We infer Amazonian climate change events from the presence of fresh craters with excess volumes of ejecta. Using the Mars Orbiter Laser Altimeter data, the geometrical properties of 572 fresh impact craters with rim diameters between 2.5 and 102 km were compiled in lowland and highland plains. The data reveal a class of fresh craters with anomalously high ejecta volumes preferentially located in Utopia Planitia. These fresh, “excess ejecta” craters have volumes of material above the preimpact surface larger than the crater cavity volumes by factors of 2.5 to 5.8. The excess volume corresponds to an excess thickness of ejecta of about 20 to 100 m averaged over the continuous ejecta blanket. The excess material cannot be accommodated by ejecta bulking alone and requires an external process to increase the apparent volume of the ejecta and/or the uplifted surface. On the basis of the geologic setting, ejecta morphology, and calculations of increased ice stability with burial, we conclude that the most likely origin of the excess ejecta volume is the presence of an ice-rich layer tens of meters thick at the time of impact. The icy layer or a lag deposit is partially preserved beneath the ejecta blanket today. In this scenario, the icy layer has since been removed from areas unprotected by ejecta blankets, creating an apparent preimpact surface lower than the original elevation. The statistical occurrence of excess ejecta craters is consistent with climate model predictions of recent glacial periods on Mars.
 In this work, we investigate the relationships between the fresh impact cratering record and the recent history of water on Mars. We employ crater geometry measurements that demonstrate regional variability in surface properties [Stewart and Valiant, 2006] and focus on the relationships between crater forms and the history of water in the Utopia Planitia region. The history of water in Utopia has been varied and exciting. Utopia Planitia is a 3380-km-diameter impact basin, formed in the early Noachian at approximately 4.1 Ga [Frey, 2006]. The Noachian aged basement of the lowland plains has since been buried by up to several kilometers of volcanic and sedimentary material [Head et al., 2002; Watters et al., 2007]. Water from catastrophic outflow channels emplaced sedimentary deposits in the Late Hesperian; this water disappeared by the end of the Hesperian, allowing rebound of the basin floor [Thomson and Head, 2001]. In the early Amazonian, lava flows from Elysium Mons surged almost 2400 kilometers into Utopia Planitia; Russell and Head  suggest that some of these flows might have been water-mobilized lahars. The smoothness and homogeneity of the Vastitas Borealis Formation, which covers most of the northern plains, support the possibility that it was deposited by a large body of water [Boyce et al., 2005; Kreslavsky and Head, 1999]. Recent observations of ice-related features and theoretical work on the dynamical history of Mars have illuminated how obliquity variations have led to the periodic deposition of ice-rich layers on the surface, and in particular on the flanks of Elysium [Forget et al., 2006; Head et al., 2003; Laskar et al., 2004; Mischna et al., 2003]. Observations of deep impact crater geometries have been used to infer the presence of a strong Noachian layer beneath the surficial fluvial and volcanic deposits in Utopia [Boyce et al., 2006; Stewart and Valiant, 2006]. Fresh crater morphologies have also been used recently to extract information about Amazonian water related events [e.g., Black and Stewart, 2005; Boyce et al., 2005; Kreslavsky and Head, 2006; Meresse et al., 2005].
 We begin by summarizing the methods used for geometric measurements of fresh crater cavity and ejecta volumes using the HMars toolkit [Stewart and Valiant, 2006]. We identify and critically examine the observation of a population of midlatitude craters with anomalously high volumes of uplifted and ejected materials. After consideration of several possible formation mechanisms for these craters, we conclude that deflation of a volatile-rich surface or near-surface layer is the most likely explanation. Our model [Black, 2005; Black and Stewart, 2005] is similar to the models conceived independently by Meresse et al.  and Barlow . In this work, we study the properties of specific craters with excess ejecta and consider the plausibility of the deflation model by examining the rates of sublimation of ice covered by an insulating ejecta blanket, the temperature field after an impact event, and the statistics of impact cratering during glacial periods.
2. Crater Measurements
 This investigation uses two main tools to examine fresh impact craters: (1) crater geometry measurements, collected with the HMars program [Stewart and Valiant, 2006] using the Mars Orbiter Laser Altimeter (MOLA) topography data set, and (2) crater imagery using visible image mosaics from the Viking orbiters, infrared and visible spectrum images from the Thermal Emission Imaging System (THEMIS) on Mars Odyssey, and high-resolution visible images from the Mars Orbiter Camera (MOC) on Mars Global Surveyor. The crater geometry database forms the primary quantitative data set, and the images are used to aid in identification of fresh craters, location of ejecta blankets, and geomorphologic analyses.
 The geometric measurements methods are described in detail by Stewart and Valiant  and summarized here. The geometric properties of impact craters are measured using the HMars interactive toolkit, a graphical interface to the MOLA Precision Experiment Data Records (PEDR) altimetry profiles. Digital elevation maps (DEMs) are generated on the fly from the altimetry profiles using the Delaunay triangulation algorithm [Barber et al., 1996] at the user-specified spatial resolution for the region of interest. The primary crater measurements are illustrated in Figure 1. Measurements of the crater rim diameter, DR, and rim height, HR, are derived from altimetry data points, which are spaced by about 300 m along each orbital track. The maximum altimetry point on each track crossing the crater rim systematically underestimates the true rim height because it is much more probable that a point will fall on the slopes of a crater rim, rather than hitting the exact crest. For that reason, interpolated rim radius and rim height are also calculated based on the intersection of quadratic fits to altimetry points on the crater wall and on the outer rim wall.
 Using the DEM and imagery information, the user identifies points on the background, preimpact surface. These points are interpolated across the crater cavity and ejecta blanket using a Delaunay triangulation to define the regional preimpact surface. VCavity is the measured crater cavity volume below the preimpact surface, and VAbove is the measured volume of material above the preimpact surface. VAbove is the sum of the ejecta volume and volume of uplifted preimpact surface. The uplift profile is not known precisely, but constrained by impact cratering models, experiments, and observations. Here, the uplifted surface height, hu, is approximated by hu = 0.5HR (r/RR)−n, where r is the radial distance from the crater center, RR is the crater rim radius, and n is constrained to be between 3 and 5.5 [Stewart and Valiant, 2006]. VEjecta is calculated by subtracting the uplifted volume from VAbove. Note that the measurement of VAbove is much more accurate than the estimate of VEjecta because of the uncertainty in the uplift profile. The profile of the uplifted surface is discussed further by Stewart and Valiant . Other measurements relative to the preimpact surface, i.e., the surface depth, ds, and the surface diameter, Ds, are also based on the DEMs. The crater depth from the rim, dr, is the sum of the surface depth and rim height. For all variables, the subscripts R and S refer to the rim and preimpact surface, respectively.
 Occasionally, sparse altimetry data or preimpact topography may interfere with accurate measurements of the volume of uplifted and ejected material. In these cases, one or more user-defined pie-shaped wedges are employed to exclude an area from integration, and the measured volumes are adjusted to compensate for the missing area assuming axisymmetry. The user also traces the edge of the ejecta blanket and, if present, inner ejecta ramparts. The ejecta blanket morphology is categorized following the recommendations of Barlow et al. . In total, about 30 measurements and observations are recorded in a data structure for each crater.
 Each MOLA altimetry point covers a spot on the surface about 168 m in diameter with a vertical accuracy of about 1 m, and the along-track spacing between points is about 300 m [Smith et al., 2001]. Crater measurements are conducted at a minimum DEM resolution of 40 pixels per crater rim diameter and usually at the limit of the MOLA data.
 To determine the accuracy and precision of the HMars toolkit, Stewart and Valiant  measured sets of simulated fresh impact craters (2 ≤ DR ≤ 50 km) on different background terrains and at altimetry track densities representative of low, middle, and high latitudes. There were no systematic offsets in the measurements, and the results were reproducible by different users within the measurement errors. The measurement precision was best for ≥6 km rim diameter craters, with very little latitude dependence. Reasonable measurements are possible on populations of fresh axisymmetric craters as small as 2 km rim diameter given sufficient track resolution.
2.2. Study Regions and Definition of Freshest Craters
 In this study, we focused on a comprehensive database of fresh crater geometry measurements in southern Utopia Planitia, including the early Amazonian Elysium flow units [86–163°E, 18–55°N]. Utopia impact basin is of particular interest because its low elevation in the northern plains forms a natural catchment for volcanic and sedimentary processes, and previous studies of Utopia have described a rich geologic history [Russell and Head, 2003; Tanaka et al., 2003, 2005, available at http://pubs.usgs.gov/sim/2005/2888/; Thomson and Head, 2001]. All fresh craters with rim diameters in the range of 4 to 50 km in southern Utopia Planitia were measured. Occasionally, smaller craters were well resolved, and small regions of Utopia Planitia extending north of Elysium Mons were investigated. Craters with insufficient altimetry track coverage were not included in this study. Craters at latitudes above 55°N were not included because most have undergone significant modification.
 Fresh crater populations in Lunae Planum [286–300°E, 6–20°N], Solis Planum [261–284°E, 15–30°S], Isidis Planitia [81–99°E, 5–25°N], and southern Acidalia Planitia [308–355°E, 27–54°N] provide comparisons with the properties of craters in Utopia. This work includes all fresh craters measured in Isidis and Utopia, whereas Stewart and Valiant  only considered craters within restricted regions defined by the estimated diameters of the impact basins. All studied regions are relatively smooth plains where very accurate crater geometry measurements are possible. Isidis was chosen for its proximity to Utopia, and southern Acidalia Planitia provides a comparison lowland terrain at the same latitude as southern Utopia. Lunae Planum and Solis Planum are Hesperian highland plateaus at low and middle latitudes, respectively.
 A total of 572 potentially fresh craters were measured. The freshest craters are identified quantitatively by deep crater cavities and high rims, as discussed below. Qualitative confirmation of the freshness of each crater was obtained by visual inspection of the following features: rim sharpness, ejecta degradation, interior deposits, and thermal inertia relative to the background surface, as formulated by Barlow . As a crater becomes degraded due to weathering and other erosive processes, crater rims are worn down and crater cavities are infilled. Obviously infilled and eroded craters were not measured. The characteristic rim height to diameter and crater depth to diameter ratios vary by region because crater shapes are sensitive to the local compositional and structural properties of the surface [Stewart and Valiant, 2006]. Following the method presented by Stewart and Valiant , in each region, we fit a maximum crater depth versus DS function and a maximum rim height versus DS function with two power laws, one for the strength regime and one for the gravity regime. The power laws are shown in Figure 2 and tabulated by Stewart and Valiant . The set of freshest craters satisfies three quantitative criteria: (1) cavity depths deeper than a cutoff fraction c of the maximum depth function for the region; (2) rim heights taller than the same cutoff fraction c of the maximum height function for the region; and (3) a minimum of 4 rim points in the measurements. The gradual transition from the strength to the gravity regime occurs between 7 ≤ Ds ≤ 10 km, and in the transition zone, the smaller of the two values of the cutoff criteria were applied. In some cases, craters satisfy only one of the geometric criteria and are not included in the freshest set, as illustrated in Figure 2.
 All Martian craters are degraded to some extent, and there is no rigorous distinction between fresh and degraded. Hence the definition of freshness is relative to the other craters within each region, and any correlation with absolute age depends on the geologic history of the region. In this work, we define the nominal subset of freshest craters by a cutoff value c = 0.64. In choosing a nominal cutoff value, we are guided by the following bounding constraints: (1) the criterion should provide large enough numbers of craters in the freshest subset to derive robust fits to size-dependent characteristics, and (2) the criterion should preserve the observed regional differences in geometric properties. For values of c between 0.55 and 0.75, the number of craters identified ranges from 292 to 190. The dependence on the value of c is uniform across each region, with 30–40% fewer craters identified when raising c from 0.55 to 0.75. For our nominal value of c, the depths of the freshest craters in the strength regime show a clear separation between lowland and highland terrain (Figure 2a). In each region, the freshest crater population is expected to be Amazonian in age (<2–3.2 Ga [Hartmann, 2005]). The derived crater densities in Utopia Planitia are consistent with an early/middle Amazonian population (discussed in section 4.4).
 In this work, the overall results are not sensitive to the precise cutoff value. The locations of the freshest craters defined with c = 0.64 are shown in Figure 3. This set of 260 freshest craters includes 121 craters in Utopia Planitia (blue circles), 28 craters in Isidis Planitia (squares), 30 craters in Acidalia Planitia (red pluses), 48 craters in Lunae Planum (red triangles), and 33 craters in Solis Planum (diamonds). About half of the craters in this subset have single layer ejecta (SLE, see nomenclature definitions by Barlow et al. ), one quarter have double layer ejecta (DLE), and one quarter have multiple layer ejecta (MLE) morphologies.
3. Measurement Results: Excess Ejecta Craters
3.1. Identification of Excess Ejecta Craters
 We identify a population of impact craters with anomalously high volumes of ejecta and uplifted material. A formal definition of excess ejecta craters is presented below. The measured ejecta volumes as a function of surface diameter are shown in Figure 4. The ejecta volume values are calculated using a preexisting surface uplift exponent n = 5.5 in order to avoid over subtraction of material near the crater rim [Stewart and Valiant, 2006]. Power law fits in both the strength and gravity regime for each region show that the lowland regions have slightly higher ejecta volumes compared to the highland regions. In particular, there is a population of craters in Utopia with significantly larger apparent ejecta volumes compared to all the other regions. Craters with large ejecta volumes are identified in Figure 4b, which presents the excess ejecta volume, dVEjecta, after subtracting the fit to VEjecta in Lunae Planum. In the regions studied here, the craters in Lunae Planum have the least amount of scatter in the geometrical measurements of the freshest craters, so they are used as the reference set for comparisons between regions [Stewart and Valiant, 2006]. Craters with negative dVEjecta are not plotted. The 1σ error bars, based on measurement uncertainty, are only plotted for excess ejecta craters for clarity of presentation; error bars are similar in the other measurements of freshest craters. Note that power law fits presented in this section are tabulated by Stewart and Valiant .
 Precise measurements of the ejecta volume are difficult, however, because of the uncertainties in the profile of the uplifted background surface that must be subtracted to infer the ejecta volume. Therefore a more robust measurement is the total volume of uplifted and ejected material, VAbove (see Figure 1). Measurement validation tests show that the standard deviation in the measurement of VAbove is about 20 to 40%, depending on the roughness of the background terrain [Stewart and Valiant, 2006]. As shown in Figure 5, fits to the measurements of VAbove show that the freshest craters in lowland plains have larger volumes of ejected and uplifted materials compared to the freshest highland craters. As seen in the measurements of ejecta volume, there is a group of craters in Utopia with significantly larger amounts of material above the level of the background surface. These craters have values of VAbove more than twice the average for the freshest craters in Utopia.
 To investigate the origin of the large values of VAbove, we compare VAbove to the observed volume of the crater cavity below the background surface, VCavity, and to estimates of the maximum values of VCavity (denoted by Max VCavity) in each region (Figure 6). Neglecting changes in the bulk density of the material, mass conservation requires that the cavity volume be approximately equal to the volume of uplifted and ejected material immediately after impact and before any degradation. In Figure 6, the mean ratios of VAbove/VCavity are shown with a horizontal line for each region. As expected, the mean ratios are nearly one, and there is very little dependence on the crater diameter. Note that we found no obvious systematic difference of VAbove/VCavity values between SLE, DLE, and MLE type craters.
 Surprisingly, we find a subset of the freshest craters with volumes above the surface up to 5.8 times larger than their cavity volumes. Excluding craters in Utopia, the mean ratio VAbove/VCavity is 0.99 ± 0.41 for all the other freshest craters presented here. Therefore values of VAbove/VCavity greater than 2.5 are more than 3 standard deviations from the mean and fall significantly outside the natural variability within the freshest crater data set.
 We define the set of excess ejecta (EE) craters as craters that satisfy the geometric criteria for the freshest craters in their region and have values of VAbove/VCavity > 2.5. These craters are identified by the green squares. Ten craters in our data set have VAbove/VCavity ratios greater than 2.5 (Figure 6). Of these 10 excess ejecta craters, nine are located in Utopia Planitia and the remaining one is in Acidalia Planitia. The properties of the 10 excess ejecta craters are detailed in Tables 1 and 2. If we relax the VAbove/VCavity cutoff to 2.0, which is >2 standard deviations from the mean, 9 moderately excess ejecta (MEE) craters are found in the lowland plains regions. Of the 9 moderately excess ejecta craters, 5 are found in Utopia, 3 in Acidalia, and 1 in Isidis. These craters are denoted with green diamonds. The comparison to the estimated maximum crater cavity volumes (Figure 6b) is discussed below.
Table 1. Geometrical Measurements of Excess Ejecta Craters and Typical Fresh Utopian Cratera
Errors are ±1σ . Percent indicates HMars toolkit systematic measurement errors on populations of similarly sized craters; absolute indicates derived from natural variability of each crater and typically larger than systematic errors [see Stewart and Valiant, 2006].
Here 1–10 are excess ejecta craters and A is the typical fresh Utopian crater.
Here 1–10 are excess ejecta craters, and A is the typical fresh Utopian crater.
Here tuplift + ejecta, thickness of uplift and ejecta (from VAbove), and texcess, excess ejecta thickness (from VAbove - VUplift ), are averaged over 2π (REjecta2 - RRim2 ); n is the stratigraphic uplift exponent in equation (1).
 THEMIS and Viking imagery of the 10 EE craters, along with an example fresh, normal ejecta Utopia crater (crater “A”) for comparison demonstrate that the four smallest craters all have fresh rims and cavities, with little or no evidence of infill (Figure 7). In general, the THEMIS images depict an EE population that is relatively fresh overall, with fairly pristine ejecta. Some inner ejecta blankets do appear softened, which may indicate the presence of volatiles, as discussed in section 4.1. Altimetry profiles nearest the crater center (Figure 8) suggest that craters 1, 2, 3, 4, and 8 are the least infilled. The MOLA data coverage of each EE crater is good, and the geometrical measurements of each crater using the HMars toolkit are presented in Figure 9.
3.2. Verification of Excess Ejecta Craters
 Here, we critically examine the measurements of excess ejecta craters. First, we address the question of whether or not the observed VAbove/VCavity ratios could be accounted for by the natural variability of impact crater geometries. Specifically, we consider the effects of the observed departures from axisymmetry. On the basis of traces of the crater rim and the edge of the continuous ejecta blanket, azimuthal variations of the EE craters are typically <10% (Table 1 and Figure 7). These variations are less than the errors in the measurements of VCavity and VAbove and cannot account for the observed excess volumes greater than a factor of 2.
 Next, we consider an upper limit to the expected cavity volume for comparison to the observed VAbove. We estimate the maximum cavity volume for each region by using the fit to the maximum crater depth used to identify the freshest crater population (tabulated by Stewart and Valiant ). For simple craters, the fresh cavity shape is paraboloid, and the maximum crater volume is given by (π/8)ds,maxDs2. Complex craters have a broader cavity profile with an upper bound cylindrical volume of πds,maxDs2 (Figure 8). Applying these formulae to all crater measurements, the ratios of VAbove/Max VCavity are shown in Figure 6b and Table 2. The mean value for VAbove/Max VCavity using all craters except those in Utopia is 0.75 ± 0.28, much lower than the observed mean value of 0.99. A mean ratio less than 1 indicates either that the maximum VCavity model provides a real upper limit for VCavity or that all craters have experienced both significant erosion of the ejecta blanket and cavity infill in order to normalize their observed VAbove/VCavity ratios. Comparisons between the freshest Martian crater population to fresh craters on the Moon and Earth indicate that the latter is not the case [Stewart and Valiant, 2006]. Hence the maximum VCavity model does give an upper limit for each crater's cavity volume, providing a reasonable test to account for natural variability in cavity volume.
 As expected, the ratios of VAbove/Max VCavity for the excess ejecta craters are lower than the observed VAbove/VCavity values. However, all 10 excess ejecta craters have ratios significantly higher (2.6σ to 10.9σ) than the mean (Table 2). The moderately excess ejecta crater population, on the other hand, could be a result of natural variability and partial infill. Their VAbove/Max VCavity ratios group within the general freshest crater population. Therefore, while the moderately excess ejecta population is interesting, we cannot rule conclusively that the excess ejecta measurement for that set is robust.
 We note that the mean values of VAbove/VCavity and the number of EE craters are weakly sensitive to the definition of the freshest crater population. Some EE craters lie near the nominal geometric cutoff criteria of c = 0.64 for the freshest crater population, as defined in section 2.2. If the cutoff value were raised to 0.75, then craters number 1, 3, 4, and 5 would no longer be included in the freshest subset of craters. Lowering the cutoff value to 0.55 increases the number of EE craters to 13 by slightly lowering the mean value of VAbove/VCavity for each region. The additional 3 craters are drawn from the MEE population (defined by the nominal criteria), which do not show definitive excess volumes.
 As described in section 3.1, we rely on comparisons of VAbove with VCavity to measure excess crater volumes. In addition to this method, one could also compare the ejecta volumes to the expected excavated volume of material. Comparing VEjecta to VExcavated has higher margins of error than VAbove/VCavity because of the necessary assumptions about the volume of the transient crater. VExcavated can be approximated by a cylinder with a diameter of 0.7DR and depth of 0.07DR [Melosh, 1989; Stewart and Valiant, 2006]. Using this model, we find that the observed ejecta volumes for EE craters are larger than the estimated excavated volumes.
 Next, we consider whether the observation of values of VAbove/VCavity much greater than one could be due to infill. Over time, slumping, relaxation, and/or sediments may fill the crater cavity and increase the ratio of VAbove/VCavity. The cavity volumes for the freshest craters are shown in Figure 10. There is considerably less scatter in the cavity volume compared to VAbove and VEjecta. The measurement errors of VCavity are also smaller, only 10–20% [Stewart and Valiant, 2006]. In Figure 10, the excess ejecta craters do not have the largest crater cavity volumes, but neither do they have anomalously small crater cavities compared to the other freshest craters in Utopia. The offset in VCavity values between lowland and highland strength-dominated craters is attributed to differences in the effective strength of the surfaces between regions [Stewart and Valiant, 2006]. Note that all excess ejecta craters have cavity volumes larger than the average for Lunae Planum of the same diameter (Figure 10). Therefore the excess ejecta craters do not have anomalously small crater cavities.
 Another check for infill is performed by individually modeling fresh crater cavities for each EE crater. For the simple EE craters (all but craters 8 and 9), a second-degree polynomial fit to the MOLA altimetry points along the crater walls defines a surface of revolution model for the crater cavity, shown by the thick grey lines in Figure 8. Complex crater cavities (craters 8 and 9) are modeled as regular cylinders of the observed maximum crater depth to provide an upper limit to the original fresh cavity volume. For craters 1–7 and 10, infill is estimated to be no higher than 22% (in the case of crater 7), and for most craters is 10% or less. Craters 5 and 6, which appear to have moderate deposits of infill, are estimated to have only 10.6% and 8.2% infilling by volume respectively. As shown in Table 2, the VAbove/Fit VCavity ratios are only slightly lower than the observed VAbove/VCavity and higher than VAbove/Max VCavity, calculated above. We consider the VAbove/Max VCavity ratio to be a very conservative estimate for the excess uplift and ejecta volumes. The small difference between the observed and fit crater cavity volumes demonstrates that infill is a negligible component in the excess values of VAbove/VCavity.
 The depth and rim heights of the excess ejecta craters provide a geometric check for relative freshness (Figure 11). Recall that the definition of freshness is based on the deepest crater cavities and tallest rims in each region. Although the excess ejecta craters do not have the deepest cavities in the northern lowlands, neither do they have anomalously shallow cavities. Excess ejecta crater cavities are normal on a global scale as well; for example, all excess ejecta craters and most moderately excess ejecta craters have cavity depths and volumes larger than similarly sized craters in the highland plateaus. Similarly, the maximum rim heights of excess ejecta craters have average values within the freshest crater data set.
 We also consider whether or not bulking of the ejecta and uplifted surface could account for the high values for VAbove. Using a Maxwell's Z model fitted to Martian craters to describe the excavation flow of the transient cavity, Stewart and Valiant  calculate the relative contribution of uplift and ejecta to the total volume above the preimpact surface. They find that the unbulked ejecta accounts for about 40% of the total VAbove for the size range of craters studied here. Typical increases in volume due to fracturing of the uplifted surface will be less than the 5–10% bulking observed in the breccia lens within small terrestrial crater cavities [Grieve and Garvin, 1984]. Around simple craters, bulking of the near-rim ejecta will be similar to the breccia lens and increase with radial distance from the crater center. The ejecta around the 1.8-km rim diameter Lonar crater, which formed within the Deccan Traps, India, display features similar to the ground-hugging flow around Martian craters [Fudali et al., 1980; Maloof et al., 2007; Stewart et al., 2005]. At Lonar crater, the volume of the distal ejecta is on average about 20% larger than the bulk volume of the undisturbed country rock [Maloof et al., 2007]. From these observations of ejecta bulking on Earth, we infer that the final ejecta volume should be much less than a factor of 2 times the original volume. Therefore bulking of the ejecta and uplifted surface cannot account for more than a small fraction, comparable to the standard deviation of the measurements, of the observed ratios of VAbove/VCavity.
 For comparison, the same methods were used to examine the VAbove/VCavity ratio for a fresh normal crater, crater A, located at 107.4 E, 24.5 N in Utopia. Crater A has no significant infill in its deep cavity (Figure 8), which is indicative of a high strength near-surface layer compared to highland regions on Mars [Boyce et al., 2006; Stewart and Valiant, 2006]. The measured VAbove/VCavity ratio is 0.95, essentially identical to the ∼1 value expected for fresh craters. When each of the validation checks described above is applied to crater A, the VAbove/VCavity remains constant and close to one, indicating that there is no systemic preference in our measurements or checks toward high VAbove/VCavity ratios (Table 2). In addition, independent measurements of the EE and MEE craters by two or three HMars users verified the reproducibility of the geometric properties within the measurement errors.
 Hence we conclude that the identified set of excess ejecta craters is not an artifact of degradation, natural variability, or measurement error. The EE craters have fresh crater morphologies with anomalously high volumes of uplifted and ejected materials.
3.3. Quantity of Excess Ejecta
 Now that a set of EE craters has been identified, the quantity of excess material may be investigated in detail. First, the average thickness of uplifted and ejected material over the continuous ejecta blanket, tuplift + ejecta, is provided in Table 2. For simplicity, the volume VAbove is distributed equally over π(Rejecta2 − RR2) , where REjecta is the mean radius of the continuous ejecta blanket and RR is the mean crater rim radius. Note that in double layered ejecta blankets, the proximal ejecta is much thicker than the distal ejecta (see Figure 8 and Boyce and Mouginis-Mark ), and the excess thickness is an average over the combined inner and outer ejecta layers. The values of tuplift + ejecta range from 41 to 134 m. The ratios of VAbove/VCavity imply that more than half of this thickness is excess material, given by texcess = tuplift + ejecta (1-Vcavity/VAbove). The excess thickness values range from 27 to 108 m, with a mean value around 50 m (Table 2).
 Another method for estimating the excess thickness uses the expected ejecta volume rather than the crater cavity volume. In order to compare the observed VEjecta with predicted values, the uplifted volume must be subtracted (see Figure 1). The uplift profile, hU, was modeled by
where hU is the uplift height at a distance r from the center, HR is the crater rim height, and RR is the crater radius. The uplifted surface comprises about half the crater rim height and the decay exponent, n, ranges from 3.0 to 5.5 [Stewart and Valiant, 2006]. On the basis of terrestrial impact craters, the ejecta thickness, hEj, at a distance r from the crater rim is estimated by
for simple craters and
for complex craters [Kring, 1995]. The expected ejecta volume, VExpected ejecta, is calculated assuming axisymmetry around the ejecta blanket. The inferred ejecta volume, VAbove − VUplift, is calculated for n = 5.5 and n = 3.0. The excess ejecta is now estimated by
 The ratio of inferred ejecta to expected ejecta is given in Table 2.
 The excess volume ratios based on expected ejecta are also much greater than one, supporting the presence of excess volume based on comparisons to crater cavity volume. The excess thickness of material, averaged over the continuous ejecta blanket, is bounded by the two uplifted volume estimates (equation (1) with n = 5.5 and n = 3.0). The range of excess thickness agrees remarkably well with the excess thickness calculated from VCavity for both simple and complex craters (Table 2).
 Using the expected ejecta volumes, the mean excess thickness is around 40 and 50 m for uplift profiles with n = 3.0 and n = 5.5, respectively. In contrast, the estimated excess thickness for crater A was only –3 and 18 m for n = 3.0 and n = 5.5, respectively (Table 2). The tight agreement in average VExcess among EE craters using two independent methods increases our confidence in the derived values. The range of excess thickness, from 16 to 108 m, includes some of the uncertainty in volume measurements using the MOLA data set. The dispersion in values may also reflect a heterogeneous origin for the excess material and some of the natural variability in crater formation. The thickness of any icy deposits would probably have differed substantially depending on the location and moment of crater formation. On the basis of two independent estimates, we conclude that the detection of excess material, on average several tens of meters thick, is robust.
 We note that the presence of excess material biases the original measurements of the geometry of EE craters. For a nominal change in the surrounding surface of 50 m, the average error in VCavity is 12% (ranges from 6 to 19%, with the largest error for the smallest craters). This is comparable to our measurement error on VCavity. The unbiased values of VCavity are closer to the regional values (Figure 10), which confirms that the value of VAbove has the primary contribution to the high observed VAbove/VCavity values. The observed surface depths, rim heights, and surface diameters are slightly offset from the original geometry of the EE craters.
4.1. Geologic Context
 The excess ejecta morphology is not spatially random. Figure 3 identifies the locations of the excess ejecta craters. The EE craters are found in a latitude band between 32°N and 44°N, and the MEE craters lie between 17°N and 50°N. The concentration of EE craters in southern Utopia allows for closer examination of the possible origins of the excess ejecta morphologies. In Figure 12, normal fresh craters (open circles and squares) and excess ejecta craters (yellow squares and diamonds) are identified on Kreslavsky and Head's  kilometer-scale surface roughness map. In the roughness map, brighter colors denote a rougher surface, with 0.6−, 2.4−, and 19.2-km-scale roughness baselines represented by blue, green, and red tones, respectively. In southern Utopia, the surface roughness is dominated by early to middle Amazonian lava and lahar flows from Elysium Mons [Russell and Head, 2003; Tanaka et al., 2005, available at http://pubs.usgs.gov/sim/2005/2888/]. Russell and Head  interpret the rougher light blue units on the Elysium-Utopia flows as lahars and the smoother dark blue unit as lava flows. EE craters are found both on and off the Elysium-Utopia flows. To the north of Utopia is the circumpolar Vastitas Borealis Formation (VBF), a smooth layer ∼100 m in thickness. The VBF could have been deposited uniformly as it settled out of suspension in a northern ocean, or it could have been deposited preferentially in depressions (such as crater interiors) as a result of flooding [Boyce et al., 2005].
 The stratigraphic relationships between EE craters and the Elysium-Utopia flows combined with examination of nighttime thermal inertia data from THEMIS [Fergason et al., 2006] provide information about the relative ages and state of preservation of the EE crater ejecta blankets. In general, EE crater ejecta are identified in nighttime infrared imagery with bright (higher thermal inertia) crater cavities and dark (lower thermal inertia) ejecta blankets that contrast with the surrounding terrain. The thermal inertia (TI) pattern is indicative of fresh craters with rockier exposures in the crater wall and unconsolidated ejected materials, with craters 3 and 7 appearing most modified. EE craters exhibit some variation in states of preservation and age, as discussed in more detail below.
 The evidence for variable age is based on differing relationships with the Elysium-Utopia flows. Russell and Head  suggest that the flows were emplaced during the early to middle Amazonian. We considered relative ages among EE craters qualitatively, based on these spatial relationships with the Elysium-Utopia flow units and also visual appearance and thermal signatures. We do not attempt to quantify the relative ages or to suggest a specific order of formation for these particular craters.
 EE crater 1 is located near the western boundary of the Elysium-Utopia flows, on the Distal Unit, identified by Russell and Head  as lava flows mantled by a climatically controlled, 1- to 10-m-thick, volatile- and dust-rich layer. The ∼75-m-thick, inner ejecta blanket of this DLE crater is clear in visible imagery (Figure 7), but the thin, outer ejecta do not have a clear terminus. In the nighttime infrared (THEMIS image number I01594006), the outer ejecta, extending to about 3.4 crater rim radii (Figure 9), is clearly present and reasonably well preserved, with modification from erosion or mantling. Therefore the crater is younger than the lava flows within the Distal Unit. Nearby, similar and slightly smaller sized craters are infilled (to the –5000 m datum) with lower thermal inertia deposits suggesting that crater 1 (cavity depth to the –5540 m datum) has not been modified by flooding of the basin related to the formation of the VBF, which would have preferentially infilled the lowest available depressions [Boyce et al., 2005].
 The ∼100-m-thick ejecta blanket around EE crater 2 is more eroded and without a clear rampart terminus. There is a thermal inertia contrast to the surrounding plains that is slightly larger than the area of the apparent visible ejecta blanket, which extends to 3.2 crater rim radii and provides a tentative DLE classification. While crater 2 has a high TI cavity with depth to the –5520 m datum, a smaller crater due east has been infilled with low TI material to the –4930 m datum (THEMIS I10544010). Craters 2 and 8 are located in Adamas Labyrinthus, an unmantled region dissected by polygonal troughs, adjacent to the eastern front of the Elysium-Utopia flows (Figure 12). A common feature among crater ejecta in the region is the appearance of erosion by retreat of the distal ejecta (Figure 7). Several smaller craters in the region have similar morphologies (with both infilled and fresh crater cavities), but are not well resolved by the MOLA topography. For example, a 4.5-km rim diameter crater at 103.8°E, 36.5°N appears to have VAbove/VCavity ∼ 2.
 EE crater 3 is located at the base of the Elysium Rise, on top of the Smooth Lobate Unit lava flows [Russell and Head, 2003]. The southern ejecta blanket is cut by a lahar deposit (Rough Lobate Unit). The apparent background level for the crater is the Smooth Lobate Unit. Parts of the southern crater cavity and ejecta blanket are covered by a high-albedo, low thermal inertia, presumably dusty mantle. The edge of the inner layer of the DLE crater is not well defined and the distal ejecta edge extends to 4.4 crater rim radii. Nearby craters all have low TI deposits within the crater cavities (THEMIS I10493013).
 EE crater 4 has formed on the Elysium Rise Unit AHEe [Tanaka et al., 2005, available at http://pubs.usgs.gov/sim/2005/2888/]. The DLE crater is reasonably well preserved (THEMIS I09831020) with a ∼100-m-thick inner ejecta layer. Most nearby craters are infilled by Elysium lava flows. EE crater 5 is a DLE crater that formed on the Smooth Lobate Unit, near the boundary to the Distal Unit and EE crater 1. The nighttime TI is characteristic of a fresh crater with partial infill by low TI materials (THEMIS I18743009). The ejecta blanket is partially eroded, but displays clear radial grooves from the inner ejecta (∼100 m thick) to outer ejecta blanket.
 EE craters 6, 7, and 9 formed on the Etched Unit of the Elysium-Utopia flows, which are interpreted as lava flows covered by lahar or fluvial deposits [Russell and Head, 2003]. The nighttime TI shows a relatively fresh crater signature for crater 6 with partial infill of low TI materials and partial removal of the distal ejecta (THEMIS I10306010). The inner ejecta of the DLE crater are about 100 to 150 m thick. The nighttime TI for DLE crater 7 is less distinct, but shows high TI crater walls (THEMIS I04914002), and the inner ejecta are about 100 to 150 m thick.
 EE craters 8 and 9 have similar size, ejecta morphologies, and fresh crater nighttime TI signatures (THEMIS I04865005 and I04533002). Both are craters with typical DLE morphologies [Boyce et al., 2006], with softened inner ejecta layer, slightly eroded outer ejecta layers, and radial grooves that cut continuously across the inner to outer ejecta. The inner ejecta layers are about 150 m thick, with ramparts up to 250 m tall. Some craters near crater 8 are infilled to around the –4700 to –4900 m datum and have ejecta blankets with indistinguishable TI from the background plain.
 EE crater 10 is located in Acidalia Planitia on a region of the Vastitatis Borealis interior unit characterized by polygonal troughs, just south of Acidalia Mensa [Tanaka et al., 2005, available at http://pubs.usgs.gov/sim/2005/2888/]. The nighttime TI pattern is for a fresh DLE crater (THEMIS I17400010). Nearby craters display similarly large ejecta blankets, but are partially infilled.
 The locations of EE and MEE craters share similar surface roughness traits. The Utopian excess ejecta craters are situated on areas with high small-scale roughness, in the midst of a band of 0.6-km-scale (blue toned) roughness in the Utopian midlatitudes [Kreslavsky and Head, 2000]. Previous researchers have identified a smoothed terrain interpreted as an originally ice-rich mantle deposit with a thickness of 1–10 m, which is continuous and well preserved above 60° latitude but is undergoing desiccation and erosion in the region between 30° and 60° latitude [Head et al., 2003; Mustard et al., 2001]. The EE and MEE craters occur between 30° and 60°, where this terrain is undergoing degradation. At subkilometer and kilometer scales, surface roughness can be influenced by deposits only several meters thick, because vertical topography in the northern plains is often limited, with slopes in the range of 1° or less [Head et al., 2003]. Thus the 0.6-km-scale roughness band mentioned above could be a result of uneven desiccation of this volatile-rich layer. On the basis of superposed crater densities, Head et al.  suggest a maximum age for these deposits of 10 Ma, and devise a Martian ice age model in which ice transported from polar regions during high-obliquity intervals would mix with windborne dust to form the icy mantle deposits. After a return to lower obliquities, the gradual sublimation of the ice content would weaken this layer, making it susceptible to partial erosion and removal.
 We identified another water-related feature at several of the EE craters, in the form of Martian gullies. The latitudes where excess ejecta craters are observed also coincide with the latitude bands from roughly 30° to 70° north and south where Martian gullies have been found [Bridges and Lackner, 2006; Heldmann and Mellon, 2004; Malin and Edgett, 2000]. Among our EE craters, high-resolution MOC images reveal gullies originating on the walls of craters 1, 2, 5, 7, 8, 9, and 10. Craters 4 and 6 did not appear to have fresh gullies, but do have evidence for some mass wasting on the crater walls. No high-resolution MOC or THEMIS images were available for crater 3. Among the craters with gullies, craters 8, 9, and 10 displayed particularly well-defined examples, with channels originating at particular layers near the top of the crater walls and extending down to the crater floor. Gullies are among the youngest geologic features on the planet [Malin et al., 2006], and while the mechanism(s) that generate Martian gullies are not yet clearly understood, one proposed mechanism is related to glacial deposits of ice [Christensen, 2003]. Hence future work could examine the relationships between EE craters and gullies in more detail including a search in the southern hemisphere for EE craters.
 Another interesting phenomenon in the Utopia impact basin is the presence of a population of impact craters with anomalously deep cavities. Layering and high target strength have been advanced as explanations for this phenomenon [Boyce et al., 2006; Pike, 1980; Stewart and Valiant, 2006]. Note that the strong mechanical strength characterizes a buried layer at least a few kilometers thick [Stewart and Valiant, 2006]. The thermal inertia data suggest that Utopia Planitia shares the dominant Martian surface of duricrust, sand, rocks and bedrock [Putzig et al., 2005]. We note that EE craters are not the deepest craters in Utopia (Figure 11) and may have formed in slightly weaker areas or experienced more gradation compared to the Utopian craters with the deepest cavities.
 In the THEMIS imagery, most of the EE craters have clear double layer ejecta structures and most of the inner layers have a softened appearance. This trait could be an indicator of volatile enrichment leading to ice creep [Squyres and Carr, 1986]. The blotchiness exemplified by the inner ejecta of craters 4, 8 and 9 resembles the dissected terrain described by Mustard et al. , which is proposed as an area where ice deposits have sublimated, loosening the surface terrain. The observed blotchiness could thus be a sign of recent or persisting subsurface volatiles. MOLA altimetry profiles of the EE craters (Figure 8) show few identifiable patterns totally unique to EE craters. Several of the profiles have an outer rim syncline (or moat) and intermediate flat segments that could represent terraces on the outside of the rims. Craters 1, 2, 3, 4, 5, and 9 exhibit this trait to a greater or lesser extent. The observed ejecta topography is similar to those produced by impact cratering simulations into surfaces with layers of weak and strong materials [Senft and Stewart, 2007a, 2007b].
 As described above, many of the EE craters either overlap or are close to the Elysium lava flows and the lahar units described by Russell and Head . Despite this proximity, we do not believe the lahars or the flows are responsible for the excess ejecta volumes we have observed. Several of the EE craters we have identified are not located near either lava flows or lahars. Furthermore, our measurements are based on original background levels to account for surface features like the Elysium flows. Among the EE craters, crater 9 is the only case where original background levels are less certain, due to its position partly on and partly off a lava flow. Measurements of crater 9 using different background reference levels are within the reported error bars.
 On the basis of stratigraphic relationships with the Elysium lava flows, the EE craters appear to have formed during and after the early to mid-Amazonian flow events. Finally, the texture and topography of the ejecta blankets around EE craters and their latitudinal association with gullies and dissected terrain suggest some relationship with near-surface ice deposits.
4.2. Interpretation of Excess Ejecta Craters
 The standard model of impact crater formation, with cavity excavation, uplift, and ejecta emplacement, is insufficient to explain the excess ejecta craters observed in our data. If the crater cavity is the only source for material, the ejecta mass cannot surpass the mass of material excavated from the cavity. As discussed above, comminution alone cannot explain the excess volumes. In order for VAbove to be significantly greater than VCavity, material from another source must be included in the measurements of VAbove.
 There are two options for inflating VAbove: (1) mantling, where material is added to the ejecta blanket (and perhaps the surrounding terrain), or (2) deflation, where material was removed from the terrain surrounding the ejecta blanket, lowering the reference level dividing VCavity and VAbove (Figure 1) and raising the ratio of VAbove/VCavity. Cavity infill has been ruled out in section 3.2 as an option for inflating the VAbove/VCavity ratio. There are several processes that offer potential explanations for the excess ejecta; each possibility is evaluated below. The minimum requirements of any hypothesis are that it preserves rim height and cavity depth while increasing the apparent volume of ejecta.
 An ice-rich mantle of material may be deposited at midlatitudes during periods of high obliquity. On the basis of climate models and geologic studies of surface deposits, the thickness of ice-rich layers may range from about 10 m to about 1 km thick [Forget et al., 2006; Milliken et al., 2003; Mustard et al., 2001]. In a study of high-latitude impact craters in the Vastitas Borealis Formation (north of 52°N), Kreslavsky and Head  find that an Amazonian icy mantle has smoothed the ejecta blanket by filling in the topographic lows. Because of the larger surface area of the ejecta blanket compared to the crater cavity, the mantle would inflate the apparent volume of the ejecta blanket to a larger extent than decreasing the crater cavity volume and produce a large VAbove/VCavity ratio. For the lower-latitude craters in this study, mantling alone cannot explain the observations. Although some of the ejecta blankets may be partially mantled, craters 5, 8 and 9 display clear radial grooves, which are typically less than 20 m deep [Boyce and Mouginis-Mark, 2006]. The inferred excess thickness of material is much larger (Table 2). The topographic profiles across the ejecta show that the topographic lows are not all infilled (Figure 8), and the thermal inertia signatures of the craters are consistent with primary ejecta.
 Alternatively, the apparent excess in ejecta volume may be due to deflation of the surrounding terrain. A mechanism for generation of episodic, transient surface layers is obliquity variations. Recent research suggests that Martian obliquity variations in the last tens of millions of years could have led to glacial deposits of ice at low and middle latitudes [Forget et al., 2006; Hauber et al., 2005; Head et al., 2003, 2005; Levrard et al., 2004; Smith et al., 2001]. Icy surface layers would leave behind a lag deposit, of suspended dust and rock, after sublimation of the ice fraction. Erosion of the lag deposit (e.g., by aeolian processes) would expose the original surface.
 We find that formation of craters within a layer of material that has subsequently been removed from the surrounding terrain is most consistent with the observations. In our proposed scenario, summarized in Figure 13, an impact crater forms at a time when an icy surface layer is present. The ejecta blanket, a mix of material from the original surface and the icy layer, covers the icy layer surrounding the crater cavity. Then, from obliquity-driven climate change, the ice fraction sublimates and the lag deposit is susceptible to erosion. The ejecta blanket mantle protects the icy material surrounding the crater cavity. The ejecta blanket also insulates the icy layer and prolongs the time period for sublimation of the ice. The terrain around the observed ejecta blanket is lower than the original surface at the time of impact. The excess material in the apparent ejecta blanket may be a pure lag deposit or may possibly contain some of the original ice fraction.
 We also considered the deflation of a dry (non-icy) layer of material. We prefer the deflation of an ice-rich layer of material for the following reasons. The surface morphology of the inner ejecta blankets suggests softening by partial deflation. Partial deflation is inconsistent with an originally dry layer of material but is easily explained by partial sublimation of ice from beneath the ejecta. In addition, the EE craters are associated with the latitudinal zone of mantled and desiccated materials formed by recent icy deposits [Mustard et al., 2001]. The stratigraphic relationships of the EE craters indicate that they did not all form at once, during the same depositional circumstances. Instead, the observed excess materials were probably generated by icy layers deposited at different times for different craters. Glacial and interglacial periods provide a mechanism for this episodic formation of EE craters.
 We note that the temperature field following crater formation would preserve an ice-rich surface layer. Melting and vaporization of the surface rock is confined to a small volume compared to the observed crater cavity. Melting of ground ice is confined primarily to material excavated from the crater cavity [Stewart et al., 2004]. Numerical simulations of crater formation in layered terrains on Mars indicate that the ejecta blanket is not significantly heated [Senft and Stewart, 2007a, 2007b]. Most of the continuous ejecta blanket experiences peak pressures less than the elastic strength of basalt (about 5 GPa [Nakazawa et al., 1997]); hence most of the shock deformation can be accommodated elastically and there is negligible heating. The maximum effect of frictional heating after ballistic emplacement can be estimated from the kinetic energy of the ejecta blanket. For a nominal 10-km-diameter crater formed in basalt, about 50% of the continuous ejecta blanket experiences a peak pressure ≤1 GPa, and about 90% a peak pressure ≤5 GPa. A peak pressure of 1 GPa corresponds to a particle velocity of about 250 m s−1. If all of the kinetic energy is converted to heating the same parcel of material (an unlikely scenario), the maximum temperature increase is several tens of kelvins (78 K for a heat capacity of 800 J kg−1 K−1). So, under nominal present-day average surface temperatures around 200 K, the ejecta blanket should remain below freezing temperatures and cool through radiative, conductive, and convective heat loss processes.
 In our model, the ejecta blanket would include material derived from the proposed icy layer. The ice component of the ejecta would also sublimate and decrease the apparent volume of the ejecta blanket. Similarly, sublimation of ice that collapsed onto the original crater floor would slightly increase the apparent value of VCavity. Both effects are within the error in the measurement of VEjecta and VCavity (Table 1).
 The observed average excess thickness (tExcess in Table 2) should then be considered an upper limit to the mean thickness of the ice that has sublimated from the deflated layer. The lower limit on the thickness of ice is a factor of two less, in the case where half of the sublimated ice was derived from the overturned icy layer within the ejecta blanket. The volume fraction of ice within the ejecta blanket depends on the thickness of the icy layer, the size of the crater, and the radial distance from the crater rim. The actual thickness of the original icy layer also depends on the fraction of dust and rock. In section 4.3, we consider a wide range of effective burial depths and ice to dust/rock ratios in the original icy layer.
 In this paper, we propose an evolutionary sequence for midsized midlatitude craters that formed in a glacial environment, but the interaction of volatiles with the cratering record may vary as a function of latitude and crater size. At latitudes above 52°N, Kreslavsky and Head  find evidence for a mantling layer filling topographic lows on top of ejecta blankets. Barlow  proposes a deflation model to explain the small (<6 km DR) pedestal craters found between 35°N and 60°N. Although our study is more general, and does not make assumptions about the genetic relationship of the EE craters to other morphologies, Barlow's model is similar to the sequence we propose for our EE craters: an impact follows high-obliquity ice deposition; the armoring of volatiles by ejecta preserves the terrain under pedestal craters as the surrounding topography undergoes up to 1 km of deflation.
 Next, we consider the effect of an insulating ejecta layer on the stability of ground ice and the longevity of a buried icy layer.
4.3. Stability of Ground Ice Beneath Impact Crater Ejecta Blankets
 To determine the plausible longevity of a buried icy layer on Mars and the effect of variable burial depths, we constructed a simple model for volatile sublimation. The flux of vapor molecules through a porous medium, Fv, can be modeled with Fick's law, where
in molecules m−2 s−1. D is the diffusion coefficient and δN/δZ is the vertical gradient of the vapor number density. Under present atmospheric pressures of ∼700 Pa, the mean free path of a gas molecule is of the same order (8.2 μm) as the pore size; consequently, the diffusion regime is indeterminate. We apply Knudsen diffusion [Moore et al., 1996], achieving similar diffusion coefficients to the range of diffusion coefficients used by previous workers [Hudson et al., 2004; Mellon and Jakosky, 1993; Schorghofer and Aharonson, 2005; Stewart and Nimmo, 2002]. In Knudsen diffusion through a porous medium, the diffusion coefficient for the water vapor is
where ϕ is porosity, τ is tortuosity, mw is the molecular weight of water, T is the temperature of the ice, R is the gas constant, and rp is the pore size [Heer, 1972]. As nominal values, we take porosity to be ϕ = 0.1 and tortuosity to be τ = 5 [e.g., Stewart and Nimmo, 2002]. We assume that the air just above the ice layer is saturated. δN is the difference between the top of the icy layer and the surface. At the surface, the number density is given by
where Pw is the partial pressure of water in the Martian atmosphere and NA = 6.022 × 1023 is Avogadro's number. The water saturation vapor density curve N is given by a standard exponential,
where k = 1.38 × 10−23 J K−1 is Boltzmann's constant [Moore et al., 1996]. Because the depth Z changes as the ice sublimates, δN/δZ is calculated iteratively and the vapor flux Fv evolves as the depth to the top of the icy layer increases.
 At each step, the time to sublimate a given thickness h is given by
where mw is the molecular weight of water. The time needed to sublimate the ice is the sum of the individual sublimation times calculated for a discrete thickness h.
 Many of the parameters used in equations (5) through (9) can vary widely, reflecting variations in Martian environmental conditions from equator to pole, from month to month, and from glacial to interglacial periods. In cases where parameters were uncertain, a range of values was explored. Sublimation time is inversely proportional to pore radius; values presented in Table 3 are based on a pore radius of rp = 10 μm, consistent with the parameters assumed by Mellon et al. . Under current conditions for most of the latitudes in question, 210 K is a fair approximation of the surface temperature; 190 K is representative of prevailing temperature in the midlatitudes during a high-obliquity episode.
Table 3. Sublimation Time Scales for a 50-m-Thick Icy Layer as a Function of Temperature, Initial Burial Depth, and Initial Ice Fraction
Initial Burial Depth, m
Sublimation Time, Ma
TS = 190 K
TS = 190 K
 Midlatitude ice that experiences one of these obliquity shifts could therefore be preserved longer as a result of lower temperatures during the high-obliquity period. How much longer it would take to sublimate such ice depends on the duration of that high-obliquity period. According to Armstrong et al.'s  1 Ga climate model, Mars has spent about 6% of the last billion years at obliquities higher than 40°, so this effect is probably moderate.
 The parameters used in these calculations are consistent with thermal parameters reported by previous workers. Our approach yields diffusion coefficients of D = 6.6 × 10−5 m2 s−1 at 210 K and D = 6.3 × 10−5 m2 s−1 at 190 K. Mellon and Jakosky  suggest a theoretical diffusion coefficient in the comparable range of D ≈ 1 × 10−4 to D ≈ 2 × 10−3 m2 s−1 at 200 K with 10-μm pore radii, while experimental results fall into a similar range between D ≈ 5 × 10−4 to D ≈ 1.5 × 10−3 m2 s−1 [Hudson et al., 2004]. Mellon et al.  estimate a higher cumulative sublimation rate for nonequilibrium ice; their more complex model suggests that a 200 m equatorial icy layer would completely sublimate in 19 Ma. Our results are comparable and provide an upper limit to the stability of buried ice held at a constant temperature.
 Sublimation rates, shown in Figure 14, are roughly proportional to burial depth. Burial in more than 1 meter of dust and rock can greatly increase the stability of ice. Dust deposition rates on Mars are uncertain; Aharonson et al.  suggest that they are fairly low, perhaps 0.4 g m−2a−1. Glacial lag deposits may be more efficient in achieving substantial burial depths. The lag deposit helps to bury and insulate the ice, further increasing its stability as sublimation progresses. However, we note that due to the Martian geotherm, ice stability begins to decrease below a few hundred meters. This geothermal effect is negligible for the thickness of ejecta blankets examined in this work.
 The estimated sublimation times affirm that buried ice can remain stable for millions of years and that burial depth is an extremely significant factor. The sensitivity of ice stability to environmental variables is shown in Table 3. Here we consider a nominal layer that is 50 m thick with varying ice content, similar to the average thickness of excess material in Table 2. At a constant temperature of 210 K, a 50-m-thick 50% ice layer by volume, buried under 10 m of ejecta (10% porous, 10-μm pores), would sublimate in around 34 Ma. By contrast, the same ice layer unprotected by ejecta would sublimate in less than 19 Ma. Therefore an icy layer under an ejecta blanket could persist for millions of years more than exposed icy deposits. Sublimation timescales are inversely proportional to porosity and pore size; however, the timescales vary substantially with temperature (which could vary to about 190 K with obliquity) and ice fraction. At a constant temperature of 190 K, the 50-m-thick, 50% ice layer buried at 10 m could be stable for several hundred millions of years. As Mars passed through glacial-interglacial periods, the sublimations rates would vary with temperature. Note that, under no or little initial depths of burial, very high ice fractions are less stable than small ice fractions because high ice fractions leave thinner lag deposits.
 Several complicating factors are neglected in this analysis. The insulation effect from an ejecta blanket over an icy layer depends on the composition and porosity of the ejecta as well as how much physical mixing occurs during ejecta emplacement. The extent of mixing of the icy layer into the ejecta deposits is sensitive to whether or not the icy layer is initially buried or at the surface [Senft and Stewart, 2007a, 2007b]. Significant mixing of the icy layer will reduce the stability timescales because of the net reduction in insulating layer thickness. In addition, more realistic models of vapor diffusion of the sublimating ice through a porous layer should also account for a pore/fracture size distribution and adsorption of water and plugging of pore spaces [e.g., Mellon et al., 1997; e.g., Schorghofer and Aharonson, 2005]. Finally, because of the strong sensitivity to temperature, more accurate stability analyses should consider time-varying surface temperatures.
 Nevertheless, these simple estimates for ice longevity demonstrate that unless surface ice is being deposited very recently or presently, the upper meters are depleted very quickly by sublimation (Figure 14). Hence ice is not expected to be present in the upper ∼1 m of EE ejecta blankets today. Our calculations demonstrate that sublimation rates would be much faster in the outer ejecta blanket, where the insulating layer is thinner. However, ice buried beneath an insulating layer tens of meters thick could persist through glacial-interglacial periods. Recent climate models suggest that ice deposition could be very rapid in small regions, up to several tens of millimeters per year, at 45° obliquity [Forget et al., 2006]. Plausibly, over multiple glacial periods, icy layers at middle and low latitudes could accumulate to several tens of meters or more [Milliken et al., 2003]. If the Amazonian climate has been dominated by glacial periods with low-latitude ice deposition, then episodes of ice accumulation and sublimation could form the EE crater structures observed today. Next, we consider the likelihood of forming the number of observed EE craters based on the frequency of impact cratering events and glacial periods.
4.4. Cratering Frequency and Glacial Periods
 Recent climate simulations and models of Martian obliquity provide overall results that support the viability of remnant icy layer deposits. The high-precision dynamical model of Laskar et al.  suggests that there have been high-obliquity periods (of more than ∼45°) on Mars in the past 20 Ma. Laskar et al. also derive a mean obliquity of 35° over the past 250 Ma, with a standard deviation of 10°. Mischna et al.  predict that during a 35° obliquity period, substantial amounts of ice would accumulate in roughly the region of the excess ejecta craters: on the flanks and to the west of Elysium Mons, and in the Acidalia Planitia lowlands. Depending on the length of time between high-obliquity episodes, Milliken et al.  suggest that successive glaciations could have a cumulative effect. Milliken et al.'s viscous flow features are concentrated within the 30°–50° latitude range; they argue that viscous flow features are caused by flow and deformation of ice and dust layers, which may or may not continue to be active. While they assume an icy layer about 10 m thick, they point out that with a typical glacial period deposit of 1–10 m [Milliken et al., 2003; Mustard et al., 2001] repeated at 100-ka intervals, a 1-km-thick icy layer could accrete after 10 Ma. This episodic emplacement of volatile-rich materials could help explain the tens of meters of excess material we observe. In addition, the texture of Mustard et al.'s  intact ice-dust mantle and the dissected terrain where the ice has sublimated resemble patches of smooth, lobate ejecta juxtaposed with “softened” ejecta, as seen in some of our EE craters (e.g., crater 4 in Figure 7).
Forget et al.  also predict glacial deposits on the flanks of Martian volcanoes, including Elysium Mons. Significantly, they find that hundreds of meters of ice could accumulate very rapidly, within several thousand years. With frequent high-obliquity intervals lasting roughly 40,000 years [Head et al., 2003], there would be ample opportunity for impacts to occur on icy surfaces. The most recent 45° obliquity period was ∼5 Ma, prior to which high-obliquity periods would have been common [Forget et al., 2006]. Head et al.  propose that an ice age could have occurred as recently as 0.4 Ma. Such an ice age would be too recent to accumulate significant craters, but it is indicative of the ongoing volatile presence in the midlatitudes.
 Following the method of Hartmann , we estimate the apparent retention age of the fresh crater population in Utopia Planitia. We used the region bounded by 20°N, 40°N, 95°E, and 135°E, a quadrangle that includes most of Utopia Planitia, as our sample area. As Figure 15 shows, the number density of the freshest crater population in Utopia lies between the 1 and 2 Ga isochrons. The errors are derived by varying the cutoff fraction c from 0.55 to 0.75 in the geometric definition of the set of freshest craters (section 2.2). In cases where the cutoff criteria did not change the number of craters in an individual size bin (e.g., when all the craters satisfied c = 0.75), the error bar represents ±1 crater. The apparent age of the freshest crater population is in excellent agreement with the early to mid-Amazonian age of the Vastitas Borealis Formation and the Utopia-Elysium flows [Hartmann, 2005; Tanaka et al., 2003]. Although it is not a statistically significant sample, the excess ejecta crater population has an apparent age of approximately 100 to 200 Ma. In both sets of data, the number density of smaller craters (DR < 8 km) is incomplete and limited by the resolution of the MOLA data.
 Because of the stratigraphic relationships between the EE craters and the Elysium-Utopia flows, we do not interpret the apparent age of the EE craters as the actual formation age. Instead, the apparent age indicates that the time interval(s) required to form the EE population is a small fraction of the formation age of the sediment deposits and volcanic flows in Utopia basin. The formation time of the EE crater set seems to span the Amazonian; hence we interpret their low number density to imply an association with a rare and perhaps periodic phenomenon. In our proposed formation model, we expect excess ejecta craters to form during or in the aftermath of several different glacial episodes.
 It is important to note that many potential EE craters may have been buried or eroded, disqualifying them from this study. Other craters that might have excess ejecta were excluded because they were infilled. We do not claim to have identified every EE crater, because of our conservative criteria for resolution and freshness, including our constraints on infill. For example, we believe that the latitudinal dependence of obliquity-related volatile deposits makes it highly probable that there are additional EE craters in Acidalia. These craters may be difficult to identify because they have suffered a different gradation history than craters in Utopia. Based only on VAbove and VEjecta, several craters in Acidalia appear to be EE candidates, however their cavity volumes indicate infill. For example, two Acidalia craters at 348.2°E, 46.3°N and 351.0°E, 43.4°N have VAbove and VEjecta substantially higher than most Acidalia craters and close to values for Utopia EE craters, but they are moderately and highly infilled, respectively, disqualifying them from EE status. In Utopia, we also identify fewer small EE craters than we expect. This may be caused by the heightened sensitivity of smaller craters to infill. A small amount of infill has a larger effect on smaller craters, causing them to be more easily excluded. In addition, smaller craters are harder to resolve using the MOLA data. For these reasons, we believe that there are more EE craters than the 10 examples we have presented. Further study and identification of infilled EE craters in Acidalia and smaller impact craters using new topography data sets from Mars Express could provide new perspectives on the EE morphology.
 In summary, crater densities and climatic modeling support the conclusion that 10 (or more) craters with diameters greater than 1 km could plausibly have formed during a glacial period in the Utopia-Elysium region. The age of the fresh crater population is estimated to be 1–2 Ga, more than 10 times the interval required to form the EE craters. Laskar et al.  predict that in the past billion years, Mars probably spent about half its time at obliquities greater than 35°. Mischna et al.  predict that at obliquities of 35° or higher, surface ice will begin to accumulate at midlatitudes on Mars, in particular, in the vicinity of Elysium Mons. An icy layer deposited during a high-obliquity glacial interval is therefore a viable hypothesis to help explain the genesis of excess ejecta craters.
 Analyses of the geometric properties of the population of freshest craters in Lunae Planum, Solis Planum, Utopia Planitia, Isidis Planitia, and Acidalia Planitia revealed the existence of excess ejecta (EE) craters in the lowland plains at midlatitudes. Several percent of the freshest craters in Utopia and Acidalia planitiae have volumes of material above the background surface (Vabove) that are significantly larger than the cavity volume below the background surface (Vcavity).
 We define a population of excess ejecta craters with VAbove/VCavity ratios greater than 2.5 (>3σ from the mean). A critical examination of the morphology and geometry of the anomalous craters confirms that the detection of excess ejecta is robust. The identified excess ejecta craters have undergone relatively little degradation. Independent calculations of the ratio of observed to expected ejecta volumes, recorded in Table 2, confirm the existence of unexplained extra volumes of ejecta. The mean excess thickness of material averaged over the continuous ejecta blanket ranges from about 20 to 100 m. Because mass conservation dictates that, before degradation, the volume of uplift plus ejected material should be roughly equal to cavity volume, a process unrelated to crater formation is invoked to explain VAbove/VCavity ratios significantly greater than one.
 After consideration of several possible mechanisms for the formation of excess ejecta, we advocate a model based on the deflation of an ice-rich surface or near-surface layer of material. Our model, developed independently, is similar in concept to, but more detailed than, the deflation events proposed by Meresse et al.  and Barlow . The geometric analyses of the excess material strongly support a model for deflation of the surrounding terrain leading to the apparent excess ejecta thickness. Here, we argue that the removed layer was likely ice-rich. In our model (described in Figure 13), the preimpact terrain possesses one or more layers of an ice-rich material at or near the surface. As the climate changes from obliquity variations, the icy material sublimates and leaves behind a lag deposit. However, the ejecta blanket from the impact crater insulates the icy layer, extending the longevity of the ice deposit. Subsequent erosion of the lag layer unprotected by the ejecta blanket deflates the surrounding surface. The lag deposit is preserved beneath the ejecta blanket, producing the excess thickness observed today.
 We find that the ages of EE craters span the Amazonian period, indicating episodic formation. The range of latitudes where EE craters are found corresponds with the latitudinal band of viscous flow features associated with glacial episodes [Head et al., 2003; Milliken et al., 2003; Mustard et al., 2001]. The correlation suggests an association with high-obliquity periods. Many of the EE crater ejecta blankets appear softened, as if partially deflated from sublimation from ice located below the surface. These observations support the hypothesis that an ice-rich layer of material has been deflated.
 The plausibility of the model is tested by considering the sublimation timescales for a buried ice deposit and the frequency of impact cratering events and glacial periods. We demonstrate that burial beneath ejecta tens of meters thick significantly extends the timescale for stability of an ice deposit. The frequency of high-obliquity periods on Mars over the last several hundred millions of years is consistent with the observed minimum number of EE craters as a proportion of the total number of fresh craters.
 If this model is correct, it is plausible that ice deposits could be present beneath the ejecta blankets of some EE craters. EE craters can provide a means for estimating the volumes of material involved in low-latitude glacial deposits. More detailed studies of the structure and composition of the crater walls and ejecta layers could provide information about the properties of midlatitude icy deposits during glacial periods.
 This work was supported by NASA grant NAG5-13474 and the Harvard College Research Program. Nadine Barlow, Joseph Boyce, and Robert Carlson provided insightful reviews that improved this manuscript. The authors thank Vaidya G. Rajagopalan, Zain Khalid, and Greg Valiant for their assistance with crater measurements.