Within the renewed interest in the study of the Moon, in 2006 the European Space Agency approved a feasibility study for the European Student Moon Orbiter (ESMO) mission. In order to accomplish the ESMO mission objectives, a Microwave Radiometric Sounder (MiWaRS) was selected as a possible payload for flight on the ESMO satellite. This work summarizes the results of a numerical analysis of MiWaRS sounding capabilities. An (inhomogeneous) multilayer model of the microwave emission from the Moon's subsurface is presented, focusing the attention on the Moon's morphological, thermal, and dielectric properties. These properties have been determined and parameterized after a thorough investigation of available measurements and models. To this end, a radiative transfer numerical model, neglecting volume scattering, is coupled with a nonlinear thermal equation to simulate the microwave emission of the Moon's subsurface. Numerical simulations between L and Ka bands are performed to investigate the capability of MiWaRS to sound the characteristics of the Moon's regolith subsurface and detect the presence of rocks and ice under the near-surface regolith layer. Under these forward model assumptions, results show that the Moon's brightness temperature response allows the detection of discontinuities within regolith media down to 2 and 5 m depth when channels at 3 and 1 GHz are used, respectively. Lunar near-surface temperature may be also estimated within an accuracy less than a few kelvins. The discrimination of ice from rock by MiWaRS is hardly practicable and is limited to the presence of ice in the upper layers (with a depth less than 20 cm) beneath the lunar crust.
 There is nowadays a renewed and growing interest in returning to the Moon, as in the 1960s when the final achievement was the walk of the Apollo astronauts on its surface. The current target of the envisaged lunar missions is to establish a permanent presence in order to start a new era of Moon exploration and exploitation [see NASA Office of Public Affairs, 2006]. The experimental observation of the Moon began in 1959 with Lunar 2, a space orbiting probe launched by the Soviet Union [Harvey, 1996]. Prior to that launch, the only available means of investigation had been observation from the Earth by telescopes, principally at visible and infrared wavelengths [Heiken et al., 1991]. These have been the preferred wavelengths for lunar remote observation, even though microwave band was also used occasionally [Keihm and Langseth, 1975]. Unlike visible and infrared waves, microwaves can penetrate to a certain depth, depending on the frequency and the dielectric properties of the subsurface [Ulaby et al., 1982].
 The first theoretical studies of the lunar microwave brightness temperature began in the 1940s, thanks to the contributions of Dicke and Beringer  as well as Piddington and Minnett . These studies were basically continued until the 1980s, exploiting the available lunar measurements through the work of Keihm and Langseth , Keihm and Cutts  and Keihm . After 2 decades of relatively little research activity, the increasing interest to explore the Moon and its environment has motivated several space agencies to design new lunar remote sensing missions [Bhandari, 2005; Houghton et al., 2006]. In 2006 the European Space Agency (ESA) approved the feasibility study of the ESMO (European Student Moon Orbiter) mission. The major objective of the ESMO mission is to acquire high-resolution visible images of the Moon's surface using an orbiting platform [Walker and Cross, 2010]. The Microwave Radiometric Sounder (MiWaRS) was also selected as a possible candidate payload for the ESMO satellite [Montopoli et al., 2007]. It is worth mentioning that the Chinese Chang'e-1 satellite, launched at the end of 2007, includes a multifrequency microwave radiometer as well [Sun and Dai, 2004]. This passive instrument operates at the following frequencies: 3, 7.8, 19.35 and 37 GHz and the nominal lunar penetration depths are of the order of 30, 20, 10 and 1 m, respectively.
 This work illustrates the results obtained during the MiWaRS feasibility study, focusing on the development of a multilayer, inhomogeneous, subsurface forward model to simulate the microwave brightness temperature (TB). These numerical simulations are intended to investigate the expected performance of MiWaRS in order to accomplish the following objectives: (1) mapping of the surface microwave emissivity, (2) probing of subsurface thermal properties, and (3) sounding of the regolith layer and its possible anomalies. In order to keep the instrument as small as possible, we have also confined our analysis to frequencies between 1 and 24 GHz, thus including both MiWaRS and Chang'e-1 radiometer specifications.
 The paper is organized as follows: Section 2 gives an introduction to the Moon's stratigraphy and the adopted electromagnetic (EM) models. Section 3 summarizes a simplified thermal model of the Moon's subsurface, whereas in section 4 the incoherent approach used to simulate the brightness temperature TB emitted by the Moon's stratified subsurface is described and a numerical sensitivity analysis performed. In section 5 conclusions are drawn.
2. Moon's Subsurface Characterization
 The first step to build the electromagnetic (EM) model of the Moon's subsurface is to understand the Moon's stratigraphy and its characteristics [Keihm and Cutts, 1981; Strangway et al., 1977]. In this section, the properties of the Moon's stratigraphy and models describing the vertical profile of EM and physical properties, defined in terms of dielectric constants and material density, respectively, are briefly described.
2.1. Moon's Stratigraphy
 After the Apollo missions in the 1970s, many samples of lunar material have been collected and many experiments have been carried out, so that a better view of the lunar stratigraphy has been obtained Heiken et al. . A simplistic but effective view of the lunar stratigraphy, as shown in Figure 1a, consists of a series of layers of material ejected from individual meteorite craters over a period of time, and embedded lava flows and pyroclastic deposits [Heiken et al., 1991]. The upper layer is composed of regolith (or lunar soil, often used as synonyms), which is the result of meteorite impacts.
 The regolith ideally consists of two zones: (1) a near-surface reworked or gardened zone, typically from a few cm to few meters thick, in which all layers have been homogenized or mixed together, and (2) an underlying slab or sequence of slabs in which the original layering is still undisturbed. As the distance under the lunar surface increases some intrusion of rocks can be found. Dust, unconsolidated rock, glass and fragments constitute the lunar surface regolith, which has been subjected to continuous impacts by meteoroids of variable size during its long geologic history. Based on current knowledge, the regolith thickness varies from several meters to tens of meters. Due to the impacting process throughout the geological history of the Moon, it is plausible that the thickness of the regolith is correlated with the lunar surface age: the greater the age, the thicker the regolith [Jin and Fa, 2009; Heiken et al., 1991]. Additionally, it should be mentioned that the average thickness of the regolith depends on the region, where maria (i.e., region where there are large, dark, basaltic plains formed by ancient volcanic eruptions) and highlands (i.e., areas that are heavily cratered and mountainous) regions have thickness of about 4–5 m and 10–15 m, respectively. The latter consideration has been used by Jin and Fa  to derive a model for the regolith thickness from a lunar digital elevation model. In this paper a maximum thickness of 5 m is considered in order to match the sounding capabilities of the MiWaRS with its feasibility for space exploration.
Figure 1 suggests the presence of lunar subsurface fragments within the regolith layer. Data about the abundance and size distribution of subsurface fragments are quite sparse [Heiken et al., 1991]. A power law parameterization of their size distribution between 0.5 and 50 cm was proposed by Keihm , summarizing the Surveyor available data into 9 rock population models with volume fraction between 0.05 and 0.20.
2.2. Electromagnetic Characterization of Moon Materials
 Following the Moon stratigraphic model, introduced in section 2.1 and in order to maintain the treatment as simple as possible, we have decided to describe the vertical profile of the Moon's subsurface composition by means of three homogeneous media: regolith, rock and ice, placed, in different order at different depths beneath the Moon's surface. Although there is no direct evidence of ice beneath the lunar surface, we have considered an ice slab in the following simulations in order to investigate the sensitivity of TB to this kind of inhomogeneity. Several Moon's subsurface scenarios will be then analyzed in section 4.
 The simulation of the Moon's EM response, in terms of brightness temperature, requires a detailed knowledge of the EM properties of the Moon's materials at the frequencies of interest. This is not a simple task, especially at frequencies greater than 1 GHz, since most of measurements made on lunar samples have been accomplished at frequencies below 1 GHz. Fortunately, as reported by Heiken et al. , a few measurements on Moon regolith and rock samples at 10 and 30 GHz were performed and these allowed us to extract some models to describe the permittivity of the lunar regolith and rock. In greater detail, the lunar complex relative permittivity ɛr (or complex relative dielectric constant) may be indicated as follows:
where ɛr′ and ɛr″ are the real and imaginary part of ɛr, respectively, tanδ indicates the loss tangent as a function of frequency f and depth z, expressed in GHz and cm, respectively. To be consistent with the reference system shown in Figure 1, it should be noted that the variable z is negative below the Moon's surface. It should be recalled that the loss tangent is a measure of the degree of power dissipation in the considered material. From Heiken et al.  and Ji et al. , it emerges that, for lunar material, ɛr′ is quite insensitive to frequency variations and its vertical profile can be described as:
where d is a constant and ρ(z) is the material density expressed in g cm−3 as a function of the depth z. On the other hand, the imaginary part of ɛr, ɛr″ = ɛr′[tanδ(f) ], is modeled for both regolith and rock through the loss tangent Heiken et al. , by means of the following power law relation:
where a, b, c are empirical regression coefficients, and pch is the percentage of the chemical composition (i.e., dioxide of titanium TiO2 and oxide of iron FeO) in regolith or rock, chosen in the range [0%, 30%]. Indeed, the frequency dependence of the loss tangent in (3) was not mathematically explicit in the work by Heiken et al. , but we have performed a linear approximation of the available experimental data of tanδ (see Heiken et al. [1991, p. 546, Figures 9.60–9.61] at frequencies between 1 and 30 GHz). Doing so, we can express the coefficient a as follows:
Concerning the regolith density ρ, needed for example to obtain a value using equation (2), heat flow experiments deployed on the Apollo 15 and 17 missions indicated that the bulk density must be approximately 1.3 g cm−3 at the Moon's surface [Keihm and Langseth, 1973; Langseth et al., 1976]. As a result of the aforementioned experiments, a hyperbolic relationship between density and depth is proposed in Heiken et al.  and here reported for convenience:
From equation (5), at z = 0, the surface density is exactly 1.30 g cm−3 and, when z reaches the value of 50 cm below the lunar crust and beyond, the maximum density of 1.92 g cm−3 is approached, in agreement with available measurements. At depths greater than 3 m, we have no direct measured data about the density of the lunar regolith. Note that the minus sign in equation (5) is due to the chosen reference system in which the positive z axis points upward in the zenith direction.
 For lunar rock material to our knowledge there is no empirical or theoretical model available, but it is quite well established that typical rock density variations lie between 2 and 3.3 g cm−3 [Heiken et al., 1991]. Thus, we have arbitrarily decided to fix a density value of 2.5 g cm−3 for rock. On the other hand, for ice, we have considered a density constant value of 0.92 g cm−3 and a permittivity of 3.15 − j × 5 × 10−4.
Table 1 summarizes the values adopted to describe dielectric constants and material densities in equations (1)–(4), whereas Figure 2 shows the trend of the dielectric constant as a function of density for regolith (bold lines in Figure 2 (top)) and rock (bold lines in Figure 2 (bottom)) at the frequencies of 1, 3, 12 and 24 GHz. On the other hand, for rock and ice ɛ′ and ɛ″ assume values equal to 3.15 and 5 × 10−4, respectively. Given these assumptions, for regolith density of 1.5 g cm−3 the values of loss tangent vary between 4.5 × 10−3 and 1.5 × 10−2 over the range of frequency used in this work, whereas for rock density of 2.5 g cm−3 the loss tangent is limited to the range [5 × 10−3, 1.5 × 10−2]. These intervals seem to be consistent with measurements made on lunar samples of regolith and rock, at frequencies below 1 GHz listed by Heiken et al. [1991, Table A9.16].
 Noting the large values of ɛr′ with respect to those of ɛr″ in Figure 2, both lunar regolith and rock seem to show low power losses as dielectrics typically have, although this is particularly true for rock. However, this property is slightly less verified at high frequencies where ɛr″ increases, resulting in larger power losses for a wave propagating in the considered media.
3. Moon Spatiotemporal Thermal Model
 As will be clear when the EM emission is introduced, a physical thermal profile model of the Moon's subsurface is a very important ingredient to accurately simulate the brightness temperature TB for each layer in which the Moon's stratigraphy can be discretized. The physical temperature T, at a certain depth z, has been calculated by solving the heat flow equation in a one dimensional framework:
where ρ is the media density (g cm−3), cp the specific heat (J g−1 K−1) and kT is the thermal conductivity (J s−1 cm−1 K−1), and the time dependence t has been included considering the temporal heat flux variations due to the Sun. As briefly discussed in Appendix A, the heat flow equation is solved following the method described by Di Carlofelice and Tognolatti , based on transmission line formalism. This method is able to take into account nonlinear phenomena, due to blackbody radiation (through the Stefan-Boltzmann “fourth power” law) and the temperature dependence of conductivity for superficial layers, through the Harmonic Balance method [Rizzoli et al., 1988].
 Concerning the thermal conductivity kT, its vertical profile has been described following the model formulated by Keihm  based on the surface and subsurface temperature data at the Apollo 15 heath flow site (see Keihm and Langseth  for details):
In equation (7), for z ≤ 2 cm, k1 = 6 × 10−6 and k2 = 3.78 × 10−13, whereas for z > 2 cm, k1 = k3 + k4 exp(0.25(2 − z)) with k3 = 8.25 × 10−5, k4 = 7.65 × 10−5 and k2 has been set to zero for simplicity. k1, k3, k4 are expressed in (J s−1 cm−1 K−1) and k2 in (J s−1 cm−1 K−4). The thermal conductivities of the ice and rock are kT = 2.5 × 10−2 and kT = 2 × 10−4, respectively. In addition, cp has been set to 0.55 [J g−1 K−1].
Figure 3 shows the results of the temperature simulations in terms of surface time series (shown in Figure 3 (left)) and vertical profiles (shown in Figure 3 (right)) for z between 0 and 5 m over a lunation period at lunar equator.
Figure 3 shows that the Moon's surface temperature (i.e., at z = 0) varies over a large range from about 90 K to 370 K. These extreme values are well established in the Moon literature [e.g., Heiken et al., 1991; Keihm and Cutts, 1981] as well as the increase in T with increasing depth z which is mainly due to the internal lunar heat flow. It may also be noted that the temperature does not change with time below about 50 cm in agreement with the findings from Apollo 15 data reported by Heiken et al. .
4. Moon Subsurface Emission Model at Microwaves
 In this section the EM model, based on an incoherent approach, will be described after a brief introductory background on the radiative transfer theory.
4.1. Radiative Transfer Equation
 In order to simulate the microwave radiation emitted by the Moon, in terms of azimuthally isotropic brightness temperature TB (K), the radiative transfer problem has to be addressed. The radiative transfer equation (RTE) for vertically stratified media to be solved is expressed by [Ulaby et al., 1982; Tsang et al., 1985]:
where ka (cm−1) is the absorption coefficient depending on frequency f and depth z, p stands for the polarization state (p = h or p = v for horizontal or vertical polarization, respectively) and θ is the zenith angle (see Figure 1). As is known, equation (8) is obtained under the Rayleigh-Jeans approximation, valid at microwaves, and neglecting the scattering effects [Ulaby et al., 1982]. It should be noted that at S and L band volume scattering may be ignored, but at higher frequencies, i.e., at Ka band and above, volume scattering can play a role depending upon the particle distribution albedo [Keihm, 1982; Tsang et al., 1985; Marzano, 2006]. The lack of information about the vertical profile of subsurface fragment size distribution has led us to neglect the effect of volume scattering [Heiken et al., 1991]. In regions which contain an average concentration of centimeter and larger-sized fragments, the scattering effect is expected to be 20%–30% of that due to heat flow [Keihm, 1982]. Indeed, at higher frequencies the microwave penetration capability is limited to upper layers and this tends to limit the effects of our approximation on the final results. Finally, the possible roughness of surface and layer interfaces is also neglected even though it could be included as a proper emissivity parameterization representing the surface height standard deviation through a small perturbation approach [Ulaby et al., 1982].
 In section 4.2 we will introduce the formulation for a numerical solution of equation (8), derived from Tsang et al. , and apply this approach to model the Moon's radiation properties at microwave frequencies. It should be noted that equation (8) requires, among other information, the knowledge of the physical temperature of the considered media (or the temperature vertical profile for the case of stratified media). This information will be provided by the thermal model which has been described in section 3.
4.2. Incoherent Layered Model of Moon Electromagnetic Emission
 Due to the conjectured stratified behavior of the Moon's subsurface, it appears plausible and, from a computational point of view, relatively simple to model the Moon's subsurface as a set of n flat homogeneous layers, as depicted in Figure 1b. Beneath the lunar crust (i.e., z < 0), the lth layer, into which the Moon medium is discretized, is characterized by electromagnetic parameters, such as the complex relative dielectric constant, indicated by ɛrl = ɛr(−zl) = ɛrl′ − jɛrl″, and the magnetic relative permeability, indicated by μrl = μ(−zl) = 1, and by thermal parameters, summarized by the physical temperature indicated by Tl = T(−zl). Hereafter the subscript “l” indicates a quantity at a given depth z = −zl where l can range from 0 to n. As already mentioned, the main hypothesis of the incoherent model considered here is that the scattering contribution is negligible with respect to that due to absorption. This hypothesis introduces a limitation in the general description of the Moon's EM emission, but, on the other hand, it strongly reduces the forward model complexity. The solution of equation (8), in terms of brightness temperature, for the case of stratified media in the Cartesian geometry shown in Figure 1b, can be expressed in the following form:
where we have distinguished between the downward TBdl and upward TBul contribution for the lth layer. The quantities kal, θl and p indicate the absorption coefficient, the upward and downward incidence angle at lth layer interface (see Figure 1b) and the polarization state, respectively.
 The parameters Al and Bl are coefficients derived from the boundary conditions at each interface. These relations take into account the radiation conditions at the first and last semi-infinite layers and the continuity of tangential components of the EM field. Equations (9a) and (9b) and their coefficients Al and Bl can be derived by solving the following linear system [Tsang et al., 1985]
where Γpl is the Fresnel reflectivity at the boundary z = −zl and the terms Ll, Ll+1* indicate the quantities exp(kalsecθlzl) and exp(kal+1secθl+1zl), respectively. Note that the quantity L at l and l+1 has the same coordinate zl and this is due to the fact that the reflection coefficients from below and above zl are the same. Once the coefficients Al and Bl are computed, it is straightforward to obtain the vertical profile of TB through (9a) and (9b) where the absorption coefficient kal (cm−1) is given by:
where λ0 (cm) is the operating wavelength in vacuum. Equation (11) substantially expresses that kal is twice the attenuation coefficient experienced by plane waves propagating through media [Ulaby et al., 1982]. The angles θl for l∈[0,n] can be calculated by propagating the observation angle θ0 at l = 0 up to the nth interface by means of Snell's law: nlsin(θl) = nl+1sin(θl+1) where θl and θl+1 are the angles of incidence and transmission, respectively, of the EM radiation at the boundary between the lth and (l+1)th layer and nl is the refractive index defined as (ɛ′rlμrl)0.5.
5. Numerical Results
 Simulations of upwelling TB(z, θ, f, p) at the Moon's surface (i.e., z = 0) have been performed at frequencies f of 1, 3, 12 and 24 GHz, horizontal polarization (p = h) and at the Moon's equator by using the thermal equation (6) and radiative transfer equation (9b). Referring to the stratified medium as in Figure 1b, the upwelling TB has been computed for an observation angle θ0 of zero degrees (θ0 = θ = 0), considering a maximum depth zn equal to 5 m and discrete step Δz equal to 0.01 m. Not all of the considered frequencies may be of interest for a deep radiometric sounding, but an exhaustive preliminary analysis requires an investigation of a range of plausible frequencies to select the best ones as a compromise between scientific and technological requirements.
5.1. Subsurface Radiometric Sounding
 As a first investigation of the capability of the proposed radiometer to sound the Moon soil, Figure 4 shows the time series of TB simulated for different scenarios and frequencies.
 The scenarios considered are basically of three types: (1) a macrolayer depth dRe m of regolith (indicated as dRe/0/0); (2) two subsequent macrolayers with depths of dRe and dRo m composed of regolith and rock (dRe/0/dRo), respectively; and (3) three subsequent macrolayer with depths dRe, dIc, and dRo m composed of regolith, ice, and rock (dRe/dIc/dRo), respectively. In Figure 4, dRe and dRo are varied from 1 to 1.4 m and 3.6 to 4.7 m, respectively, in steps of 0.1 m and dIc has been held constant at 0.2 m. All cases derived by varying the depths dRe, dIc, dRo are compared with the reference scenario, represented by the case 5/0/0 and shown with a dark gray color in Figure 4, so that ΔTB may be deduced therefrom. From Figure 4 it is evident that as the frequency increases from 1 to 24 GHz the difference in TB of the subsurface scenarios from the reference case are less and less evident (i.e., dark gray curves tend to be closer and closer to the other ones) and that these variations cannot be successfully detected anymore, for example, from a Moon-orbiting spacecraft equipped with a microwave radiometer. At lower frequencies, such as 1–3 GHz, there is a clear separation among the three proposed types of Moon scenarios. For example, in case of the scenario 1/0/4, there is a difference of about 4 K with respect to the case 5/0/0, as opposed to the case of scenario 0.1/0.2/4.7 where the difference is 7 K and increases to 8 K for the case of scenario 0.3/0.2/4.5. As a last consideration in Figure 4, we can observe that, at higher frequencies, TB tends to approach the surface physical temperature (shown in Figure 3 (left)) since at higher frequencies the main contribution to TB comes from thin layers just beneath the lunar surface (compare Figure 3 (left) to Figure 4 (bottom right)). However, due to the large variation of the thermal conductivity in the first 2 cm of moon regolith, at 24 GHz the excursion of TB is less than that of T at the surface.
 As a further investigation of the properties of simulated TBs, the dependence on the polarization as a function of observation angle θ0 is shown in terms of surface emissivity, in Figure 5, where the case of regolith and rock (1/0/4) has been taken as reference. It should be recalled that apparent surface emissivity is defined as the ratio between TB and T at the surface [Al Jassar et al., 1995]. In Figure 5, in order to consider time information, the temporal average (solid lines) of surface emissivity is shown, together with its maximum and minimum (see dotted lines) over the entire synodic period. It is straightforward to conclude, from Figure 5, that small differences exist between vertical and horizontal polarizations and that, if the observation angles are limited to less than or equal to 20 deg for all frequencies, the variation of emissivity from the value calculated at θ0 = 0 is negligible. Different Moon scenarios from that considered in Figure 5 and labeled as (1/0/4), have been tested but they have not led to different conclusions.
5.2. Sensitivity to Subsurface Inhomogeneity
 The last point to discuss is the quantitative evaluation of the sensitivity of a microwave radiometer to distinguish among the variations which can occur at few meters beneath the lunar surface. To this aim, in Figure 6 we have considered the difference of brightness temperature, emerging from the Moon 13 terrestrial days after lunar noon, between the type of scenario 5/0/0 and the other two types of scenarios dRe/0/dRo (Figure 6, left) and dRe/dic/dRo (Figure 6, right) as a function of dRe which varies from 0.01 m to about 5 m (see nested plots in Figure 6).
 At 24 GHz no stratigraphy change is sensed for depths larger than about 0.15 m as opposed to the trend at 1 GHz where at depths of about 5.0 m, variations due to regolith-rock and regolith-ice, cause TB variations of about 0.5 K. This, in principle, means that a radiometer with a sensitivity of about 0.5 K is able to detect those discontinuities at the just aforementioned depths and shallower. Of course, a reduction of the radiometer sensitivity results in a decrease in the sounding capabilities of deeper layers of the Moon. The use of higher frequencies reduces the sounding capabilities as well and in the case of 3 and 12 GHz, with sensitivity of about 0.5 K, the penetration depths are 2 and 0.2 m for both of the considered Moon scenarios.
 Finally, it should be mentioned that the results presented in this paper are in contrast with those of Sun and Dai , where at 3 GHz a penetration of 30 m with a radiometric sensitivity of 0.5 K is stated. It seems that there is a factor of about 15 in penetration depth between their findings and our results summarized by Figure 6 (left). The difference might be explained by the different subsurface thermal and dielectric models, which strongly influence the brightness temperature response, or by an alternative way to define the penetration depth. Indeed, recently, the lunar mission Change'e, described by Sun and Dai , has been successfully launched on 24 October 2007 and the remote sensing community is looking forward to receiving the first data from a microwave radiometer observing the Moon's surface. In addition, it should be noted that, in order to sound the subsurface temperature, a large penetration depth may be not required since the temperature itself tends to be time independent below about 50 cm. On the other hand, morphological and dielectric sounding of the lunar subsurface would be more effective if deeper penetration would be achieved.
 Even though Figure 6 clearly shows the capability of MiWaRS to detect discontinuities down to a few meters below the lunar surface, it does not give information about the ice discrimination ability. To this aim we have verified that, when the ice slab is positioned within 0.2 m depth with thickness larger than 0.2 m and TBs at 1 and 3 GHz are used for inversion purpose, ice detection may be possible. When the frequency channel at 1 GHz is not considered, the presence of rock tends to completely mask the ice contribution in terms of TB. Due to the tradeoff between the system complexity increase when including a 1 GHz channel and the unlikely scenario of ice within the near-surface layers, we can conclude that ice detection from MiWaRS in its current configuration may be questionable.
 A numerical investigation of the capability of a multifrequency microwave radiometer to acquire thermal and dielectric information as a function of depth beneath the lunar surface has been presented. The lunar regolith has been subjected to frequent impacts by meteoroids of variable size during its long geologic history and its thickness may vary from several meters to tens of meters in a way probably correlated with the lunar surface age. The motivation for studying the microwave emission from the Moon is the possibility to include a multifrequency microwave radiometer as a possible payload of the foreseen ESMO mission which is one of the small satellite missions promoted by ESA.
 A detailed investigation of the properties of Moon materials, in terms of thermal and dielectric properties, has been carried out by inspecting past work either concerning experiments made on lunar samples brought back to Earth after the Apollo missions in the 1960s and 1970s or performed through radio observations from the Earth. The information so collected has been used as input to an incoherent stratified EM model coupled with a thermal model. Within the proposed coupled forward model, vertical depth inhomogeneity has been thoroughly characterized, but layer interface roughness and subsurface fragment volume scattering have been neglected mainly due to the lack of input measured data. Numerical simulations have been performed using this emission model, yielding results of brightness temperatures emitted by the Moon at 1, 3, 12 and 24 GHz. Results have provided evidence for the capability, supposing a radiometric sensitivity of 0.5 K for the radiometer instrument, to obtain information from subsuperficial Moon layers at depths up to 5 m (at 1 GHz). Discontinuities, due to the presence of regolith-rock or regolith-ice steps lying in the aforementioned range of depths, may also be detected under the assumptions of the proposed EM thermal model. Ice discrimination by MiWaRS is limited to scenarios where ice is present up to 20 cm beneath the lunar surface and with a thickness larger than 20 cm. The channel frequency of 1 GHz has to be used for ice detection, posing some constraints on MiWaRS system in terms of weight and size. Concerning the MiWaRS operational frequencies, the final choice is driven by a tradeoff between scientific and technological requirements. Frequencies below 3 GHz seem to provide the best performance in terms of penetration depth, but in this case it would be difficult to accomplish the payload size requirements. On the other hand, at higher frequencies the penetration depth is reduced but with the advantages of a small instrument.
 The EM thermal forward model, developed in this study, is an essential step to design a multifrequency radiometric retrieval algorithm to infer the lunar surface temperature estimation, the discrimination of the subsurface material type and the estimation of the near-surface regolith thickness. On the basis of the obtained results, the probability of detecting the presence of discontinuities beneath the lunar crust should be quite high, whereas the near-surface regolith thickness estimation may depend on the actual radiometer instrumental error. The Moon's near-surface temperature from microwave passive measurements may be accomplished with an uncertainty of less than a few kelvins, dependent on the choice of radiometric frequencies to be used; this microwave product may provide useful complementary information to thermal infrared radiometric data whose temperature estimation is confined within the first few millimeters of lunar regolith.
 Spaceborne radiometry of the Moon's subsurface is an appealing goal and the previous numerical analysis has shown some potential for lunar stratigraphy characterization. The coupled forward model should be further refined in order to include both interface roughness and volume scattering effects. However, at present this numerical study confirms the capability of the MiWaRS payload aboard of ESMO satellite to provide global maps of the temperature of the Moon up to a few meters beneath its surface and to sound its possible anomalies. Indeed, these results should be evaluated within a more general analysis of MiWaRS system constraints such as: ground spatial resolution, transmission data rate and radiometric resolution. Spatial resolution is constrained by the antenna beam width and satellite altitude: in case of a beam width of 15° and an altitude of 200 km, linear dimensions of the radiometer footprint are about 30 km. For example, the Chang'e satellite radiometer exhibits ground resolutions between 35 and 50 km, depending on the frequency channels. These spatial resolutions may be larger than typical lunar surface inhomogeneities, having effects on measured TBs due to the antenna spatial filtering. Data rates are usually determined by requiring contiguous footprints which, in the case of no scanning, are aligned along track. Considering a flight velocity of u = 1.58 × 103 m s−1 at 200 km in circular orbit, an integration period of less than 22.15 s has to be used. In order to represent the TB range values (of the order of 300 K at most), a quantization of 10 bits is needed if a radiometric sensitivity of 0.5 K is assumed. This means that the minimum data rate is equal to about 0.5 bit/s per channel, leading to constraints on the design of the onboard storage unit and Earth-satellite link capacity. Note that antenna deconvolution techniques may be applied as well in order to increase the spatial resolution but this results in a reduction of the along-track sampling period. In this case acquisitions at higher rate would be required.
Appendix A:: Heat Flow Model
 The one-dimensional heat flow, given in equation (6), can be rewritten as:
where ΦT(z,t) is the thermal flux equal to the term −kT(∂T/∂z) in equation (6). For the stratified medium geometry of Figure 1, the boundary conditions at z = 0 and z = −∞ to be imposed to equation (A1) are related to the downward heat flux due to the Sun (ΦTsun) and to the sky (ΦTsky), and to upward heat flux due to the lunar soil (ΦTmoon):
In equation (A3)S0 = σ(Tsun)4 is the solar constant with σ the Stephan-Boltzmann parameter, A is the Moon's surface albedo, Tsun the Sun temperature, Ω = 2π/Δt is the synodic pulsation with Δt its period (equal to 29.53 terrestrial days), Tsky is the sky temperature. For the solution of equation (6), due to the absence of the atmosphere on the Moon, the term ΦTsky has been neglected since it does not appreciably influence the thermal solution. If the endothermal source from the Moon interior is also taken into account, an additional flux, ΦTendo, should be considered.
 In order to show the analogy of equation (A1) with the transmission line differential equation in terms of line voltage V and current I, we can Fourier transform equations (A1) and (A2) into the temporal frequency domain and define the following quantities:
where indicates the Fourier transform. By applying the transform (A4) to (A1) and after simple algebraic manipulation, we obtain:
where we defined Z as per-unit-length series impedance and Y as per-unit-length shunt admittance. From (A5) we recognize that V and I satisfies the classical line transmission differential equation where Z is purely resistive and Y is purely capacitive.
Equation (A1) can be then solved following the method described by Di Carlofelice and Tognolatti , based on transmission line formalism. This method is able to include the endothermal source as well. It takes into account nonlinear phenomena by using the Harmonic Balance numerical technique [Rizzoli et al., 1988]. Within this approach, the heat fluxes ΦTsun and ΦTsky are equivalent to alternating current (AC) and direct current (DC) ideal current sources, respectively, at the line input, whereas ΦTendo is a DC current generator, at the line termination. Note that the surface Moon upwelling heat flux ΦTmoon(0,t) can be modeled as a shunt nonlinear resistor whose characteristics is given by I = σV4.
 We gratefully acknowledge Roger Walker, project manager for educational satellite projects at ESA, and his staff for the opportunity given to us to design a microwave radiometer for exploration of the Moon. In addition, we thank all of the students of the University of L'Aquila and the Sapienza University of Rome who contributed to the progress of the present work through their theses.