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In this paper, we present a frequency-domain volume integral method to modelthe microseismic wavefield in heterogeneous anisotropic-elastic media. Theelastic wave equation is written as an integral equation of theLippmann-Schwinger type, and the seismic source is represented as a generalmoment tensor. The displacement field due to a moment tensor source can becomputed using the spatial derivative of the elastodynamic Green's function.The existing matrix-based implementation of the integral equation iscomputationally inefficient to model the wavefield in a three-dimensionalearth. An integral equation for the particle displacement is, hence, formulatedin a matrix-free manner through the application of the Fourier transform. Thebiconjugate gradient stabilized method is used to iteratively obtain thesolution of this equation. We apply the numerical scheme to three differentmodels in order of increasing geological complexity and obtain the elasticdisplacement fields corresponding to the different types of moment tensorsources. The volume integral method has an advantage over the time domainmethods in regard to adding multiple sources since it can work with discretefrequencies, one by one, and limit the computational cost. The generatedsynthetic data can be useful in inversion for the microseismic source and modelparameters.

Integral equation method for microseismic wavefield modelling in anisotropic elastic media