In von Karmam-type random media, when the center wavenumber is higher than the corner wavenumber of the PSDF and the phase shift across the correlation length is not small enough, the conventional Born approximation is not applicable. To overcome this difficulty, Chapter 7 presents an ongoing study of a hybrid MC simulation using the spectrum division method for scalar/vector wavelet propagation. Taking the center wavenumber as a reference, we divide the PSDF into two parts, then we serially use the Born wide-angle scattering for the high-wavenumber spectral component and the Eikonal narrow-angle ray bending and travel distance fluctuation for the low-wavenumber spectral component. For the elastic case, the synthesized three-component RMS velocity amplitude time traces are compared with the ensemble average of the FD simulation results in realized random elastic media.

Chapter 4 introduces phenomenologically the radiative transfer equation of the directional distribution of the energy density for a given anisotropic scattering coefficient of scalar waves in random media. We solve the radiative transfer equation analytically by using the Legendre expansion for isotropic radiation from a point source. By probabilistically interpreting the Born scattering coefficient and the Eikonal angular spectrum function and the traveling distance fluctuation for scalar waves, we construct the corresponding pseudo-random number generators, where the rejection sampling method is introduced. Then, we synthesize the space–time distribution of the energy density for isotropic radiation from a point source using the MC simulation and compare it with the analytical solution of the radiative transfer equation.

Chapter 2 presents the isotropic scattering model, which is mathematically tractable. We introduce the basic concepts of the probability density function of scattering angles and the inverse transform sampling method for constructing the corresponding pseudo-random number generator of scattering angles. We then present the MC simulation of the space–time distribution of the energy density for isotropic radiation from a point source, which is compared with the analytical solution of the radiative transfer equation.

Chapter 3 investigates the applicable conditions of the Born and Eikonal approximations for scalar wave scattering in random velocity fluctuations characterized by the power spectral density function. The Born approximation leads to the anisotropic scattering coefficient, which represents the directional scattering power per unit volume. The Eikonal approximation leads not only to the angular spectral function that governs narrow-angle ray bending but also to travel–distance fluctuation.

Chapter 5 presents the realization of random media for a given power spectral density function for FD simulation of scalar wavelet propagation. We verify the MC simulation results with the ensemble average of the FD simulation results in realized random media for a given power spectral density function.

Chapter 1 introduces the historical background of the RTT approach with random media modeling for the study of the medium heterogenetity of the solid Earth. It also presents a summary of recent measurements of the random inhomogeneity spectra and scattering characteristics of seismic waves in the solid Earth.

Chapter 6 studies vector wave scattering in random elastic media. The Born approximation leads to PP, PS, SP, and SS scattering coefficients, from each of which we construct the corresponding PRNG of scattering angles. Using these in MC simulations, we synthesize three-component RMS velocity amplitude time traces for the radiation from a point shear dislocation (PSD) source. The simulation results are compared with the ensemble average of FD simulation results in random elastic media for a given power spectral density function.

In the epilogue, we discuss unresolved issues and possible future developments.