Radiative transfer in 1D and connection to the O’Doherty-Anstey formula

There is a growing interest in incorporating multiply scattered waves into modeling the Earth’s interior using radiative transfer. We examine radiative transfer in a layered medium with general scattering and directional properties of the source. This allows us to demonstrate in detail the nature of energy propagation in the presence of strong scattering. At its most basic level, radiative transfer predicts that, after a distance known as the mean free path, the wavefield breaks into a coherent, or wave-like part and an incoherent, or diffusive flow. The dynamic properties of both aspects are linked. For 1D point scatterers, or thin beds, we derive the equivalence of the exponential decay of the transmitted wave predicted by the O’Doherty-Anstey formula with the coherent, or direct, wave obtained from the radiative transfer equation. The equivalence shows an underlying relationship between mean field theory and radiative transfer. Turning to the incoherent wave intensity, we make the well-known diffusion approximation to the late-time radiative transfer behavior. A finite-difference simulation of the wave equation with random scatterers corroborates the theoretical results for the incoherent energy.