Hagfors' law revisited

Hagfors' scattering law, σHo(θ), is in wide use in connection with the study of backscatter data from planetary surfaces because it provides good agreement with a variety of observations. The surface root-mean-square slope inferred on the basis of σHo(θ) is customarily taken as C-1/2, where C is the shape parameter in σHo(θ). The relationship between the surface slope and C is indefinite, however, because of the indeterminateness of the surface scales contributing to the scattering process. Moreover, the horizontal scale of the inferred slope obtained is not specified. As a consequence of limitations in the Kirchhoff approximation on which it is predicated, σHo (θ) does not conserve energy. The use of a fractional Brownian fractal surface model leads to a scattering law with the same functional form as σHo(θ) when the Hurst exponent characterizing the fractal model is 1/2. Fractal-based scattering laws, derived by applying the Kirchhoff approximation, suffer the same deficiency with regard to conservation of energy. In contrast to σHo(θ), slope information for fractal-based laws is explicit with respect to horizontal scale. Both σHo(θ) and fractal-based laws require that theilluminated surface area exceeds a certain value, which is a function of the electromagnetic wavelength and surface parameters, in order to reduce the surface radar cross section overestimation error, introduced by a mathematical approximation, below some specified value. This requirement may be necessary to take into account in experiments where the radar resolution cells are comparable in size to the wavelength, such as in Mars Advanced Radar for Subsurface and Ionosphere Sounding (MARSIS). Copyright 2006 by the American Geophysical Union.