# A Near-Field Treatment of Aperture Synthesis Techniques using the Murchison
Widefield Array

S. Prabu1 International Centre for Radio Astronomy Research, Curtin
University, Bentley, WA 6102, Australia  [ S.J. Tingay1 A. Williams1

###### Abstract

Typical radio interferometer observations are performed assuming the source of
radiation to be in the far-field of the instrument, resulting in a two-
dimensional Fourier relationship between the observed visibilities in the
aperture plane and the sky brightness distribution (over a small field of
view). When near-field objects are present in an observation, the standard
approach applies far-field delays during correlation, resulting in loss of
signal coherence for the signal from the near-field object. In this paper, we
demonstrate near-field aperture synthesis techniques using a Murchison
Widefield Array observation of the International Space Station (ISS), as it
appears as a bright near-field object. We perform visibility phase corrections
to restore coherence across the array for the near-field object (however not
restoring coherence losses due to time and frequency averaging at the
correlator). We illustrate the impact of the near-field corrections in the
aperture plane and the sky plane. The aperture plane curves to match the
curvature of the near-field wavefront, and in the sky plane near-field
corrections manifest as fringe rotations at different rates as we bring the
focal point of the array from infinity to the desired near-field distance. We
also demonstrate the inverse scenario of inferring the line-of-sight range of
the ISS by inverting the apparent curvature of the wavefront seen by the
aperture. We conclude the paper by briefly discussing the limitations of the
methods developed and the near-field science cases where our approach can be
exploited.

###### keywords:

techniques:interferometric – radio continuum, techniques: image processing,
instrumentation: interferometers

S<EMAIL_ADDRESS>
##  1 Introduction

Using three basic assumptions, conventional aperture synthesis theory derives
a 2D Fourier relationship thompson2017interferometry (25, 10) between the
visibilities sampled by an interferometer in the aperture plane and the sky
brightness distribution. The three assumptions are: that a narrow bandwidth is
used; that the object being observed is in the far-field; and that a narrow
Field of View (FOV) is being imaged. Techniques such as Multi-Frequency
Synthesis sault1994multi (22, 2, 21) and W-Stacking offringa-wsclean-2014 (13,
14) have been developed to to overcome the bandwidth and FOV limits, in this
paper we develop techniques/tools to observe objects in the near-field of the
instrument, building upon previous work. We demonstrate near-field imaging
techniques using the Murchison Widefield Array (MWA) Tingay2013TheFrequencies
(27, 31), whose long ($6$ km) baselines see objects in Low Earth Orbit (LEO)
in the near-field (for the frequency range that the MWA operates in).

The MWA is a radio interferometer built as a precursor to the low-frequency
component of the Square Kilometre Array (SKA) and is located in the radio-
quiet region of the Murchison Shire in Western Australia, at Inyarrimunha
Ilgari Bundara, the CSIRO Murchison Radio-astronomy Observatory. The MWA is
capable of observing between $80-300$ MHz with an instantaneous bandwidth of
$30.72$, MHz (each element in the interferometer is an aperture array composed
of dual-polarised bow-tie antennas arranged in a 4x4 format, a so-called
“tile”).

###  1.1 Previous work

Aperture synthesis studies in the near field have been performed by many
groups with varying objectives. While often signals in the near-field are
considered a source of interference in astronomical observations
wang2021satellite (30, 28), 2023arXiv230104188F (4) has recently developed
techniques to use satellite signals to perform calibration of the instrument.
Spacecraft enthusiasts among the VLBI community have successfully been able to
track the near-field transmissions from satellites within the solar system
duev2012spacecraft (3, 9). In the ‘ultra’111distances that are comparable to
the size of the aperture array. near-field, RFIs have also been localized to
nearby transmission
lines222https://www.ursi.org/proceedings/procGA05/pdf/JE.5(01141).pdf,
electric cars being
charged333https://www.astron.nl/dailyimage/main.php?date=20210802, and new LED
lamps being installed in nearby
farmhouses444https://www.astron.nl/dailyimage/main.php?date=20200512.

In this paper, we expand on our preliminary near-field work discussed in
2022AdSpR..70..812P (19), and develop more versatile tools that are capable of
being used for a wide range of science cases. As our previous work
Tingay2013OnFeasibility (26, 32, 20, 18) has demonstrated the MWA to be
capable of detecting FM reflections from LEO satellites, we use a single $2$
min observation of the International Space Station (ISS) as the near-field
target of choice in this work. Using our developed near-field correction
tools, we then infer the range of the ISS across multiple time steps by
inverting the focal distance that provided the maximum signal to noise on the
source.

This paper is structured as follows. We describe the data used and our methods
in Section 2, and our results in Section 3. The discussion and conclusions are
in Section 4 and 5, respectively.

Figure 1: The Topocentric Cartesian Coordinate (TCC) system used in this work
to calculate near-field corrections. Figure 2: MWA images for four different
focal distances. In all four panels, the green arrows show the location of the
ISS signal and the white arrow shows the location of a background astronomical
point source. We note that at large focal distances, the objects in the near-
field (e.g ISS) appear de-correlated and at much closer focal distances the
background far-field sources appear de-correlated. An animation of this Figure
is available at https://www.youtube.com/watch?v=sqieJJYJCAo.

##  2 Observations and Methods

###  2.1 Data Pre-Processing and Calibration

We use a single $2$ min Phase 3 MWA observation (obs ID 1333747192, and
henceforth referred to as the target observation) of the ISS in the FM band.
We downloaded the data from the All-Sky Virtual
Observatory555https://asvo.mwatelescope.org/ (ASVO), with $0.25$ s time-
averaging and $40$ kHz frequency averaging. The backend of ASVO uses
Birli666https://github.com/MWATelescope/Birli to convert MWA correlator output
files to the CASA defined measurement set format 2007ASPC..376..127M (11).

We calibrate the instrument by using an observation (obs ID 1333747784) of the
radio galaxy Hercules-A, also downloaded from ASVO using the same parameters
as the target observation. We perform RFI flagging on the calibration
observation using the AOFLAGGER tool 2015PASA…32….8O (16). Using the calibrate
tool 2016MNRAS.458.1057O (15) we perform a preliminary round of calibration
using the source model777https://github.com/StevePrabu/MWA-
ORBITAL/blob/master/models/model-HerA-27comp_withalpha.txt of the Hercules-A
radio galaxy. As Hercules-A is the brightest source in the FOV of the
observation, we obtain a reasonably good initial amplitude and phase
calibration solution. Using WSCleanoffringa-wsclean-2014 (13, 14) and the
preliminary calibration solution, we perform a round of self-calibration to
obtain better calibration solutions. The final set of calibration solutions
are then transferred to the target observation.

Figure 3: The top three panels show the reconstructed image, visibility phases
plotted against baseline length, and histogram distribution of the visibility
phases for the frequency differenced visibilities obtained from differencing a
channel with ISS FM signal from an adjacent channel with no FM signal. The
bottom three panels show the same but for frequency difference visibilities
obtained for two channels, neither of which had an ISS reflection signal. An
animation of the figure for a wide range of focal distances can be obtained at
https://www.youtube.com/watch?v=hqPI-iFX6bY

###  2.2 Near-field Correction using LEOLens

The radiation wavefronts originating from near-field objects appear curved
when viewed using the long baselines of the MWA. During correlation most
interferometers, including the MWA, assume the sources to be in the far field
of the instrument, thus resulting in a loss of coherence in the near-field
signal due to de-correlation. However, we can recover near-field phase
coherence by re-arranging the fringes (phase rotation of visibilities)
projected by the interferometer in the plane of the sky. We apply this near-
field correction to the target visibilities using a python
casacore888https://github.com/casacore/python-casacore tool we call
LEOLens999https://github.com/StevePrabu/LEOLens, explained in the following
paragraphs.

In Figure 1, we show a Topocentric Cartesian Coordinate (TCC) system centered
at the geometrical centre of an interferometer, such that $\hat{x}$ points
East, $\hat{y}$ points towards the zenith, and $\hat{z}$ points North. Having
obtained the coordinates of every MWA tile in this coordinate system, we next
obtain the coordinates of the near-field object. The TCC coordinates of the
near-field object ($f_{x}$, $f_{y}$, $f_{z}$) can be obtained from its azimuth
angle ($\theta$), elevation angle ($\phi$), and range ($f_{dist}$) from the
origin,

$\begin{array}[]{l}f_{x}=f_{dist}\times cos(\phi)\times sin(\theta),\\\
f_{y}=f_{dist}\times sin(\phi),\\\ f_{z}=f_{dist}\times cos(\phi)\times
cos(\theta).\end{array}$ (1)

For every baseline between antennas $A_{i}$ ($X_{i},Y_{i},Z_{i}$) and $A_{j}$
($X_{j},Y_{j},Z_{j}$), the delay (or the w-term) for the baseline can be
obtained using Equation 2.

$\begin{array}[]{l}r_{i}=\sqrt{(f_{x}-X_{i})^{2}+(f_{y}-Y_{i})^{2}+(f_{z}-Z_{i})^{2}},\\\
r_{j}=\sqrt{(f_{x}-X_{j})^{2}+(f_{y}-Y_{j})^{2}+(f_{z}-Z_{j})^{2}},\\\
w_{near-field,i,j}=r_{j}-r_{i}.\end{array}$ (2)

As the voltage streams from both antennas are already correlated using far-
field delays ($w_{far-field,i,j}$) by the correlator, LEOLens updates every
visibility measurement using the near-field delay.

The capacity to define a delay (or w-term) for a baseline comes from having
set a phase reference in the sky plane. Hence, any changes in the w-term (such
as due to the near-field corrections done here) will rotate the baseline
phase. This phase correction is applied by LEOLens to the visibility using
Equation 3.

$\Delta\phi_{i,j}=\exp^{i2\pi\frac{\Delta w_{i,j}}{\lambda}},$ (3)

where $\phi$ is the interferometer phase, $\lambda$ is the wavelength of
radiation, and $\Delta w_{i,j}=w_{near-field,i,j}-w_{far-field,i,j}$.
$\Delta\phi_{i,j}$ is the phase difference applied to the visibility phase on
baseline formed by antennas $A_{i}$ and $A_{j}$. We illustrate the impact of
near-field corrections in the aperture plane and the sky plane in A.

##  3 Results

Having described our near-field correction method, we present our results in
three steps. In Section 3.1, we show the MWA near-field images made for a wide
range of focal distances, followed by a null test in Section 3.2. Having
developed confidence in the method using the null test, we then proceed to
demonstrate results from our range estimation method in Section 3.3.

###  3.1 Near-Field Images

Using a single time-step during which the ISS was detected through FM
reflection, we show near-field images at varying focal distances (Figure 2).
We focus the array over a wide range of distances ($10,000$ km, $1000$ km,
$500$ km, and $50$ km) using LEOLens and then create images using WSClean. A
distance of $10,000$ km is in the far-field of the instrument, and hence the
image is not noticeably different from the image produced without any near-
field corrections. As we bring the focus of the array to much closer
distances, at about $500$ km we see a streak-like signal from the ISS, as
coherence is obtained. The ISS signal is again de-correlated as we bring the
focal distance to $50$ km.

In the 10,000 km image of Figure 2, we also see a point source (background
radio galaxy) whose location in the sky is shown using the white arrow. We
note that the point source is de-correlated as we bring the focal distance to
smaller distances, in accordance with our expectations. Due to the source
being un-resolved, all baselines respond equally to the point source across
the aperture plane. As the Phase 3 MWA extended array has predominantly long
baselines which undergo significant delay correction (or rotation of fringes
in the sky) as we change the focal distance, the point source is de-
correlated. Conversely, the overall phase structure of any extended source in
the observation is expected to remain preserved for a wider range of focal
distances, as the structure of an extended source is sampled by the shorter
baselines that do not undergo significant delay corrections101010an animation
of near-field corrections applied to an extended source can be found here
https://www.youtube.com/watch?v=sqieJJYJCAo. As the observation used in the
animation is an MWA Phase 1 observation, and was processed differently to the
other observations used here, we do not put it in the main body of the
manuscript. .

###  3.2 The Near-Field Null Test

We build confidence in our near-field techniques by performing a null test
described below. We select a fine-frequency channel ($\nu_{1}$) containing the
ISS FM reflection signal and difference the visibilities with an adjacent
fine-frequency channel ($\nu_{2}$) that did not have any FM reflection. Doing
so isolates the signal of interest (ISS FM reflection) from the background
astronomical sources and we use it to perform the null test. Due to the close
proximity of the two selected channels, the instrument’s response to the
background sky ($S_{sky}$ below) can be considered to be identical, the
difference subtracting the sky’s contribution to the measured visibilities.
Also, due to the closeness of the two channels, the instrument’s response is
not noticeably chromatic and does not leave behind artifacts while
differencing. Mathematically the frequency differenced visibilities can be
represented as follows

$\begin{array}[]{l}V(\nu_{1})=\epsilon(\nu_{1})+S_{ISS}+S_{sky}(\nu_{1}),\\\
V(\nu_{2})=\epsilon(\nu_{2})+S_{sky}(\nu_{2}),\\\ \Delta
V=V(\nu_{1})-V(\nu_{2}),\\\ \Delta
V\approx\epsilon_{1,2}+S_{ISS}\quad[\because S_{sky}(\nu_{1})\approx
S_{sky}(\nu_{2})].\\\ \end{array}$ (4)

In the above equations, due to the noise [$\epsilon(\nu_{1})$ and
$\epsilon(\nu_{2}$)] in the two channels being un-correlated, the ISS signal
($S_{ISS}$) should be detected above a Gaussian noise distribution. We focus
the differenced visibilities to $500$ km (approx. range of the ISS) and show
the corresponding image and phase-distrbution of the visibilities in the top
three panels of Figure 3. Due to the ISS signal being at the phase centre of
the image, the phases of the differenced visibilities cluster near
$0^{\circ}$deg and $180^{\circ}$deg, as expected.

In order to test the reliability of our near-field correction technique, we
create a different set of frequency differenced visibilities, but this time
neither of the two channels show significant ISS signal, and we focus the
differenced visibilities to $500$ km as before. As neither of the two channels
have ISS signal in them, we expect the differenced visibilities to show noise-
like properties. The image and the phase distribution of this new set of
differenced visibilities are shown in the bottom three panels of Figure 3.
From Figure 3 we see that the phases of the visibilities are randomly
distributed (as would be expected for noise) and show no coherent signals in
the reconstructed image, thus building confidence in the near-field
techniques/software developed in this work.

Figure 4: The top-left panel shows the image with visibilities focused in the
far field for one of the time steps considered. The image is also phase-
centred at the default pointing centre of the observation. The insert panel
(top-middle) shows the phase-tracked image of the ISS. In the top-right panel,
we show the SNR of the ISS signal when focusing the array to a wide range of
focal distances for a single timestep. Having performed this across multiple
timesteps, we plot the estimated line of sight range in the bottom-left panel.
Close to the actual range of the ISS, we attempt focussing at every 5km
intervals, and hence we use $5$ km as the error in the bottom-left plot. We
also show the Using the estimated azimuth, elevation, and range of the ISS, we
are able to track its trajectory in 3D as shown in the bottom-right panel. An
animation of the Figure can be obtained from
https://www.youtube.com/watch?v=99vksNf1viA

###  3.3 Near-Field Object Range Inference

In Section 3.1, we demonstrated being able to focus on a near-field object
using prior knowledge of its distance from the geometric centre of the MWA. In
this section, we demonstrate the inverse problem of inferring the line of
sight range to the object. We focus the array at a wide range of trial focal
distances. The distance that provides the maximum coherence (measured as
signal-to-noise ratio (SNR) in the images) on the source is assumed to be the
range to the object.

We demonstrate our ranging method using the frequency differenced visibilities
described in Section 3.2. For every time-step that the ISS was detected, we
change the phase-centre of the visibilities to have the ISS signal at the
centre of the image. We then focus the differenced visibilities to a wide
range of trial focal distances and plot the SNR of the ISS signal in the top-
right panel of Figure 4. As we obtain maximum coherence on the ISS signal when
the assumed focal distance matches the true distance, we use the peak of the
SNR vs focal distance curve as a proxy for the range measurement. In the
bottom-left panel of Figure 4, we plot our range measurements for the ISS
across multiple time-steps and compare it to the range predicted from the Two
Line Element (TLE) data for the ISS published by space-track.org during the
epoch of the observation. As we know the azimuth, elevation, and range of the
ISS from our observations, we are able to track the ISS trajectory in 3D space
with respect to the MWA, shown in the bottom-right panel of Figure 4.

####  3.3.1 Modelling from First Principles

The top right panel of Figure 4 shows that the signal recovered as a function
of the assumed focal distance is quite distinctive. Thus, while identifying
the peak signal to noise of this function provides an estimate of the optimal
focal distance, it would be better to model the entire function and use all of
the measurement data. We briefly consider the plausibility of forming such a
model in this section.

A model to describe the data shown in Figure 4 needs to take into account the
interferometric response of the MWA, as a function of assumed focal distance
to an object. We have tested a model that utilises the known relative
positions of the MWA tiles, and calculates the interferometric response of
each baseline (pair of tiles) in the array for an object at an arbitrary
location with respect to those tiles, for different values of the focal
distance. In this case, errors in the delay and delay rate, relative to the
true values, will cause a loss of coherence in an observation that is averaged
in time and frequency. These errors vary as a function of the length and
orientation of the baseline, relative to the direction of the object.

As such, across a given time-step, the model requires three parameters to
describe the starting position of the object in the array’s frame of reference
as well as three parameters to describe it’s finishing position. When
described in spherical coordinates, the parameters can be expressed as a
range, azimuth, and elevation. The six parameter model can be fit to
observational data, such as in Figure 4. In practise, the parameter space for
the fit has many local minima and least squares methods struggle to approach
the global minimum. Thus, we have attempted a brute force grid search of the
parameter space, which is computationally expensive. For example, implemented
in python, a single trial for a single grid point takes $\sim$10 seconds on a
single CPU. The resolution of the grid needs to be high, as the array will be
coherent when all differential delays are correct to within a fraction of a
wavelength ($\sim$3 m), and the grid range also needs to be large, as the
inherent accuracy of our starting point from a TLE is $\sim$1 km. Thus, that
translates to $\sim$1018 trials, or $\sim$1019 seconds, or 100 billion years
on a single CPU, which is clearly not feasible. Optimised code would assist to
reduce the compute time.

We have implemented trials using a far coarser grid as a test to see if
promising regions of parameter space can be identified for further
investigation, which are being run on large high performance computing
clusters, and produce output that closely resembles the behaviour of the
observational data. Further refinements are a work in process.

##  4 Discussion

###  4.1 Visibility amplitudes

While we have successfully demonstrated the recovery of phase coherence in the
near-field, we have not made any comments about the amplitude of the
visibilities. For astronomical sources, the amplitude of the electric field
seen by both antennas of a baseline can be assumed to be identical due to the
large distance to the radiating source. However, for near-field objects, the
antennas see different amplitudes due to different path lengths between the
object and the antennas, and is derived below.

Consider again a baseline between two antennas ($A_{i}$ and $A_{j}$) observing
an isotropically radiating source of luminosity $L$ (J/s) at distances $r_{1}$
and $r_{2}$ from the antennas, respectively. If the effective collecting area
of the two antennas are $A(l_{i},m_{i})$ and $A(l_{j},m_{j})$, where $l$ and
$m$ are orthogonal direction cosines of the near-field source with respect to
the antenna, the powers measured by the antennas are,

$\begin{array}[]{l}P_{i}=L\times\frac{1}{4\pi r_{i}^{2}}\times
A(l_{i},m_{i})\\\ P_{j}=L\times\frac{1}{4\pi r_{j}^{2}}\times
A(l_{j},m_{j})\end{array}$ (5)

If the radiating source is observed in a direction that is not orthogonal to
the baseline, there is a propagation path length difference ($\Delta r$)
between the source and the two antennas. However, due to geometrical reasons,
the path length cannot exceed111111not true for sources very close to the
instrument with curved wave-fronts but is a good first-order guide for the
magnitude of $\Delta r$. the actual physical distance ($b$) between the two
antennas (for example, the MWA’s longest baseline is $6$ km long, and $\Delta
r$ for the baseline $<=6$ km). The two different propagation path lengths are
defined as follows,

$\begin{array}[]{l}r_{i}=r\\\ r_{j}=r_{i,j}+\Delta
r_{i,j}\quad(\mbox{where}\;\Delta r_{i,j}<=b)\end{array}$ (6)

Using Equations 5 and 6, the ratio of powers measured by the two antennas is
given by,

$\begin{array}[]{l}P_{ratio}=P_{i}/P_{j}\\\
P_{ratio}=\frac{A(l_{i},m_{i})}{A(l_{j},m_{j})}\times\frac{r_{j}^{2}}{r_{i}^{2}}\\\
P_{ratio}=\frac{A(l_{i},m_{i})}{A(l_{j},m_{j})}\times\left[\frac{r+\Delta
r}{r}\right]^{2}\par\end{array}$ (7)

Figure 5: updated figureWe show the ratio of powers seen by a baseline with
($A_{1}$ not equal to $A_{2}$) and without ($A_{1}=A_{2}$) holographic
effects. For the above plot, we assume $\Delta r=2$ km (approx. delay in a $6$
km baseline observing a source $20$ degrees from the zenith).

For astronomical sources,
$\lim_{r\to\infty}P_{ratio}\approx\frac{A(l_{i},m_{m})}{A(l_{j},m_{j})}$, and
the two antennas only see holographic effects, but for the objects in the
near-field (where $\Delta r$ is comparable to $r$) the two antennas see
significantly different powers from the radiating source. We do not account
for this effect in our ISS analysis and find it to be about $P_{ratio}=0.986$
(r=$500$ km, $\Delta r$=$3.35$ km) for the longer baselines.

###  4.2 Confusion from near-field sources

In most radio-astronomy observations, objects in the near-field constitute
Radio Frequency Interference (RFI). A common practice to mitigate RFI is to
flag the data, channels and/or times, and/or baselines, that respond to the
RFI signal ford2014rfi (5). However, with the advent of satellite mega-
constellations, the radio sky is getting increasingly polluted and we may
reach a time in the future when flagging is no longer an affordable option due
to the constant presence of satellite signals in the data. An alternative RFI
mitigation strategy would be to subtract or ‘peel’ the RFI’s signal
contribution perley2003removing (17) from the visibility matrix, demonstrated
recently by 2023arXiv230104188F (4) using a Bayesian framework to subtract
satellite RFI from MeerKAT jonas2016meerkat (7) simulated data. However, based
on the near-field aperture synthesis understanding developed in this work, we
comment that the peeling of near-field objects without any near-field
corrections could leave behind residual sidelobe confusion noise in the
images, explained further below.

We select five different fine channels (one with a strong ISS reflection
signal and the others without any noticeable ISS signal), focus the
visibilities to a wide range of focal distances, and plot the corresponding
residual noise maps in Figure 6. In a perfectly cleaned image, the residual
map should just contain contributions from thermal noise (due to the
instrument operating at non-absolute zero temperature) with a random
distribution of phases, much like the bottom panels of Figure 3, and the noise
level would not be expected to change with any phase-rotation that may be
applied to the visibilities. We see from Figure 6 that for the channel with
the ISS signal the noise in the residual map is the lowest when the fringes
are rotated to the correct focal distance of the near-field RFI, and for all
other fringe positions (or focal distances) the cleaning process leaves behind
residual noise due to imperfect subtraction of the near-field source. In the
other four channels with no ISS signal, no astronomical sources are detected
in $0.25$ s time-averaged fine-channel data (due to very large thermal noise),
and the residual map noise levels do not noticeably change with focal
distance, thus demonstrating the importance of near-field considerations while
subtracting near-field RFI objects from the visibility matrix. To properly
reach the lowest noise levels in the data when mitigating RFI, the peeling
process has to be performed in the near field. We also note that the lowest
noise in the channel with the ISS signal is higher than the other channels,
possibly due to increased system noise in this channel (as the ISS signal is a
few thousand Jy).

Figure 6: Noise levels in residual maps for five different channels focused to
a wide range of focal distances. Note that the noise is lowest in the channel
with the ISS when the focal distance is approx. $500$ km (also the line of
sight range to ISS)

###  4.3 Fringe washing

While longer baselines have better range resolution using our method (as they
see more change in wavefront curvature), as previously discussed in
2022AdSpR..70..812P (19), the signals from fast-moving objects get blurred
into smears (fringe-washed) when they move more than a synthesized beam during
the visibility integration time. The phase measurement of a source by a
baseline depends on the source’s position in the fringe pattern projected by
the baseline in the sky plane. However, for fast-moving objects such as
satellites, the source moves through many fringes within the time-averaging
interval of the correlator, and hence we obtain an averaged phase measurement.
Hence, even though we have developed techniques in this work to account for
the near-field curvature perceived by the long baselines, the reduction in SNR
due to fringe-washing is not recoverable. For example, for a satellite at
$400$ km moving with an angular speed of $1^{\circ}/s$ near the zenith, apprx.
$75\%$ of the MWA extended configuration baselines are fringe-washed, while
even the longest baseline of MWA’s compact configuration would not be
affected. For a more detailed discussion on fringe-washing, we direct the
reader to 2022AdSpR..70..812P (19).

###  4.4 Future Work

Near-field tools and techniques can be used in a wide range of science cases.
We currently have plans to use LEOLens to search for low-frequency intrinsic
radio emission from meteors, previously only detected by obenberger2014limits
(12) using the Long Wavelength Array 2012JAI…..150004T (23).
Zhang2018LimitsMWA (32) previously attempted to detect the meteor emission
using a $322$ hour MWA observation campaign, but no candidates were
identified. The study discarded all the longer MWA baselines and just used the
short baselines to mitigate the near-field effect and hence was performed at a
much-reduced sensitivity. Given that we now have tools to achieve coherence on
near-field objects using the long baselines, we aim to perform a more
sensitive search for intrinsic meteor emission with the MWA.

A natural extension of the discussion on residual confusion noise from near-
field objects in Section 4.2 is to develop a near-field RFI peeling capability
in conjunction with LEOLens. Our preliminary peeling test with the ISS
observation used here shows promising results and we aim to develop this
peeling method into a tool for the wider community in the future.

##  5 Conclusions

In this paper, we explore the near-field aperture synthesis techniques using a
single observation of the ISS using the Murchison Widefield Array. For desired
focal distances, we calculate the appropriate delays for every baseline which
would bring the near-field signal into focus of the reconstructed image. We
illustrate the effect of near-field corrections in the aperture plane that
results in ’curving’ the array through addition delays such that the incoming
near-field wavefront falls coherently on the array. This delay correction in
the aperture plane, translates to rotating fringes at different rates in the
sky plane. As longer baselines see more of the near-field curvature, they
undergo more delay corrections (or fringe rotation) when compared to the short
baselines.

Having developed a python tool that performs near-field corrections to the
input interferometric dataset, we use it to demonstrate the inverse problem of
inferring the range of the near-field event from the apparent curvature in
radiation as seen by the array. We do so by trialing many focal distances to
make a focused image of the near-field signal. When the assumed focal distance
is equal (or approximately equal) to the true distance, the re-constructed
image has the highest SNR. We demonstrate this using the ISS observation, and
obtain ranges are in agreement with the distances predicted by its Two Line
Elements.

We conclude the paper by discussing the limitations of the near-field methods
used in this work. For objects that are very close (distances comparable to
its longest baseline length) to the array, the near-field signal undergoes
different amounts of propagation fading resulting in different powers seen by
the two antennas of the baseline. We also find that the peeling of near-field
RFI from interferometric data can leave behind residual confusion noise when
not accounting for the near-field effects.

This scientific work uses data obtained from Inyarrimanha Ilgari Bundara / the
Murchison Radio-astronomy Observatory. We acknowledge the Wajarri Yamaji
People as the Traditional Owners and native title holders of the Observatory
site. Establishment of CSIRO’s Murchison Radio-astronomy Observatory is an
initiative of the Australian Government, with support from the Government of
Western Australia and the Science and Industry Endowment Fund. 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.
This work was supported by resources provided by the Pawsey Supercomputing
Research Centre with funding from the Australian Government and the Government
of Western Australia. Steve Prabu would also like to thank Adrian Sutinjo for
the valuable comment on amplitude fading during the CIRA Journal Club talk on
near-field aperture synthesis.

Figure 7: In the above four panels we show the absolute delay correction
performed by LEOLens for four different focal distances. An animation of the
figure can be obtained from https://www.youtube.com/watch?v=9LIN6fErVbI

### Sofware

We acknowledge the work and the support of the developers of the following
Python packages: Astropy theastropycollaboration_astropy_2013 (24, 1), Numpy
vanderwalt_numpy_2011 (29), Scipy jones_scipy_2001 (8), matplotlib Hunter:2007
(6), and python-casacore 121212https://github.com/casacore/python-casacore.
The work also used WSCLEAN offringa-wsclean-2014 (13, 14) for making fits
images and DS9131313ds9.si.edu/site/Home.html for visualization purposes.

##  Appendix A Near-Field Correction in Aperture Plane vs Sky Plane

We illustrate the effect of near-field correction in the aperture plane
(u,v,w) and the sky plane (l,m) using a single FM band fine channel data for
one of the time steps that the ISS was detected through reflection.

In the aperture plane, the near-field correction can be thought of as
‘curving’ the array to match the near-field wavefront such that the near-field
signal falls coherently on the array. We show this in the four panels of
Figure 7. For an arbitrarily chosen four different focal distances ($10,000$
km, $1000$ km, $500$ km, and $50$ km) we show the absolute delay correction
performed by LEOLens to approx. $8000$ instantaneous baselines of the MWA. We
note from Figure 7 that for the wide range of focal distances considered, the
short-baselines do not go through much delay correction as they would still
see the objects in the far field. On the contrary, the long baselines have
larger delay and delay rate (slope of delay vs focal distance) corrections as
we bring the focal distance closer to the instrument.

In the sky plane, the near-field corrections result in rotating the fringes at
different rates across the sky as we bring the focal distance from a faraway
distance to shorter focal distances. We illustrate this using an animation
provided at https://www.youtube.com/watch?v=3HlUgVY_nfU. In the top-left and
top-middle panels of the animation, we show the fringe produced by a long
baseline and a short baseline (the baseline lengths were arbitrarily chosen to
help demonstrate better). As we bring the focal distance of the array from
far-field to much closer distances, the fringes are rotated at different rates
in the sky. The bottom-left and bottom-middle panel shows the delay
corrections performed to the long and short baseline as we change the focal
distance. The MWA has about $8000$ instantaneous baselines and in the bottom-
right panel we show the combined effect of all the rotated fringes as we
change the focal distance. When the assumed focal distance matches the range
of the ISS (about $500$ km), all the fringes coherently produce the image of
the ISS with the maximum possible SNR.

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