Three-Dimensional Random Wave Coupling Along a Boundary and an Associated Inverse Problem

Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 39-65, March 2024.
Abstract. We consider random wave coupling along a flat boundary in dimension three, where the coupling is between surface and body modes and is induced by scattering by a randomly heterogeneous medium. In an appropriate scaling regime we obtain a system of radiative transfer equations which are satisfied by the mean Wigner transform of the mode amplitudes. We provide a rigorous probabilistic framework for describing solutions to this system using that it has the form of a Kolmogorov equation for some Markov process. We then prove statistical stability of the smoothed Wigner transform under the Gaussian approximation. We conclude with analyzing the nonlinear inverse problem for the radiative transfer equations and establish the unique recovery of phase and group velocities as well as power spectral information for the medium fluctuations from the observed smoothed Wigner transform. The mentioned statistical stability is essential in monitoring applications where the realization of the random medium may change.