The radial difference between the crust-mantle density interface and the surface topography (ftp://ltpftp.gsfc.nasa.gov/projects/tharsis/marscrust2/), is represented by spherical harmonic coefficients to degree 85. The largest single component of the crustal thickness is the degree one zonal, or (1, 0) term, which represents a sinusoidal component of the latitudinal structure, or pole-to-pole slope; this term accounts for more than 80% of the degree 1 variance and for the ∼3-km center-of-figure offset to the south relative to the center of mass. The remainder of the degree 1 power, in the (1, 1) terms, represents a substantial thickening of crust in the direction of the Tharsis construct (262.4°E). The degree 2 components, which represent much of the non-hydrostatic character of Tharsis [Zuber and Smith, 1997] and its antipode, are an order of magnitude smaller in variance than the degree 1 terms.
5.1. Crustal Thickness Variations
 Assuming 600 kg m−3 crust-mantle density contrast, our model crustal thickness varies from 5.8–102 km, with the smallest values near the center of Isidis and the northwestern floor of Hellas Planitia. The crustal thickness certainly cannot be less than 0. Our chosen value of 45 km mean crustal thickness results in roughly 6 km of crust at the center of Isidis. Since the relief of Isidis' 5-km-high rim has been largely obliterated in the northeast, and its floor is buried by several km of Syrtis Major flows [Frey et al., 2000], 5 km of crust might result from subsequent infill of the basin interior. The global mean thickness could be as low as 43.5 km [Zuber et al., 2000], but much thinner crust than assumed in our model seems unlikely. This places a lower bound of 6 × 109 km3 for the volume of the crust. A similar argument applies to the floor of the 10-km-deep Hellas basin, whose interior has been extensively modified [Moore and Wilhelms, 2001] and exhibits <7 km thickness in our model.
 The amplitude of crustal variations in our model scales inversely in magnitude with the Moho density contrast Δρ = 600 kg m−3. If in our model we had assumed a crustal density of 2700, a similar model with 5 km thickness in Isidis would have required only 35 km mean thickness. A crustal density of 3100 (smaller Δρ), on the other hand, requires 65 km mean thickness to match the Bouguer signal and maintain a minimum thickness of 5 km. These extremes are consistent with the value of 50 ± 12 km advocated by Wieczorek and Zuber  on the basis of a combination of several geophysical and geochemical studies.
 An equal-area histogram of crustal thickness (Figure 5) shows that the global crustal structure is resolved into two major peaks, at approximately 32 km and 58 km. The crustal dichotomy is demarcated by the local minimum in the histogram at 40 km. The heavily cratered highlands contribute the largest peak, with nearly all the crust thicker than 60 km confined to the southern hemisphere. The lowlands peak is broader than that of the uplands, owing to the quasi-sinusoidal character (i.e., degree one zonal component) of the crustal thickness distribution. While the southern highlands have a relatively uniform thickness, the northern lowlands and Arabia Terra region of the southern hemisphere collectively show a latitudinally dependent crustal thickness structure. The crust thickens southward of Lunae Planum and Sinai Planum, contributing to a wide shoulder to the histogram beyond 70 km, with the thickest crust corresponding to the Claritis and Thaumasia structures on the Tharsis rise. Alba Patera, Olympus and Ascraeus Mons are the areas in the north with the thickest crust. The relatively younger northern lowlands comprise the majority of thinner crust, with the Hellas and Argyre Basins contributing the only southern crust that is thinner than 30 km. The major impact basins collectively contribute a small histogram peak from 5 to 20 km.
 Crustal structure profiles along longitudes (Figure 6) through the dichotomy boundary show thickening beneath the highland and Tharsis provinces and thinning beneath the lowland plains. Transitions in thickness between crustal provinces in eastern Mars are abrupt, corresponding to the relatively steep topographic signature of the dichotomy boundary [Frey et al., 1998] while elsewhere the crust thickens more gradually, in a manner similar to the topographic pole-to-pole slope [Smith and Zuber, 1996]. Figure 6c shows that in both the Arabia and Elysium regions the dichotomy boundary is compensated by a crustal thickness variation, with more pronounced relief along the Moho than at the surface. In Arabia, however, the edge of the transition in Moho depth lies northward of the steepest portion of the topographic scarp of the dichotomy boundary. Similarly, the transition in mantle depth beneath Chryse and Acidalia Planitia occurs several degrees eastward of the Lunae Planum and Tempe Terra topographic rises. The associated uncompensated mass deficit produces linear, negative gravity anomalies [cf. Smith et al., 1999a; Zuber et al., 2000; Phillips et al., 2001].
Figure 6. Crustal structure showing the pole-to-pole variations along three longitudinal great circle transects. Regions at the poles and through Arsia Mons that have been modeled with local density anomalies are shown in lighter shades. Vertical exaggeration 60:1.
Download figure to PowerPoint
 Where the profile at 300°E (Figure 6b) crosses Valles Marineris over a narrow portion of Coprates Chasma, crustal thinning is resolvable at a 300 km wavelength, about 5° of latitude, although the chasm itself is somewhat narrower. The model is only partially resolved, owing to the filters applied to mitigate noise, but crustal thinning is apparently insufficient to compensate the Eos and Coprates deeps, resulting in ∼300–500-mGal negative gravity anomalies [Lemoine et al., 2001; Yuan et al., 2001].
 Hellas and Utopia show prominent crustal thinning, as with the comparable-sized lunar South Pole-Aitken basin [Neumann et al., 1996; Wieczorek and Phillips, 1998]. Hellas' crustal thickness variation is dramatic along the edges of the basin (Figure 6a). The crustal thickness is fairly uniform within 500 km of the center of both Hellas and Utopia (Figures 6a and 6b), increasing at greater distances. In contrast, the crustal thickness increases rapidly within 350 km of the center of Isidis and Argyre.
 The Tharsis province consists of greatly thickened crust, consistent with its predominantly volcanic mechanism of formation [Solomon and Head, 1981], although Figure 6 shows that the bulk of the Martian crust lies in the southern highlands. The broad Alba Patera rise overlies thickened crust, and the thickest crust of Mars lies below the Arsia Mons volcanic construct in the southern Tharsis Plateau. Tharsis is encompassed by a roughly circular bulge ∼6000 km in diameter straddling the equator and covering more than 1/6 of the planet [cf. Zuber and Smith, 1997]. This province includes Alba Patera, Tempe Terra, Lunae Planum and the Olympus Mons aureoles in the north, and the Tharsis Montes, Daedalia, Syria, Sinai, Solis, and Icarius Planae in the south. Restricting our consideration to this circular region, the volume of this province is 1.5 × 109 km3, of which 0.36 × 109 km3 was required to form the Tharsis rise beyond that of the southern highlands. This volume places an upper bound on the amount of volcanic material emplaced by Tharsis formation [cf. Phillips et al., 2001].
 Table 3 gives thicknesses at historical and selected landing sites. It is evident that the regions explored thus far, as well as the Isidis Planitia site targeted by the Beagle Lander, lie within the lowland portion of the dichotomy. The Mars Exploration Rovers landed in moderately thickened crust, while the thickest terrains remain inaccessible to landed probes because of their latitude and inadequate atmospheric density at their higher elevation.
Table 3. Selected Model Crustal Thicknesses
 Figure 7 shows the thickness model in map view. The crustal dichotomy boundary, as determined by the 40-km contour, broadly parallels the geologically mapped boundary but places much of the heavily cratered Arabia Terra within the thin crustal province, In contrast, the geologically inferred dichotomy boundary passes north of Arabia and south of the Tharsis Montes. As noted by Zuber et al. , Arabia Terra has a crustal thickness representative of the northern lowlands rather than southern highlands. The lowlands crustal province excludes the entire Tharsis region, except small portions of Mareotis Fossae, Kasei Valles and Coprates Chasma. The excursions of thin crust into the Tharsis region have a linear character that suggests tectonic control of these channeled regions [Anderson and Grimm, 1998].
 The highland provinces to the south of the 40-km contour comprise roughly five eighths of the surface area, but make up 75% of the volume of the primordial crust. South of 60°S, the cratered highlands exhibit very uniform crustal structure, with a mean of 59 km and standard deviation of 2.9 km. Thinning of crust in the 300-km-diameter Newton and Copernicus basins in the southern highlands stands out against the modest background variations, while an association of thinned crust with some of the larger, more topographically muted basins cannot be confirmed. In spite of its size and preservation, the south polar Prometheus Basin does not exhibit crustal thinning; the annular rings about the pole in our model are likely an artifact of unreliable high-degree zonal potential coefficients.
 After correction for the density of polar ices, our model reveals the underlying thin crust common to much of the northern lowlands. The crustal thickness in the lowlands, despite benign topography, varies appreciably more than the relatively uniform highland crust. The greater topographic variance likely reflects modification of the primary lowlands crust by impacts and subsequent volcanotectonic activity. Crustal anomalies in the vicinity of Korolev and Lomonosov at 290°E, 72°N and 170°E, 75°N are more extensive than the craters themselves, and these anomalies may be older and unrelated to the visible craters. The most prominent anomaly “E–NE” at 63°E, 71°N is thinner than 10 km. This polar anomaly has no topographic or geological expression and is overlain by Hesperian northern plains volcanics and tectonic lineations [Withers and Neumann, 2001; Head et al., 2002]. The prevalence of large north polar anomalies suggests that this terrain preserves ancient impacts that predate the visible record.
 The crust is thinned beneath nearly all major impact basins and some large craters, as noted in the Bouguer anomaly map. Table 4 gives the minimum model thickness for the four largest basins. The thinned crust beneath Hellas is asymmetric N–S, and its structure is elongated by about 12% in the E–W direction, while others are roughly circular. Modification via volcanic emplacement or crustal flow in a weaker lithosphere to the south might account for such asymmetry, although alternative explanations have been suggested [e.g., Schultz, 1997].
Table 4. Basin Minimum Crustal Thickness, Moho Relief, and Superisostatic Uplift in Kilometers
|Name||Centera|| ||Diameterb||Min. Thickness||Reliefc||Upliftd|
 Only a few of the highly degraded basins previously inferred from geological mapping [Schultz et al., 1982; Schultz and Frey, 1990] express a resolvable geophysical signature. For example, a circular structure in Arcadia [Schultz et al., 1982; Schultz and Frey, 1990; Fuller and Head, 2002] (190°E30°N) suggests a multiring basin. Many of these previously mapped features are near the resolution of the gravity field. High topographic resolution and inadequate sampling of gravitational power at medium wavelengths can result in spurious features in the crustal thickness model [Smith and Zuber, 2002]. Consequently, a crustal thickness anomaly may be indicative of surface topography rather than deeper variations in crustal structure, and caution is needed before ascribing deeper structure such as buried impacts to small anomalies.
5.2. Regional Mantle Depth Variations
 Figure 8 shows our preferred model of Moho depth. As a consequence of our correction for the density of the polar ice, the depth to the Moho is relatively constant at each pole, rather than deepening as in the model of Zuber et al. . The depth to mantle increases from the north pole, 38 km below an ellipsoidal datum, to the south, reaching 68 km depth. Since the average depth is assumed, the precise depth to Moho is only constrained in a relative sense; nevertheless, half of the deepening occurs at the hemispheric dichotomy between roughly 10°N/S, while the remainder follows a gradual pole-to-pole slope. The deepest mantle occurs in southern Tharsis near Syria and Solis Planum, and exceeds 80 km depth beneath the Thaumasia rise.
Figure 8. Moho relief (5 km contours), in Mercator and polar stereographic projections as in Figure 3. The degree 2 zonal component is removed. Polar masks of layered terrain show regions where a lower density is applied to compute the residual Bouguer anomaly. Similarly, a higher density is applied within the enclosing contours of the Tharsis volcanoes (not shown).
Download figure to PowerPoint
 Neither Olympus, Ascraeus, Pavonis Mons nor any of the smaller volcanos exhibit local compensation by deepening of the mantle. Their surface expressions were explicitly modeled as having denser surface composition; otherwise, they would exhibit crustal thinning. Their interiors may indeed be denser than we have assumed.
 The broad Arsia, Alba and Elysium volcanic features, or domal rises, are supported mainly by depression of the crust-mantle boundary, and therefore did not require very thick lithospheric or dynamic support, unlike the Olympus Mons edifice [McGovern et al., 2002]. On the basis of surficial distribution of tectonic features, McGovern et al. [2001, 2002] suggested that these features were formed via intrusive sills with superposed volcanics.
 The shallowest Moho lies beneath Isidis Planitia, closely followed by Hellas, Utopia, “E–NE”, and to a lesser extent Argyre and “NWO”. The cratering process on the Moon and other planets is thought to remove large quantities of crust, replaced by uplifted mantle [e.g., Wieczorek and Phillips, 1999], and in the case of lunar mascon basins, this mantle uplift is super-isostatic and persists to the present day. Surficial fill by mare also contributes to the positive lunar gravity anomalies, although this fill appears in many cases to have been minimal or absent. The extent to which impact excavation of crust occurs in martian basins and is preserved over geologic time may be quantified by the relief of Moho in our model relative to the deepest immediately surrounding mantle (Table 4). These values are taken from radially averaged profiles of surface and Moho relief, which do not take into account the topographic asymmetries of these basins nor local effects from surrounding volcanos and impacts, but nevertheless reveal appreciable differences in basin relief.
 A profile illustrating mantle super-isostatic uplift is shown in Figure 9. If the surface equipotential topography were fully compensated, disregarding possible basin fill of anomalous density, the Moho relief would mirror that of the surface, scaled by the ratio of crustal density to density contrast at the crust-mantle interface, or 2900/600. An example is the Hellas basin, whose Moho relief is very nearly in isostatic equilibrium. The other three basins, most notably Isidis, have varying degrees of super-isostatic loading or Moho uplift.
Figure 9. Profiles of surface height and model depth of Moho, relative to an areoid, along a great circle through Tharsis, Argyre, Hellas, Isidis, and Utopia. The mantle relief required to locally compensate the surface topographic load, using our model density contrast, is shown by a dashed curve.
Download figure to PowerPoint
5.3. Impact Redistribution of Crust
 We now contrast the crustal morphologies of the four major basins. The degree of impact modification is greatest in the case of Hellas, whose crust has been displaced outward to form a broad rim [Zuber et al., 2000]. Utopia's rim is most clearly seen in map view as a ring of 10-km thicker crust (Figure 7), overprinted by the Isidis Basin and Elysium Mons. The mantle plugs of Hellas and Utopia are flat-topped (Figure 6). The northeastern Hellas rim appears to form part of the dichotomy boundary, while to the southwest the rim is indistinct and overprinted by the Amphitrites Patera complex. Hellas has a relatively flat floor, sloping somewhat northward, with most of its mantle relief developed toward the margins. Isidis and Argyre have relatively flat floors, but their mantle relief is paraboloidal (Figure 9). The excavation of crust by the impact is concentrated at the center of Isidis and Argyre. In contrast, the mantle relief of Utopia and Hellas develops near the basin walls; relief is modest within 500 km of their centers. Such morphology would result if the mantle was exposed by the impact over a ∼1000-km-wide region, rose isostatically, and was subsequently covered by a few km of crustal material over the basin floor.
 Utopia's preserved internal structure reflects the least amount of crustal excavation of the four major basins (Table 4). This basin has been filled by materials associated with the resurfacing of the northern lowlands [Zuber et al., 2000], resulting in the present-day mascon. The crust thickens slightly toward the center, perhaps from the deposition of many km of light, water-deposited sediments [Thomson and Head, 2001; Buczkowski and Cooke, 2004]. Its surficial rim has been eroded and embayed by the Vastitis Borealis formation north of the depression, while in the south the surrounding surface is covered by Isidis Planitia. However, >10 km of crustal thickening remains at a radius of 2150 km, analogous to features preserved in association with large impact basins on the Moon [Zuber et al., 1994]. If Utopia's annulus of thickened crust is indicative of its size, it is then the largest discernible basin on Mars (but slightly smaller than originally proposed by McGill  on the basis of geological mapping). Utopia's moderate Moho relief could have resulted from post-impact viscous flow, but such flow would have removed the steeper mantle relief near the edges of the basin. Modest relief more likely resulted because there was simply less target crust available for redistribution.
 The Hellas impact event appears to have redistributed crust radially over nearly the same diameter as Utopia, but the surface and Moho have substantially greater relief. The Hellas crustal rim is diffuse and extends radially over several hundred km. Argyre and Isidis also have diffuse annuli of thickened crust and depressed mantle. Argyre, the smallest of the four basins, has greater relief than Utopia, the largest, as do Hellas and Isidis. This paradox may be explained by the fact that, like Hellas, Argyre is located in thick southern highland crust.
 Isidis is positioned at the hemispheric dichotomy boundary, surrounded by thick crust, and is sufficiently close to Hellas and Utopia that the pre-impact crust would have been influenced by the impact events for both of these structures. The Moho relief of Isidis is nearly as great as that of Hellas despite its smaller size.
 If one were to attempt to reconstruct the original depth and diameter of excavation from the variations in crustal thickness, as was done for lunar basins by Wieczorek and Phillips , the apparent depths of excavation would be less than 50 km. The actual depths might have been substantially greater but would have penetrated the mantle, limiting the amount of crust removed. The inferred diameters of excavation would be a significant fraction of the crustal thickness diameters in Table 4 [Melosh, 1989]. Thus it would appear that their transient depth/diameter ratios were considerably less than the 0.1 value obtained by Wieczorek and Phillips  for smaller lunar basins up to the size of Orientale or Crisium. The basins in Table 4 more closely resemble the anomalously shallow Serenitatis and Imbrium basins. These basins have undergone varying and diverse surface modification due to erosion, and volcanic and sedimentary infilling. Further structural modeling, as has been done for the Moon [cf. Wieczorek and Phillips, 1999], is needed to address the relationship between the current surface expression and subsurface structure of these basins, including estimation of the depth of excavation. Well-resolved comparisons with smaller lunar basins will require refinements in the gravity field of Mars, such as to be provided by the upcoming Mars Reconnaissance Orbiter Mission.
 A further measure of impact modification is the degree to which the Moho uplift exceeds that required to compensate present-day surface relief. The Hellas Moho almost exactly mirrors the equipotential topography, scaled by the mantle density contrast, whereas the Isidis, Argyre, and Utopia mantle uplifts overcompensate the basin depressions (Figure 9). Such variations in local isostasy were also found on the Moon, and where ages were known, the Moho relief and amplitude of super-isostatic uplift was inversely correlated with age but not significantly with size [Neumann et al., 1996; Wieczorek and Phillips, 1999]. Basins situated in areas on the Moon which were thermally enhanced also tend to be near isostatic equilibrium [Wieczorek and Phillips, 2000]. Hellas, like South Pole-Aitken on the Moon, might therefore have impacted a region of thermally enhanced crust, as suggested by lithospheric studies [Zuber et al., 2000; Nimmo and Stevenson, 2001; McGovern et al., 2002]. Mars' surface evolution has been long and complex, and detailed modeling of the thermal evolution of its basins may prove fruitful.
5.4. Impact Constraints on Average Crustal Thickness
 At 400+ mGal amplitude, the Noachian-age Isidis impact basin forms one of the largest mascon gravity highs in the solar system [Lemoine et al., 2001]. The main topographic ring is 1500 km in diameter [Frey et al., 2000], and scaling laws [Melosh, 1989, equation 5.54] would suggest 60–100 km of excavation. For lunar basins up to half the size of Isidis, like Orientale and Crisium, 26–52 km of crust were removed [Wieczorek and Phillips, 1999]. Assuming that nearly all of the pre-Isidis crust was excavated, gravitational restoring forces would produce a central mantle uplift. Such a structure would not result in a mascon, as the gravitational effect of the uplift of denser material balances the mass deficit of the cavity. An additional source of excess mass is required. The present-day gravity anomaly may have resulted from prompt mantle rebound beyond the level required for local isostatic adjustment. Such a phenomenon has been clearly demonstrated on the Moon, where several of the mascon basins appear to have minimal or no infilling by mare basalts [Konopliv et al., 1998]. Whether such prompt super-isostatic rebound could survive in the early Martian thermal environment is uncertain. Alternatively, the basin mascons could result from volcanic or other loading after the crust cooled and the lithosphere thickened [Comer et al., 1985]. We consider an end-member scenario, whereby viscous relaxation occurred following impact, followed by surface loading on a thickened lithosphere, sufficient to produce the present-day plains and to match present-day gravity and topography.
 Figure 10 shows such a model. On the left we show the hypothetical state of the crust relatively soon after impact. A transient crater excavates to the depth of the mantle, followed by gravitational rebound. The mantle near the surface is covered by minimal amounts of melt sheets and ejecta fallback. The mass deficit in a central vertical column is in Airy isostatic balance with a deeper, positive mass excess, resulting in a weakly negative gravity anomaly, similar to that seen over Hellas [Smith et al., 1999a].
Figure 10. Radially averaged model of Isidis equipotential topography, Moho, and gravity before (left, dots) and after (dashes) surface loading and flexure of a 150-km-thick elastic lithosphere.
Download figure to PowerPoint
 Modification of the resultant ∼8-km-deep basin continued long after the period of heavy bombardment. Bolides undoubtedly produced many 100-km-diameter craters on this surface, as seen on the basin rim, but such large craters are completely buried by volcanic flows and by mass wasting of the Isidis rim to the northeast. The relicts of such craters seen as quasi-circular depressions in much of the lowlands are also absent [Frey et al., 2002]. The amount of material transferred to the basin interior was evidently substantial.
 The Isidis gravity high is surrounded by a ring of negative anomaly that suggests flexure due to a central load and/or thickening of the crust surrounding the impact [Neumann et al., 1996; Wieczorek and Phillips, 1999]. Using flexural graben (Nili and Amenthes Fossae), Comer et al.  inferred an effective elastic thickness Te > 125 km. The right side of Figure 10 shows a model where the mantle lithosphere has deformed under a set of concentric cylindrical loads, producing the present-day topographic surface; the flexure due to this load was used to calculate the initial surface topography and isostatic Moho relief shown on the left. We employ the approximations of Comer et al. , whereby several cylindrical loads from 100–500 km radius are superposed to calculate flexure of a thin elastic shell, accounting for fiber and membrane stress, with Young's modulus E = 1 × 1011 Pa and Poisson's ratio v = 0.25 [Johnson et al., 2000]. We use the finite-amplitude Cartesian formulation of Parker  to calculate gravity, an adequate approximation given the large uncertainties in model thickness and density.
 Volcanic loading by a 3200 kg m−3 density basalt, 6.5 km thick, results in 2.7 km flexure at a time when lithosphere has thickened to 150 km. The resulting structure, shown on the right of Figure 10, conforms to the present-day topography and closely matches the radially averaged gravity. In this model, the impact excavated nearly all of the crust at the impact center. This example requires a significant high-density surface load, equivalent to 2.1 × 106 km3 of basaltic fill, to reproduce the present-day mascon, but does not require super-isostatic mantle uplift. Similar modification of Utopia and Argyre by lesser thicknesses of fill could produce residual mascon anomalies with the shallow Moho depths listed in Table 4.
 The 45-km average crustal thickness assumed provides a plausible model for the Isidis mascon. A thinner crust would require super-isostatic uplift and/or unusually high mantle density to explain the post-impact crustal structure, while a thicker global crust would require a substantially greater volcanic load and a cooler geotherm (thicker lithosphere) to resist flexural relaxation and achieve the same mascon amplitude. We do not consider models incorporating a layered crust that, albeit more realistic, would require nearly the same amount of crustal excavation and only introduce more parameters.