Hamiltonian Monte Carlo Inversion of Seismic Sources in Complex Media

Abstract

We present a probabilistic seismic point source inversion, taking into account 3‐D heterogeneous Earth structure. Our method rests on (1) reciprocity and numerical wavefield simulations in complex media and (2) Hamiltonian Monte Carlo sampling that requires only a small amount of test models to provide reliable uncertainty information on the timing, location, and mechanism of the source. Using spectral element simulations of 3‐D, viscoelastic, anisotropic wave propagation, we precompute receiver side strain tensors in time and space. This enables the fast computation of synthetic seismograms for any hypothetical source within the volume of interest, and thus a Bayesian solution of the inverse problem. To improve efficiency, we developed a variant of Hamiltonian Monte Carlo sampling. Taking advantage of easily computable derivatives, numerical examples indicate that Hamiltonian Monte Carlo can converge to the posterior probability density with orders of magnitude less samples than the derivative‐free Metropolis‐Hastings algorithm, which we use for benchmarking. Exact numbers depend on observational errors and the quality of the prior. We apply our method to the Japanese Islands region where we previously constrained 3‐D structure of the crust and upper mantle using full‐waveform inversion with a minimum period of 15 s.

Trans‐Dimensional Surface Reconstruction With Different Classes of Parameterization

“…This comes at the expense of requiring calculation of gradients. Regardless of this extra cost, Hamiltonian Monte Carlo has recently been used in some nonlinear geophysical inverse problems (Fichtner & Simutė, 2018;Fichtner et al, 2019;Sen & Biswas, 2017).…”

“…Hamiltonian Monte Carlo proposals uses an auxiliary variable technique and calculation of the gradient of the posterior to generate new model proposals far away from the current model (Duane et al, 1987;Fichtner & Simutė, 2018;Neal, 2011;Sen & Biswas, 2017).…”

Hamiltonian Nullspace Shuttles

“…For instance, repeated runs of the nullspace shuttle with randomly chosen initial momenta generates the test models of an HMC scheme. Selecting or rejecting samples based on a Metropolis rule produces samples of the posterior distribution e − χ (Betancourt, ; Fichtner & Simute, ; Fichtner et al, ; Neal, ). In contrast, vanishing initial momentum produces a family of gradient descent methods, as shown in equation .…”

Bayesian Elastic Full‐Waveform Inversion Using Hamiltonian Monte Carlo

“…With this work, we explore nonlinearity and uncertainty quantification in FWI with a high‐dimensional model space using a sampling method that has recently been popularized in geophysics, known as Hamiltonian Monte Carlo (HMC) (Betancourt, ; Duane et al, ; Fichtner et al, ; Fichtner & Simute, ; Sen & Biswas, ). Exploiting derivative information, HMC may solve high‐dimensional problems where widely used variants of the Metropolis‐Hastings algorithm (Chib & Greenberg, ) tend to fail.…”