Abstract

Realistic anelastic attenuation can be incorporated rigorously into finite difference and other numerical wave propagation methods using internal or memory variables. The main impediment to the realistic treatment of anelastic attenuation in 3D is the very large computational storage requirement imposed by the additional variables. We previously proposed an alternative to the conventional memory-variable formulation, the method of coarse-grain memory variables, and demonstrated its effectiveness in acoustic problems. We generalize this memory-efficient formulation to 3D anelasticity and describe a fourth-order, staggered-grid finite-difference implementation. The anelastic coarse-grain method applied to plane wave propagation successfully simulates frequency-independent Qp and Qs. Apparent Q values are constant to within 4% tolerance over approximately two decades in frequency and biased less than 4% from specified target values. This performance is comparable to that achieved previously for acoustic-wave propagation, and accuracy could be further improved by optimizing the memory-variable relaxation times and weights. For a given assignment of relaxation times and weights, the coarse-grain method provides an eight-fold reduction in the storage requirement for memory variables, relative to the conventional approach. The method closely approximates the wavenumber-integration solution for the response of an anelastic half-space to a shallow dislocation source, accurately calculating all phases including the surface-diffracted SP phase and the Rayleigh wave. The half-space test demonstrates that the wave field-averaging concept underlying the coarse-grain method is effective near boundaries and in the presence of evanescent waves. We anticipate that this method will also be applicable to unstructured grid methods, such as the finite-element method and the spectral-element method, although additional numerical testing will be required to establish accuracy in the presence of grid irregularity. The method is not effective at wavelengths equal to and shorter than 4 grid cell dimensions, where it produces anomalous scattering effects. This limitation could be significant for very high-order numerical schemes under some circumstances (i.e., whenever wave-lengths as short as 4 grids are otherwise within the usable bandwidth of the scheme), but it is of no practical importance in our fourth-order finite-difference implementation.

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