Multilevel Monte Carlo Acceleration of Seismic Wave Propagation under Uncertainty

TL;DR

This paper applies the Multilevel Monte Carlo method to seismic wave propagation models with uncertain parameters, significantly reducing computational costs in seismic inversion problems using synthetic data.

Contribution

It introduces MLMC for seismic wave simulations with uncertainty, demonstrating substantial cost reductions for two different quantities of interest in 2D models.

Findings

Cost reduction of up to 97% for QoI_E

Cost reduction of up to 78% for QoI_W

Applicable to 3D models with further efficiency gains

Abstract

We interpret uncertainty in a model for seismic wave propagation by treating the model parameters as random variables, and 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. Targeting source inversion problems, where the source of an earthquake is inferred from ground motion recordings on the Earth's surface, we consider two QoI that measure the discrepancies between computed seismic signals and given reference signals: one QoI, $\hbox{QoI}_E$, is defined in terms of the $L^2$-misfit, which is directly related to maximum likelihood estimates of the source parameters; the other, $\hbox{QoI}_W$, is based on the quadratic Wasserstein distance between probability distributions, and represents one possible choice in a class of such misfit functions that have become increasingly popular to solve seismic inversion in recent years. We simulate seismic wave propagation, including seismic attenuation, using a publicly available code in widespread use, based on the spectral element method. Using random coefficients and deterministic initial and boundary data, we present benchmark numerical experiments with synthetic data in a two-dimensional physical domain and a one-dimensional velocity model where the assumed parameter uncertainty is motivated by realistic Earth models. Here, the computational cost of the standard Monte Carlo method was reduced by up to 97% for $\hbox{QoI}_E$, and up to 78% for $\hbox{QoI}_W$, using a relevant range of tolerances. Shifting to three-dimensional domains is straight-forward and will further increase the relative computational work reduction.