The statistical properties of echoes diffracted from rough surfaces

Abstract

This work aims to provide the statistical theory behind recent wide-angle pulsed radar experiments to determine the topography and roughness of (a) polar glacier beds from measurements near the snow surface, and (b) the lunar surface from satellites in close orbit. W e analyse the statistical properties of the echo received back at the source when a quasimonochromatic pulse of radiation, isotropic in direction, is reflected from a rough surface which is a random perturbation of a horizontal plane. Kirchhoff diffraction theory is employed, so that multiple scattering and shadowing are neglected. But the r.m.s. surface height is unrestricted, and closed formulae are obtained which are valid through the transition from coherence (flat-mirror surface) to incoherence (surface of high relief, to which geometrical optics is applicable). For fixed source-receiver height the average echo wavefunction, and the average echo power, are calculated as functions of delay time, and the echo autocorrelation function is calculated as a function of the separation of two source-receiver points at fixed delay time. The duration of the echo is calculated, and the long-wave limit, the smooth-surface limit, the Fraunhofer limit, the continuous wave limit and the geometrical-optics limit are examined. A method is suggested for inferring the statistics of the surface from measurements of the average echo power. The theory of random noise is applied to the fluctuations about the calculated averages which occur as the source-receiver is moved horizontally. These fluctuations constitute 'spatial fading', and we calculate several measures of the spatial fading rate for the echo wavefunction and the time-smoothed echo power, as well as a measure of the degree of spatial periodicity of the fading. Finally, an estimate is made of the smallest detectable horizontal displacement of the source-receiver relative to the rough surface.