Fig 1.: Solution to the 2D acoustic wave equation in a medium with constant density and randomly varying velocity. 

Posted by Manuel Alejandro Jaimes Caballero

As waves propagate in random media, they exhibit complicated patterns due to the interaction with inhomogeneities (see Fig 1). Examples of random media include volcanoes, well logs, and rock samples. In random media, it is not always possible to compute the amplitudes and phases of wavefields because one may not know the exact location of the inhomogeneities. For instance, one could think of two samples of granite with mineral inclusions, which look geologically similar but whose inclusions slightly change in location and shape. One alternative to describe wave propagation in random media, is to use energies rather than amplitudes and phases, if one knows the autocorrelation function which describes the inhomogeneities. To describe the energy propagation one can make use of the radiative transfer equation, which is based on energy conservation. Fig 2 illustrates, heuristically, the basis of the radiative transfer equation. In this equation one assumes that there is a background velocity v, and that the autocorrelation function which characterizes the random medium is encoded in the loss and gain terms. Developing numerical solutions for the radiative transfer equation is of interest to understand wave propagation in random media and has applications in the estimation of scattering properties and coda wave interferometry for imaging time-lapse changes. In this article  Dr. Roel Snieder and I construct a time-stepping algorithm to solve the acoustic radiative transfer equation in two dimensions for a medium with arbitrary (smooth) initial conditions, angle-dependent scattering, and non-uniform scatterer density.

 Fig 2.: Heuristic illustration of the 2D acoustic radiative transfer equation. In the absence of anelastic attenuation, the wave energy is conserved, and it only changes direction.