Synthesis of scattered energy density for nonspherical radiation from a point shear-dislocation source based on the radiative transfer theory

Abstract

It is well known that S-coda wave amplitudes of local earthquakes smoothly decay with increasing lapse time and asymptotically converge to a common master curve irrespective of their epicentral distances. This property is thought to be the conceptual basis for the conventional coda normalization method, which has often been used for the estimation of amplitude attenuation and site amplification factors around the world. We note that it has also been widely accepted that the common decay curve is independent of the radiation pattern especially at long lapse times. However, there have been few theoretical attempts to clarify the way how S-coda wave envelopes neglect the radiation pattern of the source with increasing lapse time. Focusing on this problem, we try to formulate the multiple scattering process in a scattering medium for an impulsive radiation from a point-shear dislocation source on the basis of the radiative transfer theory. We assume that the inhomogeneous medium in a 3-D space is represented by a homogeneously random distribution of point-like scatterers and, for simplicity, that scattering is isotropic. The master equation is written in the form of a convolution integral equation for the energy density, which corresponds to the mean square seismogram amplitude. Introducing the spherical harmonics expansion for nonspherical radiation from the source, in addition to the Fourier transform in space and the Laplace transform in time, we are able to solve the integral equation in each spherical harmonics mode analytically. The inverse Fourier-Laplace transformation gives the spatiotemporal change in energy density for each spherical harmonics mode. As the order of spherical harmonics mode increases, the corresponding energy density decreases more rapidly with increasing lapse time. The energy density faithfully reflects the nonspherical radiation pattern of a point-shear dislocation immediately after the direct wave arrival. However, the direction dependence diminishes as lapse time increases, and the energy density asymptotically converges to that for a spherically symmetric radiation, which corresponds to the lowest spherical harmonics mode. At a distance of the mean free path from the source, the difference between the maximum energy density and the minimum energy density at the same lapse time becomes less than 3% when the lapse time exceeds twice the travel time of the direct wave.