Multilevel Monte Carlo Acceleration of Seismic Wave Propagation under Uncertainty
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We interpret uncertainty in the parameters of a model for seismic wave propagation by treating the parameters as random variables, and we apply the Multilevel Monte Carlo (MLMC) method to reduce the cost of approximating expected values of selected, physically relevant, quantities of interest (QoI) with respect to the random variables. Aiming to solve source inversion problems, where we seek to infer the source of an earthquake from ground motion recordings on the Earth's surface, we consider two QoI which measure the discrepancy between computed seismic signals and given reference signals: one QoI is defined in terms of the L2-misfit, which is directly related to maximum likelihood estimates of the source parameters; the other is based on the quadratic Wasserstein distance between probability distributions, and represents one possible choice in a class of such misfit functions which have become increasingly popular for seismic inversion. We use a publicly available code in widespread use, based on the spectral element method, to approximate seismic wave propagation, including seismic attenuation. With this code, using random coefficients and deterministic initial and boundary data, we have performed benchmark numerical experiments with synthetic data in a two-dimensional physical domain with a one-dimensional velocity model where the assumed parameter uncertainty is motivated by realistic Earth models. In this test case, the MLMC method reduced the computational cost of the standard Monte Carlo (MC) method by up to 97 based on the quadratic Wasserstein distance, for a relevant range of tolerances. Extending the method to three-dimensional domains is straight-forward and will further increase the relative computational work reduction, allowing the solution of source inversion problems which would be infeasible using standard MC.
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