This article was corrected on 9 FEB 2015. See the end of the full text for details.
A two-layer model of Mars' regolith/bedrock media with a cratered rough surface/subsurface is presented for radar echo simulation of planetary exploration research. The numerical approach of geometric ray tracing for the scattering of rough surfaces, which is digitized by the triangulated network, is applied to the calculation of the scattering and imaging simulation of radar range echoes. Numerical simulations of a cratered rough surface generated by the Monte Carlo method are used to analyze the functional dependence of radar range echoes at 1–50 MHz center frequencies upon the surface/subsurface feature and the parameters of the layering media, that is, layer depth and dielectric properties. The radar range echoes from two areas of the real Mars surface, which is described by digital elevation model data with a resolution of 1 m × 1 m and a vertical error of less than 1 m, are also simulated and analyzed. Based on these simulations, this study presents a numerical imaging test of radar sounder at center frequencies 1–50 MHz for exploration of different dielectric regolith and bedrock media. The channel 50 MHz with high resolution might be an optimal frequency. Finally, inversion of the dielectric constants of the two-layer media and the regolith layer thickness are developed.
Radar sounder technology has been applied to planetary exploration, such as on the Moon and Mars [Picardi et al., 2005]. The MHz radar wave can penetrate through the regolith media, and radar range echoes demonstrate many media features, including the surface/subsurface layering structure and the dielectric properties of the regolith/bedrock media. In the recent decades, there have been two most important missions of satellite-borne radar sounders for Mars explorations. One is the MARSIS (Mars Advanced Radar for Subsurface and Ionosphere Sounding) on board European Space Agency's Mars Express launched in June 2003. It is a multiband (0.1–5.5 MHz) radar sounder with a 1 MHz bandwidth. Its primary objective is to probe the layering structure of the 5 km depth below the surface [Picardi et al., 2004]. The other is the SHARAD (SHAllow RADar) on board NASA's Mars Reconnaissance Orbiter launched in August 2005. It operates at 20 MHz central frequency with a 10 MHz bandwidth, which makes the penetration depth about 1 km [Seu et al., 2004]. Some parameter retrieval and analysis based on MARSIS and SHARAD measurements has been reported [Mouginot et al., 2010; Nouvel et al., 2004; Zhang et al., 2008; Zhang and Nielsen, 2011].
However, theoretical modeling of extraterrestrial stratified media, echo data/image simulation and validation, geophysical parameter inversion/analysis, and so on remains to be further studied. To explore the regolith thickness and dielectric inhomogeneity of the underlying media, a two-layer model of the regolith/bedrock media was studied for space explorations of the Moon and Mars [Kobayashi et al., 2002; Campbell and Shepard, 2003; Jin et al., 2008].
In this paper, a two-layer model of the cratered rough surface/subsurface is presented to study radar echoes from the Martian regolith/bedrock media. The radar range echo from the layered media with rough interface is numerically simulated based on the calculation of rough surface scattering using ray tracing geometric optics. The cratered surface is first randomly generated using the Monte Carlo (MC) method. Numerical simulation of radar range echoes at 1–50 MHz shows functional dependences of the media parameters, for example, dielectric properties and regolith thickness. As an example, two areas of the Mars digital elevation model (DEM) data are specifically chosen for simulation of the radar range echoes. Based on these simulations, inversions of the Mars surface/subsurface parameters, such as the regolith thickness and dielectric properties of the layering media, are then developed.
2 Radar Echoes From the Layered Structure
As shown in Figure 1, a radar sounder at height H transmits a short pulse with a beam width θ upon the top surface. The regolith and bedrock consist of the inhomogeneous media with rough surface/subsurface. In this two-layer model, the reflected wave from the surface nadir point A returns to the radar antenna with the time delay of 2H/c, where c is the speed of light in a vacuum. A portion of the incident energy is transmitted through the surface layer to reach the subsurface point, B, with some attenuation. The reflected wave from the subsurface point B passes through the surface with another time delay of to reach the radar. Since the surface is generally rough, the radar sounder also receives the off-nadir echoes, such as echoes from point C. The echo from the subsurface point B is usually weak due to attenuation, while the off-nadir echoes may be stronger and mask the echoes from the subsurface.
If the dielectric constant ε1 of layer 1 (regolith) becomes known, the layer thickness d can be inverted from the echo range, that is, via . The echo intensity can also be analyzed for estimating the interface reflectivity and attenuation of the intervening layer.
Using the ray tracing of geometric optics for rough surface scattering, as illustrated in Figure 1, the surface is first numerically divided into discrete meshes. All the diagonal lines of each mesh are connected, and the surface is finally formed using the triangulate network, as shown in Figure 2. The normal vector of each triangle mesh and its area are calculated by the coordinates of the triangle's vertices.
Using the Stratton-Chu integral formula of Huygens' principle [Kong, 2000], the scattering from all meshes can be calculated, and total scattering fields from these meshes, whose ranges fall in the same range bin rn (the nth range bin) can be obtained and summed. Based on the Kirchhoff approximation (KA) [Fa and Jin, 2010], the scattering field from the surface (sur) Esur(rn) and subsurface (sub) Esub(rn) received by the radar sounder can be written as
where denotes the vertical or horizontal polarization (v-pol or h-pol), and are the incident and scattering wave vectors, and are v-pol and h-pol reflection coefficients between the boundary of the 0–1 media, as shown in Figure 1, respectively. The distance rpq (p,q = 0,1,2,3) denotes the ray length linking the points on the media surface (i.e., r10, r12, r23, and r30 as indicated in Figure 1), and is the normal vector of each local mesh. Derivations of equations (1) and (2) can be found in Fa and Jin  and Kong .
Thus, the total field received by the radar sounder is
To make enough range resolution and transmitted energy, the linear frequency modulation (LFM) pulse is usually employed as the transmission signal [Kobayashi et al., 2002; Curlander and McDonough, 1991]. Taking the transmission signal into account, the electric field E(t) received by the radar sounder is synthesized as
where τn = 2rn/c is the time delay, and the transmission signal Tr(t, τn) is expressed as
where W(t), f0, K, and T denote, respectively, the pulse envelope, the initial frequency, the frequency modulation rate, and the pulse length. The bandwidth of the transmission pulse is written as BW = KT.
The mixed signal of the received electric field is written as
where the reference signal Sref(t) is expressed as
where τref and S0 are the starting time and the amplitude of the reference signal, respectively.
Taking a Fourier transform of the mixed signal Emix(t) gives
According to the property of the LFM signal, the relation between the frequency f and range r is written as [Curlander and McDonough, 1991]
3 Simulation of Radar Echoes From the Cratered Rough Surface/Subsurface
To simulate the cratered rough surface of Mars, 500 craters in a region of 25 km × 35 k m were generated using one realization of the MC method (Figure 3). The radii and positions of these craters were randomly produced [Biccari et al., 2001; Baldwin, 1964]. The area outside the craters can be generated as a random rough surface with certain height variance and correlation length. The thickness of the top regolith layer was set at d = 0.2 km, and the bedrock media were, alternatively, assumed to be different media as (7.1 + i 0.0355) (e.g., the basalt) and (3.15 + i 0.01575) (similar to the water ice), and the loss tangents were both assumed to be 0.005 in the numerical examples. It was found that the water ice layer might exist with a thickness of 1.3 km in the polar region [Grima et al., 2009].
To make smaller illumination areas only for numerical simulation, it was supposed that the radar flies along the arrow direction at a speed of 2 km/s at an altitude of 50 km, as shown in Figure 3. This gave the radius of the illuminated area to be about 12.5 km, that is, a beam width of 28° (in the 50 MHz simulation, the radius was changed to 5 km, that is, a beam width of 11.5°, to reduce the mesh number in the calculations). The other parameters chosen were the pulse repetition frequency 20 Hz, the antenna length L = λ/2, the antenna impendence 50 Ω, and the transmitted power of the radar sounder 800 W. Each mesh size was made to be λ/6, that is, 10 m at 5 MHz, 2.5 m at 20 MHz, and 1 m at 50 MHz. The length of the transmission pulse was T = 2.76 × 10− 5 s, and the envelope weighting function was chosen as [Kobayashi et al., 2002]
Numerical simulations of the radar range echoes (radargram) are presented in Figures 4-6, where the vertical ordinate is the echo range minus the radar height (i.e., the average range of the surface is 0 km), and the horizontal ordinate indicates the observation position on the radar ground track with an interval of 100 m. The nadir echoes from the top surface at around the zero range and the echoes from the flat subsurface at around several hundred meters appear as a straight line. As the radar moves forward, the ranges from the radar to each crater change and make the echoes from a single crater appear as an arc curve.
It is noted, for example, that the range 0.533 km indicated by Figure 4a satisfies the regolith thickness km as initially proposed.
Because the dielectric constant of basalt is larger than that of water ice, higher surface reflectivity, and stronger off-nadir echoes are produced in Figure 4a than those in Figure 4b. Meanwhile, larger imaginary parts of the basalt caused larger attenuation in Figure 4a.
It can be seen from the comparisons of Figures 4-6 that the image resolution becomes finer as the higher center frequency and bandwidth are adopted. However, higher frequency caused more attenuation through the media and reduced the subsurface echoes.
4 Simulation of Radar Range Echoes From a Martian Surface
A region of 5 km × 23 km around (26°S, 35°W) of Mars from the elevation data of the High Resolution Imaging Science Experiment database [http://hirise.lpl.arizona.edu/dtm] was chosen. Following the surface elevation data, the surface was numerically divided into right triangular meshes with two right lengths of 1 m, which are less than λ/6 of the radar frequency. Suppose a flat subsurface is at −1850 m indicated in Figure 7. The radar flies from the south to the north at a speed of v = 4 km/s at H = 100 km, as shown in Figure 7.
Suppose that the radar illumination radius is 2.3 km, that is, the beam width is about 5.6°. Other parameters are the same as the example of Figures 4-6. The images of the radar range echoes at f = 5, 20, and 50 MHz are shown in Figures 8-10.
As the radar keeps moving, the nadir echoes directly received from the surface form a range curve following the DEM; meanwhile, the echoes from the subsurface bend to the opposite direction because their effective ranges gradually become larger as the top layer thickens. It can also be seen that the echoes from the subsurface at small depths appear significant due to less attenuation.
Generally, there is a slope of about 2–3° on the averaged undulated surface. Extending the band may cause fluctuation of the echo range. Such a band in Figures 8a, 9a, and 10a seems wider than that in Figures 8b, 9b, and 10b, because of the low wave speed in the high dielectric medium.
As another example, a region of 5 km × 20 km around a deep Zumba Crater (29°S, 133°W) is chosen, as shown in Figure 11. Suppose that a flat subsurface is located at −1700 m indicated in Figure 11, the radar flies from the south to the north at v = 4 km/s at H = 100 km, and the radar illumination radius is 2.3 km, that is, the beam width is 5.6°. Other parameters are the same as in the previous example. Figures 12-14 give the images of radar range echoes at f = 5, 20, and 50 MHz.
It can be seen that the surface nadir echo range is around 0 km while the subsurface nadir echo range is around 1.3 km (top layer is basalt) or 0.85 km (top layer is water ice). Different from Figures 8-10, the nadir echoes from both the surface and subsurface appear straight, because the surface fluctuation is small except for the ground track of 2 km and 10 km where there are two craters (see Figure 11).
The range echoes from a small crater in the ground track of 2 km are actually totally masked by the surface echoes in Figures 12 and 13 and can only be seen near the surface echoes in Figure 14, which has a high resolution at 50 MHz.
The subsurface echoes form a straight line with large range, which is broken due to the existence of a big crater in the ground track of 10 km.
The echoes from this big crater can be divided into four parts:
The echoes from the crater rim appearing around a bend of the range of −0.2 km to 0 km (e.g., indicated by the number 1, especially in Figure 14b).
The echoes from the crater floor appearing around the range of 0.5 km (see the number 2 in Figure 14b); these echoes gradually become thicker and then reduce.
The echoes from the crater inner wall at around 0 km to 0.5 km; because this crater wall is steep and rather smooth, backscattering might be weak, and those echoes are actually spreading and not strong (indicated by the number 3 in Figure 14b.
The regolith thickness in the crater is only about 30 m, the range from the subsurface of the crater inner floor is about 0.58 km (a short line indicated by the number 4 in Figure 14b), which is much less than the subsurface range outside the crater; because the subsurface is so close to the crater inner floor, its echoes are masked by the surface echoes, especially in Figures 12 and 13. A large crater with a steep rim and wall make it more difficult to recognize the echoes from the surface and subsurface.
5 Parameter Inversions of the Two-Layer Media
5.1 The Real Part of the Surface Dielectric Constant ε1
As the transmitted power of the radar sounder is Pt, the received echo power from the illuminated area is written as [Ulaby et al., 1986]
where G is the antenna gain, Rt and Rr denote the distances between the radar and illuminated surface in transmission and receiving ways, respectively, and σ is the radar cross section of the illuminated surface and is written as
where γ is the backscattering coefficient and A is the projection of the illuminated area along the incidence direction. Based on the Huygens principle [Fa and Jin, 2010], the scattering field received by the radar sounder is written as
where is the dyadic Green function of the free space. Using the tangent plane approximation, the scattering field in equation (13) is derived as [Fa and Jin, 2010]
where F(r, α, β) is a function of the h-pol and v-pol local surface reflection coefficients, Rh and Rv, respectively, and local surface slopes (α, β), and so on.
Except for the steep slopes on the crater rims and edges, most of Mars' surfaces are rough with mildly local slopes. Under the approximation of a rough surface with mean zero slope, α = β = 0, to make F(r, α, β) ≈ F(r, 0, 0), equation (14) can be expressed as
Thus, local reflection coefficients Rv, Rh in F(r, α, β) of equation (14) can be moved with F(r, 0, 0) from surface integration. In the backscattering direction leads Rv(θi = 0) = Rh(θi = 0) ≡ R0. The copolarized backscattering coefficient at nadir direction (θi = 0), including both coherent and incoherent contributions, is written as
Substituting equation (16) into equations (11) and (12), the received backward power is derived as
It can be seen that the received backward power is linearly proportional to the reflectivity as , where (kh, kℓ) denotes the surface roughness. In other words, R0(ε1) might be directly inverted from the copolarized echo power Pr, and finally yields the surface permittivity ε1.
The inversion is designed as follows. Making a presupposed permittivity , the backward power is numerically simulated as a test based on the DEM rough surface data. Suppose the measured power from this rough surface is now Pr(ε1), and the ratio of radar echoes in the nadir direction gives
Thus, solving R0(ε1) of equation (8), the dielectric constant ε1 is inverted as
In order to apply equations (15) and (19), the inverted region with zero mean slope, especially away from a steep slopes such as crater rims or edges, should be specially chosen.
Now, suppose , we can calculate the peak power of surface echo based on our two-layer model. Using equations (18) and (19), we obtain the inversed , which fits well the true value of basalt in Figures 8a–10a.
5.2 The Regolith Layer Thickness d
The speed of the electromagnetic wave in the top layer is
Using the difference between time delays of the surface and subsurface echoes, the top layer thickness d is easily calculated as
To extract the echo range accurately requires high frequency with good resolution. However, high frequency causes more attenuation and reduces the penetration depth.
As an example of Figure 7 and Figure 10a with 1.9 m resolution, the inverted depth d is shown in Figure 15, and its average is < dinv > ≈ − 1842 m with variance of about 8 m. It can also be seen that most of the inversed d is above the true value (dashed line in Figure 15), because the mesh in the exact nadir direction might not be the nearest place from which to echo the radar. It might underestimate the regolith thickness.
5.3 The Imaginary Part of Surface Dielectric Constant ε″1
Similar to equation (11), the radar echo from the surface/subsurface is written as
where is the imaginary part of the wave number in the regolith layer and are reflectivity and transmitivity of the interfaces m n (i.e., m n = 01, 10, or 12), respectively.
Two locations with different d and d + δ, but under the assumption of topography similarity, are specifically chosen. Their power ratio may be yielded as
Thus, this ratio is only related to and δ and can yield
It is noted that the derivation of equation (23) is based on the topography similarity of the two locations. Figure 16 gives the surface slopes for the two location candidates of Figure 7.
It should be noted that even if the roughness of two locations is similar, different topographies can still cause different powers in numerical calculations. As shown in Figure 17, most echoes from a concave area contribute to the radar, but most echoes from a raised area become dispersive and do not reach the radar. Thus, carefully choosing two locations is necessary for good inversion of equation (24).
We selected the powers from each of five observations at two locations as P1 and P2 to see how the is inverted by equation (24) due to a different layer thickness d and surface slope α. It was found that
Small d difference and large α difference: choosing P1 at a ground track of 2–3 km and P2 at a ground track of 7–8 km in Figure 16, the d difference is 35 m and the α difference is 1°, yielding the inversed = 0.16.
Large d difference and large α difference: choosing P1 at a ground track of 14.4–15.4 km and P2 at a ground track of 7–8 km in Figure 16, the d difference is 211 m and the α difference is 1°, yielding the inverted = 0.055.
Small d difference and small α difference: choosing P1 at a ground track of 14.4–15.4 km and P2 at a ground track of 10–11 km in Figure 16, the d thickness difference is 126 m and the α is almost equal, yielding the inverted = 0.048.
Large d difference and small α difference: choosing P1 at a ground track of 14.4–15.4 km and P2 at a ground track of 2–3 km in Figure 16, the d difference is 176 m and the α is almost equal, yielding the inverted = 0.035.
It can be seen that a large d difference and a small α difference can give more accurate inversions.
5.4 The Real Part of the Subsurface Dielectric Constant ε″2
Similar to the inversion of , the echo powers from the subsurface Psub for the bedrock ε2 and as the measurement and test can be obtained using a two-layer model and DEM data. The ratio of Psub(ε2) and is
From the simulated measurement Psub(ε2), where ε2(=3.15) is to be inverted, and simulated , where is a test value assumed as 4.0, Figure 18 gives at different locations. This ratio is dispersive mainly due to the previous error of inverted .
Using the linear regression method to reduce the influence due to the inverted error, equation (26) gives . It leads the inversion of .
If is known, it gives
If is known, it gives
Compared with the true value 3.15, the result of equation (28) is good. It needs a priori knowledge to decide which one is adopted.
A two-layer model of Mars' cratered rough surface/subsurface is presented. Using ray tracing of geometric optics and KA of the rough surface scattering, images of radar range echoes are numerically simulated for center frequencies 5 MHz, 20 MHz, and 50 MHz, especially. The relationship between the radar range echo and the physical parameters of the cratered rough surface/subsurface features (such as dielectric properties of the media and regolith thickness) is quantitatively presented.
Low frequency, such as 1–5 MHz, has good penetration depth. However, this low frequency and narrow bandwidth leads to low resolution and might produce inaccurate inversion of the layering parameters. Thus, the frequency of 5 MHz or even lower might be good for ionosphere sounding rather than subsurface exploration [Mouginot et al., 2010]. On the other hand, too high a frequency, for example, higher than 50 MHz, has very small penetration depth (a hundred meters or even less, depending on the media dielectric properties). It is suggested that using both center frequencies of 20 MHz and 50 MHz with appropriate bandwidth can be a balance for the Chinese Mars exploration program with good resolution to reach the 1 km subsurface, and conveniently comparable with the SHARAD data.
On the basis of numerical simulations, inversions of the dielectric constants of the two-layer media and the regolith layer thickness are developed. These approaches are feasible, but the inversed regions must be carefully selected.
This work was supported by the National Science Foundation of China 41071219.
In the originally published version of this article, the equation d = 0.533 / square root of (7.1) = 0.2 was incorrectly typeset. The equation has since been corrected and this version may be considered the authoritative version of record.