Using an improved understanding of the Lunar Prospector Gamma-Ray Spectrometer (LP-GRS) spatial footprint, we have derived a new map of global thorium abundances on the lunar surface. This map has a full-width, half-maximum spatial resolution of ∼(80 km)2 and is mapped on the lunar surface using 0.5° × 0.5° pixels. This map has allowed the identification and classification of 42 small-area (<[80 km]2) thorium features across the lunar surface. Twenty of these features, all of which are located in the nearside Procellarum KREEP terrane, show a thorium-iron anticorrelation that is indicative of mixing between mare basalts and thorium-rich mafic impact-melt breccias (MIB). However, there exists at least one example of a farside location (Dewar crater) that appears to have abundances similar to the thorium-rich MIBs. This new map has also allowed the identification of mare basalts having high thorium abundances (>3 μg/g) in southwestern Mare Tranquillitatis, near the Apollo 11 landing site. With our better understanding of the LP-GRS spatial footprint, we have been able to constrain the surface thorium abundance at the Compton/Belkovich thorium anomaly to 40–55 μg/g, which is higher than any other measured location on the lunar surface and higher than most samples. Finally, using 1 km/pixel FeO abundances from Clementine and LP-GRS spatial footprint information, we have been able to obtain plausible thorium distributions around Kepler crater at a resolution of 1 km/pixel. The materials around Kepler crater appear to be a relatively simple mixing of thorium-rich MIB compositions and high-thorium mare basalts.
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 Data from the Lunar Prospector Gamma-ray Spectrometer (LP-GRS) have been used to determine the thorium abundances over the entire lunar surface. These measurements included an initial determination of relative thorium abundances on 150 km × 150 km sized pixels [Lawrence et al., 1998] and later absolute thorium abundances on 60 km × 60 km sized pixels [Lawrence et al., 1999, 2000]. The thorium distribution seen with these measurements is highly asymmetric with relatively high thorium abundances (5–12 μg/g) in and around Imbrium basin, moderately low thorium abundances (2–5 μg/g) within South-Pole Aitken (SPA) basin on the farside, and very low thorium abundances (<2 μg/g, yet with some exceptions) over most of the lunar highlands regions. Measurements of surface thorium abundances are important because thorium is an indicator of a lunar material called KREEP (potassium (K), rare earth elements (REE), phosphorus (P)), which is thought to have crystallized late in lunar formation. In particular, KREEP is enriched in incompatible elements that are not readily incorporated into most of the lunar minerals that make up the lunar crust and interior.
 There are significant implications regarding this new understanding of the thorium distribution across the lunar surface. For example, a number of studies have investigated the extent to which the heat producing elements (i.e., radioactive thorium, uranium, and potassium) on the nearside may have influenced the asymmetric production of lunar mare basalts [e.g., Wieczorek and Phillips, 2000; Hess and Parmentier, 2001; Arkani-Hamed and Pentecost, 2001]. Other studies have sought to understand how the thorium-rich region was formed in the first place [e.g., Loper and Werner, 2002; Parmentier et al., 2002]. Studies by Korotev  and Korotev and Gillis  have argued that KREEP is not a general highlands-type material, but rather is restricted to nearside regions as delineated by the global thorium data. Finally, a study by Jolliff et al.  has advocated, partially on the basis of the orbital gamma-ray data, that instead of the standard mare/highlands dichotomy, the Moon should be thought of as being made up of three different provinces: the Procellarum KREEP Terrane (PKT), the Feldspathic Highlands Terrane (FHT), and the South Pole-Aitken Terrane (SPAT). As explained by Jolliff et al. [2000, p. 4197], the term “terrane” implies that each “represents a geologic province that has distinctive character laterally and at depth, and each has a distinctive and unique geologic history.” Because this three-terrane paradigm is now being extensively used, we will also utilize the three-terrane concept in this paper.
 There are, however, some limitations regarding the thorium abundance data of Lawrence et al. . First, the pixels of this data set have a size of 60 km × 60 km, which is close to the size of the full-width, half-maximum (FWHM) spatial footprint size for the LP-GRS data. It is therefore difficult to use these data to study small-area surface features that have a size of ∼(60 km)2 or smaller. For the remainder of this paper, we will define “small-area” features as having an area on the order of (60–80 km)2 or less. In particular, there are a number of different types of studies that would benefit from having spatial information that is smaller than the 60 km × 60 km pixels (e.g., studies of the thorium abundances inside and outside medium sized craters [Warren, 2001]; studies of varied mare basalts and other features within SPA basin [Pieters et al., 2001]). In more general terms, if we can better identify small-area thorium features and how they may be associated with other features on the lunar surface, we will develop a better understanding of the local geology than with broad regional abundances. Finally, in addition to giving us a better understanding of the thorium abundances on the lunar surface, the identification and characterization of small-area features will help us to better understand the LP-GRS data set and details about the technique of planetary gamma-ray spectroscopy. In particular, what we learn about the LP-GRS data from studying small-area features is fundamental to the development of spatial deconvolution algorithms that can substantially improve the final spatial resolution of the LP-GRS data. The LP data set is ideal for carrying out such studies because six months of data were collected at a low altitude of 30 km, there is very little hydrogen on the lunar surface (large amounts of hydrogen can greatly complicate the analysis of orbital gamma-ray spectroscopy [Evans et al., 1993]), and there are large composition contrasts on the lunar surface. The results of these studies will help us understand not only the Moon and LP data, but also inform us about how to understand and apply other gamma-ray and neutron data, such as gamma-ray and neutron data from the Mars Odyssey mission [Boynton et al., 2002; Feldman et al., 2002; Tokar et al., 2002] and the upcoming missions to Mercury [Gold et al., 2001; Anselmi and Scoon, 2001] and the asteroids Vesta and Ceres [Prettyman et al., 2003], which all have gamma-ray and neutron spectrometers.
 A second limitation to currently published data concerns the accuracy of the absolute abundances, especially for low abundances (<2 μg/g). As described by Lawrence et al. , the current data set was derived using an energy window technique. With this technique, the total number of gamma-ray counts within an energy window around the 2.6 MeV thorium line were summed to arrive at a total counting rate. In order to derive absolute abundances, a background counting rate needs to be determined. At the time, the calculation needed to properly estimate the background counts was not available, so the counting rate background was estimated by assuming that the 60 km × 60 km pixel with the lowest counting rate had a thorium abundance of 0 μg/g. This approximation allowed a relatively accurate measure of thorium abundances to be determined in high-thorium regions. However, for low-thorium regions ([Th] < 2 μg/g), it was recognized that the determination of absolute abundances could have relatively large uncertainties [Lawrence et al., 2000]. Such uncertainties have been recognized by others [e.g., Gillis et al., 2000; P. H. Warren, “New” lunar meteorites: II. Implications for composition of the global lunar surface, of the lunar crust, and of the bulk Moon, submitted to Meteoritics and Planetary Science, 2002 (hereinafter referred to as Warren, submitted manuscript, 2002)] as being systematic biases. In particular, Warren (submitted manuscript, 2002) has suggested that for low-thorium regions, the data of Lawrence et al.  could be high by up to a factor of two. Such large biases could greatly limit the utility of the global LP-GRS data in particular studies. For example, an important parameter of lunar composition is its global thorium content. Since the lunar highlands, which have a low thorium content, cover most of the surface area on the Moon, a factor of two bias in the highlands thorium content translates into a large bias in the estimation of the bulk moon thorium content. These biases for low-thorium regions have recently been minimized using the complete LP-GRS analysis of Prettyman et al. [2002a, 2002b].
 The goals of this paper are therefore the following: 1) We plan to understand the systematics of the LP-GRS spatial response function and how it affects our understanding of the LP-GRS spatial resolution; 2) Using the knowledge of the spatial response function, we will optimize the spatial information that can be obtained with the LP-GRS data without losing information to statistical noise. Furthermore, using simulated data, we will seek to understand the effects of this optimization regarding how our measurements relate to the true thorium distribution on the lunar surface; 3) We will present a new thorium map that has an optimized full-width, half-maximum (FWHM) spatial resolution of ∼(80 km)2 mapped onto 0.5° × 0.5° pixels. This map will also use the newly reduced thorium abundances of Prettyman et al. [2002a, 2002b], which have an appropriate background determination, to obtain better estimates of the thorium abundances in low-thorium regions; 4) Finally, we will use this new map to investigate and classify a number of small-area thorium features on the lunar surface.
2. Data Selection and Mapping Issues
 As described by Lawrence et al.  and D. J. Lawrence et al. (Gamma-ray measurements from Lunar Prospector: Time series data reduction for the Gamma-Ray Spectrometer), manuscript in preparation, 2003) (hereinafter referred to as Lawrence et al., manuscript in preparation, 2003), the LP mission was divided into two major phases of high-and low-altitude data collection. For the high-altitude data (duration = 331.6 days), the average altitude was 100 km; for the low-altitude data (duration = 220.5 days), the average altitude was 30 km. Since the spatial resolution of the LP data improves with decreasing altitude, the focus of this study is the low-altitude data because of its significantly better spatial resolution.
 We are also using the energy band technique for measuring variations in thorium abundance. As was done in previous studies [Lawrence et al., 1998, 1999, 2000], we measure the variations in total counting rate in a 0.2 MeV energy band around the 2.6 MeV radioactive thorium line. The advantages of this technique are: 1) this technique is relatively simple to carry out because no spectral fitting procedures or complicated background corrections need to be completed; and 2) the energy band technique is easily carried out on low statistics data such as one or two spectral measurements. However, a major disadvantage of the energy band technique is that it is difficult to obtain an accurate background determination. As noted above, an accurate background determination is particularly important for measurements of low-thorium abundance regions such as the lunar highlands. In contrast to the energy band technique, Prettyman et al. [2002a, 2002b] has carried out a comprehensive spectral fitting and modeling analysis of LP-GRS data. Included in this analysis is a more accurate determination of absolute thorium abundances (especially for low-thorium regions). However, because of the need to carry out spectral fits with a statistically significant number of spectra, these abundances were determined on 60 km × 60 km and 150 km × 150 km pixels. The spectral fitting and energy band techniques are therefore complementary in how they are used and the types of information that they provide. For this study, we will use the energy band technique to determine relative thorium abundances on 0.5° × 0.5° pixels (15 km × 15 km at the equator) that have a small number of spectra per area. We will then use the absolute abundances from the Prettyman et al. [2002a, 2002b] study to obtain absolute thorium abundances for the global thorium map presented here.
 A detailed description of how LP-GRS time series data are reduced and mapped on the lunar surface is given by Lawrence et al. (manuscript in preparation, 2003). While the reader is referred to that paper for details regarding mapping procedures, a summary of the relevant ideas is given here. First, the most basic LP-GRS map is obtained when all 32s spectra are binned into 0.5° × 0.5° equal-degree pixels on the lunar surface. The size of these pixels was chosen because they are reasonably small compared to the size of any LP-GRS spatial footprint. However, the number of spectra per pixel is quite small, having a value of 1.89 spectra per 0.5° × 0.5° pixel for the low-altitude data (Lawrence et al., manuscript in preparation, 2003). To illustrate, Figure 1 shows a global map of the low-altitude 2.6 MeV energy band data on 0.5° × 0.5° pixels. While spatial information is clearly seen in the map, the data also show a significant amount of scatter. Two ways for reducing this scatter are described by Lawrence et al. (manuscript in preparation, 2003): 1) Bin the data into larger, equal-area pixels. This approach has been carried out in some studies of both LP-GRS and LP Neutron Spectrometer (LP-NS) data [Feldman et al., 1998a, 1998b; Lawrence et al., 1998, 1999, 2000; Maurice et al., 2000; Prettyman et al., 2002a, 2002b]. 2) Carry out an image restoration using an equal-area smoothing algorithm. Studies using this smoothing approach are given by Feldman et al. , Lawrence et al. , and this study. Here, we go into more details about the justifications and limitations of the smoothing technique in reducing the scatter of the data and improving our understanding of the spatial resolution.
3. Image Restoration Through Equal-Area Smoothing
 In this section, we describe an image restoration technique we call equal-area smoothing. The purpose of this technique is to find an optimum map that both reduces the statistical scatter inherent in the raw mapped data (Figure 1) and preserves the maximum amount of spatial information. The trade-off for reducing the scatter is that because this is a smoothing technique, the final spatial resolution will be necessarily poorer than the intrinsic spatial resolution of the unsmoothed data. However, this technique will help provide an understanding of the LP-GRS spatial response function (see below) that is required for proper implementation in the future of spatial deconvolution algorithms that do improve spatial resolution [Jansson, 1997]. In this section, we will first set up and describe the mathematical formulation and justification for the equal-area smoothing technique. Second, we will define and describe the modeled spatial response function that will be used in the technique. Third, we compare high- and low-altitude LP-GRS data with the modeled spatial response function.
3.1. Equal-Area Smoothing Algorithm
 When gamma-rays from the lunar surface are measured from orbit at a given point in time, they do not all originate from the sub-spacecraft location. Rather, they arrive at the detector from a distribution of locations that is determined by the physics of gamma-ray production, transport, and detection. Specifically, the spatial response function (also known as the point spread function), is the spatial distribution that would be observed from orbit for a gamma-ray point source on the lunar surface. For the raw data map of Figure 1, information about the spatial response function is ignored and all gamma-ray counts from a given 32s spectrum are assigned to a single 0.5° × 0.5° pixel. One way to reduce the scatter in Figure 1 is to represent the measured counting rate at a given pixel as a weighted average of all measurements in the vicinity of the sub-spacecraft point. Formally, this smoothing procedure can be represented using the following relation:
Here, Cunsmoothed(φ,λ) and Csmoothed(φ,λ) represent the unsmoothed and smoothed counting rate maps, respectively, at the longitude, latitude location (φ,λ). w(D[φ, λ; φ′, λ′]) is a normalized spatial response function and is computed as a function of the distance D along the surface from (φ′,λ′) to the sub-spacecraft point (φ,λ). The reason this technique is called “equal-area” is because the integral is carried out over equal areas, dA′, and not equal longitude/latitude coordinates. The total area of integration is limited because w(D) typically goes to zero as D increases to ∼5h–10h (where h = spacecraft altitude). On the basis of previous calculations [e.g., Reedy et al., 1973], earlier studies of LP-GRS and LP-NS data have assumed that w(D) can be approximated by an axially symmetric, two-dimensional Gaussian function [Feldman et al., 2001; Lawrence et al., 2002]. However, now that models for the entire gamma-ray production, transport, and detection process have been completed [Prettyman et al., 2002a, 2002b], we can determine a more accurate estimate of the true spatial response function.
3.2. Modeled Spatial Response Function
 Here, we use the results of Prettyman et al. [2002a, 2002b] for a model of the gamma-ray spatial response function. Included in these results is the angular dependence of gamma-rays that are detected from the lunar surface. The following assumptions were made for this model. First, the response function has been assumed to be axially symmetric such that its value only depends on the distance from the sub-spacecraft point. Second, effects due to the spacecraft speed (∼50 km/32s) have been ignored because this effect only introduces a smearing effect of <15 km (see Appendix A). If needed, these effects can be accounted for in later studies.
Figure 2 shows a subset of the modeled spatial response functions as a function of distance, D, from the sub-spacecraft point. Figure 2a shows the response for four different spacecraft altitudes at the equator. As seen, the width of the functions increases for increasing altitude. Figures 2b and 2c show the equatorial response functions for h = 30 and h = 100 km respectively. Also shown are Gaussian fits to the functions (dashed lines). While a Gaussian function fits a portion of the response function, the tails of the modeled response are clearly not well fit. The modeled response, however, can be fit by a related function called a kappa function [Kivelson, 1995]:
Here, the kappa function is parameterized by both a width, σ, and a tail parameter κ. As κ approaches infinity, the kappa function approaches a Gaussian. The solid gray lines in both Figures 2b and 2c show a very good fit of equation (2) to the modeled response functions.
Figure 3 shows how the fit parameters, σ and κ, vary with spacecraft altitude for λ = 0°. Figure 3a shows the linear fit parameters between σ and h. As might be expected, σ varies linearly with altitude having larger values for greater altitudes. κ also shows some variation with altitude being ∼8% larger for h = 30 km compared to h = 100 km. While the linear trend with h is not as strong for κ compared to σ, we have also fit κ to a linear function (Figure 3b). Equation (3) gives the modeled variation of σ and κ with altitude:
 The spatial response functions were also modeled for various spacecraft latitudes. The functions for all altitudes and latitudes were well fit by equation (2). Figure 4 shows how both the σ and κ parameters vary with respect to latitude and altitude. First, for most latitudes, σ is nearly constant. Only for latitudes approaching the poles is there a significant change, with σ shown to be larger compared to midlatitude regions. The most likely reason this occurs is because of the changing aspect ratio of the asymmetric cylindrical anticoincidence shield of the LP-GRS instrument and the location of the photomultiplier tube on the BGO scintillator [Feldman et al., 1999]. The κ parameter shows a similar trend with latitudes being relatively constant for midlatitudes and increasing in value near the poles. While there is a κ variation with height, it is muted compared to the σ dependence with height. In fact, at high-latitudes, the κ height dependence appears to reverse direction. We will see in the next section that the LP data show general trends that are consistent with these modeled parameters in respect to both height and latitude.
3.3. Comparison of Modeled Response to LP Data
 In principle, the best way of testing the response function model is to know, a priori, of a gamma-ray point source on the lunar surface and measure this point source with good statistics from orbit (here, when we say “point source,” we mean a surface feature that has a spatial extent that is substantially smaller than the FWHM spatial resolution of the LP-GRS footprint). However, while there do exist locations that appear to be point sources (see section 5.3.1), we do not want to make the a priori assumption that such a point source exists in order to demonstrate the validity of the response function models. Rather, we would prefer to demonstrate the validity of the modeled response functions without making any assumptions about the surface thorium distribution. In this section, we describe a technique where we make use of both the high- and low-altitude LP data and show that these data are consistent with the modeled response functions. Central to this technique is the idea that the high-altitude data have a much poorer spatial resolution than the low-altitude data. If we can smooth the low-altitude data to match the poor spatial resolution high-altitude data using a known response function, then this should show that we understand how the instrument response smoothes the high-altitude data. This, in turn, will give us confidence that we understand the spatial response function for all altitudes.
Figure 5 shows a conceptual flowchart of how we will use the high- and low-altitude data to validate the modeled response functions. Here, we summarize the technique. Subsequent sections give details about the results. First, on the lunar surface there is a real thorium abundance distribution (A) that is being measured using both high- (D) and low-altitude (E) data. While the source distribution is the same for both measurements, the measurements will differ on the basis of their different spatial footprint spreads (B and C). Since the low-altitude data will have much better spatial resolution than the high-altitude data, we should be able smooth the low altitude data to match the high-altitude data if we use an appropriate spreading function. Using a minimization technique, we can “measure” which function (F) smoothes the low-altitude data to achieve a match (G) with the high-altitude data. Next, we can set up a “simulated” lunar surface (H) where we assume an abundance distribution and spread out this distribution using both the modeled high- (I) and low-altitude (J) response functions to get simulated high- (K) and low-altitude (L) data. Finally, we can spread the simulated low-altitude data using the “measured” spreading (M) we determined from the LP data (step (F)). If the simulated, low-altitude smoothed data (N) matches the simulated high-altitude data (K) using the measured smoothing function, this will be evidence that the LP data are consistent with the modeled response functions.
3.3.1. Properly Normalize the High- and Low-Altitude Data
 In an empirical comparison between the high- and low-altitude data, it was noticed that the two data sets have a nonunity slope and nonzero offset. The source of these differences is not clear. However, in order to properly carry out the comparison between the high-and low-altitude data, all slope and offset differences between the two data sets need to be removed to ensure both data sets have the same absolute counting rate scale. The best way to measure any slope/offset differences is to rebin both data sets on pixel sizes that are substantially larger than the high-altitude spatial footprint. Figure 6a shows a plot of the correlation coefficient between the two data sets versus the rebinned pixel size (in units of equivalent degrees at the equator). For small pixel sizes, it is observed that the correlation coefficient is relatively low. This reflects the fact that although both data sets are measuring the same thorium spatial distribution, the information contained in the data sets is different because of the differing spatial footprints. In contrast, for large pixels where spatial differences are minimized, the correlation is almost perfect between the two data sets. To illustrate this more clearly, Figure 6b shows a plot of high- versus low-altitude data. The gray points show data binned on 2° × 2° pixels while the black points show data binned on 36° × 36° pixels. The 2° × 2° data clearly show deviations at high counting rates such that higher values are seen in the low-altitude data compared to high-altitude data. In contrast, there is an almost perfect correlation for the 36° × 36° data. As seen by the linear fit to the 36° × 36° data, there is indeed a slope and offset difference between the two data sets. For the subsequent minimization analysis, we have therefore used a revised high-altitude data set as reflected by the following equation:
3.3.2. Minimize Differences Between High- and Low-Altitude Data Sets
 Here we describe the technique we use to find the optimum smoothing function that will match a smoothed low-altitude data set to the high-altitude data set (i.e., step (F) in the flowchart of Figure 5). First, we assume the smoothing function has the form of equation (2) and the σ and κ values are parameterized with height using equation (3). For the sake of simplicity, we also use response functions only for the equator (λ = 0°) and ignore the possible latitude dependencies on σ and κ shown in Figure 4. If such dependencies turn out to be important, they will likely be reflected in the results given below.
 In order to determine an optimum smoothing function, we employ a χ2 minimization technique [e.g., Press et al., 1988, chap. 14]. Specifically, the optimum smoothing function will be the one that minimizes the χ2 value as defined by
where Chigh,revised and Clow,smoothed are the measured high- and low-altitude counting rates, respectively. si is the measured standard deviation of the counting rate in each pixel, and the summation is carried out over N total pixels. For this analysis, we have rebinned all data into pixel sizes of 60 km × 60 km, which are substantially smaller than the expected FWHM high-altitude footprint of 150 km. The total number of 60 km × 60 km pixels over the lunar surface is 11,306 (Lawrence et al., manuscript in preparation, 2003).
 Before reporting the results of the minimization analysis, a few considerations need to be explained. First, care needs to be taken regarding which surface regions are used for the analysis. For example, if the procedure is carried out over a large region that has uniform composition (e.g., the low-abundance highlands), then one would expect there to be a poorly constrained minimum χ2. In contrast, for regions that have a large composition contrast over small areas, one would expect to have a very well constrained minimum χ2 value. In addition, because of the possible latitude dependent σ and κ values, we might expect such dependencies to be reflected in the minimization analysis.
 To account for these considerations, we have divided the lunar surface into various regions as shown in Figure 7. Overlain on a lunar albedo map are four different major regions. The inside of the small area white line is a region of high thorium abundances we call the Fra Mauro region because it contains the high-thorium Apollo 14 Fra Mauro landing site. The inside of the black line contains all of the nearside high-thorium abundance areas. We call this the Procellarum-KREEP Terrane (PKT) after Jolliff et al. . The area outside the large area white line we call the highlands region. With some exceptions, this region contains mostly low thorium abundances. Finally, the 14 squares show how the PKT has been divided up into 30° pixels for a study of latitude dependencies.
 In order to carry out the optimization analysis, the low-altitude data were smoothed using equation (2) as a response function for eight different values of σ ranging from 20 km to 130 km. As given by equation (3), once σ is specified, a corresponding κ is determined. The range of σ values were chosen so that the resulting smoothed maps would either be undersmoothed (low σ) or oversmoothed (high σ) compared to the high-altitude map.
Figure 8 shows the resulting χ2 values (closed circles) for the eight different σ values using data from four different regions. A cubic spline (solid line) is used to interpolate between the discrete σ values. The four different regions are the whole Moon (Figure 8a); the highlands region (Figure 8b); the Fra Mauro region (Figure 8c); and the PKT region (Figure 8d). All plots show clear minimum χ2 values that occur for σ values between 52 and 60 km. This demonstrates that there is indeed a range of σ values that will optimally match the smoothed low-altitude data to the high-altitude data. The one-sigma and three-sigma uncertainty values on the optimum σ values are determined by finding the σ values at χ2min + 1 and χ2min + 9, respectively [Press et al., 1988]. These are listed in Figure 8. Upon inspection, there are clear differences among the regions, most notably is that the whole Moon and highlands data show flatter profiles than the two nearside regions. This shows that the regions with greater compositional contrast more sharply define a minimum χ2 than regions with lesser compositional contrast.
 One of the benefits of the χ2 minimization technique is that in addition to the qualitative assessments done above, quantitative measures of “goodness of fit” and uncertainties can be explicitly determined. First, in a general sense, a moderately good fit is one such that the minimum χ2 is approximately equal to or greater than the degrees of freedom, ν, in the fit. In our case, ν = N − 1, since we are optimizing only the one parameter σ. When we apply this to the four regions of Figure 8, we find that for the whole Moon and highlands cases, the minimum χ2 values are less than ν. In such a case, this usually indicates that the given standard deviations, si, are overestimated in relation to the data. For these regions, it is likely that part of the reason why χmin2 is so low is because the thorium abundances appear to be uniformly low over large areas in both the low- and high-altitude cases, therefore making the resulting fit unrealistically good.
 For the Fra Mauro and PKT regions, the minimum χ2 is close to ν, but slightly higher. As described by Press et al. , a goodness of fit parameter can be explicitly determined using the following relation:
where the fraction on the right side of equation (6) is the incomplete gamma function. If the errors have a Gaussian distribution, then for a reasonably good fit, G should generally be greater than 0.1. However, the fit can still be considered acceptable if G is greater than 0.001 when the errors are not exactly Gaussian. As described by Lawrence et al. , the uncertainties in the LP-GRS thorium data generally follow a Gaussian distribution.
 The goodness of fit values for the two nearside thorium-rich regions are GFM = 0.29 and GPKT = 0.022 (Table 1). While the goodness of fit for the PKT region is poorer than for the Fra Mauro region, it is still within the acceptable range as described above. One reason the PKT fit might be poorer is that non-Gaussian errors due to latitude dependencies may be affecting the fit.
Table 1. List of χ2 Values and Goodness of Fit Values
Fra Mauro (ν = 296)
PKT (ν = 1915)
7.3 × 10−12
2.2 × 10−4
 We have tested the uniqueness of the fit by carrying out the same procedure using two types of response functions other than a kappa function. The first response function is an axially symmetric step function such that for distances less than σ, the function has a value of 1 and for distances greater than σ, the function has value of 0. Table 1 shows that the goodness of fit values for both regions in the step function case are substantially lower than for the kappa function case. In particular, for the PKT region, the goodness of fit parameter of 7 × 10−12 shows that a step function is not an acceptable response function. The second type of response function is an axially symmetric Gaussian function, which has been used as a response function in earlier studies [Feldman et al., 2001; Lawrence et al., 2002]. The last row in Table 1 shows the χ2min and goodness of fit values for the Gaussian response function. While the goodness of fit value for the Fra Mauro case is acceptable, it is nevertheless lower than for the kappa function. For the PKT region, the fit is much poorer, which indicates that the kappa function better reflects the data than a Gaussian function.
 Finally, Figure 9 shows a plot of σbest fit values for the 14–30° × 30° regions outlined in Figure 7. The error bars represent one-sigma uncertainties in the σbest fit values. This plot shows a trend of higher σbest fit values for higher latitudes. This is consistent with the modeled trend of Figure 4a and suggests that the spatial resolution of the LP-GRS data may indeed be somewhat dependent on latitude. However, since Figure 4a shows that strong σ variations only occur close to the poles for a small portion of the lunar surface, we will ignore any latitude dependent spatial resolution in this global study of thorium abundances. For future studies of smaller defined regions (especially near the poles), a latitude dependent spatial resolution should be studied in more detail.
3.3.3. Compare Measured and Simulated Response Functions
 We are now in the position to make a comparison between the measured and simulated response to check for consistency of all response functions. This is carrying out steps (H)–(N) diagrammed in Figure 5. First, we set up a simulated thorium abundance. We have defined this as a point source with an arbitrary intensity value of 1 (step (H)). Next, we convolve this point source through both the modeled high- and low-altitude response function of equation (2) (steps (I) and (J)). The results are simulated high- and low-altitude data (steps (K) and (L)). Finally, using the measured parameters determined in the previous section (Table 1), we apply an additional convolution to the simulated low-altitude data (step (M)). If the two sets of simulated data, (K) and (N), show a good match, this is evidence that the simulated data (using measured response function parameters) are consistent with the real data. On the other hand, if the two data sets do not match, this suggests that the simulated data and response functions do not represent the real data.
Figure 10 shows the results of this comparison for the Fra Mauro and PKT regions, where the solid line is the simulated high-altitude data (K) and the closed circles are the simulated low-altitude, smoothed data. The gray region outlines the spread from the three-sigma uncertainty of the measured σbest fit value (Table 1). For the Fra Mauro region, the two data sets show a relatively good match, however, near the peak, there is a mismatch that is beyond the spread due to the three-sigma uncertainty in the σbest fit value. For the PKT region, there is an extremely good match between the two data sets as the modeled high-altitude data overlaps the spread in the smoothed low-altitude data.
 For comparison, a similar exercise was carried out using a Gaussian response function instead of a kappa function. Estimates of model Gaussian parameters were determined by fitting Gaussian functions to the response functions of Figure 2. For the measured σbest fit values, we used the numbers given in Table 1. Figure 11 shows simulated data using the Gaussian response function. In contrast to the relatively good match for the kappa functions seen in Figure 10, the simulated data in Figure 11 show a clear mismatch. This shows that the simulated data with a Gaussian response function are not consistent with the measured σbest fit value.
3.3.4. Summary of the Comparison of Modeled Response to LP Data
 In comparison to a Gaussian response function, it appears that a response function having the form of equation (2) is more representative of the real data. This conclusion is based on both the goodness of fit values given in Table 1 and the analysis described in section 3.3.3. As a consequence, it appears that a kappa function provides a good representation of the true LP-GRS spatial response. While there may still be some latitude dependencies that have not yet been accounted for, we have shown that such variations will likely be important for only a small portion of the lunar surface. Therefore, for the remaining portion of this paper, we will use equation (2), with σ = 22.5 km and κ = 0.626, as the spatial response function for all smoothing operations on the low-altitude data. Figure 12 shows the final map where the unsmoothed data of Figure 1 have been smoothed using equation (1) and the spatial response function of equation (2). Note, however, that while this smoothing procedure reduces scatter from Poisson statistics, it also increases the FWHM spatial resolution of the final map to a value of ∼(80 km)2. This implies that while spatial information can be distinguished on scales smaller than 80 km, spatial features are fully resolved when they are more than 80 km apart. The scale bar is given both in units of counts per 32s and absolute thorium abundances. The next section describes how the absolute scale of μg/g thorium is determined.
4. Absolute Thorium Abundances
 The final step in the production of the smoothed thorium map is to convert the units of counts per 32sec to absolute thorium abundances in units of μg/g. In previous studies [Lawrence et al., 1999, 2000], this was done by making assumptions about the background such that the pixel with the lowest counting rate had a thorium abundance of 0 μg/g. As described elsewhere [Gillis et al., 2003; Prettyman et al., 2002a, 2002b; Warren, submitted manuscript, 2002], this assumption can introduce relatively large biases in the data for low-abundance regions. The recent analysis of Prettyman et al. [2002a, 2002b] has now derived absolute thorium abundances using high-altitude data on 150 km × 150 km pixels by properly accounting for the background counting rate using instrument and spectral modeling information that was not available for the Lawrence et al.  analysis. As described below, these new results have eliminated the discrepancies between sample and LP-GRS data identified by Warren (submitted manuscript, 2002) and Gillis et al. . In particular, for low-abundance regions, the orbital data are consistent with what is expected from thorium abundances in feldspathic lunar meteorites thought to be from low-thorium highlands regions [Korotev et al., 2003]. Furthermore, for high-abundance regions, the results of Prettyman et al. are close to earlier derived values.
 Here, we choose to scale the smoothed low-altitude counting rate data to the high-altitude abundance data of Prettyman et al. [2002a, 2002b] for two reasons. First, the absolute abundances derived by Prettyman et al. [2002a, 2002b] are based on solid physical principles of gamma-ray production, modeling, and spectral fitting. For the global data set given here, we therefore think it is fully justified to tie the smoothed thorium counting rate data to the larger pixel absolute abundances of Prettyman et al. [2002a, 2002b]. The alternative is to tie the absolute abundances to ground truth sample data at selected landing sites. However, because of the large mismatch in size between the LP-GRS footprint and Apollo and Luna sample sites, we have previously argued against such a calibration [Lawrence et al., 2000]. This argument against calibrating the LP-GRS data with ground truth sample data, of course, does not preclude comparing the absolute abundances with sample data after the LP-GRS absolute abundances have been determined (this has already been done by Warren (submitted manuscript, 2002) and Gillis et al.  using the Lawrence et al.  abundance determinations). Second, when we carry out such a comparison of LP-GRS data with low-abundance regions (Figure 15), we find there is a good correspondence between the expected highlands thorium abundances based on lunar meteorites and the measured LP-GRS data. This agreement further justifies our decision to use the Prettyman et al. [2002a, 2002b] abundances for calibrating the smoothed thorium map.
Figure 13 shows a plot of smoothed, low-altitude counting rate data versus high-altitude absolute abundance data. As in section 3.3.1, both data sets have been rebinned to large 36° × 36° pixels to eliminate spatial footprint differences. As expected, there is an excellent correlation (R = 0.99) between the two data sets and the linear fit parameters connecting the low-altitude counting rate to absolute abundances are given in the figure. When the smoothed data are scaled in absolute abundances and compared to the 150 km × 150 km data of Prettyman et al. [2002a, 2002b], we still find some residual differences that are biased by latitude (Figure 14). These differences are most likely the result of differing techniques for making latitude corrections to the counting rate data [Prettyman et al., 2002a, 2002b; Lawrence et al., manuscript in preparation, 2003]. To ensure consistency between this data set and the data set of Prettyman et al. [2002a, 2002b], we have made a final latitude dependent correction to the smoothed thorium data as empirically determined by the solid line in Figure 14.
Figure 15 shows histograms comparing the high-altitude, 150 km × 150 km data of Prettyman et al. [2002a, 2002b] to the final low-altitude, smoothed data of this paper. Generally, the distributions are very similar. However, for high abundances, the smoothed map shows larger values than the high-altitude data. This is just the result of the smaller spatial footprint of the low-altitude data providing less spatial averaging than for the high-altitude data. For low abundances, the peak in both distribution is ∼0.4 μg/g, which is close to the expected value for the lunar highlands on the basis of a comparison with feldspathic lunar meteorites [Korotev et al., 2003].
5. Global Survey of Small-Area Thorium Features
 In this section, we report on a survey we have conducted to identify small-area features having both high- and low-thorium abundances. We have multiple reasons for conducting such a survey. First, we want to better understand the limit of spatial information in the LP-GRS data; i.e., where does the abundance signal end and noise begin? To address this type of question, it can be helpful to compare the thorium data (in particular small-area features) with other information and data sets to see if the thorium data can be plausibly related to other features on the lunar surface. If a positive correspondence can be established between the thorium data and other data sets in at least some cases, this will go far to solidifying our confidence for the entire thorium data set and response function analysis. In addition to giving us better confidence in the global data set, relating the thorium data to other data sets can also help to constrain the true thorium abundance on the surface. Examples of such constraints will be given in section 5.3. Finally, by identifying and classifying these features, we will provide directions for future work, since a detailed study of all these features is beyond the scope of this paper.
 This section proceeds as follows. First (section 5.1), we will summarize the global survey by listing and identifying features and describing their characteristics. Second (section 5.2), we will give examples of five difference classes of features and give descriptions about each example. Finally (section 5.3), we will describe two of the features (Compton/Belkovich and Kepler crater) in more detail, with a goal of demonstrating the kind of information that can be learned when gamma-ray spatial resolution information is combined with surface feature and elemental abundance information from other data sets.
5.1. Summary of Global Survey
 For the survey of small-area features, the global thorium map was searched for features using the following criteria: 1) The size of the features should be close to the size of the full-width, full-maximum (FWFM) LP-GRS spatial footprint (∼240 km). Here FWFM is defined as the point at which the spatial response function goes down to ∼1% of its maximum value. In particular, the features should be spatially distinguishable with good signal to background from larger, more spatially homogeneous areas; 2) Features can either have higher or lower thorium abundances compared to the surrounding regions. However, because of lower counting rate statistics, low-thorium regions may be more difficult to clearly identify than high-thorium regions. In particular, there may be a bias in the selection of small-area features toward high-abundance areas that are inherently easier to identify.
 The global thorium map was manually searched to identify features using the above criteria. Table 2 lists 42 features that have been identified using these criteria. It should be noted, that while most thorium features have likely been identified, some may have been missed due to marginal signal to background (especially for possible low-abundance features). Figure 16 shows the location of all 42 features.
Except for Rima Billy Apennine Bench and Mons Vinogradov, the thorium features are named after the nearest crater or basin even if there does not appear to be a correspondence between the thorium feature and crater.
 In addition to the features themselves, Table 2 lists various characteristics of the features. For example, the fifth column lists whether the feature has high- or low-thorium abundances with respect to the surrounding area. 34 out of the 42 features (81%) have high-thorium abundances with respect the surrounding regions. The next two columns summarize comparisons made with two other data sets, namely a high-resolution airbrush map based on photographic data [Rosiek and Aeschliman, 2001] and the 1 km/pixel FeO data of Lucey et al.  taken from Clementine spectral reflectance data. While other data sets could be used (and in the future should be used), these two are good representative data sets for identifying both morphological and compositional surface features.
 A “yes” is given in the “Correlated With Surface Feature?” column if the thorium feature is spatially correlated, as based on the airbrush map, to a surface feature of comparable size. 34 of the 42 (81%) thorium features show some correlation with a feature in the airbrush map. If the features are spatially correlated with iron abundances, this is noted in the column labeled “Correlated With FeO Feature?” 14 of the 42 (33%) features show no spatial correlation with iron abundances. Of the remaining, 8 (19%) show positive correlations (i.e., high-thorium with high-iron or low-thorium with low-iron) and 20 (48%) show anticorrelations (i.e., high-thorium with low-iron or low-thorium with high-iron). Finally, the last column lists the terrane location of each feature. Most of the features (31 features or 74%) are located on the nearside either in the PKT (25 features) or on the border of the PKT (6 features). This is not surprising given the very high thorium abundances on the nearside. However, there are also 11 small-area features (26%) located outside of the PKT either in the FHT (9 features), the eastern mare (1 feature) or SPA (1 feature). A study of these features will be particularly important for understanding the distribution of thorium outside the thorium-dominated PKT.
5.2. Classification of Small-Area Features: Examples
 Each of the 42 features listed in Table 2 can be classified as one of five different types of features. Figure 17 shows eight examples of different features. The left column of Figure 17 shows regional thorium maps with an overlay of the airbrush map of Rosiek and Aeschliman . The right column of Figure 17 shows regional FeO maps at 1 km/pixel taken from Lucey et al.  using Clementine Spectral Reflectance (CSR) data. In all maps, a circle having a diameter of 240 km (the FWFM of the mapped LP-GRS spatial footprint at 30 km altitude) is superposed at roughly the centroid of the thorium feature. This allows a direct comparison of the feature size to the footprint size, as well as a comparison between the location of the thorium feature and any corresponding FeO composition features. Examples of the five classes will now be described.
5.2.1. Thorium-Iron Anticorrelation (Th-Fe AC)
 There are 20 features that show a thorium-iron anticorrelation trend along with a correlation to a morphologic feature. All of these features are located in the PKT. These results are consistent with the study of Gasnault et al. [2002a] that identified 15 of these features as having anticorrelations between thorium and fast neutrons. As described elsewhere, fast neutrons provide a measure of iron and titanium [Maurice et al., 2000], or more specifically atomic mass [Gasnault et al., 2001] of the lunar surface. For these features, the thorium abundances range from 3 to 11 μg/g, while the FeO abundances generally range from 7 to 15 FeO wt.%. Furthermore, when the thorium abundances are plotted versus the iron abundances in these regions (e.g., see section 5.3), there are often strong anticorrelation trends that suggest the existence of both physical and footprint mixing between mare and nonmare lithologies. We also note that the high-thorium features of this class (after accounting for the footprint smoothing) have thorium and iron abundances similar to the mafic impact-melt breccias (MIB) described by Korotev . In particular, Korotev describes that the principle lithologic carriers of thorium and the KREEP signature are 1) nonmare impact-melt glasses and breccias that are more mafic (∼10 FeO wt.%) than materials characteristic of the FHT; and 2) those breccias have high concentrations of thorium and other incompatible elements such as those of KREEP. While others have suggested that this type of lithology is a representative composition of the lower crust [e.g., Lucey et al., 1998], Korotev suggests that the MIBs are a special product of the PKT. It is therefore notable that all instances of this class are found in the PKT. However, there does exist at least one highlands location (Dewar crater, see below) that shows compositional similarities to MIBs.
Figures 17a–17f give three examples of features showing a thorium-iron anticorrelation. Figures 17a and 17b show the thorium and FeO abundances at Kepler crater (38°W, 8.1°N, 31 km dia.). The crater and ejecta show high thorium and low iron abundances in contrast to the low thorium and high iron abundances of the surrounding mare basalt. In section 5.3, we give a more detailed description of Kepler crater.
Figures 17c and 17d show the thorium and iron abundances at Timocharis crater (13.1°W, 26.7°N, 33 km dia.) located in SW Imbrium basin. A thorium enhancement is clearly related to both the crater and a relative low in iron abundances. A thorium enhancement at Timocharis crater was first noted by Etchegaray-Ramirez et al.  using Apollo Gamma-ray (AGR) data. Etchegaray-Ramirez et al. (as well as Hawke and Head ) suggested that Timocharis excavated high-thorium material underlying the Imbrium mare basalts that is similar to the high-thorium material of the Apennine Bench formation. The Apennine Bench formation is seen in Figures 17c and 17d as the high-thorium (∼9 μg/g), lower-iron (10–12 FeO wt.%), rough terrain to the east of Timocharis crater, and is possibly one of the few examples of nearside KREEP volcanism [Spudis, 1978; Hawke and Head, 1978].
 Here, with higher resolution thorium data, we find similar results. For example, let us assume: 1) that the high-thorium material at Timocharis crater has a surface abundance of 9 μg/g, which is the measured abundance at the larger Apennine Bench formation; 2) that the high-thorium material is uniformly distributed across the lower-iron ejecta blanket, which is roughly 70 km diameter; and 3) that the background thorium abundance is 3 μg/g (the approximate value in the Imbrium mare basalts to the NW). When such a feature is convolved through the instrument and smoothing response functions, the derived abundance is 7 μg/g, which is the maximum measured value at Timocharis crater. Therefore, using these reasonable assumptions, we find that the thorium abundance at Timocharis crater is likely very similar to the thorium abundance at the Apennine Bench formation.
Etchegaray-Ramirez et al.  and Hawke and Head  also suggested that other nearby craters (Lambert, 21°W, 25.8°N, 30 km dia.; Pytheas, 20.6°W, 20.5°N, 20 km dia.; and Euler, 29.2°W, 23.3°N, 27 km dia., not seen in Figures 17c and 17d) may have excavated high-thorium material underlying the Imbrium mare basalts. Even though these craters are smaller than Timocharis, if they excavated materials with thorium abundances of 9 μg/g or higher, we should be able to see evidence of such materials. The thorium data (Figure 17c), however, show that these craters have not excavated high-thorium material similar to what is seen at Timocharis. Since Timocharis is the eastern-most of these craters and closest to the expected surface expression of KREEP basalt (at the Apennine Bench formation), these data indicate that the KREEP basalt was not emplaced much further to the west than Timocharis. Alternatively, if such KREEP basalt was emplaced further to the west of Timocharis, then it must be covered by a thicker mare basalt layer than at Timocharis, since all these craters have similar sizes and would have excavated similar amounts of material.
 Finally, Figures 17e and 17f show the thorium and iron abundances around Plato crater (9.4°W, 51.6°E, 109 km dia.). Here, the thorium abundances are low and iron abundances are high within the mare basalt fill of the crater compared to the lower iron regions around Plato crater, where the thorium abundances show high values. Since we might expect the entire mare basalt fill within Plato to have low-thorium abundances (i.e., similar to the mare basalts in the Imbrium basin interior to the southwest), the thorium map of Figure 17e provides an excellent example of how the broad spatial footprint of the LP-GRS data defocuses the sharp boundaries of even a large 100 km diameter crater.
 The second class includes features that show a positive thorium-iron correlation along with a correspondence to a morphologic feature. There are eight features in this class and only two (Herschel and Copernicus) are located in the PKT. Five of the features are scattered around the FHT and one (Langrenus) is near the eastern PKT/FHT boundary. In contrast to the similarities seen among the features of the Th-Fe AC class, the features of this class have more diversity. Two of the features are associated with areas of farside mare basalt (Orientale and Moscoviense). The remaining features are associated with craters. As with the Th-Fe AC class, these results are consistent with the study of Gasnault et al. [2002a] that found positive correlations between thorium and fast neutrons for many of these features. Figures 17g–17j show two examples of these types of features.
Figures 17g and 17h shows the thorium and iron abundances near Dewar crater (165.5°E, 2.7°S, 50 km dia.), which is located NW of South Pole-Aitken (SPA) Basin. The thorium abundances are relatively low (2–2.5 μg/g) and while Dewar crater is near the enhancement, it does not appear to be co-located with the centroid of the enhancement. Furthermore, because the thorium abundances here are low (which implies a low signal to noise), some of the irregularities seen in the thorium enhancement may be due to statistical noise. Finally, despite the low abundances, the thorium enhancement is clearly associated with the high-FeO regions that have an abundance range of 6–14 FeO wt.%.
 Ideally, we would like to know the thorium abundance around Dewar crater on spatial scales similar to what is seen in the FeO data. However, because we do not know how the thorium is distributed, we cannot uniquely determine the surface thorium abundance. On the basis of the large variations seen on small spatial scales in the FeO abundances, it is certainly possible that the sources of the thorium enhancement also have small areas. If this is true, then surface abundances could be significantly higher (e.g., up to 6–7 μg/g, based on a convolution through the instrument response and smoothing algorithm) than the measured values of 2–2.5 μg/g. If this is the case, then this region would represent one of the few locations outside of the PKT that has abundances similar to the nearside MIBs. This is consistent with conclusions made by Gasnault et al. [2002a].
Figures 17i and 17j show the thorium and iron abundances at Tycho crater (11.1°W, 43.4°S, 102 km dia.). Despite the low spatial resolution, the iron abundances for this region are taken from the LP-GRS [Lawrence et al., 2002] instead of the high spatial resolution CSR data. This was done because data from both the LP-GRS and LP-NS suggest that the iron abundances at Tycho crater are quite low compared to the higher abundances derived with the CSR data [Lawrence et al., 2002]. We should note that low thorium abundances were also seen at Tycho crater using the earlier data set of 60 km × 60 km pixels [Lawrence et al., 2000]. However, with the larger pixel data, Tycho crater was covered by only two pixels. Here, with the smoothed data, we see that the thorium abundances are indeed low in and around Tycho crater. Furthermore, the low-thorium distribution is similar to the low-iron distribution seen around Tycho. All of this information is consistent with earlier conclusions [Lawrence et al., 2002; Lucey et al., 2002] that the layer of high-thorium material seen elsewhere on the nearside was absent at Tycho crater.
5.2.3. Thorium-Morphology Correlation (Th-M)
 The third class consists of thorium features that correlate with a morphological feature, but not with an iron composition feature. Six such features have been identified and interestingly, none of them are located within the PKT. Most of them have relatively low thorium abundances with low signal to noise and therefore exhibit spatial irregularities that may be indicative of statistical noise. Furthermore, their connection to the named crater may be entirely coincidental. However, all of these features have enough contiguous pixels with similar abundance values to warrant further study. A typical example of this class of feature is shown in Figures 17k and 17l. The thorium enhancement shown in Figure 17k is centered on the crater Romer (36.4°E, 25.4°N, 39 km dia.), which is located at the northeastern edge of Serenitatis Basin. The centroid of the thorium enhancement appears to be located near Romer crater with an abundance of almost 3 μg/g. Furthermore, the broader portion of the thorium enhancement appears to be correlated to some extent with some of the rougher terrain further distant from Romer crater and possibly with some of the lower iron abundances to the southwest of Romer crater. However, the irregular nature of the moderate thorium abundances (1–2 μg/g, reddish colors) suggests that statistical noise is the cause of the small (i.e., smaller than the FWFM) spatial scale variations. In the FeO map (Figure 17l), the main thorium enhancement is located on the boundary between a moderately high (10 FeO wt.%) and low (4 FeO wt.%) iron region. There is no clear counterpart in the iron map that can be associated with the thorium enhancement.
 The final class of thorium features are those with no correlation to either a morphologic or composition feature. There are seven such features and all are located in either the PKT or near the PKT. As with some of the other classes, there is no dominant type or location (except for being within the PKT) for these features. Some are located within mare basalt regions (Arago, Seleucus 2) and others are located in rougher terrain (e.g., Dechen, Keldysh). The main characteristic these features have in common is that they have no discernable counterpart that appear to be the source of the thorium enhancement. For example, Figures 17o and 17p show the thorium and iron abundances around Arago crater (21.4°E, 6.2°N, 26 km dia.), located in southwestern Mare Tranquillitatis. In our previous study using 60 km × 60 km pixels [Lawrence et al., 2000], it appeared that the thorium enhancement was directly associated with Arago crater, even though Arago only covered a fraction of one 60 km × 60 km pixel. Now with the higher spatial resolution thorium data, it is clear that the enhancement is not directly associated with Arago. Rather, the centroid of the enhancement is ∼40 km to the west of Arago in a region of high-iron (>20 FeO wt.%) mare basalt. Furthermore, the spatial extent of this enhancement is significantly larger than the FWFM spatial footprint, which indicates that the thorium on the surface has a spatial extent larger than the LP-GRS footprint. Finally, a close inspection of both the Clementine visual and FeO abundance data shows no clear correlation of any feature with the thorium enhancement. Rather, the iron abundances are almost universally high in many portions of the high-thorium regions. In previous studies using orbital data, there have been indications of unsampled mare basalt regions (e.g., areas of western Procellarum) having higher thorium abundances than typically seen in the sample collection [Etchegaray-Ramirez et al., 1983; Jolliff et al., 2001; Gillis et al., 2002b; Flor et al., 2002; Gasnault et al., 2002b]. If the high-thorium abundances are intrinsic to the mare basalt, this is an indication that these unsampled mare basalts may have formed differently than the more abundant lower thorium mare basalts. However, of all the sample collection sites, the Apollo 11 site (0.7°N, 24.3°E) shows mare basalts with some of the highest measured thorium abundances [Taylor et al., 1991; Korotev and Gillis, 2001]. Since the Apollo 11 site is just at the edge of the observed thorium enhancement in Mare Tranquillitatis, it is certainly possible that the higher thorium mare basalts sampled from the Apollo 11 mission are derived from this large area of observed high thorium abundances.
5.3. Example Analyses of Small-Scale Features
 In this section, we will describe in more detail two of the features discussed in the previous section. In addition to the intrinsic importance of these features, one of the goals of this discussion is to illustrate what information and constraints can be determined when the smoothed thorium data are combined with response function information and other data sets.
5.3.1. Compton/Belkovich Thorium Anomaly
 The Compton/Belkovich thorium anomaly was first identified using the low-altitude LP-GRS data as described by Lawrence et al. . Further descriptions of the Compton/Belkovich thorium anomaly have been given by Lawrence et al. , Elphic et al. , and Gillis et al. [2002a]. The following gives a summary of these and other results regarding the Compton/Belkovich thorium anomaly. The Compton/Belkovich thorium anomaly is a thorium enhancement in the northeastern highlands located at (100°E, 61°N) between the craters Compton (103.8°E, 55.3°N) and Belkovich (90.2°E, 61.1°N). Various measured elemental abundances at the anomaly are given in Table 3 along with the elemental abundances for various thorium-rich lithologies. As previously reported, the abundances of potassium, SiO2, MgO, CaO, Al2O3, and an independent measure of thorium have been determined from the recent full analysis of high-altitude LP-GRS on 150 km × 150 km pixels [Prettyman et al., 2002a, 2002b]. While the Compton/Belkovich anomaly is seen and distinguishable with the low- and high-altitude thorium, potassium, and samarium data, the anomaly is not clearly seen with the other elemental data.
Table 3. List of Elemental Compositions for Various Thorium-Rich Lithologies
 In addition to the compositional information, Gillis et al. [2002a] have also used Clementine imaging data to identify a bright albedo feature (∼15 × 30 km) that is co-located with the thorium enhancement. While there is no proof that the albedo feature is the source of the thorium enhancement, the existence of the albedo feature can nevertheless be used to provide plausible constraints on the surface thorium abundance. It should be noted, however, that the thorium feature does show an asymmetry to the east with no clear compositional albedo or morphological counterpart. Therefore, even if the albedo feature is the source of the thorium at the surface, it is likely not the only source.
 With the smoothed thorium map and a good understanding of the response function, we can now attempt to provide better constraints on the surface thorium abundance of the Compton/Belkovich thorium enhancement. In order to constrain the surface abundance, we need to know three pieces of information: 1) the measured thorium abundance; 2) the full gamma-ray response function including the intrinsic instrument response and smoothing algorithm; and 3) the true surface thorium distribution. At this point, our only unknown is the surface thorium distribution. Using additional information, constraints can be put on the effective size of the surface distribution, and hence the surface abundance.
 A plausible constraint on the minimum size of the feature is that it is probably no smaller than the observed albedo feature of ∼15 × 30 km. This assumption is plausible for two reasons: a) If the feature were substantially smaller than the albedo feature, this would lead to surface thorium abundances that are surprisingly (and maybe unreasonably) high (i.e., >100 μg/g). Furthermore, there is no other corresponding feature to use for a minimum size, so without the albedo feature, there would be few criteria to define a minimum size. b) Because of the observed asymmetry in the measured thorium distribution, it is likely that the thorium is distributed on the surface more widely than just at the albedo feature. If this is true, then the albedo feature truly provides a lower size limit.
 To obtain a limit on the maximum size of the feature, we can use the measured thorium data. Specifically, we will make use of the unsmoothed thorium data (Figure 1). While the large amount of scatter hinders the usefulness of unsmoothed data in many cases, here we can use these data to infer a maximum feature size. Figure 18 shows a north-south profile of the thorium counting rate through the center of the thorium enhancement. The peak of the enhancement is clearly seen. The error bars represent the one-sigma statistical uncertainty. The solid line in Figure 18 is a fit of the profile to the kappa function (equation (2)) that represents the spatial response function. We should note that this good fit, along with the σ and κ parameters that are similar to the calculated parameters (equation (3)), gives further evidence that the calculated response function is a good approximation of the true response function.
 To obtain an upper limit on the feature size, we can use the following approximate relation (which is strictly true for a convolution of Gaussian functions):
where Smeasured is the measured FWHM diameter of the feature, Sfootprint is the FWHM width of the instrumental response function, and Sfeature is the approximate scale size of the feature. Using the unsmoothed data, we made 14 different profile cuts through the center of the enhancement (avoiding the asymmetry to the southeast) to obtain a measured size of Smeasured = 46.3 ± 4.4 km. If we are conservative and use a 3-sigma uncertainty at the large size, then Smeasured ranges from 46.3 to 59.5 km. With Sfootprint = 46.2 km, equation (7) implies that the feature has a size ranging from 3 to 37.5 km. This is within the size range given by the size of the albedo feature.
 Now with the lower and upper size ranges, we can determine a corresponding range in surface abundances. To carry this out, we have simulated two features: one the size and shape of the albedo feature, as determined by the 1 km/pixel CSR 415 nm data, and the other a 38 km diameter circle. Each feature is placed on a thorium background of 0.63 μg/g, as taken from Figure 17, and convolved through the instrument spatial response function and smoothing algorithm. Statistical variations due to the limited number of measurements have also been taken into account with this analysis. The thorium abundance of the simulated features is then adjusted so that the simulated data match the measured data. The resulting surface abundances are given in Table 3 (fourth column, labeled “Surface Estimate”) and range between 40 and 55 μg/g. This is higher than any other LP-GRS measured thorium abundance on the lunar surface and higher than most sample abundance values. Furthermore, after completing the survey of small-area thorium features, there does not appear to be any other location on the Moon that has this combination of high thorium abundances and low iron abundances. In short, on a scale size of ∼40 km, the Compton/Belkovich thorium anomaly is a compositionally unique location on the lunar surface.
 The earlier studies of Lawrence et al. , Elphic et al. , and Gillis et al. [2002a] concluded that of known rock types, the Compton/Belkovich region was most consistent with an evolved igneous rock type such as alkali anorthosite. Now, with more complete elemental compositions, we find the data are still most consistent with an alkali anorthosite-type lithology, albeit with some caveats. Compared to a granite or KREEP norite type composition, the SiO2, FeO, CaO, and Al2O3 abundances at Compton/Belkovich are all most similar to an alkali anorthosite (Table 3). The MgO abundances, on the other hand, are a good bit higher than for alkali anorthosite. However, there is still significant scatter in the LP-GRS MgO abundances and therefore these data may not be representative of the small Compton/Belkovich region. The samarium abundances, while also showing the anomaly, may be biased so that the Sm/Th ratio can easily be close to the alkali anorthosite/KREEP norite values. However, as discussed by Elphic et al. , if this region were dominated by a granite-type lithology, the samarium abundances would probably be too low to be detectable.
 Finally, the surface thorium abundances are higher than what is typically seen with alkali anorthosites. However, most measured alkali anorthosite samples show significant variation in thorium abundances (the thorium abundances for different alkali anorthosite samples range from 0.1 to 26 μg/g (R. Korotev, personal communication, 2002)). Furthermore, since most of these samples are small and coarse grained, it is not at all clear that the measured thorium abundances are representative of a generalized alkali anorthosite rock type. As a consequence, very little is known about this rock type and it may indeed be possible that the thorium abundances of an alkali anorthosite type rock at Compton/Belkovich could be as high as 40–55 μg/g.
5.3.2. Kepler Crater
 Kepler crater, a 31 km diameter crater located at (38°W, 8.1°N), is one of the best examples of the Th-Fe AC class. As seen in Figures 17a and 17b, there is a thorium-iron anticorrelation across the entire 20° × 20° field of view and not just at the crater. Kepler is a relatively young Copernican age crater [Wilhelms, 1987] that has excavated high-thorium and moderately high iron material lying underneath high-iron mare basalts of western Procellarum. Kepler, however, did not excavate deep enough to penetrate the high-thorium material and excavate more anorthositic material, as was apparently done by Copernicus crater to the east [e.g., LeMouélic et al., 1999].
Figure 19 shows a plot of thorium versus iron abundances for the 20° × 20° field of view around Kepler. There is a strong anticorrelation trend (the correlation coefficient is −0.82) for these data. In fact, of all selected thorium enhancements in Table 2, Kepler shows the strongest and simplest anticorrelation trend between thorium and iron. One other region, however, that shows a similar anticorrelation trend, but is not listed in as a small-area thorium enhancement, is the Fra Mauro Formation/mare basalts around the Apollo 12 and 14 landing sites [Gasnault and Lawrence, 2002]. Similar thorium-iron anticorrelations have also been noted with Apollo 12 sample data [Korotev et al., 2000].
 The simple trend seen in Figure 19 for Kepler suggests that in this region there is a simple mixing of two end-member compositions. This mixing can be physical mixing of regolith materials (vertical and/or lateral) as well as instrumental footprint mixing from the thorium data. In addition to the orbital data, Figure 19 also shows various rock compositions that can be possible mixing end-members. At high-thorium, low-iron abundances, the circle and asterisk show the compositions for two different MIB groups described by Korotev [1998, 2000]. These clearly lie close to the trend seen in the orbital data. For low-thorium and high-iron compositions, there is not as clear an end-member composition from the sample collections. For example, Apollo 12 and 14 mare basalts [Korotev, 1998] lie below the trend line given by the orbital data, with lower thorium and/or iron compositions than would be expected for the observed orbital composition. However, on the basis of orbital CSR and LP-GRS data [Lawrence et al., 2002; Staid and Pieters, 2001], there are numerous unsampled mare basalt regions that have FeO abundances high enough (20–25 FeO wt.%) to reach up to the trend line. In addition, as was suggested earlier in this paper (section 5) and by others [Etchegaray-Ramirez et al., 1983; Jolliff et al., 2001; Gillis et al., 2002b; Flor et al., 2002; Gasnault et al., 2002b], there are indications that there exist high-thorium (>2 μg/g) mare basalts that are either underrepresented or absent from the sample collection. Both the high-iron and high-thorium trends are given by arrows in Figure 19, suggesting a likely range of possible end-member compositions. In summary, it appears that this region around Kepler consists of various mixtures of a KREEP-like MIB composition and high-iron and/or high-thorium mare basalt.
 With this relatively simple region of two-component mixing of plausible end-members, one question we can ask is does this two-component mixing trend extend down to smaller scales of 1 km? For example, there are small (20–40 km) low-iron (∼8 FeO wt.%) regions to the north and south of Kepler crater that appear to have high-thorium counterparts in the smoothed thorium map of Figure 17a. Could these be regions of relatively high thorium abundances (i.e., >12 μg/g) that are smoothed out by the broad LP-GRS footprint? With our knowledge about the LP-GRS footprint, we can try to address this question; i.e., does the thorium-iron anticorrelation trend hold up at small scales? If not, can we determine why?
 To answer this, we have carried out the following analysis. First, we have smoothed the CSR FeO data in the Kepler region using both the instrument response and smoothing algorithm (Figure 20a) and mapped the result as 1/FeO. Next, using a scatterplot of the smoothed FeO data and thorium data (Figure 20b), we have derived a scale and offset factor that converts the 1/FeO map to units of μg/g thorium. We can now compare the thorium and smoothed 1/FeO map (in units of μg/g thorium) to see if they agree or disagree (Figure 20c).
Figure 21 shows a histogram of the residuals between the two maps. As seen, over 70% of the pixels have residuals of less than 0.1. Furthermore, pixels with larger residuals are not randomly distributed but are concentrated in areas to the north and west of Kepler crater in various portions of the ejecta blanket (Figure 20c).
 While the match between the two maps is not perfect, it is good enough to lead us to a final step of making a map, at 1 km/pixel resolution, of FeO in units of μg/g thorium (Figure 22). If this map reflects the true thorium abundances at the surface, then there are indeed numerous locations that have high-thorium abundances (up to 15 μg/g) with very little mixing from mare basalt materials. For example, the triangle in Figure 19 shows the average thorium abundance in the lower abundance iron regions (9–10 FeO wt.%) to the north and south of Kepler crater. Since the southern regions, in particular, are located in areas of low residuals, this is evidence that the 1km/pixel 1/FeO map closely approximates the true thorium abundances. Figure 19 shows that since these regions are close in abundance to the thorium-rich MIBs, they may represent exposures of MIB-type materials with very little mixing from mare basalts. Furthermore, even in regions where the match between the thorium and smoothed 1/FeO map is not so good, we can learn information. For example, if the region north of Kepler crater shows large residuals because the thorium-iron anticorrelation trend breaks down, the regional trend may not be valid because the CSR and LP-GRS data sample to different depths. Specifically, the CSR data is only sensitive to composition variations from the top few microns while the LP-GRS data senses composition variations down to depths of 30–50 g/cm2. If the ejecta blanket is relatively thin in this region (i.e.,<30 cm), then the LP-GRS data may simply be sensing the low-thorium, high-iron mare material that lies beneath a higher-thorium ejecta blanket.
 Of course, there may be other reasons for the areas of larger discrepancies. For example, some of the regolith material may not follow the regional thorium-iron trend for geochemical reasons unrelated to surface thickness. Second, the response function used for the smoothing may not be perfect. Finally, the missing data in the CSR map to the west and north may also be affecting the smoothing analysis so as to increase the residuals. All of these possibilities should be studied in further detail.
6. Conclusions and Future Work
 In this paper, we have derived a map of thorium abundances on the lunar surface with a known spatial resolution (∼[80 km]2 FWHM) that is better understood than results from any other orbital gamma-ray study. In particular, as part of this study, we have gained a good understanding of the orbital gamma-ray spatial footprint using modeling confirmed by both high- and low-altitude LP-GRS measurements. With our improved understanding of the spatial footprint, we have further been able to study, classify, and characterize a variety of small-area thorium features on the lunar surface.
 Further work that should be done as a result of this study includes the following. First, in the area of better understanding the technique of gamma-ray spectroscopy, the results of this study should be used to carry out spatial deconvolution analyses of the LP-GRS data. If the spatial response functions can be adequately known, spatial deconvolution analyses can possibly improve the spatial resolution by up to a factor of two to three [Jansson, 1997]. The confirmation from this study that we have a good understanding of the spatial response function is an excellent starting point for carrying out such work. Work should also be done for understanding the spatial response of other elemental abundances from the LP-GRS data, notably iron abundances from the 7.16 MeV gamma-ray line. Finally, more work can be done to better understand the latitude dependencies of the spatial response that were suggested here by this study.
 In lunar science areas, there is substantial work to be done on better understanding the 42 small-area thorium features listed in Table 2. Specific tasks to carry out include photogeology studies of each region and more detailed remote sensing composition measurements using both CSR and LP-GRS data. In particular, using our understanding of the gamma-ray spatial footprint, we can try to constrain our understanding of the surface composition for all these regions.
 The uncertainty in spatial resolution caused by the spacecraft motion of 50 km/32s can be calculated using the following relation:
where σ2 is the variance, P(x) is a normalized probability, and the integral is calculated over the ground track x. Here P(x) takes the form
 The authors would like to thank Randy Korotev and Mark Robinson for helpful reviews. This work was supported by NASA through PG&G grant W-19-849 and conducted under the auspices of the U. S. Department of Energy.