3D frequency-domain elastic wave modeling with the spectral element method using a massively parallel direct solver

摘要

Complex topography, the free-surface boundary condition, and anelastic properties of media should be accounted for in the frame of onshore geophysical prospecting imaging, such as full-waveform inversion (FWI). In this context, an accurate and efficient forward-modeling engine is mandatory. We have performed 3D frequency-domain anisotropic elastic wave modeling by using the highly accurate spectral element method and a sparse multifrontal direct solver. An efficient approach similar to computing the matrix-vector product in the time domain is used to build the matrix. We validate the numerical results by comparing with analytical solutions. A parallel direct solver, the sparse direct multifrontal massively parallel solver (MUMPS), is used to solve the linear system. We find that a hybrid implementation of message passing interface and open multiprocessing is more efficient in flops and memory cost. The influence of the deformed mesh, free-surface boundary condition, and heterogeneity of media on MUMPS performance is negligible. Complexity analysis suggests that the memory complexity of MUMPS agrees with the theoretical order O(N-4) (or O(N-3.5) with an efficient matrix reordering method) for an N-3 grid when nontrivial topography is considered. With the available resources, we conduct a moderate scale modeling with a subset of the SEAM Phase II Foothills model, where 60 wavelengths in the x-axis are propagated. Computing one gradient of FWI based on this model using the frequency-domain modeling is shown to require similar or fewer computational resources than what would be required for a time-domain solver, depending on the number of sources, while larger memory is necessary. An estimation of the increasing trend indicates that approximately 20 Tb of memory would be required for a 50 x 50 x 50 wavelength modeling. The limit of MUMPS scalability hinders the application to larger scale applications.