GEOMETRIC OPTICS AND ENHANCED BACKSCATTER FROM VERY ROUGH SURFACES

A simple explanation is given for the phenomenon of enhanced backscatter from randomly rough surfaces in the geometric-optics or infinite-wave-number limit. The problem treated is that of a scalar wave reflected at a rough boundary, for which either a Dirichlet or Neumann boundary condition is satisfied and the surface is assumed to have a Gaussiun correlation function and to obey Gaussian statistics. Only the one-dimensional case is treated, although a similar approach is possible for two-dimensional surfaces. By considering the governing integral equation on the surface, it is shown by stationary phase arguments that there is an order-one correction to the Kirchhoff approximation, in the limit as the incident wave number tends to infinity, if the surface roughness is not small compared with a typical horizontal scale (taken here to be the correlation length). This finding is confirmed by numerical ray-tracing experiments. These, in turn, are validated by comparison with Monte Carlo solutions of the governing integral equation for moderate surface roughness and finite, but large, incident wave number. Direct consideration of the mechanisms for the scattering shows that the backscatter phenomenon is caused by multiple scattering, with the most significant contribution given by increasingly high orders of scatter as the rms surface slope is increased. For exact backscatter to occur, a ray must be normally incident upon the (local) surface at the m(th) reflection, where m = 1.2, .... With this condition, it is possible to derive an elementary formula for the surface slope at the point where the ray first intersects the surface, which will give rise to direct backscatter. Using these values and the probability density of surface slopes, I give an approximate model for the variation of backscatter with rms surface slopes and incident angle.