Lunar Far-Side Radio Arrays: A Preliminary Site Survey–A

# Lunar Far-Side Radio Arrays: A Preliminary Site Survey

Zoe A. Le Conte  1,2,3 Martin Elvis E-mail<EMAIL_ADDRESS>1
Philipp A. Gläser4
1Center for Astrophysics | Harvard & Smithsonian  60 Garden Street  Cambridge  MA 02138  USA
2Centre for Extragalactic Astronomy  Department of Physics  Durham University
South Road  Durham DH1 3LE  UK
3Faculty of Engineering and Physical Sciences  University of Southampton
Southampton  SO17 1BJ  UK
4Technical University of Berlin  Kaiserin-Augusta-Allee 104-106  Berlin  10553
Germany

(Accepted 16/06/2023. Received 25/05/2023; in original form 15/12/2022; 2023)

###### Abstract

The origin and evolution of structure in the Universe could be studied in the
Dark Ages. The highly redshifted HI signal between $30<z<80$ is the only
observable signal from this era. Human radio interference and ionospheric
effects limit Earth-based radio astronomy to frequencies $>30$ MHz. To observe
the low-frequency window with research from compact steep spectrum sources,
pulsars, and solar activity, a 200 km baseline lunar far-side radio
interferometer has been much discussed. This paper conducts a preliminary site
survey of potential far-side craters, which are few in number on the
mountainous lunar far-side. Based on LRO LOLA data, 200 m resolution
topographic maps of eight far-side sites were produced, and slope and
roughness maps were derived from them. A figure of merit was created to
determine the optimum site. Three sites are identified as promising. There is
a need to protect these sites for astronomy.

###### keywords:

Moon – planets and satellites: surfaces – site testing – radio lines: general

##  1 Introduction

The far-side of the Moon is sheltered from anthropogenic low-frequency radio
emissions and is considered the best location for a low-frequency radio
telescope to investigate a broad scope of research areas, including the cosmic
Dark Ages, compact steep spectrum (CSS) sources and solar radio bursts. This
paper seeks to establish the best sites on the far-side for an array that
maximally extracts information from the highly redshifted 21 cm signal and
surveys the low-frequency radio sky.

The Astro2020 Decadal Survey of Astronomy and Astrophysics identified the Dark
Ages as the discovery era for cosmology (See National Academies of Sciences,
Engineering, and Medicine, 2021). Studying the Universe during the so-called
Dark Ages can deepen our understanding of the evolution of large-scale
structure and the theorised Epoch of inflation (Adams et al. (1993)). The Dark
Ages can be probed using the signal from a highly redshifted 21 cm transition
of neutral hydrogen. This forbidden transition is produced by the emission of
a photon with rest frequency 1420 MHz, by the process of the hydrogen’s proton
and electron spin-spin coupling, and results in a hyperfine transition spin-
flip and subsequently the emission of a photon (Muller & Oort (1951)).

The 21 cm signal can be measured as a sky-averaged global signal or spatial
fluctuations. The 21 cm absorption trough traces the density field of the
universe during the Dark Ages epoch. Only one detection of the 21 cm
absorption profile, in the Cosmic Dawn, in the sky-averaged spectrum, has been
claimed (but this has been disputed (Singh et al. (2022))). This detection
comes from the Experiment to Detect the Global Epoch of Reionization Signature
(EDGES) experiment and is centred at 78 ± 1 MHz corresponding to a redshift
$z\approx 17$ (Bowman et al. (2018)). The EDGES trough is $\sim 3$ times
deeper than the standard model, and adiabatic cooling is insufficient to
explain the result (see, e.g., Burns, 2021; Bowman et al., 2018; Mebane et
al., 2020).

Spatial fluctuations in the highly redshifted 21 cm signal test the structure
formation predictions of the standard cosmological model. Measuring the
absorption of galaxy precursor neutral hydrogen gas clouds against the CMB
increases the number of observable modes enormously. The billions of galaxies
in galaxy surveys were potentially constituted from hundreds of millions of
cold hydrogen gas clouds; therefore, trillions of independent modes can be
measured (Silk (2018)). They provide unprecedented constraints of the early
Universe. In particular, on the matter power spectrum spectral index (Mao et
al. (2008)), non-gaussianity (Muñoz et al. (2015)) of the inflationary era,
the mass of the neutrino and the curvature of the Universe (Mao et al.
(2008)). 21 cm observations have two other major advantages over the CMB
(Furlanetto et al. (2006)): (i) They are unaffected by Silk damping (the
smoothing phenomenon of primordial density fluctuations) and so spatial
fluctuations hold at much smaller mass scales in comparison to the CMB (Silk
(1967)); (ii) They can be used to construct the hydrogen structure in a 3D
volume (Furlanetto et al. (2006)).

An ultra-long wavelength radio interferometer is required to measure high
redshift spatial fluctuations. The 21 cm signal is observed by measuring the
neutral hydrogen spin temperature against the CMB radiation temperature. The
evolution of the transition spin temperature is explained by Loeb &
Zaldarriaga (2004), limiting the redshift range to $30<z<80$. For this
redshift range, the 21 cm signal lies in the meters wavelength range (6.5 -
17.0 m) and the tens of MHz frequency range (17.6 - 46.1 MHz). Koopmans et al.
(2021) specifies the requirements for a lunar-based instrument to extract the
faint 21 cm signal with S/N > 10 (Refer to Section 7 of Koopmans et al., 2021,
and Figure 8 for S/N predictions). The cosmological radio interferometer
relies on a densely packed (order of $10^{4}$ dipoles) core to reach the
sensitivity of the 21 cm signal and a baseline of a few kilometres.

Very low-frequency radio astronomy has been recognized to have several other
potential uses described below. Currently, the largest radio telescope, LOFAR
(Low-Frequency Array, van Haarlem et al. (2013)), operates at the longest
wavelengths observed from Earth. The LOFAR Low Band Antenna (LBA) Sky Survey
(LoLSS, de Gasperin et al. (2021)) observes the frequency range 42 - 66 MHz
with high-sensitivity $\sim$ 1 mJy beam-1, and high-resolution $\sim$ 15
arcsecs. To reach these exceptional parameters, the baseline of the LBA is
$\sim$ 100 km. Baselines up to $\sim$ 200 km can be attained on the lunar far-
side, and applying the same imaging capabilities as LOFAR would achieve 10 MHz
with an angular resolution of $\sim$ 25 arcsecs. The multi-research
capabilities of a lunar far-side radio interferometer would extend the
capabilities of LoLSS.

There is a variety of low-frequency radio-loud active galactic nuclei (AGN)
sources, from CSS at a few hundred years to radio galaxies on the scale of
Mpcs and aged at $\sim$ 1 Gyr (e.g., Dabhade et al., 2020). Jets, driven by
synchrotron processes, propagate beyond the host galaxy and are classed as
Fanaroff and Riley radio galaxies (FRIs and FRIIs, Fanaroff & Riley (1974))
with signature double-lobed structures that can span up to hundreds of
kiloparsecs. CSS, by contrast, have radio propagation extents $\sim$< 20 kpc
and some are thought to grow in strength and evolve into FRIs and FRIIs (e.g.,
Fanti et al., 1990). Alternatively, the phenomenon "frustration" describes the
inability of jets in CSS sources to penetrate the interstellar medium (ISM)
due to a dense ISM or intrinsically weak jets (e.g., van Breugel et al.,
1984). High-resolution and sensitive low-frequency measurements of compact
objects could assess the extent of the physical processes responsible for the
termination of jets (e.g., McKean et al., 2016).

An ultra-long wavelength interferometer would open a window for new low-
frequency pulsars studies Stappers, B. W. et al. (2011). Despite pulsars being
intrinsically brightest at low radio frequency ranges (< 300 MHz), past
surveys were conducted at $\sim$ 350 MHz or $\sim$ 1.4 GHz due to
observational biases. Pilia et al. (2016) investigates the evolution of pulsar
profiles over a frequency range < 200 MHz with respect to magnetospheric
origins and dark matter-induced variations. A specific trend cannot be
identified with decreasing radio frequency, and the distribution of the
average spectral index is compatible with different pulsar behaviours.

Solar activity, such as coronal mass ejections (CME), solar flares and
sunspots, can be observed across the low-frequency radio spectrum; from Type
III bursts extending to tens of KHz to the observation of Type I bursts up to
hundreds of MHz. Thermal or non-thermal, coherent or incoherent emission
mechanisms produce solar radio bursts which trace energetic phenomena in the
interplanetary medium and solar corona. Type II radio bursts are emitted from
highly energetic accelerated electrons from shock waves driven by supersonic
CMEs (e.g., Kouloumvakos et al., 2021). Monitoring solar activity below 10 MHz
would advance studies in CME release mechanisms and assess the impacts of CMEs
on Earth.

For a radio interferometer applicable to a wide range of research, this paper
adopts 200 km as the minimum interferometer baseline distance.

Earth-based low-frequency/long-wavelength radio astronomy has been constrained
by the uncontrollable production of human-created radio frequency interference
(RFI) over recent decades. Shortwave broadcasting, emitted by
telecommunication and radio communication satellites in Earth orbits and
ground-based transmitters, floods radio telescope instruments with noise below
30 MHz (e.g., Maccone (2021)). Radio observations are further restricted by
ionospheric effects, with frequencies below 10 MHz being absorbed and
distorted, becoming effectively opaque (Kaiser & Weiler (2000)).

A solution is constructing radio astronomy instruments on the far-side of the
Moon. The 3,400 km thickness and tidal locking of the Moon shield the far-side
surface from RFI by as much as 90 dB and creates a Radio Quiet Zone (RQZ)
(e.g., Kim (2021)). The location of this zone is determined by the attenuation
of radio waves around the lunar limb. There are three factors to consider: (1)
geometry; (2) libration; (3) diffraction. (1) Geometry refers to the RFI
produced from the surface of the Earth or by a Geosynchronous Equatorial Orbit
(GEO) satellite. Lunar longitude is measured from the sub-Earth point, so the
limb as seen from Earth is at $\pm 90$°. The longitude at which RFI from the
satellite diminishes to 90 dB is $\sim 6$° beyond the lunar limb. (2)
Diffraction further constrains the lunar RQZ. At 10 MHz with a -80 dB
threshold, waves are attenuated $\sim 4$° beyond the geometric limit, reducing
the RQZ to $\sim 160$° on the lunar far-side (see Bassett et al., 2020). (3)
Optical librations occur from a shift in viewing angle due to the non-circular
and inclined orbit of the Moon and have three forms: latitude, parallactic,
and longitude. Longitude is the most significant libration in which the RQZ
varies by $\approx$ 8° (Ratkowski & Foster (2014)). Combining these three
effects, the -80 dB RQZ in longitude begins at $\pm 108$° stretching 4,366 km
across the anti-Earth in longitude and latitude begins at $\pm 79$°, centred
at the anti-Earth, spanning 4791 km.

Concepts for radio interferometer arrays on the Moon have been developed. The
FARSIDE (Farside Array for Radio Science Investigations of the Dark Ages and
Exoplanets) mission concept is a recent example (see Burns et al., 2021). The
development of radio interferometer arrays on the Moon will be an evolutionary
process, starting with the few-kilometre scale, which has a large choice of
locations. Instead, the 200 km-class arrays that are ultimately needed are
much harder to find a site. Any such site must be traversable by a rover.
Rovers for the Moon and Mars have limited capabilities to deal with steep or
rough terrain (Table 2). For example, NASA’s Perseverance is a 26 cm diameter
wheeled Mars rover with a 30° safety incline limit. Similarly, NASA’s
Volatiles Investigating Polar Exploration Rover (VIPER), which will explore
the South Pole, has $\sim$> 40% slip on slopes > 15° and can traverse objects
only up to 10 cm in height. These considerations greatly reduce the suitable
site selection as the lunar far-side is extremely mountainous, without the
large, smooth maria of the near side.

This paper examines the topography of eight promising sites for a multi-
purpose 200 km-scale low-frequency radio telescope on the Moon. The contents
of this paper are observations and data reduction in Section 2; Candidate
selection in Section 3; Data analysis methods in Section 4; Map products in
Section 5; Site comparisons in Section 6; Discussion in Section 7; Conclusions
in Section 8.

##  2 Observations and Data Reduction

###  2.1 Lunar Elevation: Lunar Reconnaissance Orbiter

The data used in this paper was taken by the NASA Lunar Reconnaissance Orbiter
(LRO). LRO orbited the Moon in a nominal circular 50 km altitude polar orbit
from September 15, 2009, until moving to a fuel-conserving, 1800 km semi-major
axis, elliptical orbit on December 11, 2011 (Chin et al. (2007)). After this
date, the change of LRO orbit led to the loss of much data from the spacecraft
while LRO was at higher altitudes, limiting the data used for this study
(Barker et al. (2021)). One of the LRO mission objectives is to find potential
lunar landing sites for crewed missions. The mission also searches for
potential resources and characterises the radiation environment. LRO directly
produced topographic maps and stereo images from onboard instruments, which
can be used to derive digital terrain maps (DTMs). Images taken by the Wide
Angle Camera (WAC), part of the LRO Camera (LROC) experiment, were used to
derive a global topographic map of the Moon at a 100 m image resolution
(Robinson et al. (2010)). LRO is still operating. The Lunar Orbiter Laser
Altimeter (LOLA) (Smith et al. (2010)) is one of the six scientific
instruments on LRO. LOLA measures topographic elevations to metre elevation
accuracy but is limited to 5 m spatial resolution in the best cases and, more
often, to 57 m.

###  2.2 The Lunar Orbiter Laser Altimeter (LOLA)

LOLA provides one of the highest-resolution global surface elevation data sets
for the Moon. LOLA uses LIDAR (Smith et al. (1997)) to do so. LIDAR measures
the round trip Time Of Flight (TOF) for a laser shot reflected off the lunar
surface. The LOLA LIDAR consists of a laser transmitter (Nd: YAG in the LOLA
case), emitting a wavelength of 1064 nm and a receiver (an aluminium
Cassegrain telescope). TOF halved gives the distance from LOLA to the lunar
surface from the light travel time, in conjunction with the spacecraft’s well-
determined orbit. A similar instrument package was used onboard the earlier
Clementine spacecraft (Council (1997)). LOLA measured the slope, roughness and
the 1064 nm reflectance of almost the entire lunar surface on scales of 500 m,
200 m, 50 m and 5 m.

LOLA operates by propagating a single laser pulse (laser 1 and 2 alternate
monthly) through a Diffractive Optical Element (DOE), splitting the pulse into
five separate beams, and rotating 26° to the down-track direction. LOLA’s five
pulse spot pattern is illustrated in Figure 1. LOLA emits short pulses at the
rate of 28 Hz (considering the five beams, this is 140 measurements per
second) from two lasers (of energies 2.7 mJ and 3.2 mJ for laser 1 and laser
2, respectively). These five beams illuminate the lunar surface in a cross
pattern in the far field. Each beam has a diameter of 5 m at the surface. Each
beam backscatters off a spot on the lunar surface (yellow in Figure 1). The
receiver telescope detects the returning pulses, and the five-spot pattern is
imaged. The laser has a pulse width of $<10$ ns, and the width of each
returned pulse is recorded, measuring the height variation (roughness) within
the 5 m footprint of the laser on the surface. Consecutive five-spot patterns
are displaced by 57 m due to the orbital motion of LRO. The detector’s field
of view is shown in blue. The spot pattern calculates slope and surface
roughness along a range of azimuths.

Figure 1: Two LOLA shots encircled in dashed grey with five labelled spots as
blue circles. The pattern angle is at a 26° angle to the down-track direction.
The distance between shots is 57 m, and the smallest spot-to-spot length is 25
m. Yellow circles display the five illuminated 5 m diameter spot footprints,
and blue circles show the field of view of each detector. Orange lines show
the six possible slope calculations between spot 3 and the surrounding spots.

####  2.2.1 LOLA Pulse Width Measure

LOLA also directly measures the roughness within the 5 m footprint of a single
spot using the measured pulse width (Section 2). However, despite the LOLA
data being extensively filtered, by the selection of pulses only above 0.15 fJ
energy levels and recorded at less than 5° off-nadir angle, the pulse width
roughness maps suffered from erroneous values (Gläser (2014)). Pulse width
roughness maps will not contribute to site assessment.

###  2.3 LOLA Data and Corrections

NASA’s Planetary Data System (PDS) is a public online archive system where
laboratory results, planetary missions and observational data are stored in
common descriptions. We used the calibrated, geolocated and aggregated, time-
ordered Reduced Data Record (RDR) files (G.A. (2009)) to create LOLA DTMs for
the lunar sites of interest. LOLA raw data includes two instrumental
signatures; their removal is required to make accurate elevation maps. The two
signatures are that (1) the LIDAR pulses are not symmetric, and (2) there are
biases in the TOF calculation. These biases vary for each of the five
detectors. Both effects have been modelled by Riris & Cavanaugh (2009). They
find (1) pulse asymmetry: a variable bias dependent on the received pulse
strength arises from an induced range bias and asymmetry of the detector’s
electronic impulse response leading to a distortion of the received pulse. (2)
A TOF bias: each channel has different cable lengths, leading to a fixed
offset. The fixed and variable offset corrections are applied to the pulse
width datasets; the fixed offsets can be found in Riris & Cavanaugh (2009).

The receiver pulse width error calculated by standard deviation remains below
800 ps overall energies, equivalent to 0.24 m in elevation. From 10 ns to 25
ns, the received pulse widths have residuals compared with the emitted pulse
width at all pulse widths of $\approx 1$ ns, equivalent to 0.3 m elevation
error (Riris & Cavanaugh (2009)). The residuals are larger for pulse widths
$<10$ ns, $\approx 5$ ns, 1.5 m.

The co-registration techniques of Gläser et al. (2013) were applied to the
individual LOLA tracks that are incorporated in the DTMs. Co-/Self-
registration is an algorithm identify misaligned measurements and tracks from
the inputs of LOLA data and DTMs to which laser altimeter profiles are
registered (Scholten et al. (2012, 2011); Gläser et al. (2017)). A 0.13 m to
several meters (depending on the number of measured laser shots) positional
accuracy and 0.18 m residual heights can be achieved for data sets. This can
be compared with the pre-registration accuracy of the LOLA data, Araki et al.
(2009); Bussey et al. (2003) found positional errors of $\sim$4 m radially and
$\sim$77 m horizontally. There are negligible for the $\sim 200$ m scale being
probed here.

###  2.4 Data Loss from a Thermal Anomaly

There is a thermally-induced anomaly in the LOLA instrumentation. All five
channels work nominally when the spacecraft is over the lunar day side;
however, only two of the five receiver channels acquire significant data when
LRO observes during the cold lunar night. This ‘thermal anomaly’ was found in
ground testing due to the thermal blanket’s contraction in the cold. This
contraction pulls the transmitter beam out of focus with the receiver (Smith
et al. (2017)). Detectors 1, 2 and 5 in Figure 1 do not operate on the ‘night
side’. Detectors 3 and 4 align to emitted spots 2 and 5 by a fortunate
coincidence, enabling continuous observation (Gläser (2014)).

For instance, the effects of the LOLA thermal anomaly can be observed in the
Mare Moscoviense region. The variation in the number of spots detected per
shot means LOLA is less effective for this site. Only $47\%$ of shots yield
detections in all five spots; another $41\%$ of the area is illuminated by two
spots per shot; $9\%$ is illuminated by a single spot per shot; $2\%$ is
covered by four spots per shot; $1\%$ is covered by three spots per shot. The
impact of this anomaly is important when constructing slope and roughness maps
because there are inconsistencies between the number of spots per fitted plane
(Section 4), which limits the spatial resolution of these maps.

The smallest lunar site used in this study is Daedalus. Figure 2 shows LOLA
track coverage maps of the smaller site for elevation and pulse width
roughness data. The larger region (Figure 2(a)), $181x196$ km in size,
consists of 4,614,796 data points. When zooming in on the site to a $91x106$
km region (Figure 2(b)) the number of data points decreases to 1,138,030 and
is covered by 203 LOLA tracks. The small $91x106$ km region measured 431,842
pulse width roughness points and is covered by 179 LOLA tracks. The coverage
maps show that the centre of Daedalus has data gaps with maximum horizontal
separation $\approx 6$ km. As a result, the pulse width roughness map has
worse coverage and alternative methods to derive roughness are discussed in
Section 4.3.

(a)

(b)

(c)

Figure 2: Topographic maps of the lunar crater Daedalus. (a) A DEM of a 181 x
196 km region where a colour scale represents the elevation in km. Data points
from the 5 m footprints are coloured and white means there is no data at those
locations. (b) A zoomed-in DEM of a 91 x 106 km region where a colour scale
represents the elevation in km. (c) A zoomed-in pulse width roughness map of a
91 x 106 km region where a colour scale represents the roughness in cm. The
pulse width data is measured by broadening the received pulse to the
transmitted pulse on the 5 m footprint scale.

##  3 Candidate Site Selection

The search for a suitable site began with the visual selection of eight large
($>100$ km diameter) maria111Maria are identified as large darker surfaces
formed by mare volcanism (Shultz & Spudis (1983)). and craters within the RQZ
(longitude $>108$°) because they have a higher probability of having a smooth
crater floor. The locations of the eight lunar far-side sites are shown in
Figure 3. Their properties are listed in Table 1. The third column of Table 1
gives the distance in degrees of the lunar sites past the RQZ boundary.
Elevation maps of the eight sites of interest are shown in Figure 4 to give a
visual impression of their topography. A 4°$x$4° ($122x122$ km) grid is
overlaid on each site map, showing their relative sizes.

In addition, three comparison equatorial regions were studied on the lunar
far-side. At longitudes +30°, 0° and -30° from the anti-Earth point. These
equatorial regions are evidently rougher terrain than the eight candidate
sites (Figure 4).

Figure 3: Topographic map of surface height on the lunar farside using LOLA data (image available on NASA’s PDS (NASA/GSFC, 2010)). The data shows the mountainous terrain and distribution of impact craters. The sites of interest assessed in this paper are marked with black stars and labelled. Table 1: A table of the chosen sites and their properties, labelled by name. Their central coordinates are listed in the Selenographic coordinate system. Degrees into the Radio Quiet Zone (RQZ) is the distance in degrees past 147° East or West. The diameter of each site is the longest horizontal length within the crater boundaries. The minimum and maximum elevation is determined from LOLA data and is in reference to the Moon’s radius, 1,736 km. Below the eight chosen sites are three equatorial positions mapped to show how mountainous the far-side is.  Site | Coordinates | Degrees into RQZ | Diameter (km) | Min Elevation (km) | Max Elevation (km)
---|---|---|---|---|---
Candidate Far-Side Radio Array Sites
Apollo | 36.1°S 151.8°W | 49.6°W | 537 | -8.6 | -1.4
Daedalus | 5.9°S 179.4°E | 77.2°E | 93 | -0.4 | 4.1
Hertzsprung | 1.4°N 128.7°W | 26.5°W | 570 | -1.5 | 5.3
Korolev | 4.0°S 157.4°W | 55.2°W | 437 | -0.1 | 8.7
Mare Ingenii | 33.7°S 163.5°E | 61.3°E | 282 | -5.8 | 0.6
Mare Moscoviense | 27.3°N 147.9°E | 45.7°E | 276 | -5.6 | 4.2
Mendeleev | 5.7°N 140.9°E | 38.7°E | 313 | -4.2 | 5.8
Tsiolkovsky | 20.4°S 129.1°E | 26.9°E | 180 | -2.2 | 4.2
Comparative Equatorial Regions
Equatorial Central | 0°N 180°E | 77.8°E | 445 | -1.6 | 6.6
Equatorial East | 0°N 135°E | 32.8°E | 445 | -3.6 | 7.5
Equatorial West | 0°N 135°W | 32.8°W | 445 | -1.5 | 9.4

Figure 4: Interpolated digital elevation maps of the eight sites of interest
and three equatorial sites, where a pixel is 200 m long. Elevation in km is
shown by a colour scale below each map and the site name is above each map.
The maps are overlaid by a 4° in length grid ($\sim$ 122 km).

##  4 Data Analysis Methods

###  4.1 Digital Elevation Maps (DEM)

More quantitative topographic information is derived from constructing Digital
Elevation Maps (DEMs) from the LOLA data. DEMs display surface topography by
mapping the height variation of a region. DEMs were generated by the Generic
Mapping Tools software (GMT) (Wessel et al. (2019)). GMT is a free and open
source code222https://docs.generic-mapping-tools.org/6.4/ that allows for
manipulating data parameters to produce sophisticated illustrations for Earth,
ocean and planetary sciences. The motivation for creating DEMs is to determine
whether a site has approximately 200 km regions of smooth, low-slope terrain
and to note any physical barriers, such as significant mountainous or crater
features.

The LOLA data sets (Longitude, Latitude, Elevation) are projected onto a grid
and interpolated to create a regular grid. Given a discrete data set,
interpolation derives a polynomial function which passes through the provided
data, enabling intermediate values to be estimated. The generated grid files
can be read into Python and MATLAB to produce elevation maps. These tools
enable 3D interactive elevation created to be used to visualise these sites
from differing and exaggerated perspectives.

Non-interpolated DEMs were also constructed, where elevation values are
assigned to a colour scale and plotted on a longitude-latitude plane. These
images are useful as they highlight the non-uniform track coverage of LOLA.
These maps are also convenient for laser spot detection analysis where the
LOLA thermal anomaly can be observed for each site (Section 2.4). The DEMs can
then be used to generate slope and roughness maps.

###  4.2 Slope

The slope is defined as the terrain height variance over a specified distance
and so is a function of the distance over which the slope is measured, the
length scale. Several measures of surface roughness have been defined in
literature (e.g., Kreslavsky & Head III, 2000; Shepard et al., 2001). These
include both one-dimensional and two-dimensional slopes.

####  4.2.1 One-Dimensional Slope

Discussed here are three one-dimensional methods to derive slopes from LOLA
data: (1) RMS slope; (2) median absolute slope; (3) median differential slope
for a range of lengths.

(1) The RMS slope is defined in one-dimension as the Root Mean Square (RMS)
difference in height, $\Delta z$ between two points (also called the
deviation, $v$) over the distance between the points, $\Delta x$:

$s(\Delta x)=\frac{\Delta v(x)}{\Delta
x}=\frac{\langle[z(x_{i})-z(x_{i-1})]^{2}\rangle^{\frac{1}{2}}}{\Delta x}$ (1)

where angular brackets indicate the mean of the bracket contents (Rosenburg et
al. (2011)).

(2) The median absolute slope can be derived on the smallest scales, $\sim 25$
m, the slope of the central spot and one of the four edge spots, shown in
Figure 1 (Rosenburg et al. (2011)).

(3) The median differential slope (Kreslavsky & Head III (2000)) is derived as
follows. For a given baseline, $L$, through five points, find the difference
in elevation for a point half the baseline ahead, $z_{\frac{1}{2}}$, and a
point half the baseline behind, $z_{-\frac{1}{2}}$. Calculate the elevation
difference for points a baseline ahead, $z_{1}$, and a baseline behind,
$z_{-1}$. Subtract half the latter derived elevation difference from the first
elevation difference.

$\tan(\alpha)=\frac{\left(z_{\frac{1}{2}}-z_{-\frac{1}{2}}\right)-\frac{1}{2}\left(z_{1}-z_{-1}\right)}{L}$
(2)

The resultant elevation difference over the baseline is the tangential slope,
$\alpha$.

Each method has strengths and weaknesses: (1) RMS is an established method
because it is also used to measure the scatter of radar reflection. However,
the method is sensitive to outliers because of its dependence on the deviation
squared. (1,2) The RMS and median absolute slope are one-dimensional slope
methods derived in the down-track direction. Therefore, both underestimate the
surface gradient if the steepest slope diverges from this direction. (3)
Median differential slope removes small-scale and large-scale surface
roughness features. Arguably, the median differential slope can be described
as an intuitive parameter because small-scale roughness is measured with
respect to the long wavelength roughness profile. The median differential
slope method is a better measure compared to the RMS slope or median absolute
slope method because natural surface slope-frequency distributions are
commonly non-Gaussian with long tails.

####  4.2.2 Two-Dimensional Slope

A two-dimensional slope is preferred because slopes that diverge from the
down-track direction are included. The two-dimensional slope can be derived
from multiple spot points within a LOLA shot, e.g., between spots 1-3-4 in
Figure 1. A total of six slope measurements can be derived from spot 3 in
Figure 1 in the directions shown in orange. Vector geometry computes the plane
through three spots, identifying slope magnitude and the azimuth of the slope.
The baseline in this method is determined as the square root of the area of
the triangle. As discussed in Section 2.4, LOLA experiences a thermal anomaly
which reduces the number of spots from which signals are detected. An
interpolated grid has to be used to provide a consistent distance over which
slopes are calculated, and a vectorized method is the most computationally
efficient.

###  4.3 Roughness

Roughness can be defined as the Root Mean Square (RMS) deviation from a
specified plane (Gläser (2014)), i.e., the scatter about a one- or two-
dimensional slope. Roughness, similarly to slope, is dependent on the length
scale and is calculated here on a 200 m scale because smaller scales show
missing LOLA data tracks. Therefore, slope and roughness cannot be reliably
determined. The plane fitting method used in Section 4.2.2 measures roughness.

The pulse width measurements from LOLA are not used in this study (Section 2);
instead, the $Sigma-Z$ method was used. This method calculates the standard
deviation, $\sigma$, of the LOLA elevation, $Z$, spot measurements to a plane.
Alternatively, a method known as roving-window analysis can be used, in which
a $3x3$ kernel is scanned over the interpolated DEM maps and assigns the
deviation in elevation of surrounding pixels to a given pixel. This method is
more flexibly applied to various scales.

##  5 Map Products

###  5.1 Equatorial Regions

The three equatorial regions are mapped to show the typical mountainous
surface of the lunar far-side. Figure 5 shows the slope maps for three 445 km
squared regions displaced by 45° along the lunar equator. The maps are
overlaid by a 4°$x$4° grid ($122x122$ km), which shows that within these
grids, a traversable surface in the length of hundreds of kilometres is not
obtainable. The anti-Earth region (Figure 5(b)) shows no traversable path with
inclines < 15°, and at inclines < 25°, two perpendicular paths on the order of
hundreds of kilometres cannot be achieved. The East and West regions are
similar to the central region, but they both show an area of large impact
craters which is a smoother surface (dark purple) in Figures 5(a) and 5(c).
These areas could be accessible to a rover on inclines < 25° and on the scales
of an interferometer. However, the surface is disrupted by many smaller
craters with high slopes (< 25°), so deploying a full-sized 200 km array
becomes challenging. Instead, sites for baselines a few kilometres in length
are quite easy to find.

(a)

(b)

(c)

Figure 5: Slope maps of three 445 km squared regions centred on the lunar far-
side equator (Latitude = 0°). The slope is in the degree range of 0-40° over a
200 m scale, represented by a purple colour scale. Dark purple is a smooth
surface, and yellow is an inclined surface. (a) The West equatorial region
centred at coordinates (-135°,0°). (b) The central equatorial region centred
at coordinates (180°,0°). (c) The East equatorial region centred at
coordinates (+135°,0°).

The highly variable slope of the far-side demonstrates a need for the
identification of large smooth marina and craters.

###  5.2 Digital Elevation Maps

The results for the Mare Moscoviense are described in detail for each map type
below. We have chosen to present the results of Mare Moscoviense because the
site is a sizable candidate location for an interferometer for its lack of
obscuring terrain. The corresponding maps for all eight sites are listed in
Table 1, and presented the Appendix A.

Mare Moscoviense has a mare floor 3 to 4 km below the lunar equatorial radius
(1737.4 km, (Williams (2021))), spanning the longest length of 280 km. The
mare edge becomes extremely mountainous, reaching up to 4 km above the lunar
equatorial radius and forming a steep border to the mare floor.

Figure 6 (left) shows a non-interpolated DEM of Mare Moscoviense. The maximum
separation between tracks is 1.8 km. Figure 6 (middle) shows the interpolated
DEM of Mare Moscoviense produced in GMT. The right of Figure 6 (generated
using MATLAB) shows the interpolated DEM in an exaggerated perspective view
showing highlights and shading for a given illumination angle (-45, 30)
applied to all of the MATLAB figures. The vertical range is 11 km.

(a)

(b)

(c)

Figure 6: DEMs of the lunar site Mare Moscoviense, a $487x445$ km region.
Elevation in km is represented by a colour scale with 0 km at the lunar
equatorial radius. (a) Single LOLA tracks colour-coded by height. White
stripes on the image display areas lacking LOLA coverage. (b) 200 m scale
interpolated DEM of the same region. (c) The perspective view of a smaller
region (300 km on a side), with a factor of six exaggerated vertical axes, is
shown using the interpolated 200 m scale DEM.

###  5.3 Slope Maps

Three visualisations of the slope maps were created for each site for the
recognition of inaccessible terrain. All of the interesting information for
rover deployment of the radio array is at low slopes, up to $\sim 25$°, with
$<15$° being especially important.

Figure 7 shows three slope maps of the site Mare Moscoviense with decreasing
slope range scales. The mare floor does not have slopes greater than 20°. The
middle map highlights the accessibility of the site, with all areas in black
being less than 20° but identifies the smaller crater within the mare. The
right map colours slope greater than 30°, inaccessible to all wheeled
vehicles. Mare Moscoviense does not present challenges by a rather generous
30° criterion at the 200 m scale for wheeled vehicles. Even for a threshold of
15°, Mare Moscoviense is traversable except for the lower West region of the
crater.

(a)

(b)

(c)

Figure 7: Slope maps of the lunar site Mare Moscoviense, a $487x445$ km
region. A colour scale represents the slope in degrees. (a) Slope over the
range 0 - 40°. (b) Slope over the range of 20 - 40°. The slope below 20° is
black. (c) Slope over the range of 30 - 40°. The slope below 30° is black.

###  5.4 Roughness Maps

Maps on different scales were created for different topographic goals. 500 m
scales measure hills and mountains and indicate possible flat terrain areas.
200 m and 100 m scales locate smaller site features, such as small impact
craters but are subject to errors due to poor interpolation between tracks.

Mare Moscoviense demonstrates the value of RMS roughness maps on these
different scales. Figure 8 presents roughness maps of Mare Moscoviense on 500
m, 200 m and 100 m scales. A significant increase in low roughness (coloured
green and blue in the maps) shows that smoother areas appear at smaller
scales, as do erroneous tracks. The 500 m scale shows that Mare Moscoviense is
a site to be studied in more detail because of a lower roughness area, though
these still involved 50 m -scale roughness that would be impassable to a rover
unless smaller scales show paths through. This terrain spans $\approx 280$ km,
but smaller rough terrain within the area is present.

The 200 m scale shows an increase in less rough terrain (less than 20 m
elevation changes) with isolated rough crater features.

The 100 m scale shows the prominent rough features at the mare walls. However,
at this scale, track features due to interpolation are visible. The areas with
low roughness (< 20 m roughness in 200 m scale are blue in Figure 8) are more
patchy than the slope < 20° areas. The low roughness area in the lower left
region of the crater is divided by a linear feature of high roughness ($\sim$
20 - 60 m). Whether this feature is traversable will require higher-resolution
mapping.

The Gini coefficient can quantify the concentration of roughness levels. The
Gini coefficient is dependent on the mean of the absolute difference between
pairs of individual measures. The Gini coefficient ranges between zero and one
for the clumping of roughness measurements. If the absolute difference between
neighbouring measures of roughness is small (large), then the terrain is
described as having constant (inconstant) roughness, and the Gini coefficient
will be close to one (zero).

Topographic features such as these imply that rovers deploying radio antennae
will have to take more circuitous paths to reach the full extent of the array.
As a result, the rovers will have a larger payload and deployment time will
increase.

(a)

(b)

(c)

Figure 8: RMS Roughness maps, ranging from 0 - 100 m, of the lunar site Mare
Moscoviense on three scales, a $487x445$ km region, 500 m, 200 m, and 100 m
scaled interpolated DEMs. Roughness in m is represented by a colour bar. (a)
500 m scale roughness map. (b) 200 m scale roughness map. (c) 100 m scale
roughness map.

##  6 Site Comparison

###  6.1 Threshold Slope Maps

Maps are tailored to characterise the suitability of the sites for a radio
array produced from the maps described in Section 5. A threshold was applied
to the slope maps to highlight inaccessible and highly accessible areas
clarifying the differences between the sites for deploying an array.

The slope from roughness measurements is calculated by projecting the
roughness value across the scale. A series of slope thresholds are applied to
data sets which mask data exceeding the limit. The result is a slope map
presenting the terrain likely traversable by a rover and indicating the
accessibility of a site.

The slope threshold boundaries were chosen by the maximum slope capability of
successful rover missions listed in Table 2. The maximum incline wheeled
vehicles are designed for is 30° therefore, this is the maximum slope value in
the scale. A conservative 25° is determined as the inaccessible slope limit
because it allows several sites to qualify. A threshold of 15° is a safe
choice, as all rovers can handle inclines up to this value. VIPER is planned
to land on the Moon in late 2024 and has demonstrated complications traversing
slopes > 15° (Shirley et al. (2022)). 15° is the lower limit to the threshold
slope range.

Table 2: Mars and Moon missions which included the deployment of rovers. The
mission, rover name, location and landing date are listed. The rover
properties: wheel diameter in cm; the maximum speed of the rover on flat
terrain in $cms^{-1}$; the maximum allowed incline in degrees are listed.

Mission | Rover | Location | Landing Date | Wheel Diameter (cm) | Flat Speed ($cms^{-1}$) | Slope (degrees)
---|---|---|---|---|---|---
Luna 17 | Lunokhod 1333 | Moon | 17/11/1970 | 26 | 3 |
American Apollo Program | Lunar Roving Vehicle444 | Moon | 1971 | 41 | 500 | 15
Luna 21 | Lunokhod 2555 | Moon | 16/01/1973 | 26 | 28 |
Chang’e 3 | Yutu666 | Moon | 12/14/2013 |  |  |
Chang’e 4 | Yutu-2777 | Moon | 01/03/2019 | 15 | 6 | 20
Mars Pathfinder | Sojourner888 | Mars | 04/07/1997 | 6 | 1 |
Mars Surface Exploration | Spirit999 | Mars | 04/01/2004 | 13 | 5 | 30
Mars Surface Exploration | Opportunity101010 | Mars | 24/01/2004 | 13 | 5 | 30
Mars Science Laboratory | Curiosity111111 | Mars | 05/08/2012 28 | 4 | 30 |
Mars 2020 | Perseverance121212 | Mars | 18/02/2021 | 26 | 2 | 30
CLPS | VIPER131313 | Moon | 2024 | 50 | 20 | 15

  * •

3 http://selena.sai.msu.ru/Home/Spacecrafts/Lunokhod1/lunokhod1e.htm. 4
https://www.hq.nasa.gov/alsj/lrvhand.html. 5
http://www.orbitalfocus.uk/Diaries/Luna/Luna21.php. 6/7
https://spaceflight101.com/change/change-3/. 8
https://www.jpl.nasa.gov/missions/mars-pathfinder-sojourner-rover. 9
https://mars.nasa.gov/mer/. 10 https://mars.nasa.gov/mer/mission/rover-
status/$\\#opportunity$. 11 https://mars.nasa.gov/msl/home/. 12
https://mars.nasa.gov/mars2020/. 13 https://www.nasa.gov/viper.

Figure 9 shows slope maps produced in Section 4.2.2 for all eight lunar sites.
A colour scale represents the slope in increments of 5° rather than masked.
Orange terrain represents inaccessibility (slope > 25°). Accessible and easily
traversable terrain in blue. All the sites have large areas with slopes < 15°.
Several of the sites have areas of rough terrain that will have to be avoided.
Mare Ingenii is a striking example where the full extent of the site can only
be accessed by passing through two narrow passes between ridges. Korolev
appears less traversable, given the large areas of rough terrain. The varying
difficulty of the terrain at each site is a factor in ranking the suitability
of the sites.

Figure 9: Site slope maps on the 200 m scale created from interpolated DEM.
Slope in degrees is represented by a colour scale in increments of 5°. Blue
areas are accessible. Green and yellow areas are challenging areas and orange
is inaccessible to rovers. Special features are labelled.

###  6.2 Figure of Merit (FoM)

An objective comparison of these lunar sites can be helped by creating a
suitable FoM that combines four factors: (i) Slope constrained size; (ii)
Point Spread Function (PSF); (iii) roughness; (iv) terrain. These are
discussed in turn below.

####  6.2.1 Factors

Four factors were considered to assess the suitability of the sites of
interest.

  1. 1.

Slope-constrained site size. The size below a given slope threshold is the
length of accessible terrain to a wheeled vehicle. A wheeled vehicle requires
connected accessible terrain with routes around special features and the
general roughness of a surface to deploy an array of dipoles. A larger site
provides more flexibility in array design and a smaller PSF.

  2. 2.

PSF (angular resolution). Baseline lengths are the longest linear distances
achievable on a site, representing the baselines of possible arrays. The
length does not represent the true path length which considers avoiding
objects and excessive terrain sloping, resulting in a greater distance
traversed by the rover. At increasing slope thresholds, a site becomes more
accessible and the achievable baseline increases. For the simplest array
design, orthogonal baselines are required, but this is not attainable in some
sites; bisecting baselines within 20° were used for this FoM.

  3. 3.

Roughness. If a surface is rough with rapidly varying elevation, then the Gini
coefficient will be close to zero, and if the terrain is smooth with little
elevation variation, the Gini measure is close to one. A highly ranked site
will have a high Gini coefficient, indicating a more concentrated smooth
surface.

  4. 4.

Terrain obstacles. Special features are identified by using interactive three-
dimensional slope maps. Significant mountain regions, large obscuring craters,
and the passageways around such features are special features. The questions
asked are: Do these obstacles lie in the path of a dipole? Is extra distance
traversed and extra cable (and greater mass) required to avoid these
obstacles?

####  6.2.2 Calculating Figures of Merit

The FoM factors are listed in Table 3. For a site at a given slope threshold,
a point for each factor, between one (worst) and five (best), is awarded
contingent on the value of each factor, as in Table 3. Low slope and roughness
on the meter scale are most important for a cosmology radio interferometer
with a high concentration of dipoles in the central region, meaning the
surface must be flat. Hence, the points for baseline (slope dependent) and
roughness are doubled. A high overall score indicates a good site; the highest
possible score is 30. Site size, baseline lengths and length ratios depend on
the slope threshold, i.e. the awarded merit typically increases for increasing
threshold slope. Gini is independent of the slope threshold, i.e. the awarded
merit is the same value for all slope threshold maps. An example: Apollo, with
an applied 20° slope threshold, has the longest length baseline $>200$ km
(compare Figure 10) and a bisecting baseline between the lengths 100 to 150
km, so the points awarded respectively are ten (factor awarded double points)
and three from Table 3.

Table 3: Factors contributing to the figure of merit and their awarded points for each slope threshold. Factors with double points are annotated with $\ast$. | i. | ii. | ii. | iii.
---|---|---|---|---
Points Awarded | Site Size (km) | Baseline (km)$\ast$ | Baseline Ratio | Roughness (Gini)$\ast$
5 | >300 | >200 | 1:1 | > 0.4
4 | 250 - 300 | 150 - 200 | 1:2 | 0.3 - 0.4
3 | 200 - 250 | 100 - 150 | 1:3 | 0.2 - 0.3
2 | 150 - 200 | 50 - 100 | 1:4 | 0.1 - 0.2
1 | <150 | 0 - 50 | 1:5 | 0.0 - 0.1

The figure of merit is calculated for a given site by summing the points
awarded for each factor for a given slope threshold. Two baseline points are
awarded for the two baselines which bisect. The maximum number of points
awarded by the figure of merit is 30. Below, each factor is analysed and
followed by the figure of merit result in Table 5.

Figure 10 shows how the maximum baseline (dotted line) and the maximum
bisecting baseline change (dashed line) for increasing slope thresholds from
15° to 25°. For some sites, the longest achievable baselines depend strongly
on threshold slope, e.g., Apollo’s second baseline. Sites are shown in
different colours. The 22° slope threshold for a bisecting baseline has only
one site, Apollo, with a length greater than 200 km, but by increasing the
slope threshold to 24°, four sites have lengths with the required 200 km
length. The area between 22° and 24° is shaded in light orange to highlight
the changes in baselines, and baselines $>200$ km are shaded in green.

Figure 10: The maximum baseline (dotted line, A) and the maximum bisecting
baseline (solid line, B) change for increasing slope thresholds, from 15° to
25°. Different coloured lines represent lunar sites. Lengths greater than 200
km are shaded in green. Slope threshold 22° - 24° is lightly shaded in orange,
and slope thresholds greater than 24° are heavily shaded in orange.

The simplest radio interferometer design would involve two bisecting
orthogonal dipoles, each 200 km in diameter. For such a design, it is
important for the dipole lengths to be close to a 1:1 ratio. The site Korolev
does not achieve a 1:1 ratio, whereas all other sites are close to a 1:1 ratio
with the most tolerant slope limit, 25°.

Roughness and slope maps of Korolev suggest that the site is one of the
roughest and least desirable sites for an interferometer by eye. The Gini
coefficient validates this assumption by being the minimum, 0.23. Similarly,
sites showing the most promise from observing their topographic maps are
Apollo, Mare Ingenii, and Daedalus. These sites have the highest Gini
coefficients $>0.3$, implying more consistent terrains.

With the FoM characteristics considered, the weighting of each characteristic
is summed, and the total merit for each site is calculated. The highest-ranked
site is determined by having the greatest total points. Tables 4 and 5 show
the figure of merit results by presenting the measured factor value and the
sum of points awarded at each slope threshold, respectively. The most feasible
locations for a 200 km array are Apollo, Mare Ingenii and Mare Moscoviense.
The threshold slopes 15°, 20°, 22° and 24° were chosen. The lower boundary,
15°, is not sufficient to traverse a 200 km baseline, except in Apollo. The
following thresholds show increases in the number of sites achieving a 200 km
baseline. Figure 11 shows the 15° threshold map of the top-ranked sites
Apollo, Mare Ingenii and Mare Moscoviense. A restrictive limit allows for one
baseline to be traversed. However, a near perpendicular bisecting baseline
cannot be traversed.

Sites | Site Size (km) | Baseline (km) | Baseline Ratio | Roughness (Gini)
---|---|---|---|---
| 15° | 20° | 22° | 24° | 15° | 20° | 22° | 24° | 15° | 20° | 22° | 24° |
Apollo | 228 | 270 | 320 | 385 | 67 | 107 | 301 | 352 | 3.36 | 2.52 | 0.94 | 0.91 | 0.40
Mare Ingenii | 82 | 143 | 202 | 277 | 34 | 129 | 129 | 218 | 2.41 | 1.11 | 1.56 | 1.27 | 0.37
Mare Moscoviense | 129 | 182 | 203 | 249 | 68 | 109 | 184 | 230 | 1.90 | 1.67 | 1.10 | 0.93 | 0.34
Hertzsprung | 55 | 185 | 248 | 278 | 16 | 135 | 171 | 214 | 3.35 | 1.36 | 1.45 | 1.30 | 0.32
Tsiolkovsky | 61 | 135 | 135 | 136 | 30 | 88 | 121 | 125 | 2.00 | 0.73 | 1.00 | 0.96 | 0.36
Daedalus | 36 | 121 | 135 | 135 | 23 | 42 | 55 | 106 | 1.60 | 0.33 | 0.40 | 0.78 | 0.40
Mendeleev | 30 | 60 | 121 | 182 | 0 | 38 | 76 | 171 | - | 1.60 | 1.60 | 1.06 | 0.27
Korolev | 30 | 81 | 162 | 297 | 0 | 27 | 68 | 129 | - | 2.95 | 2.40 | 2.31 | 0.23
Table 4: Figure of merit for all eight sites at a 15°, 20°, 22° and 24° slope thresholds. The factor values in this table determine the merit points awarded (points awarded between one to five shown in Table 3). Table 5 gives the sum of points awarded for each of the eight sites at each slope threshold. Table 5: Figure of merit results. The figure of merit is calculated for each site at the slope thresholds 15°, 20°, 22° and 24°. The maximum figure of merit value is 30. The table is ordered by the highest-ranked site at 20° first to the lowest-ranked site last.  Site | Threshold Slope
---|---
| 15° | 20° | 22° | 24°
Apollo | 25 | 26 | 30 | 30
Mare Ingenii | 19 | 24 | 27 | 30
Mare Moscoviense | 20 | 23 | 27 | 28
Hertzsprung | 14 | 22 | 25 | 26
Tsiolkovsky | 19 | 22 | 24 | 24
Daedalus | 17 | 19 | 21 | 24
Mendeleev | 9 | 11 | 14 | 19
Korolev | 6 | 10 | 15 | 19
Figure 11: Left to the right, slope maps of the three highest ranked lunar
sites Apollo ($338x306$ km), Mare Ingenii ($306x213km$) and Mare Moscoviense
($306x306$ km). A colour scale with a 15° slope threshold represents the slope
in degrees. White areas cannot be traversed and purple are the smoothest
surfaces.

##  7 Discussion

We conducted a preliminary lunar far-side site survey to determine the optimum
location for a 200 km radio-class, low-frequency (10s MHz) interferometer. The
lunar far-side has few smooth craters and mare regions where a 200 km radio
interferometer could be situated. LOLA Digital terrain maps and slope and
roughness maps derived from them were used to examine eight potential sites.
The surface of the Moon is a strenuous environment for wheeled vehicles. The
engineering limits of the existing and proposed vehicles influence the
threshold slope of terrain maps. Since we measured an average slope on a 200 m
scale, slope thresholds of 15°, 22° and 24° were used. 22° and 24° show an
increase in sites with 200 km baselines. It should be noted that a 15° slope
threshold is not sufficient to achieve the goal of a 200 km baseline
interferometer.

We examined four factors: slope-constrained size, PSF, roughness and terrain
obstacles. From the DTMs, slope maps and roughness maps, measurements of these
factors created the FoM to describe and rank the sites. Terrain features
affect the ease of deployment of the antennas of the array. Figure 9 shows the
significant features of each site and their accessibility.

Apollo is ranked top in the FoM because it is the largest site, having the
largest baseline through terrain with minimal slope. However, the largest
baseline is challenging because of the mountainous terrain bisecting the
crater. The roughness Gini coefficient of Apollo is high because the crater
floor is a smooth U-shaped region. Notably, within the U-shape of accessible
terrain, a highly rough region in the lower right region of the crater poses a
challenge to a rover (see also Figure 11).

Mare Moscoviense has few obscuring objects within the mare floor but does show
rough regions, though these appear avoidable.

Mare Ingenii has an entirely flat, $<5$°, surface separated by a steep
mountain wall $>25$°. Between crater walls are flat passageways, which create
accessible routes for a rover.

If the interferometer maximum baseline could be reduced whilst achieving the
scientific goal, Tsiolkovsky and Daedalus become suitable locations for arrays
smaller than 150 km. The two sites have the most accessible terrain within
their crater floors and show one avoidable mountainous feature greater than
the threshold slope.

The sites Hertzsprung, Korolev and Mendeleev show very rough and disrupted
terrain with features inclined greater than 25°. Routes through rougher sites
are not identifiable and challenging because the 200 m scale on which the maps
were interpolated implies that non-mapped features would be present.

Limitations to this study arose from both computational resources and data
anomalies. The topographic maps shown in this paper were produced with the
best achievable scale of 200 m per pixel. The LOLA instrument has a nominal
ranging precision of 10 cm and a vertical precision of < 1 m (Smith et al.
(2017)). LOLA can calculate slope on the smallest scale $\sim 25$ m. The
anomaly affects the instrument’s operation by the common case of two spots out
of five emitted being detected. To overcome this challenge, interpolated data
was created to estimate the surface between track coverage and the greater
scale was chosen for computational efficiency. Using large scales means we do
not have a detailed understanding of the lunar surface at these sites, and
unobserved obstacles could make a site inaccessible. Fernandes & Mosegaard
(2022) generated $\sim$ 1 m per pixel scale topographic maps of the site Mare
Ingenii using images from the LRO Camera, which capture shadows cast from
sunlight, and they relate this to the gradient. Future work includes using
high-resolution topographic maps to determine realistic traverses and
resultant cable length requirements. Improvements to the FoM can be made. For
example, different weights can be assigned to each factor depending on how
they affect mission design requirements (e.g., traverse length, available
baseline given realistic slope capabilities).

###  7.1 Site Protection

This study has identified the importance of a few lunar far-side sites with
the likelihood that these locations would be highly competitive in the next
decades. The lack of smooth terrain on the far-side increases the value of
such exceptional sites. In this case, preventing harmful interference at the
sites will form disputes over entitlement to access sites regardless of the
local resources (Elvis et al. (2021)). Currently, profitable lunar sites
correspond to ‘common-pool resources’ (Edwards & Steins (1998)) in which ‘no
single nation has a generally recognized exclusive jurisdiction’ (Wijkman
(1982)). The impact of the SpaceX Starlink Satellites has been observed on the
Zwicky Transient Facility Survey observations (Mróz et al. (2022)). Growing
concerns over the impact on astronomical observations from low Earth orbit
satellite constellations demonstrate the urgency to protect future research
from similar experiences (Lawrence (2021)). We expect to see developments
within the field as a recently established group within the International
Academy of Astronautics (IAA), the Moon Farside Protection Permanent
Committee, aims to call attention to lunar interference corruption141414See
https://iaaspace.org/about/permanent-committees/#1658152007849-7e3d453e-d9ba.

##  8 Conclusions

We have conducted a preliminary study of lunar far-side radio array sites.

  1. 1.

Eight sites of dimension > 100 km were investigated by generating topographic
maps on the scale of 200 m with a ranging precision of 10 cm and a vertical
precision of < 1 m;

  2. 2.

Only the site Apollo is traversable (meaning a linear 200 km baseline can be
accessed by a rover) when a 15° slope threshold is applied to the region;

  3. 3.

Four sites (adding Mare Ingenii, Mare Moscoviense, and Hertzsprung) are
traversable for a 24° slope threshold;

  4. 4.

The sites Tsiolkovsky and Daedalus would be good sites for smaller arrays
$\sim$ 100 km;

  5. 5.

A figure of merit was created using size, slope, roughness, and topographic
features to compare the sites objectively.

The ongoing work to achieve higher-resolution topography maps would provide a
more rigorous site study. The rarity of good sites points to a need for their
protection.

## Acknowledgements

This work was supported by STFC [ST/X508354/1]. This research used
GMT,151515https://www.generic-mapping-tools.org/ a free, open-source software
licensed under the LGPL (Wessel et al., 2019). This research also made use of
MATLAB,161616https://www.mathworks.com/products/matlab.html?s_tid=hp_products_matlab
a programming and numeric computing platform (MATLAB, 2021). This research
also used Astropy,171717http://www.astropy.org a community-developed core
Python package for Astronomy (Astropy Collaboration et al., 2013, 2018). I
would like to thank Diego Altamirano and Jeremy Drake for their support of
this project.

## Data Availability

The data in this research can be found in the PDS Geosciences Node Lunar
Orbital Data Explorer (ODE)181818https://ode.rsl.wustl.edu/moon/index.aspx.

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##  Appendix A Topographic Maps

Here the measured elevation and pulse width and the derived slope and
roughness maps are presented for all eight candidate sites on the lunar far-
side.

Figure 12: Elevation maps of all eight sites from LOLA data. Elevation in km
is shown by a colour scale. The site names are labelled at the top left of
each map. A 4°$x$4° ($122x122$ km) grid is overlaid on each site map, showing
their relative sizes. Figure 13: Slope maps of all eight sites from LOLA data.
Slope in degrees is shown by a colour scale. The site names are labelled at
the top left of each map. A 4°$x$4° ($122x122$ km) grid is overlaid on each
site map, showing their relative sizes. Figure 14: Pulse width roughness maps
of all eight sites measured by LOLA. Roughness in cm is shown by a colour
scale. The site names are labelled at the top left of each map. A 4°$x$4°
($122x122$ km) grid is overlaid on each site map, showing their relative
sizes. The pulse width measurements are not used in this study because of the
lack of LOLA data due to a thermal anomaly. Figure 15: RMS Roughness maps of
all eight sites derived from LOLA data. Roughness in metres is shown by a
colour scale. The site names are labelled at the top left of each map. Blue
areas can be traversed by a rover and brown areas would not be traversable by
a rover. Figure 16: 22° slope threshold maps of all lunar sites of interest
derived from LOLA data. The slope in degrees is shown by a colour scale. The
site names are labelled at the top left of each map. Terrain not traversable
by a rover is shown by white space, where slopes are greater than 22°.
Potential baselines are shown in cyan and their lengths are labelled in km,
note that the rover path may be significantly longer than the baseline
illustrated.