Equatorial locations of water on Mars: Improved resolution maps based on Mars Odyssey Neutron Spectrometer data

Equatorial locations of water on Mars: Improved resolution maps based on Mars Odyssey Neutron Spectrometer data

Accepted Manuscript Equatorial locations of water on Mars: Improved resolution maps based on Mars Odyssey Neutron Spectrometer data Jack T. Wilson, V...
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Equatorial locations of water on Mars: Improved resolution maps based on Mars Odyssey Neutron Spectrometer data Jack T. Wilson, Vincent R. Eke, Richard J. Massey, Richard C. Elphic, William C. Feldman, Sylvestre Maurice, Lu´ıs F.A. Teodoro PII: DOI: Reference:

S0019-1035(16)30602-9 10.1016/j.icarus.2017.07.028 YICAR 12549

To appear in:

Icarus

Received date: Revised date: Accepted date:

21 November 2016 23 June 2017 31 July 2017

Please cite this article as: Jack T. Wilson, Vincent R. Eke, Richard J. Massey, Richard C. Elphic, William C. Feldman, Sylvestre Maurice, Lu´ıs F.A. Teodoro, Equatorial locations of water on Mars: Improved resolution maps based on Mars Odyssey Neutron Spectrometer data, Icarus (2017), doi: 10.1016/j.icarus.2017.07.028

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Highlights

been improved from 520 to 290 km.

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• The resolution of the Mars Odyssey Neutron Spectrometer data has

• In addition to the polar deposits the boundaries of equatorial hydrogen reservoirs are made clear.

• Western lobes of the Medusae Fossae Formation contain up to 40 wt.

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% water equivalent hydrogen.

• The results are consistent with buried water ice existing in small regions

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close to Mars’ equator.

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Equatorial locations of water on Mars: Improved resolution maps based on Mars Odyssey Neutron Spectrometer data Jack T. Wilsona,1 , Vincent R. Ekea , Richard J. Masseya , Richard C. Elphicb , William C. Feldmanc , Sylvestre Mauriced , Lu´ıs F. A. Teodoroe a

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Institute for Computational Cosmology, Department of Physics, Durham University, Science Laboratories, South Road, Durham DH1 3LE, UK b Planetary Systems Branch, NASA Ames Research Center, MS 2453, Moffett Field, CA,94035-1000, USA c Planetary Science Institute, Tucson, AZ 85719, USA d IRAP-OMP, Toulouse, France e BAER, Planetary Systems Branch, Space Sciences and Astrobiology Division, MS 245-3, NASA Ames Research Center, Moffett Field, CA 94035-1000, USA

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Abstract

We present a map of the near subsurface hydrogen distribution on Mars,

trometer.

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based on epithermal neutron data from the Mars Odyssey Neutron SpecThe map’s spatial resolution is approximately improved two-

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fold via a new form of the pixon image reconstruction technique. We discover hydrogen-rich mineralogy far from the poles, including ∼10 wt. % wa-

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ter equivalent hydrogen (WEH) on the flanks of the Tharsis Montes and >40 wt. % WEH at the Medusae Fossae Formation (MFF). The high WEH abundance at the MFF implies the presence of bulk water ice. This sup-

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ports the hypothesis of recent periods of high orbital obliquity during which water ice was stable on the surface. We find the young undivided channel 1

Present address: The Johns Hopkins Applied Physics Laboratory, 11100 Johns Hopkins Road, Laurel, MD 20723, USA

Preprint submitted to Icarus

August 1, 2017

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system material in southern Elysium Planitia to be distinct from its surroundings and exceptionally dry; there is no evidence of hydration at the

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location in Elysium Planitia suggested to contain a buried water ice sea. Finally, we find that the sites of recurring slope lineae (RSL) do not corre-

late with subsurface hydration. This implies that RSL are not fed by large, near-subsurface aquifers, but are instead the result of either small (<120 km

diameter) aquifers, deliquescence of perchlorate and chlorate salts or dry,

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granular flows. Key words:

Mars ; Mars, surface ; Neutron spectroscopy ; Image reconstruction

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1. Introduction

The main goal of the Mars Odyssey Neutron Spectrometer (MONS) is to

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determine the major near-surface reservoirs of hydrogen on Mars (Feldman

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et al., 2004b). Knowing the present distribution of water in the Martian

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near-subsurface is important for several reasons: it allows inferences about

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the past and present climate to be drawn, which, in turn, give information

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about the dynamic history of Mars and the possibility of the past, or present,

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existence of life. Additionally, understanding the small-scale distribution of

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water is important for landing site selection for missions looking for signs

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of life or exploring in-situ resource utilisation (Squyres, 2011). MONS data

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have been used to map the hydrogen content of the Martian near-subsurface

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(Feldman et al., 2002; Tokar et al., 2002; Feldman et al., 2004b) on ∼ 550 km

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scales (Teodoro et al., 2013). Hydrogen rich deposits, with WEH content >

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25 wt. %, were found poleward of ±60◦ , which were interpreted as water-ice 3

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buried under a layer of desiccated soil (Feldman et al., 2002). Additional

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low-latitude hydrogen deposits were observed at Arabia Terra and Elysium

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Planitia (Feldman et al., 2002) with 9.5±1.5 wt. % WEH (Feldman et al.,

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2004a). Water ice should not be stable equatorward of ± 30◦ (Mellon and

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Jakosky, 1993), which has led to the suggestion that these equatorial hydro-

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gen deposits are in the form of hydrated minerals (Feldman et al., 2002).

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The ∼ 550 km spatial resolution of the MONS instrument suppresses

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smaller-scale features in the MONS data, and the inferred hydrogen distri-

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bution. It also results in a reduction in the dynamic range of the data, leading

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to an underestimate in the wt. % WEH content of small hydrogen-rich areas.

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The previously inferred WEH abundances for equatorial features will have

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been underestimated because of this effect. In this paper we will develop

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and apply an image reconstruction technique based on the pixon method,

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which has been used to successfully reconstruct planetary data (Lawrence

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et al., 2007; Elphic et al., 2007; Eke et al., 2009; Wilson et al., 2015), to

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improve the resolution of the global MONS data set in a way that is robust

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to noise. This will be the first application of a Bayesian image reconstruction

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technique to a globally defined remotely-sensed planetary data set.

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We will focus on a few locations that have been proposed to contain

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water in the equatorial regions of Mars deposited in the geologically recent

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past. This water is hypothesised to have been deposited during past periods

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of high orbital obliquity when the water ice currently at the poles becomes

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unstable and is ultimately deposited elsewhere (Forget et al., 2006). Evidence

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for equatorial hydration is both morphological (Head and Weiss, 2014) and

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compositional (Feldman et al., 2004b), and is seen at both the Medusae

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Fossae Formation (MFF) and the Tharsis Montes. The MFF is a discontinuous geological unit of easily erodible material

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that stretches ∼ 1000 km across equatorial latitudes, along the boundary of

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the northern lowlands and southern highlands, located in both Elysium and

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Amazonis Planitiae. The origin of the MFF is uncertain. Proposed expla-

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nations include consolidated pyroclastic deposits (Scott and Tanaka, 1982)

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and aeolian sediments with ice-rich material (Head and Kreslavsky, 2004),

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similar to that found in the polar layered deposits (Schultz and Lutz, 1988),

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laid down during periods of high orbital obliquity. Radar sounding using the

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MARSIS instrument onboard ESA’s Mars Express has been used to measure

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the dielectric constant of the MFF material and found it to be consistent

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with the MFF containing a large component of water ice or anomalously low

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density soil (Watters et al., 2007). It may be possible to distinguish between

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these two mechanisms using neutron derived hydrogen abundances.

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Evidence, in the form of surface morphology and cratering, for late Ama-

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zonian tropical mountain glaciers on the north-western flanks of the Tharsis

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Montes and Olympus Mons has been detailed extensively using observations

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from the Mars Express, Mars Global Surveyor and Mars Odyssey orbiters

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(Head and Marchant, 2003; Shean et al., 2005, 2007). The production of

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such glaciers at equatorial Mars today is impossible given the current cli-

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matic conditions. Thus, the existence of these glaciers is interpreted to be

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the result of the migration of volatiles (chiefly water) from the poles to the

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equator during past periods of high orbital obliquity. Climate models pre-

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dict the accumulation of ice on the north-western slopes of these mountains,

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during hypothesised periods of high obliquity, due to the adiabatic cooling

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of moist polar air (Forget et al., 2006; Madeleine et al., 2009). The extent to which these equatorial deposits remain and the form in

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which they are present are not yet settled questions. Campbell et al. (2013)

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see no evidence for buried ice at Pavonis Mons using the Shallow Radar

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(SHARAD) instrument onboard the Mars Reconnaissance Orbiter. However,

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morphological evidence is presented by Head and Weiss (2014) in the form of

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fresh ring-mold craters at both Pavonis and Arsia Mons, which is suggestive

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of the presence of buried ice today or in the very recent past.

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The utility of this data set in constraining the distribution of water on the

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scale of the tropical mountain glaciers has been limited by its relatively poor

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spatial resolution due to the ∼ 550 km full width at half maximum, FWHM,

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footprint of the MONS. Jansson’s method was used by Elphic et al. (2005) to

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perform image reconstruction on the MONS data at Tharsis. However, it is

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known that this method amplifies noise and may introduce spurious features

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(Prettyman et al., 2009).

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We will use our improved resolution map of the epithermal neutron count

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rate across the entire surface of Mars to look for evidence of buried, non-

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polar hydrogen in the form of both hydrated minerals and water ice at the

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Tharsis Montes and the Medusae Fossae Formation. In addition, we will

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examine several non-polar sites where water is suggested to be present in some

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form: Meridiani Planum, Elysium Planitia and the southern-mid latitudes,

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where RSL are observed. The increase in resolution over the unreconstructed

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MONS data allows better correspondence between structures in the neutron

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data and features in surface imagery, thus enabling a more robust geophysical

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interpretation.

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In the next section we describe the modifications made to the pixon

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method and the new algorithm that has been developed. In section 3 we

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discuss the MONS data used in this study, along with the properties of the

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instrument, before presenting the results of the reconstructions in section 4.

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Small scale features that emerge in the reconstructions are described in sec-

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tions 5 and 6. Finally, we conclude in section 7.

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2. Methods

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In this section we will describe a new version of the pixon image recon-

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struction technique, adapted for use on the sphere. The pixon method is

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a Bayesian image reconstruction technique, in which pixels are grouped to-

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gether to form ‘pixons’ (Pina and Puetter, 1993; Eke, 2001). It is motivated by considering the posterior probability ˆ ˆ ˆ M |D) = P (D|I, M )P (I|M )P (M ) , P (I, P (D)

(1)

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where Iˆ is the reconstructed image and M is the model, which describes the

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relationship between Iˆ and the data, including the point spread function,

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PSF, and the basis in which the image is represented. As the data are already

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taken, P (D) is not affected by anything we can do and is therefore constant.

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Additionally, to avoid bias, P (M ) is assumed to be uniform. This assumption

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ˆ M ), is the likelihood of the data given a leaves two terms: the first, P (D|I,

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particular reconstruction and model, calculated using a misfit statistic. The

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ˆ ), is the image prior. The grouping of pixels into pixons second term, P (I|M

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greatly reduces the number of degrees of freedom in the reconstructed image,

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improving the image prior. The most likely reconstruction is that containing

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fewest pixons that is still able to provide a sufficiently good fit to the data.

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2.1. Locally adaptive pixon reconstruction Previous versions of the pixon method imposed a strict maximum entropy

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constraint on the solution (Pina and Puetter, 1993; Eke, 2001), in order to

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simplify the maximization of the posterior by reducing the parameter space

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that is to be explored. Here we make no such requirement, which will allow

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the creation of previously inaccessible and more likely reconstructions.

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As in the maximum entropy pixon method the reconstructed count-rate

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ˆ is constructed by convolving a ‘pseudoimage’, H, with a Gaussian map, I,

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kernel, K, whose width, δ, may vary across the image, i.e.

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ˆ I(x) = (Kδ(x) ∗ H)(x).

(2)

Initially the smoothing scale, or pixon width, δ(x), is large. The final value

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is determined for each pixel using an iterative procedure, which includes the

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calculation of a newly defined local misfit statistic. In practice a finite set

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of distinct pixon sizes is used as this keeps the time of a single calculation

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of the global misfit statistic, ER , down to O(npixels log npixels ) (i.e. that of a

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fast Fourier transform), where npixels is the number of pixels in the image,

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whereas if the pixon size were allowed to vary continuously then the time

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would be O(n2pixels log npixels ). In order to generate Iˆ we interpolate linearly

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between the images based on the two pixon sizes closest to those required for

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each pixel.

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The following subsection describes the structure of the locally adaptive

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pixon algorithm before the details of the misfit statistic and prior calculation

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are detailed.

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2.1.1. Structure of the locally adaptive pixon algorithm Our goal is to maximize the posterior probability (equation (1)) whilst

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ensuring that the residual field should be indistinguishable from a random

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Gaussian field. We do this using an iterative procedure with two stages in

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each iteration (figure 1). Firstly, for a given pixon size distribution the values

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in the pseudoimage are adjusted to minimize the global misfit statistic (see

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section 2.1.2) using the Polak-Ribi`ere conjugate gradient minimization algo-

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rithm (Press et al., 1992). Secondly, we calculate the value of the posterior

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and, for each pixel in the data-grid, the local misfit statistic (section 2.1.3).

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These local misfit statistics are then ordered to form their cumulative dis-

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tribution function (CDF). The difference, in each pixel, between the actual

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and expected, χ2 -distributed, CDF is then used to modify the pixon size in

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the same pixel. The change in pixon width is directly proportional to this

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difference, with the constant of proportionality determined empirically such

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that structure appears in the image sufficiently slowly to avoid introducing

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spurious features.

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The combination of global misfit statistic minimization and pixon size

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modification is repeated, starting with large pixons and consequently a flat,

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constant reconstructed image everywhere, until the posterior probability

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stops increasing.

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2.1.2. The global misfit statistic, ER

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The misfit statistic is derived from the reduced residuals between the data

and the blurred model, R(x) =

D(x) − (Iˆ ∗ B)(x) , σ(x) 9

(3)

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Input

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Start

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Set initial pixon sizes and pseudoimage

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Calculate local misfit. Adjust pixon distribution based on local misfit.

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Minimize ER with respect to pseudoimage.

Calculate posterior, is it increasing?

no 10 Output

Stop

yes

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where σ(x) is the anticipated statistical noise in pixel x. Rather than using P χ2 = R2 , we adopt ER from Pina and Puetter (1992) as the misfit statistic. This statistic is defined as

ER =

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AR (y)2 ,

y

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(4)

R

where AR is the autocorrelation of the residuals, AR (y) =

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for a 2-dimensional pixel separation, or lag, of y. The benefit of minimiz-

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ing ER , over the more conventional χ2 , is that doing so suppresses spatial

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correlations in the residuals, preventing spurious features being formed by

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the reconstruction process. Pina and Puetter (1992) recommend that the

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autocorrelation terms defining ER should be those corresponding to pixel

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separations smaller than the instrumental PSF. For a well-sampled PSF this

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means many different pixel lags. However, for the reconstructions we have

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attempted, there is negligible difference between those including different

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numbers of pixel lags. We therefore use only terms with adjacent pixels

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(including diagonally adjacent pixels in 2D) to speed up the computation.

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Although each pixel is adjacent to eight others the symmetry implies that

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AR (y) = AR (−y), so only the four distinct terms need be included when

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calculating ER .

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Prior to this work the pixon method has been implemented only on flat

planes. When calculating ER on a spherical surface we decided to map the

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R(x)R(x + y)dx

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sphere to a flat (i.e. Gaussian curvature equal to zero) torus. This is done

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via an equirectangular projection of the sphere to a square Cartesian plane

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(by extending the regular parameterization of the sphere to 0 < θ < 2π,

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0 < φ < 2π) and identifying opposite edges. This mapping leaves the details 11

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of the calculation, including the use of FFTs for fast convolution, unchanged

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from the flat 2d case. However, as the lag terms included in ER cover a certain

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number of pixels this mapping has the side effect of mixing the physical scales

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over which the statistic is defined.

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2.1.3. Local misfit statistic, ER0

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To create a local ER statistic, ER0 , we begin by localizing AR by multiplying with a kernel, K, such that A0R (x, z)

=

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Z

R(y)R(x + y)K(z − y) dy,

(5)

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which we can rewrite in a form resembling a convolution, by defining Cx (y) =

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R(y)R(x + y), as

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A0R (x, z)

=

Cx (y)K(z − y) dy.

(6)

We are free to choose the kernel and for ease of calculation select a gaussian,

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the width of which should be larger than the largest pixon width so that it

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is the pixon size, and not its gradient, that sets the smoothing scale in the

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reconstruction. For use in determining pixon widths we need to know the

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distribution of A0R . Noting that Cx (y) has a normal product distribution with

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width one and applying the central limit theorem allows one to deduce that

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A0R must have a standard normal distribution so ER0 obeys a χ2 distribution.

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When convolving in angular space we should define (using equation (21)).   Cx 0 AR (x, ω(z)) = ∗ K (ω(z)), (7) a

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and ER0 (z)

X A0 2 (y, z) R = , σA2 0 (z) y

(8)

R

where σA2 0 can be deduced by analogy with a random walk i.e. R

σA2 0R (z)

2.1.4. Prior calculation

(9)

Counting arguments show that, in the pixon method, the prior is given

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K 2 (z − y) dy   1 2 ∗ K (ω(z)). = a =

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by N n



 n  X n−1 1 1 ln(2π) − Ni + + ln(N ) − ln(Ni ), 2 2 2 i=1 (10)

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ln(P (I|M )) ≈ N ln



where N is total amount of information in the image,Ni the information

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content of pixon i and n is the number of pixons (see e.g. Eke (2001) for

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a full derivation). In the maximum entropy pixon method the content of

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each pixon was defined to be equal, so its explicit numerical calculation is

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never necessary (i.e. Ni = N/n). However, our new pixons are not only

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inhomogeneous but there needn’t be an integer number, with pixon number

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defined as

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n=

X x

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i.e. the number of independent degrees of freedom in the image. For the

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case of a non-integer number of pixons we choose to weight the contribution

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of the final non-integer pixon, such that the natural logarithm of the prior

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(using Stirling’s approximation) becomes   1 n−1 N + ln(N ) − ln(2π) ln(P (I|M )) ≈ N ln n 2 2     bnc  X 1 1 − Ni + ln(Ni ) − (n − bnc) Nn + ln(Nn ) , 2 2 i=1

(12)

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where Nn is the information in the final fraction of a pixon scaled up to the

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size it would be if it were filling a whole pixon.

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3. Data

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Approximately 3.5 Martian years of the time-series MONS prism-1 (Mau-

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rice et al., 2011) observations from JD 2452324.125388 (just before the start

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of Mars Year 26) until JD 2454922.577436 are used in this study. Prism-1 is

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the nadir facing detector on MONS and is most sensitive to epithermal neu-

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trons. In this paper and the next we restrict our attention to the ‘frost-free’

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data, for which we use the same definition as Maurice et al. (2011) i.e. those

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data taken at a location (in time and space) in which the CO2 area density

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in the NASA Ames GCM is less than 0.2 gcm−1 .

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As Mars Odyssey is in a near-polar orbit we chose to use an equirect-

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angular pixelization, which results in approximately the same number of

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observations in each pixel. The density of the pixelation is chosen so that

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the PSF is well sampled. In this work we use a square, 0.5◦ ×0.5◦ , grid of

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pixels in a equirectangular projection. The statistical errors within a pixel

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are determined, empirically, by calculating the scatter between repeat ob-

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servations in the observed count rate within the pixel. It was necessary to

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estimate the noise amplitude per pixel empirically as the various corrections 14

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applied to the data, for atmospheric thickness, cosmic ray flux, and space-

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craft latitude and altitude, resulted in the data no longer possessing Poisson

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statistics (Maurice et al., 2011).

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3.1. Assumed instrumental properties

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As a starting point for our estimation of the MONS PSF we have taken

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the calculated prism-1 spatial response of the MONS instrument to a circular

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feature with 5 degree diameter from Prettyman et al. (2009). This calcula-

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tion of the response involved the creation of a detailed numerical model of the

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Martian surface and atmosphere and the Mars Odyssey spacecraft. The pro-

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duction of neutrons via cosmic ray interaction with the Martian atmosphere

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and surface and their subsequent transport within the surface, atmosphere

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and exosphere and detection within the spacecraft was performed using the

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Monte Carlo N-Particle eXtended (MCNPX) transport code. To convert this

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‘disk spread function’ to the instrumental PSF, we calculated the result of

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convolving a 5◦ diameter disk with various trial PSFs, assuming that the PSF

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is azimuthally symmetric about the spacecraft’s nadir point. The profile re-

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sulting from these convolutions that most closely matches that obtained from

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the simulations is taken to be the correct PSF. We make the assumption that

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the PSF can be described by a kappa function, defined as  −κ f (s) s2 = 1+ 2 , f (0) σ

(13)

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where κ and σ are parameters to be determined, s is the arc length from

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the sub-orbital point and f (s) is the response of the instrument to a source

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at s. κ functions are often used to model the response of spacecraft-borne

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gamma ray and neutron detectors (Maurice et al., 2004). The results of this 15

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calculation are shown in Fig. 2, from which the best fitting PSF has σ = 440

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km and κ = 2.3, giving a FWHM of 520km.

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4. Results

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The global frost-free MONS data and our reconstruction of the under-

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lying field are shown in the two panels of figure 3. Locally adaptive pixon

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reconstruction of the data leads to an increase in dynamic range of the count

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rate of nearly 50%, from ∼1-11 to ∼0-16 counts per second. In addition

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to the improvement in dynamic range, there is an enhancement in spatial

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resolution, quantified below. All of the conversions from epithermal neutron

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count rate to weight % water equivalent hydrogen (wt. % WEH) in this

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work use the single layer model in Feldman et al. (2004b), i.e. the surface is

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assumed to be made up of a semi-infinite layer with a constant wt. % WEH

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content.

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4.1. Improvement in resolution

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To estimate the resolution of the reconstructed data we consider the two-

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point correlation function of the reconstruction. Comparing the separation

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required for the correlation function to drop to a given level, allows the

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effective resolution of the reconstruction to be calculated. The correlation

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between the count rate at two points (with angular separation θ) on the

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surface of the planet can be calculated using the relation

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C(θ) =

∞ 1 X (2l + 1)Cl Pl cos θ, 4π l=0

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(14)

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300 350 400 450 500 550 600 650 σ / km

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Figure 2: Constraints on the parameters in the PSF. The parameters are those in equation 13. The white diamond shows the best-fit reconstruction with minimum χ2 . Coloured regions enclose probabilities of 68%/95%/99.7%, determined using ∆χ2 = χ2 − χ2min .

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∆χ2

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Figure 3: Top. The global frost-free MONS prism-1 data, used in the reconstruction. Bottom. A global locally-adaptive pixon reconstruction of the MONS prism-1 data. Underlaid on both panels is a MOLA shaded relief map. The colour bar also shows the conversion

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to wt. % water equivalent hydrogen from Feldman et al. (2004b).

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where Pl are the Legendre polynomials and Cl is the power spectrum of the

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count rate, defined by l X 1 |alm |2 , Cl = 2l + 1 m=−l

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(15)

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with alm being the spherical harmonic coefficients defining the count rate

279

field.

This function is plotted in figure 4 for both the data and the recon-

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structed image. A separation equal to the FWHM of the data (520 km

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or log(separation) = 2.72) corresponds to a normalized correlation value of

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log(C(theta)) = -0.06. The reconstructed image reaches this normalized cor-

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relation value at a separation of 290 km (or log(separation) = 2.46). Thus

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the effective FWHM of the map is 290 km rather than 520 km i.e. variations

286

on scales smaller than 290 km are unlikely to be found in our reconstruction.

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This three fold decrease in effective detector footprint leads to the enhanced

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dynamic range shown in the reconstruction in figure 3. The value of C(0)

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i.e. the power on the smallest scales, increases by 40% between the data and

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reconstruction.

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4.2. Estimation of errors in the pixon method

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applied to the MONS data, we created a set of 10 mock data sets. These

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mock data were generated by blurring the reconstructed image (figure 3 lower

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panel) with the PSF and then adding Gaussian noise with σ as determined for

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the real data. Running the pixon reconstruction algorithm on each of these

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mock data sets enables the errors in the reconstructed count rate, both sta19

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log10(C(θ))

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0.10 0.12

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Figure 4: The two point angular correlation function of the MONS data and the reconstruction, normalized to unity at zero separation. The vertical dashed line shows the MONS PSF FWHM scale. The solid vertical line at log10 (s) = 2.46 (or 290 km) shows the

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separation required for the power in the reconstruction to drop by the same proportion that the power in the data drops at a separation equivalent to the FWHM of the PSF.

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tistical and systematic, to be estimated. The upper panel of Figure 5 shows

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the RMS scatter in each pixel over the 10 reconstructions, a measure of the

300

statistical error in the reconstruction due to noise in the data. Comparison

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with figure 3 shows the noise to be greatest where the count rate is large or

302

varies quickly, such as at the south polar cap. The global mean statistical

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error in the reconstructed epithermal count rate is 0.26 counts per second.

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The average difference between a reconstruction based on mock data from

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one based on the real data is shown in figure 5 (lower panel). This provides

306

a measure of the systematic errors in the reconstruction method and has a

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global average of 0.47 counts per second. The regions with highest offset are

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mostly co-located with those with the largest random errors. The feature at

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around 0◦ N, 150◦ E is particularly discrepant as it features regions of very

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low and high count rate in close proximity. The inability to completely re-

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move the smoothing effect of convolving with the PSF causes these regions

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to blend together. The mottling in these images is an artefact related to the

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size of the pixons.

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4.2.1. Limit of detection of increased hydration

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To place a limit on the smallest source region that could be detected using

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the new pixon method we created a series of mock data sets containing circu-

317

lar regions of 100 wt% WEH with various diameters, located in some regions

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considered in this work, i.e. the southern mid-latitudes, Valles Marineris

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and the northern mid-latitudes. We then performed reconstructions of these

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mock data sets and examined the reconstructions to see how large the 100

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wt% source region must be before a statistically significant depression in the

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count rate is seen. The variation of count rate with source size is shown in

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Figure 5: Top. The standard deviation of the reconstructed count rate based on 10 mock

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data sets. Bottom. A plot of the average difference, in each pixel, between the mock reconstructions and the true image.

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figure 6, where the error bars show the scatter in a set of ten mock data sets.

324

It can be seen that sources smaller than approximately 60 km in radius do

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not lead to statistically significant decreases in count rate so are not reliably

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resolved using this method. Therefore, we cannot detect near-subsurface

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water sources smaller than 120 km in diameter.

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4.3. Comparing reconstructions with surface measurements

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Our reconstructions also enable us to examine the region around Gale

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crater, the landing site and science target of the Curiosity rover. The Cu-

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riosity rover has an active neutron sensor onboard, the Dynamical Albedo

332

of Neutrons (DAN) instrument (Mitrofanov et al., 2012). Unlike for pas-

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sive, orbital remote sensing, DAN works by illuminating the surface with a

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pulsed neutron source and then measuring the returned neutron flux to infer

335

composition. This gives us a ground truth against which to compare our

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reconstructions.

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The inferred hydrogen abundance at Gale crater changes little between

338

the raw data and the reconstruction (lower panels in figure 7), being around

339

8.0 ± 0.4 wt% WEH in both cases. However we find a gradient in the higher

340

resolution reconstruction, with hydration increasing towards the south-east.

341

It is worth noting that these results are discrepant with those obtained by

342

DAN, which suggest an average abundance of 2.1 to 2.7 wt% WEH (Mitro-

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fanov et al., 2014) and those from the High Energy Neutron Detector (HEND)

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instrument in the Mars Odyssey Gamma Ray Spectrometer suite with 5 wt%

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WEH (Mitrofanov et al., 2016). It has been suggested that this discrepancy

346

may be the result of the greatly differing sizes of the response functions of

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the ground and space based detectors (3 m for DAN, 300 km for HEND

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Figure 6: The difference in count rate between reconstructions with 100 wt% WEH sources and those without as a function of source radius. 1σ error bars are calculated from the standard deviation of 10 reconstructions with different noise realizations. The colours

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correspond to the regions where the sources were inserted. The horizontal line is at count rate difference equal to zero.

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Figure 7: The MONS data (left) and reconstruction (right) in the area around Gale Crater. Underlaid is a MOLA shaded relief map. The colour bars also show the conversion to wt. % water equivalent hydrogen from Feldman et al. (2004b).

and 520 km for MONS) and the presence of the rover in a locally dry re-

349

gion (Mitrofanov et al., 2014). However, if this were the correct explanation

350

we may expect that improving the resolution would go some way towards

351

alleviating this discrepancy and we find no evidence of this in our recon-

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struction. For this explanation to hold, the variation in abundance must

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be on a scale smaller than the scales accessible to the reconstruction (i.e.

354

290 km). Alternatively, the differing mean depths from which the detected

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neutrons originate may be responsible: DAN is sensitive only to the compo-

356

sition of the top 60 cm of the surface (Mitrofanov et al., 2014), whereas the

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orbital detectors see up to 1 m into the surface, depending on the soil water

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abundance. If the deeper regions are more hydrated than the surface then

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we would expect the orbital detectors to measure larger WEH values than

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DAN.

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Figure 8: Global MONS reconstruction outlining the regions examined in later sections. The Tharsis Montes, examined in section 5.2, are outlined in black, Elysium planitia and

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part of the Medusae Fossae Formation, in red, the remaining section of the Medusae Fossae Formation, in orange, and Meridiani Planum in violet. The locations of both confirmed

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(black) and candidate (grey) RSL are also shown as identified in McEwen et al. (2011) (squares), Stillman et al. (2014) (circles) Stillman et al. (2016) (triangles) and Dundas and

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uity

We showed, in section 4.1, that pixon reconstruction of the MONS data

yields a near two-fold improvement in linear spatial resolution and a 50%

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5. Possible sites of remnant hydration from periods of high obliq-

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McEwen (2015) (star).

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increase in dynamic range. Here, we will use these results to examine a

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series of science targets that have sizes close to or smaller than the MONS

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PSF and are suggested, or believed, to contain contemporary water ice or

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hydrated minerals. The selected sites are outlined in figure 8. 26

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In this section we will focus on a few locations that have been proposed

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to contain water in the equatorial regions of Mars deposited during periods

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of high orbital obliquity and use the results of the previous section to infer

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their present hydration.

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5.1. The Medusae Fossae Formation: ice or dry, porous rock?

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The topography and MONS reconstruction of the MFF are shown in fig-

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ure 9. The western lobes, Aeolis, Zephyria and Lucus Plana, of the MFF

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are greatly enriched in hydrogen, with >10 wt. % WEH. At Aeolis Planum

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the neutron data imply a WEH abundance > 40 wt. %. This is too great

378

an abundance to be explained as hydrous silicates, which have WEH abun-

379

dances of 10-20 wt. % (Feldman et al., 2004b), so must be buried water ice

380

or hydrated salts. WEH abundances above ∼ 26 wt. % cannot be accom-

381

modated as ice within soil pore spaces, so if ice is present at the MFF it

382

must be present as bulk ice (Feldman et al., 2011). However, the eastern

383

lobes of the MFF contain little, if any, increased hydration. Geologically, the

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western lobes are associated with the lower members of the deposit, whereas

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the eastern lobes corresponds to upper and middle members only (Scott and

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Tanaka, 1986; Greeley and Guest, 1987). The region of enhanced hydration

387

at 10◦ S, 165◦ W is not coincident with the MFF, but does lie on the di-

388

chotomy boundary (the line separating the Martian northern lowlands and

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southern uplands).

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At present water ice is unstable at any depth near the Martian equator

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(Schorghofer and Aharonson, 2005; Schorghofer, 2007). However, Martian

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orbital obliquity has varied greatly over the past 20 Ma (Laskar et al., 2004).

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During high obliquity periods the polar water ice caps become unstable and 27

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Figure 9: (Top) Raw MONS data and (Bottom) pixon reconstruction around the MFF. A MOLA shaded relief map is underlayed. The contours outline the young undivided channel material (black) and middle and lower members of the MFF (white solid and

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dashed, respectively) identified in Scott and Tanaka (1986).

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ice may be deposited down to equatorial locations (Forget et al., 2006). Pro-

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tection of this deposited ice by dust, perhaps containing cemented duricrust

396

layers, could lead to its continued presence today (Feldman et al., 2011; Wang

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and Ling, 2011).

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Our result is consistent with the lower member of the MFF containing

399

ice rich material, which lends weight to the theory that, at least part of, the

400

MFF is a polar layered-like deposit. Salt hydrates could also explain the

401

observations as they have up to 50 wt. % WEH, but their stability in the

402

Martian regolith has not been demonstrated (Bish et al., 2003). No detec-

403

tions of hydrated minerals within the MFF are reported in either CRISM or

404

OMEGA data sets (Carter et al., 2013).

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5.2. Tropical mountain glaciers at the Tharsis Montes

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The Tharsis region, shown with a black rectangle in figure 8, is presented

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in more detail in figure 10. In our pixon reconstruction in the right panel

408

of figure 10, the line of the Tharsis Montes is seen to separate regions of

409

higher and lower WEH abundances, where the north-west side of the line

410

is enhanced in hydrogen with abundances >7 wt% WEH on the north-west

411

flanks of the Tharsis Montes. This hydrogen concentration is greater than

412

that in the raw data (figure 10 left panel) and than in Elphic et al. (2005a).

413

The area with the highest WEH concentration is north-west of Olympus

414

Mons (at 18.65◦ N, 134◦ W) and coincides with the Lycus Sulci region where

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the WEH content reaches 11 wt%.

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In addition to these hydrogen enhancements, Forget et al. (2006) suggest

417

that ice should condense west of Elysium Mons (25.02◦ N, 147.21◦ E) during

418

periods of high orbital obliquity. In figure 8 we also find a decreased epither29

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Fig. 2: Left: Gridded MOND prism-1 data at Tharsis. Right: Pixon reconstruction of the MOND prism-1 data at Tharsis, showing the WEH conversion of [15]. Underlayed is a MOLA shaded relief map. The white contours show fan shaped deposits on the slopes of the Tharis Montes (and are based on those in [16]).

Figure 10: Left: Gridded MONS prism-1 data at Tharsis. Right: Pixon reconstruction

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of the datais at the the WEH conversion of Feldman et al. vary onlyMONS on largerprism-1 scales. This doneTharsis, to create showing an mining precise spatial resolution of the reconstrucimage that has a spatially constant information content, tion at this location. (2004b). Underlaid is a MOLA shaded map.WEH Theconcentrations white contours fan ofshaped which has the effect of precisely maximizing the relief enin the show top metre soil suptropy of the reconstructed image. port the past existence of glaciers on the north-western deposits on the slopes of the Tharsis Montes (and are based on those in Fastook et al. We use the prism-1 data (a measure of epithermal slopes of the Tharsis montes, as are predicted to form neutrons) in periods of high orbital obliquity. However determin(2008). from the MOND instrument of the three martian years from 2001, which has been corrected for ing whether this hydrogen is in the form of relict ice or altitude and look direction of the spacecraft and variahydrated minerals necessitates further study. tion in environmental conditions [14]. Conclusions: We have created a high resolution mal count rate,neutron implying enhanced at this location. The neutron conversion from epithermal count rate map ofWEH, the epithermal neutron count rateForget across the to WEH is done using the relation in Feldman et al. entire surface of Mars. Using this map we see that the et al.[15]. (2006) also note that if the sourceWEH for content the atmospheric waterof during (2004) on the western slopes the Tharsis Results: The Tharsis region, shown with a black Montes is enhanced to ~7 wt%, which is higher than periods of high obliquity polar cap alone included rectangle in Fig. 1, is presented in morewere detail not in Fig.the 2. north otherwise suggested. This isbut consistent with thean past In the reconstruction in the right panel of Fig. 2 the presence of glaciers north-west of the Tharsis Montes, additional, cap,during thena period ice would be deposited line of the Tharsishypothetical, Montes is seen to south separate polar regions water emplaced of high orbital obliquity. of higher and lower WEH abundances, where the References: [1] Head J. W. and Marchant D. R. ◦ north-west side of the Hellas line is enhanced hydrogen (2003) Geology, 31, Sean D. E. for et al. in the Eastern basin in(110 E, 40◦ S). However, we641-644. see no[2]evidence with abundances >7 wt% WEH on the north-west (2005) JGR, 110, E5. [3] Sean D. E. et al. (2007) JGR, flanks Tharsis8. Montes. This hydrogen concen112, E03004. [4] Forget F. et al. (2006) Science, 311, this ofinthefigure tration is greater than that in the raw data (Fig. 2 left 368-371. [5] Madeleine J. -B. et al. (2009) Icarus, 203, panel)That and thanwe in [9]. area with highest WEH 390-405. [6] Campbell B. A. so et al. (2013) JGR, seeThe a peak in the the count rate in the reconstruction closely coin-118 concentration is north-west of Olympus Mons and co436-450. [7] Head J. W. and Weiss D. K. (2014) Planincides with the Lycus Sulci region where the WEH et. & Space Sci., 103, 331-338. [8] Elphic, R. C. et al. cident with the position of Olympus Mons in MOLA topography (figure 10) content reaches 11 wt%. LPC XXXV, Abstract #2011. [9] Elphic R. C. et al. LPC Implications of the observed hydration: That we XXXVI, Abstract #1805. [10] Prettyman T. H. et al., strengthens our faith in the other features shown in the reconstruction. The see a peak in the count rate in the reconstruction so JGR, 114, E08005 . [11] Teodoro L. F. A. et al. LPC closely coincident with the position of Olympus Mons XLIV, Abstract #2623. [12] Pina R. K.◦ and Puetta ◦ R. C. of enhanced content Mons (107 W,V.12 N) inregion MOLA topography (Fig. 2)hydrogen strengthens our faith in around (1993) Ascraeus PASP, 104, 1096-1103. [13] Eke R. (2001) the other features shown in the reconstruction. Mon. Not. R. Astron. Soc., 324, 108-118. [14] Maurice, is The larger the hydrogen glacial content deposit identified Parsons Head (2005). region than of enhanced around S. et al.by (2011) JGR, 116, and E11008. [15] Feldman W. C. Ascraeus Mons is larger than the glacial deposit identiet al (2004) JGR, 109, E09006. [16] Fastook J. L. et al. This may. This be an result, although itsParsons significance will J. fied by [17] may interesting be an interesting result, alIcarus,establishing 198, 305-317. [17] R. L. and Head though establishing its significance will require deterW. LPC XXXV, Abstract #1776.

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require determining the precise spatial resolution of the reconstruction at

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this location.

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6. Other locations with evidence of present water

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In this section we will examine in our pixon reconstruction of the MONS

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data several locations that are suggested on the basis of other spectral or

436

imaging data to show evidence of past or present hydration. The MONS

437

reconstruction will be used to reveal whether the top metre of the surface is

438

hydrated at these locations today.

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6.1. Elysium Planitia: ice or lava?

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The Cerberus fossae are a 1600 km long set of parallel fissures at Elysium

441

planitia, in equatorial Mars. The fissures stretch from 154.43◦ E, 16.16◦ N to

442

174.72◦ E, 6.23◦ N. The fault is believed to be related to the Elysium Montes,

443

located to the northwest, and there is morphological evidence that it has

444

been the source of both water and lava floods (Berman and Hartmann, 2002;

445

Burr et al., 2002; Head and Marchant, 2003) within the last 2-10 Myr. It

446

had been assumed that the water had sublimated away, leaving only the

447

fluvial channels of Athabasca and Marte Valles as evidence of the water flows.

448

However, Murray et al. (2005) identified, in Mars Express High Resolution

449

Stereo Camera (HRSC) images, plate-like features that they argued must be

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the remains of a buried frozen sea 800×900 km across and up to 45 m deep,

451

centred at 5◦ N, 150◦ E. Their argument is based on three observations: the

452

geomorphological similarities of the HRSC images with those taken above

453

Antartica showing the break up of pack-ice, discrepancies between the ages

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454

of the plates and inter-plate regions, and the reduction in the volume of

455

craters that have been filled by the flow. The epithermal neutron flux is sensitive only to the top metre of the

457

surface, which, if the frozen sea were present, may consist of sublimation lag

458

or pyroclastic deposits from subsequent eruptions from the Cerberus fossae.

459

Nonetheless, if the outflow occurred within the last 10 Myr we may still

460

expect to see excess hydration due to partial protection of the water by a

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dust cover lag, perhaps containing hydrated mineral layers that provide a

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barrier protecting a deeper, higher-humidity environment (Feldman et al.,

463

2011; Wang and Ling, 2011).

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The locally adaptive pixon reconstruction of the region around the pro-

465

posed, buried sea is shown in figure 9. The location of the water ice sea,

466

5◦ N, 150◦ E, corresponds to one of the locations with the highest epithermal

467

neutron flux on the entire surface of Mars. This suggests that the top few

468

tens of centimetres of soil, in this region, are unusually dry with < 1 wt.

469

% WEH. The dry feature in the reconstruction extends beyond the region

470

identified by Murray et al. (2005) and covers much of the smooth plains that

471

are believed to be young basaltic lavas from Cerberus fossae (outline with a

472

black contour in figure 9). Additionally, Boynton et al. (2007), using Mars

473

Odyssey Gamma Ray Spectrometer data, find this region to be enhanced

474

in Fe. Taken together, these result suggests that the plate-like features at

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Elysium Planitia are unlikely to be a buried water ice sea but are, instead,

476

young, Fe-rich, volatile-poor basalts. Although no enhancement in the near-

477

subsurface hydrogen content is seen in this region the very high epithermal

478

neutron flux suggests that it is compositionally unique. Previous work using

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Figure 11: Locally adaptive pixon reconstruction of the MONS epithermal data around

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Meridiani Planum.

MONS data has considered only the effects of H and CO2 on the epithermal

480

neutron flux. To understand the effect of changing the abundances of other

481

elements, such as iron, would require new neutron transport simulations.

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The results of section 4.2 indicate a systematic error of up to 1.5 counts

483

per second and a random error of 0.6 counts per second in this region. These

484

errors are amongst the highest on the surface, however even in the worst case

485

scenario that the random and systematic errors were in the same direction

486

southern Elysium planitia remains an exceptionally dry region. Thus, our

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conclusions will not be affected, qualitatively, by errors in the reconstruction

488

technique.

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6.2. Meridiani Planum Meridiani Planum is a plain centred at 0.2◦ N, 2.5◦ W and near the di-

491

chotomy boundary. The plains became an area of intense interest after they

492

were identified as being one of the few locations containing, and the only large

493

deposit of, crystalline, grey hematite on Mars (Christensen et al., 2001). The

494

favoured explanation for the origin of this mineral is chemical precipitation

495

from aqueous Fe-rich fluids. This process requires the presence of liquid

496

water, on the surface of Mars, for substantial periods of time, early in its

497

history (Christensen et al., 2001). For this reason the region was selected as

498

the landing site for NASA’s Mars Exploration Rover, Opportunity.

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The grey hematite deposits are centred on 4◦ W, 2◦ S. Comparison with

500

figure 11 shows this area is not enriched in H in the MONS reconstruc-

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tion. However, an area of the Planum ∼600 km to the north, at 5◦ W, 4◦ N,

502

does show hydrogen enrichment with ∼15 wt. % WEH, consistent with the

503

presence of hydrated minerals. This region was shown, using the OMEGA

504

visible-near infrared hyperspectral imager, to contain hydrated minerals in-

505

cluding kieserite, gypsum, and polyhydrated sulfates (Gendrin et al., 2005).

506

Why the hematite deposits appear less hydrogen rich than hydrated mineral

507

deposits to their north is not clear.

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6.3. Recurring slope lineae

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Recurring slope lineae are narrow, dark markings that appear and grow

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on Martian steep slopes during warm seasons and fade in cold seasons. They

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have been found at equatorial and mid-latitudes and were first identified using

512

HiRISE images (McEwen et al., 2011). Contemporary liquid water activity

513

on the surface of Mars has recently been confirmed at sites showing recurring 34

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slope lineae, using infra-red spectral data from the Compact Reconnaissance

515

Imaging Spectrometer for Mars onboard the Mars Reconnaissance Orbiter

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(Ojha et al., 2015).

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The origin and recharge mechanism of these liquid brines detected in

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recurring slope lineae (RSL) has important implications for the current Mar-

519

tian water cycle and budget. However the exact mechanism that recharges

520

the RSL, which may vary across the surface, remains poorly understood.

521

At least three possible scenarios exist: feeding from shallow briny aquifers

522

(McEwen et al., 2011; Chevrier and Rivera-Valentin, 2012); melting of shal-

523

low ice during warm seasons (McEwen et al., 2014), which provides good

524

temporal agreement with RSL formation but may not be able to provide

525

suffient water over long periods (Stillman et al., 2014; Grimm et al., 2014);

526

and absorption of atmospheric water vapour by deliquescent salts. The first

527

two of these hypotheses require some subsurface store of water but the third

528

does not. Additionally, a dry hypothesis involving granular flow is possible

529

(McEwen et al., 2011).

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adaptive pixon reconstruction of the MONS epithermal neutron data.

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6.3.1. Correlation of RSL position with H abundance

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Comparison of the high resolution hydrogen map with the locations of

RSL shows no positive correlation (figure 8). To test this comparison and

535

find whether RSL occur in special regions in the WEH map we will exam-

536

ine the probability density function of the count rate. Figure 12 shows the

537

probability density function of the reconstructed count rate in the southern

538

mid-latitudes (SML), along with the probability distribution of the count

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rate restricted to sites within the SML containing RSL. We compared these

540

two distributions using a two-sample Kolmogorov-Smirnov test (Press et al.,

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1992), which measures the maximum difference between the cumulative dis-

542

tribution functions of two samples. This tests the null hypothesis that the

543

data sets are drawn from the same distribution. Large values of the test

544

statistic imply that the two samples are not drawn from the same under-

545

lying distribution. To carry out this calculation the data were repixelated

546

more coarsely so that adjacent pixels are effectively uncorrelated. We find

547

a Kolmogorov-Smirnov test statistic of 0.2 and corresponding p-value of 0.2

548

(i.e. test statistics at least this extreme are expected 20% of the time). This

549

is not sufficient to imply that the count-rates (and consequently water abun-

550

dances) where RSL are present are different from those locations where RSL

551

are absent. In turn, this result implies that the occurrence of RSL is in-

552

dependent of the WEH of the top few 10s of cm of soil on large scales. It

553

should be noted that this analysis assumes that the distribution of discovered

554

RSL reflects their true distribution across the surface in an unbiased way. If

555

there are strong selection effects that correlate with WEH abundance then

556

the results may change, for example if the low albedo regions in which RSL

557

are found also have anomalous hydrogen abundances.

559

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M

ED

PT

CE

558

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539

That WEH abundance in the soil is not correlated with RSL activity can

be explained either by reference to the resolution of the neutron data, even at 290 km the resolution is too coarse and the regions containing RSL are

561

blurred into those without, or we must conclude that there are no sizeable

562

stores of subsurface water close to the RSL thus their formation and renewal

563

is not driven by subsurface aquifers or water ice. Water buried beneath

AC 560

36

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ACCEPTED MANUSCRIPT

0.8

All pixels RSL only

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0.6 0.5 0.4

M

0.3 0.2 0.1 5

6 7 8 9 10 11 12 13 Epithermal neutron count rate / s−1

CE

PT

0.04

ED

Probability density

0.7

Figure 12: The probability density function of the count rate between 53◦ S and 30◦ S and

AC

just those pixels containing RSL. The translucent regions indicate 1σ errors.

37

ACCEPTED MANUSCRIPT

the level accessible to the MONS data (i.e. deeper than about 1 m) could

565

not act as a source for the RSL because thermal inertia prevents ice buried

566

beneath 20 cm from ever being melted (Stillman et al., 2014). It was shown

567

in section 4.2.1 that water deposits smaller than 60 km in radius do not

568

produces statistically significant decreases in epithermal neutron count rate,

569

so RSL sources smaller than this are not ruled out by this work.

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564

If the RSL water were sourced from a subsurface reservoir too small to

571

be seen in MONS data then we may expect that improving the resolution

572

would go some way towards alleviating this discrepancy. However we find

573

no evidence of a systematic increase in water abundance at the RSL with

574

reconstruction. If the water source for the RSL is not subsurface then it

575

must be either atmospheric or the sublimation of deeply buried ice deposits.

576

But, both of these mechanisms require that the deliquescence of chlorate and

577

perchlorate salts (the presence of which has been confirmed by CRISM and

578

in-situ measurements (Hecht et al., 2009; Glavin et al., 2013; Ojha et al.,

579

2015)) produces sufficient quantities of liquid brines to darken the soil and

580

produce the RSL.

581

7. Conclusions

M

ED

PT

We have investigated several regions of interest across the Martian sur-

face that have previously been hypothesised to contain water or hydrated

AC

583

CE

582

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570

584

minerals. A consequence of blurring by the detector footprint is that ear-

585

lier analyses of the neutron count rates had led to underestimation of the

586

dynamic range of hydrogen abundance in regions with local variation.

587

At low and equatorial latitudes we found evidence, in the reconstruction 38

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of the MONS data, for buried water ice in the Medusae Fossae Formation

589

and on the western slopes of the Tharsis Montes and Elysium Mons. This

590

supports the hypothesis that Mars in the recent geological past rotated on

591

an axis highly inclined to its current one. On the other hand, the Cerberus

592

fossae region containing plate-like features, suggested to be a buried water

593

ice sea, was found to be exceptionally dry. Although this region seems not to

594

contain hydrogen within the top few tens of cm of the surface, its unusually

595

high epithermal neutron flux is consistent with the area being unusually high

596

in Fe and low in H. Further work using Monte Carlo particle transport codes

597

is required to learn more about the composition of this feature.

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588

Recurring slope lineae are transient features recently discovered on Mar-

599

tian steep slopes during warm seasons. Their narrow, dark shapes resemble

600

water courses – and the presence of liquid water has recently been spectro-

601

scopically confirmed near some RSL sites. Yet, if this is what they are, then

602

their source remains unknown. We find that the hydrogen abundance, on

603

scales of ∼290 km, at RSL sites is statistically no different from anywhere

604

else at similar latitudes. This implies that RSL are not supplied by large

605

(> 160 km diameter), near-surface briny aquifers. Preferred hypotheses re-

606

main for RSL to either be fed by water vapour, either from sublimating deeply

607

buried water ice or from the atmosphere, or by small (< 160 km diameter)

608

or deeply (> 1 m) buried aquifers.

AC

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M

598

609

Acknowledgement

610

JTW is supported by the Science and Technology Facilities Council [grant

611

number ST/K501979/1] and Cosmiway [grant number GA 267291]. VRW is 39

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supported by the Science and Technology Facilities Council [grant number

613

ST/L00075X/1]. RJM is supported by a Royal Society University Research

614

Fellowship. This work used the DiRAC Data Centric system at Durham

615

University, operated by the Institute for Computational Cosmology on be-

616

half of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment

617

was funded by BIS National E-infrastructure capital grant ST/K00042X/1,

618

STFC capital grant ST/H008519/1, and STFC DiRAC Operations grant

619

ST/K003267/1 and Durham University. DiRAC is part of the UK national

620

E-Infrastructure.

621

A. Convolution with an azimuthally symmetric bandlimited func-

AN US

622

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612

tion

The convolution between a circularly symmetric (or zonal) function (i.e.

624

one that depends only on arc-length from a particular point, taken to be the

625

north pole), G, with an arbitrary function, f , defined on the sphere can be

626

written:

ED

M

623

627

i.e. an integral over the 2-sphere.

G(ξ) =

G(ξ · ξ 0 )f (ξ 0 )dξ 0

(16)

∞ X l X

glm Ylm (ξ).

(17)

l=0 m=−l

As G is circular, alm = 0 for m 6= 0 thus ∞ X G(ξ) = gl0 Y0l (ξ)

AC 629

S2

The spherical harmonic (SH) representation of G is

CE

628

PT

(G ∗ f )(ξ) =

Z

l=0

G(ξ) =

∞ X l=0

r

2l + 1 0 g Pl (ξ), 4π l

40

(18)

ACCEPTED MANUSCRIPT

630

where Pl is the lth Legendre function. substituting this back into equa-

631

tion (16) gives ∞ Z X

r

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2l + 1 0 gl Pl (ξ · ξ 0 )f (ξ 0 )dξ 0 4π 2 l=0 S r Z ∞ X l X 4π 0 m Ylm (ξ 0 )f (ξ 0 )dξ 0 = gl Yl (ξ) 2l + 1 S2 l=0 m=−l r ∞ X l X 4π 0 m m = g f Y (ξ), 2l + 1 l l l l=0 m=−l

(19)

AN US

(G ∗ f )(ξ) =

632

which can be seen to be a simple scaling of the SH coefficients of the func-

633

tion f . These scaling factors can be calculated empirically. In the case of

634

convolution with a Gaussian function, of width σ, the appropriate SH coeffi-

635

cients can be calculated by solving for the fundamental solution of the heat

M

equation, on the sphere. We find that the appropriate solution is r 2l + 1 −l(l+1)σ2 m gl = e 2 . 4π

(20)

ED

636

2 This algorithm has a time complexity of O(lmax ), where lmax is the maxi-

638

mum order used in our spherical harmonic decompositions, so in practice will

639

be dominated by the time taken to perform the spherical harmonic transforms

640

3 of the function being convolved, which in our implementation has O(lmax )

641

complexity. This method is therefore O(lmax ) faster than the ‘fast’ method

CE

2 described in the previous section and O(lmax ) faster than the brute force

AC

642

PT

637

643

method. As, in this work, lmax will typically be O(102 ), care will be taken to

644

formulate the deconvolution techniques such that all convolutions are with

645

azimuthally symmetric functions.

41

ACCEPTED MANUSCRIPT

646

B. Integration of functions defined in pixel space It should be noted that the integrations in this and the previous section

647

dx = Npix Z ∂x = dΩ ∂Ω Z 1 = dΩ, a (Ω)

(21)

(22)

(23)

AN US

system to the other, note that Z

CR IP T

are done over solid angle rather than pixels. To convert from one coordinate

where a (Ω) is the area of the pixel at position Ω, which is given by

M

     ∆θ ∆θ a (Ω) = a (θ, φ) = ∆φ cos θ − − cos θ + . 2 2 2π , X

(24)

where X is the

648

∆φ is the width of the pixels in longitude, i.e. ∆φ =

649

number of pixels in the longitudinal direction, similarly, ∆θ =

650

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