Acoustic wavefield propagation using paraxial extrapolators

Modeling by paraxial extrapolators is applicable to wave propagation problems in which most of the energy is traveling within a restricted angular cone about a principle axis of the problem. Frequency domain finite-difference solutions are readily generated by using this technique. Input models can be described either by specifying velocities or appropriate media parameters on a two or three dimensional grid of points. For heterogeneous models, transmission and reflection coefficients are determined at structural boundaries within the media. The direct forward scattered waves are modeled with a single pass of the extrapolator operator in the paraxial direction for each frequency. The first-order back scattered energy can then be modeled by extrapolation (in the opposite direction) of the reflected field determined on the first pass. Higher order scattering can be included by sweeping through the model with more passes.

The chief advantages of the paraxial approach are 1) active storage is reduced by one dimension as compared to solutions which must track both up-going and down-going waves simultaneously, thus even realistic three dimensional problems can fit on today's computers, 2) the decomposition in frequency allows the technique to be implemented on highly parallel machines such the hypercube, 3) attenuation can be modeled as an arbitrary function of frequency, and 4) only a small number of frequencies are needed to produce movie-like time slices.

By using this method a wide range of seismological problems can be addressed, including strong motion analysis of waves in three-dimensional basins, the modeling of VSP reflection data, and the analysis of whole earth problems such as scattering at the core-mantle boundary or the effect of tectonic boundaries on long-period wave propagation.