fitting cases to all models. We
also compared the joint $T_{\rm Moon}$ estimates from all the sky models. The
variation in the joint $T_{\rm Moon}$ estimates is about $1-2$ K between
different models. Although all cases are consistent with each other, the
GSM2016 model provides the best estimates of the uncertainties. Table 8 (Left)
shows the $T_{\rm Moon}$ estimates from all sky models. Since the Earthshine
significantly contaminates the FM band, a rectification of the data,
discarding the FM and other RFI-affected bands, can be made to check whether
it improves our estimates or not. Although the FM frequencies are expected to
cover a crucial portion of the CD-EoR phase transition, we compared the
estimates of full-band with the FM-removed cases as a test case scenario for
the GSM2016 sky model. We found that for the individual epochs, the fitting
uncertainties on the Galactic Temperature and Moon Temperature have variations
of only about $1-2$ K, while the mean temperature varies from $1-10$ K. Also,
the joint estimate of the same yields similar levels of uncertainty. The
constraints from full-band and test-case can be considered up to a similar
level since both are consistent within the estimated uncertainty. However, it
can be seen that discarding the FM band neither improves the estimates and nor
significantly underestimates the occulted sky temperature. Table 8 (Right)
shows the fitted values of $T_{\rm Moon}$ and $T_{Gal150}$ for the FM-removed
scenario.

At present, we used three different nights of MWA-phase I data. However, it
would be valuable the see whether the same method can be used to assimilate
multiple nights and perform foreground mitigations. We leave this to future
works, where we aim to include more observations from different observing
nights and with varying Moon elevations with respect to the Galactic plane.

###  7.1 Limitations

In this part, we address various limitations of our present work, which can be
dealt with in future works. Our present work includes MWA-phase I observations
from multiple nights. However, in the future, the first requirement would be
to check the nature of the observations before incorporating multiple datasets
into the data processing. The location of the Moon in the sky during the
observation can significantly alter the quality and usefulness of the data. It
would be preferred to have the Moon situated near the zenith during the ON-
Moon observation (if the analysis is done based on the image plane). One
limitation that we faced during our data processing was the drift and shift
method of the MWA observation. As the Moon was not actively tracked by MWA, it
resulted in the beam-former settings creating the beam slightly away from the
location of the Moon. As a result, we had to correct the beam response in the
images and perform an additional rectification on the data. We chose only
datasets where the beam response was greater than that at FWHM, which limited
the total integration time. The EoR signal is significantly weak compared to
the foregrounds; therefore, one must deal with precise calibration of the
foreground, which can create huge issues if done incorrectly. The presence of
bright foreground or some missing sources in the main or side-lobes can
completely contaminate the images. Therefore, better and more accurate sky
models are required to get good calibration results. Also, there is a
possibility that the Moon can obscure a bright foreground source during some
ON-Moon observations, which can result in calibration anomalies; therefore, a
careful selection of the observation is required prior to data processing. We
see that the angular size of the specular RFI (number of pixels acquired by
the specular reflection) at a given frequency does vary with LST. Since we
used the same specular RFI mask in the modelling process, it underestimated
the quasi-specular component. LSTs having a wider angular size of the specular
reflection leaked a number of pixels which still were dominated by the
specular RFI outside the model. This can be seen in the image residuals (see
fig.1) having $\approx 10-15\%$ flux density to the respective quasi-specular
models. We used a catalogue of FM transmitters to model the reflected power
received at the MWA from all the FM stations at the time of the ON-Moon
observation. We made two basic assumptions in order to generate the
Earthshine. First, we assumed that the FM stations always remain active and
transmit isotropically. Second, we counted all the stations from where the
Moon was above the horizon at the time of the observation. Although these
assumptions were made due to incomplete information on the beam patterns and
operating hours of the FM stations, it certainly limits our analysis of the
Earthshine mitigation using simulation (it can be seen in fig.5 that the
observed flux density fluctuates significantly within the assumed bandwidth of
the FM stations which otherwise is constant in the simulation). In reality,
the FM transmitters beam the signal along the horizontal direction and have a
significant signal loss $(\approx 20\rm dB)$ within $10^{\circ}$ tangential to
the beam direction. This directivity and beam pattern of the station can
significantly reduce the station counts and allow only stations having a Moon
altitude of approximately $10^{\circ}$ (near the horizon). Taking into account
this effect would replace the $4\pi$ factor from the eq.5 with the beam angle
$\theta$, which can significantly alter the simulated flux density. Using
that, we can also calculate the specular reflections using a vector ray
tracing algorithm. We can assume a $\approx 16^{\prime\prime}$ region around
the centre of the Moon to act as a smooth FM reflector and isolate the FM
stations which satisfy the reflection criteria. Therefore, one can argue that
the accuracy of the FM simulation would significantly alter based on the
assumptions and number of factors contributing to the Earthshine.

###  7.2 Earthshine Avoidance

As we have seen in our simulation results (see fig. 13), increasing station
count increases the overall reflected flux density from the Moon; thus, it can
be said that the reflected FM from the Moon correlates with the Earth’s
terrestrial area (i.e. the land area) being exposed to the Moon. In contrast,
a minor reflected FM contribution would be there if the marine parts of the
Earth were facing the Moon. Therefore, in Earthshine avoidance, we can use
guided simulations to choose suitable observing windows when the FM station
count is minimised. This would help us to reduce the strong FM RFI from the
Moon, which otherwise hugely contaminates the flux received from the Moon.
Fig. 12 shows a two-month simulation between November to December 2023 of the
FM station count as a proxy for the reflected RFI from the Moon. The
simulation targeted only the nighttime between $20:00-04:30$ hrs. as it is
relevant for our work and used the location of the Moon as seen by the MWA and
FM stations. We counted only those time frames when the Moon was above the
horizon at both the MWA and the FM station during the observing window
(between $20:00-04:30$ hrs) every night. There are missing data points in the
middle of figure 12, which indicate that the Moon is below the horizon at the
MWA site during the observing window. The station counts in the figure show
repetitive behaviour over a month’s time. During certain days, the station
counts are significantly less than others; hence, we can utilise such time
windows for scheduling future observations of the Moon.

Figure 12: Simulation of the station counts between November and December
2023. The variation of station counts roughly repeats over a month. The
individual spikes in the plot correspond to a single night’s observation
between 20:00-4:30 hrs, with a time separation of 30 mins. The zoomed figure
on the top right shows the variation of station counts over a single night
from 13th December 2023.

###  7.3 Future prospects for 21-cm signal measurements

Our current estimates motivate us to explore other techniques of Earthshine
mitigation in our future work. The RFI avoidance technique might be helpful to
deal with the strong reflected RFI and, therefore, can provide much-improved
estimates of the observable quantities. However, we also wish to investigate
whether the inclusion of the station beam patterns improves our modelling and
Earthshine mitigation. In our immediate future work, we aim to analyse our
latest observation of the Moon using MWA-phase II extended from 2022. MWA-
phase II extended has a higher angular resolution, and it can be helpful in
improving the total SNR in the FM band. Based on the results, it would be
valuable to check whether we can perform foreground mitigation techniques on
the phase-II data. We can utilise a similar approach as followed by the SARAS
project to quantify the acceptable regimes of the global 21-cm models or
alternatively explore Principle Component Analysis (PCA), Gaussian Process
Regression (GPR), and Machine Learning (ML) techniques Tauscher_2021 (51, 26,
61) to isolate the EoR component from our data. In addition to that, we can
explore new techniques based on the simulations of the FM station counts
during the given nights. We can choose two distinct nights where the behaviour
of the station counts matches (hence the reflected RFI), train machine
learning models with the datasets from one night, and use the trained models
to mitigate the Earthshine on the other. However, in such methods, the models
are required to be trained up to very high accuracy. It would be worth
checking the goodness of such trained models if operated on the datasets from
other nights or different instruments (e.g. EDA-2).

##  8 Conclusion

The lunar occultation technique aims to utilise radio interferometers to
estimate the sky-average global 21-cm EoR signal. However, one must accurately
mitigate the Galactic foreground and reflected Earthshine in order to reach
the final goal of detecting the global 21-cm signal. In this work, we used two
approaches to model the FM Earthshine from the Moon. We started with six
nights of LST-locked ON and OFF Moon observations at $40$ kHz fine channel
resolution and generated difference images of the Moon between $\approx
70-180$ MHz. We observed that the individual observations around the FM band
were highly contaminated by the Earthshine, which otherwise gets suppressed at
coarse channels. We assumed two Earthshine components (the diffuse disk and
quasi-specular) in the Moon’s emission Ben_2018 (28) and used the relation
between the specular and diffuse components from Evans_1969 (15) to estimate
the flux density of the Moon. We took two different approaches to get the
final estimates of the flux density of the Moon. In the first approach,
similar to Ben_2018 (28), we utilised the data itself to mitigate the
Earthshine from the data, whereas, in the second approach, we used FM
simulations to mitigate the Earthsine from the data. We used an FM catalogue
of radio transmitters across the Earth to estimate the simulated diffuse
Earthshine from the disk of the Moon. Using these methods, we estimated the
flux density of the Moon (fig. 4 bottom panel) and converted it into the
brightness temperature (ref. eq. 1). The brightness temperature is the measure
of the difference between the intrinsic temperature of the Moon $(T_{\rm
Moon})$ and the Galactic foreground temperature $(T_{\rm Gal}(\nu))$. We have
assumed that the Moon had a constant temperature over our desirable
frequencies; therefore, we fit the observed brightness temperature from eq.10
with a modified Galactic power-law equation which had an additional factor of
constant temperature offset $(T_{\rm offset})$ (see eq.11). The offset
temperature measured the $T_{\rm Moon}$. We jointly estimated the $T_{\rm
Gal}(\nu)$, $T_{\rm Moon}$ and spectral index $\alpha$ by combining all of the
three observation epochs while putting a restriction on the $T_{\rm Moon}$. In
our joint analysis, we were able to recover the Galactic spectral index
$(\alpha)$ of the occulted sky within $5-7\%$ level of the GSM2016 estimates.
Also, our estimates of the $T_{\rm Moon}$ ($184.40\pm{2.65}\rm K$ and
$173.77\pm{2.48}\rm K$ for the first and second Earthshine mitigation methods,
respectively) provided tighter constraints and are consistent with the
previous results from Ben_2018 (28).

This shows that we can include multiple nights of data and can strengthen the
total SNR of our analysis. The next step would be to check the performance of
the foreground subtraction techniques on the data. The estimation of the
global 21-cm signal is the ultimate goal of our work. Tackling the prior
difficulties of these approaches can put us one step closer to understanding
the CD-EoR.

This project is supported by an ARC Future Fellowship under grant FT180100321.
This research was partially supported by the Australian Research Council
Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D),
through project number CE170100013. The International Centre for Radio
Astronomy Research (ICRAR) is a Joint Venture of Curtin University and The
University of Western Australia, funded by the Western Australian State
government. The MWA Phase II upgrade project was supported by Australian
Research Council LIEF grant LE160100031 and the Dunlap Institute for Astronomy
and Astrophysics at the University of Toronto. This scientific work makes use
of Inyarrimanha Ilgari Bundara, the CSIRO Murchison Radio-astronomy
Observatory. We acknowledge the Wajarri Yamatji people as the traditional
owners and native title holders of the Observatory site. Support for the
operation of the MWA is provided by the Australian Government (NCRIS), under a
contract to Curtin University administered by Astronomy Australia Limited. We
acknowledge the Pawsey Supercomputing Centre, which is supported by the
Western Australian and Australian Governments. Data were processed at the
Pawsey Supercomputing Centre.

##  9 Data Availability

The data used in this work will be made available upon reasonable request.

## References

  * (1) Zara Abdurashidova  “First Results from HERA Phase I: Upper Limits on the Epoch of Reionization 21 cm Power Spectrum”, 2021  arXiv:2108.02263 [astro-ph.CO]
  * (2) N. Aghanim  “Planck 2018 results. VI. Cosmological parameters” [Erratum: Astron.Astrophys. 652, C4 (2021)]  In _Astron. Astrophys._ 641, 2020, pp. A6  DOI: 10.1051/0004-6361/201833910
  * (3) R. Barnett et al.  “Observations of the Lyman series forest towards the redshift 7.1 quasar ULAS J1120+0641”  In _Astron. Astrophys._ 601, 2017, pp. A16  DOI: 10.1051/0004-6361/201630258
  * (4) G. Bernardi et al.  “Bayesian constraints on the global 21-cm signal from the Cosmic Dawn”  In _mnras_ 461.3, 2016, pp. 2847–2855  DOI: 10.1093/mnras/stw1499
  * (5) Somnath Bharadwaj and SK. Ali  “On using visibility correlations to probe the HI distribution from the dark ages to the present epoch. 1. Formalism and the expected signal”  In _mnras_ 356, 2005, pp. 1519  DOI: 10.1111/j.1365-2966.2004.08604.x
  * (6) Judd D. Bowman and Alan E.. Rogers  In _nat_ 468.7325 Springer ScienceBusiness Media LLC, 2010, pp. 796–798  DOI: 10.1038/nature09601
  * (7) Judd D. Bowman, Alan E.. Rogers and Jacqueline N. Hewitt  “Toward Empirical Constraints on the Global Redshifted 21 cm Brightness Temperature During the Epoch of Reionization”  In _apj_ 676.1, 2008, pp. 1–9  DOI: 10.1086/528675
  * (8) Judd D. Bowman et al.  “An absorption profile centred at 78 megahertz in the sky-averaged spectrum”  In _nat_ 555.7694, 2018, pp. 67–70  DOI: 10.1038/nature25792
  * (9) Jack O. Burns, Stuart Bale and Richard F. Bradley  “Dark Cosmology: Investigating Dark Matter and Exotic Physics using the Redshifted 21-cm Global Signal with the Dark Ages Polarimeter Pathfinder (DAPPER)”  In _American Astronomical Society Meeting Abstracts #234_ 234, American Astronomical Society Meeting Abstracts, 2019, pp. 212.02
  * (10) J.R. Carson  “Notes on the Theory of Modulation”  In _procire_ 10.1, 1922, pp. 57–64  DOI: 10.1109/JRPROC.1922.219793
  * (11) Angelica de Oliveira-Costa et al.  “A model of diffuse Galactic radio emission from 10 MHz to 100 GHz”  In _mnras_ 388.1, 2008, pp. 247–260  DOI: 10.1111/j.1365-2966.2008.13376.x
  * (12) David R. DeBoer  “Hydrogen Epoch of Reionization Array (HERA)”  In _Publ. Astron. Soc. Pac._ 129.974, 2017, pp. 045001  DOI: 10.1088/1538-3873/129/974/045001
  * (13) Jayce Dowell et al.  “The LWA1 Low Frequency Sky Survey”  In _mnras_ 469.4, 2017, pp. 4537–4550  DOI: 10.1093/mnras/stx1136
  * (14) Michael W. Eastwood et al.  “The 21 cm Power Spectrum from the Cosmic Dawn: First Results from the OVRO-LWA”  In _AJ_ 158.2 American Astronomical Society, 2019, pp. 84  DOI: 10.3847/1538-3881/ab2629
  * (15) J.. Evans  “Radar Studies of Planetary Surfaces”  In _ARAnA_ 7.1, 1969, pp. 201–248  DOI: 10.1146/annurev.aa.07.090169.001221
  * (16) John V. Evans and Tor Hagfors  “Study of radio echoes from the Moon at 23 centimeters wavelength”  In _Journal of Geophysical Research_ 71, 1966, pp. 4871–4889  URL: https://api.semanticscholar.org/CorpusID:129777556
  * (17) Xiaohui Fan  “A Survey of z ¿ 5.7 quasars in the Sloan Digital Sky Survey. 2. Discovery of three additional quasars at z ¿ 6”  In _Astron. J._ 125, 2003, pp. 1649  DOI: 10.1086/368246
  * (18) Steven Furlanetto, S. Oh and Frank Briggs  “Cosmology at Low Frequencies: The 21 cm Transition and the High-Redshift Universe”  In _Phys. Rept._ 433, 2006, pp. 181–301  DOI: 10.1016/j.physrep.2006.08.002
  * (19) Steven R. Furlanetto, Aaron Sokasian and Lars Hernquist  “Observing the reionization epoch through 21-centimetre radiation”  In _mnras_ 347.1, 2004, pp. 187–195  DOI: 10.1111/j.1365-2966.2004.07187.x
  * (20) Simon Haykin  “An intro to analog and digital communications”  Wiley New York, 1989, pp. xvi652 p.  URL: https://nla.gov.au/nla.cat-vn431517
  * (21) C.. Hogan and M.. Rees  “Spectral appearance of non-uniform gas at high z.”  In _mnras_ 188, 1979, pp. 791–798  DOI: 10.1093/mnras/188.4.791
  * (22) C.. Jordan et al.  “Characterization of the ionosphere above the Murchison Radio Observatory using the Murchison Widefield Array”  In _mnras_ 471.4, 2017, pp. 3974–3987  DOI: 10.1093/mnras/stx1797
  * (23) E. Komatsu  “Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation”  In _ApJS_ 192, 2011, pp. 18  DOI: 10.1088/0067-0049/192/2/18
  * (24) V D Krotikov and V S Troitskiĭ  “RADIO EMISSION AND NATURE OF THE MOON”  In _Soviet Physics Uspekhi_ 6.6, 1964, pp. 841  DOI: 10.1070/PU1964v006n06ABEH003615
  * (25) J… Line et al.  “PUMA: The Positional Update and Matching Algorithm”  In _pasa_ 34 Cambridge University Press, 2017, pp. e003  DOI: 10.1017/pasa.2016.58
  * (26) T. Makinen et al.  “deep21: a deep learning method for 21 cm foreground removal”  In _jcap_ 2021.4, 2021, pp. 081  DOI: 10.1088/1475-7516/2021/04/081
  * (27) J.. Mather et al.  “Measurement of the Cosmic Microwave Background Spectrum by the COBE FIRAS Instrument”  In _apj_ 420, 1994, pp. 439  DOI: 10.1086/173574
  * (28) B McKinley et al.  “Measuring the global 21-cm signal with the MWA-I: improved measurements of the Galactic synchrotron background using lunar occultation”  In _mnras_ 481.4, 2018, pp. 5034–5045  DOI: 10.1093/mnras/sty2437
  * (29) B McKinley et al.  “LOW-FREQUENCY OBSERVATIONS OF THE MOON WITH THE MURCHISON WIDEFIELD ARRAY”  In _AJ_ 145.1 American Astronomical Society, 2012, pp. 23  DOI: 10.1088/0004-6256/145/1/23
  * (30) Matthew McQuinn et al.  “Cosmological parameter estimation using 21 cm radiation from the epoch of reionization”  In _ApJ_ 653, 2006, pp. 815–830  DOI: 10.1086/505167
  * (31) F.. Mertens  “Improved upper limits on the 21-cm signal power spectrum of neutral hydrogen at z 9.1 from LOFAR”  In _mnras_ 493.2, 2020, pp. 1662–1685  DOI: 10.1093/mnras/staa327
  * (32) F.. Mertens, B. Semelin and L… Koopmans  “Exploring the Cosmic Dawn with NenuFAR”, 2021, pp. 211–214  DOI: 10.48550/arXiv.2109.10055
  * (33) “Understanding the Epoch of Cosmic Reionization”  In _Understanding the Epoch of Cosmic Reionization: Challenges and Progress_ 423, Astrophysics and Space Science Library, 2016  DOI: 10.1007/978-3-319-21957-8
  * (34) Raul A. Monsalve, Alan E.. Rogers, Judd D. Bowman and Thomas J. Mozdzen  “CALIBRATION OF THE EDGES HIGH-BAND RECEIVER TO OBSERVE THE GLOBAL 21 cm SIGNATURE FROM THE EPOCH OF REIONIZATION”  In _ApJ_ 835.1 American Astronomical Society, 2017, pp. 49  DOI: 10.3847/1538-4357/835/1/49
  * (35) A.. Offringa, B. McKinley and Hurley-Walker  “WSClean: an implementation of a fast, generic wide-field imager for radio astronomy”  In _mnras_ 444.1, 2014, pp. 606–619  DOI: 10.1093/mnras/stu1368
  * (36) A.. Offringa et al.  “The Low-Frequency Environment of the Murchison Widefield Array: Radio-Frequency Interference Analysis and Mitigation”  In _pasa_ 32 Cambridge University Press (CUP), 2015  DOI: 10.1017/pasa.2015.7
  * (37) Gregory Paciga  “A refined foreground-corrected limit on the HI power spectrum at z=8.6 from the GMRT Epoch of Reionization Experiment”  In _mnras_ 433, 2013, pp. 639  DOI: 10.1093/mnras/stt753
  * (38) Gregory Paciga et al.  “The GMRT Epoch of Reionization experiment: A New upper limit on the neutral hydrogen power spectrum at z 8.6”  In _mnras_ 413, 2011, pp. 1174–1183  DOI: 10.1111/j.1365-2966.2011.18208.x
  * (39) Aaron R. Parsons  “New Limits on 21cm EoR From PAPER-32 Consistent with an X-Ray Heated IGM at z=7.7”  In _ApJ_ 788, 2014, pp. 106  DOI: 10.1088/0004-637X/788/2/106
  * (40) Nipanjana Patra, Ravi Subrahmanyan, Agaram Raghunathan and Udaya Narayana Rao  “SARAS: a precision system for measurement of the Cosmic Radio Background and signatures from the Epoch of Reionization”  In _Experimental Astronomy_ 36, 2012  DOI: 10.1007/s10686-013-9336-3
  * (41) L. Philip et al.  “Probing Radio Intensity at High-Z from Marion: 2017 Instrument”  In _JoAI_ 08.02, 2019, pp. 1950004  DOI: 10.1142/S2251171719500041
  * (42) Jonathan Pober et al.  “The Precision Array for Probing the Epoch of Reionization”  In _American Astronomical Society Meeting Abstracts 217_ 217, American Astronomical Society Meeting Abstracts, 2011, pp. 432.06
  * (43) Jonathan R. Pritchard and Abraham Loeb  “21-cm cosmology”  In _rpp_ 75, 2012, pp. 086901  DOI: 10.1088/0034-4885/75/8/086901
  * (44) M. Remazeilles et al.  “An improved source-subtracted and destriped 408-MHz all-sky map”  In _mnras_ 451.4, 2015, pp. 4311–4327  DOI: 10.1093/mnras/stv1274
  * (45) P.. Shaver, R.. Windhorst, P. Madau and A.. de Bruyn  “Can the reionization epoch be detected as a global signature in the cosmic background?”  In _aap_ 345, 1999, pp. 380–390  arXiv:astro-ph/9901320 [astro-ph]
  * (46) D.. Sigel et al.  “Deployable antenna concepts for the Dark Ages Radio Explorer mission”, 2013, pp. 1–1  DOI: 10.1109/USNC-URSI-NRSM.2013.6525025
  * (47) Saurabh Singh et al.  “On the detection of a cosmic dawn signal in the radio background”  In _nat_ 6, 2022, pp. 607–617  DOI: 10.1038/s41550-022-01610-5
  * (48) Saurabh Singh et al.  “First Results on the Epoch of Reionization from First Light with SARAS 2”  In _ApJ_ 845.2 American Astronomical Society, 2017, pp. L12  DOI: 10.3847/2041-8213/aa831b
  * (49) Saurabh Singh et al.  “SARAS 2 Constraints on Global 21 cm Signals from the Epoch of Reionization”  In _apj_ 858.1, 2018, pp. 54  DOI: 10.3847/1538-4357/aabae1
  * (50) Marcin Sokolowski et al.  “BIGHORNS - Broadband Instrument for Global HydrOgen ReioNisation Signal”  In _pasa_ 32, 2015, pp. e004  DOI: 10.1017/pasa.2015.3
  * (51) Keith Tauscher et al.  “Global 21 cm Signal Extraction from Foreground and Instrumental Effects. IV. Accounting for Realistic Instrument Uncertainties and Their Overlap with Foreground and Signal Models”  In _The Astrophysical Journal_ 915.1 American Astronomical Society, 2021, pp. 66  DOI: 10.3847/1538-4357/ac00af
  * (52) S.. Tingay et al.  “The Murchison Widefield Array: The Square Kilometre Array Precursor at Low Radio Frequencies”  In _pasa_ 30 Cambridge University Press (CUP), 2013  DOI: 10.1017/pasa.2012.007
  * (53) Cathryn M. Trott  “Comparison of Observing Modes for Statistical Estimation of the 21 cm Signal from the Epoch of Reionisation”  In _pasa_ 31, 2014, pp. e026  DOI: 10.1017/pasa.2014.23
  * (54) Cathryn M. Trott  “Deep multiredshift limits on Epoch of Reionization 21 cm power spectra from four seasons of Murchison Widefield Array observations”  In _mnras_ 493.4, 2020, pp. 4711–4727  DOI: 10.1093/mnras/staa414
  * (55) Cathryn M. Trott et al.  “Assessment of Ionospheric Activity Tolerances for Epoch of Reionization Science with the Murchison Widefield Array”  In _ApJ_ 867.1 The American Astronomical Society, 2018, pp. 15  DOI: 10.3847/1538-4357/aae314
  * (56) van Haarlem and others  “LOFAR: The LOw-Frequency ARray”  In _AnA_ 556, 2013, pp. A2  DOI: 10.1051/0004-6361/201220873
  * (57) H.. Vedantham et al.  “Lunar occultation of the diffuse radio sky: LOFAR measurements between 35 and 80 MHz”  In _mnras_ 450.3 Oxford University Press (OUP), 2015, pp. 2291–2305  DOI: 10.1093/mnras/stv746
  * (58) Tabitha C. Voytek et al.  In _ApJ_ 782.1 American Astronomical Society, 2014, pp. L9  DOI: 10.1088/2041-8205/782/1/l9
  * (59) Randall Wayth et al.  In _pasa_ Cambridge University Press, 2017, pp. e034  DOI: 10.1017/pasa.2017.27
  * (60) H. Zheng et al.  “An improved model of diffuse galactic radio emission from 10 MHz to 5 THz”  In _mnras_ 464.3, 2017, pp. 3486–3497  DOI: 10.1093/mnras/stw2525
  * (61) Shifan Zuo, Xuelei Chen, Reza Ansari and Youjun Lu  “21 cm Signal Recovery via Robust Principal Component Analysis”  In _aj_ 157.1, 2019, pp. 4  DOI: 10.3847/1538-3881/aaef3b

##  Appendix A Noise Estimation

###  A.1  For $S_{m}(\nu)$

Our analysis used 34 full-band observations (72-230 MHz), with each full-band
observation comprising 5 coarse-band observations (in the context of MWA, each
coarse-band observation is at 1.28 MHz resolution and 30.72 MHz wide). We
produced the beam-corrected images at every 40 kHz fine channel for each of
the observations. As a result, we had $768\times 5$ beam-corrected images for
each full-band observation. We evaluated the RMS noise for both disk and
quasi-specular components $(\Delta S_{\rm disk},\Delta S_{\rm spec})$ using
eq.4 and propagated these errors through eq.6 to estimate $\Delta S_{m}(\nu)$.
In eq. 6, we used the values of $S_{\rm disk}(\nu_{0})$ to get $S_{\rm
diffuse}(\nu_{0})$. We fitted a line to $S_{\rm disk}(\nu)$ and obtained
$S_{m}(\nu_{0})$ as the fitted value at $\nu_{0}$.

$S_{\rm diffuse}(\nu_{0})=S_{\rm disk}(\nu_{0})-Y_{\rm fit}(\nu_{0})$

The errors on the fitting parameters, namely the slope and intercept $(\Delta
m,\Delta c)$ of the fitted line, were used to obtain the fitting error of the
line at $\nu_{0}$.

$\Delta Y_{{\rm fit}}(\nu_{0})=\sqrt{(\Delta m\times\nu_{0})^{2}+\Delta
c^{2}}$

therefore, $\Delta S_{\rm diffuse}(\nu_{0})$ was evaluated as,

$\Delta S_{\rm diffuse}({\nu_{0}})=\\\ \sqrt{\Delta S_{\rm
disk}^{2}(\nu_{0})+\Delta Y_{\rm fit}^{2}(\nu_{0})}$

Please note that, as our T-tests satisfy the null hypothesis at $\nu_{0}$ in
all three epochs, we used the same value of uncertainty $\Delta S_{\rm
diffuse}(\nu_{0})$ in both methods of Earthshine mitigation. From there, we
estimated $\Delta R_{e}(\nu)$,

$\Delta R_{e}(\nu)=\left(\frac{\nu}{\nu_{0}}\right)^{0.58}\times\\\
\sqrt{\left(\frac{\Delta S_{\rm diffuse}(\nu_{0})}{S_{\rm
diffuse}(\nu_{0})}\right)^{2}+\left(\frac{\Delta S_{\rm spec}(\nu_{0})}{S_{\rm
spec}(\nu_{0})}\right)^{2}}$ (12)

and $\Delta S_{\rm diffuse}(\nu)$ was obtained using,

$\Delta S_{\rm diffuse}(\nu)=R_{e}(\nu)S_{\rm spec}(\nu)\times\\\
\sqrt{\left(\frac{\Delta R_{e}(\nu)}{R_{e}(\nu)}\right)^{2}+\left(\frac{\Delta
S_{\rm spec}(\nu)}{S_{\rm spec}(\nu)}\right)^{2}}$ (13)

Finally, $\Delta S_{m}(\nu)$ was estimated as,

$\Delta S_{m}(\nu)=\sqrt{\Delta S_{\rm disk}^{2}(\nu)+\Delta S_{\rm
diffuse}^{2}(\nu)}$ (14)

As a final result, we had $S_{m}(\nu)$ and $\Delta S_{m}(\nu)$ for all 34
full-band observations and propagated them as quadrature in the mean. We put
the estimated errors in eq.1 to estimate the occulted sky temperature and
errors therein.

###  A.2  For GSM $T_{\rm Gal}(\nu)$ and $T_{\rm refl-Gal}(\nu)$ models

We used GSM models to generate the sky maps in the middle of each 34 full-band
observations. The maps were generated to match the observed frequency range
with a frequency resolution of 5 MHz. In order to get the Galactic temperature
$(T_{\rm Gal150})$ and Galactic spectral index $(\alpha)$, we fit our model
with a power-law equation

$T_{\rm Gal}(\nu)=T_{\rm Gal150}\left(\frac{\nu}{150\rm MHz}\right)^{\alpha}$
(15)

We used the GSM2016 model to represent the Galactic temperature. We used the
model’s $5\%$ intrinsic map estimation error in the fitting and measured the
values of $T_{\rm Gal150}$ and $\alpha$ for all 34 full-band observations. The
fitted parameters $(T_{\rm Gal150},\alpha)$ and the uncertainties on the
fitted parameter $(\Delta T_{\rm Gal150},\Delta\alpha)$ values were propagated
in the quadrature rule. In a similar fashion, we estimated the uncertainties
in the $T_{\rm refl-Gal}(\nu)$ using the eq. 9 (to distinguish between the
spectral index parameter of the reflected Galactic power-law equation and the
Galactic power-law equation, we denoted the reflected spectral index parameter
in eq. 9 as $\beta$).

###  A.3 Reflected flux density variation with station count

We obtained the reflected FM flux density corresponding to every ON-Moon
observation comprising the FM band. The variation of the reflected FM flux
density changes with the number of FM stations. The increasing number of FM
stations can be understood as the more terrestrial land area being exposed to
the Moon. Fig. 13 showing the variation in the reflected flux density at
three-time stamps during the second observing epoch.

Figure 13: Reflected FM flux density at three different time stamps during the
second observing epoch.

###  A.4 T-test results

We estimated the $p$-values from the T-test at all fine-frequency channels .
As we already mentioned the limitation of our simulations, our T-tests
performed poorly on more than half of the fine-frequency channels. The null
hypothesis got rejected at those channels. However, we only required a single
value of $S_{\rm diffuse}$ at $\nu_{0}$ to estimate $R_{e}(\nu)$ and since our
T-test accepted the null hypothesis at $\approx\nu_{0}$ at all three epochs,
we utilised the values of simulated reflected flux density $S_{\rm
FM}(\nu_{0})$ as $S_{\rm diffuse}(\nu_{0})$. Fig. 14 showing the T-test
results on the dataset from the first epoch.

Figure 14: T-test results on the first epoch datasets. X-axis shows the FM
frequency band, and the Y-axis shows the corresponding $p$-values. The
horizontal line is at $p=0.05$, above which the null hypothesis is considered
to be accepted. The data points shown in blue colour show the accepted, and
the grey colour show rejected frequency channels. The $p-$value at
$\approx\nu_{0}$ is shown with lime colour.

###  A.5 Correlation between the fitted parameters

| $T_{\rm Gal150}$ | $\alpha$ | $T_{\rm offset}$
---|---|---|---
$T_{\rm Gal150}$ | 1 | 0.99 | 0.99
$\alpha$ | . | 1 | 0.99
$T_{\rm offset}$ | . | . | 1
Table 5: Correlation between the parameters of eq. 11 when fitted with the dataset from the first epoch using the first Earthshine mitigation method.  | $T_{\rm Gal150;~{}Aug.}$ | $T_{\rm Gal150;~{}Sept.}$ | $T_{\rm Gal150;~{}Dec.}$ | $\alpha_{\rm fitting;~{}Aug.}$ | $\alpha_{\rm fitting;~{}Sept.}$ | $\alpha_{\rm fitting;~{}Dec.}$ | $T_{\rm offset}$
---|---|---|---|---|---|---|---
$T_{\rm Gal150;~{}Aug.}$ | 1.000 | 0.965 | 0.954 | 0.914 | 0.797 | 0.806 | 0.992
$T_{\rm Gal150;~{}Sept.}$ | . | 1.000 | 0.932 | 0.911 | 0.819 | 0.790 | 0.973
$T_{\rm Gal150;~{}Dec.}$ | . | . | 1.000 | 0.897 | 0.770 | 0.848 | 0.958
$\alpha_{\rm fitting;~{}Aug.}$ | . | . | . | 1.000 | 0.753 | 0.761 | 0.936
$\alpha_{\rm fitting;~{}Sept.}$ | . | . | . | . | 1.000 | 0.653 | 0.804
$\alpha_{\rm fitting;~{}Dec.}$ | . | . | . | . | . | 1.000 | 0.812
$T_{\rm offset}$ | . | . | . | . | . | . | 1.000
Table 6: The correlation between the parameter of eq. 11 when fitted jointly
with combined datasets from all three epochs.

Our final estimates of occulted sky temperature $T_{\rm Gal150}$, Galactic
spectral index $\alpha$, and offset temperature $T_{\rm offset}$ (which
represents the $T_{\rm Moon}$) show a very high correlation with each other.
When checked on the single night dataset (from fig. 8), it showed a near unity
level of correlation, see table 5.

The sky position of the Moon is different at all three observing epochs;
hence, the galactic spectral index and occulted sky temperature are also
different at all three epochs. In the joint analysis, we considered these
parameters as independent parameters while restricting $T_{\rm offset}$
(assuming the Moon to have the same temperature across the epochs). The
correlation table (6) between all parameters still shows a high correlation,
however, the spectral index shows less correlation compared to the temperature
parameters.

$\rm Epoch$ | $\rm Sky-model$ | $T_{\\{\rm Gal150;~{}model\\}}\rm~{}(K)$ | $\alpha_{\rm model}$ | $\rm Fitting$ | $T_{\\{\rm Gal150;~{}fitting\\}}\rm~{}(K)$ | $\alpha_{\rm fitting}$ | $T_{\\{\rm Moon\\}}\rm~{}(K)$
---|---|---|---|---|---|---|---
| $\rm GSM$ | $250.4\pm{12.5}$ | $-2.540\pm{0.002}$ | $\rm Method~{}1$ | $199.2\pm{5.4}$ | $-2.70\pm{0.05}$ | $188.3\pm{4.6}$
|  |  |  | $\rm Method~{}2^{\rm~{}(FM~{}sim.)}$ | $183.5\pm{5.1}$ | $-2.79\pm{0.05}$ | $174.2\pm{4.4}$
| $\rm GSM2016$ | $242.7\pm{12.1}$ | $-2.621\pm{0.003}$ | . | $195.5\pm{5.3}$ | $-2.72\pm{0.05}$ | $186.1\pm{4.5}$
$\rm Aug.$ |  |  |  | .. | $179.2\pm{5.0}$ | $-2.82\pm{0.05}$ | $171.4\pm{4.3}$
| $\rm LFSM$ | $313.9\pm{15.7}$ | $-2.689\pm{0.003}$ | . | $202.8\pm{5.3}$ | $-2.70\pm{0.05}$ | $188.5\pm{4.5}$
|  |  |  | .. | $189.0\pm{5.1}$ | $-2.78\pm{0.05}$ | $176.2\pm{4.4}$
| $\rm Haslam$ | $253.7\pm{12.7}$ | $-2.603\pm{0.003}$ | . | $199.2\pm{5.3}$ | $-2.71\pm{0.05}$ | $187.9\pm{4.6}$
|  |  |  | .. | $183.7\pm{5.1}$ | $-2.80\pm{0.05}$ | $174.0\pm{4.3}$
| $\rm GSM$ | $253.4\pm{12.7}$ | $-2.540\pm{0.002}$ | . | $177.6\pm{4.4}$ | $-2.54\pm{0.04}$ | $189.4\pm{4.2}$
|  |  |  | .. | $185.6\pm{5.1}$ | $-2.43\pm{0.05}$ | $198.5\pm{4.9}$
| $\rm GSM2016$ | $241.0\pm{12.0}$ | $-2.585\pm{0.003}$ | . | $174.4\pm{4.3}$ | $-2.56\pm{0.04}$ | $187.4\pm{4.4}$
$\rm Sept.$ |  |  |  | .. | $182.2\pm{5.0}$ | $-2.47\pm{0.05}$ | $196.0\pm{4.8}$
| $\rm LFSM$ | $295.3\pm{14.8}$ | $-2.689\pm{0.003}$ | . | $180.8\pm{4.4}$ | $-2.56\pm{0.04}$ | $189.2\pm{4.2}$
|  |  |  | .. | $188.6\pm{5.0}$ | $-2.46\pm{0.04}$ | $198.0\pm{4.8}$
| $\rm Haslam$ | $250.7\pm{12.5}$ | $-2.603\pm{0.003}$ | . | $177.4\pm{4.4}$ | $-2.56\pm{0.04}$ | $188.9\pm{4.2}$
|  |  |  | .. | $185.2\pm{5.0}$ | $-2.45\pm{0.05}$ | $197.8\pm{4.8}$
| $\rm GSM$ | $395.8\pm{19.8}$ | $-2.496\pm{0.002}$ | . | $235.6\pm{5.4}$ | $-2.66\pm{0.04}$ | $173.8\pm{5.0}$
|  |  |  | .. | $239.5\pm{4.1}$ | $-2.60\pm{0.03}$ | $177.4\pm{3.8}$
| $\rm GSM2016$ | $380.4\pm{19.0}$ | $-2.497\pm{0.003}$ | . | $232.1\pm{5.4}$ | $-2.68\pm{0.04}$ | $172.2\pm{5.0}$
$\rm Dec.$ |  |  |  | .. | $235.7\pm{4.0}$ | $-2.62\pm{0.03}$ | $175.5\pm{3.7}$
| $\rm LFSM$ | $453.4\pm{22.7}$ | $-2.557\pm{0.003}$ | . | $238.7\pm{5.4}$ | $-2.68\pm{0.04}$ | $173.7\pm{5.0}$
|  |  |  | .. | $242.4\pm{4.0}$ | $-2.62\pm{0.03}$ | $177.2\pm{3.7}$
| $\rm Haslam$ | $414.2\pm{20.6}$ | $-2.603\pm{0.003}$ | . | $236.0\pm{5.4}$ | $-2.68\pm{0.04}$ | $173.4\pm{5.0}$
|  |  |  | .. | $239.7\pm{4.0}$ | $-2.62\pm{0.03}$ | $176.8\pm{3.7}$
Table 7: Table showing the best-fit parameters from all sky models for every
individual case.

$\rm Sky-model$ | $T_{\\{\rm Moon\\}}\rm~{}(K)$ | $\rm Fitting$
---|---|---
$\rm GSM$ | $186.5\pm{2.7}$ | $\rm Method~{}1$
| $176.7\pm{2.5}$ | $\rm Method~{}2^{\rm~{}(FM~{}sim.)}$
$\rm GSM2016$ | $184.4\pm{2.6}$ | .
| $173.8\pm{2.5}$ | ..
$\rm LFSM$ | $186.6\pm{2.7}$ | .
| $178.7\pm{2.5}$ | ..
$\rm Haslam$ | $186.1\pm{2.7}$ | .
| $176.4\pm{2.5}$ | ..

$\rm Epoch$ | $T_{\\{\rm Gal150\\}}\rm~{}(K)$ | $T_{\\{\rm Moon\\}}\rm~{}(K)$ | $\rm Joint~{}Fitting$
---|---|---|---
$\rm Aug.$ | $196.9\pm{6.1}$ | $187.6\pm{5.3}$ | $\rm Method~{}1$
| $179.5\pm{5.6}$ | $171.3\pm{4.8}$ | $\rm Method~{}2^{\rm~{}(FM~{}sim.)}$
$\rm Sept.$ | $174.8\pm{4.8}$ | $187.7\pm{4.6}$ | .
| $181.3\pm{4.9}$ | $195.1\pm{4.7}$ | ..
$\rm Dec.$ | $223.3\pm{5.3}$ | $164.5\pm{4.9}$ | .
| $236.2\pm{4.3}$ | $175.9\pm{4.0}$ | ..
$\rm Joint~{}Aug.$ | $192.6\pm{3.4}$ | $184.5\pm{2.9}$ | $\rm Method~{}1$
$\rm Sept.$ | $171.5\pm{3.2}$ | ” | .
$\rm Dec.$ | $243.2\pm{3.3}$ | ” | .
$\rm Joint~{}Aug.$ | $178.8\pm{3.0}$ | $173.3\pm{2.6}$ | $\rm Method~{}2^{\rm~{}(FM~{}sim.)}$
$\rm Sept.$ | $159.1\pm{2.8}$ | ” | ..
$\rm Dec.$ | $231.5\pm{2.9}$ | ” | ..

Table 8: Left: Joint-fit estimates of $T_{\rm Moon}$ from all sky models.
Right: GSM2016 model fit estimates for removed FM-band.