Most probable ballistic waves in random media: a weak-fluctuation approximation and numerical results

Abstract

A statistical approach to describe the time-space pulse evolution in single realizations of two- and three-dimensional random media is presented. For this purpose, we construct the ensemble-averaged wavenumber of a plane wave. The wavenumber is obtained by a combination of the Rytov and Bourret approximations for the log-amplitude and phase increment, both related by the Kramers-Kronig relations. The real part of the wavenumber is related to the phase velocity, whereas its imaginary part denotes the attenuation coefficient of the ballistic wavefield. The validity range is limited by the weak-fluctuation regime, but has practically no restrictions in the frequency domain. We show that this wavenumber is a partially self-averaged quantity. This means that the wavenumber is a Gaussian-distributed quantity exhibiting decreasing relative fluctuations while the wave travels a finite distance such that it is still in a weak-fluctuation regime. Then, the ensemble-averaged wavenumber carries the information of maximum probability realizations. The Green's function constructed with this wavenumber corresponds to the most probable ballistic wave. Using this Green's function, it is possible to characterize single-wavefield realizations in single realizations of random media. We verify the concept and analytic results with the help of finite-difference simulations of waves propagating in two-dimensional random media. We demonstrate the partial self-averaging property of the wavefield attributes by computing their relative standard deviations as a function of propagation distance L. Within the weak-fluctuation regime, these relative standard deviations decrease as 1/√L, indicating that the process of self-averaging takes place. Reconstructing the joint probability density of amplitudes and phases of ballistic waves allows us to identify most probable ballistic waves. They show a good agreement with the analytic results.