Efficient Simulation of Constant Q using Coarse-Grained Memory Variables

Steven M. Day

Published August 1998, SCEC Contribution #391

Improvements in computing speed have progressively increased the usable bandwidth of seismic wave-field simulations computed with time-stepped numerical schemes (e.g., finite difference, finite element, pseudospectral). As computational bandwidth increases, anelastic losses become increasingly significant for some important applications such as earthquake ground-motion modeling, whole earth seismogram simulation, and exploration seismic profile modeling, and these losses need to be included in the simulations. As bandwidth increases, however, the memory variables necessary to incorporate realistic anelastic losses account for an increasing proportion of total computational storage requirements, a consequence of the broad relaxation spectrum of typical earth materials. To reduce these storage requirements, we introduce a new method in which the memory variables are coarse grained, that is, redistributed in such a way that only a single relaxation time is represented at each node point (and therefore a single memory variable per stress component is required). Guided by a perturbation analysis, we effect this redistribution in such a way that spatial variability of this single relaxation time simulates the full relaxation spectrum. Such coarse graining reduces memory-variable storage requirements by a factor of 8 for 3D problems or a factor of 4 for 2D problems.

In fourth-order finite-difference computations for the 3D acoustic-wave equation, the method simulates frequency-independent Q within a 3% tolerance over 2 decades in frequency, and it is highly accurate and free of artifacts over the entire usable bandwidth of the underlying finite-difference scheme. These results should also hold for the elastodynamic equations. The method is readily generalized to approximate specific frequency-dependent Q models such as power laws or to further reduce memory requirements. In its present implementation, the main limitation of the method is that it generates artifacts at wavelengths equal to 4 grid cell dimensions and shorter, which may, in some limited circumstances, overlap the usable bandwidth of very high-order finite-difference and/or pseudospectral schemes.

Key Words
numerical models, three-dimensional models, finite difference analysis, stress, statistical analysis, elastic waves, relaxation, two-dimensional models, attenuation, finite element analysis, Q, ground motion, anelasticity, propagation, seismic waves, earthquakes

Day, S. M. (1998). Efficient Simulation of Constant Q using Coarse-Grained Memory Variables. Bulletin of the Seismological Society of America, 88(4), 1051-1062.