A Time-Domain Finite-Difference Method With Attenuation By A Recursive Algorithm

Abstract

A recursive algorithm to incorporate attenuation into a time-domain finite-difference calculation is developed. First, a rheological model of the generalized Maxwell body is chosen. The discrete relaxation frequency and the peak strength of these Maxwell bodies are jointly determined by fitting to an arbitrary Q law in the frequency band of the interest. A conjugate gradient technique and a randomly chosen starting model are used to determine optimum fitting. Examples of constant and frequency dependent Q models are shown. Second, in order to include the attenuation into a finite-difference staggered-grid scheme, the convolution integral of stress and strain is evaluated directly. The convolution integral can be expressed recursively. This is possible because the time-domain viscoelastic modulus function is exponential. The implementation of 1-D wave propagation in a constant Q medium is shown as a example. At the distance of 50 wavelengths and with three relaxation frequencies, the finite-difference results are in very good agreement with the analytic solutions.