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%\authorrunninghead{KARUNATILLAKE ET AL.}
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%\titlerunninghead{Calculated Martian in situ compositions}
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514 Space Sciences Building, Cornell University, Ithaca, NY, 14853, USA.
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\title{Representative Martian in situ chemical compositions subject to Mars Odyssey Gamma Ray Spectrometer constraints}
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 \authors{Suniti Karunatillake, \altaffilmark{1}
 John M. Keller, \altaffilmark{2}
 Steven W. Squyres, \altaffilmark{1}
 William Boynton, \altaffilmark{2}
 Johannes Br\"uckner, \altaffilmark{3}
 Daniel M. Janes, \altaffilmark{2}
 Olivier Gasnault, \altaffilmark{4}
 and Horton E. Newsom \altaffilmark{5}
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 \altaffiltext{1}
 {Department of Astronomy, Cornell University, Ithaca, NY, USA.}

 \altaffiltext{2}{Lunar and Planetary Laboratory, University of Arizona, 1629 E University Blvd., Tucson, AZ, USA.}

 \altaffiltext{4}{Centre d'Etude Spatiale des Rayonnements - Centre National de la Recherche Scientifique - Universit\'e Paul Sabatier Toulouse, Toulouse, France.}

\altaffiltext{3}{Abteilung Geochemie, Max-Planck-Institut f\"ur Chemie, Postfach 3060, Mainz, Germany.}

\altaffiltext{5}{Institute of Meteoritics and Department of Earth and Planetary Sciences, University of New Mexico, Albuquerque, New Mexico, USA.}


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The Mars Odyssey Gamma Ray Spectrometer (GRS) is the first instrument suite to return elemental distributions throughout the midlatitudes of Mars. Concentrations of Cl, Fe, H, K, Si, and Th have been determined as mass fractions with reasonable confidence.  Comparing such data with, or normalizing them to in situ compositional data is difficult due to issues such as dramatic differences in spatial resolution; difficulties in convolving densities, abundances, and compositions of different regolith components; and a limited number of elements observed in common. We address these concerns in the context of the GRS, using Si at Pathfinder to normalize remote data. In addition, we determine representative in situ compositions for Spirit (both with and without Columbia Hills rocks), Opportunity, and Viking 1 lander sites using GRS-derived H content to hydrate the soil component. Our estimate of the Si mass fraction at Pathfinder, with \SI{13}{\%} areal fraction of rocks, is \SI{21}{\%}. The composition of major elements, such as Si and Fe, is similar across the four lander sites, while minor elements show significant variability. Areal dominance of soil at all four lander sites causes representative compositions to be driven by the soil component, while proportionally large uncertainties of bulk densities dominate the net uncertainties. GRS compositional determinations compare favorably with the in situ estimates for Cl and K, and for Si by virtue of the normalization. However, the GRS-determined Fe content at each lander site is consistently higher than the in situ value.
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\section{Introduction}
\label{I}

Mars Odyssey is the first Mars mission to provide elemental concentrations (as mass fractions) of global extent. These data have been generated with the Gamma Ray Spectrometer (GRS) instrument suite, which consists of the Gamma Subsystem (GS), Neutron Spectrometer (NS), and the High Energy Neutron Detector (HEND). Our discussion focuses on the GS. It consists of a passively cooled, n-type, reverse biased, high-purity Ge detector mounted on a \SI{6}{m} boom \citep[][]{Boynton2004}. The boom and detector mounts are relatively free of geochemically significant elements in order to minimize interference from spacecraft components. At present, the GS determination methods have provided mass fraction data of global extent for the elements Cl, Fe, H, K, Si, and Th with reasonable uncertainties \citep[][]{Boynton2007}.

In analyzing the GS data, a model composition is used to predict expected $\gamma$ spectral peak intensities for each element subject to atmospheric corrections. The ratio of expected to observed intensities enables scaling of the model composition to represent the actual regolith composition. Intermediate steps and assumptions therein introduce the potential for systematic errors, the presence or absence of which may be verified by comparing remotely determined elemental mass fractions with corresponding in situ values. Instruments similar to the Mars Odyssey GS (i.e., Ge based) are already en route to orbit Mercury \citep[e.g.,][]{McNutt2006} and planned for the Moon \citep[e.g.,][]{Kobayashi2005}. The future may hold similar missions to other solar system bodies and surface missions as well. Consequently, an effective approach to compare and normalize remote data with in situ estimates of a representative regolith composition could be useful even beyond Mars. In essence, such comparisons act as ``reality checks'' for remote observations, making the computation of representative in situ compositions a primary goal of our discussion.

As described below, the effective instrumental spatial resolution of the GS is $\approx\ang{3.7} (\approx\SI{220}{km})$ arc radius.  In situ measurements at lander sites, on the other hand, involve samples of only centimeter and tens of centimeter spatial extent. Furthermore, lateral spacing of samples is only on the order of kilometers even for the Mars Exploration Rover (MER) mission, the most mobile mission to date. Additional concerns include dramatic differences in the sampling depths between remote instruments (the GS, for example) and surface instruments (the Pathfinder Alpha Proton X-ray Spectrometer, for example), a limited set of elements observed in common between them, and difficulties in convolving bulk densities and areal fractions of different regolith components with their compositions.

Given the underlying issues, is it reasonable to compare the GS data with in situ data? In spite of all the caveats, there is a correspondence between these data sets in that the $\gamma$ photons are produced in the upper few tens of centimeters of the Martian surface, which includes the surficial materials investigated by the landers and rovers.  Furthermore, the nature of surficial material at the lander sites can be extrapolated to the GS spatial resolution scales with remote sensing data such as thermal inertia -- sensitive to  thermophysical properties of the upper surface such as thermal conductivity, bulk density, and heat capacity -- mineralogic information \citep[e.g.,][]{Bibring2006}, and future visual observations at fine spatial resolution \citep[cf.,][]{Mustard2005,Bibring2006a,Shkuratov2005}. However, the massive difference in spatial resolution implicitly applies representative in situ compositions over GS footprint spatial scales whenever the GS data are compared or normalized with in situ values. We address this issue by using a weighted mean of rock and soil component compositions at lander sites, considered the ``ground truth'' composition.

GS estimates of mass fractions of most elements are from intensities of $\gamma$ spectral peaks above the continuum from scatter and capture neutron-nuclear reactions. The rates of these two reaction types, particularly capture, are strongly affected by the energy distribution of neutrons in the regolith. Presence of neutron moderators such as H at elevated concentrations can dramatically alter the neutron energy distribution, consequently affecting $\gamma$ spectral estimates of elemental abundances. At present, these effects have not been modeled accurately for the extreme latitudes where H is compositionally layered as buried \ce{H2O} ice. Therefore, the GS global datasets are constrained to the midlatitudes of Mars as delineated with a mask that excludes the polar regions where the H content begins to increase rapidly. However, data are available at the extreme latitudes for the radiogenic elements, K and Th, since their $\gamma$ photon production is independent of neutrons. \citep[][]{Boynton2007}

Elemental mass fractions determined with the GS are spatially binned on a uniform latitude-longitude grid, typically at $\ang{5} \times \ang{5}$ resolution. Since the composition is estimated with the relative intensities of spectral peaks, only mass fractions, not absolute masses, of elements and oxides are estimated directly. This is different from wet chemical analysis of terrestrial rocks, for example, where ideally the total mass of all elements present must equal the known mass of the sample.  In this case the oxidation states of important elements such as Fe may be determined directly. Other types of analyses, such as electron induced X-ray microanalysis (e.g., electron microprobe analysis at the micron scale), and remote and in situ spectral observations do not allow such rigor, and generally require reasonable oxidation states and absence of X-Ray invisible elements/minerals to be assumed for mass fraction normalization, as done by missions involving Alpha Particle X-Ray Spectrometers (APXS) of the Mars Exploration Rovers (MERs) \citep[e.g.,][]{Gellert2006}, Alpha Proton X-Ray Spectrometer (APXS) of the Mars Pathfinder Sojourner rover \citep[][]{Rieder1997}, and X-Ray Fluorescence Spectrometers (XRFS) of the Viking Landers \citep[][]{Clark1977}.

Where systematic differences between in situ and remote compositions are evident, it is possible to normalize the remote data with in situ values. Insufficient information, on the galactic cosmic ray (GCR) flux, neutron energy distribution, and $\gamma$ photon production factors \citep[][]{Evans2006,Masarik1996}, to rigorously model all nonlinear relationships between composition and $\gamma$ photon production necessitates this normalization for the GS. Abundances of radiogenic elements such as K and Th are unaffected by such issues and do not require normalization with in situ measurements.

Normalization with in situ data would be reasonable only for elements that do not show much variability in mass fractions across lander sites. As such, Si, the most abundant element that is in situ and GS detectable, is the best choice for the GS normalization. An additional benefit of normalizing Si is that it has both capture and scatter derived $\gamma$ peaks (refer to the work of \citet{Evans2006} for identification of individual peaks), while Cl, Fe, and H are primarily estimated with capture peaks. As discussed in Section~\ref{KGSDPS}, the relationship between the normalized scatter-derived Si concentration and that derived via capture can indirectly resolve issues with capture-derived elemental concentrations. Given the importance of normalization to an in situ result, selecting a suitable value for the Si normalization is also a key goal of our work.

We calculate representative in situ compositions at the Pathfinder Sojourner (at Ares Vallis), Spirit (at Gusev crater), Opportunity (at Meridiani Planum), and Viking 1 (at Chryse Planitia) lander sites. The Viking 2 lander site is excluded, given its proximity to the extreme latitude regions with elevated H and consequent exclusion from reported GS results.

\section{Key GS Data Processing Steps}
\label{KGSDPS}
There are numerous intermediate steps to generating mass fractions with the GS, which are discussed in detail by \citet[][]{Boynton2007}, \citet[][]{Evans2006}, and \citet[][]{Kim2006a}. We focus only on those immediately relevant for the comparison between remote and in situ compositions. Key among them, in the general order in which they are applied, are: model composition grid, scatter corrections, spatial filter, normalization of Si mass fractions $(w(\ce{Si}))$ to an in situ value, and capture corrections.

\subsection{Model Composition Grid and Instrumental Spatial Resolution}
The model composition grid is the starting point for GS determination of the Martian regolith composition. The grid consists of $\ang{0.5}\times\ang{0.5}$ cells, with each assumed to be compositionally homogeneous at the few tens of centimeter sampling depth scale of the GS (the model composition is described by \citet[][]{Boynton2007}). The $\gamma$ photon count rates in orbit due to each cell and a given nuclear reaction may be predicted with neutron production simulations, stochastic parameters, regolith attenuation coefficients, and atmospheric attenuation coefficients \citep[][]{Boynton2007, Evans2006, Kim2006a}. Even though the Martian atmosphere acts as a weak collimator, since the GS is an uncollimated instrument, it receives $\gamma$ photons from limb to limb at the nominal \SI{400}{km} altitude mapping orbit \citep[e.g.,][]{Boynton2004}. Therefore, the set of cells within which \SI{>99}{\%} of the signal originates is used for predictive modeling, corresponding to a nadir-centered signal contribution region (SCR) of \ang{\approx 17} (i.e., \SI{1000}{km}) arc radius \citep[][]{Kim2006a}.

For a given nuclear reaction of each element, the ratio of predicted $\gamma$ photon counts to the observed is computed within the SCR. This ratio scales the model composition, yielding an estimate of the actual composition. The estimated composition is assigned to the cell at nadir, and the process repeated for each cell of the $\ang{0.5}\times\ang{0.5}$ grid. Such a fine grid is appropriate for data processing and display, but does not represent the actual footprint of the GS, which is defined by the set of cells that contribute more than \SI{50}{\%} of the signal. While dependent on photon energies, the typical footprint is \SI{\approx440}{km}, corresponding to \ang{\approx7.4} arc diameter \citep[e.g.,][]{Boynton2004, Boynton2007}.

When photons of differing energies are emitted by a single element, a weighted mean of corresponding mass fractions is generally used to represent the true composition of the element \citep[][]{Boynton2007}. The signal-to-noise ratio is maximized by the use of cumulative spectra over durations of one to two Martian years called epochs (we use the epoch from 08 Jun 2002 to 02 Apr 2005). The large number of cells within each SCR and the use of epochal $\gamma$ spectra yield statistically rigorous uncertainties that are primarily functions of counting statistics and smaller than model composition uncertainties \citep[][]{Boynton2007,Evans2006}.

The regolith model may introduce systematic errors for all elements, while the model composition may do so for elements determined with capture or scatter processes. The GS regolith model for the midlatitudes assumes compositional homogeneity at GS sampling depths. It also assumes lateral compositional homogeneity within each cell. Significant deviations from lateral homogeneity have the potential of seriously weakening the accuracy of GS compositional estimates, as would the presence of compositional layering \citep[e.g.,][]{Squyres1992}. We consider such deviations and explain what we classify as rocks and other material in Section~\ref{ST}.

\subsection{Scatter Corrections, Spatial Filter, and Effective Spatial Resolution}
Scatter corrections constitute another important step in the reduction of GS data. Si mass fractions derived from the \SI{1779}{k~eV} $\gamma$ flux are subject to the scatter corrections, increasing the initial Si estimates by $\approx \times 3$ on average. These correction factors, which account for variations in the fast neutron flux driven mainly by Fe and H content in the regolith, precede the spatial filter application, Si normalization to an in situ value, and capture correction determination \citep[][]{Boynton2007}.

The third processing step, spatial filter application, is important in the context of instrument detection efficiency and resolution. The choice of a gamma ray spectrometer type is a trade off between the high efficiency of scintillators and the high energy resolution of solid state detectors \citep[][]{Metzger1990,Pirard2005}. The latter option was appropriate for the Mars Odyssey GS, where the ability to resolve and identify many of the lines that characterize the Martian spectrum optimized the science return \citep[][]{Boynton2004,Boynton2007}. Nevertheless, GS data have low signal-to-noise ratios at fine spatial scale. This concern is addressed by first smoothing with a constant arc radius mean filter. The mean filter is applied simultaneously, not iteratively, on a $\ang{0.5}\times\ang{0.5}$ grid with the arithmetic mean of values within the filter window assigned to the central cell. \citet[][]{Boynton2007} discuss and illustrate the scatter of values before and after the smoothing.

To first order, the filter radius is greater for elements with smaller signal-to-noise ratios. For nonradiogenic elements in particular, the spatial filter is applied at several processing steps \citep[][]{Boynton2007}. The filter arc radii for different elements at the final processing steps are: \ang{5} for K; \ang{10} for Cl, Fe, H, and Th; and \ang{15} for Si. An unfortunate outcome of filtering is the increase in spatial uncertainty in the form of spatial autocorrelation. As the mutual dependence of spatially adjacent data, spatial autocorrelation leads to fewer degrees of freedom than the number of data \citep[e.g.,][pp. 273--324]{Haining2003}. Subsequent to the filter application, the data are binned on fairly coarse latitude-longitude grids, typically at $\ang{5}\times\ang{5}$. Rebinning further increases the signal-to-noise ratio and provides the necessary oversampling to account for the difference in shape between the response function of the GS, which is circular, and of cells, which are rectangular. However, since the filter window is much larger than an individual $\ang{5}\times\ang{5}$ spatial bin, spatial autocorrelation is aggravated in the GS data. Nevertheless, the filter enables a significant increase in numerical precision as evident in relative root-mean-square uncertainties of \SI{10}{\%} for Cl, \SI{8}{\%} for Fe, \SI{11}{\%} for H, \SI{7}{\%} for K, \SI{2}{\%} for Si, and \SI{10}{\%} for Th.

A combined effect of the spacecraft footprint, the SCR, and the spatial filter is a dramatic difference in spatial resolution between the GS and in situ instruments. This difference necessitates careful selection of compositional attributes from in situ missions for comparison with and normalization of the GS data. It is particularly important as the GS is insensitive to the substantial variability at small lateral spatial scales, in composition, density, and texture as seen by Viking, Pathfinder, and MER missions.

\subsection{Si Normalization, Capture corrections, and Feedback Effects}
As described earlier, we select only one element, Si, for the normalization of GS data. The normalization is achieved in several steps. First, the scatter-derived \SI{1779}{k~eV} $\gamma$ photons from Si are used to estimate $w(\ce{Si})$ subject to the scatter corrections. These mass fractions are then smoothed with the mean filter. Next, the filtered values are scaled by a constant so that the GS-derived $w(\ce{Si})$ at the coordinates of a chosen lander site equals the representative in situ $w(\ce{Si})$. For reasons discussed in Section~\ref{PGSN}, we chose the Pathfinder lander site for the Si normalization.

Capture corrections are subsequently determined as the ratio of the capture-derived $w(\ce{Si})$ to the \SI{1779}{k~eV} scatter-derived (also scatter corrected and normalized) $w(\ce{Si})$ at each cell \citep[e.g.,][]{Boynton2007}. These ratios act as scaling factors for mass fractions of all capture-derived elements (Cl, Fe, and H) \citep[][]{Boynton2007}, increasing initial estimates by $\approx\times 4$ on average. Such scaling is necessary to account for variations of the thermal neutron flux in the regolith due to the presence of neutron moderators, such as H, even at low concentrations \citep[][]{Boynton2007}.

A slight feedback effect exists between the scatter and capture correction steps, since Fe and H content, which are subject to the capture correction, affect the scatter correction. Furthermore, as described in Section~\ref{OH}, we renormalize the calculated in situ composition to allow for the presence of GS-determined H, which in turn affects the Si normalization. However, these feedback effects among the scatter correction, capture correction, and Si normalization are so subtle that relative variations in estimated elemental mass fractions during iterative processing was $<\SI{0.1}{\%}$.


\section{Summary of the Technique}
\label{ST}
As outlined in Section~\ref{I}, since we compute a weighted mean of in situ rock and soil compositions, it is important to classify regolith components into these two categories. Throughout our discussion, ``rocks'' generally refer to \SI{10}{cm} scale and larger materials (e.g., fragments, breccia, and exposed outcrops) that have high thermal inertia, have areal fractions consistent with the Viking Infrared Thermal Mapper (IRTM) data \citep[e.g.,][]{Christensen1986,Golombek2005}, and are immobile under current eolian conditions. ``Soil'' refers to all other, typically unconsolidated, material including those sufficiently fine-grained to be mobilized by wind \citep[e.g.,][]{Banin1992,Jerolmack2006}. Soil consequently encompasses a variety of regolith components identified at the five lander sites. Typical examples are: bedform armor, clasts, concretions, drift, dust, rocky fragments, sand, and soil \citep[e.g.,][]{Fergason2006,Golombek2005,Sullivan2005,Yen2005}.

While our classification of the regolith into rocks and soil is very broad, it suits the GS data context, with the \SI{\approx440}{km} instrument footprint and upper few tens of centimeter sampling depth as described in Section~\ref{KGSDPS} \citep[e.g.,][]{Boynton2007}. For example, the GS would be insensitive to the variability of rock areal fraction by factors of two to four \citep[e.g.,][]{Squyres2004a} as observed in the \SI{}{km} scale traverse of Spirit. Satisfactory agreement between average in situ estimates and IRTM orbital estimates of the areal fraction of rocks further support our simple classification \citep[e.g.,][]{Golombek2005}. Furthermore, the mean free path of neutrons is \SI{\approx 10}{cm} \citep[e.g.,][]{Squyres1992}, ensuring that what we consider as rocks appear distinct to the GS. However, our rock size threshold assigns smaller rock fragments to the soil component, potentially making the areal fraction we assign to rocks less than their in situ value. In addition, since the daily temperature variations do not penetrate very far into rock, and less into soil, even a thin ($>\SI{1}{cm}$) soil layer can conceal rocks from the IRTM and lead to an underestimation of the rock abundance.

For a regolith with two key components, rock and soil materials, distributed as a homogenous mixture within a sample volume, we may compute the representative mass fraction of an element or oxide as

\begin{equation}
C_{\ce{m}} = \frac{c_{\ce{r}} A_{\ce{r}} \rho_{\ce{r}} + c_{\ce{s}} (1 - A_{\ce{r}}) \rho_{\ce{s}}}{A_{\ce{r}} \rho_{\ce{r}} + (1 - A_{\ce{r}}) \rho_{\ce{s}} }
\label{e mean composition}
\end{equation}
with $C_{\ce{m}}$ the representative in situ concentration of the element (or oxide) as a mass fraction, $A_{\ce{r}}$ the areal fraction of rocks, $\rho_{\ce{r}}$ the density of rocks, $\rho_{\ce{s}}$ the density of soil, $c_{\ce{r}}$ the mass fraction of the element (or oxide) in rocks, and $c_{\ce{s}}$ the mass fraction of the element (or oxide) in soil.
Uncertainties are propagated for $C_{\ce{m}}$ and all other computed values with the standard formula \citep[e.g.,][pp. 96--101]{Young1962}:

\begin{equation}
s_{\ce{m}} = \sqrt{\sum_i\left|\frac{\partial f}{\partial x_i}\right|^2 s_{m,x_i}^2}
\label{e error propagation}
\end{equation}
where $x_i$ is the $i^{\ce{th}}$ variable of the function $f$, $s_{\ce{m},x_i}$ the standard error (i.e., net uncertainty defining \SI{68}{\%} statistical confidence under normality assumptions) of $x_i$, and $s_{\ce{m}}$ the standard error of the function. Therefore, under gaussian assumptions, $f$ would be \SI{68}{\%} probable to be within $1s_{\ce{m}}$. Following the notation within companion papers of the Mars Odyssey Special Issue \citep[e.g.,][]{Boynton2007,Karunatillake2006,Keller2006,Taylor2006}, we state uncertainties to $1s_{\ce{m}}$. In contrast, the MER team reports APXS data uncertainties to $2s_{\ce{m}}$ \citep[e.g.,][]{Gellert2006,Rieder2004}.

For a given soil type at each MER site, we estimate $c_{\ce{s}}$ in Equation~\ref{e mean composition} as the arithmetic mean $\left(\hat{\mu}\right)$ of several samples identified in Sections~\ref{Spirit} and~\ref{Opp}. The corresponding standard error $(s_{\ce{m},x_i}=s_{\ce{m},\ce{c}_{\ce{s}}})$ for use in Equation~\ref{e error propagation} is derived from the standard deviation $s = \sqrt{\frac{\sum_i\left(c_{\ce{s},i}-\hat{\mu}\right)^2}{N-1}}$ as $s_{\ce{m},\ce{c}_{\ce{s}}}=\frac{s}{\sqrt{N}}$, where $N$ is the number of samples and $c_{\ce{s},i}$ is the concentration in the $i^{\ce{th}}$ sample \citep[e.g.,][pp. 35--41, 62--63, 106--110]{Mandel1964}. The alternative of computing the maximum likelihood mean and corresponding $s_{\ce{m},\ce{c}_{\ce{s}}}$ using the reported uncertainties is less defensible in the rover context, since such samples generally present statistically inconsistent \citep[e.g.,][pp. 147--152]{Taylor1982} data even though they are spatially indistinct from the GS perspective. We similarly utilize the standard error of the arithmetic mean along with calibration uncertainties for Viking 1 data. In all other cases, we use the reported standard error as $s_{\ce{m},x_i}$, which is primarily driven by counting statistics for the GS data. In summary, the standard error of $C_{\ce{m}}$ is mostly a function of the following: (1) statistical uncertainties in rock and soil densities as described in Section~\ref{BDAF}, (2) reported uncertainty in rock areal fraction (Table~\ref{table areal fractions}), (3) reported uncertainties of in situ element/oxide mass fractions (with the exception of MER soil data as described above), and (4) the standard error of GS-derived $w(\ce{H2O})$ (Table~\ref{table H2O} and Section~\ref{OH}).

A difficulty with Equation~\ref{e mean composition} is that density, areal fraction, and element concentration measurements are not simultaneously available for a given lander site. In addition, Equation~\ref{e mean composition} is valid only where the regolith is a homogenous mixture of rocks and soil across the $\gamma$ photon sampling depth, which, as discussed in Section~\ref{KGSDPS}, is also a primary assumption of all GS datasets. While this assumption may be roughly true for the Gusev, Pathfinder, and Viking  landing ellipses \citep[e.g.,][]{Moore1977,Golombek2005}, Meridiani is a clear exception as noted in Section~\ref{Opp}. The effect of rock/soil mixing geometries on mass fraction estimation has been modeled before \citep[e.g.,][]{Squyres1992}. Additional studies are underway to quantify these effects for rocks embedded in a soil matrix \citep[e.g.,][]{Kim2006}.


\section{Bulk Densities and Areal Fractions}
\label{BDAF}
An estimate of rock and soil bulk densities is key to evaluating Equation~\ref{e mean composition}. However, none of the in situ missions have been equipped with instruments to directly estimate bulk densities. Consequently, the most reliable density estimates to date are from reasonable terrestrial analogues that utilize thermophysical, textural, and compositional information of the Martian regolith \citep[e.g.,][]{Fergason2006,Grant2004}. While bulk densities of Martian rocks would be consistent with those of Shergottite-Nakhlite-Chassignite (SNC) meteorites, it is important to note that SNC meteorites may not be petrologically representative of the general Martian regolith \citep[e.g.,][]{McSween1994,McSween2002}.

In the case of the MER, Rock Abration Tool (RAT) currents and M\"ossbauer contact plate forces provide additional estimates of rock and soil properties, respectively \citep[e.g.,][]{Arvidson2004, Arvidson2004a,Squyres2006}. In addition, the Panoramic Cameras (Pancams) \citep[e.g.,][]{Farrand2006,Bell2004} and Microscopic Imager (MI) \citep[e.g.,][]{Sullivan2005,Cabrol2006,Herkenhoff2006} yield much enhanced textural information. Overall, the MER mission has provided evidence of much greater variability in bulk properties than recognized on earlier missions \citep[e.g.,][]{Fergason2006}. Nevertheless, in the context of remote missions sensitive only to average physical properties at coarse spatial resolution, detailed modeling of in situ bulk densities is unlikely to yield useful insight. Therefore, we utilize the Viking regolith property observations and corresponding terrestrial analogs as our primary sources of density information, supplemented with evidence for heterogeneity as observed by the MER mission.

The density of ``average'' Martian rocks, considered physically analogous to terrestrial dense basalts \citep[e.g.,][]{Fergason2006,McSween2004}, is $(\num{2.6}\pm\num{0.5})\times\SI{E3}{kg~m^{-3}}$ \citep[][]{Moore1977,Moore1989,Olhoeft1989}. To account for significant variability in rock bulk densities, we quote and substitute the standard deviation $(s)$ instead of the standard error of $\rho_{\ce{r}}$ in $s_{\ce{m},x_i}=s_{\ce{m},\ce{\rho}_{\ce{r}}}$ of Equation~\ref{e error propagation}. Two soil components were identified at the Viking 1 site: ``drift'' with a density of $(\num{1.15}\pm\num{0.15})\times\SI{E3}{kg~m^{-3}}$ and ``blocky soil'' with a density of $(\num{1.6}\pm\num{0.4})\times\SI{E3}{kg~m^{-3}}$ \citep{Moore1989}. While the near-field analysis by \citet[][]{Moore1989} identified only a ``crusty to cloddy soil'' with a density of $(\num{1.4}\pm\num{0.2})\times\SI{E3}{kg~m^{-3}}$ at the Viking 2 site, the far-field analysis by \citet[][]{Moore1991} verified the presence of ``drift'' as well. Since soil densities at each site overlap within the uncertainties, we use their far-field areal fraction weighted mean as the density of the soil in Equation~\ref{e mean composition}. The mean density of the soil, $\rho_{\ce{s}}$, may then be expressed as:

\begin{equation}
\rho_{\ce{s}} = \frac{A_{\ce{d}}\rho_{\ce{d}} + A_{\ce{s}}\rho_{\ce{s}}}{A_{\ce{d}}+A_{\ce{s}}}
\label{e mean density}
\end{equation}
where d represents ``drift'' and s represents ``soil'' (``blocky'' type at Viking 1 and ``crusty to cloddy'' type at Viking 2). The remaining symbols are analogous to those of Equation~\ref{e mean composition}, with the standard error of $\rho_{\ce{s}}$ obtained by substituting $f=\rho_{\ce{s}}$ in Equation~\ref{e error propagation}.

Since Viking 1 and 2 reported different relative areal fractions (Table~\ref{table areal fractions}) and densities for the ``drift'' and ``soil'' components, $\rho_{\ce{s}}$ differs slightly between the two soil proxies. For the Pathfinder and MER lander sites where density estimates have not been reported, the maximum-likelihood mean \citep[e.g.,][pp. 131--135]{Mandel1964} of the two, $(\num{1.34}\pm\num{0.13})\times\SI{E3}{kg~m^{-3}}$, was used as the mean density, $\rho_{\ce{s}}$, of the soil component. This approximation is further justified by the favorable comparison of soil properties across Viking, Pathfinder, and MER lander sites \citep[e.g.,][]{Arvidson2004}. On the other hand, $\rho_{\ce{s}}$ at the Viking 1 lander site was computed directly from Equation~\ref{e mean density} to be $(\num{1.5}\pm\num{0.3})\times\SI{E3}{kg~m^{-3}}$.


In addition to the densities of the rock and soil components, Equation~\ref{e mean composition} requires the areal fraction of rocks, $A_{\ce{r}}$. As explained in Section~\ref{ST}, we rely on the global rock areal fraction map at $\ang{1}\times\ang{1}$ resolution generated with thermal inertia data from the IRTM \citep{Christensen1986}. We compute the mean of rock areal fractions within a \ang{15} arc radius circle, representative of the GS spatial filter and centered at the coordinates of each lander site: Viking 1 (\ang{22.27} latitude, \ang{-47.94} east longitude) \citep{Mayo1977}, Pathfinder (\ang{19.13}, \ang{-33.22}) \citep{Golombek1999}, Spirit (\ang{14.57}, \ang{175.47}) \citep{Squyres2004a}, and Opportunity (\ang{-1.95}, \ang{-5.53}) \citep{Squyres2004}. The absolute, not relative, uncertainty of each mean rock areal fraction is taken to be \SI{5}{\%} \citep[e.g.,][]{Christensen1986}. These are summarized in Table~\ref{table areal fractions}.

\section{Oxidation and Hydration}
\label{OH}
We compare compositions across lander sites while conforming with wet chemical analysis normalization of mass fractions at each site to unity. This requires the oxidation state of a given element to be the same across all lander sites for both rock and soil. Reported values satisfy this condition for most elements, with the exception of Fe and S. We adjust the oxidation states of these two elements in three steps. First, we stoichiometrically (atomic masses by \citet[][]{Loss2003}) convert reported oxide mass fractions of all elements into the unoxidized (i.e., elemental) mass fractions. Second, we convert the elemental mass fractions back into oxide mass fractions assigning the same oxidation state to a given element across all regolith components and lander sites. Third, we separately renormalize these mass fractions to unity for rock and soil.

Only Pathfinder data needed the oxidation state adjustment for S, by converting rock composition from \ce{S} to \ce{S^6+}. The adjustment for Fe is much more complicated. To date, the MER M\"ossbauer spectrometer is the only instrument to measure in situ Fe oxidation states. While the MER data demonstrate significant variability in the molar fraction of \ce{Fe^3+}, once the outcrop and hematitic soils in Meridiani and the Columbia Hills rocks in Gusev are excluded, the molar fractions are in rough agreement between the two sites \citep{Klingelhofer2004,Morris2004}.  It is important to note however, that the Fe oxidation states observed in the Columbia Hills indicate much variability, with the molar fraction of \ce{Fe^3+}, $\frac{\ce{Fe^3+}}{\ce{Fe_{total}}}$, varying from approximately \num{0.2} to \num{0.9} \citep[][]{Morris2006}. Columbia Hills rocks are also older than the plains rocks \citep[e.g.,][]{Squyres2006}, and may be comparable in age to the bedrock beyond the crater rim. While these concerns indicate that using a mean of Fe oxidation states observed across lander sites may be simplistic, as explained in Section~\ref{SMCU}, its effect on the computed in situ composition is negligible due to the normalization of oxides. In contrast, a major systematic error could occur if Fe and/or S existed in sulfide or elemental form instead of as oxides.

Given the evidence for significant variability of Fe oxidation states between and within the only two sites where such data are available, we use the Gusev plains material to estimate a mean molar fraction of \ce{Fe^3+} solely as a formalism.  The arithmetic mean and standard deviation of the \ce{Fe^3+} molar fraction of all measurements in the Gusev plains is $\num{0.28} \pm \num{0.01}$ \citep{Morris2004}. Consequently, we use a mean Fe oxidation state of \num{+2.28} as the standard in all our representative in situ estimates. The renormalizations of reported compositions to \ce{FeO_{\cmath{1.14}}} and \ce{SO3} preceded all other calculations.

While GS measurements include $w(\ce{H})$ (reported as stoichiometrically equivalent $w(\ce{H2O})$), H content has not been directly determined in situ \citep[][]{Rieder2003,Gellert2006}, with the exception of the Viking landers \citep[e.g.,][]{Anderson1979}. We model the rock component to be devoid of \ce{H2O}, and propagate the GS-derived $w(\ce{H2O})$ into the soil component. This is achieved by inverting Equation~\ref{e mean composition}:

\begin{equation}
c_{\ce{s}} = \frac{C_{\ce{GS}}\left\{A_{\ce{r}}\rho_{\ce{r}}+\left(1-A_{\ce{r}}\right)\rho_{\ce{s}}\right\}}{\left(1-A_{\ce{r}}\right)\rho_{\ce{s}}}
\label{e soil H2O}
\end{equation}
where $c_{\ce{s}}$ is the estimated in situ $w(\ce{H2O})$ in soil, and $C_{\ce{GS}}$ the GS-derived $w(\ce{H2O})$ at the lander site coordinates. $C_{\ce{GS}}$ values and corresponding standard errors are listed in Table~\ref{table H2O}, and we compute the standard error of $c_{\ce{s}}$ by substituting $f=c_{\ce{s}}$ in Equation~\ref{e error propagation}. The in situ oxide and elemental mass fractions in soil are renormalized to $1-c_{\ce{s}}$ to allow for the presence of \ce{H2O}. In spite of exceptions \citep[e.g.,][]{Clark2005} discussed in Section~\ref{Opp}, assuming the rock component to be free of \ce{H2O} remains a reasonable approximation for the majority of lander sites. As a concluding step, we convert the representative in situ compositions, renormalized for both uniform oxidation states and the presence of H, into stoichiometrically equivalent elemental mass fractions.

In general, $C_{\ce{GS}}$ values (Table~\ref{table H2O}) are consistent with in situ estimates of $w(\ce{H2O})$ at Pathfinder, Spirit, and Viking 1. At pathfinder, the stoichiometric $w(\ce{H2O})$ is \SI{\approx2}{\%} in soils and $\SI{0.1}{\%}--\SI{4.3}{\%}$ in rocks as calculated via mass balance and subject to Fe and S oxidation state assumptions \citep[][]{Foley2003}. The re-analysis of Viking 1 data indicated that the soil at Viking 1 may contain $w(\ce{H2O})$ in the range $\SI{1.1}{\%}--\SI{3.0}{\%}$ \citep[][]{Anderson1979}. Mineralogic modeling of \ce{H2O} content at Spirit is difficult due to the large mineralogic variability of regolith components that contain H. Nevertheless, current modeling tentatively suggests $w(\ce{H2O})\approx\SI{0.7}{\%}$ as Goethite $(\alpha-\ce{FeOOH})$ in the most altered Gusev rocks \citep[][]{Morris2006} and $w(\ce{H2O})<\SI{11}{\%}$ as hydrated sulfates in subsurface soils \citep[][]{Wang2006}. Furthermore, recently analyzed salt deposits in the Columbia Hills, such as Paso Robles, may contain much higher amounts of bound \ce{H2O} (e.g., Johnson, J. R. et al., Mineralogic constraints on sulfur-rich soils from Pancam spectra at Gusev Crater, Mars, manuscript in preparation to Geophys. Res. Lett., 2007; \citet[][]{Lane2007}; \citet[][]{Wang2007}; and Yen, A. et al., Soil geochemistry with an emphasis on Paso Robles class materials, manuscript in preparation to J. Geophys. Res., 2007).


\section{In Situ Specifics}
In spite of the generally similar methodology to estimate representative in situ compositions, some differences arise in the calculation details at different lander sites. These differences reflect the compositional diversity on Mars and differences in instrumental capabilities. For example, the elemental composition of the Viking 1 lander site is not as well characterized as the others, since the XRFS had much more limited elemental detection capabilities than the APXS used in subsequent missions. Moreover, the MER mission provides additional information due to a comprehensive suite of instruments and mobility.

MER instruments are sensitive to geochemical  variations at spatial scales vastly exceeding the sensitivities of remote (e.g., the GS) missions. For example, the MERs have revealed striking morphological, textural, and compositional variations \citep[e.g.,][]{Arvidson2006,Cabrol2006,Grotzinger2005} at centimeter to tens of centimeter scales with elemental and mineral detection instruments -- the APXS \citep[e.g.,][]{Gellert2006}, Miniature Thermal Emission Spectrometer (Mini-TES) \citep[][]{Fergason2006}, and Miniature M\"ossbauer Spectrometer (MIMOS II) \citep[][]{Morris2006} -- and at sub-millimeter scales with the MI \citep[e.g.,][]{Herkenhoff2006}. As an additional complexity, Spirit has observed chronological variability, including the removal of a thin dust cover by dust devils \citep[e.g.,][]{Arvidson2006,Greeley2005}. Consequently, while we use all the samples of earlier in situ missions, only a carefully selected subset of the MER data is used for remote comparisons.

\subsection{Viking 1}
All Viking data may be used in representative in situ estimations since major compositional heterogeneities, which typify Gusev and Meridiani, were inevident most likely due to instrumental limitations. The absence of rock compositional data poses a challenge at the Viking site. Given the general geological similarities and spatial proximity of the regolith at Viking 1 and Pathfinder, we address this by using the Pathfinder mean soil-free rock composition \citep{Wanke2001} as a proxy for the rock composition at Viking 1. However, the apparent geological similarities may conceal significant compositional differences, with the potential to invalidate our choice of a proxy rock composition. Nevertheless, relative to Pathfinder, rock compositions at either MER site are likely to be even more inappropriate as proxies to rocks at Viking.

Besides the lack of rock analyses, the Viking 1 XRFS was also incapable of detecting Cr, Mn, Na, and P. Consistent with the approach by \citet[][]{McSween2000}, for example, we account for this issue by normalizing Viking 1 rock and soil compositions to \SI{97}{\%}, the net mass fraction of Pathfinder soil-free rock attributable to elements and oxides other than \ce{Na2O}, \ce{Cr2O3}, \ce{MnO}, and \ce{P2O5}. Furthermore, the mean K content based on the work of \citet{Clark1982} is below the XRFS detection limit. We use the arithmetic mean of values reported by \citet{Clark1982} to estimate the Viking 1 soil composition.

\subsection{Spirit}
\label{Spirit}
We utilize the reported APXS data for Spirit \citep[][]{Gellert2006}. The choice of a suitable rock composition in the context of the GS footprint is somewhat complicated at Gusev. The uncontaminated composition of rocks in the Gusev crater floor could be represented effectively by the APXS measurements following the second Rock Abration Tool application (RAT 2) on the plains rock ``Humphrey'' (Figure~\ref{fig SpiritContext} and Table~\ref{table sols}) \citep[][]{Gellert2004}. However, rocks in the ``Columbia'' Hills are chemically altered, vary significantly in composition leading to at least five compositionally distinct classes, and are inferred to be geologically older (possibly Noachian) than the potentially Hesperian plains rocks \citep[e.g.,][]{Arvidson2006,Cabrol2006,Squyres2006}.

%\begin{figure*}[p]
%\begin{landscapefigure*}
%\noindent\includegraphics{GusevSamples}
%\caption{test}
%\end{landscapefigure*}
%\label{fig SpiritContext}
%\end{figure*}

While the Columbia Hills constitute \SI{<}{5\%} of the area within the Gusev crater, the crater itself, with a diameter of \SI{\approx160}{km} \citep{Squyres2004a}, accounts for only \SI{\approx13}{\%} of the GS footprint. The bedrock beyond the crater is more likely to be of a geological age comparable to the Columbia Hills than the Gusev plains. Consequently, using only Humphrey RAT 2 data to represent the rocks of Gusev may be more inappropriate than an areally weighted mean of Gusev plains and Columbia Hills rock compositions. We consider both options, as they reflect the complexity of Martian surface composition and the difficulty of selecting a particular composition to represent even a single location at the spatial scale of a GS footprint. Though found only as float, the composition of ``Wishstone'' class brushed and abraded rock (Figure~\ref{fig SpiritContext} and Table~\ref{table sols}) is used to represent the Columbia Hills rock composition as it has been fairly common in the Columbia Hills, and similar in elemental composition to the ``Watchtower'' class of outcrop \citep{Squyres2006,Gellert2006}.

Compositional differences between the typical crater floor soils (sampled with the ``Boroughs'' and ``Big Hole'' trenches, Figure~\ref{fig SpiritContext} and Table~\ref{table sols}) and ``hollows'' (sampled with the ``Road Cut'' trench, Figure~\ref{fig SpiritContext} and Table~\ref{table sols}) complicate the choice of a representative soil composition at Gusev. As discussed for Equations~\ref{e mean composition} and~\ref{e error propagation}, we use four samples from Boroughs and Big Hole trenches to represent the typical soil, and two samples from the Road Cut trench to represent hollows soil. Hollows appear to be relatively young eolian infill of small impact craters, while the surrounding plains contain mature soil material \citep[e.g.,][]{Haskin2005,Wang2006}. Furthermore, in spite of the ubiquity of hollows \citep[][]{Golombek2006}, plains soils are areally dominant along the rover traverse \citep[][]{Arvidson2004,Grant2004}. In light of these issues, we compute a 9 to 1 ratio mean of plains to hollows as the representative composition of soil at Gusev, which is supported in part by areal evidence presented by \citet[][]{Golombek2006}.

In spite of compositional complexities, trench data sampling up to \SI{\approx 11}{cm} deep in the making of Boroughs are much more relevant at GS sampling depth scales than any surface soil measurement. The MER trench campaigns were motivated primarily by this consideration \citep{Wang2006}, and have revealed the presence of significant compositional layering at tens of centimeter depths \citep[e.g.,][]{Haskin2005,Soderblom2004,Wang2006}. This is not weakened by the lack of evidence in GS data for Cl layering at large lateral spatial scales \citep[][]{Keller2006a}, since Br and Cl in particular have shown highly localized vertical variations at both Gusev \citep[e.g.,][]{Wang2006} and Meridiani \citep[e.g.,][]{Rieder2004}.

\subsection{Opportunity}
\label{Opp}
In contrast to Gusev and other lander sites, the surface at Meridiani consists of a layer of soil with variable thickness at points exceeding \SI{\approx 1}{m} \citep[][]{Soderblom2004}, overlying sedimentary bedrock of aqueous origin \citep[e.g.,][]{Grotzinger2005,Jolliff2006,McLennan2005,Squyres2006a}. Therefore, the soil and bedrock can each be deeper than the tens of centimeter GS sampling depths, sometimes acting as a semi-infinite layer and sometimes not, complicating the analysis. However, bedrock exposures are sparse in the rover traverse and limited mostly to craters (e.g., Victoria) and fractures (e.g., Anatolia) \citep[][]{Squyres2004,Squyres2006b}. 

With the exception of a lag deposit of hematite spherules derived mostly from the bedrock, the soil is predominantly basaltic sand of potentially external origin \citep[][]{Squyres2004,Weitz2006,Soderblom2004}. Another difference from the other lander sites is that the bedrock at Meridiani may contain $w(\ce{H2O})$ of up to \SI{22}{\%}, possibly as hydrated sulfate minerals, while the soil is unlikely to contain more than \SI{4}{\%} \citep[][]{Clark2005}. If applicable at GS footprint scales, this would attribute much of the GS-derived H content to the outcrop rather than the soil. Furthermore, with the exception of basaltic sands, the unique minerals at Meridiani may make the densities of its regolith components differ significantly from density estimates based on Viking observations (Sullivan, personal communication, 2005).

Such issues considerably weaken the relevance of Equations~\ref{e mean composition} and~\ref{e soil H2O} in estimating a representative composition at Meridiani. However, as a first-order approximation to the representative Meridiani composition, we use Equation~\ref{e mean composition} assuming a rock-free surface (i.e., $A_{\ce{r}} = 0$) dominated by the hydrated (hematite-free) basaltic component of the soil material. Even though the areal fraction of soil as determined with IRTM data is not \SI{100}{\%} (Table~\ref{table areal fractions}), it is sufficiently close to unity for our first-order approximation. As at Gusev, the areally dominant soil component -- in the context of the GS sampling depths -- was sampled by trenching experiments which also revealed the presence of finer grains at depth in addition to the volumetrically abundant basaltic grains \citep[][]{Weitz2006,Sullivan2005}. We represent this soil with the arithmetic mean (refer Equation~\ref{e error propagation} and related text for uncertainty estimation) composition at depth in two trenches, ``BeagleBurrow\_Trench'' and ``Left\_of\_peanut\_TrenchFloor'' (Figure~\ref{fig OpportunityContext} and Table~\ref{table sols}), which, as evident in the low areal fractions of hematite and jarosite \citep[][]{Klingelhofer2004,Morris2006a}, effectively excludes the lag deposit and other contributions from the outcrop. We use the recalibrated APXS-derived mass fractions (available at \url{ http://pds-geosciences.wustl.edu/geodata/mer1_mer2-m-apxs-5-oxide-sci-v1/merap_2xxx/data/apxs_oxides_mer1.csv}) as an update to those given by \citet[][]{Rieder2004}.

%\begin{figure*}[p]
%\begin{landscapefigure*}
%\noindent\includegraphics{MeridianiSamples}
%\caption{test}
%\end{landscapefigure*}
%\label{fig OpportunityContext}
%\end{figure*}


Complicating matters further, outcrop sampled in the current \SI{}{km} scale traverse of Opportunity may not be representative of bedrock in the greater Sinus Meridiani region \citep[e.g.,][]{Edgett2005} and by extension, within the GS's \SI{\approx440}{km} instrumental and \SI{\approx1800}{km} filter footprints. In addition, accurately modeling the compositional effect from rocks would require a checkerboard surface model, given the distinctly layered nature to the regolith at Meridiani. Such a model would consist strictly of weighting by the areal fractions of outcrop and soil without the use of bulk densities, since each layer would usually appear infinitely thick to the GS. However, a checkerboard model is unlikely to significantly improve the accuracy or precision of our calculated composition in the context of GS comparisons, particularly since the variability in soil depth is poorly known.


\subsection{Pathfinder and the GS normalization}
\label{PGSN}
The reported geochemistry of rock and soil material at Pathfinder shows less heterogeneity than those at either Gusev or Meridiani. In part this may be due to the short Pathfinder traverse of about \SI{100}{m} (since Sojourner rover remained within \SI{\approx10}{m} of the lander throughout the mission) \citep[][]{Golombek1999} relative to MER's \SI{}{km} scale traverse. Pathfinder also lacked the capability to brush rock surfaces or abrade them, as the MER rovers have done with the RAT, which may have potentially concealed compositional variations. For example, complex alteration processes have produced both brushable surface accumulations and abradable alteration zones on rocks at both MER sites \citep[e.g.,][]{Hurowitz2006}, the compositional effects of which cannot be removed by the S extrapolation method used at Pathfinder \citep[][]{Wanke2001}.

More important, the Pathfinder landing ellipse in Ares Vallis is larger \citep{Golombek1997} than those of either MER \citep{Arvidson2003,Cabrol2003,Golombek2003}. While the landing ellipse is an engineering constraint, missions with smaller landing ellipses are more capable of reaching geologically unique areas. Consequently, the much larger Pathfinder landing ellipse -- located in a region less diverse than either MER site -- and its immediate neighborhood present a geologically less heterogeneous area that is better suited for comparison with the GS given its coarse spatial resolution. Ironically, the advantage offered by the MER mission with smaller landing ellipses at geologically unique localities is lost to the GS since both ellipses are within geologically heterogeneous regions at GS footprint scales. For example -- much as Lunokhod-2 was the first rover to traverse a geologic contact on the moon \citep[e.g.,][]{Basilevsky1977,Basilevsky1996} -- Spirit is the first rover to traverse a geologic contact on another planet \citep[e.g.,][]{Arvidson2006,Crumpler2005}.

The apparent compositional homogeneity at Pathfinder enables us to use revised values for all samples, as reported by \citet{Wanke2001}, to estimate the mean compositions of soil and rocks. \citet{Bruckner2003} provide details of the refined calibration that was used by \citet{Wanke2001}. Data from both these papers are essentially in agreement. \citet{Foley2003} describe Pathfinder APXS data that were derived from an independent calibration and are very similar to those quoted by \citet{Wanke2001} and \citet{Bruckner2003} for the major elements, while discrepancies exist for minor elements.

The representative composition at Pathfinder estimated from the work of \citet{Foley2003} is $(\num{50}\pm\num{20})\SI{}{\%}$ higher for Cl and $(\num{30}\pm\num{10})\SI{}{\%}$ lower for K than that based on values by \citet{Wanke2001}. In spite of this, since all three works overlap within net uncertainties for the major elements Fe and Si, we do not anticipate significant changes to GS estimates should the data by \citet{Foley2003} be used. The Cr estimate has a minor caveat as well, since soil-free rock has been modeled without Cr \citep[e.g.,][]{Bruckner2003,Foley2003,Wanke2001}. As a result, our estimate of Cr assumes a modeled zero content of Cr in rocks, with only the uncertainty of $w(\ce{Cr})$ in soil incorporated into Equation~\ref{e error propagation}.

\section{Sensitivity of Representative Compositions to Unknowns}
\label{SMCU}
The accuracy and precision of representative compositions are important considerations given the limited spatial sampling of in situ instruments, IRTM modeling of rock areal fractions, and Viking density estimates. As expected, the content of major elements, such as Si, is consistent across the lander sites and APXS measurements have rigorously quantified uncertainties. Rock areal fraction estimates from rovers and landers have been consistent with IRTM modeling as well. In addition, the density estimates are based upon reasonable parallels with terrestrial material. Collectively, these suggest that the representative compositions are sufficiently accurate for use at GS footprint scales and sampling depths, barring unknown systematic errors in instrument calibrations.

However, there is some variability in rock abundance due to the massive difference in sampling area between the surface instruments and GS. Density uncertainties are poorly constrained, since none of the missions had the means to determine densities. An additional uncertainty is introduced by the oxidation state of Fe, though again, a reasonable value was used in our calculations. A final uncertainty affecting the Gusev representative composition is the areal ratio of typical soil to hollows soil. We estimated the effects of these uncertainties by varying rock abundance, rock density, and soil density values by \SI{50}{\%}; varying the Fe oxidation state as +2, +2.5, and +3; and using a typical soil to hollows soil areal fraction ratio of 3 to 7. The alternative typical soil to hollows soil ratio was selected as a potential value at GS footprint scale, which is much larger than the Gusev crater, as well as to investigate compositional effects if hollows were to be areally dominant.

The effect of the Fe oxidation state was minimal, changing the representative composition by less than \SI{1}{\%}. Rock areal fraction and density variations caused representative compositions to vary by less than \SI{7}{\%} at most. While affecting only the Gusev representative composition the last issue, areal fraction ratio of typical soil to hollows soil, altered the S content by as much as \SI{26}{\%}, even though Si varied by only \SI{6}{\%}. This was a consequence of the elevated S levels at mid-depth in the Boroughs trench wall \citep[e.g.,][]{Wang2006}. In summary therefore, with the exception of Gusev, poorly constrained parameters in the representative composition calculation are unlikely to cause more than a \SI{10}{\%} relative uncertainty. While Gusev may be subject to a higher uncertainty, major elements such as Si are still unlikely to vary by more than \SI{15}{\%}. We report standard errors -- computed with Equation~\ref{e error propagation} -- in Table~\ref{table mass fractions} to one significant figure with the exception of those with 1 or 2 as the leading digit where we report to two significant figures. We accordingly retain the last significant figure of the result to be of the same order of magnitude as the uncertainty \citep[e.g.,][pp. 15--21]{Taylor1982}.

\section{Discussion of Results}
The representative in situ compositions as computed above are listed in Table~\ref{table mass fractions}. As evident in Figure~\ref{fig key elements}, the concentrations of major elements vary less and have smaller relative uncertainties than those of many minor elements. Consider for example, the difference in variability and uncertainty between K (dynamic range approximately a factor of 3) and Si (dynamic range approximately a factor of 1.2). Furthermore, the variability of K as represented by the standard deviation of the mass fractions is \SI{49}{\%} of the (arithmetic) mean $w(\ce{K})$ of all lander sites, while the variability is only \SI{9}{\%} for Si and Fe. The similarity in major elements across the four sites suggests that normalizing $w(\ce{Si})$ to Pathfinder is unlikely to introduce systematic errors. On the other hand, normalization of minor elements to in situ values would have been more prone to error, given the variability and associated uncertainties of their mass fractions across the lander sites. The geochemical implications of representative in situ compositions, particularly in terms of their comparison with GS data, are discussed by \citet[][]{Boynton2007}, \citet[][]{Karunatillake2006}, \citet[][]{Keller2006}, \citet[][]{newsom2007}, \citet[][]{Taylor2006}, and \citet[][]{Taylor2006a}.

Comparison of the two representative in situ compositions estimated for Gusev, as described in Section~\ref{Spirit}, provides insight into the impact on the representative composition from the compositional variability of different rock types at a single lander site. That the impact is minor is evident in Figure~\ref{fig key elements}, where the representative concentrations of most elements are similar even though Humphrey and Wishstone are compositionally distinct even with respect to major elements such as Fe \citep[e.g.,][]{Gellert2006}. The primary reason for the subdued effect is the areal dominance of the soil component which, due to the general absence of striking differences in soil composition between the plains and Columbia hills \citep[e.g.,][]{Arvidson2006}, was taken to be compositionally uniform both within and beyond the Gusev crater. Since the soil component is areally dominant throughout much of Mars \citep[e.g.,][]{newsom2007,Karunatillake2006}, the GS should also be more sensitive to compositional variations of the Martian soils than of rocks. GS data indicate significant chemical variations in soil \citep[][]{newsom2007} in spite of evidence for compositionally similar soils across lander sites \citep[e.g.,][]{Yen2005}. 

The GS-determined K content at Pathfinder is higher at the $1s_{\ce{m}}$ level relative to the K content at the other sites, though not at the $2s_{\ce{m}}$ level \citep[][]{Boynton2007}. Si does not appear enriched at Pathfinder even at $1s_{\ce{m}}$ \citep[][]{Boynton2007}, which is consistent with the absence of Si enrichment in surface type 2 material \citep[][]{Karunatillake2006}. The GS estimates are compared with the representative in situ estimates in Figure~\ref{fig GS elements}. In addition to the Si mass fractions, which approximate in situ values by virtue of the normalization, Cl and K also compare favorably. The favorable comparison for Cl (which overlaps with the GS value within $2s_{\ce{m}}$) lends additional support for the Si normalization \citep[][]{Boynton2007}. The favorable comparison for K -- for which the GS and in situ values overlap within $1s_{\ce{m}}$ at Opportunity and Viking -- is specially significant, since K is a radiogenic element unaffected by capture and scatter processes.

However, the GS estimate of Fe content appears to be consistently higher, even though the relative difference, varying from a low of \SI{10}{\%} at Meridiani to a high of \SI{20}{\%} at Gusev, is always less than $3s_{\ce{m}}$. Since the disparity exists across all lander sites in the same direction, it is unlikely to be an effect of the simplifications and sample selections we made in calculating representative in situ compositions at the four sites. \citet{Boynton2007} discuss the implications of these comparisons in depth.


\section{Conclusions}
Our study estimating the representative in situ compositions at four lander sites is an example of the synergy between remote sensing and surface missions \citep[cf.,][]{Bibring2006a}. For example, the GS determination of H and IRTM determination of rock abundance can be combined with surface data to derive representative in situ chemical compositions across large areas even though data from surface instruments are inherently local in scope. This method of estimating representative compositions at lander sites is applicable to future remote sensing missions as well. In addition, the representative in situ compositions that we derive (Table~\ref{table mass fractions}) may be used in the normalization and ``ground truth'' comparisons of future remote sensing missions that have depth and lateral spatial resolutions comparable to the GS. As expected at such spatial scales the concentration of major elements tends to be similar across lander sites. However, the compositional diversity evident at the MER sites should be utilized for remote sensing missions with fine lateral spatial resolution.

Future constraints on Martian regolith mechanical properties would help increase the precision of our estimates, since bulk density uncertainties (refer Section~\ref{BDAF}) dominate the standard errors in Table~\ref{table mass fractions}. Furthermore, as revealed by the extreme cases in Section~\ref{SMCU} and given the areal dominance of soil, improved determination of densities and areal fractions of regolith components would enhance the accuracy of our calculations much more than improved determination of elemental oxidation states. In this respect, the future availability of high resolution images and spectral data (such as the Mars Reconnaissance Orbiter data) for the different lander sites would be very useful, since it would greatly enhance the estimates of relative amounts of different regolith components over the entire GS footprint at each lander site.  In addition, improved remote sensing data may help reveal and establish the GS footprint most representative of a given lander site, even when it is not centered at the lander coordinates. Such improvements would be further enhanced by more realistic modeling of material mixing geometries \citep[e.g.,][]{Kim2006} by revealing the compositional effect of rocks in spite of their low areal abundance. Better constraints on the surface GCR flux with the Mars Science Laboratory Radiation Assessment Detector \citep[][]{Hassler2006} may help refine the GS forward modeling as well.

\begin{figure*}[htb]
\center{\includegraphics[width=20pc]{GusevSamples}}
\caption{\footnotesize Images of rocks and trenches at Gusev that were used to obtain APXS data for our study. The width of trenches and wheel tracks is $\SI{\approx20}{cm}$ and of the RAT holes is $\SI{\approx5}{cm}$. The labeled traverse maps in the background are from \citet{Arvidson2006}. True color images courtesy of NASA/JPL/Cornell as posted at \url{http://marswatch.astro.cornell.edu/pancam_instrument/true_color.html}, and other images as posted at \url{http://marswatch.astro.cornell.edu/pancam_instrument/mosaics.html}. \citet{Bell2003}, \citet{Bell2006a}, and \citet{Bell2006} describe the image processing techniques and Pancam instrumentation in detail.

The images in sol order are as follows:
(Sol 47) Approximate true color mosaic, made with Pancam's \SI{753}{nm}, \SI{535}{nm}, and \SI{432}{nm} filters, of the ``Road Cut'' trench in the ``Laguna Hollow'' area.
(Sol 60) True color image of ``Humphrey'' following abrasion with the RAT (identifier Sol060A\_P2597\_1\_True\_RAD.jpg). Humphrey is $\approx\SI{60}{cm}$ tall.
(Sol 116) Approximate true color mosaic, made with Pancam's \SI{753}{nm}, \SI{535}{nm}, and \SI{482}{nm} filters, of the ``Big Hole'' trench in typical soil.
(Sol 142) Approximate true color mosaic, made with Pancam's \SI{753}{nm}, \SI{535}{nm}, and \SI{482}{nm} filters, of the ``Boroughs'' trench in typical soil. Boroughs is $\approx\SI{11}{cm}$ deep.
(Sol 342) True color image of ``Wishstone'' (identifier Sol342A\_P2571\_1\_True\_RAD.jpg). Note wheel tracks for scale.
}
\label{fig SpiritContext}
\end{figure*}


\begin{figure*}
\noindent\includegraphics[width=20pc]{MeridianiSamples}
\caption{\footnotesize Images of trenches at Meridiani that were used to obtain APXS data for our study. The width of each trench is $\SI{\approx20}{cm}$. The labeled traverse map in the background courtesy of NASA/JPL/OSU as posted at \url{http://marsrovers.nasa.gov/mission/tm-opportunity/images/MERB_481_br2.jpg}. Arrows from the images to the traverse indicate the approximate locality of the trenches. \citet{Bell2003}, \citet{Bell2006a}, and \citet{Bell2006} describe the image processing techniques and Pancam instrumentation in detail.

(a) Approximate true color mosaic of the sol 81 ``BeagleBurrow\_Trench'' area. Two imprints from Opportunity's M\"ossbauer spectrometer are visible. The trench is $\SI{\approx11}{cm}$ deep. Image taken with the Pancam's \SI{430}{nm}, \SI{530}{nm}, and \SI{750}{nm} filters on sol 81 courtesy of NASA/JPL-Caltech as posted at \url{http://marsrovers.jpl.nasa.gov/gallery/press/opportunity/20040428a/09-SM-02-TrenchPan-B093R1.jpg}
(b)Part of the sol 368 ``Left\_of\_peanut\_TrenchFloor'' area in true color showing a M\"ossbauer imprint within. Image courtesy of NASA/JPL/Cornell as posted at \url{http://marswatch.astro.cornell.edu/pancam_instrument/images/True/Sol373B_P2552_1_True_RAD.jpg}
}
\label{fig OpportunityContext}
\end{figure*}


\begin{figure*}
\noindent\includegraphics[width=39pc]{AllElements2006JE002859}
\caption{The variation in chemistry at the different landing sites is illustrated by plotting the ratio of the representative in situ compositions at each site to that at the Pathfinder site (data from Table~\ref{table mass fractions}). Deviation of the ratio from unity illustrates the difference of the in situ value from the Pathfinder value. Error bars represent the standard error, $1s_{\ce{m}}$, of the ratio as obtained by substituting the ratio for $f$ in Equation~\ref{e error propagation}. The text related to Equation~\ref{e error propagation} identifies the uncertainties that have been included in the error propagation. Note that Cr, Mn, Na, and P data are unavailable for Viking 1. SpiritHW uses an area-weighted mean of Humphrey and Wishstone compositions while SpiritH only uses the composition of Humphrey.}
\label{fig key elements}
\end{figure*}


\begin{figure*}
\noindent\includegraphics[width=39pc]{GSElements2006JE002859}
\caption{Representative in situ compositions compared with corresponding GS estimates for the four elements determined with the GS \citep[][]{Boynton2007}. Comparison as the ratio of the calculated in situ value to the GS estimate at each lander site. Deviation of the ratio from unity illustrates the difference of the in situ value from the GS estimate. Error bars represent the standard error $(1s_{\ce{m}})$ of the ratio as obtained by substituting the ratio for $f$ in Equation~\ref{e error propagation}. Note that the Si ratio for Pathfinder is unity by virtue of the GS normalization. SpiritHW uses an area-weighted mean of Humphrey and Wishstone compositions while SpiritH only  uses the composition of Humphrey.}
\label{fig GS elements}
\end{figure*}


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\begin{acknowledgments}

We thank the Mars Odyssey Project for support. Robert Sullivan was instrumental in preventing a simplistic treatment of regolith properties at the two MER sites and in ensuring that we select the appropriate surface samples for comparison with the GS datasets. We are grateful for the time and effort he spent in aiding us. James Dohm advised us on geological properties, while Phillip R. Christensen provided Viking IRTM data. We also thank Jeff Taylor for identifying important caveats to the compositional results presented here and helping us better articulate those limitations. Our reviewers helped refine our overall presentation in general, and discussion of uncertainties in particular.

\end{acknowledgments}
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%	TABLES
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\begin{table}
\caption{
Reported and computed areal fractions, as \SI{}{\%}, of rock ($A_{\ce{r}}$) and soil ($A_{\ce{s}}$) components at the lander sites. The relative fractions of ``drift'' ($A_{\ce{d}}$) and ``soil''($A_{\ce{s}}$) values as reported by the Viking mission \citep[][]{Moore1991} are used in mean soil density estimation. Areal fraction of rocks is estimated with IRTM data \citep[][]{Christensen1986} as the mean within \ang{15} arc radius of each lander site. Standard error of the rock areal fractions ($s_{\ce{m}}$) is from the work by \citet[][]{Christensen1986}. Note that the calculated composition at Opportunity assumes a rock-free surface as described in Section~\ref{Opp}.
}
\label{table areal fractions}
\begin{flushleft}
\begin{tabular}{llccc}
\tableline
Lander/Rover & $s_{\ce{m}}$ & Rock areal fraction & \multicolumn{2}{c}{Soil areal fraction}\\
& & & $A_{\ce{d}}$ & $A_{\ce{s}}$ \\
\tableline
Opportunity &  & \num{9} & \multicolumn{2}{c}{\num{91}}\\
Pathfinder &  & \num{13} & \multicolumn{2}{c}{\num{87}}\\
Spirit & \num{5} & \num{6} & \multicolumn{2}{c}{\num{94}} \\
Viking 1 &  & \num{13} & $\num{18.}\pm\num{4.}$ & $\num{66.}\pm\num{13.}$ \\
Viking 2 &  & \num{16} & $\num{30.}\pm\num{6.}$ & $\num{54.}\pm\num{11.}$ \\
\tableline
\end{tabular}
\end{flushleft}
\end{table}


\begin{table}
\caption{
GS-derived $w(\ce{H})$ at each lander site, expressed as the stoichiometrically equivalent $w(\ce{H2O})$ ($C_{\ce{GS}}$), and corresponding standard error ($s_{\ce{m}}$). Values stated as \%. Proximity to the polar H regions led to the exclusion of Viking 2.
}
\label{table H2O}
\begin{flushleft}
\begin{tabular}{llc}
\tableline
Lander/Rover & $C_{\ce{GS}}$ & $s_{\ce{m}}$ \\
\tableline
Opportunity & \num{5.4} & \num{0.6}\\
Pathfinder & \num{2.9} & \num{0.4}\\
Spirit & \num{7.4} & \num{0.6}\\
Viking 1 & \num{2.5} & \num{0.4}\\
\tableline
\end{tabular}
\end{flushleft}
\end{table}

\begin{table}
\caption{Identification information on APXS samples at both MER locations extracted from the work of \citet[][]{Gellert2006} for Spirit and \url{ http://pds-geosciences.wustl.edu/geodata/mer1_mer2-m-apxs-5-oxide-sci-v1/merap_2xxx/data/apxs_oxides_mer1.csv} for Opportunity. ``R'' refers to abrasion with the RAT, while ``T'' refers to excavation with the wheels. Sols are subsequent to landing at each site.
}
\label{table sols}
\begin{flushleft}
\begin{tabular}{lllll}
\tableline
Rover & Sol & Sample type &	Integration time (h)	& Target name \\
\tableline
Opportunity	&	81	& T soil          &	\num{4.0}	&	BeagleBurrow\_Trench\\
Opportunity	&	368	& T soil	        &	\num{3.0}	&	Left\_of\_peanut\_TrenchFloor\\
Spirit			&	49	& T hollows soil	&	\num{6.5}	&	Road Cut\_Floor3\_\\
Spirit			&	50	& T hollows soil	&	\num{7.6}	&	Road Cut\_WallMIonl\\
Spirit			&	60	&	R unaltered rock&	\num{5.0}	&	Humphrey\_RAT2\\
Spirit			&	114	&	T typical soil	&	\num{9.9}	& Bighole\_RS2\\
Spirit			&	115 & T typical soil	&	\num{4.0}	&	Bighole\_Trico\\
Spirit			&	140 & T typical soil	&	\num{8.3}	&	Boroughs\_Mills\_bottom\\
Spirit			&	141 & T typical soil	&	\num{4.0}	&	Boroughs\_Hellskitchen\_
side\\
Spirit			&	335	&	R altered rock	&	\num{3.3}	& Wishstone\_chisel\_RAT\\
\tableline
\end{tabular}
\end{flushleft}
\end{table}

\begin{table}
\caption{
Computed representative in situ mass fractions (as \%) at four lander sites. The net uncertainty of each mass fraction is given parenthetically to one standard error ($1s_{\ce{m}}$). Refer to Equation~\ref{e error propagation} and related text for details on the uncertainty estimates. SpiritHW uses an area-weighted mean of Humphrey and Wishstone compositions while SpiritH only uses the composition of Humphrey.
}
\label{table mass fractions}
\begin{flushleft}
\begin{tabular}{llllll}
\tableline
Element & Opportunity & Pathfinder & SpiritH	&	SpiritHW & Viking 1\\
\tableline
Al	&	\num{4.69}(\num{0.23})	&	\num{4.5}(\num{0.3})	&	\num{4.38}(\num{0.17})	&	\num{4.6}(\num{0.3})	&	\num{4.4}(\num{1.1})\\
Ca	&	\num{4.53}(\num{0.03})	&	\num{4.8}(\num{0.4})		&	\num{3.99}(\num{0.16})	&	\num{4.07}(\num{0.22})	&	\num{4.6}(\num{0.6})\\
Cl	& \num{0.466}(\num{0.006})	&	\num{0.49}(\num{0.07})	&	\num{0.71}(\num{0.08})	&	\num{0.72}(\num{0.07})	&	\num{0.71}(\num{0.17})\\
Cr	&	\num{0.303}(\num{0.007})	&	\num{0.16}(\num{0.08})	&	\num{0.26}(\num{0.02})	&	\num{0.23}(\num{0.02})	&	-\\
Fe	&	\num{13.8}(\num{0.7})		&	\num{14.6}(\num{0.9})		&	\num{12.3}(\num{0.3})	&	\num{11.8}(\num{0.3})		&	\num{13.2}(\num{1.3})\\
K		&	\num{0.336}(\num{0.016})	&	\num{0.63}(\num{0.07})	&	\num{0.25}(\num{0.02})	&	\num{0.284}(\num{0.014})	&	\num{0.22}(\num{0.13})\\
Mg	&	\num{4.18}(\num{0.13})	&	\num{4.2}(\num{0.6})		&	\num{5.36}(\num{0.18})	&	\num{5.02}(\num{0.24})	&	\num{3.3}(\num{0.6})\\
Mn	&	\num{0.273}(\num{0.011})	&	\num{0.40}(\num{0.08})	&	\num{0.252}(\num{0.008})	&	\num{0.238}(\num{0.006})	&	-\\
Na	&	\num{1.58}(\num{0.05})	&	\num{1.0}(\num{0.3})		&	\num{1.70}(\num{0.03})	&	\num{1.88}(\num{0.16})	&	-\\
P		&	\num{0.32}(\num{0.02})	&	\num{0.42}(\num{0.07})	&	\num{0.295}(\num{0.010})	&	\num{0.49}(\num{0.15})	&	-\\
S		&	\num{2.1}(\num{0.4})	&	\num{2.2}(\num{0.5})		&	\num{3.4}(\num{0.5})	&	\num{3.4}(\num{0.5})		&	\num{2.7}(\num{0.7})\\
Si	&	\num{20.2}(\num{0.6})		&	\num{21.0}(\num{1.9})		&	\num{18.5}(\num{0.5})	&	\num{18.4}(\num{0.5})		&	\num{22.6}(\num{1.6})\\
Ti	&	\num{0.677}(\num{0.017})	&	\num{0.55}(\num{0.10})	&	\num{0.469}(\num{0.016})	&	\num{0.59}(\num{0.08})	&	\num{0.41}(\num{0.05})\\
\tableline
\end{tabular}
\end{flushleft}
\end{table}

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