Corresponding author: Walter S. Kiefer, Lunar and Planetary Institute, 3600 Bay Area Boulevard, Houston, TX 77058, USA. (firstname.lastname@example.org)
 The Marius Hills, the Moon's largest volcanic dome field, has more than 250 basaltic domes and cones in an area 200 × 250 km across. It is a major free-air gravity anomaly, 236 mGal in the north and 150 mGal in the south. In the northern half of the structure, the topography can only explain about half of the gravity anomaly, and in the south, there is virtually no topographic relief associated with the gravity anomaly. High-density material must be present at depth, most likely as mare basalt intruded into the underlying porous feldspathic highland crust. The gravity anomaly is modeled using two spherical caps. The northern cap is 160–180 km in diameter and at least 3.0 km thick. The southern cap is 100–140 km in diameter and at least 6.2 km thick. The intruded basalt may have served as the magma chambers that fed the overlying surface volcanism. Magma crystallization within these chambers provided a source of crystal-rich, high viscosity lava that fed the volcanic domes. The volume of intruded basalt is 1.6 × 104 km3. The total volcanic volume, including both intruded and extruded material, is 2.6 × 104 km3, indicating that the Marius Hills is a major volcanic center. Intrusion of hot magma may cause thermal annealing of the porous feldspathic host rock, significantly reducing the host rock porosity. This would allow a large volume of magma to be intruded into the crust with little change in overall crustal volume.
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 The Marius Hills in central Oceanus Procellarum is the largest volcanic dome complex on the Moon. The volcanic complex is about 200 × 250 km across and contains 250–300 volcanic domes and cones and 20 sinuous rilles. The individual domes and cones are up to 25 km across and 500 m high, although most are less than 15 km across [Whitford-Stark and Head, 1977; Srisutthiyakorn et al., 2010]. Figure 1 shows a representative portion of the dome field, including numerous volcanic domes and several sinuous rilles, and illustrates the typical density of volcanic structures in this region. Srisutthiyakorn et al.  showed that the concentration of domes in the Marius Hills is about 30% of the concentration of volcanic constructs in the Snake River Plains of the western United States and similar to the concentration of volcanoes in several volcanic shield fields on Venus. Volcanic activity in the Marius Hills is Upper Imbrian in age (3.2–3.8 Ga) [Wilhelms, 1987]. Mare material in adjacent parts of Oceanus Procellarum has a broad range of ages, from 3.7 Ga to possibly as young as 1.2 Ga [Hiesinger et al., 2003].
 Early geologic mapping showed that the volcanic structures in the Marius Hills can be subdivided into low domes with typical flank slopes of 2–3°, steep domes with flank slopes of 6–7°, and narrow, steep cones [McCauley, 1967a; Whitford-Stark and Head, 1977]. The differences in flank slope were interpreted as a possible indicator of differences in magma composition, with the low domes having a basaltic composition and the steeper domes possibly having a more silica-rich composition. The differences in magma composition were suggested to be a product of differentiation within the magma chamber [McCauley, 1967a, 1967b]. On the other hand, Guest and Murray  noted the absence of albedo variations across this area and favored a similar composition for all of the domes and the adjacent plains. An alternative explanation for the morphology of the steeper domes is that the erupting magma was basaltic in composition but was relatively crystal rich, resulting in a high magma viscosity. The various volcanic rilles in the area must have formed from relatively fluid lava, probably basaltic in composition [Greeley, 1971]. Volcanic sources within the Marius Hills were a major source for mare volcanism in the adjacent parts of southwest Oceanus Procellarum, particularly the stratigraphic unit termed the Hermann Formation [Whitford-Stark and Head, 1980]. Based on regional-scale topography that resembles a shield volcano (see Figure 3 below) and the concentration of small volcanic constructs, Spudis et al. [2011, 2013] interpreted the Marius Hills as an example of a lunar shield volcano.
 Spectral observations show that both the Marius Hills and the surrounding Oceanus Procellarum plains are basaltic, with modest variations in composition [Weitz and Head, 1999; Heather et al., 2003; Heather and Dunkin, 2002]. The highest spectral resolution data come from the Moon Mineralogy Mapper. The dominant spectral unit in the Marius Hills in the M3 data is interpreted as high-calcium pyroxene basalt, and a younger, less abundant unit is interpreted as more olivine-rich basalt [Besse et al., 2011]. High circular polarization ratios measured by radar at 12.6 and 70 cm imply that the Marius Hills have a rough, blocky surface texture, covered by at most a few meters of regolith [Campbell et al., 2009].
 On the entire rest of the lunar mare, only about 200 additional volcanic domes are known [Head and Gifford, 1980]. The next largest lunar volcanic dome complex is the Rümker Hills, which is about 80 km across and contains 30 low domes [Smith, 1974; Whitford-Stark and Head, 1977]. These observations emphasize the unusual nature of the Marius Hills, which makes it a worthy target for geophysical study. Previous studies of lunar gravity anomalies have generally emphasized large-scale structures, particularly the mascons [e.g., Neumann et al., 1996; Wieczorek and Phillips, 1999; Hikida and Wieczorek, 2007; Namiki et al., 2009]. However, our knowledge of the Moon's gravity and topography is now sufficient to resolve structures on the scale of the Marius Hills, and this will continue to improve with the ongoing Gravity Recovery and Interior Laboratory (GRAIL) mission. Previous studies have shown that gravity modeling can provide a useful probe of subsurface volcanic structures such as dike swarms and cumulate chambers on both the Earth [e.g., Kauahikaua et al., 2000; DeNosaquo et al., 2009; Flinders et al., 2010] and Mars [Kiefer, 2004].
 In this paper, I use high-resolution gravity and topography observations of the Marius Hills, together with recent laboratory measurements of the density and porosity of lunar rocks, to constrain the structure of the magmatic plumbing that underlies the Marius Hills. The Marius Hills gravity anomaly consists of two distinct lobes. The north lobe has a peak anomaly at spherical harmonic degree 110 of 236 mGal, of which only about half can be explained by the surface topography. The south lobe has a peak anomaly of 150 mGal but virtually no associated topography. Thus, both structures must contain significant, high-density subsurface material to explain the observed gravity anomalies. Numerical models show that the north lobe is 160–180 km across and for a density contrast of 440 kg m-3 is 3.0–3.3 km thick. The south lobe is narrower and thicker, 100–140 km across and at least 6.2 km thick, but contains a similar total volume as the northern structure. These structures are most likely dense mare basalt that has intruded as dikes and sills into the less dense feldspathic highland crust and in effect function as magma chambers that fed the surface volcanic fields. Cooling and crystallization of magma within these structures provides a source for the high viscosity, crystal-rich magma required to explain the volcanic dome morphology. Thus, the gravity observations serve to map the volcanic plumbing of the Marius Hills. The total volume of basalt inferred in this study is 2.6 × 104 km3, of which 1.6 × 104 km3 is intruded into the crust and the remainder is extruded at the surface. The total thickness of basalt in this region is significantly larger than found in remote sensing and geologic mapping studies of the region, further demonstrating gravity's importance as a subsurface mapping tool.
 Measurements of the Moon's gravity field are made by Doppler tracking of spacecraft in lunar orbit. Here, I use gravity model LP165P, which was constructed primarily using tracking of the Lunar Prospector spacecraft but also includes tracking data from Lunar Orbiter, Apollo, and Clementine [Konopliv et al., 2001]. The low altitude (10–30 km) Lunar Prospector Doppler tracking currently provides the highest-resolution gravity observations for the Moon's near side. Although the LP165P model contains spherical harmonic coefficients up to degree 165, the coefficients are thought to be well determined on the lunar nearside only up to degree 110, which corresponds to a half-wavelength spatial resolution of 50 km [Konopliv et al., 2001]. The overall gravity anomaly mapped below in Figure 2 is approximately 250 × 200 km across. This is significantly larger than the gravity field resolution, indicating that the Marius Hills anomaly is resolved by this spherical harmonic model. A recent reanalysis of the Lunar Prospector tracking data [Han et al., 2011] has extended the resolution of the gravity field somewhat (maximum spherical harmonic degree 150, half-wavelength spatial resolution 37 km) but does not fundamentally alter the results shown here. Relay satellite tracking of the three spacecraft of the SELENE mission has recently provided a dramatic improvement in our knowledge of the gravity of the Moon's far side [Matsumoto et al., 2010], but the SELENE gravity model SGM100h does not incorporate the lowest-altitude Lunar Prospector tracking and thus does not provide the best resolution on the Moon's near side. Ongoing mapping of the lunar gravity field by NASA's GRAIL mission will decrease the estimated uncertainty in the long-wavelength portion of the gravity anomaly shown in Figure 2 and also significantly improve our knowledge of the short-wavelength portion of the lunar gravity field.
 The gravity results in this paper are presented in terms of the free-air gravity anomaly [Lambeck, 1988]:
 GM = 4.9028 × 1012 m3 sec-2 and R = 1738.0 km for the LP165P gravity model [Konopliv et al., 2001], n and m are the spherical harmonic degree and order, φ is the latitude, and θ is the longitude. Pnm is the normalized associated Legendre polynomial, and Cnm and Snm are the nondimensional spherical harmonic coefficients of the gravity model. The factor (n – 1) in equation ((1)) is appropriate when the gravity anomaly is calculated on an equipotential surface. Alternatively, evaluating equation ((1)) on a spherical surface leads to a factor (n + 1). Based on a covariance analysis, the estimated uncertainty in the gravity anomalies on the near side is 30 mGal through harmonic degree 110 [Konopliv et al., 2001]. Fn is a cosine taper filter, which is applied to smoothly taper the highest harmonics used in the model to zero to minimize ringing in the gravity and topography results shown here:
 Here, nmin = 100 is the highest harmonic degree which is not tapered and nmax = 111 is the lowest harmonic degree that is excluded from the mapping. Fn = 1.0 for all degrees less than nmin. This filter is applied to all observed and model gravity profiles shown in this paper.
 Figure 2 shows that there are two distinct gravity highs in the Marius Hills region. The northern anomaly, centered at 14°N, 307.5°E, reaches a maximum amplitude of 236 mGal and encompasses most of the domes in this region (the shaded relief base map in Figure 2 shows the locations of the larger domes). The southern anomaly, centered at 8.5°N, 308.5°E, reaches a maximum amplitude of 150 mGal. There are a smaller number of volcanic domes contained within the southern gravity anomaly—the maximum dome concentration in the south is about 15% of the maximum concentration in the north [Srisutthiyakorn et al., 2010]. The two gravity anomalies are in contact at their periphery, with an amplitude of 73 mGal at the transition between the two anomalies (see profile line CC’ in Figure 7). Based on the geographic proximity of the two anomalies and the presence of volcanic domes in both locations, I interpret both the northern and the southern gravity anomalies to be part of the overall Marius Hills structure. Alternative interpretations of the gravity anomalies are considered in Section 4. Outside the two Marius Hills anomalies, the surrounding region has relatively small gravity anomalies, mostly between ±50 mGal. There is a small, strongly negative gravity anomaly to the northwest of the Marius Hills. It is apparently related to the drop off in topography in the northwest corner of the study region (Figure 3), but it will not be considered further in this study.
 The topography model used here is spherical harmonic model LRO_LTM03, which is derived from more than 1 year of measurements by the Lunar Orbiter Laser Altimeter (LOLA) on the Lunar Reconnaissance Orbiter [Smith et al., 2010]. LOLA measurements have a resolution of 20 m along track and 1–2 km cross-track and a vertical uncertainty of about 10 m. At the scale appropriate for gravity modeling, the topography is essentially perfectly known. The topography model was initially derived in a spherical reference frame, but it is expressed in Figure 3 as topography with respect to an equipotential surface by subtracting the LP165P geoid. The results in Figure 3 are shown for spherical harmonic degrees 1–110 using the same cosine taper filter as was applied to the gravity anomaly map.
 Figure 3 shows that the Marius Hills is a local topographic high that is very similar in shape and location of maximum elevation to the northern gravity anomaly. There is about 1.2 km in relief across the volcanic field. Note that because this map is limited to spherical harmonic degree 110, this elevation range refers to the broad topographic swell of the entire volcanic complex rather than to the elevation change across any particular volcanic dome. One significant difference between the gravity anomaly and topography maps is that there is no unusual topographic elevation in the region where the southern gravity anomaly is located.
3 Gravity Models
3.1 Topographic Load Models
 The observed gravity anomaly contains contributions both from the surface topography and its compensating root and from internal density anomalies that are unrelated to the topography. These internal density anomalies will be referred to as subsurface loads in this paper. To constrain the distribution and magnitude of the subsurface loads, it is therefore necessary to remove the topographic contributions from the observed gravity field. This requires having reasonable estimates of the density of the near-surface crust and of the thickness of both the crust and the elastic lithosphere; the latter two parameters are the primary controls on the compensation state of the topography.
 Figure 4a shows the observed topography for profile line AA’ across the northern part of the Marius Hills. One can think of the topography in the Marius Hills as the sum of two components. The first is a long-wavelength topographic low related to the regional topography in Oceanus Procellarum. The second is a shorter-wavelength topographic high associated with Marius Hills volcanism. Results are shown for all wavelengths up to spherical harmonic degree 110 (solid lines), for long wavelengths (spherical harmonic degrees 1–20; dashed lines), and short wavelengths (spherical harmonic degrees 21–110; dotted lines). Note that the short-wavelength topography shown here reflects regional volcanic loading in the Marius Hills (wavelengths > 100 km) and that individual volcanic domes are too narrow to show up in Figure 4a.
 Figure 4a shows that the long- and short-wavelength topography in the Marius Hills have very different behaviors. We can use this to develop an estimate of the gravity anomaly that is due to the topography by considering the likely compensation states of the two segments of the topography. The gravity anomaly for a flexurally supported topography is [Turcotte et al., 1981]
 Note that although equation ((1)) was written with the spherical harmonic coefficients in the usual nondimensional format, both equations ((3)) and ((4)) are written with gravity and topography harmonics in dimensional form for clarity. The spherical harmonic coefficients both for the gravity anomaly due to flexurally supported topography and for the observed topography, hnm, implicitly include both the cosine and the sine terms at each value of degree n and order m. Because the Turcotte et al.  flexure model includes a correction for the geoid height, the topography coefficients hnm used in equation ((3)) are expressed with respect to a spherical reference surface with the mean lunar radius of 1737.15 km. d is the mean crustal thickness (50 km) [Wieczorek et al., 2006] and ρC is the density of the crust. The numerical value of the crust density depends on the geologic context being modeled and is discussed below. Cn is a compensation factor derived for elastic flexure of a thin spherical shell, including the effects of both membrane and bending stresses. It is calculated using equations (6), (7), and (27) of Turcotte et al. . In addition to previously defined quantities, Cn also depends on the elastic shell thickness (or elastic lithosphere), Young's modulus (1011 N m-2), Poisson's ratio (0.25), and the mantle density (3300 kg m-3) [Bills and Rubincam, 1995]. Cn is 1 for isostatic compensation. As the lithospheric thickness and the amount of flexural support increase, Cn goes to zero, which makes the effect of the crustal root less important (the second term in the brackets in equation ((3))) and increases the gravity anomaly for a given amount of topography. Cn should not be confused with the spherical harmonic coefficient Cnm. Equation ((3)) uses a mass sheet approximation to relate gravity and topography, which is an acceptable approximation given the limited topographic relief (~1 km) in this region (Figure 3). For loading at a wavelength of 400 km, appropriate for the broad-scale Marius Hills load, vertical attenuation changes by only 1.5% over an altitude range of 1 km, justifying the neglect of the topography's finite amplitude in equation (3).
 Based on the partial burial of small impact craters, the mare basalts in Oceanus Procellarum are interpreted to be typically at most only a few hundred meters thick [DeHon, 1979; Hörz, 1978]. Moreover, because of the large density difference between mare basalts and feldspathic rocks, if there were large thicknesses of mare basalt beneath most of Oceanus Procellarum, one would expect large gradients in the gravity anomaly at the edge of the mare that are not observed. Because the basalt layer in most of Oceanus Procellarum is probably thin, most of the long-wavelength topography (harmonic degrees 1–20) is probably controlled by variations in the thickness of the highland crust that lies beneath the mare. The highland crust is assumed to be a porous, plagioclase-rich rock [Taylor, 2009] with a density of 2500 kg m-3 [Huang and Wieczorek, 2012; Kiefer et al., 2012a, 2012c]. Most of the Moon's highland crust formed very early in lunar history, when the elastic lithosphere was thin. Crosby and McKenzie  used a spherical shell flexure model of the southern near-side highlands and estimated an elastic lithosphere thickness of no more than 7.5 km when the highland crustal topography formed. As will be shown later in this paper, there is a significant subsurface density anomaly in the Marius Hills, and placing a lower bound on the magnitude of this subsurface structure is one of the primary objectives of this paper. In turn, this requires setting an upper bound on the gravity anomaly caused by the surface topography. Because the long-wavelength topography is negative, the associated gravity anomaly is maximized (or least negative) if the topography is isostatically compensated. This model gravity result is shown for profile line AA’ in Figure 4b as the long dashed line for harmonic degrees 2–20. Changing the compensation from isostatic to an elastic lithosphere thickness of 7.5 km (Crosby and McKenzie's  value for the southern highlands) only changes the gravity anomaly slightly, from +2 to -5 mGal at longitude 308°E.
 The short-wavelength topography, degrees 21–110, appears to be primarily due to volcanic loading within the Marius Hills. Most of this volcanism is Upper Imbrian in age (3.8–3.2 Ga) [Wilhelms, 1987]. Thus, it is considerably younger than the long-wavelength topography and was emplaced on a significantly thicker lithosphere. The best existing estimates of lithospheric thickness in the relevant age range come from studies of the tectonics of mare basins [Solomon and Head, 1980]. In particular, the elastic thickness at the time of basalt emplacement in a mare basin controls the radial distance from the basin center at which various types of tectonic features are observed. The lithospheric thickness estimate that is geographically closest to the Marius Hills is derived from the tectonics of the Imbrium basin, where extensional rilles on the periphery of the basin suggest an elastic thickness of 50–75 km in the age range 3.6–3.8 Ga [Solomon and Head, 1980]. One of the major chemical factors distinguishing various lunar basalt compositions is the overall abundance of titanium [Neal and Taylor, 1992; Papike et al., 1998], which in turn has an important effect on density [Kiefer et al., 2012a]. Lunar Prospector Gamma Ray Spectrometer observations suggest an intermediate abundance of titanium in the Marius Hills basalts (5–9% TiO2) [Prettyman et al., 2006]. Adopting a representative grain density of ~3400 kg m-3 (intermediate between low and high Ti basalts) and a typical mare basalt porosity of 7%, the bulk density of the basalt is taken to be 3150 kg m-3 [Kiefer et al., 2012a]. Most measured mare basalts have porosities between 5% and 9% [Kiefer et al., 2012b], corresponding to bulk densities between 3090 and 3230 kg m-3. The maximum topographic contribution to the gravity is achieved if the topography is uncompensated or fully supported by elastic flexure (Cn = 0 for all n in equation ((3))). The short dashed line in Figure 4b shows the resulting model gravity for degrees 21–110 along profile line AA’. However, the result is only weakly sensitive to the lithospheric thickness; changing the elastic thickness to 50 km only reduces the peak gravity anomaly from 147 to 144 mGal. Choosing to place the transition between isostatically compensated long-wavelength topography and flexurally compensated short-wavelength topography between spherical harmonic degrees 20 and 21 is based on the half-wavelength of the transition, 260 km, being similar to the lateral dimension of the Marius Hills. Although the precise choice of transition wavelength is somewhat arbitrary, changing the transition between degrees 18 and 24 only changes the maximum gravity due to topography by 15 mGal, which is a small fraction of the total gravity anomaly.
 The total model gravity anomaly calculated using this hybrid compensation model for spherical harmonic degrees 2–110 from Figure 4b is shown as the black dashed line in Figure 5 and is the maximum possible gravity anomaly that can be produced by the Marius Hills surface topography. The parameter values used for the topographic loading models in Figures 5-7 are summarized in Table 1. For comparison, the solid black line shows the observed gravity anomaly for degrees 2–110; both profiles use the same cosine taper filter (equation ((2))). The surface topography can account for only about half of the observed gravity anomaly along profile AA’, so it is clear that some additional dense subsurface material must be present to fully explain the observed gravity in the northern part of the Marius Hills. A similar conclusion is even more dramatically true for the southern part of the Marius Hills, where there is a strong positive gravity anomaly (Figure 2) but no short-wavelength topography (Figure 3). Figure 6 shows gravity profiles across profile line BB’ in the southern part of the Marius Hills, and Figure 7 shows gravity profiles along north–south profile line CC’. Figures 5-7 all use a similar format: the solid black line is the observed gravity anomaly and the dashed black line is the maximum gravity anomaly that can be derived from the surface topography alone. For both the northern and the southern portions of the Marius Hills, it is clear that a significant volume of subsurface, high-density material must be present to explain the observed gravity anomalies.
Table 1. Topographic Load Parameters
2500 kg m-3
3150 kg m-3
3.2 Subsurface Load Models
 The foregoing analysis demonstrates that significant, high-density subsurface loads must be present in the Marius Hills region. We can develop quantitative models of the density distributions in these structures using the DISKGRAV modeling program [Kiefer, 2004]. Subsurface loads are modeled as finite thickness spherical caps, which are sometimes referred to here as disks for simplicity. This relatively simple geometry was chosen because it provides an excellent representation of the anomalies in the Marius Hills and uses only a small number of adjustable parameters to define each cap. Each cap is defined by its outer radius, Rdisk, and by a taper zone width, Tdisk. Each disk is assumed to have a constant thickness and a density contrast that varies with distance from the center of the disk. Moving from the center of the disk horizontally outward, the density contrast between the disk and the surrounding terrain is a constant value, δρ, from the center of the disk out to a horizontal distance of (Rdisk – Tdisk). From (Rdisk – Tdisk) out to Rdisk, the density contrast decreases linearly with distance from δρ to zero at a horizontal distance of Rdisk. At all horizontal distances larger than Rdisk, the density contrast between the spherical cap and the reference crust is 0. Physically, if one is modeling a magma chamber that is the aggregate of numerous small sills and dikes, the decrease in density near the edge of the disk may represent a decrease in the volumetric filling factor of the dikes near the outer edge of the chamber. An alternative approach to specifying the disk structure would be to assume a constant density and a thickness that thins to zero at the disk edge. Modeling the disk in this way would introduce additional nonuniqueness to the problem, because there might be many model disk shapes that fit the data equally well. Parameterizing the change in disk shape with distance from the center would thus introduce additional unconstrained model parameters without providing additional insight into the actual mass distribution. For clarity, note that the radius Rdisk is measured horizontally across the surface from the center of the disk and is not the same thing as the spherical radii in the integral in equation ((4)). This density structure is expanded into spherical harmonics, δρnm. The taper zone width is used to control the rate at which the gravity anomaly caused by a subsurface load decays with distance from the center of the load but also serves to minimize the effects of ringing due Gibbs phenomenon in the spherical harmonic expansion due to a step-function discontinuity in density at the disk edge.
 Given these assumptions about the geometry of the subsurface load, the associated gravity anomaly is [Kiefer, 2004]
 Equation ((4)), as in equation ((3)), is presented in terms of dimensional gravity and density contrast harmonics, and it is implicitly assumed that both cosine and sine terms are included in the spherical harmonic expansions for δρnm and . Although Kiefer  refers to the “buried load” geometry in terms of vertical cylinders, the subsurface loads in that paper were actually finite thickness spherical caps, just as used here.
 The integration limits in equation ((4)), r1 and r2, can be adjusted to control the thickness and depth below the surface of the subsurface load. For any such choices of load geometry, the inclusion of the wavelength- and depth-dependent attenuation factor and the integration over depth ensures that the gravity anomaly calculation correctly accounts for finite amplitude effects. The subsurface loads in this study will be interpreted as magma chambers, suggesting that they likely occur in the shallow subsurface. In this work, they are assumed to begin just below the lunar surface, as this minimizes the effects of attenuation and thus sets a lower bound on the total mass of required high-density material. It is possible to stack multiple spherical caps at different crustal depths in the same geographic location. This capability can be used to place a cap which serves as a crustal root at depth. Depending on the relative thickness and density contrast of the near-surface and deep disks, such an arrangement of spherical caps can be used to simulate either isostatic or flexural compensation of the near-surface disk. In the present study, based on the evidence cited above for a relatively thick lithosphere at the time the Marius Hills volcanic load was emplaced, I assume that the subsurface loads, like the short-wavelength topography, are uncompensated, which minimizes the mass of the subsurface loads that is required to fit the observations.
 The total model gravity anomaly is the sum of the gravity from the flexurally supported topography and from the various subsurface disks,
 When the model gravity is mapped into the spatial domain, all of the components that contribute to the model spherical harmonic coefficients in equation ((5)) are filtered using the same cosine taper as is used for the observed gravity anomaly (equations ((1)) and ((2))). In principal, the number of subsurface disks, i, can be as large as necessary to fit the observations. In this work, based on the shape of the observed gravity anomaly in the Marius Hills region (Figure 2), two subsurface disks are used to fit the observed gravity field. These will be referred to as the north disk and the south disk, and the relevant model parameters for the two disks are given in Table 2. The disk centers are based on the locations of the observed gravity maxima. The preferred disk radii in Table 2 are based on minimizing the two-dimensional root-mean-square (RMS) misfit between the observed and the model gravity anomalies. The trade-off between disk thickness and assumed density contrast is discussed in Section 4. The red lines in Figures 5-7 show the combined gravity model (sum of the gravity due to the topography and to the 2 subsurface disks) as solid red lines for profile lines AA’, BB’, and CC’. The subsurface disks have a cylindrical symmetry, but the topographic contributions to the gravity make the combined gravity profiles asymmetric. The combination of the two east–west profiles (AA’ and BB’) and north–south profile line CC’ demonstrates that the two subsurface disk model is, on average, an excellent fit to the gravity observations. There are small deviations between the observed and the modeled gravity near the peaks of both gravity highs (near 307°E on AA’ and 308°E on BB’), which can be accounted for by small variations in the thickness of the subsurface layer. These deviations typically correspond to changes in the subsurface layer thickness of a few hundred meters, which is a small fraction of the total subsurface layer thickness (Table 2) and constitutes a geologically reasonable variation in the thickness of the intruded magma layer.
Table 2. Subsurface Disk Load Parameters
Taper width (km)
Density contrast (kg m-3)
Disk mass (kg)
2.1 × 1016
2.9 × 1016
 Profile line CC’ shows a larger discrepancy, up to 50 mGal, between the observed and the model gravity anomalies near 11°N. This corresponds to the “bridge” of high-density subsurface material between the north and the south disks, which is evident in the observed gravity field (Figure 2). It would be possible to introduce a third, narrow disk in the region between the two main disks, but the parameters of the third disk are unlikely to be well defined given the current resolution of the gravity field; thus, a quantitative model has not been attempted. In any case, this bridge region represents only a few percent of the total mass and volume of the intruded magma. There are also larger mismatches between observed and model gravity east of 305°E on profile BB’ and west of 312°E on both AA’ and BB’ (Figures 5 and 6). These regions are far enough from the subsurface disks that the model gravity is controlled almost entirely by the topographic compensation model. Moreover, these regions are outside of the bounds of the Marius Hills volcanic field, as defined both by the volcanic dome distribution and by the shape of the high gravity anomaly, and thus will not be considered further here.
 Figure 8 explores the uncertainty in the size of the northern and southern subsurface disks. Decreasing the diameter of a disk has an obvious effect of decreasing the width of the model gravity anomaly. Smaller disk diameters also place more of the gravity power at high spherical harmonic degrees. Because these models are filtered with a cutoff at harmonic degree 110 (equation ((2))), decreasing the disk diameter also has the effect of decreasing the maximum gravity amplitude. Because of this, the disk diameter and disk thickness were simultaneously adjusted to obtain the minimum RMS misfit at each gravity anomaly. Results were obtained by varying the diameter and thickness for one disk while holding the parameters for the second disk fixed and then iterating between the disks to obtain the best overall solution. The misfit for the northern anomaly was determined over the region 12–16°N, 305–309°E. The misfit for the southern anomaly was determined over the region 7–10°N, 307–310°E. For both disks, the misfits shown in Figure 8 are less than the estimated uncertainty of ~30 mGal in the observed gravity field [Konopliv et al., 2001]. The north disk has a preferred diameter in the range 160–180 km. For the preferred density contrast of 440 kg m-3 (discussed more fully below), this corresponds to a subsurface basalt layer that is 3.0–3.3 km thick. As one would expect from the observed gravity map (Figure 2), the southern disk is somewhat narrower, with a best-fit diameter of 100–140 km. A narrower disk cannot be ruled out based on the resolution of the LP165P gravity field but may be better tested with forthcoming GRAIL data. To achieve the required gravity amplitude with such a narrow disk, the disk must be 6.2–12.9 km thick. Given these diameters and thicknesses, the two disks have similar total masses and volumes (Table 2).
 Figure 9 shows the residual gravity anomaly for the study region, calculated as the difference between the observed free-air gravity anomaly (Figure 2) and the model gravity anomaly parameterized as in Tables 1 and 2. The black circles show the location of the two subsurface disks and are identical to the circles in Figure 2. Figure 9 shows the model parameters derived here provide a very good overall explanation for most of the Marius Hills gravity anomaly. As noted in the discussion of Figure 7, there is a region between the north and the south disks, which requires a small zone of dense subsurface material that is not well modeled by the two disks derived here (referred to as the “bridge” region in the discussion of Figure 7). The westernmost extent of the Marius Hills, just west of the north disk, is also not well modeled here. This is because the modeling code uses spherical caps, whereas the actual structure is somewhat elongated in the east–west direction. The residual anomalies in these two regions could be reduced by inserting two additional disks with smaller radii, a task that would be easier to constrain with high-resolution GRAIL gravity data. Because of these residuals, the total disk mass of intrusive volcanic material for the Marius Hills as a whole is somewhat larger than shown in Table 2. In the northeast corner of Figure 9, the large positive residual reflects the southern extent of the gravity anomaly associated with the Aristarchus Plateau, which is itself an important volcanic complex. Other residual anomalies include a positive anomaly in the southeast corner of Figure 9 (5°N, 300°E) and a negative anomaly near 17°N, 306°E; both are associated with local topographic lows in Oceanus Procellarum (Figure 3).
 The gravity results presented above demonstrate that a significant amount of subsurface, high-density material must be present in the Marius Hills region. A full interpretation of these results depends on a variety of factors, including the free-air gravity anomaly, the topography, and the likely densities of the various rock types in the study region. Both the Marius Hills and the surrounding Oceanus Procellarum are basaltic, with modest variations in composition [Weitz and Head, 1999; Heather et al., 2003; Heather and Dunkin, 2002; Besse et al., 2011]. Thus, any density difference between the basalts is probably small. However, based on the partial burial of small craters, the basalts in Oceanus Procellarum are thought to be thin, averaging 400 m thick [DeHon, 1979] and possibly even less [Hörz, 1978]. Moreover, as noted in Section 3.1, if there were thick lenses of basalt in much of Oceanus Procellarum, one would expect large changes in the free-air gravity anomaly across the highland–mare boundary that are not observed. Below this thin veneer of basalt, the crust is presumably anorthosite-rich feldspathic rocks, as observed in the lunar highlands [Taylor, 2009]. Thus, if the subsurface basalt layer in the Marius Hills is thick, the relevant density contrast is between basalt under the hills and brecciated anorthosite elsewhere (Figure 10a). This density difference is ultimately the cause of the gravity anomaly in Figure 2.
 The gravity models developed in this paper assume the maximum possible gravity anomaly from the surface topography and still require subsurface high-density layers that are a minimum of several kilometers in thickness (quantified below). The observed topographic relief across the Marius Hills is presumably a combination of topography created by volcanic construction by extrusive lava flows and by uplift over subsurface intrusions. The observed long-wavelength topography is about 1 km over the northern gravity anomaly and essentially zero over the southern gravity anomaly (Figure 3). Thus, an important interpretive consideration is developing a model for intruding several kilometers of subsurface material that produces little or no topographic uplift. One possible mechanism for minimizing the topographic expression of the basalt in the disk loads is if the basalt filled preexisting craters of appropriate sizes (Figure 10b). In this case, the appropriate density contrast is between basalt of intermediate Ti abundance [Prettyman et al., 2006] and the anorthosite-rich highland crust. As discussed in Section 3.1, a reasonable estimate of the basalt bulk density is 3150 kg m-3 [Kiefer et al., 2012a]. Plausible bulk densities for the feldspathic highlands crust are 2200–2600 kg m-3 depending on the mafic mineral content of the rock and on the porosity [Jeanloz and Ahrens, 1978; Kiefer et al., 2012a, 2012c; Huang and Wieczorek, 2012]. The maximum possible density contrast is thus about 950 kg m-3, which minimizes the required layer thickness. This corresponds to a basalt layer thickness of ~1.6 km for the north disk. For the south disk, the thickness is 2.7 km for a 140 km diameter disk and 5.3 km for a 100 km disk. For the most likely feldspathic crust density near 2500 kg m-3, the density contrast is 650 kg m-3, implying a basalt layer thickness of 2.2–2.4 km for the north disk and 4.1–8.1 km for the south disk. The power law relationship between depth and diameter for large fresh impact structures on the Moon predicts crater depths of 4.0 km at 100 km diameter and 4.5 km at 180 km diameter [Williams and Zuber, 1998]. Although this basin depth range overlaps with the required basalt layer thicknesses, the complete absence of basin rim structures in the topography of this region (Figure 3) indicates that this model is unlikely to be the correct interpretation of the gravity data.
 If the basalt is intruded into the subsurface as a volcanic sill, one would anticipate that the intruded basalt volume would lead to significant surface topography uplift (Figure 10c). The precise magnitude of the uplift will depend on the details of the elastic lithosphere thickness at the time of basalt intrusion. As discussed earlier, at the time that the surface volcanism was active (presumably the same time as the volcanic intrusions), the lithosphere was probably 50 km or more in thickness. However, if the subsurface load is close to surface, as one would expect for a sill or magma chamber, most of the elastic layer would be below the subsurface load and only a thin elastic layer would be above the subsurface material. In this geometry, the elastic lithosphere would act to resist subsidence under the load and thus prevent isostatic compensation of the load, but it would not be very effective in resisting topographic uplift. Even if the elastic layer could suppress topographic uplift over the intruded sill, stresses of order δρgh would develop, where δρ is the density contrast, g is the gravitational acceleration (1.62 m s-2), and h is the layer thickness. In this case, δρ and h are similar to the filled basin model (950 kg m-3 and 2–5 km), resulting in stresses of order 3–8 MPa (30–80 bar). Changing the value of δρ would require a corresponding change in h to match the gravity data, leaving the expected stress magnitude unchanged. The magma system would need to be overpressured by this amount to fill the sill to the required thickness against the resisting flexural stresses. It is not clear if this amount of overpressurization is possible in the Moon's shallow, highly fractured subsurface. Partly because of this, and primarily because most of the elastic layer is likely to be below the intruded, high-density material and thus unable to resist topographic uplift caused by the sill, I do not favor this mechanism, although additional quantitative modeling of this mechanism may be justified.
 Another possible mechanism for intruding large volumes of basalt into the feldspathic highland crust without inducing significant surface uplift is to make use of the preexisting pore space within the feldspathic crust. A similar process has been proposed at Kilauea volcano in Hawaii, where temporal changes in the gravity anomaly between 1975 and 2008 without accompanying surface uplift were proposed to be due to intrusion of magma in subsurface void space, primarily in the form of a connected network of cracks. A new eruptive vent opened near the summit of Kilauea in 2008 near the gravity anomaly peak, suggesting that the 2008 eruption was fed from this subsurface magma reservoir [Johnson et al., 2010].
 To estimate the thickness of the high-density material, it is first necessary to constrain the density contrast between the volcanic unit and the surrounding crust. Assuming that the pore space in the feldspathic highland crust is completely filled with basalt over some depth range, the density difference between the feldspathic crust within the Marius Hills and outside the volcanic field is
where Pcrust is the average porosity of the unfilled crust and ρB is the density of the basalt that fills the pore space. Equation ((6)) assumes that the only difference between the crust in the Marius Hills and in the surrounding region is that the pore space is at least partially filled with basalt in the Marius Hills. Based on the discussion earlier in section 3.1, ρB is assumed to be 3150 kg m-3. The porosity of feldspathic lunar rocks can be up to 20% [Jeanloz and Ahrens, 1978]. Based on the location of the Marius Hills near the Imbrium basin rim, it is likely that Imbrium ejecta also forms a portion of the upper crustal column. Imbrium basin ejecta, as sampled by the Fra Mauro Formation at the Apollo 14 landing site, has a porosity of 18–22% [Kiefer et al., 2012a]. Taking P = 20% as an upper bound on the crustal porosity sets an upper bound of δρ = 630 kg m-3 for the density contrast in this model. In turn, this would imply a lower bound on the subsurface layer thickness of 2.3 km for the northern disk (for 160 km diameter) and 8.4 km for the southern disk (for 100 km diameter).
 However, it is not clear that such a large porosity can be created and maintained over a crustal column that is several kilometers thick. Recent laboratory measurements of the porosity of feldspathic lunar rocks give a mean porosity of 8.6 ± 5.3% [Kiefer et al., 2012a, 2012c]. Spectral admittance modeling of Kaguya gravity data for the Moon's far side imply a typical porosity in the upper few km of the Moon's far side crust of 7.7 ± 2.8%, with an upper bound of about 14% [Huang and Wieczorek, 2012]. Considering P = 14% as a reasonable upper bound (one sigma above the mean value of Kiefer et al. [2012c] and the maximum observed value of Huang and Wieczorek ) results in δρ = 440 kg m-3. This implies a northern disk thickness of 3.0–3.3 km for disk diameters of 180 and 160 km, respectively, and a southern disk thickness of 6.2–12.9 km for disk diameters of 140 and 100 km. These are the values of density contrast and disk thickness used in Table 2. Adopting a smaller porosity of P = 10% results in δρ = 315 kg m-3. In turn, this increases the northern disk thickness to 4.7 km (for D = 160 km) and the southern disk thickness to 20.3 km (for D = 100 km). If vertical attenuation of the gravity signal did not occur, decreasing δρ from 630 to 315 kg m-3 would require doubling of the layer thickness to maintain the gravity signal. The northern disk thickness increases only slightly more than this because attenuation is a small factor in this relatively thin disk. The narrower and thicker southern disk is much more sensitive to attenuation.
Kiefer  hypothesized that the gravity anomaly associated with the shield volcano Syrtis Major on Mars was due to dense cumulate minerals such as olivine or pyroxene filling the magma chamber. This case can be modeled by changing ρB in equation ((6)). As an example, for an olivine cumulate similar to that used in the Syrtis Major model (ρB = 3590 kg m-3) and P = 14%, δρ is 503 kg m-3, resulting in a best-fitting northern disk thickness of 2.9 km for D = 160 km. For this model to be appropriate, one must assume that the less dense components of the magma (presumably plagioclase rich) that are left over after loss of the cumulate material must escape laterally from the cumulate zone. If the remnant basalt remains in the same vertical column as the cumulates, then the aggregate value of ρB that is used in equation ((6)) should be that of the original basalt rather than just the cumulate minerals.
 Apollo seismic observations show a rapid increase of seismic velocity with depth in the upper 20–25 km of the crust. These observations are interpreted as being due to the closure of porosity with increasing pressure, although the precise rate at which pores close with depth is not well constrained by the available observations [Todd et al., 1973; Toksöz et al., 1973]. Accounting for pore closure with depth will increase the inferred thickness of these disks. For δρ of 315–630 kg m-3 (porosity 10–20%), the northern disk is sufficiently thin that all of the inferred intrusive material can fit into crustal porosity even when accounting for pore closure with depth. The southern disk, however, is much thicker, and the total amount of basalt that can be stored in the southern disk may well be limited by the rate of pore closure with depth. This may favor either a relatively large diameter for the southern disk or a large near-surface porosity, because both of these choices limit the required thickness of the southern disk.
 Cracks in lunar rocks are commonly quite narrow [Simmons et al., 1975]. Given that the required disk thicknesses are at least 3.0 km and possibly much more, it is not clear that basalt would be able to freely fill the preexisting pore space while traveling such large distances through narrow cracks. An alternative possibility is that the basalt is intruded as dikes or sills and thermally anneals the surrounding host rock, reducing or possibly eliminating the porosity in the host rock. In this case, the volume increase due to the sill intrusion may be largely or entirely offset by the volume decrease due to loss of porosity in the feldspathic host rock (Figure 10d). Numerous Apollo samples have granulitic textures, interpreted as due to annealing at relatively high temperature. In many cases, this annealing may be due to proximity to impact melt in moderate size impact craters [Cushing et al., 1999], but thermal annealing would be equally possible in the region adjacent to an intruded dike or sill. This process can be very effective at reducing porosity. For example, Apollo 15 anorthositic norite sample 15418 experienced such a thermal history and now has a porosity of just 3.2 ± 0.9% [Nord et al., 1977; Kiefer et al., 2012a]. In some cases, this annealing occurred at temperatures of 1000–1100 °C, as determined from two-pyroxene geothermometry [Cushing et al., 1999]. However, the total amount of sintering or annealing that occurs is a function of both the maximum metamorphic temperature a rock experiences and the time it stays at that temperature. Experimental studies indicate that lunar rocks experienced significant annealing in less than 1 year if the temperature exceeded 800 °C [Simonds, 1973; Uhlmann et al., 1975]. Mare basalts would intrude into the crust at a liquidus temperature that is most likely in the range 1150–1320 °C [Longhi et al., 1974; Walker et al., 1977; Wieczorek et al., 2001]. Cooling of the basalt, including its latent heat of fusion, would provide a significant amount of energy for heating and possibly annealing the adjacent country rock. In this model, the disk thickness shown in Table 2 represents the aggregate thickness from many smaller intrusive events. If the intruded material was in the form of dikes or sills with half-widths exceeding 5 m, the cooling time would exceed 1 year [Turcotte and Schubert, 2002], consistent with the time required to anneal the surrounding host rock. In this model, the porosity used in equation ((6)) should be the amount of porosity lost due to thermal annealing and may be less than the total original porosity. If the volume of intruded magma exceeds the volume of lost pore space, then some surface uplift is likely to occur over the intrusion. This is a possible explanation for some of the short-wavelength topography in the northern part of the Marius Hills but evidently did not occur in the southern part of the Marius Hills, where there is no short-wavelength topography (Figure 3). The depth in the lunar crust at which the vertical propagation of dikes stalls and transitions to horizontal magma emplacement as sills depends in part at the depth at which the dike reaches neutral buoyancy with the surrounding crust [Head and Wilson, 1992]. Because the thermal annealing process results in a denser crust, it will also allow dikes to propagate to shallower depths before stalling.
 The total mass in the two subsurface disks is 5.0–5.6 × 1016 kg, with the southern disk being slightly more massive than the northern disk (Table 2). Based on the assumed basalt density of 3150 kg m-3, this is equivalent to an intruded magma volume of 1.6–1.8 × 104 km3. To obtain the total volcanic volume in the region, we must also add in the volume of the extrusive volcanism, which can be estimated from the regional topography. In the northern disk, the topography at the center of the volcanic field has a maximum of slightly more than 1 km and an average elevation of about 0.5 km above the regional base level, averaged over a region about 80 km in radius. This implies an approximate volume of 104 km3 for the surface volcanic complex. There is little change in topography across the southern disk, implying that the extrusive volcanic volume in the south is small. Thus, the total magmatic volume for the Marius Hills is about 2.6–2.8 × 104 km3, of which more than 60% is intrusive material. This volume demonstrates that the Marius Hills is a major volcanic complex.
 An alternative approach to estimating volcanic volumes is to use the exposure of basalt and highland materials in the ejecta blankets of various size craters to estimate the thickness of the basalt layer. Based on the lack of exposure of highland material in craters in this region, Heather et al.  determined that the minimum basalt thickness is 152–1100 m in different parts of the region, with a minimum volcanic volume of 5320 km3. They recognized that this is a minimum value, as this approach can only “see” as deeply as the craters permit. In particular, the largest crater in the region is Marius, which coincidentally lies on the eastern edge of the volcanic field, where the amount of intruded basalt is significantly less than near the center of the field. In the south, there are no large craters available to sample into the very thick intruded basalt layer. Thus, it is not surprising that remote sensing estimates of volcanic volume may be significantly smaller than the true volume. Gravity, on the other hand, can detect the full thickness of the high-density material and should yield a more accurate volcanic volume.
 As noted in the introduction, the morphology of the volcanic domes in the Marius Hills requires a relatively high magma viscosity. Because of the spectral evidence that the domes are basaltic in composition, the high viscosity is likely a consequence of a high degree of crystallinity within the erupting magma. The subsurface, basalt-rich layers mapped in this study in effect served as magma chambers for the Marius Hills volcanic field, providing a location for the magma to cool and crystallize. Eruption of a crystal-rich magma onto the lunar surface is unlikely to occur strictly due to magma buoyancy alone. The flexural deformation due to volcanic loading in the center of major mare basins such as Imbrium can create stresses that favor magma eruption in the region surrounding the basin [McGovern and Litherland, 2011] and might serve to help drive crystal-rich magmas from subsurface reservoirs onto the surface. These stresses will decline in amplitude with distance from the mare basin and might explain why the southern Marius Hills, which is farther from Imbrium, has many fewer domes than the northern part of the Marius Hills. If recharge of the magma chambers occurred during the time in which eruptions were occurring, it is possible that fractional crystallization within the magma chamber created a form of cyclic geochemical layering within the sills, similar to the layering observed in the Snake River Plains in the western United States [Shervais et al., 2006].
 The Marius Hills volcanic complex is the largest volcanic field on the Moon, with more than 250 basaltic domes and cones in a region 250 km across. The prominent free-air gravity anomaly in this region helps to map out the volcanic plumbing system that fed the surface volcanism. In the north, only about half of the gravity anomaly can be explained by the observed surface topography. In the south, there is essentially no topography associated with the gravity anomaly. As a result, in both the north and the south, there must be a significant volume of high-density material in the subsurface to explain the gravity anomaly. The most likely explanation for this dense material is intrusion of mare basalt as dikes and sills into the porous feldspathic highland crust, which underlies the Marius Hills. This network of dikes and sills may have functioned as a magma chamber system for the surface volcanic field, providing a place for magma to cool and partially crystallize before erupting on the surface as high viscosity, crystal-rich magmas that formed the volcanic domes. Numerical models show that the gravity anomaly can be well represented by a pair of spherical caps that represent the northern and southern parts of the Marius Hills. The northern disk is 160–180 km across and at least 3.0 km thick. The southern disk is 100–140 km in diameter and at least 6.2 km thick. The total volume of intruded magmatic material is 1.6–1.8 × 104 km3. The total volcanic volume, including both intrusive magmatism and extrusive volcanism, is 2.6–2.8 × 104 km3, demonstrating that the Marius Hills is a major volcanic complex. This is about five times the minimum amount of volcanic material inferred by remote sensing studies and demonstrates the value of gravity modeling as a subsurface mapping tool. The intruded basalt may have heated and thermally annealed the surrounding host rock. The resulting loss of porosity in the highland crust could at least partially balance the volume of intruded magma, allowing the intrusive volcanism to occur with little net change in crustal volume and thus with little resulting uplift of the surface in this region.
 I thank Don Bogard for suggesting that I consider thermal annealing and Tracy Gregg, Jim Kauahikaua, and Mark Wieczorek for helpful reviews of the manuscript. This work was supported by NASA grant NNX07AF95G and NASA Cooperative Agreement NNX08AC28A. The Lunar and Planetary Institute is managed by Universities Space Research Association. LPI Contribution 1699.