Thermal inertia, a parameter calculated from surface temperatures obtained from spacecraft, has long been used to quantify the amount of loose, fine-grained material on the Martian surface. With little “ground truth” available, studies often refer to Martian dune fields to calibrate thermal inertias. The well-understood physical properties of dune sand make it an ideal basis for comparison to more complex surfaces. However, higher-resolution data sets available from the TES (Thermal Emission Spectrometer onboard Mars Global Surveyor) and THEMIS (Thermal Emission Imaging System onboard Mars Odyssey) show spatial variations in the thermal properties within dune fields, calling into question their effectiveness as controls for thermal inertia studies. In order to explain these variations, we apply a thermal model developed for TES data to a commonly investigated dune field in Noachis Terra, that on the floor of Proctor Crater. We show that in this dune field, the thermal variations on the scale of 30 J m−2 s−0.5 K−1 are present and correlate spatially with aeolian features in the dune field. These variations correspond to three types of surfaces observed in the Mars Orbital Camera Narrow Angle (MOC NA) images: (1) dune sand, (2) interdunes exposing the surface underlying the dune field, and (3) sand-covered interdunes, or dune troughs. Both the interdunes and the dune troughs have cooler nighttime temperatures than the dune sand, corresponding to lower thermal inertia values. The dune troughs may be sand-covered areas with either minor amounts of dust accumulation or a mean sand grain size lower than that of dune sand. Because fine sand grains tend to preferentially accumulate on dune crests rather than in dune troughs, the second hypothesis is considered less likely than the first. This has implications for the recent sedimentary history of the dune field: Dust accumulation in dune troughs may imply that sand saltation is not prevalent enough to scour away all of the dust settling out from suspension in the atmosphere; however, it is prevalent enough to keep the dunes crests themselves clear of dust.
 Thermal inertia is a measure of a material's thermal response to the diurnal heating cycle. Loose (unconsolidated) fine-grained sediments heat more quickly after sunrise and cool more rapidly after sunset than coarse-grained sediments and solid rock. The finer materials, therefore, have a lower thermal inertia (lower resistance to temperature change) than the coarser materials. Increasingly consolidated sediments and coarser-grained materials (i.e., sand, gravel, boulders, and bedrock) lead to successively weaker diurnal extrema from higher heat retention and therefore exhibit progressively higher thermal inertia values. In ideal situations where all other factors are accounted for, and only under Martian atmospheric pressures in which thermal conductivity varies with particle size (e.g., see discussion by Edgett and Christensen ), thermal inertia can be used to estimate the average grain size of a particulate surface material [Kieffer et al., 1977; Presley and Christensen, 1997].
 Dune fields are composed of well-sorted unconsolidated particulates of a naturally controlled grain size, making them ideal targets for particle size estimates and therefore for verification of thermal models as well. Elsewhere on the planet, thermal inertia only defines the “effective particle size,” which may represent any combination of loose grains, rocks, and consolidated materials within each pixel or footprint. In some sense, dune fields may be regarded as “ground truth” for thermal inertia calculations, because their thermal properties and uniformity are so well constrained. Because of this predictability, the thermal inertia of Martian dunes has been studied since the first thermal models were produced for Mars, [e.g., Christensen, 1983; Zimbelman, 1986; Edgett and Christensen, 1991]. In particular, the Proctor Crater dune field located just west of the Hellespontus Montes in Noachis Terra (located at 47°S, 30°E; see Figure 1a) has been used as a basis for comparison between different models [Edgett and Christensen, 1991, 1994] and between different dune fields [Herkenhoff and Vasavada, 1999].
 In comparison with earlier data sets, thermal data from TES (Thermal Emission Spectrometer onboard Mars Global Surveyor) and THEMIS (Thermal Emission Imaging System onboard Mars Odyssey) provide a much higher spatial resolution view of the surface of Mars. From TES we use surface temperatures derived from the thermal bolometer, which is sensitive to the wavelength region from 5.1 to 150 μm [Christensen et al., 2001]. The bolometer consists of an array of 3 × 2 detectors, each sampling the surface with a resolution of 3 × ∼8 km in the mapping orbit of the Mars Global Surveyor. TES orbit numbers are counted in “ocks” (orbit counter keepers). Part of the THEMIS experiment consists of a thermal infrared multispectral imager with nine spectral channels ranging from 6.5 to 15 μm [Christensen et al., 2004]. From the Mars Odyssey mapping orbit, THEMIS acquires infrared images with a resolution of 100 m.
 With the TES and THEMIS data sets, we look once again at the thermal properties of the Proctor Crater dune field. We find that there are thermal variations present within the dune field that are only discernable with THEMIS data. Some variations correspond to materials underlying the dunes, exposed in interdunes, but others occur even in areas of complete sand cover. We put forth a hypothesis that may explain the observed thermal variations within the dune sand and discuss potential implications of the local sedimentary history.
2. THEMIS Thermal Model Description
 The thermal model used to derive thermal inertias from THEMIS observations is the same as that developed for single-point TES thermal inertia by Jakosky et al.  and Mellon et al. . Using the thermal model, a lookup table was generated on the basis of a number of varying input parameters (albedo, surface pressure, dust opacity, latitude, local time, and season) in addition to thermal inertia. The observed surface temperature and these other parameters corresponding to a given point (either a TES or a THEMIS pixel) are interpolated through the lookup table until a suitable match to thermal inertia is found.
 In the case of THEMIS images, surface brightness temperatures are estimated from the 12.57-μm band (Band 9) radiances for each pixel, and they can be considered a reasonable approximation of the surface kinetic temperature [Bandfield et al., 2004]. These surface temperatures are contained in a lookup table of surface radiance convolved with the spectral response function of Band 9. No atmospheric correction is made to the radiances: it is assumed that Band 9 radiances represent surface emission only. These temperatures are used to derive corresponding thermal inertias. These thermal inertias are then mapped to the surface temperatures in the image, producing a thermal inertia image.
 The thermal model uses nighttime surface temperatures to estimate thermal inertia in order to minimize the influence of albedo on surface temperature, which strongly affects daytime surface temperatures [e.g., see Mellon et al., 2000, Figure 1]. However, the albedo does still influence nighttime surface temperatures and cannot be ignored. MOC (Mars Orbiter Camera) and THEMIS VIS images clearly show that there are albedo variations within the dune field, but slopes and shading caused by the topography of the dunes make these variations difficult to quantify. Daytime TES albedos of the dune field have an average value of 0.115, and those of the surface adjacent to the dunes have an average value of 0.140 [see Fenton et al., 2003, Figure 9f]. Thermal inertias were estimated with both albedo values, so that the thermal effects of albedo variations could be constrained.
3.1. THEMIS Surface Temperatures
Figure 1a shows a MOC Wide-angle image of the Proctor Crater dune field as context for the following figures, as well as its location on the planet. Figure 1b shows three overlapping nighttime THEMIS IR images (shown in color to emphasize temperature changes) that pass over the Proctor Crater dune field. Each image was obtained at a different time during the same Martian summer, leading to seasonal differences in surface temperatures. In Figure 1c, temperatures in two of the images have been linearly offset to make them appear to have the same range of temperatures as the third (i.e., accounting for seasonal shifts in temperature by assuming all surfaces exhibit similar temperature shifts). Although the application of this offset has little quantitative value, the result is of great qualitative usefulness: spatial variations in surface temperature from one surface feature to another become much easier to identify.
 The temperatures within the dune field follow a striking pattern. The many very straight, almost vertical lines (along-track oriented banding) in the images are caused by uncorrected streaks in the THEMIS data set, and they may be considered artifacts in the images. The remaining pattern appears to follow the NNW-SSE trend of dune ridges that are visible in Figure 1a. In general, higher nighttime temperatures correspond to dune crests and lower nighttime temperatures correspond to dune troughs and interdunes. In this work, we define an “interdune” as an area between dunes that exposes the surface underlying the dune field, and a “dune trough” as an area where sand cover is deep enough to span the low-lying areas between dune crests (i.e., a sand-covered interdune).
 The narrow angle MOC images of the dune field have the resolution to determine which surface features correspond to which set of temperatures. Figure 2a shows one narrow angle image crossing the dune field, and Figure 2b shows the corresponding THEMIS IR region (the region is indicated in Figure 1c). The lowest temperatures correspond to interdunes, but some regions of low temperature correspond to dune troughs as well. To characterize the distribution of each type of low-temperature area, all interdune areas were marked on MOC narrow angle images crossing the dune field. These areas are shown speckled with black dots in Figure 1d. Most of the interdunes are located near the edges of the dune field in these images. In contrast, cooler areas in the center of the dune field correspond to sand-filled dune troughs, rather than interdunes exposing the underlying surface. Table 1 shows that in two THEMIS images, typical spot temperatures of the dune ridges are 2.3–2.5 K higher than those of the dune troughs and interdunes.
Table 1. Temperature Differences Within the Dune Field
Troughs/Interdunes Mean TB
Dune Ridges Mean TB
3.2. THEMIS and TES Thermal Inertias
Figure 3 shows a comparison between TES and THEMIS thermal inertia calculations. On the left (Figures 3a, 3c, and 3e) are nighttime surface temperatures during the summer, and on the right (Figures 3b, 3d, and 3f) are thermal inertias from the same areas. Figure 3a shows three TES ground tracks crossing the Proctor Crater dune field. The temperatures are offset from one another because they were obtained during different times of the year: ock 5781 occurred at Ls = 296.7° ock 6108 occurred at 312.5°, and ock 6435 occurred at Ls = 327.7°, all during the same year. Figure 3b shows the corresponding TES thermal inertias calculated by Mellon et al. . The resulting values fall between 260 and 360 J m−2 s−0.5 K−1 (units assumed throughout). Error bars correspond to the 6% error estimated by Mellon et al. . Using the thermal inertia relation from Presley and Christensen  at a surface pressure of 5 mbar, these values correspond to an effective particle size ranging between 500 μm to 2 mm, or coarse to very coarse sand [e.g., see Fenton et al., 2003, Table 1].
Figures 3c and 3e show surface temperatures from two of the three THEMIS images shown in Figure 1b. These images overlap spatially near the center of the dune field, roughly along the same path as the TES ground tracks in Figure 3a. Figures 3c and 3e show surface temperatures from every pixel in a 10-km-wide swath across the dune field where the two THEMIS images overlap.
 There is an offset in temperatures from one THEMIS swath to the next. This is caused by a seasonal shift, much like that observed in the TES ground tracks: THEMIS I00967002 was obtained at Ls = 336.4°, and THEMIS I01304006 was obtained at Ls = 351° (this effect is also visible in Figure 1b). There is a consistent scatter of roughly 3 K in each THEMIS image. This is caused partly by the spatial temperature variations visible in Figure 1c, and partly by pixel-to-pixel noise with a 1-σ random error of ∼1 K (provided by J. L. Bandfield, unpublished data, 2004). A running mean and the ±1-σ random error are plotted over the measured temperatures.
Figures 3d and 3f show thermal inertias along the 10-km THEMIS swaths in each image. For both swaths, the scatter in thermal inertias is on the order of 40 units. However, there is a shift in value of ∼30 units from one THEMIS image to the next. The reason for this offset in thermal inertia is unclear. In the TES tracks, surface temperatures vary because of a seasonal shift, and yet the thermal model accounts for this and produces thermal inertias that are statistically similar (see Figure 3b). This type of offset is known in modeling thermal inertias for THEMIS data, but no source has yet been found [Putzig et al., 2004]. Although the scatter in both temperatures and thermal inertia values is influenced by pixel-to-pixel noise, the strong spatial correlation of thermal variations with surface features suggests that variations in surface properties are present in the dune field. We propose that the 40-unit variation within the 10-km swath in each THEMIS image represents the relative variation of thermal inertia across the dune field, regardless of the absolute thermal inertia values.
 Because albedo impacts derived thermal inertias, it is possible that some of the 40-unit scatter in thermal inertia is caused by spatial variations of albedo. The thermal inertias shown in Figures 3d and 3f were calculated using a uniform albedo of 0.115, thus assuming that every THEMIS pixel sampled dark dune sand. However, it is clear that the higher-albedo interdunes and dune troughs scattered throughout the dune field correspond with lower nighttime temperatures (see Figure 2). In ideal circumstances, relatively brighter regions in MOC NA images could be identified and separated in the THEMIS images, so that the thermal inertias of bright and dark regions could be calculated with appropriate albedos. Unfortunately, image registration between the two instruments is quite difficult. Furthermore, sunlit and shaded slopes in MOC NA images strongly influence the apparent albedo of dunes, interdunes, and dune troughs, precluding the use of an unbiased or automatic routine to identify relatively bright and dark regions. A simpler method of estimating the effect of albedo variations is to use a different albedo in thermal inertia estimates of THEMIS pixels with lower temperatures. Assuming that all pixels with surface temperatures below the 1-σ random error correspond to interdunes and dune troughs with a higher albedo than the remainder of the dune field, thermal inertias for these pixels were calculated using an albedo consistent with the terrain adjacent to the dune field (0.140).
Figure 4 shows the resulting thermal inertias for each THEMIS image after accounting for two different albedos in the dune field. Applying a higher albedo to the colder surfaces raises the thermal inertia for these pixels, which otherwise had an artificially lowered thermal inertia. The resulting effect on the thermal inertias of the dune field is to lower the scatter of thermal inertias from 40 units to ∼30 units. However, even though an albedo correction has been applied, the brighter surfaces still correspond to the lower end of the thermal inertia estimates, suggesting that their surface properties are slightly different from those of the dark sand.
 The lower temperatures of the interdunes correspond to a surface with a lower thermal inertia than that of the dune sand. This measurement is consistent with previous work, in which small bright bed forms such as those visible in these interdunes (see Figure 2a) have a lower thermal inertia than dune sand [Fenton et al., 2003; Zimbelman, 2003] However, the lower temperatures from the sand-covered troughs between dunes are more difficult to explain.
 First, it must be established that the difference in thermal inertia is caused by a change in the properties of the surface material. For example, ground ice or nighttime frost in the lower portions of the dunes could be responsible for lower temperatures in the dune troughs. However, daytime surface temperatures in the summer soar to above 260 K, well above the sublimation temperature of 198 K. If the dune field traps water ice, then it is stable at a level too deep to be sampled by the THEMIS and TES instruments (greater than a few centimeters, the thermal skin depth). Ice at the surface is also not a likely factor influencing surface temperatures. Smith  estimated that the Martian atmosphere contains a water column abundance of 15–40 pr-mm at the latitude of Proctor Crater during the summer season (the same time of year when the THEMIS images were obtained). If all of this water condensed out on the surface as frost, it would be less than 0.04 mm thick. As with the signal from deeper ground ice, the signal of such thin surface ice would be overwhelmed by the several top centimeters of the surface that are sampled by THEMIS.
 Another possible cause for the observed temperature variations within the dune field are the effects of slope. Dune fields are composed of broad surfaces that are tilted in many possible orientations with slopes ranging from a few degrees up to ∼35°. Thermal emission from a sloped surface will transfer energy to any nearby surface tilted less than 180° relative to the emitting surface. The net effect of a hilly area is to heat up low-lying areas that feel the cumulative effects of emitted heat from adjacent hills, while allowing the hilltops to radiate away internal heat. This process causes topographic peaks such as dune crests to cool faster than low areas such as dune troughs and interdunes. However, THEMIS images show that the dune crests are in fact warmer than the dune troughs and interdunes, opposite to that created by slope effects. Thus other processes or surface characteristics that influence nighttime temperatures in the dune field far outweigh the effects caused by local slopes.
 A third potential cause of temperature variations is a spatial variation in sand induration. Most estimates of the grain size of Proctor Crater dune sand are in the range of coarse sand or larger, ≥0.5 mm (see references in section 1). It is possible that the dune sand is in fact composed of finer sand but that the dunes are somewhat indurated, artificially elevating the grain size estimates. Malin and Edgett  discuss dunes that show characteristics of induration, such as dunes with downslope movement on slipfaces reminiscent of erosional landslide scars (see their Figure 41D) and wind-grooved dune surfaces (see their Figure 42). If induration causes the variation in surface temperatures, than an explanation must be found for why it affects the dune crests but not the dune troughs. Without an obvious reason for the spatial variation of induration, we conclude that, although it may play a role in what is observed in the images, induration alone is not responsible for the variation in surface temperatures.
 We propose that differences in the material properties of the surface are responsible for the observed temperature variations. Two explanations for lower temperatures in low-lying areas are that the dune troughs contain relatively finer sand grains or that they have accumulated small amounts of dust.
 Finer sand grains will have a lower nighttime temperature than coarser grains. However, in terrestrial dunes the finest grains tend to accumulate at dune crests, rather than between dunes [Lancaster, 1995]. Finer sand grains preferentially move onto the dune crests where saltation is most efficient; coarser grains tend to remain behind in dune troughs where the wind is less effective, sometimes too weak to transport the larger grains at all [Lancaster, 1995]. Without an explanation for why these dunes would accumulate finer sand grains in troughs, we find this hypothesis unlikely.
 The second hypothesis is that the dune troughs have accumulated a small amount of dust. The term “dust” is used, as is typical in aeolian work, to represent fine particles that travel in suspension in the atmosphere, as opposed to “sand,” which travels in saltation along the ground. The MOC NA image in Figure 2a shows that these troughs typically have a slightly higher albedo than the sand on the dune ridges. The combination of higher albedo and lower thermal inertia is consistent with a dusty surface. Terrestrial dunes that are partially active tend to show activity mainly at dune crests; a similar process may be acting here in which the dune ridges are active enough to clear off annual dust fallout, but the dune troughs are not active enough to keep up with the dune ridges and thus accumulate dust.
 Using thermal inertias from TES ock 5781, chosen for being a midrange value of the four sets of thermal inertias, an estimate of the percentage of fine material covering the dune troughs can be made. The mean and standard deviation is 308 ± 18, leading to a range of 290–326 for most of the measured pixels. If the bright dune troughs are represented by ∼290 (assuming these pixels represent a mixture of sand and dust) and the undusted dune sand is represented by ∼326 (assuming these pixels represent pure dune sand), then the bright dune troughs can be modeled as a linear mixture of two different particle sizes and the areal percentage of dust cover can be estimated. For example, the bright dune troughs may contain some small bright grains that are horizontally mixed on the surface, perhaps in centimeter-sized bright patches, or trapped between larger dark sand grains, or as a single thin uniform layer over dark sand. Atmospheric dust grain sizes estimated from TES data range from 1.3 to 1.8 μm [Snook, 2002], with an effective thermal inertia of 61–66 at 5 mbar [Presley and Christensen, 1997]. With a linear mixture of 1.5-μm grains at 64 and pure sand at 326 units, the areal percentage (f) of 1.5-μm grains can be estimated by assuming the dune trough radiance can be represented as a linear sum of two blackbody curves,
 At Ls = 341°, midway between the season of the two THEMIS images, at a local time of 3 H (∼3 AM), the thermal model predicts TI=64 = 167.03 K, TI=290 = 192.12 K, and TI=326 = 190.09 K, leading to a dust cover estimate f of 9.7%. This fraction is enough to increase the visible albedo of the dune troughs somewhat, but not enough to obscure the presence of dark sand.
 The dune troughs with sand cover are located in the center of the dune field, where the dunes are larger and more widely spaced than at the edges. Fenton et al.  found a ∼50-m-high mound in the center of the dune field using MOLA elevations, which they interpreted as an accumulation of sand similar to those seen in terrestrial dune fields located in convergent wind regimes (e.g., see comparison of the Proctor Crater dune field with the Kelso dune field of the southwestern U. S. by Fenton ). It may be that over time, this sand mound has grown and dust has slowly accumulated on it, perhaps as deposits interfingered with sand layers. This mound may contain layers that record recent dust accumulation rates, and perhaps dust storm activity.
 If the lower thermal inertias in the low-lying troughs are indeed due to the presence of accumulated dust, then these areas may depress the average thermal inertia of the dune field by 30 units or more. It is important that pure, unconsolidated sand is considered in thermal inertia and thermal model comparisons, so we recommend that only dune crests be sampled in further studies.