Radiative transfer in layered media and its connection to the O’Doherty-Anstey formula

We examine radiative transfer theory, which accounts for the multiple scattering of waves, in a layered medium composed of randomly placed thin beds excited by a 1D source. At its most basic level, radiative transfer predicts that the wavefield separates into a coherent, or wavelike, part and an incoherent, or diffusive, flow after a length scale known as a mean free path. The dynamic properties of the coherent and incoherent wavefield are linked. For 1D Rayleigh scatterers, or thin beds, we show that the exponential decay of the coherent wave predicted by radiative transfer corresponds to the decay predicted by the O’Doherty– Anstey formula. This equivalence reveals an underlying relationship between radiative transfer and mean field theory. Finite-difference simulations of the scalar wave equation with randomly placed thin beds demonstrate the diffusive behavior of the incoherent energy at late times.