Parabolic approximations to the time-independent elastic wave equation

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Abstract
A splitting matrix method is used to derive two parabolic-approximation partial differential equations to the three-dimensional, linear, elastic wave equation in isotropic, inhomogeneous media. The derivation is valid for media whose Lamé parameters vary slowly on the length scale of the wavelength of the elastic waves. Next an integral form of the full wave equation is derived based on the splitting matrix and the parabolic approximation solution. Iteration of this equation gives a three-dimensional vector-valued series which generalizes the one-dimensional Bremmer series (which was used in the study of second-order ordinary differential equations). These results are expected to have applications to geophysical modeling and nondestructive evaluation.

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