Formulation of the multiple non-isotropic scattering process in 2-D space on the basis of energy-transport theory

Abstract

SUMMARY For the synthesis of high-frequency seismogram envelopes, the energy-transport theory has provided a framework for the multiple isotropic scattering process in distributed scatterers. By introducing the angular distribution of energy density, we here propose a formulation of the multiple non-isotropic scattering process in 2-D space. The energy-transport equation is written as simultaneous linear equations by using the Fourier transform in space, the Laplace transform in time and the Fourier series expansion with respect to scattering angle. When the non-isotropic scattering is written in a finite number of terms of the Fourier series, we obtain the spatio-temporal distribution of the energy density by solving linear equations with a finite number of unknowns. The lowest term of the Fourier series expansion with respect to scattering angle corresponds to isotropic scattering. This formulation is valid for any type of non-isotropic scattering. In the case of strong forward scattering, the energy density is found to be spatially rather uniform around the source at large lapse time. This is different from the spatial distribution for multiple isotropic scattering, which has a concentration of energy around the source.