We propose a simple technique to estimate a roughness parameter, RMS gradient, of a planetary surface. The technique is practiced on A-scope data of radio sounding observation of planet with a simple formula. The formula was derived analytically for the case of Gaussian random rough surface using a geometrical optics model. The technique was examined by simulations of planetary sounding observations which is based on physical optics, and demonstrated to estimate the RMS gradient of the planetary surface in the range from some fraction of degrees to about ten degrees. Finally, the upper limit of valid range of RMS gradient was formulized as a function of observation altitude and the maximum detection range.
If you can't find a tool you're looking for, please click the link at the top of the page to "Go to old article view". Alternatively, view our Knowledge Base articles for additional help. Your feedback is important to us, so please let us know if you have comments or ideas for improvement.
 In these subsurface sensing by radio sounder, planetary surface echo is regarded as clutter that is nothing but an obstacle to detection of weak subsurface echo. However, surface clutter itself contains information concerning the surface feature. For example we know that a rougher surface gives stronger surface clutters, i.e., clutter intensity carries information of surface roughness. This suggests that we could estimate surface roughness if we apply an appropriate data processing technique to the surface clutter data.
 In this short paper, we investigate the relationship of surface roughness and the surface clutters using A-scope data of planetary soundings. To do this, we first attempt to model a sounding observation of planetary surface on the basis of geometrical optics approximation. Then, introducing a quantity, range surplus, we analytically derive a simple formula that expresses the range surplus as a function of planetary surface roughness, RMS gradient, for the case of Gaussian random rough surface. The analytic result is examined by a numerical technique. Computer simulation of planetary sounding observations is conducted based on physical optics. Some discussions are given on the valid range of the technique and on its application.
2. Mean A-Scope Range and Range Surplus
 As mentioned above, we know the intensity of the surface clutter, or off-nadir surface echo, is directly connected to the surface roughness. The question paused is how to evaluate observed surface clutter in order to estimate the surface roughness.
 We evaluate the surface clutter in terms of its contribution to the mean A-scope range, a quantity that we introduce in this study. The mean A-scope range, μ(R), is the mean value of weighted surface echo (clutter) ranges in which the weight is the intensity of the surface echo (clutter), and is defined as
where Rn is the range of the nth range bin and P(Rn) is the received echo (clutter) power of the nth range bin.
P(Rn) in (1) may be regarded as a distribution function of Rn from the statistical point of view. Also, one may extend (1) to a continuous function of a continuous variable R, which we do in section 3.
 We ignore possible subsurface echoes among the surface clutter because first, they are weak in general and, second, they are not likely to be found at many ranges at the same time therefore their contribution to the mean A-scope range would not be significant.
 Let us define another quantity, the range surplus. The range surplus, Rsurplus, is defined as the difference of the mean A-scope range, μ(R), and the nadir range, Rnadir, as
Rnadir and Rsurplus are used to estimate the surface RMS gradient.
3. Geometrical Optics Model
 We model a radio sounding observation of a planetary surface with geometrical optics approximation. Some assumptions are made to construct the model.
 1. Global curvature of the surface is not considered, or the mean surface of the planet is a flat plane: which means that we consider a flat planet of the infinite extension rather than a spherical body.
 2. The planetary surface has the topographic feature of a Gaussian random rough surface. A Gaussian random rough surface is characterized by two statistical properties of its RMS height and the correlation length, and is defined by its power spectrum [Ogilvy, 1992]
where k, σ0, and λ0 are wave number, RMS height, and correlation distance, respectively, and the suffix denotes the direction.
 3. We assume the surface is isotropic
 4. Though we consider the local surface gradient, we do not consider the local height of the surface. A scattering point is assumed to be on the mean surface.
 5. Roughness of planetary surface is defined as the RMS gradient, αRMS, of the surface that has a very simple form of
for the case of a two-dimensional Gaussian random rough surface [Ogilvy, 1992].
 6. No ionosphere is considered for the simplicity of the model.
 7. Sounder pulse is a spherical wave pulse that is transmitted from a point antenna.
 The configuration of the model is schematically shown in Figure 1. The observation point, T, is on the z axis at the altitude Rnadir.
3.2. Mean A-Scope Range and Range Surplus
 We calculate the mean A-scope range under these assumptions with geometrical optics approximation, then formulize the estimation of the surface RMS gradient.
 Interestingly, if the surface is a Gaussian random rough surface, not only its height distribution but its surface gradient distribution also follows a Gaussian statistics [Ogilvy, 1992]. The distribution function of the surface gradient of an isotropic Gaussian random rough surface is written as
where h is the height. Since the problem is axial symmetric to the nadir line, as Figure 1 shows, a cylindrical coordinate has an advantage to treat this problem analytically. Thus we transform (5) to a cylindrical coordinate to obtain
 In this geometrical optics model, all the surface clutters observed are specular back scattering echoes of surface objects. The specular back scattering condition is simply written as
where r is the ground range (radial distance on the mean surface from the suborbital point, O) of the scattering point, and Rnadir is the altitude of observation point. Applying this condition to (6) gives a probability density function
Equation (8) gives the probability density of the specular back scattering event at the ground range r. However, all the received surface echoes are exclusively specular back scattering echoes. Thus what probability density function we need is the conditioned probability density function of the specular back scattering event, which is obtained by normalizing (8). The normalizing factor A is calculated by integrating (8) over the entire surface as
thus the conditioned probability density function of specular back scattering event is
With (10), we calculate the mean values of ground range, 〈r〉, as
 Through (9) and (11) we carried out integrals over the entire surface, which is equivalent, in an sounder observation, to open receiving time window for infinitely long period of time so that any echo from infinitely far range is received. However, of course, in an actual observation, a receiving time window is finite and a sounder observation is done within a limited detection range. Therefore corresponding integrals of (9) and (11) should be definite integrals of finite intervals if they should be rigorously consistent with an actual observation. However, for following two reasons we leave those integrals of (10) and (11) as infinite integrals: first, the infinite integral of a Gaussian function has a simple analytic form, which makes the problem easy to be treated, and secondly, clutter contributions from very far range in an actual sounder observation is not significant unless the surface roughness is very large, thus is approximated by the infinite integrals.
 Finally, we calculate the mean A-scope range, μ(R), as
If the RMS gradient of the surface is not very large (i.e., smaller than a few tens of degrees), then
The first term of the right-hand side of (15) is the nadir range or the range to the mean surface, a smooth flat plane, and contains no information of surface roughness. The information of surface roughness is contained in the second term.
 Let us call the second term the range surplus, Rsurplus, because this is an additional range term to the nadir range to make the mean A-scope range:
Once the range surplus is obtained, then, recalling (4), the RMS gradient αRMS of the planetary surface is estimated as
4. Physical Optics Model
 In this section we conduct computer simulation of planetary sounding observations by using the KiSS code. The KiSS code was developed based on Kirchhoff approximation theory, or physical optics approximation, for the purpose of simulating radio sounder observation of planetary surface/subsurface from orbit in HF band. For the detail of the code, readers are referred to Kobayashi et al. .
4.1. Simulation Condition
 We set some simulation conditions as follows.
 1. The altitude of observation point is assumed to be 300 km above the mean surface.
 2. The center frequency of transmitted sounder pulse is 5 MHz and the frequency band width is 1 MHz.
 3. The maximum detection range is 330 km.
 4. The planetary surface is assumed to be a Gaussian random rough surface as we assumed for the geometrical model in section 3.2.
 5. Seven values of RMS height, σ0, (60, 19, 6.0, 1.9, 0.6, 0.19, and 0.06 m) and five values of correlation distance, λ0, (4243, 1342, 424, 134, and 42 m) are chosen in a logarithmic manner to the reference values of the sounder pulse wavelength, 60 m, and the first Fresnel zone radius, 4242.6 m, respectively.
 6. Surface realizations of all the combinations of those parameter values are investigated; thus the total of 35 realizations are simulated.
 7. Ten sounding observations are made while the observation point moves along the orbit above the surface.
 8. The spatial interval of observation points along the orbit is 10,000 m, which is larger enough than correlation distance λ0.
4.2. Simulation Results
 Simulation results were obtained as A-scope data set. Figure 2 shows some examples of A-scope data.
 From these A-scope data, we calculate the mean A-scope range, μ(R), by (1). We determine the nadir range, Rnadir, by choosing the range at which the surface echo has the maximum power of the A-scope data.
 The range surplus is then obtained by (2), and the RMS gradient of the planetary surface is estimated by (17).
5. RMS Gradient
5.1. Range Surplus
Figure 3 compares the range surplus after the geometrical optics model (hereinafter referred as GO model) with one after the simulation (hereafter referred as PO model) as functions of RMS gradient of the planetary surface. The range surplus of the GO model is shown as the thick solid line while those range surplus of the PO model are plotted as open circles.
 It is obvious that the results of two models show good agreement in the range of RMS gradient from some fractions of degree to a little larger than 10 degrees allowing statistical fluctuation in PO model results.
5.2. Lower Limit of Valid Range of RMS Gradient
 In the range of RMS gradient smaller than 1 degree, discrepancy exists: while GO model predicts monotonous variation of the range surplus as a function of RMS gradient at the constant rate, the range surplus of PO model converges to about 30m as RMS gradient becomes smaller. This discrepancy is attributed to that the planetary surface in the KiSS code was described by the collection of discrete definition points which caused discretizing noise. Therefore, even if the surface is defined as a smooth plane where the RMS gradient is 0, the discretizing noise in the simulation behaves as if it were surface clutter, as the result we obtain finite value of the range surplus. The discretizing noise is seen in the A-scope at the bottom in Figure 2 as the noise floor of which power is ∼−30 dB with reference to the nadir echo. This phenomenon implies that, in an actual sounder observation, the lower sensitivity limit of the technique to the surface roughness is determined by the system noise of the sounder.
5.3. Upper Limit of Valid Range of RMS Gradient
 Discrepancy also occurs in the range of large RMS gradient. This is because of the failure of the assumption that the sounder observation with the finite detection range can be approximated by an infinite integral of GO model. While surface roughness is small, the clutter contribution comes only from near range regions, thus the infinite integral of GO model can well approximate PO model. As the surface roughness becomes larger, more clutter contribution comes from farther range. Finally, nonnegligible contribution would come from beyond the maximum detection range so that the assumption fails.
 Let us estimate the upper limit of the valid range of RMS gradient, below which (17) is valid. We apply very well known 3σ rule to this problem, in which an event whose absolute value is larger than 3σ is ignored, where σ is the standard deviation of the value of the event. In our problem, the σ is the RMS gradient of the surface (note that this σ does not denote RMS height, σ0).
 In a one-dimensional surface case, the event is the global incident angle θ of Figure 4. Therefore, having θmax be the maximum global incident angle, (1) is regarded equivalent to (11), so far as
holds. For a given surface, the maximum value of global incident angle is found at the maximum detection range rmax as
Thus, from (18) and (19), the upper limit of the valid range RMS gradient, αmax is given as
 As to the present study, applying the parameter values to (20), the upper limit of the valid maximum αRMS is obtained as 8.7°. The result is extended to the two dimensional surface case to obtain the valid maximum αRMS as 12°. It should be noted that (20) implies that the upper limit of the valid RMS gradient can be controlled by either design parameter of the system(the maximum detection range) or observation condition (orbit altitude, Rnadir).
5.4. RMS Gradient
 Finally, we apply (17) to estimate the RMS gradient of the surface of the simulation. Figure 5 presents the comparison of estimated RMS gradient of the surface and that of the model. We can see from this result that the RMS gradient of the planetary surface can easily estimated in the range from some fraction of degrees up to 12°, the upper limit of valid RMS gradient.
 The proposed method will be applied to SELENE Lunar Radar Sounder (LRS) observation [Ono et al., 2000]. LRS is an HF sounder that is one of onboard science missions of SELENE, a Japanese lunar exploration program. The center frequency of LRS pulse is 5 MHz and the frequency bandwidth is 2 MHz. SELENE is planned to be put in a polar orbit of which altitude is 100 km. The maximum detection range is 125 km. Therefore the upper limit of valid RMS gradient range is estimated as large as 20°.
 SELENE will be operated for 1 year period, and LRS is planned to carry out its observation as many as possible through the operation period so that whole lunar surface is observed. By the end of the operation period RMS gradient map over the entire lunar surface will be made.
 On the Moon, where erosion process of neither air nor water exists, meteorite bombardment and the relaxation process of lunar material have played important roles to make the present surface gradient distribution. We should be able to extract, from the surface gradient distribution, those information relevant to the selenologic history that is responsible for present surface topography. Therefore, along with optical images, global mapping of RMS gradient is expected to provide an important information for reconstructing the selenologic history of lunar surface.
 The proposed method may be applied to Mars observation of which soon MARSIS data and SHARAD data will be available. Martian surface topography has been measured by Mars Orbiter Laser Altimeter (MOLA) [Smith et al., 2001] of the Mars Global Surveyor project and global mapping of RMS gradient was done by [Aharonson et al., 2001]. Although MOLA measurement was done with high accuracy, its along track spatial resolution is 300 m under which scale of the surface feature MOLA is blind. On the other hand, the wavelength of MARSIS sounder pulse is 60 m (5 MHz)[Picardi et al., 2004] and that of SHARAD is 15 m (20 MHz)[Seu et al., 2004], both of which are much smaller than 300 m thus these sounders are capable of observing the roughness of such a small scale that MOLA could not. This advantage would be best realized in northern hemisphere where MOLA found a vast flat plane.
 One difference of Martian sounder observation from lunar sounder observation is that Mars has an ionosphere which is not thick but cannot be ignored. The propagation delay must be compensated for the case of MARSIS if not always for SHARAD before the range surplus is determined. The MOLA results will be made use of to verify the ionospheric delay compensation.
 We have proposed a technique to estimate the RMS gradient of a planetary surface from A-scope data of HF sounder observations. The technique was first considered by the geometrical optics model of HF sounder observation. We introduce a quantity, the mean A-scope range, then the range surplus, the difference of the mean A-scope range and the nadir range. A formula was analytically derived, which directly connects the RMS gradient of the surface and the range surplus which is known from observation data. Then we applied the formula to the results of simulation that was conducted based on physical optics. Good agreement between the estimation and the true value of RMS gradient was seen in a moderate RMS gradient range. The valid range of estimation is considered and explained. The valid range is extended to the smaller RMS gradient range by the observation with less system noise and is extended to the larger RMS gradient range by changing either design parameter of the system (the maximum detection range) or observation condition (orbit altitude).
 The technique can be applied to the radio sounder observation of Mars as well as that of the Moon by SELENE LRS.
 The KiSS code was developed in collaboration with the Information Synergy Center, Tohoku University.