Numerical simulation of attenuated wavefields using a Pade approximant method

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 dissipative. 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.