Dual-frequency bistatic radar experiments were conducted with Mars Express at a rate of one to two per month during 2005. Each observation provided power measurements of orthogonally polarized surface echoes at one specular point; the ratio of these components yielded values of the dielectric constant in the range 2.0 < ɛ < 4.0. Doppler sorting of X-band (wavelength λ = 3.6 cm) echoes was used to achieve one-dimensional surface resolutions of about 20 km. Although much weaker, simultaneous S-band (13-cm) echoes yielded dielectric constants that are 10–50% higher than 3.6 cm echoes, consistent with deeper surface penetration.
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 Bistatic radar is an effective method for remotely probing planetary surfaces at scales of interest to landers and rovers. The centimeter wavelengths used in today's spacecraft telecommunications systems interact most strongly with surface structure having dimensions of 1–100 cm. When a spacecraft high-gain antenna (HGA) is aimed toward the specular point on a planetary surface (Figure 1), the power reflected toward an Earth-based receiver is maximized. Frequency dispersion in the quasi-specular echo is proportional to the root-mean-square tilt of reflecting facets having dimensions of a few centimeters to a few meters. Amplitude of the echo is proportional to the Fresnel reflectivity of the surface material, which is a function of the dielectric constant and can be related, through modeling, to the density of the top few centimeters of regolith material [e.g., Gold et al., 1970]. Bistatic radar was first described in detail by Fjeldbo ; principles and results from its first 30 years were reviewed by Simpson .
 Four of the earliest Mars Express (MEX) experiments will be discussed here (Table 1). In Table 1, tr is the time at which an observer on Earth would have “seen” the spacecraft antenna illuminate the specular point, given by its latitude and longitude (β, Λ). The corresponding transmit time is tt = tr − ∣rR − rT∣/c, where rT is the position of the spacecraft at transmission with respect to the center of Mars, rR is the position of the receiver at reception, c is the speed of light, and propagation directly from spacecraft to Earth is assumed. The flight time for a photon traveling the carom path is about 5 ms longer, a difference that is not important here. These experiments were conducted at moderate to high spacecraft altitudes, incidence angles 50–70°, latitudes 50–70° from the equator (putting them largely beyond the reach of conventional Earth-based planetary radar systems), and Earth-Mars distances beyond about 1 AU.
Table 1. Summary of Mars Express Bistatic Radar Observations
Earth Receive Time tr, UTC
Earth-Mars Distance RR, AU
Latitude β, °N
Longitude Λ, °E
Incidence Angle i, deg
Slant Range, km
12 May 2004
27 Feb 2005
3 Apr 2005
6 Jul 2005
 Mars Express adopted a fixed inertial attitude for these experiments rather than dynamically tracking the moving specular point, as has been customary for other bistatic radar operations [Simpson, 1993]. Hence conventional interpretations based on the methods of Fjeldbo  apply only to a small area around (β, Λ).
 In each experiment, the spacecraft generated an unmodulated spacecraft carrier signal at two frequencies (Table 2), which was reflected from the surface. The incident RCP signal was converted to both RCP and LCP during the reflection process. The power reflection coefficients for RCP and LCP received are
respectively, where the “horizontal” (RH) and “vertical” (RV) terms are
ϕi is the angle of incidence (also the angle of reflection), and ɛ is the dielectric constant of the surface material.
Table 2. Nominal Performance of Key System Elements
A targeting error on 27 February 2005 caused the spacecraft antenna bore sight to miss the specular point by about 0.3°; otherwise pointing is believed accurate to about ±0.1°.
Wavelength λ, m
Transmitted power PT, W
Transmit antenna bore sight gain GT, dB
Transmit antenna half power half beam width θT,a deg
Ground antenna bore sight gain GR, dB
Ground antenna effective aperture AR, m2
Nominal receiving system noise temperature TSYS, K
 The radar equation gives the surface echo power received on Earth as
where PT and AR are given in Table 2, and RT and RR are the magnitudes of rT and rR, respectively. GT is the gain of the transmitting antenna, which will have the boresight peak value given in Table 2; but the gain decreases away from the boresight in a pattern which is most simply characterized by the half power half beam width θT. The quantity σ is the surface radar cross section, which is often expressed as the product of a specific radar cross section σ0 (per unit surface area) and the area S mutually visible to both transmitter and receiver. In incremental terms
where σ0 can be a function of many variables including incidence angles, reflection angles, dielectric constant, and polarization. For simple surfaces (e.g., homogeneous, isotropic, and having gaussian height statistics) we can follow Hagfors  and model the specific radar cross section as
where ρi is the appropriate reflection coefficient (1), C is a parameter often interpreted as the inverse mean square surface slope, and γ is the tilt angle needed for a surface element to reflect specularly. Angle γ is measured from the local mean surface normal; at the specular point itself γ = 0, but nonzero values are required elsewhere for local surface elements to reflect specularly. For most real surfaces the probability density function for tilt decreases with angle, meaning that the source of surface echoes is concentrated about the specular point for the mean spherical surface. For rougher surfaces the distribution of properly oriented facets will be broader around the specular point than for smooth surfaces. To first order, frequency dispersion in the echo signal is proportional to the breadth of this spatial distribution, so measurements of echo bandwidth can be used to infer C. Except for the smoothest surfaces, however, the Mars Express HGA beam pattern limits the echo frequency dispersion, so estimates of surface roughness must be made less directly.
 The experiment conducted on 6 July 2005 combined echoes having good signal-to-noise with an observing geometry that is convenient for interpretation. To illustrate how the radar equation (3) and Doppler dispersion interact in generating the surface echo, we have simulated the X-band scattering processes for two instants separated by 60 s near the time of maximum echo strength. The calculations were carried out for a 401 × 401 grid with 5 km grid point spacing and a specific radar cross section σ0 = 1 (no dependence on angle, polarization, or dielectric constant). Mars was assumed to be a sphere with radius 3382.65 km. Increasing the radius by 1 km changed the location of the specular point by less than 0.01° in latitude and longitude, changed its incidence angle less than 0.01°, and changed its Doppler frequency by about 2 Hz compared to our frequency resolution of 24.4 Hz in processing; so the precise radius assumed, and the local topography around (β, Λ), are not important.
 The results in Figures 2 and 3 show clearly the importance of the spacecraft HGA pattern, which dominates the echo shape; the variations due to changes in RT are inconsequential. The instantaneous specular point was chosen as the grid origin in each case; but the specular point moved southward by only 10 km during the time separating the two simulations, a minor adjustment compared with the 1000 km covered by the grid. The largest differences between the two simulations result from the eastward sweep of the inertially fixed HGA illumination and a modest migration of Doppler contours toward the northeast.
 For actual surfaces the assumed σ0 = 1 in the simulations must be replaced by a more realistic function, such as (5). As we demonstrate below, however, the shape of σ0 cannot be determined directly from the data collected on 6 July 2005; but the relative amplitudes of σ0 in the two orthogonal polarizations provide a straightforward way to obtain the surface dielectric constant. All of the factors in (3), (4), and (5) except ρi are common to both RCP and LCP, so measuring the ratio of RCP to LCP echo power yields the ratio ρR/ρL, which can be solved for ɛ (we do this by linearly interpolating between values of ρR/ρL, precomputed for known ϕi and dielectric constant steps of Δɛ = 0.01). We calculate each PR by comparing it to the background noise power density N0 = kTSYS, where k is Boltzmann's constant. N0 is measured simultaneously with PR and is quantified through a series of straightforward, albeit time consuming, calibration steps for TSYS on each receiver channel. Success in determining ɛ therefore depends on the precision with which we can calibrate TSYS for each receiving channel.
Figure 4 illustrates the key elements of each receiver. Before and after each experiment the background sky noise power was calibrated against a resistive load at ambient temperature while the antenna was pointed to zenith. The noise diode was calibrated at the same time, then used to provide an additive noise reference at 60–90 min intervals while the antenna was tracking Mars and the spacecraft. Neglecting small changes in the gain of the microwave front end, we can monitor TSYS during the bistatic radar surface measurements with an accuracy of about 5% over time intervals of a few minutes. For comparison, the statistical uncertainty in our highest time resolution bistatic radar power spectra is about 5% (25,000 samples/s, taken 1024 points at a time, and averaged over 15 s). Other contributions to measurement uncertainty include nonlinearities at various points in the system, and occasional radiofrequency interference (particularly at S-band).
 During processing of the bistatic radar data, spectral regions B within the 25 kHz bandwidth were identified as being free of direct signal, echo, and interference. These were used for dynamic calculation of TSYS; PN = kTSYSB could then be used to scale power spectra to correct amplitudes. PN was then subtracted from the total spectrum leaving the echo signal and sometimes a residual, directly propagating carrier signal from the spacecraft. We elaborate on our signal model, calibration issues, and calculation of uncertainties in Appendix A.
Figures 5 and 6 show echo spectra as the HGA beam swept over the specular point on 6 July 2005; these correspond to Figures 2 and 3, respectively. The X-RCP echo power, integrated over all contributing frequency bins, peaked at 1163 × 10−21 W at 2319:35 Earth receive time (ERT), about 75 s after the specular point was centered in the HGA beam. The rise from and fall to the half power points took about 7 min. The integrated X-LCP echo power peaked at 535 × 10−21 W. Both S-band channels showed a broad enhancement in power, coincident with the X-band increases, although the enhancement showed no clear rise, peak, or fall. Only two of the integrated S-band echoes exceeded 35 × 10−21 W, largely a consequence of lower PT and GT at the longer wavelength.
 Ratios of the measured RCP to LCP powers are shown in Figure 7, plotted as a function of incidence angle for the associated specular point. In each case the echo powers at all frequencies in a single spectrum were summed; then the results from 365 spectra, representing 15 s in time, were averaged before the ratio was calculated. The S-band points are more widely scattered; five exceeded the plotting limit (ρR/ρL < 4) and are not shown.
 If we average the 6 July X-RCP and X-LCP echo powers over the 20 min centered on the optimum specular point condition and then take their ratio, we find ρR/ρL = 2.066. Although this procedure confuses the specular point interpretation by folding data into the calculation from higher and lower values of ϕi (and possibly resulting from nonspecular processes), this was the only practical way to obtain results for the May 2004 data. For the February and April 2005 data, we generally obtained satisfactory ratios using 5 min averages. Results for both 5 and 20 min averages are shown in Table 3, where the specular point latitude, longitude, and incidence angle are listed as though the HGA radiated isotropically. We note that the specular point theory supports this procedure, provided that the specular angle at the illuminated point is used in the calculation; we discuss this and associated issues later. For the optimum specular point geometry on 6 July, ρR/ρL = 2.066 corresponds to a dielectric constant of ɛ = 2.58. The solid line in Figure 7 shows the expected ratio for a surface with ɛ = 2.58, as computed from (1).
 Summing the power in a single echo means the associated surface footprint covers a significant area, for example, the area enclosed by the outer elliptical contour in Figure 2. Averaging these sums over time superimposes many such footprints, making the effective surface resolution still larger. When the signal-to-noise ratio is high, we can evaluate power ratios in individual frequency bins, the Doppler stripe associated with the specular point pluses in Figure 2, for example, where its east-west limits are the outer elliptical contour. Although a fixed location on the surface will drift from one frequency bin to another over time, we can calculate and remove the drift and obtain estimates at spatial resolutions commensurate with the frequency resolution in the spectra.
Figure 8 shows power ratios plotted against latitude for six consecutive 15-s spectra; we have corrected for frequency migration and the spectral bins have been associated with fixed latitudes on the surface. In each 15-s time interval the instantaneous specular point moved 2.5 km to the south and 13.3 Hz higher in frequency, relative to the direct signal frequency fD. Since a single bin corresponds to about 17 km north-south on the surface and 24.4 Hz in frequency, the net drift of a fixed point on the surface is +0.4 bins in 15 s or 2.4 bins over the time represented by these six spectra. Although only six spectra at specular point latitudes between −60.86 and −61.07 were used in this exercise, the scatter in the data points (typically less than ±15%) suggests that valid results can be obtained as much as ±1.5° from the central latitude. In Table 4 we have listed the results for ten adjacent latitudes, centered approximately on the optimum specular point. Means and standard deviations were calculated at each latitude from the six measured power ratios; dielectric constants were derived assuming ϕi = 64.65°. The constant incidence angle is justified because δϕi/δt = −0.13°/min and the north-south variation in the simulation grid is ∣δϕi/δy∣ < 0.0001°/km at 2318:00 ERT. There is also a west-to-east variation in the grid δϕi/δx ∼ 0.001°/km, which we expect to have little effect since each latitude bin contains echo from the same longitudes.
Table 4. Doppler Sorted Measurements (6 July 2005)
Latitude β, °N
Incidence Angle i, deg
Measured Power Ratio ρR/ρL (Average of Six Values)
2.081 ± 0.151
2.570 ± 0.095
2.220 ± 0.197
2.463 ± 0.109
2.200 ± 0.249
2.482 ± 0.141
2.090 ± 0.118
2.561 ± 0.073
2.106 ± 0.043
2.545 ± 0.026
2.227 ± 0.134
2.454 ± 0.073
2.200 ± 0.280
2.486 ± 0.159
2.328 ± 0.256
2.390 ± 0.129
2.410 ± 0.384
2.348 ± 0.184
2.581 ± 0.326
2.239 ± 0.134
 MEX bistatic radar experiments provide new data about the surface of Mars. Each of the targets listed in Table 1 is well outside the view of conventional range-Doppler radar systems on Earth. Only Hellas has been probed using bistatic techniques before; those measurements were made by Viking and were near the center of the basin rather than on its southern margin as here. The MEX measurements also represent the first simultaneous use of dual frequency probing and the first use of the polarization ratio method for determining dielectric constant on a regular basis.
 The numerical simulations of the 6 July 2005 experiment show good agreement with the results in terms of echo offset frequency and shape. Our use of σ0 = 1 for the surface scattering function may have seemed restrictive; but comparison of the X-band computed spectrum with the data suggests that the echo shape is largely determined by the HGA radiation pattern and not the surface roughness. The echo shape is beam-limited if the surface has RMS roughness greater than 4–5° on scales of centimeters to meters.
 If we assume that radio wave scattering is described by the Hagfors function (5), we can make a simple estimate of roughness based on the observed reflectivity. The inferred MEX dielectric constant ɛ = 2.58 was derived from a polarization ratio ρR/ρL = 2.066 at an incidence angle ϕi = 64.65°. The individual reflection coefficients are found from (1) and (2) to be
From Figures 5 and 6 we see that σ0 = 2 provides a reasonable match to the X-RCP echo and that a number slightly less than σ0 = 1 would match the X-LCP echo. Evaluating (5) for γ = 0° gives C ∼ 22 and an RMS roughness on the order of C−1/2 = 12°. This is large for Mars, where typical values are more like 3–5°; Simpson et al.  reported roughness values of 2.5–4.5° and a dielectric constant ɛ = 3.1 along a Viking Orbiter specular point track in Hellas between (47S, 296W) and (40S, 300W) at S-band. However, RMS values larger than 10° have been observed on very rough surfaces around the Tharsis volcanoes. The new Hellas roughness estimate may be reduced if it is found from further calibration that the PTGTAR product based on the values in Table 2 is too large; there is some evidence in this direction from more recent experiments not covered by this paper.
 We have presented three methods for deriving dielectric constant from polarization ratios. The simplest is to sum all of the echo in each polarization and to take the ratio of the results. This is the only method that works for very weak signals, such as when Earth-Mars separation is more than 2 AU or in many of the S-band observations to date. Because it can be applied to all of the data which have been processed, we have listed these numbers in Table 3 for consistency. However, this method has an ill-defined footprint; the HGA beam can sweep across a lot of terrain in 20 min and, although we have associated the incidence angle at the targeted specular point with these measurements, there is a wide range of facet geometries that contributes to these echoes. As noted earlier, however, there is virtually no ϕi variation in the north-south direction and only a linear drift east-west; so “corruption” introduced solely by variation of ϕi over the illuminated areas may be very modest, at least for geometries similar to those seen on 6 July 2005.
 Simulations with realistic functions for σ0 are one solution which we will be incorporating in future analyses. We note, however, that even with the simple geometry of the 6 July 2005 experiment we may have to face heterogeneous surfaces. The asymmetry and shoulder(s) on the low-frequency side of the data spectra in Figures 5 and 6 imply that the surface north of these specular points is rougher or less reflective (or both) than the surface to the south. So a specific radar cross section that varies with position may be required for more detailed analysis within the 6 July 2005 target area.
 The second method for deriving dielectric constant is to take the ratio of total powers in individual spectrum pairs. The X-band results in Figure 7 are satisfactory, although the number of points shown far exceeds the number of independent measurements because adjacent footprints overlap by more than 90%. The individual footprints are smaller than the effective footprint in the 20 min case, however. Visual inspection of the X-band points in Figure 7 suggests that the individual ratios lie more above the solid curve than below, meaning that the inferred dielectric constants would be slightly lower than ɛ = 2.58. The model curve shows an increase in ρR/ρL with increasing incidence angle, whereas the data points suggest a very slight decrease. Again, more sophisticated use of the simulation, including a spatially varying σ0 function, may provide additional insight.
 The third method is to use the power measurements in individual frequency bins to compute ratios, to associate bins with fixed locations on the surface, and to track the migration of these fixed points through the drifting spectra as a function of time. Our limited exercise shows that reproducible ratios and dielectric constant estimates can be obtained with about 10% standard deviation in each. Using Doppler sorted data from 6 July, we confirmed that dielectric constant decreases as the specular point moves south, a conclusion first drawn from the whole spectrum ratios plotted against angle of incidence (Figure 7).
 We believe that the third method gives the best estimates of surface dielectric constant when there is sufficient signal-to-noise to allow calculations at the individual frequency bin level and the frequency bin and time resolution are both meaningful. We note common details in the 15 s spectra such as notches 2–4 bins below the peak values in both X-RCP and X-LCP and approximately linear changes in echo powers over −1500 to −1200 Hz, particularly in Figure 5. Such features are diagnostic of surface heterogeneity and have been used in favorable circumstances to map surface morphology [Simpson et al., 1977]. Shoulders on the high-frequency side of X-LCP echoes are not visible in X-RCP, however.
 We have said very little about the S-band echoes. Weaker by a factor of 20 in PT and another factor of 20 in GT, as compared with X-band, the S-band signals are more difficult to detect. However, there are mitigating factors: the HGA beam is nearly 4 times broader at S-band and natural surfaces tend to have higher C values at the longer wavelengths making them easier to detect in quasi-specular scattering. Although we computed a 20-min power ratio from the S-band measurements on 6 July 2005, we were unable to define a consistent echo shape. For other targets, particularly where the surface is smoother, we expect to find the S-band echo to be more diagnostic of surface properties.
 Scatter among dielectric constant estimates made under similar conditions is less than 10%; statistical variations at the finest time resolution are about 5%, and calibrations appear to be reproducible at about the 5% level. Assuming an overall uncertainty on the order of 20%, we believe that the target to target ɛ variations listed in Table 3 are valid.
 We believe that at least part of the observed variation is a real response to heterogeneous surface properties. Although limited in scale, we believe our Doppler sorting demonstrates the viability of this high-resolution processing for future MEX bistatic data.
 Although the S-band measurements are more uncertain, we have found, in all experiments conducted to date, that the dielectric constant is smaller at X-band than at S-band, in agreement with models of surface penetration. This is the first time simultaneous measurements have been used on Mars to confirm the wavelength variation in ɛ.
 As this paper is written, we have just passed Mars opposition when bistatic echo strengths were strongest. Five recent experiments have been conducted over Mars' south pole in hopes of learning more about the radar-bright residual south polar ice cap (RSPIC) discovered by Muhleman et al. . The quasi-specular bistatic mode is not likely to trigger the anomalous scattering reported earlier; but the new data may still be helpful in understanding the phenomenon. We are also seeking opportunities to schedule bistatic radar over the equatorial “Stealth” region, which is unusual in its weak scattering response [Muhleman et al., 1991].
 Operationally, the Mars Express Project has agreed to implement specular point tracking so that we can follow the path of maximum echo strengths with the HGA rather than sweeping once across. Potentially, this will allow us to collect hundreds of data points per observing session whereas we are limited to one (plus whatever we can extract using Doppler sorting) with inertial HGA pointing.
Appendix A:: Error Sources and Attributions
 We model the output of each receiver as the sum of a random noise process and a surface echo, which has certain noise-like properties but is statistically independent. For purposes of assigning error bars, we calculate average noise power density N0 in a spectral region free of spacecraft signals (either direct or reflected) and interference. The uncertainty σN in N0 is inversely proportional to the square root of the product of the number of power spectra averaged and the number of frequency bins averaged in order to obtain N0. The uncertainty σR in echo power PR is assumed to be the value of σN scaled upward by the bandwidth BR occupied by the echo signal. There are, of course, uncertainties associated with the noise-like nature of PR; lacking a good model for these at present, we adopt σR = σNBR as the fundamental limit to measurement accuracy. Uncertainties in power ratio and dielectric constant are the propagated values of PR and PR ± σR, where the latter results are weighted by e−0.5 in computing the derived means and standard deviations.
 There are systematic sources of error which are more difficult to characterize and which are potentially larger. Repeated measurements of noise alone, lasting up to an hour, show that we can reproduce the spectral response of the 25 kHz output bandwidth to accuracies of ±0.5% across receiving stations, bands, polarizations, and time. These “filter functions” are used to “equalized,” or flatten, the data spectra before the noise pedestal is removed. Noise calibrations (e.g., determination of TSYS using Figure 4) are reproducible to within 5% before and after our experiments. We generally use the preexperiment values to scale the data spectra since they are acquired more systematically and with more time integration.
 However, stormy weather moved into the area around the receiving station during the early activities on 6 July 2005. The postexperiment calibrations proved to be more stable (and more representative of conditions during the Mars observations) in spite of a more than 100% increase in TSYS (compare N0 in Table 3 for this date against others). Less pronounced weather changes presumably affected other observations, though our dynamic referencing of echo power to simultaneously acquired values of N0 should minimize associated errors.
 Short term changes in gain of the microwave front end (Figure 4) cannot be distinguished from weather transients except by comparing RCP and LCP noise levels from the same band. There have been instances where such gain changes apparently occurred; but they are infrequent, and the magnitudes fall within what we believe to be a 20% absolute error budget on absolute echo powers and a ±10% accuracy in derived dielectric constants.
 The capable and conscientious support of many people within the Mars Express Project, the NASA Deep Space Network, and the Jet Propulsion Laboratory Radio Science Systems Group is gratefully acknowledged. This work was supported at Stanford University under NASA contract JPL 1217744 and in Germany by the Deutches Zentrum für Luft- und Raumfahrt (DLR), Bonn under grants 50QM9909, 50QM0008, and 50QM0401.