Multistatic Response Matrix: Statistical Structure

Abstract

In multistatic wave imaging, waves are emitted by a set of sources and they are recorded by a set of sensors in order to probe an unknown medium. The responses between each pair of source and receiver are collected and assembled in the form of the multi-static response (MSR) matrix. The indices of the MSR matrix are the index of the source and the index of the receiver. When the data are corrupted by additive noise, we study the structure of the MSR matrix using random matrix theory.We start this chapter by presenting an acquisition scheme, known as Hadamard technique, for noise reduction. Hadamard technique allows us to acquire simultaneously the elements of the MSR matrix and to reduce the noise level. The feature of this technique is to divide the variance of the noise by the number of sources. Then we investigate the statistical distributions of the singular values of the MSR matrix in the presence of point reflectors. In the presence of small inclusions, we find the statistical distribution of the angles between the left and the right singular vectors of the noisy MSR matrix with respect to those of the unperturbed one. Our results in this chapter will be useful for designing detection tests, estimating the number of point reflectors or inclusions in the medium, and localizing them.