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 We examine the relative importance of coherent and incoherent echo components in near-nadir radar scattering from self-affine surfaces. The coherent component, important only for observations at small incidence angles (nadir-looking sounders and altimeters), is a function of both surface roughness and scattering area. Models for the incoherent component require a minimum scattering area for validity; for smaller regions, the radar echo is overestimated. We propose a simple criterion for the horizontal scale that defines this shift in applicable scattering model and show the range of roughness parameters for which MARSIS, SHARAD, and possibly JIMO radar sounder measurements may be modeled using the two approaches.
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 Radar echoes from natural surfaces are often analyzed using theoretical models for the scattering process. In most cases, these scattering models treat the scattered field as being either “coherent” or “incoherent”. The coherent component is characterized by a deterministic phase behavior (i.e., constructively interfering surface scatterers), such that the mean electric field is nonzero and the average power is given by the squared mean field. The incoherent component assumes random interference between scatterers, such that the mean electric field strength is zero, and the average power is given by the sum of the squared local field strength across the surface.
 In practice, there is a transitional behavior between the two components, and the total reflected power is not simply their sum. An example is to consider two point sources that radiate toward the receiving antenna. If their respective complex-valued electric fields at the receiver are given by E1 and E2, then the coherent and incoherent reflected power is
where the asterisk denotes the complex conjugate. For a natural surface, the received echo is the superposition of many such reflections. The difference between the two components lies in the degree of correlation between the electric field vectors. In the case of perfect correlation, the coherent component (1) contains all of the reflected power; the calculated incoherent component does not constitute additional received signal. For a diffusely scattering surface, however, the squared terms in (1) are cancelled by the random signs of the electric field cross-products, and the total reflected power is given by the incoherent component. For backscatter data collected at incidence angles typical of imaging radar systems (>20°), the incoherent echo is safely assumed to dominate the return. The coherent return is only important when the incidence angle is small (the “near-nadir” regime), since this component contributes primarily to “mirror-like” reflections from smooth portions of the surface. In this paper, we examine models for the coherent and incoherent echo components, and discuss their applicability for analysis of near-nadir radar backscatter data.
 Our interest in near-nadir scattering processes is motivated by two upcoming Mars radar sounder instruments. The Mars Express Mission carries the Mars Advanced Radar for Subsurface and Ionosphere Sounding (MARSIS), which operates at frequencies from 1.8 to 5 MHz. The Mars Reconnaissance Orbiter, to be launched in 2005, will carry the Shallow Subsurface Radar Sounder (SHARAD), which operates at a center frequency of 20 MHz. These sounders are intended to search for Martian subsurface geologic interfaces, but will also measure the surface return from the nadir region (and potentially from favorably oriented off-nadir regions). The surface echo may obscure subsurface returns, and significant effort is directed toward understanding the dominant scattering mechanisms and possible methods for identifying and suppressing surface reflections [e.g., Peeples et al., 1978; Orosei et al., 2001; Plaut et al., 2001]. An additional motivation comes from the possible inclusion of a radar sounder instrument on the Jupiter Icy Moons Orbiter (JIMO) mission.
 This paper examines the relative importance of coherent and incoherent near-nadir radar backscattering, using a fractal description of surface roughness. Section 2 reviews the observing geometry of the MARSIS and SHARAD instruments. In section 3, we discuss existing models for these two components, and correct an error in the normalization of the coherent scattering model of Shepard and Campbell . We also characterize the dependence of normal-incidence backscatter on surface roughness and scattering area. Section 4 assesses the range of validity for models of incoherent scattering, and the conditions under which coherent surface returns may be observed, relative to the operating parameters of the MARSIS and SHARAD sounders.
2. MARSIS and SHARAD Observations
 MARSIS and SHARAD use synthetic-aperture Doppler processing techniques to achieve improved spatial resolution in the along-track direction. Combined with the delay resolution achieved with a chirped (wide-bandwidth) pulse, the sounder data may be processed to form a delay-Doppler array of “resolution cells” centered on the nadir point (Figure 1). Because the antenna beam is directed toward the nadir, the delay-Doppler array is ambiguous to the left and right of the ground track. Possible subsurface echoes will appear in the array “bin” appropriate to their round-trip delay and frequency shift. For this paper, we ignore the subsurface reflections and examine the possible range of behavior in the surface echoes.
 The term “clutter” is used in radar sounding to refer to off-nadir surface echoes that can be confused with true subsurface reflections (since both arrive at greater delay than the nadir surface point). We extend this definition here to include the nature of the surface return from the nadir region. Clutter may have “random” and “deterministic” components. The random component can be considered to represent surface roughness on scales much smaller than the resolution cells (e.g., small hills, boulder fields, dunes), such that its average behavior is well represented by an angular backscatter function. The deterministic component arises from tilted, discrete off-nadir features (e.g., large hillsides, crater walls) or from locally smooth patches in the nadir region. In general, random clutter may be treated with models for the incoherent radar scattering component. Deterministic off-nadir clutter, and the nadir return, may have a strong coherent component under conditions discussed below.
3. Radar Scattering Models
 In this work, we deal with surfaces described by self-affine, or fractal, statistics. For a fractal surface, roughness may be described by the RMS slope (the ratio of the Allan deviation to the horizontal step size) at some reference scale and the Hurst exponent, H [e.g., Shepard et al., 2001]. We use the illuminating wavelength, λ, as the reference horizontal scale, and denote the corresponding RMS slope as sλ. The RMS slope at any other scale, x, is given by
Higher values of H correspond to surfaces whose RMS slope declines more slowly with x than that of low-H surfaces. Fractal surface profiles are also described by a power law form of the surface height power-spectral density function, S(f):
where So is a scaling factor (m3), f is the spatial frequency, and fo is a reference frequency [Turcotte, 1992].
3.1. Scattering Model for the Coherent Component
 In general, coherent radar echoes are confined to situations where the reflecting region is very smooth at the scale of the illuminating wavelength, λ, and is oriented nearly perpendicular to the incident beam. Because the coherent return requires constructive interference between surface scattering elements, this mechanism is also limited to scattering areas (i.e., resolution cells) whose total RMS height deviation is much less than λ. In effect, we treat the scattering region as an antenna (or single “facet”) whose coherent gain is reduced by surface roughness. This is the basis for the reduction in effective surface reflectivity with RMS height discussed by Barrick and Peake . Our contribution in this paper is to treat the coherent scattered component from a self-affine surface of finite extent.
 The coherent scattered electric field from a rough fractal surface was initially derived by Shepard and Campbell , exploiting the fact that surface heights along concentric annuli about any chosen point have an RMS value (with respect to the mean plane surface), ξ, that is related to the radius of the annulus, r, the RMS slope at the wavelength scale, and the Hurst exponent:
The coherent backscattered field for each annulus is reduced, relative to that of a smooth plate, by the factor
where k = 2π/λ and ϕ is the radar incidence angle. The total scattered field is the integral over all annuli within a given uniformly illuminated area:
where Z is the vertical distance from the observer to the center of the target area, Eo is the illuminating field, J0 is a Bessel function of the first order, Re is an effective reflectivity, and we introduce the dimensionless variable = r/λ. max is thus the radius of the scattering area in multiples of λ (Figure 2). The dimensionless backscatter coefficient, σo, is defined by
where Ao is the scattering area [Ulaby et al., 1982]. Note that the definition of the backscatter coefficient is identical for coherent and incoherent echoes; only the scattered field changes, depending upon our surface scattering model. If we let the integral in (7) be given by K, the backscatter coefficient of the coherent component for the fractal surface is:
Note that, in the limit of a flat plate, Re2 is identical to the Fresnel normal reflection coefficient, ρo. This expression represents a correction to equations (22)–(24) of Shepard and Campbell , which did not properly normalize for the scattering area.
 We may obtain some physical insight from the behavior of a mirror-like smooth surface, which will exhibit the maximum possible normal-incidence backscatter coefficient for any given reflectivity. A finite smooth disc is characterized by sλ = 0, yielding
When ϕ = 0:
This result is in agreement with the derivation cited by Ulaby et al. [1982, equation [10.22]] for the behavior of a flat, conducting circular plate, which has a normal-incidence radar cross section (m2) of 4πAo2/λ2. The backscatter coefficient is thus 4πAo/λ2. Substituting Ao = πrmax2, max = rmax/λ, and adding the effective reflectivity of a dielectric plate, we obtain (11). Note that a fully illuminated smooth spherical scatterer with radius ≫λ has a backscatter coefficient independent of its radius (σo = ρo), since the coherently scattering region at the front cap expands at the same rate as the total projected area of the sphere [Bohren and Huffman, 1983].
 For a mirror-like surface, the normal-incidence backscatter coefficient for the coherent echo increases with the scattering area. On a rough surface (sλ > 0), σo is reduced from the maximum value given by (10), similar to the exponential decline in coherent reflectivity noted by Barrick and Peake . As the scattering area increases, even surfaces with very low sλ will eventually exhibit a negligible coherent backscatter coefficient. The reason lies in the nature of the coherent summation and the power law increase in RMS height with distance from any given point (5). For any choice of sλ and H, there is a value of max beyond which the local coherent contributions begin to destructively interfere. The analogy is to consider the area around a given point as an antenna of some surface tolerance. While we may achieve strong backscatter gains from any one such aperture, a group of antennas with arbitrary phase would have no net coherent component. This arbitrary phase is introduced as the RMS height of annuli about some center point exceeds a fraction of the wavelength.
3.2. Scattering Models for the Incoherent Component
 The incoherent component of radar scattering may be treated with a variety of methods, including the small-perturbation, integral equation, and Kirchhoff models [e.g., Barrick and Peake, 1967; Ulaby et al., 1982; Hagfors, 1964; Fung et al., 1992; Franceschetti et al., 1999]. The latter approach requires that the surface be smooth (“quasi-specular”) on horizontal scales comparable to the illuminating wavelength. For backscatter, the reflected power is the incoherent sum of echoes from surface patches oriented perpendicular to the incident beam, which in the high-frequency limit is represented by the slope distribution along a profile line [Muhleman, 1964; Simpson and Tyler, 1982].
 In the widely used Hagfors  model, the surface is described by the root-mean square of the surface tilt distribution, θrms, and the Fresnel normal reflectivity ρo. Hagfors assumed an exponential form for the surface height autocorrelation function, approximating its derivative near the origin, to obtain
where C is taken to be 1/tan2(θrms). The specific horizontal scale linked with θrms is not defined, and in some instances the surface is taken to have a scale-invariant RMS slope (i.e., H = 1 in (3)) for all length scales greater than a few times the incident wavelength. At normal incidence,
At normal incidence, only the zeroth-order term of the summation in (14) is nonzero. For H = 0.25,
for H = 0.5,
and for H = 1.0,
 The normal-incidence incoherent backscatter coefficient for a self-affine surface is thus analogous to the Hagfors model when H = 1, if we assume that C = sλ−2. In contrast, the angular scattering law derived from (14) is generally similar to the Hagfors solution when H = 0.5. These properties may be attributable to the nonfractal nature of a surface with an exponential autocorrelation function. The power spectrum of such a surface is
where Lc is the correlation length [Li et al., 2002]. For high frequencies, this approaches an H = 0.5 behavior, which may explain the correlation in scatter-law shape between the Hagfors and fractal models. The normal-incidence behavior given by (17) is typical of geometric optics models, perhaps independent of the functional form of the surface slope distribution at other angles.
3.3. Backscatter as a Function of Illuminated Area and Roughness
 Both the coherent and incoherent scattering mechanisms have a relationship to the scale of the scattering area. For the coherent return, the dependence on scale is represented by the dimensionless max term in (9). As we will show below, the maximum coherent backscatter coefficient, for some H and sλ, occurs over a narrow range of max values. Models for the incoherent component, such as (14), are predicated on the assumption that the surface is statistically well sampled, such that the predicted σo value is invariant with scattering area. In previous studies of scattering from gently undulating surfaces, this constraint is expressed as L ≫ Lc, where L is the lateral dimension of the scattering region [Ulaby et al., 1982].
 As H or sλ declines, a self-affine surface becomes more smooth with respect to a given radar wavelength, eventually approaching a mirror-like behavior. The assumptions underpinning (14) break down when the scattering region does not adequately sample the surface slope distribution (i.e., “facet” population). For any given values of H and sλ, (14) yields a value for σo(0), but for a very smooth surface this predicted backscatter coefficient may exceed that of a planar target of equal area (11). Since the planar target sets a maximum limit on the backscatter from a natural surface, we may set a lower limit on to avoid physically unrealistic situations:
 To illustrate these behaviors, we compare the backscatter coefficients of the coherent (9) and incoherent (14) components, at normal incidence, as a function of scattering area and roughness. For each pair of H and sλ values, we determine σBo(0) from (14), and a minimum scattering area from (19). We then calculate the coherent echo as a function of max, using a numerical integration of (7). These results are shown in Figure 3, where we have assumed that Re2 ∼ ρo, and set both equal to 0.15. The squares in each plot represent the incoherent backscatter coefficients, plotted at their corresponding minimum max values. The curved lines are the backscatter coefficients of the coherent echo for each value of sλ.
 Interestingly, the minimum max values from (19) are just below the scale at which the coherent echo is a maximum for any chosen roughness parameters. The coherent component declines relatively quickly for other values of max. The minimum scattering area required for validity of (14) increases for lower values of H and sλ; smoother surfaces therefore require a larger surface resolution cell. For H ≥ 0.5, and reasonable values of sλ, the required horizontal dimension of the scattering area is typically <103λ, but for lower Hurst exponents and low RMS slope the necessary scale can exceed 106λ. If a given radar observation yields a resolution cell whose maximum horizontal radius is less than that given by (19) for the particular roughness (H and sλ), the coherent model provides a more realistic estimate of the backscattered power.
4. Implications for MARSIS and SHARAD Observations of Mars
 The MARSIS and SHARAD radar sounders employ low-frequency (1.8–20 MHz) signals to penetrate the upper crust of Mars and reveal buried geologic interfaces. From the discussion above, we suggest that surface coherent returns will be strongly dependent upon the size of resolved surface elements with respect to the radar wavelength. The maximum horizontal extent of these resolved surface elements varies considerably over the illuminated area (Figure 1). For SHARAD, the free-space range resolution, Δr, is 30 m (10 MHz bandwidth), while for MARSIS Δr = 300 m (1 MHz bandwidth). Doppler processing of the radar echoes yields along-track spatial resolution, Δa, of 0.3–1.0 km for SHARAD and 5–9 km for MARSIS.
 The first sounder range “bin” contains echoes from a region of radius
where h is the spacecraft altitude. Both Mars sounders operate at minimum altitudes near 300 km, for which ro = 4.2 km for SHARAD and 13.4 km for MARSIS. The corresponding max values (ratio of resolution cell radius to radar wavelength) are 280 for SHARAD and 80–220 over the range of MARSIS wavelengths. Each succeeding range bin narrows, until at some point the projected cross-track spatial resolution becomes smaller than Δa.
 For example, the tenth range bin for SHARAD, corresponding to an incidence angle of 2.6°, has Δr = 0.7 km. The max value is related to one-half the maximum dimension of the scattering area (Figure 2), so in this case max ∼ 23. For areas further from the subradar point, the azimuth resolution will eventually set a minimum limit on max, with values of ∼10–30 (Δa = 0.3–1.0km). For MARSIS, the along-track resolution determines max for all but the first range bin, with values of 80–150 for the 60-m wavelength, and 30–54 for λ = 167 m. In contrast, the Magellan altimeter system resolved cells with max > 8,000; at these scales coherent echoes are likely negligible [Ford and Pettengill, 1992].
 Values of the Hurst exponent and wavelength-scale slope for Mars at scales of 15–170 m must, at present, be inferred from analysis of MOLA data with ∼300 m horizontal sampling. On the basis of these data, the Martian surface is characterized by an average H value of 0.7–0.8, with values of H < 0.5 confined primarily to the midlatitude northern plains [Orosei et al., 2003]. The issue of whether the power law behavior of surface roughness extends to smaller scales (e.g., 15 m for SHARAD) remains open to debate, but these values are very likely reasonable within the wavelength range of the MARSIS sounder [Campbell et al., 2003].
 Because the sounder observations may encompass surface resolution elements with relatively small max values (10–300), the validity of (14) for incoherent returns, and the contribution of coherent scattering, may vary dramatically with roughness and viewing geometry (i.e., the position of a reflecting surface area in delay-Doppler space). For this range of max, and the data shown in Figure 3, we may define some broad guidelines for the SHARAD and MARSIS observations:
 (1) H ∼ 1: the incoherent scattering model is valid for sλ > 0.5°, and coherent echoes may be significant if sλ < 1°.
 (2) H ∼ 0.5: the incoherent model is valid for sλ > 2°, and coherent echoes may be significant if sλ < 3°.
 (3) H = 0.25: the incoherent model is valid for sλ > 4°, and coherent returns may be significant if sλ < 5°.
 Synthetic fractal profiles that correspond to these upper limits on roughness for the coherent scattering model are shown in Figure 4. Note that higher values of H lead to more rapid increases in roughness with horizontal scale, so the roughness at the wavelength scale must be commensurately lower to permit significant coherent returns for a specific range of max values. Because surfaces with lower values of H have stronger coherent echoes for any value of sλ, we expect that such behavior will be most often observed in the smooth northern plains of Mars.
 We have examined the relative importance of coherent and incoherent echo components in near-nadir scattering from self-affine surfaces. These results agree with previous models that show the coherent backscatter coefficient is a function of both surface roughness and scattering area. Models for the incoherent echo from fractal surfaces are limited in their application to scattering areas large enough to sample the surface slope distribution. For smaller scattering areas, a coherent model for scattering from a slightly rough region is more realistic. The MARSIS and SHARAD sounders resolve surface areas on the order of 10–300 wavelengths in scale, in contrast to radar altimeters such as Magellan that have resolution cells of >8,000 λ. As a result, coherent echoes may be present in sounder returns when wavelength-scale RMS slope values are less than 1°–5°, with lower cutoff values for increasing H. Very smooth surfaces, though limited in occurrence on Mars, may thus exhibit near-nadir scattering behavior that varies as a function of the horizontal dimensions of delay-Doppler resolution elements.
 The authors thank P. Ford, T. Hagfors, R. Phillips, and R. Simpson for helpful discussions and comments on the manuscript. The comments of two reviewers are also appreciated. This work was supported in part by grants from NASA's Planetary Geology and Geophysics Program (BAC and MKS) and the NASA Mars Reconnaissance Orbiter project (BAC).