Incorporation of attenuation into time‐domain computations of seismic wave fields

Abstract

The only numerically tractable way yet found to incorporate attenuation into numerical time‐domain computations of seismic wave fields is to approximate the viscoelastic modulus by a low‐order rational function of frequency. The coefficients of this function can be determined by the Padé approximation. Our test computations show, however, that this approximation generally is of poor quality. Therefore, we suggest a new approach which is based on the rheological model of the generalized Maxwell body, which has a modulus of the desired rational form. We choose the relaxation frequencies logarithmically equidistant in the frequency band of interest, and determine the weight factors by simple numerical curve fitting to an arbitrary Q law. This approach is superior to the method above both in accuracy and in computational efficiency. For most practical applications, approximations of orders 2 or 3 are sufficient. The computing time and memory requirements for a finite‐difference calculation are then approximately twice those of a purely elastic calculation. As a first application of the method, we compute SH channel waves in discontinuous coal seams with Q = 50 within the coal. The results show that the high‐ frequency Airy phase is strongly attenuated. This indicates that care has to be taken in comparing the results of purely elastic model calculations of the propagation of seam waves with experimental data.