A Two-Scale Solution Algorithm for the Elastic Wave Equation

Operator-based upscaling is a two-scale algorithm that speeds up the solution of the wave equation by producing a coarse grid solution which incorporates much of the local fine-scale solution information. We present the first implementation of operator upscaling for the elastic wave equation. By using the velocity-displacement formulation of the three-dimensional elastic wave equation, basis functions that are linear in all three directions, and applying mass lumping, the subgrid solve (first stage of the two-step algorithm) reduces to solving explicit difference equations. At the second stage of the algorithm, we upscale both velocity and displacement by using local subgrid information to formulate the coarse-grid problem. The coarse-grid system matrix is independent of time, sparse, and banded. This paper explores both serial and parallel implementations of the method. The main simplifying assumption of the method (special zero boundary conditions imposed on coarse blocks in the first stage of the algorithm) leads to an easily parallelizable algorithm because very little communication is required between processors. In fact, for this upscaling implementation calculation of the load vector for the coarse solve dominates the cost of a time step. We show that for a homogeneous medium convergence is second-order in space and time so long as both the coarse and fine grids are simultaneously refined. A series of heterogeneous-medium numerical experiments demonstrate that the upscaled solution captures the fine-scale fluctuations in the input parameters accurately. Most notable for use in a seismic inversion algorithm, the upscaling algorithm accurately locates the depth of reflectors (interface changes).