Synthesis of coda wave envelopes in randomly inhomogeneous elastic media in a half-space: Single scattering model including Rayleigh waves

Seismograms of local earthquakes consist of not only direct waves but also coda waves originating from scattering due to lithospheric inhomogeneities. As a mathematical basis for interpreting these seismograms, we present a synthesis of coda wave envelope propagating through randomly inhomogeneous medium in a half-space by considering elastic conversion scattering modes of body-to-body waves, Rayleigh-to-body waves, body-to-Rayleigh waves and Rayleigh-to-Rayleigh waves. We first evaluate scattering amplitudes of above scattering modes by elastic inhomogeneity localized in a block by applying the Born approximation to the elastic wave equation in the frequency domain. Scattering amplitudes of all conversion scattering modes are represented by the Fourier spectrum of the elastic inhomogeneity multiplied by the basic scattering pattern which determines the non-isotropic scattering pattern in the long-wavelength limit. Then, considering an ensemble of random media, we derive scattering coefficients which represent the scattering power per unit volume as ensemble averages. By summing up the scattered energy from distributed scattering blocks in a half-space for a point shear dislocation source with the omega-square model, we synthesize the coda-wave envelope for each scattering mode. Numerical evaluation of three-component MS envelopes for a shallow source shows strong frequency dependence. Though scattered body waves are dominant in high frequencies, Rayleigh-to-Rayleigh scattered waves with smaller decay with increasing lapse time become dominant for low frequencies.