# Observing the Reionization : Effect of Calibration and Position Errors on
Realistic Observation Conditions

Aishrila Mazumder,1 Abhirup Datta,1 Arnab Chakraborty,1,2,3 Suman Majumdar1,4
1Department of Astronomy, Astrophysics and Space Engineering, Indian Institute
of Technology Indore, Indore 453552, India
2Department of Physics, McGill University, 3600 rue University, Montreal, QC
H3A 2T8, Canada
3McGill Space Institute, McGill University, 3550 rue University, Montreal, QC
H3A 2A7, Canada
4Department of Physics, Blackett Laboratory, Imperial College, London SW7 2AZ,
U. K E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Observation of the redshifted 21-cm signal from Cosmic Dawn and Epoch of
Reionization is a challenging endeavor in observational cosmology. Presence of
orders of magnitude brighter astrophysical foregrounds and various
instrumental systematics increases the complexity of these observations. This
work presents an end-to-end pipeline dealing with synthetic interferometric
data of sensitive radio observations . The mock sky model includes the
redshifted 21-cm signal and astrophysical foregrounds. The effects of
calibration error and position error in the extraction of the redshifted 21-cm
power spectrum has been simulated. The effect of the errors in the image plane
detection of the cosmological signal has also been studied. A comparative
analysis for array configurations like the SKA1-Low, MWA and HERA has been
demonstrated. The calibration error tolerance of the arrays, under some
assumptions about the nature of the systematic components, is optimally found
to be $\sim 0.01\%$ for the detection of the signal. For position errors, an
offset of $\gtrapprox 5\arcsec$ makes the residual foregrounds obscure the
target signal. These simulations also imply that in the SKA-1 Low performs
marginally better than the others in the image domain, while the same is true
for MWA in the power spectrum domain. This is one of the first studies that
compares performance of various radio telescopes operating under similar
observing conditions towards detecting the cosmological signal. This end-to-
end pipeline can also be extended to study effects of chromatic primary beam,
radio frequency inferences, foregrounds with spectral features, etc.

###### keywords:

methods: data analysis-methods: interferometric-radio

††pubyear: 2015††pagerange: Observing the Reionization : Effect of Calibration
and Position Errors on Realistic Observation Conditions–21

##  1 Introduction

The thermal history of the evolution of the Universe can be traced back to the
surface of the last scattering with the help of the Cosmic Microwave
Background (CMB). The evolutionary history of the Universe between the post-
recombination era until the period after the neutral intergalactic medium
(IGM) became ionized once again remains largely unexplored with observational
data. Theoretical models suggest that the Universe’s first structures- the
earliest stars and galaxies formed during this period (viz. redshifts between
$\sim$ 30 and $\sim$ 12). The first stars, generated at Cosmic Dawn (CD),
formed due to the small-scale fluctuations in matter density arising from the
gravitational instabilities. Ionizing radiations produced from these objects
ionized the IGM causing the last phase transition in the evolution of the
universe (Loeb & Barkana, 2001). This phase transition period is called Epoch
of Reionization (EoR) (for comprehensive reviews see Furlanetto et al. (2006);
Morales & Wyithe (2010); Pritchard & Loeb (2012); Loeb & Furlanetto (2013);
Barkana (2016); Dayal & Ferrara (2018)). Using quasar absorption spectrum (Fan
et al., 2006) and Thompson scattering optical depth (Planck Collaboration et
al., 2019), the extended Epoch of Reionization (EoR) is constrained between
the redshift interval of $6\lesssim z\lesssim 15$.

Understanding the conditions prevailing during these early stages of the
Universe requires a reliable probe that can trace thermal evolution history.
The hyperfine transition line of neutral hydrogen, with a rest wavelength of
21-cm, is the most promising probe into the IGM of these early cosmic epochs
(Field, 1958, 1959a, 1959b). The contrast of the brightness temperature
corresponding the 21cm line transition against the CMB gives a differential
brightness temperature which is the observable through the all sky-averaged
"global signal" (using a single total power radio telescope) or as
fluctuations in the spatial correlations (using radio interferometers) (Madau
et al., 1997). Radio interferometric observations are necessary for estimation
of astrophysical parameters and constraining the nature of reionizing sources.
Detection of an absorption trough in the global signal experiment centered at
78 MHz was reported by the Experiment to Detect Global Epoch of Reionization
Signature (EDGES) team (Bowman et al., 2018). Besides EDGES, single antenna
experiments like BIGHORNS (Sokolowski et al., 2015), SCI-HI (Voytek et al.,
2014), SARAS (Singh & Subrahmanyan, 2019), LEDA (Price et al., 2018) are also
aiming to detect the global signal but are yet to report any detection.

The most sensitive operational interferometers like the GMRT, MWA, LOFAR, HERA
have all set upper limits on the power spectrum (PS) amplitude of the signal
(Paciga et al., 2011; Beardsley et al., 2016; Gehlot et al., 2019; Trott et
al., 2020; Mertens et al., 2020; The HERA Collaboration et al., 2022) but and
no confirmed detection of the cosmological Hi 21cm signal. The cosmological
signal is prone to contamination due to bright astrophysical foregrounds
(Bharadwaj & Saiyad Ali, 2005; Jelić et al., 2008; Jelić et al., 2010; Zahn et
al., 2011; Chapman et al., 2015; Choudhuri et al., 2017; Chakraborty et al.,
2019; Mazumder et al., 2020). Other sources of contamination are Earth’s
ionosphere and several instrumental systematics. Substantial amount of focused
research has taken place over the past decade to understand and quantify each
components of these contamination, namely instrument model (de Lera Acedo et
al., 2017; Trott et al., 2017; Joseph et al., 2018; Li et al., 2018),
calibration (Offringa et al., 2015; Barry et al., 2016; Trott & Wayth, 2016;
Ewall-Wice et al., 2017; Patil et al., 2017; Dillon et al., 2018; Kern et al.,
2019; Kumar et al., 2020), foreground avoidance or removal (Datta et al.,
2010; Trott et al., 2012; Chapman et al., 2016; Vedantham et al., 2012;
Thyagarajan et al., 2015a, b; Mertens et al., 2018; Hothi et al., 2020), PS
estimation (Parsons et al., 2010; Parsons et al., 2012; Liu et al., 2014;
Choudhuri et al., 2014; Choudhuri et al., 2016; Offringa et al., 2019; Barry
et al., 2019) and Earth’s ionosphere (Jordan et al., 2017; Trott et al.,
2018). Using these advancements, highly sensitive next generation
interferometers like the Hydrogen Epoch of Reionization Array (HERA, DeBoer et
al. 2017) and the Square Kilometer Array (SKA1-Low, Koopmans et al. 2015) are
expected to detect the 21-cm signal and characterize the multi-redshift PS,
leading to tighter constrains on the astrophysical parameters in the early
Universe. The upcoming SKA1-Low is being designed to be sensitive enough to
detect the PS precisely. Additionally, it is also projected to be able to
create tomographic images of the HII regions (Mellema et al., 2014). The EoR
signal is inherently isotropic in spatial wavenumber (k) space. Additionally,
it also shows spectral structure. Thus, despite the spectrally smooth
foreground contamination can be distinguished from the foreground contaminants
(Morales & Hewitt, 2004; Morales et al., 2006; Bowman et al., 2009; Liu &
Tegmark, 2011; Parsons et al., 2012; Pober et al., 2013b; Dillon et al.,
2013). The spectrally smooth nature of the foregrounds, along with the
inherent instrumental chromaticity, keeps the contamination confined to the
"wedge" in the cylindrical Fourier space (i.e., in the 2D PS). The region
outside the wedge where the foregrounds are subdominant compared to the signal
is called the "EoR window" Morales et al. (2012). However, the interaction of
astrophysical foregrounds and the instrument causes the wedge power to leak
into the clean modes of the window- an effect called "mode mixing" (Morales et
al., 2012). If not properly mitigated, mode mixing is likely to confuse the
detection.

One of the major limiting systematic that causes the problem in cosmological
signal detection is improper calibration. In general, the most common
calibration approach for radio astronomy is "sky-based" calibration. Since
Cosmic Dawn/Epoch of Reionization (EoR) observations are limited to shorter
baselines, these observations suffer from poor angular resolutions and higher
confusion noise. This may lead to inaccurate sky models for calibration.
Errors or inaccuracies in this process propagate as residual errors and
ultimately hinder the target cosmological signal detection (Datta et al.,
2009, 2010; Barry et al., 2016; Ewall-Wice et al., 2017). The other
calibration approach being explored by various interferometers is the
redundant calibration approach . Telescopes like the HERA use the redundant
calibration approach. In this approach, repeated simultaneous measurements of
the interferometers’ redundant baselines can simultaneously solve for the
incoming sky signal and instrumental parameters. The redundant method
calibrates for real visibility by considering the prior that sky visibilities
are equal for redundant baselines, having the advantage of not requiring a sky
model for performing gain solutions for antennas. Nevertheless, this works
only for highly regular arrays and identical antenna beam patterns (Byrne et
al., 2019; Byrne et al., 2020).

It must be pointed out that irrespective of the accuracy of the calibration
approach employed, the ultimate limitation is set by the signal-to-noise ratio
(SNR). It has been shown in Datta et al. (2009), that calibration errors
$\sim$0.1% causes a drastic lowering of the dynamic range, thereby obscuring
the signal111Dynamic range is the ratio of the peak flux in the image and the
RMS noise in a source-free region. Datta et al. (2009) showed that with
calibration error of $\sim$0.1%, the dynamic range lowers to 105 compared to
the required 108. Thus, studies are necessary for estimating the extent of the
tolerance of the imperfections, which potentially contribute to the noise that
obscure the detection of the faint cosmological signal. This paper aims to
quantify this level of tolerance (or the accuracy of the calibration
algorithm) required for successful detection of the redshifted 21-cm signal
from the cosmic reionization using various sensitive telescopes.

Astrophysical foregrounds like diffuse galactic synchrotron emission, extra-
galactic free-free emission, and compact sources substantially affect the
interferometric data sets used for EoR signal recovery. Currently, most
studies focus on determining the best strategy- for avoidance, suppression, or
subtraction for handling the foregrounds. While diffuse emission is a
significant contaminant at large angular scales, EoR data sets also require
the careful handling of the extragalactic point sources. Inaccurate removal of
bright, compact sources can lead to large residuals in the data and obscures
the detection of the cosmological signal. Besides ionosphere, incomplete UV-
plane coverage gives rise to large sidelobes that contaminate the data. All
these corruption effects lead to constructing an imperfect sky model, which
causes residuals that adversely affects the cosmological signal recovery. The
study on the impact of incomplete sky model has been done by Datta et al.
(2009, 2010); Trott et al. (2012); Barry et al. (2016); Byrne et al. (2019).
However, there is still a lack of detailed study on how the effects compare
for different telescope arrays targeting the cosmological 21-cm signal from
reionization.

Our current work presents a proof of concept on developing an end-to-end
pipeline to simulate radio interferometric observations using a realistic sky
model and telescope observing parameters. The studies have been carried out
taking four array configurations - SKA1-Low core (stations around $\sim$ 2000
m of the central station), HERA 350 dish configuration, MWA Phase-1 (MWA1) 128
tile configuration, and MWA 256 tile configuration (MWA-256). This paper is
organized in the following manner: in Section 2, the detailed simulation
methodology is described. Section 3 discusses briefly the observational
effects considered in this work; Section 4 discusses the analyses formalism
which is followed by the Section 5; finally, the paper is concluded Section 7.

##  2 Synthetic Observations

This section describes the steps involved in the synthetic observation and
analysis pipeline developed. Figure 1 shows the blocks that are used for
generating the observations. The various simulation parameters and their
respective values are described in Table 1. In the following subsection, each
of the parameters in the input block is briefly described.

Figure 1: Schematic diagram of the observation pipeline. Table 1: Parameter values for the simulations Parameter | Value
---|---
Central Frequency | 142 MHz (z$\sim$9)
Bandwidth | 8 MHz
Number of frequency channels | 64
Field of view | 4∘
Number of array elements: |
SKA1-Low | 296
HERA | 350
MWA Phase-1 | 128
MWA-256 | 256
Maximum baseline (m): |
SKA1-Low |  $\sim$2000
HERA |  $\sim$880
MWA Phase-1 |  $\sim$2800
MWA-256 |  $\sim$5300
Synthesised beam (arcmin): |
SKA1-Low |  $\sim$2.5
HERA |  $\sim$8.1
MWA Phase-1 |  $\sim$2.6
MWA-256 |  $\sim$1.3
Thermal Noise (mJy $\mathrm{beam^{-1}}$): |
SKA1-Low |  $\sim$0.07
HERA |  $\sim$0.52
MWA Phase-1 |  $\sim$17.59
MWA-256 |  $\sim$8.79

The simulations have been performed using the OSKAR software (Dulwich et al.,
2009)222https://github.com/OxfordSKA/OSKAR/releases package for SKA1-Low
configuration and Common Astronomy Software Application (CASA, Jaeger
2008)333https://casaguides.nrao.edu/index.php?title=Main_Page for HERA, MWA-1
& 2\. The observations track the sky with phase center at $\alpha$=15h00m00s
and $\delta$=-30∘00′00″for 4 hours ($\pm$ 2 HA). The observing bandwidth is 8
MHz, with a channel separation of 125 kHz. For simplicity, the simulations are
noise-free. It should also be mentioned that throughout the paper, best fitted
cosmological parameters from the Planck 2018 results (Planck Collaboration et
al., 2019) are used: $\Omega_{\mathrm{M}}$ = 0.31, $\Omega_{\Lambda}$ = 0.68,
$\sigma_{8}$ = 0.811, $H_{0}$ = 67.36 km $\mathrm{s}^{-1}$
$\mathrm{Mpc}^{-1}$.

###  2.1 Telescope Model

Though several interferometers are targeting the detection of the redshifted
21-cm signal, this work undertakes a comparative analysis under identical
error conditions for four different arrays- SKA1-Low, HERA, and MWA1 & MWA2.
The telescope layouts are shown in Figure 2.

Figure 2: Telescope layouts used in the simulation. Clockwise from top left
are SKA1-Low (Low), HERA, MWA-256 & MWA-I respectively.

The SKA444https://www.skatelescope.org/, one of the most sensitive upcoming
telescopes, is expected to detect the signal statistically (Koopmans et al.,
2015) and also have just enough signal-to-noise for performing tomography
(Mellema et al., 2014). SKA will be built at two sites as two separate arrays,
a low-frequency array in Australia (frequency coverage between 50-350 MHz) and
a mid-frequency array in South Africa (frequency coverage between 350 MHz to
14 GHz). The redshifted 21-cm signal from the EoR is present at frequencies of
up to a few hundred MHz ($\sim$200 MHz at z $\sim$6) and thus will be targeted
by the former array (SKA1-Low). While the available documentation states that
the SKA1-Low will have 512 "stations" of $\sim$ 40m diameter spread around the
core or central station (having a maximum baseline length of $\sim$65 km, SKAO
2021), this work does not explore the entire layout. Since compact baselines
will be sensitive to the target signal, all stations within 2 km of the core
(i.e., a maximum baseline of 2000m around the central station) are used
(Dewdney & Braun, 2016). This results in 296 stations around the central
station, with a synthesized beam size of $\sim$2.5′at 142 MHz.

HERA555https://reionization.org/ is an interferometer designed to search for
the fluctuations in the 21-cm signal during the cosmic dawn and reionization
(DeBoer et al., 2017). Located in the South African Karoo Radio Astronomy
Reserve, the complete array would comprise 350 parabolic dishes of 14m
diameter each, spread in a highly redundant configuration (320 in a dense core
and 30 outriggers). The instrument is planned to be operational between
100-200 MHz. This work uses the entire 350 tile configuration resulting in a
maximum baseline of $\sim$880 m. At 142 MHz, this results in an
$\sim$8′synthesized beam.

MWA666https://www.mwatelescope.org/, an SKA precursor facility, is located at
the Murchison Radio Observatory in Australia (adjacent to the SKA1-Low site).
The array has an operating frequency range of 70-300 MHz. In the initial
deployment stage, the array had 128 square-shaped "tiles" (a $4\times 4$ array
of dipoles). In Phase 2, the 128 tile configuration was extended to include
additional 128 tiles (Wayth et al., 2018). The upgraded array has a compact
configuration with 56 tiles from earlier and 72 new tiles arranged in two
compact hexagons, which enhance EoR PS capacity (Wayth et al., 2018). For this
work, two MWA configurations are used- entire Phase 1 with 128 tiles and
entire Phase 2 with 256 tiles 777The MWA coordinates are available at
https://www.mwatelescope.org/telescope/configurations/phase-ii. The
synthesized beam-width at 142 MHz are $\sim$2.5′$\sim$1.3′respectively for
Phases 1 & 2 respectively.

###  2.2 The Hi 21-cm Maps

We use two semi-numerical approaches, 21cmFAST (Mesinger et al., 2011; Murray
et al., 2020) and ReionYuga (formerly known as Sem-Num) (Mondal et al., 2017;
Majumdar et al., 2014; Choudhury et al., 2009) to generate the redshifted
21-cm signal maps for our simulated observations. Both of these methods are
developed independently, and they use different approaches of excursion set
formalism to simulate the 21-cm maps from the EoR. Both of these methods have
been tested against full radiative transfer simulations and found to be
capable of simulating the signal power spectrum at large length scales with a
reasonable accuracy (Majumdar et al., 2014). They, however, show significant
differences at small length scale power spectrum. Our motivation behind using
two separate signal simulations is to test the robustness of the outcome of
our data analysis pipeline. It is going to use both extracted image and power
spectrum statistics for its evaluation.

21cmFAST generates the 21-cm maps from a matter density field obtained via
Zeldovich approximation. It uses a guided excursion set formalism to convert
the matter density field at a given redshift into an ionization field which is
finally translated into 21-cm brightness temperature fluctuations. The
excursion set approach allows it to generate multiple realizations of the
21-cm maps at a very low computing cost compared to a full radiative transfer
approach.

The ReionYuga or Sem-Num (a serial version of ReionYuga), on the other hand,
uses a N-body simulation to generate the matter distribution at a given
redshift and further uses a Friends-of-Friends algorithm to identify collapsed
halos in this matter distribution. These halos are then considered the hosts
of the first sources of lights (e.g., galaxies, quasars, etc.), which produces
the ionizing photons responsible for reionizing the Universe. The density of
the neutral hydrogen field (assuming that the hydrogen follows the dark matter
distribution) and the ionizing photon field are then estimated at a much
coarser grid compared to the original N-body simulation resolution, and using
a guided excursion set formalism, an ionization field is created. This
ionization field is then converted into a 21-cm brightness temperature field.

For our analysis in this paper, we use the output of an already existing run
of Sem-Num, precisely the fiducial light-cone map of Majumdar et al. (2016).
This light-cone volume has sky plane comoving extent of $500\times
500\,(h^{-1}{\rm Mpc})^{2}$ and the line-of-sight or redshift extent of
$7.0\leq z\leq 12.0$. The 21-cm map in this light-cone volume is saved on a
grid of size $232\times 232\times 562$. We extract a cuboid of grid dimension
$232\times 232\times 64$ from this light-cone volume and project it to a World
Coordinate System (WCS) of the same dimension. This extracted cuboid is used
as the signal model for our analysis. Similarly, we generate a 21-cm light-
cone cuboid using 21cmFast of the same volume and treat it as our other signal
model.

Figure 3 shows a slice of the signal at the central frequency (142 MHz,
z$\sim$9) for 21cmFAST (top panel) and Sem-Num (bottom panel). It is evident
from this figure that the reionization histories followed by these two models
are not the same; thus, at the same redshift, they arrive at a significantly
different level of IGM ionization. Therefore, the levels of resulting 21-cm
fluctuations are also considerably different. Figure 4 shows the power
spectrum from these two signal models as observed by four different telescope
configurations discussed earlier. Here we assume that the observational data
contains the signal alone. We follow that the resulting power spectra for the
two signal models are not drastically different at the length scales of our
interest. As 21cmFAST is a faster simulation to implement, the subsequent
analysis has been done using this signal model alone.

Figure 3: Slice through the Hi 21-cm light cone at 142MHz, projected on a WCS
using 21cmFAST (top) and SemNum (bottom). The light cones produced have
different reionization histories and are thus not identical.

Figure 4: topPower spectrum obtained for the two signal models used when
observed with the four telescope configurations. The top panel shows the
cylindrical PS with SKA1-Low (panel a-21cmFAST, b-Sem-Num), HERA (panel
c-21cmFAST, d-Sem-Num), MWA-I (panel e-21cmFAST, f-Sem-Num), MWA-256 (panel
g-21cmFAST, h-Sem-Num). bottom The spherical PS with 21cmFAST (upper subplot)
and SemNum (lower subplot). For both subplots, the layouts are SKA (maroon
dot-dash line with triangle markers), HERA (blue dashed line with diamond
markers), MWA-I (green solid line with pentagon markers) and MWA-256 (magenta
dotted lines with inverted triangle markers).

###  2.3 Foreground Models

The foreground used for this work consists of compact sources only. Once
again, two separate catalogs have been used - the 400 MHz catalog obtained
using uGMRT of the field ELAIS N1 (Chakraborty et al., 2019) and the Tiered
Radio Extragalactic Continuum Simulation (T-RECS, Bonaldi et al. (2018)) to
demonstrate the ability of the pipeline. The sources were redistributed around
a declination of -30∘. The fluxes were converted to 142 MHz values using
$S_{\nu}\propto\nu^{-\alpha}$, with $\alpha$ value of -0.8. The minimum flux
value present at 400 MHz was 100 $\mu$Jy, and the highest flux is 0.6 Jy.
T-RECS is a simulation modeling the continuum radio sky in the range 150MHz to
20GHz. It models Active Galactic Nuclei (AGNs) and Star-Forming Galaxies
(SFGs), comprising most radio galaxy populations. These simulations consider
real data from recent observations to model the sources. This gives realistic
cosmological evolution of the luminosity functions, the total intensity number
counts, polarized intensity, and clustering properties. The source catalogs
are available publicly for 1 deg2, 25 deg2, and 100 deg2. For this work, from
the 25 deg2 catalog, sources within $\sim$4 deg2 having flux cut-off the same
as EN1 catalog is chosen (the brightest source in the constructed catalogue
has a flux of 1.7 Jy at 150 MHz). Fluxes at 150 MHz were scaled to 142 MHz
using $\alpha\approx$ -0.8. The catalog comprises 2522 sources in the chosen
field of view (FoV) and flux cut-off.

It is to be noted that no diffuse emission model was used in the foreground
model to keep the present analysis simple. Future works will incorporate
models of the diffuse emission for investing their effect.

##  3 Non-ideal observing conditions

Observational effects on the cosmological 21-cm signal are numerous. However,
this work focuses on calibration errors and position errors.

Radio interferometers observe the Fourier transform of the sky intensity
distribution. The observed visibilities (i.e., spatial coherence function of
sources) are calculated by cross-correlating the detected electric field
between antenna pairs using digital correlators. The individual antenna (or
tile) measures the incident electric field modified by a complex gain factor
(arising due to the electronics). Calibration aims to solve for these complex
gains. For observations that target the cosmological 21-cm signal, either a
"traditional" sky-based technique or a "redundant" calibration technique is
used. In the former method, a sky model is used to calibrate instruments
(Taylor et al., 1999). The latter is useful for arrays that have a large
number of redundant baselines (Liu et al., 2010). However, irrespective of the
technique used, the basic equation is the measurement equation relating the
measured sky visibility to true sky visibility. The simplest form of the
equation is :

$V{{}_{ij}^{m}}=g_{i}(t)g{{}_{j}^{*}}(t)V{{}_{ij}^{t}}$ (1)

where $V{{}_{ij}^{m}}$ is the measured visibility from the antenna pairs i and
j, $V{{}_{ij}^{t}}$ is the true sky visibility and $g_{i}(t)$ and
$g{{}_{j}^{*}}(t)$ are the complex gains of the respective antennae. While
gain calibrations aim to solve for these gains, there is a limit to the
accuracy of these. The uncalibrated gains (or residual gains) that remain are
propagated in the subsequent steps and leads to errors in extracting the
target signal. The complex gain can be modeled by

$g{{}_{i}}=(a_{i}+\delta{a_{i}})exp(-i({\phi_{i}}+\delta{\phi_{i}}))$ (2)

where the phase, $\phi_{i}$ is in radian, and the amplitude, $a_{i}$ is
dimensionless. The terms $\delta{a_{i}}$ and $\delta{\phi_{i}}$ are the errors
in the gain solutions. For the ideal case, $a_{i}$ is 1 and ${\phi_{i}}$ is 0,
giving gain of 1. However, due to the presence of residual gain errors, the
error terms will give the resultant gain as

$g{{}_{i}}=(1+\delta{a_{i}})exp(-\delta{\phi_{i}})$ (3)

An "efficient" calibration algorithm will be one that can minimize
$\delta{a_{i}}$ and $\delta{\phi_{i}}$. The target signal itself is feeble,
thus attaining an SNR $>$1 becomes a challenge. The thermal noise (one of the
most fundamental limits for any instrument) can be lowered significantly with
increasing observing time. However, other noise sources like faint unmodelled
sources (Ewall-Wice et al., 2017), ionospheric variation, instrument response,
etc., cannot be reduced by increasing integration time. Thus accurate
calibration can increase SNR, but up to a certain limit as allowed by
observing conditions. This work quantifies the actual level of calibration
accuracy required for upcoming radio interferometers to obtain "good" SNR
levels.

The bright radio sources that contaminate the data sets are expected to be
removed before extracting the cosmological 21-cm signal. While the predominant
source of the bright foreground is the diffuse galactic emission, the point
source contamination also poses a problem. The problem arises mainly because
modeling and removal of each source is challenging. Any calibration
imperfections give rise to artifacts in the residual that may affect the
signal extraction adversely (Datta et al., 2009, 2010). Moreover, these
observations have limited resolution due to the signal being coherent at short
baselines only. Thus, high-resolution observations are required for accurate
modeling and removal of discrete foregrounds. Hence, there are very stringent
accuracy requirements for the sky model. Further calibration cannot minimize
any error inherent to such a model, and the same will result in contaminated
residuals. Thus position errors pose a challenge for the detection of the
cosmological signal. Moreover, due to phase errors from directional effects,
source positions may also shift. This will result in incorrect point source
removal resulting in artifacts affecting signal extraction. Most sources have
fluxes $\sim$ mJy or lower at the target frequencies for real observations.
Since the cosmological signal is feeble, compact sources with even such "low"
fluxes remaining in the residuals pose a problem. It has been shown that while
redundant calibration is potentially better for arrays with a high degree of
redundancy like HERA and the MWA Hexagonal configuration, offsets in source
positions systematically affect the phase solutions (Joseph et al., 2018). A
global sky model (having only compact sources) as observed in actual
observation is considered for this work. The effect of errors in the source
position through a systematic error in the right ascension of the sources is
investigated here.

##  4 Formalism for Analyses

The visibilities obtained from the synthetic observations described in Section
2 are used for further analysis. Figure 5 shows the simulated image of the sky
with T-RECS as foreground model for SKA1-Low, HERA, MWA-I & II respectively
(top to bottom respectively). Figure 6 shows the point spread function (PSF)
at the central frequency of observation for each of the telescopes used. It
can be seen from the figure that the best PSF response comes from SKA1-Low
(magenta curve), both in terms of width and amplitude of side lobes. PSF for
HERA (shown by green curve) has the widest response, which is, because it has
the shortest maximum baseline. However, due to the layout and configuration of
the array, it has a small side-lobe amplitude. The MWA configurations (Phase-1
in cyan and Phase-2 in blue) show high amplitude oscillatory features in the
sidelobes.

Figure 5: Simulated sky with T-RECS as foreground model for (top to bottom)
SKA1-Low, HERA, MWA-I & MWA-256. Figure 6: Normalized Point Spread Function
using robust parameter 0 at the central frequency of observation for
SKA1-Low(magenta), HERA(green), MWA-1(cyan) and MWA-2(blue).

For determination of the effect of calibration errors, the ideal visibilities
are corrupted by complex gains (as described in Equation 1) . The errors in
gain phases and amplitudes (see Equation 3) were drawn from a 0 mean Gaussian
distribution with a standard deviation equal to the percentage of error
introduced. For this work, 0.001%, 0.01%, 0.1%, 1.0% & 10.0% gain errors are
introduced in both the gain phases and amplitudes. The model visibilities are
subtracted from the corrupted ones to produce residual visibilities, i.e.,

$V{{}_{ij}^{residual}}=V{{}_{ij}^{observed}}-V{{}_{ij}^{model}}$ (4)

Here, $V{{}_{ij}^{observed}}$ is the corrupted visibility (the label
’observed’ implies actual observations the instrument records where the
visibilities come with various imperfections).

The effect of position errors (i.e., in the global sky model) is determined by
simulating observations with erroneous sky models. These inaccurate sky models
are generated, taking the PSF size of HERA at the observing frequency
(enlisted in Table 1) as a metric to quantify the offsets. A zero mean
Gaussian distribution with standard different deviations- 0.01%, 0.1%, 1.0%,
10.0% & 100.0% of 8.1′(486″) 888This means that the maximum displacement
allowed are $\sim$ 0.05″, 0.5″, 5.0″, 50″and 500″respectively for the
percentages of error mentioned. are used to generate different sets of
erroneous sky models. The position errors thus generated were added to the RA
of the sources to create a new catalog with inherent errors in the source
positions. The residuals are obtained similarly, as discussed for Equation 4.
Residuals obtained by subtracting the corrupted sky from the true one have
been used to determine the effect in both the image plane and the PS. The
formalism used to quantify the image plane effects and PS is described briefly
in the following subsections. It should be mentioned that the position error
in only the RA has been introduced for simplicity. Such errors can be present
in either one of RA or DEC or both in a point source model. Such errors will
obscure the EoR signal beyond a certain offset.

###  4.1 Image Plane

While most EoR detection experiments do not target imaging due to low signal-
to-noise ratio (SNR), the SKA1-Low is projected to produce tomographic images
with SNR$\gtrapprox$1 (Mellema et al., 2014). There are various techniques,
including machine learning approaches and the traditional CLEAN (Högbom, 1974)
algorithm that is being explored for maximum information extraction in the
image domain (for instance, see Liu & Shaw (2020) for a comprehensive
summary). However, the efficacy and robustness of these methods are not
determined for a wide variety of actual observations. Thus it is worthwhile to
investigate how the errors that have been introduced here affect the image
plane. The analysis is done by constructing a simple dirty map in CASA and
using the RMS in the image plane as the quantifying metric (note that a
comparison of various techniques for image recovery has not been made since it
is beyond the scope of the current work). The pure Hi signal used in the input
has a peak flux of $\sim$ 0.8 $\mathrm{\mu Jybeam^{-1}}$ at 142 MHz, hence to
get a significant detection, the residual RMS should lie below this level.
Thus, performance in terms of the RMS has been compared. Additionally, image
plane performance is also quantifiable using its dynamic range (DR) - the
ratio of the peak flux in the image and the residual RMS. In the case of the
presence of imperfect calibration, the theoretical DR is defined as Taylor et
al. (1999) :

$DR=\sqrt{\frac{N(N-1)}{2(a^{2}+\phi^{2})}}$ (5)

with a and $\phi$ as amplitude and phase, respectively, and N is the number of
interferometer elements. The lowest RMS obtained from the residual image near
the phase center is used to quantify the effect of the errors mentioned
earlier. Since two different foreground models are used for this work, they
are represented via the dashed (EN1) and dot-dashed (T-RECS) curves.

Figure 7 summarises the results obtained from the gain residuals. For the
dynamic range, the general trend is consistent with theoretical expectations.
The desirable calibration accuracy obtained from these simulations translates
to a dynamic range requirement of $\sim$105 or higher, which in turn implies
the necessity of a highly accurate gain solution for observations with the
SKA1-Low. Figure 8 shows the variation of RMS in the residuals with position
error as a function of angular size. In both Figures 7 and 8, the gray band
represents the maximum and minimum signal amplitudes in the image plane.

In the best case scenario, the sensitivity of the instrument will be
determined by the thermal noise in the system. The thermal noise limit is
given by (Fan et al., 2006):

$\sigma_{T}=\Big{(}\frac{1.9}{\sqrt{\Delta\nu_{kHz}t_{hr}}}\Big{)}\Big{(}\frac{T_{sys}}{A_{eff}N_{ant}}\Big{)}$
(6)

where $\sigma_{T}$ is the thermal noise in the system, $\Delta\nu_{kHz}$ is
the channel width in kHz, $t_{hr}$ is the integration time in hours, $T_{sys}$
is the system temperature, $A_{eff}$ is the antenna collecting area and
$N_{ant}$ is the number of antennas. At the frequencies of interest, $T_{sys}$
is equivalent to the sky temperature. The sky temperature, $T_{sky}$ is
calculated using $T_{sky}\sim 180\Big{(}\frac{\nu}{180MHz}\Big{)}^{-2.6}$K
(Furlanetto et al., 2006). The values of thermal noise is tabulated in Table
1. As can be seen, the least thermal noise level for a 4 hour observation is
obtained for SKA1-Low, which is shown via the solid orange line in Figures 7 &
8. In the case of SKA1-Low, Braun (2017) states that the spectral line
sensitivity at the frequency band of interest is 1258 $\mu$Jy beam-1 for a
fractional bandwidth of 10-4, and integration time $\Delta\tau$ of 1 hour; for
the specifications used in this work, the sensitivity comes down to $\sim$220
$\mu$Jy beam-1.

###  4.2 Power Spectrum

Statistical detection of the redshifted cosmological 21-cm signal is the
primary target of most low-frequency interferometric arrays. For EoR
experiments, there is a formalism to estimate the PS from both visibility (for
instance, see Trott et al. 2020) as well as from the image plane (for example
Mertens et al. 2020); this work uses visibility for its determination. The
interferometric visibility is the correlation between the signals received
between two antenna pairs, which is given by (Taylor et al., 1999) :

$V({\mathbf{U}},\nu)=\iint
A(\hat{\mathbf{s}},\nu)B(\nu)I(\hat{\mathbf{s}},\nu)e^{-i2\pi\nu\mathbf{U}.\hat{\mathbf{s}}}d\Omega,$
(7)

where, ${\mathbf{U}}$ is the baseline vector, $I(\hat{\mathbf{s}},\nu)$ and
$A(\hat{\mathbf{s}},\nu)$ are the specific intensity and antenna beam pattern
as a function of frequency ($\nu$) and $B(\nu)$ is the instrumental bandpass
response; the unit vector as, $\hat{\mathbf{s}}\equiv(l,m,n)$, where $l,m,n$
are the direction cosines towards east, north and zenith respectively with
$n=\sqrt{1-l^{2}-m^{2}}$ and $d\Omega=\frac{dldm}{\sqrt{1-l^{2}-m^{2}}}$. For
this work, $A(\hat{\mathbf{s}},\nu)$ is taken to be 1, i.e. the effect of
primary beam is not considered.

Inverse Fourier transform of $V({\mathbf{U}},\nu)$ along the the frequency
axis gives the visibility in the delay domain (henceforth represented as
$\tau$), $V({\mathbf{U}},\tau)$. Using this formalism, the cylindrical PS (as
discussed in Morales & Hewitt 2004) is given by:

$P(\mathbf{k}_{\perp},k_{\parallel})=\Big{(}\frac{\lambda^{2}}{2k_{B}}\Big{)}^{2}\Big{(}\frac{X^{2}Y}{\Omega
B}\Big{)}|V(\mathbf{U},\tau)|^{2},$ (8)

with unit $K^{2}(Mpc/h)^{3}$. In the above equation, $\lambda$ is the
wavelength corresponding to central frequency (2.1 m for these simulations),
$k_{B}$ is the Boltzmann constant, $\Omega$ is the primary beam response, B is
the bandwidth (8 MHz for this work), X and Y are the conversion factors from
angle and frequency to transverse co-moving distance (D(z)) and the co-moving
depth along the line of sight, respectively (Morales & Hewitt, 2004). The
quantity $\mathbf{k}_{\perp}$ represents the Fourier modes perpendicular &
$k_{\parallel}$ represent the k modes along the line of sight, given by

$\mathbf{k}_{\perp}=\frac{2\pi|\mathbf{U}|}{D(z)}\hskip 20.0pt\&\hskip
20.0ptk_{\parallel}=\frac{2\pi\tau\nu_{21}H_{0}E(z)}{c(1+z)^{2}}$

where $\nu_{21}$ is the rest-frame frequency of the 21 cm spin flip transition
of Hi $z$ is the redshift corresponding to the observing frequency, $H_{0}$ is
the Hubble parameter and
$E(z)\equiv[\Omega_{\mathrm{M}}(1+\textit{z})^{3}+\Omega_{\Lambda}]^{1/2}$.
$\Omega_{\mathrm{M}}$ and $\Omega_{\Lambda}$ are matter and dark energy
densities, respectively (Hogg, 1999).

From the 2D power spectrum, the 1D PS (obtained by spherical averaging of
$P(\mathbf{k}_{\perp},k_{\parallel})$)is calculated as :

$\Delta^{2}(k)=\frac{K^{3}}{2\pi^{2}}\langle P(\mathbf{k})\rangle_{k}$ (9)

where, $k=\sqrt{k_{\perp}^{2}+k_{\parallel}^{2}}$. The uncertainty in PS has
been calculated using a modified version of 21cmSense (Pober et al., 2013a)
for each array. The 3$\rm\sigma$ error bars have been put in the spherically
averaged PS to show the uncertainty in power obtained.

In the case of SKA1-Low, it is still not decided whether signal PS estimation
will be done using foreground removal, avoidance, or some other technique.
This work uses the generic method of averaging over all k-modes to determine
the 1D power. The same approach is also used for both MWA configurations and
HERA to maintain consistency (although in Trott et al. 2020 and The HERA
Collaboration et al. 2022, the upper limits have been placed using avoidance
methodology).

Figure 7:  Variation of RMS in the residual image as a percentage of
calibration error for T-RECS catalog as foreground model. The grey region
shows the amplitude of range of observed signal. The solid orange line is the
thermal noise limit for a 4-hour observation with SKA1-Low. Figure 8:
Variation of RMS in residual image with position error using T-RECS as
foreground model for the four telescope configurations considered. The RMS is
plotted as a function of the maximum displacement of the sources (in arcmin)
from the true position. The solid orange line is the thermal noise limit for a
4-hour observation with SKA1-Low.

The results obtained for each array configuration for imaging and PS
estimation are described in the following section. It should be mentioned that
both the foreground models give consistent results for all the telescope
configurations considered. The power spectrum performance plots shown in the
following subsections use T-RECS as the foreground model. A representative
case using EN1 as foreground model has been shown in the appendix for all the
configurations to emphasize that the pipeline performs consistently
irrespective of the sky model used.

##  5 Results

This section describes the results obtained from the simulations and discusses
their implications for actual observations. The results for each of the
configuration considered is described in separate subsections below.

###  5.1 SKA1-Low

The synthetic observation pipeline is primarily being developed to simulate
the effects of non-ideal conditions that are present in actual observations.
The primary motivation is to determine how these errors will affect the
performance of SKA1-Low for the sensitive low-frequency observation done for
CD/EoR science. The following subsections describe the effect of calibration
and position errors for SKA1-Low.

####  5.1.1 Calibration Error

The maroon curve shows the effect of calibration error on SKA1-Low in the
image plane in Figure 7. It is seen that for calibration inaccuracy
$\sim$10-2%, the residual RMS is well below the signal level. 0.01% represents
the borderline case where the RMS and thermal noise are nearly similar.
However, the residual RMS level exceeds the signal level by almost an order of
magnitude, indicating that the limiting factor for these observations would be
systematics (instead of thermal noise) for sufficiently large integration
times using the SKA1-Low. For even higher errors, the image RMS is high enough
to confuse or even obscure the faint cosmological signal detection.

Figure 9: Residual power spectrum for gain errors with SKA-1 Low. (left) 2D PS
of the residual visibilities for calibration errors for 0.001% (top-left),
0.01% (top-right), 0.1% (bottom-left) & 1.0 % (bottom-right). The solid black
line represents the horizon line, while the dashed black line is the FoV
limit. (right) Spherically averaged PS with calibration residuals compared
with the signal power and foreground power. The error bars are 3$\rm\sigma$
uncertainties for the k-bins including sample variance and thermal noise.

In Figure 9, the left panel shows the 2D power spectrum of the residual
visibilities for SKA1-Low. It is seen that up to 0.01% calibration error, the
wedge stays $k_{\perp}$ modes below $\sim$ 0.1 and below $k_{\parallel}\sim$
0.2, and the EoR window region have lower contamination. For cases exceeding
0.1% error, the EoR window has a very high power amplitude, implying the
residual foregrounds dominate above inaccuracies $\sim$ 0.1% in gain
amplitude.

The right panel of Figure 9 shows the 1-dimensional spherically averaged power
spectrum for gain error residuals. The zoomed-in part shows the region near
the signal power at k $\sim$0.2 and above. The green curve shows the power for
the observed sky (i.e., the sky with both signal and foregrounds). It is seen
that at 0.001% error (blue curve), the residual power overlaps with the signal
power (red curve). At 0.01%, the power spectrum starts deviation from the
signal level at k=0.5 Mpc-1 and above, but are still consistent within error-
bars. But for 1% and above errors in calibration, the foreground contribution
in the residuals overrides the signal power completely. Thus, if all other
sources of error are well understood and mitigated, the residual calibration
error can render the signal obscured if the algorithm accuracy is $\sim$1% or
worse.

####  5.1.2 Position Error

The maroon curves show the effect of position error in the image plane in
Figure 8. The top panel shows the RMS as a function of maximum displacement
from the actual position, and the bottom curve represents the same as a
function of the relative percentage of error. For errors up to $\sim$
0.048″(0.1% of the PSF size), the residual RMS stays well below the signal
level, thereby making its detection feasible. At 1% error (or
0.48″displacement), the RMS residual RMS starts getting dominated by the
residual foregrounds, which again can potentially confuse detection of the
redshifted signal. It is also seen that up to a maximum displacement of $\sim$
0.048″, the error remains below the thermal noise level for 4-hour observation
(orange line), again emphasizing for a sufficiently large integration time,
observations would be systematics limited rather than noise limited.

Figure 10: Residual power spectrum for position errors with SKA1-Low. (left)
2D cylindrical averaged power spectra of the residual visibilities for
position errors with 0.1% (top-left), 1% (top-right), 10% (bottom-left) & 100
% (bottom-right). The solid black line represents the horizon line, while the
dashed black line is the FoV limit. (right) Spherically averaged PS with
position residuals compared with the signal power and foreground power.The
error bars are 3$\rm\sigma$ uncertainties for the k-bins including sample
variance and thermal noise.

In Figure 10, the residual power for position errors with SKA1-Low is shown.
The left panel shows the 2D PS, while the right panel 1D spherically averaged
PS. It is seen for the residual 2D power spectrum that with offsets $\geq$ 10%
of PSF size (i.e., 40″and above), the residual foregrounds cause leakage into
the k-modes outside the wedge, giving a significantly high amplitude of power.
Subsequently, in the case of the 1D PS, the residual power overrides the
signal at all k modes by orders of magnitude. This is also reflected in the 1D
power spectrum. For a position error of $\sim$0.1% (i.e., 4″), the signal and
the residual foreground powers are at par with each other. Thus, it can be
seen from both the figures that the position accuracy required is
$\sim$0.01′to render the signal detectable. Hence in the case of PS
estimation, it can be seen that for achievable positional accuracy at the
observing frequency, the statistical detection of the cosmological signal is
possible.

###  5.2 HERA

HERA is one of the key EoR instruments that has already begun its observation
and has recently provided its first upper limit using the data from the
initial observing phase using 52 antennas (The HERA Collaboration et al.,
2022). For this work, the entire 350 antenna configuration (see Figure 2, top
right panel) has been used. The effect of errors seen in both image and PS
domain is described below.

####  5.2.1 Calibration Errors

The blue curves in Figure 7 show the effect of calibration residuals in the
image plane for HERA. For both the sky models, it is seen that the residual
RMS for gain solutions that have as small as 0.01% and above errors exceeds
the signal power level. Thus, under identical observing conditions and in the
presence of similar errors, SKA1-Low outperforms HERA in terms of calibration
error tolerance by an order of magnitude in the image plane.

The PS for HERA for gain error residuals is shown in Figure 11. The left panel
is the 2D PS, while the right panel is the 1D PS. In the $k_{\parallel}$\-
$k_{\perp}$ plane, the wedge can be observed in the modes within
$k_{\perp}\sim$ 0.01 and 0.3 and $k_{\parallel}\geq$ 0.3 are free from very
high amplitude of foregrounds.

The residual 2D PS shows that for calibration errors above $\sim$0.1%, the
foregrounds power becomes high and starts to dominate at all k modes. This is
also evident from the 1D power spectrum of the calibration residuals (Figure
11 right panel), where it is observed that upto 0.1%, the residual power very
slightly deviates from the signal power. However, for inaccuracies $\geq$0.1%,
the residual power exceeds the signal by a significant factor, overruling the
signal and obscuring it.

Figure 11: Residual power spectrum for gain errors with HERA. (left) 2D PS of
the residual visibilities for calibration errors for 0.001% (top-left), 0.01%
(top-right), 0.1% (bottom-left) & 1.0 % (bottom-right). The solid black line
represents the horizon line, while the dashed black line is the FoV limit.
(right) Spherically averaged PS with calibration residuals compared with the
signal power and foreground power.The error bars are 3$\rm\sigma$
uncertainties for the k-bins including sample variance and thermal noise.

####  5.2.2 Position Errors

In Figure 8, the blue curves represent the residuals of calibration error for
synthetic observations with HERA. In this case, an offset of 0.4″makes the
residual RMS low enough to detect the signal. Hence, for HERA, offsets less
than 0.4″are required for the image RMS to lie below the signal level.

The residual PS for position error residuals for HERA is shown in Figure 12.
It is observed from the left panel that for deviations of $>$1% of the
"actual" position, the power from the residual foregrounds contaminates most
of the k-modes, causing the signal to be obscured. A similar trend is seen in
the right panel, where the signal power spectrum (red curve) is being followed
(with slight deviations of a factor of $\sim$1.3) for position errors up to
$\sim$0.01″(0.1% of PSF size). At 1% of the PSF size ($\sim$0.1″), the
residual power deviates from the signal power at k modes above 0.5 Mpc -1.
Above 1% error, the deviations are orders of magnitude higher, which results
in the foreground residuals obscuring the signal.

Figure 12: Residual power spectrum for position errors with HERA. (left) 2D
cylindrical averaged power spectra of the residual visibilities for position
errors with 0.1% (top-left), 1% (top-right), 10% (bottom-left) & 100 %
(bottom-right). The solid black line represents the horizon line, while the
dashed black line is the FoV limit. (right) Spherically averaged PS with
position residuals compared with the signal power and foreground power.The
error bars are 3$\rm\sigma$ uncertainties for the k-bins including sample
variance and thermal noise.

###  5.3 MWA

The MWA is one of the precursor facilities to the SKA1-Low. In the initial
phase of operation, it consisted of 128 square tiles extended up to $\sim$3
km. The second phase of the MWA, after its upgrade, consists of 256 tiles,
which spreads out to $\sim$5 km. Both the old and the upgraded configurations
with all the tiles have been used for this work. The results for the
introduction of the errors as mentioned earlier are discussed in the following
subsections.

####  5.3.1 Calibration Error

Green and magenta curves in Figure 7 show the performance of MWA-I and MWA-256
respectively in the image domain in the presence of residual calibration
errors. For both the cases, RMS corresponding to residuals of 0.01% is below
the signal level, with values $\sim$10-1 mJy $\mathrm{beam^{-1}}$. Any
algorithm producing inaccuracies greater than this can cause the residual RMS
to override the signal by order of magnitude, thereby obscuring it. This
performance is consistently seen for both configurations.

Figure 13: Residual power spectrum for gain errors with MWA Phase I. (left) 2D
PS of the residual visibilities for calibration errors for 0.001% (top-left),
0.01% (top-right), 0.1% (bottom-left) & 1.0 % (bottom-right). The solid black
line represents the horizon line, while the dashed black line is the FoV
limit. (right) Spherically averaged PS with calibration residuals compared
with the signal power and foreground power.The error bars are 3$\rm\sigma$
uncertainties for the k-bins including sample variance and thermal noise.

The PS recovery performance for both MWA configurations is similar, which may
be attributed to the fact that baselines within a maximum of 2 km of the core
are used to determine the power spectrum (baselines of longer lengths are not
of interest for EoR statistics). Figures 13 & 14 show the PS recovered from
the residual visibility of with calibration error using MWA-I & MWA-256. For
both cases, it is observed from the left panel that for the 2D PS, gain errors
beyond 0.1%, the residual foregrounds have power from the wedge leaking into
the EoR window. Thus all the $k_{\perp}$ and $k_{\parallel}$ modes have higher
power compared to the signal power, which implies the signal being obscured.
The right hand panel for both Figures 13 & 14 show similar trends for the 1D
power spectrum. For calibration inaccuracies $\leq$ 0.1%, the residual power
follows the signal power spectrum within error bars. This is also evident from
the zoomed-in parts of both figures, where it can be seen that the residuals
follow the signal power up to $\sim$0.01%. Above 0.1% , the residual
foregrounds start to dominate over the signal exceeding the signal in orders
of magnitude, thus obscuring it.

Figure 14: Residual power spectrum for gain errors with MWA-256. (left) 2D PS
of the residual visibilities for calibration errors for 0.001% (top-left),
0.01% (top-right), 0.1% (bottom-left) & 1.0 % (bottom-right). The solid black
line represents the horizon line, while the dashed black line is the FoV
limit. (right) Spherically averaged PS with calibration residuals compared
with the signal power and foreground power. The error bars are 3$\rm\sigma$
uncertainties for the k-bins including sample variance and thermal noise.

####  5.3.2 Position Error

The green and magenta curves in Figure 8 show the residual RMS as a function
of the angular displacement from the actual source position (top panel) and
the relative percentage of error introduced (bottom) for MWA-I & MWA-256,
respectively. As seen from the green curves, in the case of MWA-I, an offset
of 1% (or $\sim$1″) or more with respect to the actual position results in the
residual RMS level exceeding the signal level, thereby obscuring it.
Similarly, for the magenta curve, i.e., MWA-256, the residuals lie within the
signal level up to an offset of $\sim$ 0.1″, beyond which it exceeds the
signal level and the thermal noise level for SKA1-Low, thereby rendering the
signal undetectable.

Figure 15: Residual power spectrum for position errors with MWA Phase I.
(left) 2D PS of the residual visibilities for position errors for 0.01% (top-
left), 0.1% (top-right), 1.0% (bottom-left) & 10.0 % (bottom-right). The solid
black line represents the horizon line, while the dashed black line is the FoV
limit. (right) Spherically averaged PS with position error residuals compared
with the signal power and foreground power. The error bars are 3$\rm\sigma$
uncertainties for the k-bins including sample variance and thermal noise.

The residual PS for MWA configurations in the presence of position error are
shown in Figures 15 & 16 \- the left panel is the cylindrical PS, and the
right panel is the spherical PS. As in the case of calibration errors, there
is no significant difference between the two MWA configurations. From both the
figures, it can be seen that in the entire $k_{\parallel}$-$k_{\perp}$ plane,
the residual power has a higher magnitude than signal power at position error
$\geq$1% (i.e., displacement $\gtrapprox$0.1′). From the 1D PS (right panel of
both figures), it is can likewise be observed that at 1% error (cyan curve),
the power spectrum of residual visibility starts deviating from the signal
level (the residuals below this value follows the signal exactly). This
implies that position accuracy better than 0.1′is required to detect the
redshifted 21-cm signal from the EoR.

Figure 16: Residual power spectrum for position errors with MWA-256. (left) 2D
PS of the residual visibilities for position errors for 0.01% (top-left), 0.1%
(top-right), 1.0% (bottom-left) & 10.0 % (bottom-right). The solid black line
represents the horizon line, while the dashed black line is the FoV limit.
(right) Spherically averaged PS with position error residuals compared with
the signal power and foreground power.The error bars are 3$\rm\sigma$
uncertainties for the k-bins including sample variance and thermal noise.

###  5.4 Discussion

After comparing the performances of the arrays considered here in terms of
both imaging and statistics (power spectrum), the following has been noted:

  * •

In the presence of residual calibration error, it is seen that for SKA1-Low,
inaccuracies of 0.001%, there is sufficient DR in the image ($\sim$106) to
detect the signal in the image domain. A careful inspection also reveals that
for SKA1-Low, up to an error of $\sim$0.07%, the residual RMS, though above
the signal level, is below the daily 4-hour thermal noise level (orange line).
The observations targeting EoR signal are expected to be systematics limited
(rather than thermal noise limited). Since the systematics are expected to be
unrelated from day to day, reducing them becomes more critical than thermal
noise. From this consideration, these simulations suggest that the image plane
performance of SKA-1 Low is marginally better than the other arrays with
identical observing strategy. However, it is worth mentioning that these
simulations do not consider thermal noise. The noise for MWA is theoretically
expected to be much higher compared to the other configurations under
identical observation conditions. Nevertheless, detailed image plane analysis
in the presence of noise will be pursued in future works.

  * •

In the presence of position error, the accuracy of the order of
$\gtrapprox$0.5″is required. At displacement 5″or more, the residual RMS lies
above both the signal and 4-hour thermal noise level (orange line). This is
consistently observed for all four array configurations. This level of
astrometric precision is in principle achievable for observations using the
SKA-1. Recent results from LOFAR and MWA show an astrometric accuracy of
$\sim$1″(for example, the LoTSS survey between 120–168 MHz (Shimwell, T. W. et
al., 2019) and the GLEAM survey between 70-231 MHz (Hurley-Walker et al.,
2016, 2019)). Thus it is expected that the SKA-1, with improved sensitivity,
should be able to produce sufficiently accurate sky models.

  * •

For PS estimation, it is seen that residual calibration error causes the EoR
window to get contaminated significantly for inaccuracies above 0.1%. This
trend is almost consistent in all the arrays considered and reflected in the
spherically averaged PS. It is observed for the 1D PS that the power spectrum
follows the signal reasonably well for errors below 0.1%. After that, the
residual amplitude in the $k_{\perp}$ \- $k_{\parallel}$ plane starts
exceeding the signal level. In the 1D PS, this manifests as deviation from the
signal power at 0.1% and above errors.

  * •

From the PS estimates, it is seen that the residual calibration error causes
significant contamination of the EoR window with calibration errors of
$\sim$0.1% and above. Below this value, the residual PS follow the "observed"
signal power, and hence can be considered as the tolerance threshold for
calibration errors.

  * •

Residual PS in the presence of position error shows that for reasonably
accurate astrometry, i.e., for a maximum position offset $\sim$0.5″, the
residual power follows the signal power. Position offset of 5$\arcsec$ appears
to be the threshold for tolerance, with some k modes lying within the signal
uncertainty limits. Thus, it can be concluded that 5$\arcsec$ is the optimal
tolerance for position error. Here, it should be reemphasized that the sky
model used for data calibration is expected to be derived from higher-
resolution observations. If they are not sufficiently accurate, the errors
propagating into subsequent analyses would potentially confuse signal
detection.

  * •

For the signal detection in PS domain, these simulations do not distinctly
favour any of the four array configurations. However, it should be pointed out
that these analyses are done under some very simplistic assumptions. In order
to compare performances under even more realistic conditions, future works
will explore various unavoidable systematics that can hamper observations
targeting the redshifted 21-cm signal from CD/EoR.

##  6 Influence of Array Specific Parameters on Calibration and Position Error

This work considers the effects of residual calibration error and position
error on the extraction of the 21-cm signal for four distinct array
configurations. However, a detailed study of the influence of different array-
specific parameters on the systematics considered here is beyond the scope of
the present study. Nevertheless, this section briefly discusses a few
parameters that vary across arrays depending on their location and
specifications.

With diameters of 40m, 14m, and 4m for SKA-1 Low, HERA, and MWA, the FoV is
$\sim$ 3∘, 8∘ and 30∘ respectively at 142 MHz. A larger FoV implies
observation of a greater number of sources. Thus, an observation incorporating
a large FoV results in more robust foreground statistics like source counts,
clustering, etc. Such robust statistics are required for precision foreground
modelling and removal. However, a larger FoV can result in the inclusion of
bright sources in the telescope primary beam. Presence of extremely bright
objects like the A-Team sources (Hurley-Walker et al., 2016) in the southern
sky and Cassiopeia A and Cygnus A in the northern sky hinder data calibration.
Thus they need to be removed very carefully from the data (Trott et al.,
2020). A larger FoV may also result in inclusion nearby bright sources in the
side lobes of the telescope beam. Their presence increase the already
difficult problem of modelling and removal of foregrounds from EoR data.
Additionally, any residual bright artifact generated by such sources can also
obscure the 21-cm signal. Thus, the different FoV of the different arrays
considered here provide both advantages and disadvantages.

Different FoV are also affected differently by ionospheric errors. The earth’s
ionosphere introduces phase corruptions to extragalactic signals at low
frequencies ($\lesssim$1 GHz). These effects are "directional", meaning they
vary across different directions, causing an apparent shift of sources from
their original positions by introducing additional phases. Lonsdale (2005)
provided a description of the different regimes for performing ionospheric
phase calibration, based on the observing FoV along with baseline lengths and
irregularity size999Lonsdale (2005) designated the different regimes of
calibration as regimes 1, 2, 3 and 4. The interested readers are directed to
Figure 1 of the same.. Variable ionospheric conditions can affect the arrays
considered here differently (Lonsdale, 2005; Intema et al., 2009; Wijnholds et
al., 2010). The compact array layout combined with the large beam size makes
MWA fall on a calibration regime where there it is susceptible to having
additional phases in the residuals due to ionosphere. This increases the
probability of higher residual errors, even after self-calibration. The
ionosphere-induced calibration inaccuracy in MWA EoR data was reported in
Jordan et al. (2017) and Trott et al. (2020) used ionospheric metric to find
usable data for providing upper limits on the EoR PS. SKA-1 Low will be
situated near the present MWA site and will face similar ionospheric
conditions. However, each SKA station is projected to have a diameter of
$\sim$40 m, making its FoV smaller than the MWA at a similar frequency. Thus,
it will fall in a different calibration regime of Lonsdale (2005), where the
smaller primary beam (i.e. FoV) is expected to introduce lesser ionospheric
phase errors. For HERA, both the ionospheric conditions and FoV are different
from SKA-1 Low and MWA, owing to its completely different location and dish
diameter. However, a detailed study for HERA has not been done, and the
analysis by The HERA Collaboration et al. (2022) also does not consider any
directional effects. Sensitive observations like those targeting the detection
of the EoR need to understand the local ionosphere and calibrate its effects.
But a detailed study into these effects is beyond the scope of the present
work and is deferred till future works.

The baseline distribution as well minimum and maximum lengths of baselines
also vary across arrays. The EoR signal is most sensitive at large angular
scales and thus shorter baselines. The smallest $k_{\perp}$ mode accessible by
an array is also determined by the minimum baseline length present. The arrays
considered here have different baseline lengths and distributions. Ewall-Wice
et al. (2017) showed that the chromaticity increases with baseline lengths,
causing longer baselines to have chromatic gain errors. So applying gain
solutions to the shorter baselines mixes the contamination from the longer
baselines to the shorter ones, which contaminates the EoR window. Thus, they
suggest up-weighting the shorter baselines compared to the longer ones. Ewall-
Wice et al. (2017) also suggest that sky-based EoR experiments use short
baselines to calibrate data used for PS estimation to prevent signal loss due
to directional effects. Here, "short baselines" is a relative term depending
on the array layout and the exact length (for example, $\sim$42 m for SKA and
$\sim$7 m for MWA) should not matter. Longer baselines are essential for
producing point source models for sky-based calibration. The recent MWA Long
Baseline Epoch of Reionisation Survey (LoBES) (Lynch et al., 2021) produced
catalogues with better angular resolution than previous ones, which
successfully removed foreground power from smaller angular scales. So it is
evident that for successful detection of EoR PS, both long and short baselines
are required for the best possible data processing.

The observing strategy also varies across arrays and is governed to a large
extent by the actual physical properties of the dishes/tiles that make up the
array. Most interferometers that are currently operational use tracking scans,
i.e., tracks the phase center throughout the observation. Conversely, HERA is
a drift scan array, meaning it does not track the sources but instead allows
the sky to drift across its beam (DeBoer et al., 2017). The feasibility of
doing drift scan observations for EoR PS estimation with the MWA has also been
explored (Trott, 2014; Patwa et al., 2021). The biggest advantage of drift
scans is that the beam is not steered to track the source, thus providing
better instrumental stability. Stable instruments are extremely important for
sensitive observations targeting EoR signal. Trott (2014) used MWA EoR
experiment specifications to show that drift scans produce slightly lower
uncertainties in the signal power and slightly higher SNR for the 1D PS
compared to tracked observations. Simulations from Trott (2014) also show that
drift scans perform better in terms of reducing cosmic variance, thus
providing better sensitivity at low k modes. But higher k modes for these
observations are thermal noise dominated causing greater uncertainties due to
the reduced coherence and subsequent increase in thermal noise.

A major challenge for drift scan observations is finding suitable calibrator
sources to perform sky-based calibrations (Kern et al., 2020). Thus, HERA has
targeted to replace sky-based techniques with redundant calibration methods.
MWA Phase 2 layout also has many redundant baselines for EoR observations.
Still, both HERA and MWA use sky-based calibration at some point. The HERA
team uses a final step of absolute calibration for breaking degeneracies of
redundant calibration (Dillon et al., 2020; The HERA Collaboration et al.,
2022). They build their models using bright sources from the GLEAM catalogue
that coincides with the HERA track (Kern et al., 2020; The HERA Collaboration
et al., 2022). If such sources are not available, then calibration for drift
scan observation constitutes a problem. It was also shown in Byrne et al.
(2019); Kern et al. (2020); Dillon et al. (2020) that the redundant
calibration technique works only for very high redundancy and identical
primary beam response. This was also seen by the HERA team while calibrating
the observed data (see Section 3.2 of The HERA Collaboration et al. 2022).
Since it is impossible to have "perfect" redundancy, non-redundancies in a
real array also lead to calibration errors. Thus, calibrating a drift scan
instrument can be challenging due to the absence of "good" calibrator sources
and imperfect redundancy. This may increase the calibration error and leak
foregrounds into the EoR window. The observation and calibration strategy for
SKA-1 Low is yet to be decided. Thus, it is worthwhile to simulate the merits
and disadvantages of each method for SKA. But it should be reiterated that
such detailed simulations are beyond the scope of the present work and are
deferred till future works.

##  7 Summary and Conclusion

Detection of the redshifted Hi 21-cm signal from the CD to EoR transitions is
the most challenging undertaking for next-generation radio interferometers.
The cosmological signal is weak, and prone to contamination by astrophysical
foregrounds and instrumental systematics. The actual observations would
require thousands of hours of data, which may be contaminated by improper
calibration or position inaccuracy. Hence,it is extremely important to
understand and quantify the effects of these systematic errors in order to be
able to detect the cosmological signal. Hence, development of an end-to-end
pipeline to check the effects of such systematic through simulations is
essential. This work presents an end-to-end pipeline dealing with synthetic
radio interferometric observations of the radio sky at low radio frequencies.
The sky model includes the redshifted 21-cm signal and astrophysical
foregrounds. Through various simulations the effects of these systematic
errors in the extraction of the redshifted 21cm power spectrum has been shown.
A comparison of results obtained for different array configurations- SKA1-Low,
MWA-I, MWA-256, and HERA has also been demonstrated. The effect of the errors
in the image plane detection of the cosmological signal has also been studied.

The simulations performed show that the optimal error in the case of
calibration errors is 0.1%, beyond which foreground domination in the
residuals overrides the signal. This translates to a DR $\sim$105 or higher,
which would require unprecedented precision in the algorithm employed. For the
case of PS recovery, the signal is overridden by residual foregrounds at
errors $>$0.1%. Since the determination of the signal power spectrum is the
primary aim for most interferometers, it can be said that the target precision
for the calibration algorithm used should be of the order of 0.1% or better.

It is also found that position errors have a significant impact only if the
displacement is $\geq$ 5″. PS derived from position error residual
visibilities show that the residual power follows the signal power amplitude
if the displacements are below $\sim$5″, valid for all the cases considered.
Thus it is concluded that for the next generation interferometers, position
errors alone should not be a major systematic limitation upto a reasonable
level. However, it is also important to note that since position errors are
not removable by calibration (being inherent in the sky model used in
calibration), unless they are treated carefully, they may become a significant
contaminant.

The simulations performed here are done using a few simplistic assumptions for
foregrounds and instruments. For such a scenario, they do not point to a
preferred configuration among the four considered. The optimal tolerance for
both calibration and position error are at a similar level for all four.
However, it is seen that in presence of calibration error, the image plane
performance of SKA and the PS performance of MWA-256 is marginally better
compared to others. This points to the requirement of more detailed
simulations with more complicated effects like primary beam chromaticity to
determine the preferred array configuration. It should also be mentioned that
for simplicity, the simulations are noise-free. Hence, the limitations set by
thermal noise have not been considered in the analysis. In our future works,
we will incorporate thermal noise which will result in a more realistic case,
to explore their impact on signal recovery.

## Acknowledgements

AM thanks Indian Institute of Technology Indore for supporting this research
with Teaching Assistantship. AM further acknowledges the NRAO staff & the
developers of OSKAR for promptly answering queries related to the use of
simulation tool in CASA and the OSKAR software respectively. AM is thankful to
Chandrashekhar Murmu and Mohd. Kamran for their help with the theoretical
aspects of Hi signal, and also to Sumanjit Chakraborty for helpful
discussions. AD would like to acknowledge the support from CSIR through EMR-II
No. 03(1461)/19. SM acknowledges financial support through the project titled
“Observing the Cosmic Dawn in Multicolour using Next Generation Telescopes”
funded by the Science and Engineering Research Board (SERB), Department of
Science and Technology, Government of India through the Core Research Grant
No. CRG/2021/004025. The authors thank the anonymous reviewer and the
scientific editor for helpful comments and suggestions that have helped to
improve the quality of the work.

## Data Availability

The simulated data for this study will be shared upon reasonable request to
the corresponding author.

$Software$ :

This work relies on the Python programming language (https://www.python.org/).
The packages used here are astropy (https://www.astropy.org/; Astropy
Collaboration et al. 2013; Price-Whelan et al. 2018),numpy
(https://numpy.org/), scipy (https://www.scipy.org/), matplotlib
(https://matplotlib.org/). Simulations have been done using OSKAR
(https://github.com/OxfordSKA/OSKAR/releases), Common Astronomy Software
Applications CASA https://casaguides.nrao.edu/index.php?title=Main_Page.

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## Appendix A

The power spectra with the 4 telescope configurations with EN1 as sky model
for 0.1% calibration and position errors. The trends observed with actual
observed sky model is consistent with that obtained with a simulated sky model
(i.e. T-RECS). The synthetic visibilities from both sky models show that the
in the limit that only calibration or only position error is present, the
optimum error should ideally be below 0.1% for calibration errors and
$\sim$1″for position.

Figure 17: Residual power spectrum with SKA-1 Low using EN1 as foreground
model. top-left 2D PS with 0.1% calibration error and top-right 2D PS with
0.1% position error. bottom Spherical power spectrum for 0.1% calibration
error (green) and 0.1% position error (blue). Figure 18: Same as Figure 17
using HERA. Figure 19: Same as Figure 17 using MWA-I. Figure 20: Same as
Figure 17 using MWA-256.

To demonstrate the capability of the pipeline to incorporate any signal model,
Figure 21 shows the spherically averaged power spectrum for observation using
the Sem-Num model in four cases - observation of pure Hi only (red curve), the
observed sky with signal and foreground (green curve), and residuals with 0.1%
calibration errors (blue curve) and 0.1% position error (black curve).
Comparison with the case for SKA1-Low (Figures 9 and 10) shows that the
results are consistent with those obtained for 21cmFAST signal model. This
shows, once again the flexibility of the developed observational simulation
pipeline to work with realistic scenarios.

Figure 21: Spherically averaged power spectrum as observed with SKA1-Low for
signal only (red curve), calibration error residual (blue curve), position
error residual (black curve) and the sky i.e. signal+foreground (green curve).