Formulation of the multiple non-isotropic scattering process in 3-D space on the basis of energy transport theory

Abstract

Summary High-frequency seismograms of local earthquakes are considered to consist of incoherent body waves scattered by random inhomogeneities in the earth medium. For the study of seismogram envelopes, we can describe the scattering process on the basis of energy transport theory by representing the random inhomogeneities as distributed point-like scatterers. By introducing the concept of directional distribution of energy density, we propose a formulation of the multiple non-isotropic scattering process in 3-D space. We suppose that the axially symmetric non-isotropic scattering is described by a directional scattering coefficient, the medium is characterized by one wave velocity, and the source radiation is spherical. In addition to the Fourier transformation in space and the Laplace transformation in time used to solve for energy density in isotropic scattering media, we use a spherical harmonic series expansion in solid angle. Then, the energy transport equation given as an integral can be written as simultaneous linear equations, where coefficients are given by the Wigner 3-j symbols and the spherical harmonic expansion coefficients of the directional scattering coefficient. The lowest term of the spherical harmonic series corresponds to isotropic scattering. When the directional scattering coefficient is written as a finite-length spherical harmonic series, the simultaneous linear equations can be solved for a finite number of unknowns and we can obtain the spatio-temporal distribution of energy density. The numerical calculation for a case with strong forward scattering shows a uniform distribution of energy density around the hypocentre at large lapse times.