Estimation of finite-frequency waveforms through wavelength-dependent averaging of velocity

The wavelength-smoothing (WS) method was introduced recently (Lomax 1994) as a method for the rapid estimation of the principal features of broad-band wave phenomena in realistic, complicated structures. The WS method is based on the concept that waves at a particular frequency and corresponding wavelength respond to a complicated velocity distribution as if the distribution were smoothed over about a wavelength. This method reproduces several finite-frequency wave phenomena, but has not been given a formal theoretical justification. Here, we use scattering theory and a local, plane-wave approximation to develop a wavelength-averaging (WA) method for modelling finite-frequency wave propagation. The new WA method is similar to the WS method in concept and implementation, but is valid only in a more limited geometry of velocity heterogeneity. In particular, the new formulation performs well for models with complex, but smoothly varying, velocity variations (‘quasi-random’ models), but does less well in models with extensive regions of slowly varying velocity that are separated by strong gradients in velocity (‘deterministic’ models). This limits application of the current formulation of the WA method to predominantly quasi-random structures, although such models may be useful in many problems, particularly for Monte-Carlo-based inversion methods requiring fast forward calculations.