Use of Operator Upscaling for Seismic Inversion: Computationally Feasible Forward and Adjoint Calculations

Abstract

To solve seismic inverse problems via the adjoint state method, we must be able to repeatedly solve both the wave equation and its adjoint efficiently. Operator upscaling applied to the wave equation imparts fine scale information to the coarse scale without requiring that we solve the full fine scale problem. We apply the algorithm to the stress-free form of the 3D elastic wave equation. This algorithm has two stages: first, we solve independent subgrid problems on the fine scale; second, we use these subgrid solutions to solve the coarse problem. Because the subgrid problems are independent, they can be solved via an embarrassingly parallel algorithm. Surprisingly, the most expensive part of the coarse grid solve is not assembling the mass matrix (which is time independent) but instead it is calculation of the load vector (which is time dependent). Thus we parallelize the load vector calculation for the coarse problem, as it dominates the time step. The most expensive parts of the algorithm (the subgrid solve and the coarse load vector calculation) exhibit near linear speedup. In the second half of the thesis we discuss using the adjoint state method to solve the seismic inverse problem. As the acoustic wave operator is self-adjoint, we chose to differentiate and then discretize the problem. The result is that the adjoint problem can be solved by the same upscaling method as the standard acoustic wave equation. The forward and adjoint upscaling algorithms differ only in the source terms and in the time stepping order.