Based on the multichannel brightness temperature observations of Chinese Chang'E-1 lunar satellite, inversions of the physical temperature profile of the lunar regolith medium and its layer thickness are discussed. The correspondence of the brightness temperature distribution to the lunar topography is demonstrated and discussed, especially around lunar polar regions under poor solar illumination. As examples, in two areas, along the lunar equator and along the line of longitude 150°W, the physical temperature profile of regolith layer and its thickness are inverted using a three-layer model of thermal radiative transfer. The results which are based on some Apollo measurements are compared with an empirical formulation of the physical temperature as a function of latitude.
 During long geological history of the Moon, the lunar surface has experienced volcanisms and high-velocity impacts of large and small meteoroids, which result in various topographic structures on the lunar surface, such as maria, mountains, cliffs, rilles and craters [Heiken et al., 1991]. The uppermost layer of the Moon surface is known as lunar regolith, which consists of fragmented materials such as surface layer dust, unconsolidated rock material, breccia, glassy fragments and so on. The average thickness of the regolith layer was estimated as about 4–5 m for maria and about 10–15 m for highlands [McKay et al., 1991]. Knowledge of the structure, composition and distribution of the lunar regolith can provide important information concerning the lunar geology and resources for future lunar exploration.
 China successfully launched its first lunar exploration satellite Chang'E-1 (CE-1) on October 24, 2007 at lunar circular orbit ∼200 km [Jiang, 2009; Fa and Jin, 2007a; Jiang and Jin, 2010]. A multichannel microwave radiometer, for the first time, was aboard the CE-1 satellite with the purpose of measuring the microwave thermal emission from the lunar surface. There are four frequency channels for CE-1 microwave radiometer: 3.0, 7.8, 19.35 and 37.0 GHz. The observation angle is 0°, the spatial resolution is about 35 km (for channels 7.8, 19.35, 37.0 GHz) and ∼50 km (for 3.0 GHz), and the radiometric sensitivity is about 0.5 K. The brightness temperature (Tb) of a footprint is measured for each 1.6 s, and the spatial velocity of the subsatellite point is about 1.4 km/s, which yields a high overlapping rate between two adjacent observations in the track direction based on the resolution of CE-1 radiometer (∼35–50 km). Since the period of one track is 127 min, it can be calculated that the observation gap between two sequential tracks at the equator is about 35 km. It yields an overlapping rate in cross-track direction about 0% and 25.1% from the equator to the near poles. During a whole track, totally about 4000 observations can be obtained. One track of CE-1 brightness temperature (Tb) data had been studied [Fa and Jin, 2010a]. During the whole CE-1 running period, a large amount of Tb data can be obtained to cover the whole lunar surface several times. For example, the CE-1 observation at lunar nighttime can cover the lunar surface for 6 times in some places at most and once at least. However, because of the different local times of the observations, only those Tb data at similar local times are collected for inversion of the lunar surface temperature. The measurements of multichannel CE-1Tb have been further applied to invert the global distribution of the regolith layer thickness, from which the total inventory of 3He (Helium-3) stored in the lunar regolith layer can be estimated quantitatively [Fa and Jin, 2007b, 2010b].
 The physical temperature of the lunar surface is a key parameter for inversions of regolith properties from thermal emission observations. Large differences of the surface physical temperature between lunar daytime and nighttime and the thermal inertia produce inhomogeneous physical temperature profile in the regolith media. Lunar cratered topography might shadow the solar illumination and makes great variations in the surface temperature. The Tb observations at high frequency channels, i.e., 19.35 and 37.0 GHz, which are especially sensitive to the top surface, are usually applied to inversion of the surface physical temperature [Fa and Jin, 2010a]. However, Tb data of two channels cannot retrieve more than two unknowns. Also, an ill-conditioned problem might happen if the Tb data at these two channels are perturbed to close each other. Thus, to invert the physical temperature over the whole lunar surface, as the physical temperatures at Apollo landing sites are first inverted as initial results and validated by the ground measurements as available, the empirical cosine formulation of physical temperature with all other latitudes was simply employed for inversions over the whole lunar surface. It means that those physical temperatures at the same latitudes were seen as the same without considering the topographic effect under different solar illumination [Fa and Jin, 2010a].
 In this paper, the primary 1307 tracks of swath data by CE-1 microwave radiometer from November 2007 to February 2008 and from May 2008 to July 2008 are collected. According to the local time and relation between the brightness temperature and the solar illumination angle in observations, the distributions of microwave Tb from lunar surface at lunar daytime are constructed using the nearest interpolation approach. To see how the cratered topography affects the Tb observations and inversion of physical temperature profile, two specific regions, i.e., the circle along the lunar equator and a line along the longitude 150°W from the southern pole to northern pole are selected.
 Derived from a simplified two-layer radiative transfer model with temperature profile [Jin, 1994; Jin and Fa, 2010], the physical temperature profile of lunar surface media, proposed as an exponential form, is inverted by the least mean square method from Tb data at high frequency channels, 19.35 and 37.0 GHz. It shows the influences of the lunar cratered topography on retrievals of the physical temperature of lunar surface media. It is then applied to retrieval of the regolith layer thickness using Tb data at lower frequency channel, 3.0 GHz. Especially, the brightness temperatures at polar regions are discussed.
2. Microwave Tb of the Lunar Polar Regions
 The gridded data to show CE-1 Tb mapping are about four pixels per degree for each latitude and longitude. A nearest neighbor interpolation method is applied to fit the places where no data available. The CE-1 Tb data of the lunar north pole (60°N–90°N) and south pole (60°S–90°S), and their DEM (Digital Elevation Model) [Ping et al., 2008], are shown in Figures 1a–1d, respectively. To map the Tb at lunar polar regions, we first construct a map of 120 × 1440 meshes dividing each latitude and longitude degree on the lunar surface into four pixels, then fill the Tb data at the very site of the pixel into each mesh, and the Tb is averaged if more than one observation available in one pixel. The microwave Tb in Figures 1b and 1d are both observed at lunar local daytime from 11:00 to 13:00 h. This local time can be calculated from the position of the observation point, the solar illumination angle and azimuth angle, which have all been provided by CE-1 data. The Tb37 (Tb at 37.0 GHz) data in Figure 1b is chosen from the observations between May and July in 2008, and the Tb37 data in Figure 1d is from November 2007 to February 2008, when the poles are all during their own summer hemisphere. It can be seen that the contours of cratered lunar topography can be identified in Figures 1b and 1d. For example, those crater rims to face equatorward are warmer than their surroundings, those rims to face poleward become cooler, and the crater floors are generally cool, especially near the poles under long time shadowing. As examples, two typical craters, the carter Peary (88.5°N, 30°E) and the crater Shoemaker (88.1°S, 44.9°E), are chosen to show Tb37 data with DEM in Figures 2a and 2b, respectively. The DEM data used in Figure 2 are from the measures of the laser altimeter onboard Japanese SELENE.
 The solar illumination intensity in lunar polar areas (latitude >85°) has been especially studied using the laser altimeter data from Japanese SELENE satellite observations [Noda et al., 2008] during January 1 2008 to March 31 2008. Figures 3a–3d present the solar illumination and Tb at 37.0 GHz in lunar north and south polar regions, respectively. Those black spots in Figures 3a and 3c are possibly the permanently shadowed regions [Noda et al., 2008], where sunlight might not be able to reach. Correspondingly, it can be seen consistently from Figures 3b and 3d that Tb at those regions are generally much lower. The IR (infrared) Tb of the lunar southern pole can also be found in the work of Paige et al. . Even if the Tb data of Figures 3b and 3d look much coarse due to the resolution of the CE-1 microwave radiometer, the contour or variation of the Tb distribution look consistently with the results from Paige et al. . And, because the microwave Tb data does not probe the extremely low temperature due to the larger footprint, the extremely low temperature in small scale might have been averaged and overlooked.
 For example, Figures 4a and 4b give Tb at 37.0 GHz, respectively, along the latitudes of the craters Peary (88.5°N, 30°E), and Shoemaker (88.1°S, 44.9°E) [Lucey, 2009]. It can be seen that poor illumination conditions on the insides of these craters cause much lower Tb compared with other places with the same latitudes. In these permanently shadowed regions, there might be a possibility to allow the stability of water ice [Paige et al., 2010]. Cratered topography does affect the solar illumination, especially in polar regions, and reduce Tb emission, consequently.
3. Data Calibration and Inversion of the Physical Temperature Profile and Regolith Thickness
 The three-layer regolith media model with physical temperature profile T2(z) and thickness d2 is shown in Figure 5. The effective dielectric constant of the lunar soil and regolith layer, ε1 and ε2, are calculated from the bulk density and FeO + TiO2 content [Jin and Fa, 2009], and that of the substratum ε3 = (10 + i0.5) is as usually assumed [Heiken et al., 1991].
 The contribution of the first layer can be expressed as [Jin, 1994]:
where κa1, κa2 are the absorption coefficients in the media 1 and 2 of Figure 5, rmn is the reflectivity at the interface of the media m and n (m,n = 0,1,2,3).
 The upgoing emission in the direction θ from a thin layer dz at z of the regolith medium is written as equation (2), when it's just below the top surface,
where κ2υ(z) is the absorption coefficient of the regolith medium at frequency υ. Note that the variable z in integral is taken as positive at this step for simplicity. And the downgoing emission in π − θ is written as
It is reflected by the underlying surface at z = −d (the p(vertically or horizontally)-polarized reflectivity is r23p(θ)) and returned upgoing after the layer attenuation as
Thus, from equations (3) and (4), the upgoing p-polarized emission at the bottom of the first layer in the three-layer model shown by Figure 5 is written as
where the first term on the right hand side is the emission from the underlying bedrock media. Finally, the p(=v, h)-polarized brightness temperature observed at observation angle θ0 on the upper surface of the first layer is obtained as
where θ and θ0 satisfy the Fresnel law, t01p = (1 − r01p) and t12p = (1 − r12p) are the p-polarized transmittivity (i.e., emissivity) between up-space and layer 1, and layer 1 and 2, respectively. Equations (5) and (6) can be numerically calculated for brightness temperature of the media with both the uniform and non-uniform temperature profile and volumetric density. Now let the medium be uniform, i.e., ε(z) = εg(ε′g + iε″g), and calculate , where c is the light speed in free space, and εg is seen as an effective permittivity of the medium, and is mainly determined by the bulk density, ρ, and its FeO + TiO2 content, which was first derived from Clementine optical data [Lucey et al., 2000; Fa and Jin, 2007a; Jin and Fa, 2009].
 To study the thermophysical properties of the regolith media, Vasavada et al. , based on a model of half-space media with a top 2 cm layer, simulated an exponential-like near surface temperature profile on Mercury and Moon.
 Thus, making the exponential form to take account of temperature as a function of depth, the temperature profile of the regolith layer at lunar daytime is proposed as
The boundary conditions yield
where T1, T3 are defined as the physical temperature at z = 0 and at the surface the substratum media, respectively. (Note that the temperature profile at lunar nighttime is treated similarly by using a negative β.)
 Solving A and B, and substituting them into equations (5) and (6), the Tb observed at θ0 = 0° (as CE-1 observation, and p-polarization is not specified) is finally obtained as
 At high frequency channels, i.e., 19.35 GHz and 37.0 GHz, the attenuation factor actually approaches zero. Thus, equation (9) at these channels can be simplified to
 On the other hand, the three-layer model can be viewed as a two-layer model for these high frequency channels, shown as Figure 6. Note that here the temperature profile has not been taken into consideration and the physical temperature of the regolith layer for 37.0 GHz channel and 19.35 G channel are T′10 and T″10, respectively. The difference of T′10 and T″10 can reflect the influence of the temperature profile in the regolith layer. The brightness temperature for such a model can be written as
Thus, the temperature profile is determined as long as T10T′10 and T″10 are obtained.
 To avoid an ill-posed problem, the least mean square method is employed to invert the physical temperatures T10, T′10 and T″10, based on the minimized difference between the simulated Tb of equations (11a) and (11b) and CE-1 Tbo data (the superscript ‘o’ denotes observation) at 19.35 and 37.0 GHz channels. It is written as
 Empirically making the variation ranges of T10 within (350°K, 400°K), T′10 (240°K, 290°K), and T″10 (230°K, 280°K) at Apollo landing sites, equation (14) is solved to find the most appropriate T′10 and T″10. The bulk density of the lunar soil layer is chosen to be 1.3 g/cm3 according to McKay et al. , and that of the regolith layer is selected as 1.5 g/cm3.
 From November 2007 to February 2008, all 13 tracks of the data at six Apollo landing sites (3 tracks for Apollo14 and 2 tracks for other 5 landing sites) observed by CE-1 radiometer at lunar daytime are collected. Figure 7 shows the inverted physical temperatures T10, T′10 and T″10 at Apollo landing sites, and the parameters of the temperature profile, β, A and B can then be calculated.
 Using these physical temperatures inverted in Figure 7 and the thickness of the regolith layer at the corresponding Apollo landing sites [Shkuratov and Bondarenko, 2001], we can simulate Tb of these 4 channels based on the three-layer model for all 13 observations at six Apollo landing sites. Comparisons of these simulations with the CE-1 observations are given in Figure 9. Deviations at lower channels, i.e., 3 GHz (black points) and 7.8 GHz, can be seen.
 Taking the deviation of 3.0 GHz channel in Figure 9, ΔTb3 = Tb3 − Tb3O, as shown in Figure 10, it can design the calibration at 3.0 GHz for different Tb3o described by a curve in the figure. Then, these calibrated Tb3 (red points in Figure 9a) match well with the simulation.
 Taking CE-1 Tb37 and corresponding DEM of Japanese SELENE data along the lunar equator, as shown in Figure 11, it can be seen that Tb on the nearside (−90°, 90°) becomes higher in mare with lower DEM, probably due to the lower albedo. However, on the lunar farside (90°, 180°) and (−180°, −90°), the Tb becomes lower where the altitude is higher and albedo is larger. The Tb at different places with the same latitude might vary with the topography due to different albedos in mare and highland. On a smaller scale, the fluctuations in Tb data might result from the topographic variation of the lunar surface, especially in the highlands with complicated topography and large shadows, which may cause different solar illuminations. In Figure 11, those tracks of data with similar solar illumination angles from 0° to 14° at the equator are collected for study, which reduce the possible influence of different solar illumination. Certainly, Tb is influenced not only by topography, but also by many other factors, e.g., dielectric properties, latitude etc. Another set of Tb data such as along a middle latitude is also tested, the correspondence between Tb and DEM looks similar.
 Assuming that T10 at the equator ranges from 350 K to 420 K, parameters T′10 and T″10 can be solved from equations (11a) and (11b). The range of the regolith thickness d2 is chosen as from 1 m to 20 m, then the Tb at 3.0 GHz channel after calibration as shown in Figure 10 is applied to inversions of T10 and d2 using the least mean square method.. The physical temperatures T10 and (T10-A), i.e., B in equation (7), inverted from the CE-1 data are given in Figure 12.
Figure 13 gives the inverted T10 with corresponding topography along the lunar equator. It can be seen that generally, T10 of highlands with high altitude is high, and T10 of maria with low altitude is also low. Fluctuation of inverted T10 partially is due to sensitivity of two parameters inversion using the least mean square method. The inverted (T10-A) of the underlying bedrock is smoothly varying around 230 K to 250 K, because the inner thermal exchange is most likely balanced with little or no dependence on the surface solar illumination.
Figure 14 gives the profile parameter β and the corresponding FeO + TiO2 content along the lunar equator, in which, the FeO + TiO2 content is calculated from the Clementine optical data [Lucey et al., 1995, 1998, 2000; Fa and Jin, 2007a]. Certainly, this attenuation coefficient, β, becomes high where there is more FeO + TiO2 content, and vice versa.
 The regolith layer thickness d2 around the lunar equator is inverted as shown in Figure 15. It can be seen that the thickness d2 in highlands is about 10–20 m, which is very similar to the results of the three-layer model with homogeneous temperatures [Fa and Jin, 2010a], and the maria have a smaller thickness only about 3–5 m. The variation tendency is also consistent with that from Fa and Jin [2010a].
 Using these inverted parameters above and equation (9) to simulate Tb, Figure 16 shows comparison between the simulated (line) and CE-1 Tb (discrete points) data. Certainly, this match has been promised in the previous approach of least mean square method.
 Now select the CE-1 Tb data along longitude 150°W. For example, Figure 17 shows Tb at 37.0 GHz with the corresponding topography. The solar incidence angles' range of the collected data is the same as Figure 11. It can be seen that Tb basically follows the latitude change in a cosine way.
along with the latitude φ. Since the temperature T10 around the lunar equator (φ = 0) has been found around 390 K, in our inversion tests, the variation range of T10 for the study regions is designed as
where the exponent, e.g., α = 0.2, and ±30°K can be tested as the proper values in inversions. The inverted T10 and T10 − A along longitude 150°W are given by Figure 18.
Figure 19 gives the correspondence of the profile parameter β and the FeO + TiO2 content along longitude 150°W. It is seen that high FeO + TiO2 content causes high β. Here, the FeO + TiO2 content is also derived from the Clementine optical data using the method proposed by Lucey et al. , the same as Figure 14.
Figure 20 presents the inverted regolith layer thickness along longitude 150°W. In the high latitudes above 75°, regolith layer thickness is not well inverted because the parameter of FeO + TiO2 content at those regions might be not reliable due to calibration problem [Lucey et al., 1998]. The inverted regolith layer thickness between the latitudes (30°∼70°) seems higher than Fa and Jin [2010a, 2010b], because the FeO + TiO2 content and inverted β of this region are both small. It might yield large thickness. It is confirmed that the FeO + TiO2 content and the bulk density in regolith layer are also key parameters for retrievals of the regolith layer thickness and other properties.
Figures 21a and 21b show comparisons of the inverted T10, T10 − A along latitudes with the empirical formulation of Lawson et al. , i.e., equation (15). It can be seen that the empirical formulation of a cosine function is quite reasonable, but the exponent for the underlying bedrock needs adjusting depending on situation.
 The Tb data at four channels are first collected from CE-1 observations during November 2007 to February 2008 and May 2008 to July 2008 at lunar polar regions. Taking the examples of Tb along the lunar equator and a line along the longitude 150°W, it well demonstrates the correspondence between Tb, DEM and solar illumination conditions. Cratered topography influences solar illumination, especially in lunar polar regions, and makes lower emission inside the craters and on the poleward crater rims due to lower solar illumination, generally.
 Based on a three-layer radiative transfer model with physical temperature profile in regolith layer, Tb observed by CE-1 multichannel radiometers can be simulated. First, observations at 3.0 GHz channel are calibrated according to the model and measurements of the regolith thickness at Apollo landing sites. Taking CE-1 Tb data especially around the equator circle and a line of longitude 150°W, the physical temperature profiles, described by three parameters T10, β, A, of the regolith media are first expressed by CE-1 Tb data at high frequency channels 19.35 and 37.0 GHz, then the regolith layer thickness and T10 is determined using the calibrated CE-1 Tb data at 3.0 GHz channel by the least mean square method. The simulated Tb from the inverted parameters T10, β, A and d2 of the three-layer model are well matched with the observed multichannel CE-1Tb data. The inversion results of physical temperature at the surface and the regolith layer along the longitude line generally correspond to the empirical formulation of an exponential form of the cosine function.
 The results demonstrate that the cratered topography affects the solar illumination and then thermal emission, especially around the lunar polar regions under poor solar illumination. It is confirmed that the lunar surface topography, the physical temperature profile, the dielectric property (closely related to FeO + TiO2 content) and the bulk density of regolith layer are key parameters for retrievals of the regolith layer thickness and other thermal properties.
 CE-1 microwave radiometer data are provided by the Center of Lunar Exploration Project and Ground Station for Data, Science and Application in China, and the DEM data of lunar surface are supplied by Japanese Aerospace Exploration Agency (JAXA). The authors would like to extend sincere thanks to them. This work was supported by NSFC 41071219 and 60971091.