Radiative transfer and diffusion of waves in a layered medium: new insight into coda Q

Abstract

This paper is devoted to the study of the seismic coda in inhomogeneous media exhibiting a discontinuity of physical properties at a given depth. We focus on the problem of a layer overlying a half-space and analyse the precise effect of a contrast of wave velocity and/or scattering strengths between them. In order to model S-coda-wave envelopes, we solve the Radiative Transfer Equation by the Monte Carlo method, thereby neglecting the polarization (i.e. the acoustic approximation). We pay special attention to the transition towards the diffusion regime. Under the assumption of an almost isotropic intensity field, a Diffusion Equation can be derived from the Radiative Transfer Equation and we accurately determine the boundary conditions associated with our models. Analytical solutions of the Diffusion Equation have been obtained and systematically compared to the numerical solutions of the Radiative Transfer equation. We identify the domain of validity of the diffusion approximation which provides a simple analytical form for the decay in the late coda. We apply our theoretical investigations to the continental lithosphere. If the scattering strengths of the mantle and the crust are assumed to be of the same order, a velocity contrast at the Moho will - according to our theory - amplify the coda signal, since part of the energy is trapped in the crust. An amplification factor is defined and given explicitly as a function of the reflection coefficients and the velocity contrast at the Moho. The shape of the long time decay is of the algebraic form t(-3/2), like that of a uniform half-space. On the other hand, if the scattering strength of the mantle is small with respect to that of the crust, the decay in the diffusive regime is predicted to be of the form t-1exp(-2πft/Qc), where Qc is a function of the reflection coefficients at the Moho, the mean free path of waves in the crust, and frequency f. The coefficient Qc quantifies the rate at which the partially trapped energy leaks from the crust into the mantle. This formula has the same form as that proposed by Aki and Chouet (1975) to fit coda observations, which has since been widely used to deduce the Qc parameter. With realistic model parameters, we find that Qc roughly equals the parameter Qc deduced from observations. This shows that the effect of partial trapping of energy in the crust may be significant. Consequently, seismic albedos of the crust may have been underestimated in previous studies. In our theory, the energy decay of seismic coda waves is determined by the layered structure of the Earth, that is a highly heterogeneous crust overlying a rather homogeneous mantle. Such structure is confirmed by geological and geochemical studies.