Density variations within the south polar layered deposits of Mars

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117,, doi: /2011je003937, 2012 Density variations within the south polar layered deposits of Mars Junlun Li, 1 Jeffrey C. Andrews-Hanna, 1,2 Youshun Sun, 1 Roger J. Phillips, 3 Jeffrey J. Plaut, 4 and Maria T. Zuber 1 Received 19 August 2011; revised 1 March 2012; accepted 5 March 2012; published 20 April [1] The south polar layered deposits (SPLD) constitute the largest known reservoir of water on Mars. Previous studies solved for the best fit uniform density of the deposits using a forward approach. Here we invert for the lateral density variations in the layered deposit using gravity data from radio tracking of Mars Reconnaissance Orbiter, topography from MOLA on board Mars Global Surveyor, and radar sounding data from MARSIS on board Mars Express. We use the gravity anomalies outside the SPLD to construct a Wiener filter, which is applied to the gravitational signature of the SPLD to remove the short-wavelength anomalies over the SPLD that are spectrally consistent with an origin in the crust or mantle. We then use a constrained inversion for the vertically averaged density within the SPLD as a function of position. The results suggest significant density variations within the SPLD. An inverse relationship between the density and thickness of the SPLD suggests that thicker portions of the cap contain less dust. Alternatively, the Dorsa Argentea Formation may extend beneath the SPLD and result in the observed high gravity anomaly in the marginal area of the SPLD. We find these conclusions to be robust against the choice of inversion constraint and perturbations to the applied filter. A synthetic test is also performed to verify the recoverability of the density variation in our approach. Citation: Li, J., J. C. Andrews-Hanna, Y. Sun, R. J. Phillips, J. J. Plaut, and M. T. Zuber (2012), Density variations within the south polar layered deposits of Mars, J. Geophys. Res., 117,, doi: /2011je Introduction [2] Today, the surface and near-surface water on Mars is primarily in the form of ice. It is estimated that volume of water in the polar caps and polar layered deposits is equivalent to a global layer about 20 m in depth [Smith et al., 2001; Plaut et al., 2007; Selvans et al., 2010]. Also, a vast quantity of water has been found within the topmost 1 m of the Martian surface at high latitudes [Boynton et al., 2002; Feldman et al., 2002, 2004; Mitrofanov et al., 2002], and impact crater morphologies suggest more water existed within the upper few kilometers of the crust over large areas of the planet in the past [Barlow and Perez, 2003]. [3] Both the south and north polar layered deposits of Mars are geologically young (<100 Ma) compared to other surface features. They have been a topic of great interest since they were first observed, as they could contain the record of relatively recent climate change [Carr, 2006]. The 1 Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA. 2 Now at Department of Geophysics and Center for Space Resources, Colorado School of Mines, Golden, Colorado, USA. 3 Planetary Science Directorate, Southwest Research Institute, Boulder, Colorado, USA. 4 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, USA. Copyright 2012 by the American Geophysical Union /12/2011JE south polar layered deposits (SPLD) are highly asymmetric with a volume estimated to be km 3 and the thickest part about 3.7 km [Plaut et al., 2007]. Many previous researchers suggested that the SPLD is primarily composed of water-ice, with central, thin CO 2 cover [Byrne and Ingersoll, 2003; Titus et al., 2003; Bibring et al., 2004; Plaut et al., 2007], though more recent work has uncovered evidence for massive CO 2 ice deposits within some portions of the SPLD [Phillips et al., 2011]. By examining the stratigraphy exposed on scarp walls from THEMIS (Thermal Emission Imaging System on board Mars Odyssey) visible images, Milkovich and Plaut [2008] inferred that the SPLD is composed of three main sequences of deposit layers, corresponding to three different geologic periods. Wieczorek [2008] used gravity data from radio tracking of Mars Global Surveyor and Mars Odyssey [Konopliv et al., 2006], and estimated the best fit constant density to be 1271 kg/m 3. He also estimated the concentration of dust content to be between 14 and 28% by volume if they were completely free of solid CO 2, depending on the assumed dust density. Alternatively, 55% CO 2 ice could be sequestrated by volume if there was no dust deposited. The effects of the CO 2 ice will be discussed in detail later in our paper. Using the data from radio tracking of Mars Reconnaissance Orbiter, Zuber et al. [2007a] assumed that the density across the SPLD is constant and found the best fit density to be 1220 kg/m 3. They also suggested that this density can be explained by water ice mixed with 15% dust. 1 of 13

2 Figure 1. (a) Surface topography (m); (b) basal topography (m) from MARSIS [Plaut et al., 2007]; (c) gravity anomaly (mgal) over the SPLD (red outline in Figure 1b). The radial free air anomaly is calculated at the reference radius R = 3396 km. The figures are oriented with 0 longitude located at the top, and the study area is about 2400 km across. [4] In this work, we inverted for the density variation within the SPLD with constraints on the density smoothness and deviation. We first use the gravity anomalies outside the SPLD to construct a Wiener filter, which is applied to the gravitational signature of the SPLD to remove the shortwavelength anomalies over the SPLD that are spectrally consistent with an origin in the crust or mantle. We also perform a synthetic test to verify the recoverability of our approach. The robustness of our inversion approach is extensively examined by setting the parameters in the inversion with different values and comparing the corresponding results. The influence of the silicate dust on the estimated permittivity of the SPLD on the inversion results is also studied. An inverse relationship between the density and thickness of the SPLD suggests that thicker portions of the cap contain less dust. 2. Methods and Results 2.1. Data Preparation [5] The gravity data from Mars Reconnaissance Orbiter tracking is in the form of spherical harmonics, extending up to degree 100, with a corresponding spatial resolution of 107 km (mromgm0020g) [Zuber et al., 2007b; Konopliv et al., 2011]. However, because the highest degrees are noisy, degrees beyond 85 are cosine tapered to reduce the noise, yielding an effective (full wavelength) spatial resolution of 125 km. Additionally, the spherical harmonic degree 2 order 0 associated with flattening is removed, as well as the rest of degrees 2 and 3, which are primarily dominated by the effects of Tharsis loading. [6] The topography and gravity anomalies for the south polar layered deposits are shown in Figure 1. Here we have removed very long wavelength residual gravity anomalies associated with the global signature of the Tharsis province. From Figures 1a and 1c we see a strong correlation between topographic features and gravity anomalies outside the SPLD. We first remove these correlated anomalies before focusing on the anomalies within the SPLD Removing Gravity Anomalies Arising From the Topography [7] The gravity anomalies shown in Figure 1 could arise from a combination of origins: 1) vertical relief along the density interface between crust and mantle (i.e., the Moho); 2) vertical relief along the density interface between the crust and the atmosphere; 3) lateral density variations arising from crustal and mantle heterogeneity; 4) vertical relief along the interface between the SPLD and the crust; 5) vertical relief along the density interface between the SPLD and the atmosphere; 6) lateral density variations arising from heterogeneity within the SPLD in terms of dust content, contrasting ice compositions and porosity; and 7) errors in the measured gravity field. In this paper, we attempt to isolate the effects of density variations within the SPLD by removing the effects of the other contributions to the gravity field through either direct observation or carefully chosen filtering of the data. [8] First, we remove anomalies originating from the Moho and crust topography. The mean crustal thickness is assumed to be 56 km [Zuber et al., 2007a]. It is not possible to solve uniquely for both the relief along the Moho and the density variations within the SPLD using the observed gravity and topography alone. To circumvent this issue, we use the observed gravity and topography outside of the SPLD to model the Moho relief there, and calculate the best fit degree of compensation. The result is then applied to the entire region both inside and outside of the SPLD in order to calculate the Moho relief. The best fit degree of compensation outside the SPLD in this region is found to be 91% [Zuber et al., 2007a]. The Moho compensation depth h c can be calculated from: r c gh l 0:91 ¼ ðr m r c Þgh c h c ¼ r ch l 0:91 r m r c where r c is the density of the crust (2900 kg/m 3 ); r m is the density of the mantle (3500 kg/m 3 ); h l is the topography of the crust (SPLD excluded). Assuming Airy compensation and applying the observed degree of compensation outside ð1þ 2 of 13

3 would not improve the model results. To compute the gravity anomalies originating from surface topography and Moho relief, the forward modeling method of Parker [1973] in Fourier space is applied:!!n#1!!!k! "!! # X!! F ½rc hln ðx; yþ F ½g ( ¼ #2pG exp #! k!z0 n! n! #ðrm # rc Þhcn ðx; yþ( Figure 2. Radial gravity anomalies associated with the crust and mantle heterogeneity, as well as the SPLD (mgal). of the SPLD to the MARSIS-derived crustal topography beneath the SPLD, we can infer the Moho topography both beneath and around the SPLD region. Zuber et al. [2007a] addressed the uncertainty in Moho topography, and concluded that the mean SPLD density is not sensitive to this error source. The lithosphere flexure due to the polar load should have a negligible effect on the density determination: Zuber et al. [2007a] investigated the possible range of the basal unit deformation under SPLD and concluded that flexure would have a small effect on the mean density (an increase in mean density of 70 kg/m3 if there is a 100 m flexural depression beneath the SPLD); also, Phillips et al. [2008] concluded that the flexure under the NPLD is small (&100 m), possibly due to a very thick lithosphere (>300 km). Wieczorek [2008] also reached a similar conclusion that the mean density of the SPLD is not sensitive to the elastic thickness of the lithosphere. [9] To forward-model the gravity anomalies arising from the different sources, we performed analyses in both Cartesian and Fourier space. The spatial scale of the SPLD is small relative to the planet and the use of global gravity and topography combined with spherical harmonic localization g ðx0 ; y0 ; z0 ; x; yþ ¼ # U ðx0 ; y0 ; z0 Þ ¼# z0 z0 ZZZ ð2þ where (x,y) is the horizontal location; G is the universal the densities of the crust gravitational constant; rc and!!rm are!! pffiffiffi2ffiffiffiffiffiffiffiffiffiffi2ffiffi and mantle, respectively;! k! ¼ kx þ ky is the amplitude of the horizontal wave number; hl and hc are the topography at the top of the crust (without the SPLD) and Moho relief, respectively; and F denotes the Fourier transform. The topography at the top of the crust (hl) is taken from MOLA topography outside of the SPLD [Smith et al., 2001], and from the relief of the base of the SPLD derived from MARSIS radar [Plaut et al., 2007]. The relief along the Moho is calculated from the topography using best fit degree of compensation as described above. [10] By removing the gravity anomalies arising from these two sources, we obtain residual gravity anomalies. The residual anomalies, shown in Figure 2, are dominated by the SPLD, but they also include contributions from density heterogeneities within the crust and mantle, from departures from the best fit degree of compensation, and from errors in the measured gravity field Inversion for SPLD Density Anomalies [11] To invert for the density anomaly distribution, we first divide the SPLD region into many rectangular prisms. After the contribution from each individual prism is isolated, sophisticated constraints on the inversion can be applied. The gravity originating from each prism is first calculated as a function of its location and the observation point, and then the modeled gravity at any particular observation point (x0, y0, z0) is simply the summation of the contributions from all prisms, each of which is 30 km by 30 km in our study. A sensitivity matrix, which characterizes the contribution from all the prisms to all the observation points by mapping the density model space to the gravity data space, is constructed. For example, an entry in the sensitivity matrix characterizing how the gravity observation at a point (x0, y0, z0) is affected by a prism located at (x, y) is calculated with the following equation [e.g., Rama Rao et al., 1999]: Gr qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dxdydz 2 ðx # x0 Þ þ ðy # y0 Þ2 þ ðz # z0 Þ > % qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi& < ð x # x Þ ð y # y Þ 0 0 B C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia þ ðy # y0 Þ ln x # x0 þ ðx # x0 Þ2 þ ðy # y0 Þ2 þ ðz # z0 Þ2 ¼ #Gr ðz # z0 Þatan@ > : ðz # z0 Þ ðx # x0 Þ2 þ ðy # y0 Þ2 þ ðz # z0 Þ2 9 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi&=!!!!x2!y2!z þ ðx # x0 Þ ln y # y0 þ ðx # x0 Þ þ ðy # y0 Þ þ ðz # z0 Þ!!! ; x1 y1 z1 % 3 of 13 ð3þ

4 Figure 3. Sensitivity kernel of the inversion for an observation point above the SPLD. The area size shown in this figure is the same as in Figure 1. The full width at half maximum (FWHM) is about 70 km. where the prism is bound by x 1,x 2,y 1,y 2,z 1,z 2. Each row in the sensitivity matrix, also called a sensitivity kernel, characterizes how the gravity at an observation point is affected by all the prisms. Figure 3 shows an example of the kernel. We can see that the gravity anomaly is most sensitive to the mass beneath it, and the sensitivity decreases rapidly to points that are displaced laterally from the vertical projection of the observation point. The calculation indicates that for an observation point about 4 km above the SPLD the sensitivity has decreased to around 1/50 of the maximum value at a horizontal distance less than 100 km away from the source of the anomaly. Therefore, the gravity anomalies that originate from the SPLD have localized origins horizontally. [12] The inversion will be affected by errors in the observed gravity data, errors in the basal topography of the SPLD, and errors in the modeled Moho topography, as well as density anomalies within the crust or mantle. Therefore, we need to implement smoothness and deviation constraints in our inversion. The inverse problem can now be written by minimizing the following expression: min kam dkþ a 2 klmkþ b 2 km m 0 k : ð4þ The first term above is the L-2 norm between the synthetic and observed data, where A is the sensitivity matrix calculated by equation (3), m is the model density distribution within SPLD for which we invert, and d is the gravity anomaly observation. The second term is the model smoothing term, where a 2 is the weighting factor and L is a five-point Laplacian operator matrix, which calculates the second derivative using four neighboring points: left-center, right-center, up-center, and down-center. The second term penalizes the density roughness of neighboring points, leading to a smoother model and more realistic representation of the density variations within the SPLD than would otherwise result. The last term controls the deviation of our modeled density from a reference model m 0 of uniform density, where b 2 is a weighting term. Here m, d and m 0 are vectors. The constant density model of 1200 kg/m 3 [Zuber et al., 2007a] for SPLD is chosen as our reference model. We first choose a 2 = and b 2 = for the inversion here, and the influence of the parameter choice on the inversion result will be discussed later in the paper. In least squares sense, equation (4) can be rewritten: A T A þ a 2 L T L þ b 2 I m ¼ A T d þ b 2 I m 0 ; ð5þ where I is the identity matrix. We then invert for the density anomaly using equation (5). The residual gravity anomalies after correcting for the topography along the crust and Moho consist of two parts: gravity anomalies from the SPLD and gravity anomalies from crust and mantle heterogeneities. Neglecting the effects of crustal and mantle heterogeneity, the results of this inversion are shown in Figure 4. In Figure 4a we see that the variation of density in SPLD is quite large, ranging from about 500 to about 2400 kg/m 3. The high and low extremes in this density distribution are unlikely for a deposit composed primarily of water ice, and would require portions of the deposits to have columnaveraged porosities approaching 40% or to be composed predominantly of rock, respectively. [13] This large variation in density results in part from the neglect of the contribution of crust and mantle heterogeneities to the residual gravity anomalies. Figure 4b shows the residual gravity anomalies after the hypothesized contribution from the SPLD has been removed. In Figure 4b we Figure 4. Inverted density distribution without removing anomalies arising from crust and mantle heterogeneities. (a) Inverted density distribution of the SPLD (kg/m 3 ); (b) residual gravity anomalies after the SPLD removed (mgal); (c) correlation between the SPLD thickness and density. 4 of 13

5 find that residual gravity anomalies within the SPLD are over-removed so that the anomalies originating from crust and mantle heterogeneities or errors in the measured gravity field are inconsistent in magnitude inside and outside the SPLD. Assuming that the gravity anomalies from the crust and mantle or from errors in the data exist and are similar in magnitude throughout this area, this suggests that gravity anomalies arising from non-spld sources are considered as anomalies arising from within the SPLD, and are mistakenly used to invert for the density variations of the SPLD. In other words, gravity anomalies that actually result from heterogeneity in the crust and mantle are modeled as arising from density variations within the SPLD. The non-spld anomalies could bias the density distribution from our inversion. Figure 4c shows the correlation between the SPLD thickness and SPLD density. The correlation coefficient is quite small ( ), suggesting that the modeled density variations do not correlate with the observed properties of the SPLD though this is in part a result of the large scatter in densities discussed above Removing Anomalies Originating From Crust and Mantle Heterogeneities [14] As discussed in the previous section, the gravity anomalies over the SPLD may originate in part from density variations within the crust and mantle, departures from the best fit degree of compensation, or errors in the gravity data, and we must remove these contributions before the true SPLD density can be recovered. Here we use the gravity residuals outside the SPLD to construct a Wiener filter to characterize the spectral nature of the residual gravity anomalies outside the SPLD that likely have a source in the crust or mantle. We adapted the approach of Pawlowski and Hansen [1990], and briefly describe our procedures as follows. [15] There are two assumptions in the processing: 1) the spectral features of anomalies originating from the SPLD are different from those originating from the crust and mantle; and 2) the crust and mantle heterogeneities are consistent and uniform across the region of study, inside and outside the SPLD. The Wiener s optimum filter theory in the application of gravity anomaly separation is to find a filter that minimizes the mean square error between the desired anomaly and actual output anomaly. The optimal Wiener filter is found by: minimize < jg s ðx; yþ gx; ð yþ hðx; yþj 2 > ð6þ where g s is the desired anomaly (in our case the gravity anomalies from the crust and mantle heterogeneities), g is the gravity observation, (the combined anomalies from crust and mantle heterogeneities as well as the SPLD), and h is the optimum Wiener filter. The * here denotes the spatial convolution. The angular bracket denotes the mathematical expectation value of the contained function. [16] The optimum filter h that satisfies equation (6) can be rewritten in the wave number domain as: Hðu; vþ ¼ < G sðu; vþg* ðu; vþ > < Gu; ð vþg* ðu; vþ > ð7þ where G(u,v) is the Fourier transform of g(x,y), G s (u,v) is the Fourier transform of g s (x,y), and H(u,v) is the Fourier transform of h(x,y). The asterisk denotes the complex conjugate. Assuming that the gravity anomalies from the SPLD are not correlated with the anomalies arising from crustal and mantle heterogeneities, and the gravity field is stationary (the spectral features remain unchanged from one area to another) and ergodic (in spite of the fact that we only have one measurement of the gravity field, we can estimate its mathematical expectation), equation (7) can be simplified as: j Hðu; vþ G sðu; vþj 2 jgu; ð vþj 2 : ð8þ To stabilize the signal separation, we further assume the anomaly is radially isotropic, and now the Wiener filter is written as: j Hðu; vþ G sðf Þj 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; where f ¼ u 2 2 þ v 2 : ð9þ jgðfþj To obtain G s, we assume crustal and mantle heterogeneity exist similarly across the whole region both beneath and around the SPLD. Under this assumption, we use the gravity anomalies around the SPLD to construct G s. Therefore, the Wiener optimum filter is calculated in terms of gravity anomalies surrounding the SPLD (G surrouding ), and the combined gravity anomalies from both the crust and mantle heterogeneities and the SPLD (G). Additionally, an area correction factor A c, which accounts for the area ratio of the surrounding area and the whole area of interest, is incorporated to correct the amplitude of the filter. The filter then can be expressed as: Hðu; vþ G surroundingðf Þ 2 A c jgðfþj 2 : ð10þ [17] Before applying this to the data, we first use a synthetic test to demonstrate the necessity of regularization and filtering. A model SPLD with an average density of 1200 kg/m 3 is used to generate the synthetic gravity. The topography and basal relief here is the same as the real SPLD. Within this model SPLD, a high-density anomaly block of 1450 kg/m 3 is surrounded by materials with a density of 1100 kg/m 3 (Figure 5a). Its gravity signal is shown in Figure 5b. A radially isotropic background noise (Figure 5c) with amplitude about 50 mgal and correlation length close to the observed noise originating from the crustal and mantle heterogeneities is added to generate a synthetic gravity model (Figure 5d). Comparing Figure 5d with Figure 4a, we consider that this synthetic test has similar noise amplitude and pattern as in the real case. [18] The inverted density model without filtering short wavelength gravity signals is shown in Figure 5e. This model is heavily biased due to the background noise, and we cannot draw any firm conclusion regarding the density distribution from the inverted model. The inverted density model using the Wiener filtered gravity signal is shown in Figure 5f. It is clear that the majority of the background noise has been successfully removed, and the reconstructed model shows a quite similar pattern of density variation as the true density model (Figure 5a). In comparison, an inversion 5 of 13

6 Figure 5. Synthetic test for regularized inversion with Wiener filtering. (a) Model SPLD with average density of 1200 kg/m 3. The density ranges from 1100 kg/m 3 to 1450 kg/m 3 ; (b) gravity signal (mgal) associated with the model SPLD; (c) background gravity noise (mgal) with a peak amplitude 50 mgal and correlation length similar to the real case; (d) noisy synthetic gravity signal (mgal) by combing Figures 5b and 5c; (e) regularized inversion without filtering first to remove short wavelength signals (kg/m 3 ); (f) regularized inversion with Wiener filtered gravity signal (kg/m 3 ). without regularization and filtering easily yields a density root mean square error (RMSE) exceeding 500 kg/m 3 ; an inversion with regularization but without filtering yields a density RMSE of 180 kg/m 3 ; an inversion with both regularization and filtering yields a density RMSE of only 100 kg/m 3. A test inversion of synthetic data representing a uniform density SPLD with the same regularization shows a density RMSE of 50 kg/m 3. This means that certain bias in density inversion is inevitable with regularization as smoothing and damping (from the mean density constraint) are enforced. Still, we need to use regularization in the inversion, or much larger RMSE in density would otherwise occur. The conclusion from the synthetic test is that using a regularized inversion subsequent to Wiener filtering, we can 6 of 13

7 Figure 6. Inversion results from Wiener filtered gravity anomalies. (a) Wiener filter in wave number domain; (b) Wiener filter output of gravity anomalies originating from crust and mantle heterogeneities (mgal); (c) gravity anomalies from the SPLD, after anomalies from the crust and mantle heterogeneities are removed (mgal); (d) density distribution of the SPLD (kg/m 3 ); (e) Residual anomalies (mgal); (f) correlation between the SPLD thickness and density. reliably extract the density distribution from the data with strong noise. 3. Results 3.1. Density Inversions for the SPLD [19] We then apply this methodology to the real data. The Wiener filter is shown in Figure 6a. Applying the Wiener filter to the gravity anomalies in Figure 4a, we can separate the anomalies originating from crustal and mantle heterogeneities, which are shown in Figure 6b. Here we see that, as we assumed, the anomalies due to crust and mantle heterogeneities are present throughout this entire region. The continuity of anomalies across the border of the SPLD suggests that the Wiener filter has successfully predicted anomalies from crust and mantle sources within the SPLD 7 of 13

8 that are consistent in magnitude and distribution with those outside the SPLD. Figure 6c shows the separated gravity anomalies originating from the SPLD. It should be noticed that there are some long wavelength features remaining outside the SPLD. These result because in constructing the Wiener filter, we assumed that the distribution of the crustal and mantle anomalies are uniform and radially isotropic, which is only an approximation. Therefore, the removal of anomalies from crust and mantle heterogeneities is actually an averaged removal. After the filtering, the signature of the SPLD dominates the residual anomaly, and we are then able to invert for the density distribution within the SPLD. [20] Figure 6d shows the density distribution within SPLD. The density ranges from 600 to 1350 kg/m 3, which is much narrower range than that without removal of crust and mantle anomalies (500 to 2400 kg/ m 3 ) with the same constraint parameters. The reason for the lower density in this solution (600 kg/m 3 ) will be discussed in detail later. We note that the density around the marginal area of the SPLD appears higher than the density in the central area. It should be noted that the high density anomaly is not caused by our inversion method, as the synthetic test has demonstrated the absence of such an anomaly if the density is not indeed higher. After plotting the density as a function of the SPLD thickness, we find a significant inverse correlation (P value 0.75, Figure 6f). This suggests that thicker regions of the SPLD may contain less dust, a lower fraction of CO 2 ice relative to H 2 O ice, or a greater porosity. In comparison, Grima et al. [2009] found that the marginal areas of Gemina Lingula in the NPLD have higher real and imaginary values of dielectric constant in radar sounding data, implying higher dust content in the PLD ice at the margins than in the thicker interior sections. [21] We also note that densities are higher in the vicinity of 270 longitude (toward the left of the SPLD in Figure 6), and also the gravity residuals are higher in that direction outside of the SPLD (Figures 6d and 7e). The correlation of these residuals outside the SPLD with the Dorsa Argentea formation (DAF) [Carr, 2006] suggests that the higher SPLD densities may also be explained in part by the possible extension of the DAF beneath the SPLD. Carr [2006] suggests that the DAF is actually an ancient, ice-rich polar deposit, analogous to the deposits that are at the pole today, and the numerous pits on the DAF are interpreted as evidence of basal melting of the ground ice [Head and Pratt, 2001; Ghatan et al., 2003; Ghatan and Head, 2002, 2004; Milkovich et al., 2002]. A higher silicate fraction within this more ancient deposit is likely, which would contribute to the larger gravity and density anomalies in this portion of the SPLD if the DAF was penetrated by the radar and included with the SPLD in the inversion. These studies also recognized the presence of small volcanoes in that area. Another small volcano at 73 S, 342 E with radially elongate pits on its flanks and irregular pits provides additional direct evidence of a connection between volcanism and the pits in the DAF. Volcanic loading in this area would contribute to the positive gravity anomalies beyond the typical background variability that could be erroneously interpreted by the inversion as greater density ice in the overlying SPLD. Alternatively, the DAF is young relative to the ancient Noachian crust, and therefore likely has a significant component of flexural support. If DAF spreads extensively beneath the SPLD, its thickness may vary with location, with the thickest part close to 270 longitude. Also, if this material was not penetrated by the radar, then this state of lithospheric support would result in higher residual anomalies in this area after applying the best fit degree of compensation [Zuber et al., 2007a] to calculate the Moho relief and subtracting the anomalies arising from the surface topography and Moho. [22] We also note that some parts within the SPLD are predicted to have anomalously low densities (600 kg/m 3 ). This might result from short wavelength positive gravity anomalies within the SPLD region that actually come from the SPLD itself (e.g., arising from thickness variations of the SPLD), but were treated as originating from the crust or mantle by the Wiener filter and therefore removed. In other words, the gravity signal originating from the crust and mantle is not totally separable from the signal from the SPLD (e.g., the synthetic test in Figure 5). It is also possible that the pattern of the crust and mantle heterogeneity underneath the SPLD does not match the overall radially averaged anomaly spectrum in this region (e.g., the heterogeneity may not be radially isotropic, or there exist real and unpredictable differences between the crust beneath the SPLD and that outside of it). In either of these two cases, the filter cannot work perfectly, and may result in some errors in the short wavelength gravity signal separation. Nevertheless, the remaining long wavelength gravity signal (Figure 6c) and the large-scale structure and density distribution of the SPLD (Figure 6d) should be reliable Robustness of Inversion and Filtering [23] We now address the influence on the results of the choice of inversion parameters a 2 and b 2 in equation (4). Here, a small a 2 results in weak spatial coherence in density, i.e., more roughness in the density model, while a larger a 2 results in greater smoothness in the inverted model. Similarly, a small b 2 allows large deviations from the reference constant density model, while larger b 2 results in a stronger resemblance between the inverted result and the reference model. Figure 7 shows the influence of a 2 and b 2 on several key descriptors of the model output. All the computations here are performed assuming the reference density m 0 is 1200 kg/m 3 [Zuber et al., 2007a]. The root mean square (RMS) of gravity anomaly residuals within SPLD (Figure 7a) is small when the constraints are loose (i.e., a 2 and b 2 are small). However, this choice of parameters causes the inversion to over-fit the gravity anomalies, and makes the RMS within the SPLD much smaller than the RMS outside the SPLD (13 mgal, mainly consisting of long wavelength features). Also, the inversion then yields the unphysical result of negative minimum densities (Figure 7c), and very large maximum density (Figure 7d). In general, we prefer to use a larger b 2 (10 3 ) in the inversion for the following reasons: 1) smaller b 2 results in a negative minimum density, which means the inversion is still overly affected by the remaining noise; and 2) the mean density (750 to 1000 kg/ m 3 ) for smaller b 2 models is contradictory the previous studies [Zuber et al., 2007a; Wieczorek, 2008]. We conclude that smaller b 2 inversions are not geologically realistic, as the predicted large density variations are not supported by the fact that the SPLD is mainly composed of layered deposits and the density changes from one 8 of 13

9 Figure 7. Dependence of RMS of gravity residuals, mean, minimum and maximum density and the correlation coefficient on inversion constraint parameters a 2 and b 2. The white star indicates the default values for a 2 and b 2 in the real data inversion. (a) RMS of the residual gravity anomalies (mgal); (b) mean density (kg/m 3 ) of the SPLD; the contour line 917 indicates the density of the pure water ice, and the contour line 1095 indicates the density of the water ice with 10% dust content; (c) minimum density (kg/m 3 ) of SPLD; the contour line 0 indicate zero density; (d) maximum density (kg/m 3 ) of SPLD; (e) correlation coefficient of SPLD; (f) correlation coefficient under the constraints a 2 = and b 2 = of 13

10 Figure 8. Dependence of RMS of gravity anomaly residuals, mean, minimum, and maximum density, and correlation coefficient on reference density m 0. (a) RMS of residual gravity anomalies on m 0 (mgal); (b) mean, minimum, and maximum density on m 0 (kg/m 3 ); (c) correlation coefficient on m 0 ; (d) correlation at m 0 = 1600 kg/m 3. place to another should be more moderate. The upper-half of the a 2 -b 2 descriptor space is more meaningful and therefore preferred in the inversion. [24] Figure 7e shows that the correlation between the SPLD thickness and density becomes stronger when b 2 becomes larger. Significantly, the correlation coefficient is negative over the vast majority of the parameter space considered, demonstrating that the inverse relationship between SPLD thickness and density is a robust result. We show another correlation plot in Figure 7f where a 2 equals to and b 2 equals to The correlation coefficient is 0.78, indicating a strong inverse relationship between the SPLD thickness and its density distribution. [25] We also study the dependence of the several aforementioned parameters on the initial reference density m 0. The results are shown in Figure 8. Figure 8a shows the dependence of the RMS of gravity anomaly residuals within the SPLD on m 0. There is a positive correlation between these two parameters, but the variation of RMS is not 10 of 13 significant. Figure 8b shows that the maximum, minimum and mean densities also have a positive correlation with m 0, as would be expected. We note that the variation of the mean density is small compared to that of the reference density, e.g., the mean density varies from 870 kg/m 3 to 1300 kg/m 3 when m 0 changes from 900 kg/m 3 to 1500 kg/m 3. This indicates that the mean density converges to a small range even if our reference model could have a larger range of plausible values. Figure 8c shows the correlation coefficient has a negative dependence on m 0, and it is sensitive to the value of m 0. However, negative correlation coefficients are calculated from all cases. Figure 8d shows the correlation between the SPLD thickness and its density when m 0 = 1600 kg/m 3. [26] We next study how the Wiener filter approximation for representing the gravity anomalies within the SPLD arising from crustal and mantle heterogeneity would affect the final inversion result. We perturb the amplitudes of our filter by multiplying it with a Gaussian noise function that

11 has a standard deviation of 1/10 the magnitude of the coefficients. These inversions assume a 2 = and b 2 = and m 0 = 1200 kg/m 3. We performed 1000 inversions using the perturbed filter to obtain a viable statistical sample. The RMS of the gravity anomaly residuals within the SPLD has a mean of 9.7 mgal, and a standard deviation of 0.20 mgal. The mean density has a mean value of kg/m 3, and a small standard deviation of 4.7 kg/m 3. The maximum and minimum densities have mean values of kg/m 3, and kg/m 3, respectively. The correlation coefficient also has a weak dependence on the inaccuracy of coefficients of the filter. We find a very small standard deviation of 0.03 and a mean value of This indicates that a strong inverse correlation between the SPLD thickness and density will still hold even if we cannot construct a completely accurate filter to remove the gravity anomalies originating from the crust and mantle heterogeneities. In general, we find that the dependence of the density and its correlation coefficient on the Wiener filter is not particularly strong. That is to say, even if we introduced some errors in constructing the Wiener filter by assuming radial average or assuming there is no spectral correlation between the gravity anomalies arising from the SPLD and the crust and mantle heterogeneities, the inversion result is still robust and reliable. 4. Discussions and Conclusions [27] In this paper, we investigate the density distribution within the south polar layered deposits of Mars. We find an inverse correlation between the thickness of the SPLD and its density, which suggests higher dust concentration in the marginal areas of the deposits. This phenomenon might be explained by the fact that the marginal area is located at lower latitude and thus has experienced more sunlight and a higher ambient temperature, which may result in more ice sublimation and higher residual dust. Katabatic winds could also conceivably contribute to volatile erosion in these areas. Our result agrees with that of Grima et al. [2009], which implied higher dust content at the margins of the NPLD. We also find that there exist positive gravity anomaly residuals associated with the Dorsa Argentea formation (DAF). These anomalies could arise due to either flexural support of the DAF, or magma intrusions within the crust in this region, as suggested by melting pits and small volcanoes on top of some pits. The high density of magma compared to the surrounding crust material may explain the positive gravity anomalies we have observed. We also test the influence of the inversion parameters as well as some possible inaccuracy of the filter coefficients on the inverted model, and find the conclusions to be robust. [28] We have noted that in the Ultimi Lingula region (150 E, 73 S) the inverted densities are relatively lower. When we compared the regularized inversion results with (Figure 6) and without (Figure 4) the filtering, we find that this region always has relatively lower densities. Excessive porosity of SPLD in this region or disproportional anomalies arising from the crust or mantle underneath that cannot be completely removed by filtering may be reasonable explanations, but there are also some artifacts in our processing that may lead to this anomalous result: 1) The removal of the gravity anomalies associated with crust and Moho topography by equation (2) introduced some unwanted noises, as we assumed constant density for the crust and mantle, as well as the uncertainties in the radar imaging of the basal unit and the estimated Moho relief [Zuber et al., 2007a]; 2) our filter assumes the gravity signal from the SPLD and other sources are largely spectrally separable, but for some regions this may not be the case and the filter incidentally removed the gravity feature associated with the SPLD (e.g., Ultimi Lingula has very outstanding thickness compared to other surrounding SPLD areas and therefore may raise spectrally similar gravity signals compared to the gravity anomalies arising from crust and mantle). [29] Recent work [Phillips et al., 2011] demonstrated the existence of a large CO 2 reservoir within the SPLD, associated with a reflection free zone observed by the SHARAD radar sounder. This CO 2 ice layer has a maximum thickness of 672 m, with a mean of 200 m (r co2 = 1589 kg/m 3 ). The maximum CO 2 thickness approximately coincides with the maximum SPLD thickness and the minimum density predicted by our inversion. The presence of this CO 2 ice has the effect of increasing the depth to the base of the SPLD relative to the model of Plaut et al. [2007], which assumed a higher dielectric constant appropriate for water ice. This would increase thickness of the SPLD by up to 200 m [Phillips et al., 2011], which would decrease the predicted density from the inversion model by 10%. The change in basal topography would also impact the modeled gravity anomalies arising from the crust and mantle, increasing the gravity anomalies attributed to the SPLD and the resulting density by a lesser amount. The net effect of accounting for the presence of this CO 2 ice in the inversion would be the prediction of a lower density in this region, which already lies in a predicted density low. This is counter to our expectations, as CO 2 ice has a greater density than water ice. However, a CO 2 ice thickness of 500 m within a 2500 m thick section of the SPLD would increase the density by only 100 kg/m 3 relative to a pure ice cap. By comparison, contamination of water ice with 10% dust at a density of 2500 kg/m 3 would increase the density by 160 kg/m 3, while a 10% porosity would decrease the density by 90 kg/m 3. The predicted inverse correlation of density with SPLD thickness in our results suggests that the increased dust content toward the margins of the SPLD has a greater effect on the mean density distribution than the CO 2 ice within the SHARAD reflection free zone. [30] In addition to being the largest surface reservoir of water on Mars [Zuber et al., 2007a], the south polar layered deposits provide an important record of the recent climate of Mars. The density variations and inferred variation in dust content derived from this geophysical inversion of gravity, topography, and radar data provide important information on the composition of the deposits. This added information can be used in conjunction with the stratigraphy of the deposits from both surface and subsurface mapping [Fishbaugh and Head, 2000; Carr, 2006; Seu et al., 2007; Phillips et al., 2008; Byrne, 2009] in trying to reconstruct the recent climate and volatile history of Mars. Appendix A: Influence From Permittivity Variation [31] On the basis of our inversion, the density distribution of the SPLD is consistent with a mixture of water ice and a 11 of 13

12 Figure A1. Final distributions of density, permittivity and basal relief correction of the SPLD after the sequential inversion. The three columns are the results when the permittivity of the silicate inclusion is 5, 7, and 9, respectively. The first row is the density (kg/m 3 ); the second row is the equivalent permittivity; the third row is the uplift of the basal topography of the SPLD (m) due to the permittivity correction (positive value means upward lifting). denser material that is likely to be silicate dust. The permittivity of the SPLD is affected by the dust content as well as the permittivity of the dust grains, which in turn affects the thickness of the SPLD calculated from MARSIS. In a previous study of the structure of the SPLD using MARSIS data [Plaut et al., 2007], a constant permittivity ɛ = 3 was assumed to convert the time delay in radar sounding to depth. As indicated in our results, the SPLD is not pure ice, thus a constant permittivity of ice (ɛ = 3) may not yield the correct thickness of the SPLD. Here we performed a sequential inversion to determine jointly the best fit density, permittivity, and thickness correction for the SPLD. The inversion process is as follows: [32] 1) Invert for the best fit SPLD density distribution using the methodology developed in this work; [33] 2) Calculate the equivalent permittivity for the dusty ice as a function of the spatially varying density. The permittivity is calculated using a mixing model [Phillips et al., 2008], assuming a mixture of ice with either silicate dust or air, for densities above and below that of pure water ice, respectively. [34] 3) Correct the SPLD basal topography and thickness with the updated equivalent permittivity and then correct the Moho relief; [35] 4) With the updated SPLD thickness model, repeat from 1) until the sequential inversion converges. [36] We tested three different values for the permittivity of the dust, namely 5, 7 or 9, representing a likely range for the silicate inclusions [Nunes and Phillips, 2006]. Note if the density is less than that of the pure ice (917 kg/m 3 ) porous ice with vacuum inclusion is assumed [cf. Phillips et al., 2008]. [37] Figure A1 shows the density, equivalent permittivity and basal relief uplift of the SPLD after the sequential inversion. It is found that the influence of the permittivity variation on the SPLD density is small, mainly because we assumed the crust is in partial Airy compensation (equation (1)) and the gravity anomaly arising from the basal topography has been compensated at the Moho. The uncertainty in SPLD thickness due to variation in permittivity has been previously discussed by Plaut et al. [2007], who estimated this uncertainty to be less than 10%. In this joint inversion of the gravity and radar data, with the change of the basal relief generally being less than 150 m, the full range of change in basal relief of 500 m for the ɛ dust = 9 case corresponds to a few to few 10 s of percent of the SPLD thickness. For the ɛ dust = 7 case, the mean and standard deviation of the SPLD permittivity are 3.48 and 0.27, and the RMS change in density relative to the baseline model is less than 5%. Our estimates of uncertainty are generally consistent with those of Plaut et al. [2007]. In all models, the same patterns of high and low density hold. Therefore, our previous discussions are still valid in spite of the permittivity variation in the dusty ice. [38] Acknowledgments. We thank Sue Smrekar and Shane Byrne for reviewing and giving constructive comments to improve our paper. We also thank Editor Mark Wieczorek for providing helpful comments on our paper. The work was supported by the NASA Mars Reconnaissance Orbiter Radio Science Gravity investigation. 12 of 13

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