A major finding in exploration of Mars is the indication that a large quantity of water has been present on Mars. To discover the distribution of this quantity of water, the Mars Express spacecraft is carrying a spaceborne radar sounder, called Mars Advance Radar for Subsurface and Ionospheric Sounding (MARSIS), in order to map Mars subsurface dielectric characteristics. The returned radar echoes from Mars will contain both surface and subsurface reflection components, but the part of the signal we are interested in is the reflected signal from subsurface layers. To retrieve these weak deep echoes from the radar signal, a signal processing algorithm needs to be developed. In this paper, we present a computationally efficient radar signal simulation based on the use of the Facet Method as a surface modeling scheme. This simulator will be used to validate the MARSIS ground processing software and to support the interpretation of MARSIS data. The first step in the simulation algorithm definition is to model the Mars surface. Many surface modeling methods have been developed and can be found in the literature, but considering the fact that Mars surface is very smooth, the modeling algorithm we have developed uses the Facet Method. In this paper, we show that the Facet Method is an efficient scheme for modeling a relatively smooth surface, such as the Mars' surface. The instrument simulation we define makes strict reference to the MARSIS radar parameters; however, it may be used to model any radar sounder.
 The science instruments on MGS and recently Odyssey spacecrafts, have found indications that a significantly large quantity of water was present on the Mars surface. The question that we face today is what quantity of water, if any, may be found in the Mars subsurface, as liquid water or ice, and, if the answer to this question is positive, to know what volume of water is present there, and where it is located.
 To answer some of these questions, in 2003, the ESA's Mars Express spacecraft will carry, for the first time, a spaceborne radar sounder instrument called Mars Advanced Radar for Subsurface and Ionospheric Sounding (MARSIS), in order to map Mars ionosphere and subsurface dielectric characteristics. Specifically, one of MARSIS' primary objectives is to map the permafrost and try to determine its thickness (up to 4000 m deep), and to search for reservoirs of liquid water or ice [Picardi et al., 2000].
 MARSIS operates by sending a long chirp (30 to 250 μs) toward the surface of Mars and then recording the returned echo. The returned echo from Mars will contain both surface and subsurface reflection components. Ideally, the surface component of the signal is only present in the early portion of the returned echo. However, depending on the characteristics of the surface, the returned echoes from the surface may last for a long time. Since the surface returns are less attenuated in comparison to the subsurface returns, they could mask the subsurface returns. This is a major problem since returned echoes from the subsurface layers are the part of the signal that we are primarily interested in. Therefore a signal processing algorithm needs to be defined to filter these surface returns and to retrieve the weak deep echoes from the signal.
 Thanks to NASA's Mars Global Surveyor mission, and more specifically, to the MOLA instrument, we now have a 500-meter resolution elevation map of nearly the entire Mars surface with an accuracy that is much better than one-tenth of the shortest wavelength of the MARSIS radar [Smith et al., 1999]. This topography map can be used to simulate MARSIS radar echo from the surface and compare with the actual MARSIS echo during the interpretation. The first step in developing such a simulator is to calculate the echoes from the surface, and to understand Mars' surface response to the MARSIS radar wave.
 The first step in simulating the radar signal is to model the Mars surface. Many surface modeling methods have been developed and can be found in the literature, but considering the fact that Mars' surface is very smooth [Aharonson et al., 2001], the modeling algorithm we used is the Facet Method. To have a complete MARSIS signal simulation, the radar wave propagation through the Mars ionosphere can also be modeled. The effect of Mars' ionosphere on the radar signal, has been studied in previous works [Nouvel et al., 2001; Safaeinili et al., 2003].
 In the following, we will define assumption and parameters used in our simulation. Then we will discuss the surface modeling algorithm and the results. Finally, we will evaluate the validity of the method presented as it applies to modeling the Mars surface.
2. A Radar Signal Simulation
2.1. Echoes Components
 The radar echo has both subsurface and surface components. The surface component of the echo itself is made of two types of signals: 1) the return from the nadir and 2) returns from off-nadir directions. The first type or the nadir echo is the coherent specular reflection of a wave at each interface and is reflected mainly on the first Fresnel zone when each interface is perpendicular to the propagation path. The second type of echo is when the signal comes from off-nadir (lateral) directions. It is composed of both coherent signals and incoherent scattering coming from the off-nadir illuminated surfaces. This signal is also known as clutter.
 In general, the nadir echo arrives first and it can be distinguished from the subsurface returns. However, for a nonspecular surface, there will be returns from off-nadir arriving at longer delay times than the specular nadir component and hence become mixed with subsurface returns. In analyzing the MARSIS signal for subsurface investigations, one is interested in the coherent part of the subsurface echo that normally (except for sloping surfaces) comes from the nadir direction and can be masked by off-nadir surface echoes.
 The clutter can be suppressed by forming a longer aperture in the along-track direction (i.e., Doppler filtering). However, this will not remove the cross-track clutter since no aperture synthesis in the cross-track direction is performed. If one wants to maximize the signal to noise ratio (SNR) in order to detect the weaker coherent signal, one needs to subtract the entire surface echo. This can only be done by defining a signal processing algorithm with the aim of filtering the entire surface response from the radar signal. Our main objective is to model the Mars surface response so it can be used to isolate the subsurface echo, and can be used for observation planning purposes.
2.2. MARSIS Instrument and Model Parameters
 In order to configure our model, we need to define a number of parameters. The specific values used here for center frequency, pulse bandwidth, burst and single pulse duration, and the pulse repetition frequency (PRF) are those from MARSIS, however the simulator described in this paper may model any radar sounder.
 MARSIS is a nadir-looking radar with the purpose of investigating Mars' subsurface to a maximum depth of 5 km. Thus, the radar frequency is selected in the range of 1.3 MHz to 5.5 MHz. MARSIS has four separate one-MHz bands that are centered at 1.8 MHz (wavelength of 166 m), 3 MHz (100 m), 4 MHz (75 m) or 5 MHz (60 m) (Table 1).
 This section focuses on the modeling of the surface echoes and the definition of the Facet Method. Its application is described in sections 4 and 5.
3.1. Surface Radar Response
 The radar signal, reflected by a reflector, can be written in the frequency domain as:
where f is the frequency of the radar, A is the modulus of S and φ is the phase of S. The phase of the radar signal is
where R is the range, c is the speed of light in the vacuum and n is the refractive index of the medium the radar wave propagates through.
 The amplitude of the reflected radar signal depends on the transmitted power, the Fresnel coefficients at surface interface and the geometric expansion of the radar signal. We assume here this amplitude is constant over the surface, and depends only on the signal frequency.
 So, integrating (1) over the surface, the Huygens principle gives the surface echo:
From each point of the surface, the range is equal to
To calculate the equation (3), we need to discuss the model of the surface.
3.2. Facet Method
 A great body of work exists on surface response calculations due to an incident radar wave [e.g., Kobayashi et al., 2001]. Here we chose to use the Facet Method because it is most relevant to our work. This is due to the fact that most of Mars appears smooth relative to the radar wavelength which is longer than 50 meters.
 The foundation of this method is an approximation of a rough surface through a series of small planar “facets”, each tangential to the actual surface (Figure 1). The Facet Method models the scattering from the combination of such facets by taking into account both their radiation patterns (modulus and phase) and the distribution of their slopes.
 Thus, each facet behaves like a reflector antenna illuminated by the incoming radar wave. From the knowledge of the facet orientation and reflectivity, related to the dielectric nature of the Mars surface, the reradiated radar signal can be computed.
 So each facet can be seen as an antenna and its radiation pattern depends on its dimensions. According to the Facet Method theory [Rees, 1990; Ulaby et al., 1982], the two criteria which must be satisfied in choosing these facets are 1) they should be much larger than one wavelength across (so that the diffraction effects are minimized) and 2) the actual and the modeled surfaces do not differ in height by more than about half a wavelength (to decrease the phase error). Provided that these two conditions are met, each facet may be considered as a quasi-specular reflector, and its contribution to the overall surface scattering can be calculated as a function of its size and orientation.
3.3. Surface Modeling
 Mars surface is approximated by a series of small planar facets. Our objective is to “discretize” the surface integral (3) in a sum of surface integrals calculated over each facet as shown below.
From the integrals' theorems, this equivalence is true, assuming integrals over each elementary surface Si converge and Si: The sum of all elementary surfaces has to cover the surface.
 We use square facets as elementary surface units. This geometry allows us to separate the variables in each surface integral and so, to obtain an analytical expression for each of them (with far field approximation, see section 4.2 for details).
 A primary objective of this work is to develop a computationally efficient modeling scheme in order to allow the calculation of modeled echoes for the entire MARSIS data set. Mars surface is curved but it can be covered by square facets if we use facets that are sufficiently small. One has to keep in mind the scale length we use is the radar signal wavelength. So to cover the surface with facets, discontinuities between two facets have to be much smaller than the wavelength.
 Considering Mars' radius (3398 km) and the radar wavelength (about 100 m), the above criterion is reached with facets of about 1 km long (case of a spherical surface). In our work, we plan to use 500 m facets since Mars digital topography maps are readily available at this scale.
 Finally, to model the surface, we define a grid of points on the Mars surface, with a resolution equal to the facet's length. Then, to define each facet, we consider five points over the grid, following the configuration introduced in Figure 2 (points number one to five). The central point (point number one) is the facet's center and the four other points give the facet's orientation.
 One can note that these four other points are themselves the center of four neighboring facets. The facet is tangent to the actual surface at its center point (point number one in Figure 2). The north/south slope is calculated from the altitudes of the two points located north and south of the central point (points number two and three in Figure 2) and the east/west slope is defined the same way, from the altitudes of the two points located east and west of the central point (points number four and five in Figure 2).
3.4. Facets Radiation Pattern
 Once we define the set of facets used to model the surface, the overall surface response is the sum of all the individual facet's responses. Considering an extreme case, the radiation pattern from an infinite plane is a delta function. Such a facet is a perfect specular reflector, so the only way for the point source to receive radiation back is to radiate rays striking the facet at a normal incidence.
 However, for a wide but finite facet, the radiation pattern is narrow and sidelobes appear even if their amplitude is low compared with the main lobe. Finally, for a narrower facet, the radiation pattern is wider and sidelobes are still present [Ulaby et al., 1982]. The qualitative meaning for the terms “narrow” or “wide” used to describe the facet size depends on the wavelength. Thus a “wide” facet must be many wavelengths across.
 The main condition for and also the main advantage of applying the Facet Method is using facets whose dimensions are much larger than the wavelength. Considering the MARSIS radar wave frequencies, we use facets that are 500 m in length. This length is a compromise that is large enough to reach this criteria (500 m is three times the wavelength at 1.8 MHz), but small enough to limit the range errors (see section 4.3.1).
 Assuming a carrier frequency of 1.8 or 5 MHz, a facet size of 500 m and a radar altitude of 300 km, a cross section of the radiation pattern for each frequency is shown in Figures 3a and 3b. Sidelobes are presents in both figures, but the radiation pattern in the case of f = 5 MHz is much more directive than the one in the case of f = 1.8 MHz.
 The width of the main lobe peak at −3 dB is three times larger with f = 1.8 MHz than with f = 5 MHz (4 degrees with f = 5 MHz and 12 degrees with f = 1.8 MHz). However, sidelobes have the same amplitude in both cases (lobes present a cardinal sinus or Sinc function geometry: the first sidelobe amplitude is 14 dB lower than the main lobe one).
4. Surface Echo Calculation
 In this section, we consider a surface without any roughness. Our objective is to test the Facet Method over this theoretical surface. The surface's dimension can be calculated as a function of the receive window size of MARSIS (130 μs, see Table 1) and the radar altitude (value set to 300 km in this test). In the following, the surface's size is fixed to 200 square-km.
 Let us use some developments of the range expression (4), from the simplest to the most complex form. The double integration shown in (3) can be done over a flat surface. That is the simplest case and it will be called the “reference” case, without any use of a facet approximation. Results from this reference case will be used as a comparison to test the modeled received echo and the Facet Method. We will call this case “Global surface summation”, see section 4.1.
 Then, we can apply the Facet Method over a spherical surface, without roughness, and integrate (3) over each facet. The surface reflection echo will be calculated by summing up individual facet contributions. This case is called “Facet elementary surface summation”, see 4.2. We will discuss results from the two cases in 4.3.
4.1. Global Surface Summation
 The surface we consider is a flat and square surface with a size equal to L. A surface point is defined by its location (x, y), each coordinate varies from −L/2 to L/2. The altitude's origin is on the surface. The spacecraft position is fixed to (xs, ys, h) and we assume the propagation support is the vacuum space.
 From this surface, the received electrical field is, from (3):
The assumption, considering the wave amplitude as constant over the surface is rough at this time, due mostly to the ignorance of the 1/R2 one-way spreading factor variation. However, due to the flat surface we consider, we expect to receive a specular reflection. This kind of return is mainly coming from the Fresnel zone, which is limited in extent (about 8 km in our case).
 We assume we can neglect the spreading factor variation on this zone, and notice the sidelobes level will be higher than the one obtained using an exact solution. From (4), the range is
Assuming h ≫ ∣x − xs∣ and h ≫ ∣y − ys∣ (i.e., a far field geometry and the Fresnel approximation), one can develop (6) following:
The far-field approximation is a coarse approximation here. It is used to compare this case with one of the facet responses summation (case developed in 4.2) where the far field approximation is used also. Validity of this approximation will be discussed in chapter 4.3.
To calculate these integrals, we use the well known Fresnel Integral:
and finally obtain:
For a 200 × 200 square-km surface and a one megahertz bandwidth signal centered at f = 5 MHz, we obtain, by FFT, the signal shown in Figure 4 (the echo presents a 20 μs offset for a better display). A Hamming window is applied before the FFT operation.
 This signal shows the received echo from a flat square surface. Since the above calculation does not use the facet approximation, it will be considered as a theoretical reference in order to assess the performance of the model describe now.
4.2. Facet Elementary Surface Summation
 As defined in section 3.3, we will discretize the surface integral (5) considering the equivalence:
Let us consider a facet's surface. The facet's size is l by l (square facet), from −l/2 to l/2 on each direction. As a parameter, we will use the look-angle α, and the geometrical equivalent cases defined in Figure 5.
 Using this geometry, the spacecraft coordinates for the kth facet are
where a = tan(αx), b = tan(αy), x varies from −l/2*cos(αx) to l/2*cos(αx) and y varies from −l/2*cos(αy) to l/2*cos(αy). So (4) becomes:
4.2.1. First-Order Development
 With and assuming ∣x∣ ≪ ∣Rk∣, ∣y∣ ≪ ∣Rk∣, a first order development of (12) gives simply: R = Rk.(1 − a.x′ − b.y′). Using this expression of R, (5) becomes:
The assumption that the wave amplitude Ak(f) is constant over each facet's surface is justified considering that the facets' size is 500 square meters. This term includes the transmitted power, the Fresnel coefficient, and the geometric expansion (proportional to 1/Rk2). Finally, to express the total received signal from all over the surface, we sum contributions of each facet following:
For a total square surface size of 200 × 200 km, a facet size of l = 500 m and a center frequency f = 5 MHz, the resulting signal is plotted in Figure 6.
4.2.2. Second-Order Development
 This signal has been obtained with a first order development of (12). With a second order development, we have
where x′ and y′ are x and y normalized to Rk, and
Here also, one can have the echo with the summation E(f) = Ek(f). For a 200 × 200 km surface, a facet size of l = 500 m and a center frequency f = 5 MHz, one obtains Figure 7.
4.3.1. Far Field Approximation
 Let us consider the error in the range calculation due to the far field approximation. In the case developed in 4.1, this error is the difference between formula (6) and (7):
while in 4.2 it is the difference between formula (12) and (14):
Figure 8 is a cross section plotting of these range errors, following x or y-axis. Equation (16) is plotted in Figure 8a. h = 300 km and x or y varies from −100 km to 100 km, nadir is fixed at the surface center. Equation (17) is the range error in the case of a second order development (12) (“Fresnel Approximation”) and it is plotted in Figure 8c, while in Figure 8b the range error corresponding to a first order development (“Fraunhofer Approximation”) is plotted. In Figures 8b and 8c, x or y varies from −250 m to 250 m on each facet, and nadir is fixed at the surface center.
 In 4.1, the range error (16) has to be calculated for x varying from −L/2 to L/2, where L is 200 km, while in 4.2, (17) is computed between −l/2 and l/2 and l is 500 m. So surfaces over which the error occurs are drastically different. Over the entire surface, the maximum range error value is 450 m (Figure 8a), while over the facet's surfaces, it is less than 11 cm for a first order approximation (Figure 8b) and less than 25 μm for a second order approximation (Figures 8c). So the far field approximation appears as a coarse approximation in the case developed in 4.1. However, we used it there also in order to apply the same approximations in 4.1 and 4.2 and then, to be able to compare results from these two cases. The main consequence of this important range error is the amplitude of the echo from the surface's edge.
4.3.2. Edge of the Surface Model
 One can observe that the main difference between the simplest case (Figure 4) and the calculation using the Facet Method (Figures 6 and 7) is the presence of a delayed peak in the echo signal. The observed delay for this peak is about 109 μs (offset of 20 μs). This delay corresponds to the delay of the signal arriving from the edge of the surface.
 The modeled surface is finite in extent. The edge of the simulation domain behaves like a reflector and causes this echo. Please note that we choose arbitrarily to end our summation at the first surface edge echo, so there are no more delayed echoes (particularly, echoes expected from the corners of the square modeled surface which are not seen).
 The remaining question to be addressed is why the surface edge echo is not observed in Figure 4. Due to the range error, the “apparent” surface in the case 4.1 is not actually flat but looks like the parabola figures in Figure 8a: From each point over the surface, the range to the spacecraft is not R, but R + ΔR. So the “apparent” surface point, “seen” by the radar, is not on the surface, but further away (i.e., subsurface).
 At the surface edge, the range error is maximum and the incident wave reaches the ground tangentially to the parabola edge. As a result, there is no returned signal from these areas (the reradiated signal is null there) and the edge of the surface is not observed in Figure 4.
 In the opposite case developed in 4.2, the range error (17) is still present but at a smaller scale. The range error is quite small over each facet surface. Consequently, the “apparent” surface seen by the radar is still flat facets and the edge of the surface is seen in Figures 6 and 7. In addition, the small range error at the edge of each facet causes the noise that one can observe in Figures 6 and 7. Figure 7 corresponds to a second order approximation, and consequently, this noise is lower than the one for a first order approximation, as shown in Figure 6.
4.3.3. Bragg Resonance Phenomenon
 In Figure 6, one can see that this noise follows a Sinc function profile. This is due to the Bragg resonance phenomenon on the facet's reflected signals [Ulaby et al., 1982]. If the range difference from the radar to two facets is a multiple of λ/2, the round-trip phase difference between the two echoes is a multiple of 2π. So both signals and range errors from these two facets add in phase (coherently, see 4.3.1). An example for this case is seen in Figure 6. With any other difference in the range, the signals add out -of -phase (incoherently) and there is no echo created.
 Now, if we calculate the echoes created by Bragg resonance, we obtain the delays 23.6 μs; 34.7 μs; 53.3 μs; 80 μs and finally 116.9 μs. These delays plotted in Figure 6 (vertical lines), obtain a significant correspondence. So the geometry observed for the noise is clearly due to Bragg resonance. This phenomenon can be seen as a grating angle effect. Indeed, we saw in section 3.2 that each facet behaves like an antenna. The resonance that occurs here is due to antenna array geometry.
5. Mars Surface Simulation
5.1. Validity of the Facet Method Applied to Mars
 We noted the criteria for using the Facet Method are that facets must be many wavelengths across and not differ in height by more than half a wavelength compared to the actual surface. Since we use a fixed facet size, this is in fact a restriction on the curvature of the actual modeled surface. Let us assume that the surface has a radius of curvature R, the facet size is 2ρ, and the angle subtended by the facet at the center of curvature is 2ϕ. ϕ is approximately equal to ρ/R, assuming ϕ is small. The deviation between the edge of the facet and the actual surface is
Once again, assuming ϕ is small, one obtains from (18):
Applying the conditions to use the Facet Method that are ρ > λ and x < λ/2, the criterion on R is simply R > λ. The radius of the curvature of the actual surface must be at least a few wavelengths in size if one wants to use the Facet Method as a modeling scheme [Rees, 1990]. This is not a very restrictive condition, but the main point is that the flatter the surface the better the facet model's agreement with the actual surface.
 Thanks to the MOLA experiment on the Mars Global Surveyor mission, we have adequate information about Mars' topography at scales that are relevant to this work. Topography generated by MOLA indicates that Mars' surface is smooth in the scales of interest to MARSIS (greater than 1 km). Even at a 300 m length scale, 93% of Mars' surface has a RMS slope lower than five degrees (Figure 9). The smoothness of Mars' surface strongly justifies the use of the Facet Method for modeling the Mars surface.
5.2. Surface Slopes Introduction
 In order to model Mars surfaces, we have to add a more realistic surface definition to the present simulation. So far our model has considered a spherical surface. Let us introduce a distribution of slopes among facets modeling the surface.
5.2.1. Slopes Definition
 In 4.2, we defined the look-angle α. It is the main parameter in the definition of the reradiated field from each facet (see Equation (15)). To introduce a slope distribution among the facets, let us add a small perturbation Δα to this parameter α. Δα is defined due to a variation of the points' elevation over the surface. These elevations are seen as a random variable that fits a Gaussian distribution with a r.m.s. value σh and a null average value. This r.m.s. value is linked to the r.m.s. slope σ and the correlation length l through the relation [Nouvel, 2003]:
where σ is expressed in radians, while σh and l are in meters. Aharonson et al.  provides some typical values for the above parameters for Mars. Figure 13 presents a few examples of surface echoes corresponding to these values of r.m.s. slope σ.
 Areas with large r.m.s. slope will have large slope average values. As seen in Figure 7, in the case of a spherical surface, there is no received echo from the surface, except in the nadir direction. As shown in Figures 3a and 3b, the reradiated signal is maximum when the radar wave impinges on the facet at normal incidence and facet radiation pattern is very directive with l = 500 m and f = 5 MHz. Consequently, when the surface is spherical, there is no signal propagating back to the radar (except in the nadir zone), while in the case of a rough surface, some facets are well oriented and reradiate radar waves towards the radar receiver. So, the rougher the surface the stronger the off-nadir radar echoes will be.
Figure 10 presents the power variation of the reflected radar signals for different surface roughness. In this figure, the clutter signal from the surface where slope r.m.s. is 5 degrees is stronger than the one from the surface where slope r.m.s. is 0.2 degrees.
5.3. Radar Signal Simulation Applied to Mars
 After the introduction of the radar signal simulation, and an application of this simulation to a random surface, we can apply this radar signal simulation to the actual Mars surface. Surface topography maps generated by MOLA provide a 1/128-degree resolution that is equivalent to 463 m resolution over the surface.
 From the MOLA digital elevation maps, we extracted a few Mars surface areas with particular geological features such as flow, crater, or rift. Figure 11 presents one of these selected areas. Altitudes vary from −900 m (crater number one) to 1900 m (light area, point number five).
 Then we used the MARSIS signal simulation technique introduced in this paper, and computed the radar echo received from this surface. We computed 100 pulses corresponding to the compressed MARSIS signal (with the duration of about 100 μs) for a spacecraft altitude of 300 km, with a 220 m ground separation and did not apply any SAR processing (Doppler Filtering). The PRF is reduced to 20 Hz in order to reduce the computation time.
 The ground track length is calculated to be approximately 22 km (shown as the horizontal white line in the center of Figure 11). Finally, the starting position is at the surface center vertical and the spacecraft displacement is from West to East.
 Echo index 1 is the first one to be received due to a surface feature close to the ground track first position. In Figure 12b, considering the delay between the first received signal and the echo index 1 (about 10 μs), one can look for a corresponding feature in Figure 11. Two features over the surrounding surface correspond to this time delay. They are two crater edges indexed number 1 (Figure 11).
 The echo index 2 is due to the small crater north of the nadir zone (Figure 11). The echo due to this surface feature has the same arrival time over the one hundred pulses calculated (Figure 12a). This is consistent with the location of this feature that normally is located in the spacecraft's movement direction.
 The echo index 3 is due to the edge of the crater located west of the nadir zone. Since the spacecraft moves from West to East, this echo's arrival time increases from one pulse to another, as seen in Figure 12a.
 Echo index 4 is due to two different surface features (two craters edge, see Figure 11). One of these craters is located west of the nadir zone, while the other one is located east of the nadir zone. So echo index 4 should be divided into two components, the one with decreasing arrival time and the other with increasing arrival time, as seen in Figure 12a.
 The agreement between expectations and results proves that we are able to recognize some surface feature locations from their corresponding surface radar echoes. At the same time, it means our echo calculation scheme is efficient to model an actual surface response to a radar wave.
 Finally, Figure 12a shows that surface clutter echoes can mask a subsurface layer. The modeling scheme presented in this paper may also be used to select Mars surface areas with less clutter signals, unlike the ones plotted in Figure 12a (Northern plains areas for example). Consequently, subsurface layers may be detected.
 In this paper, we presented a radar echo modeling scheme using the Facet Method. One of the main advantages of our method is the short computation time through using large facets. For example, it takes less than 2 minutes to compute a signal from the one plotted in Figure 12b (Intel Pentium 4, 2.2 GHz processor) in comparison to about 40 minutes when using a generic method [Kobayashi et al., 2001]. The efficiency of our algorithm will make it possible to calculate simulated radar echoes for every corresponding MARSIS observation.
 When considering the low frequencies used by the MARSIS radar sounder, the achieved computational efficiency is due to the fact that the Facet Method is applicable here. In this way, we saw in section 5.1 that the Mars surface is smooth at scales much larger than a wavelength, as shown by MOLA generated topography.
 The objective of this paper is to show the value of such a modeling method, in order to simulate the MARSIS radar sounding process. The MARSIS signal modeling presented here is currently still under development.
 Part of the research described in this paper was carried out by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautical and Space Administration. The first author would like to thank the Jet Propulsion Laboratory where part of this work was carried out during his three-month visit. Authors also would like to thank the Centre National d'Etude Spatiale (CNES) for support of the MARSIS program in the Laboratoire de Planétologie de Grenoble.