Expansion of a high-frequency time-harmonic wavefield given on an initial surface into Gaussian beams

Summary. Realistic anelastic attenuation laws are usually formulated as convolution operators, but this representation is intractable for time-domain synthetic seismogram methods such as the finite difference method. An approach based on Padé approximants provides a convenient, accurate reformulation of general anelastic laws in differential form. The resulting differential operators form a uniformly convergent sequence of increasing order in the time derivative, and all are shown to be causal, stable and dissi- pative. In the special case of frequency-independent Q , all required coefficients for the operators are obtained in closed form in terms of Legendre polynomials.
Low-order approximants are surprisingly accurate. Finite-difference impulse responses for a plane wave in a constant- Q medium, calculated with the fifth-order convergent, are virtually indistinguishable from the exact solution. The formulation is easily generalized to non-scalar waves. Moreover, this method provides a framework for incorporating amplitude-dependent attenuation into numerical simulations.