The optimum source depth distribution for reverberation inversion in a shallow-water waveguide

An approach for extracting the modal backscattering matrix from reverberation data in shallow water was proposed recently (Shang, Gao, and Tang, 2002). The kernel matrix of the inversion is constructed using the square of the modal functions. The singularity of this matrix (or the stability of the inversion) is the crucial issue to be considered. In this paper, we discuss this issue analytically for a Pekeris waveguide with the limited mode number M. The method that we used for singularity analysis is to calculate the maximum value of the determinant of this kernel matrix. We found that there is an optimum source depth distribution corresponding to the maximum value of the determinant of the kernel matrix. That means that, by choosing the optimum source depth distribution, we can get the most stable inversion. The conclusion is that, under a quite tolerant condition, the matrix is not singular, and the backscattering matrix can be inverted. Published in Russian in Akusticheskiĭ Zhurnal, 2007, Vol. 53, No. 5, pp. 682–686. The text was submitted by the authors in English.