Hybrid synthesis of scalar wave envelopes in two‐dimensional random media having rich short‐wavelength spectra

Abstract

Wave trains in high‐frequency seismograms of local earthquakes are mostly composed of incoherent waves that are scattered by distributed heterogeneities within the lithosphere. Their phase variations are very complex; however, their wave envelopes are systematic, frequency‐dependent, and vary regionally. Stochastic approaches are superior to deterministic wave‐theoretical approaches for modeling wave envelopes in random media. The time width of a wavelet is broadened with increasing travel distance mostly because of diffraction caused by the long‐wavelength components of random velocity inhomogeneity. The Markov approximation for the parabolic wave equation is effective for the synthesis of envelopes for random media whose spectra are poor in short‐wavelength components; however, we have to consider the contribution of large‐angle nonisotropic scattering if the random media are rich in short‐wavelength inhomogeneities. Multiple nonisotropic scattering can be reliably modeled as isotropic scattering by using an effective isotropic scattering coefficient given by the momentum transfer scattering coefficient, which is a reciprocal of the transport mean free path. It is mostly controlled by the short‐wavelength spectra of random media. We propose a hybrid method for the synthesis of whole wave envelopes that uses the envelope derived from the Markov approximation as a propagator in the radiative transfer integral equation for isotropic scattering. The envelopes resulting from the hybrid method agree well with ensemble average envelopes calculated by averaging envelopes from individual finite difference simulations of the wave equation for a suite of random media.