Integration of a Gaussian quadrature grid discretization approach with a generalized stiffness reduction method and a parallelized direct solver for 3-D frequency-domain seismic wave modelling in viscoelastic anisotropic media

We integrate three advanced numerical techniques-Gaussian quadrature grid (GQG) discretization, a new generalized stiffness reduction method and the latest version of an efficient parallelized direct solver to achieve accurate 3-D frequency-domain seismic wave modelling in viscoelastic anisotropic media. A GQG is employed to sample and interpolate both model parameters and wavefield quantities as well as to fit with arbitrary free-surface topography and subsurface interfaces of a geological model. A new version of the generalized stiffness reduction method is utilized to effectively remove the artificial boundary edge effects for which the common perfectly matched layer method fails. The most recent version of a multifrontal massively parallel direct solver is applied to tackle the notoriously expensive computation of frequency-domain 3-D wave modelling. We validate the 3-D modelling by comparing with the exact solutions for homogeneous viscoelastic isotropic, vertically transversely isotropic and orthorhombic media. All the results show very close matches between the numerical and analytical solutions. Then, we investigate the computational efficiency of the parallelized direct solver, compare its performance using different ordering schemes, in-core and out-of-core factorization modes and the block low-rank approximation in the factorization for different grid sizes. Our modelling results show that the ordering scheme of the so-called 'Metis' is the best for reducing computer memory and run time, and the parallelized direct solver is remarkably faster than iterative solvers for similar workloads but at the expense of higher memory requirements. The out-of-core factorization mode can effectively reduce the memory cost without a compromising on run time. The block low-rank approximation is able to significantly reduce the run time in both the factorization and solving process (up to 56 per cent in total), but will increase the memory cost when using the out-of-core factorization mode. Efficient application of this parallel direct solver should use 'Metis' as the ordering scheme and select the out-of-core factorization mode without the block low-rank approximation as the best scheme to save the memory cost, or the in-core factorization mode with the block low-rank approximation for the fastest computation. Finally, we demonstrate the excellent applicability of the 3-D wave modelling scheme for a practical and complex heterogeneous geological model.