Radiative transfer of elastic waves in two-dimensional isotropic scattering media: Semi-analytical approach for isotropic source radiation

We formulate the radiative transfer of P- and S-wave energy from an isotropicaliy radiating source in a two- dimensional infinite isotropic scattering medium. For a stable numerical calculation of seismogram envelopes, we take a semi-analytical approach: Energy densities of P and S waves are divided into three parts of the direct- wave terms, the single-scattering terms, and the multiple-scattering terms, and the first two terms and the last term are evaluated analytically and numerically, respectively. Concerning the single-scattering terms, the P- to-S and S-to-P single conversion scattering terms are expressed analytically with a complete elliptic integral of the first kind. The multiple-scattering terms are represented by a double integral with respect to frequency and wavenumber, and can be numerically evaluated by a discrete wavenumber summation and a Fast Fourier Transform. The results based on the numerical implementation are confirmed with an independent numerical calculation using the Monte Carlo method. Our formulation is also applied to consider the equilibration between P and S waves at larger lapse times. The equilibrated S-to-P energy ratio is reproduced and the equilibration time is first derived for two-dimensional cases. Our formulation will be a reference for the understanding of more complex cases.