Mean and variance of the ultrasonic signal from a scatterer beneath a rough surface: Theory

Abstract

A simple model is presented that predicts the ultrasonic signal from a weakly scattering inclusion beneath a randomly rough water–solid interface that is insonified by a normally oriented phase-sensitive transducer. The rough surface is modeled by a spatially uniform, normally distributed random process with a Gaussian autocorrelation function. The effects of roughness are introduced via the phase-screen approximation. Analytic series solutions are derived and presented for the mean, mean square, and mean absolute square of the pulse-echo signal. Rough surface effects are shown to depend on the depth of the scatterer beneath the surface. The depth dependence of the rough-surface effects is shown to arise from the statistical dependence of the phase shifts that arise due to the entry and exit of the ultrasound through the rough surface. The roughness-induced loss of the average signal is found to be greatest for scatterers immediately below the surface. The average roughness-induced signal loss is least for scatterers that are farthest from the surface for unfocused transducers or at the focal point for focused transducers and can be modeled by a simple frequency-dependent transmission coefficient in these cases. The variance in the signal increases with the frequency and is relatively large for scatterers immediately below the surface. The variance is smallest at the focal point, and in the far field for unfocused transducers. The variance is greatest for the scatterers with the smallest lateral extent and least for scatterers with the largest lateral extent. Finally, it is found that the power backscattered from a small near-surface scatterer is actually increased on the average by the rough surface.