Terrestrial crater observations and laboratory experiments demonstrate that target material beneath complex impact craters is uplifted relative to its preimpact position. Current estimates suggest maximum uplift is one tenth of the final crater diameter for terrestrial complex craters and one tenth to one fifth for lunar central peak craters. These latter values are derived from an analytical model constrained by observations from small craters and may not be applicable to larger complex craters and basins. Here, using numerical modeling, we produce a set of relatively simple analytical equations that estimate the maximum amount of structural uplift and quantify the attenuation of uplift with depth beneath large lunar craters.
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 Observations of terrestrial craters [Grieve et al., 1981; Grieve and Pilkington, 1996] and laboratory cratering experiments [Schmidt and Housen, 1987] have shown that target material beneath crater centers is uplifted relative to its preimpact position. Material is uplifted a maximum distance of one tenth of the final crater diameter for terrestrial craters [Grieve et al., 1981; Grieve and Pilkington, 1996]. For lunar craters, Cintala and Grieve  estimated uplift increases from at least one tenth to one fifth (of the crater diameter) as crater size increases from 20 km to 100 km, respectively. These estimates are derived from analytical equations that are constrained by data from relatively small complex craters and may not be good proxies for larger impact structures, including lunar basins (diameters >300 km). Those estimates also only refer to lithologies at, or near, the surface. Structural uplift can extend far beneath the crater floor; it does not, however, remain constant, attenuating with increasing depth. For example, at the Chicxulub impact structure, seismic studies suggest lower crustal rocks 5 km beneath the crater floor have been uplifted ∼10 km relative to their original depth [Christeson et al., 2001]; the Moho at a depth of ∼30 km has only been uplifted ∼1–2 km [Christeson et al., 2001, 2009]. Other observational techniques, as well as computer simulations of large-scale impact events [Collins et al., 2002, 2008; Ivanov, 2005], also indicate structural uplift attenuates with depth beneath the crater floor of terrestrial craters (see Figure S1 in the supporting information).
 To date, no scaling law has been developed to describe the attenuation of uplift beneath impact structures. In this paper, we model large complex crater- and basin-forming impact events to produce a set of relatively simple equations that can be used to estimate the maximum amount of structural uplift and its attenuation with depth beneath large impact structures on the Moon.
 The shock physics code impact-Simplified Arbitrary Lagrangian Eulerian (iSALE) [Amsden et al., 1980; Collins et al., 2004; Wünnemann et al., 2006] was used to numerically model large-scale lunar impacts (craters diameter). The iSALE code has previously been used to model individual large-scale impacts (e.g., Chicxulub and Orientale: Collins et al., 2008 and Potter et al., 2013, respectively).
 To approximate lunar target compositions, a half-space target was divided into crustal and mantle layers. Crustal thicknesses of 60 km (pre-Gravity Recovery and Interior Laboratory (GRAIL)) and 40 km (post-GRAIL) [Wieczorek et al., 2013] were used. A Tillotson equation of state for gabbroic anorthosite [Ahrens and O'Keefe, 1977] and a semianalytical equation of state (ANEOS) for dunite [Benz et al., 1989] were used to account for both thermodynamic and compressibility changes in the crust and mantle, respectively. Crustal strength and thermal parameters were calculated from data gathered from experiments on gabbro [Azmon, 1967; Stesky et al., 1974; Shimada et al., 1983]; mantle strength and thermal parameters were calculated from dunite [Shimada et al., 1983; Ismail and Murrell, 1990] and peridotite [Stesky et al., 1974; McKenzie and Bickle, 1988] experimental data, with curves for the mantle melt temperature as a function of pressure taken from Davison et al. . The strength and damage models are described by Collins et al.  and Ivanov et al. , respectively.
 The iSALE code has the option to include acoustic fluidization [Melosh, 1979; Melosh and Ivanov, 1999], a weakening mechanism that helps facilitate large-scale crater collapse. Acoustic fluidization is implemented here via the block oscillation model [Melosh and Ivanov, 1999; Wünnemann and Ivanov, 2003] with parameters chosen based on Wünnemann and Ivanov  scaling relations using the Chicxulub subsurface structure as a constraint [Collins et al., 2008]. Additionally, supersolidus material was assigned a partial melt viscosity of 1010 Pa s [see Potter, 2012]. In iSALE, the solidus marks the temperature at which a material loses all shear strength. At temperatures between the solidus and liquidus, however, the material will be a mixture of hot melt and cold clasts and, therefore, possess some resistance to shear. The rheological behavior of this supersolidus material is complex, so our partial melt viscosity is a first-order approximation of its strength.
 Three different target thermal profiles were used. Thermal profile (TP) 1 had an initial crustal temperature gradient of 10 K/km; mantle temperatures were at the solidus between depths of 150 and 350 km and ∼1670 K in the deep (> 800 km) mantle. TP2 also had an initial crustal temperature gradient of 10 K/km; mantle temperatures remained subsolidus reaching a deep mantle temperature of ∼1770 K. These profiles are the same as those in Potter et al.  and represent lunar thermal conditions ∼4 Ga during the proposed time of basin formation (the lunar cataclysm). TP3, based on a profile from Spohn et al. , represents lunar thermal conditions ∼3.5–1Ga, when the majority of complex craters ∼100–300 km in diameter formed (e.g., Hausen and Tsiolkovskiy). This thermal profile has a crustal upper mantle temperature gradient of 3.5 K/km, remains subsolidus within the mantle, and reaches ∼1620 K in the deep mantle (see Figure S2 for a comparison of the thermal profiles).
 Impact parameters were chosen to produce peak-ring craters and basins between 200 and ∼1000 km in diameter. Impactor diameter was varied between 20 and 120 km. The number of cells across the impactor was kept fixed at 40 cells for basin-forming impacts (> 300 km diameter) and 20 cells for complex crater-forming impacts (< 300 km diameter) which gave cell sizes of between 1 and 3 km. These fixed number of cells were chosen to provide a reasonable trade-off between resolution errors and computation time. The dunite ANEOS was used to represent the impactor's response to thermodynamic and compressibility changes. Impactor velocities were varied between 10 and 20 km/s. A constant gravitational acceleration field of 1.62 m/s2 was used in all models.
 Structural uplift was quantified using passive Lagrangian tracer particles. These tracers track the location of cell material throughout the crater-forming process. Uplift was calculated by comparing tracer row depth at the center and outside of the impact structure.
 As this work used the 2-D (cylindrical geometry) version of the iSALE code, all models assumed an impact angle of 90° (vertical impacts); crater features are, therefore, axisymmetric. Seismic studies at Chicxulub, however, suggest uplift within the upper crust is offset from that of the Moho; lateral asymmetry in target strength or an oblique impact [Christeson et al., 2009] could be the cause of this. Oblique impact numerical models [e.g., Wünnemann et al., 2009] suggest that, at depth, uplift is directly beneath the basin center, whereas at the surface, uplift is offset downrange. Venusian crater analysis, however, suggests no significant crater feature offsets [e.g., McDonald et al., 2008]. In any case, the vertical impact simulations used in this work are reasonable proxies for the processes investigated over a wide range of impact angles (30–90°) [Elbeshausen et al., 2009].
 Figure 1 illustrates a lunar basin-forming impact highlighting changes in target layer depth. On impact, the growing transient cavity excavates target material and displaces target layers beneath the cavity floor (Figure 1a). As the crater collapses, the crater floor begins to rise, uplifting layers within the target (Figures 1b and 1c); deeper layers are uplifted less than those nearer the crater floor. As the overheightened central uplift collapses (Figure 1d), layers near the surface move downward and outward, reducing the net uplift of this material. The outward component of the motion is greatest at the surface and attenuates with depth. Consequently, as the uppermost layers, which are (partially) molten, slide and shear off the uplift, they spread over a larger area and become thinner. Additional uplift/collapse phases may occur in the (partially) molten region (Figure 1e), but these do not affect uplift at depth. Figure 1f shows the final subsurface structure, highlighting a general attenuation of stratigraphic uplift with depth. Thinning of the uppermost layers, however, during outward collapse of the central uplift, implies that the magnitude of uplift beneath large lunar craters does not decrease monotonically with depth and the most uplifted strata are not necessarily closest to the surface.
 Figure 2 plots maximum structural uplift, Umax, against transient crater diameter, Dtc, for our modeled impacts using all thermal profiles, as well as observations and model estimates for terrestrial and lunar craters, respectively, from Cintala and Grieve . The transient crater, an important feature used to calculate impact energy/momentum, melt volume, and excavation depth, is defined here as being formed once the expanding transient cavity reaches its maximum volume. There is, therefore, some uncertainty in relating this to observational estimates of transient crater diameters. The figure shows that maximum structural uplift increases with transient crater diameter. Maximum uplift in our models also has a weak dependency on impact velocity (and therefore impact energy) hence the scatter in uplift between TP1 and TP2; lower velocities produce less uplift. Nevertheless, data from all three thermal profiles can be reasonably fit by a single equation:
 Our fit has a similar slope to the terrestrial data observed by Cintala and Grieve . Uncertainty in terrestrial transient crater estimates is likely comparable to the spread in the numerical model results. The Cintala and Grieve  lunar crater fit, however, is notably steeper than the terrestrial (and numerical model) data, despite considerable overlap between individual data points. This is likely due to methodological differences: terrestrial uplift data is observational, whereas the lunar data is part observation (central peak height) and part model (using the estimated maximum depth of impact melting). We also developed a similar relationship to that above for the final crater diameter: (see Figure S3).
 Figure 3 illustrates the attenuation of structural uplift with depth for basin-forming impacts of differing velocity, impactor diameter, and crustal thickness into the Moon with TP1 and TP2. Structural uplift, U, normalized by the maximum structural uplift, is plotted against depth beneath the crater floor, z, normalized by the transient crater radius, rtc.These data suggest that the most uplifted material (U/Umax= 1) is found at a depth equivalent to 15–30% of the transient crater radius. The results also show a general linear decrease in uplift as relative depth increases. Uplift reaches a minimum at a depth equivalent to approximately twice the transient crater radius (i.e., the transient crater diameter). An average fit to these data is
when z/rtc>0.18. These trends were also evident for the peak-ring crater-forming impacts using TP3 (see Figure S4). The TP3 data, however, showed uplift reached a minimum at depths equivalent to ∼1.1 times the transient crater radius.
 Maximum structural uplift for lunar impact craters could previously be estimated using the formula developed by Cintala and Grieve  (their equation 12). Table 1 lists maximum uplift estimates for a range of large lunar complex craters and basins using the Cintala and Grieve  equation and equation (1) of this work. These equations provide similar maximum uplift values for large complex craters (i.e., Tsiolkovskiy), but show increasing variation for larger (basin-sized) impacts, with equation (1) of this work predicting maximum uplifts of one half to one third those of Cintala and Grieve .
Table 1. Maximum Structural Uplift Estimates for Large Lunar Craters and Basins
aCalculated using equation 9 from Croft , assuming 18 km for the lunar simple-to-complex crater transition.
 The Cintala and Grieve  relationship was developed using relatively small central peak craters. Extrapolation of that relationship to the much larger basins may, therefore, be unsuitable. Cintala and Grieve  write that their formula provides only an upper limit for the maximum uplift beneath lunar basins. They also propose a separate qualitative mechanism for basin formation. These factors may help explain the differences in maximum uplift between their work and ours. Additionally, our expression takes into account the hotter thermal state of the Moon during the basin-forming epoch ∼4Ga; Potter et al.  have already shown that thermal conditions greatly affect the basin-forming process. On the other hand, many large lunar complex craters are younger than the basins and, therefore, formed in a thermally cooler Moon. This would explain the similarity in uplift estimates for large complex craters, suggesting thermal conditions had less of an effect during large complex lunar crater formation.
 Our models indicate structural uplift beneath large lunar craters reaches a maximum at a depth equivalent to 15–30% of the transient crater radius (from the original surface layer). During large-scale crater modification, the ephemeral central uplift collapses back into the crater, thinning out the upper layers as they spread laterally over the basin floor. This decreases the relative uplift of these upper target layers; hence, the maximum uplifted material (relative to its preimpact position) is not exposed on the crater floor. Below the depth of maximum uplift, strata are less affected by the lateral collapse of the central zone and structural uplift attenuates with depth. The depth of zero uplift is ∼1.1 and ∼2 times the transient crater radius for peak-ring craters and basins, respectively.
 This attenuation of uplift is similar to that seen in numerical models [e.g., Collins et al., 2002] of, and seismic data [Christeson et al., 2001, 2009] from, the large-scale Chicxulub impact structure (see Figure S1). It should be noted, however, that attenuation is sensitive to model parameters, such as the thermal gradient, and may also be affected by impact angle/direction if floor uplift is not axisymmetric. These relationships should, therefore, be applied with caution to situations that differ from those modeled in this work (i.e., nonlunar and small-scale impacts). In particular, for most of the thermal profiles considered in this work, rocks deep beneath the crater are hot and weak, which enhances uplift at depth. Hence, craters formed in colder and stronger rock targets will likely produce a smaller maximum uplift and uplift may attenuate more rapidly with depth. Some craters may have central peaks/peak rings that are initially covered in completely molten material. This material would slide off the uplift feature, exposing deeper target material at the top of these structures, thereby affecting uplift characteristics. However, because of their size and inferred target thermal conditions, this is unlikely to affect uplift associated with lunar complex craters. In small complex craters where little-to-no outward collapse of the central peak/uplift occurs, the maximum uplift will, therefore, occur at the surface and uplift is expected to decrease monotonically with depth. This is consistent with uplift attenuation observed at terrestrial craters (Figure S1).
 The fact that stratigraphic uplift beneath large craters attenuates to zero by a depth equivalent to 1–2 transient crater radii has important implications for possible impact-induced volcanism [e.g., Ivanov and Melosh, 2003; Jones et al., 2005]. Calculations of decompression melting that assume a constant amount of uplift to great depth beneath the crater [e.g., Elkins-Tanton and Hager, 2005] will grossly overestimate the degree of melting caused by uplift.
 The original depth of uplifted material is also important, particularly for mantle material. If uplifted to a shallow depth, mantle could potentially be sampled, providing insight into the Moon's bulk composition and planetary differentiation. Figure 4 plots the normalized structural uplift against the original depth of target material, zo, normalized by the transient crater radius for large complex crater-forming impacts using TP3. This illustrates that material originally at a depth equivalent to ∼80% of the transient crater radius is uplifted the most; material at a depth > 1.2 times the transient crater radius is not uplifted at all. The initial increase in relative uplift with depth can be fit by the following:
when zo/rtc≤0.8. The attenuation of uplift at depths >0.8(zo/rtc) is fit by the following:
 Our fits are reasonable as scatter and uncertainty are also present in observational estimates of transient crater radii. See Figure S5 for the relationships developed for the basin-forming impacts.
 Mantle uplift will, therefore, be negligible if crustal thickness is times the transient crater radius. Assuming an average 40 km thick crust [Wieczorek et al., 2013], mantle uplift would be minimal for transient craters ≤96 km in diameter, which corresponds to a final crater diameter of ≤130 km (using equation 9 of Croft ). This matches well with extrapolation of gravity data which suggest craters < 170 km in diameter show little-to-no positive Bouguer gravity anomaly [Baker et al., 2013] and, therefore, minimal uplift of denser materials.
 It is important to point out that in the largest basins, uplifted material is substantially shock-melted and dominantly mantle-derived. Mantle material (i.e., ultramafic lithologies) may not, however, provide the signature exposures within basins. If melt volumes are sufficiently large, they will differentiate, producing less mafic and potentially crustal-like lithologies at the surface [Morrison, 1998]. Thus, while a calculation of Umax may indicate uplift of mantle material, additional processing of that material may mask its surface signature.
 In this work, numerical impact models provide constraints on the attenuation of structural uplift, with depth, beneath large (> 200 km diameter) lunar craters. Model results indicate that maximum structural uplift does not occur at the surface (i.e., within central peaks and peak rings), but at a depth equivalent to 15–30% of the transient crater radius. The model results were also used to produce equations to estimate the amount of structural uplift beneath large lunar craters that take into account the likely thermal conditions when these impacts took place. The analytical relationships developed here should aid current (e.g., GRAIL) and future lunar, as well as cratering, studies.
 We thank Boris Ivanov, Jay Melosh, Kai Wünnemann and Dirk Elbeshausen for their work developing iSALE. We also thank Jay Melosh and an anonymous reviewer for their comments. This work was partially supported by NASA Lunar Science Institute contract NNA09DB33A (PI David A. Kring). GSC was supported by UK Science & Technology Facilities Council grant ST/J001260/1. This is LPI contribution number 1760.
 The Editor thanks H. Jay Melosh and Richard Grieve for their assistance in evaluating this manuscript.