A Bayesian linear model for the high-dimensional inverse problem of seismic tomography

Abstract

We apply a linear Bayesian model to seismic tomography, a highdimensional inverse problem in geophysics. The objective is to estimate the three-dimensional structure of the earth's interior from data measured at its surface. Since this typically involves estimating thousands of unknowns or more, it has always been treated as a linear(ized) optimization problem. Here we present a Bayesian hierarchical model to estimate the joint distribution of earth structural and earthquake source parameters. An ellipsoidal spatial prior allows to accommodate the layered nature of the earth's mantle. With our efficient algorithm we can sample the posterior distributions for large-scale linear inverse problems and provide precise uncertainty quantification in terms of parameter distributions and credible intervals given the data. We apply the method to a full-fledged tomography problem, an inversion for upper-mantle structure under western North America that involves more than 11,000 parameters. In studies on simulated and real data, we show that our approach retrieves the major structures of the earth's interior as well as classical least-squares minimization, while additionally providing uncertainty assessments.

Bayesian Spatial Modelling for High Dimensional Seismic Inverse Problems

“…For a more comprehensive description of seismic tomography for statisticians we refer to Zhang et al . () and Tian et al . ().…”

“…Parameterization of the Earth's interior is achieved through a highly irregular tetrahedral mesh of thousands of vertices spaced by 60–200 km, which is identical to the mesh that was used by Sigloch () and Zhang et al . (). Tetrahedra sizes are adapted to the locations of wave sources and receivers so that mantle volumes with denser seismic ray coverage are covered by more densely spaced grids than regions with sparser ray coverage.…”

“…The performance of this approach is examined in simulation studies in Section under various scenarios. Section applies the SPDE approach with INLA to travel time data that have previously been studied by least squares optimization (Sigloch, ) and in an MCMC approach (Zhang et al ., ). This allows for a comparison of the performance of the various estimation methods.…”

“…For high dimensional linear inverse problems, Zhang et al . () developed an efficient sampling scheme using Gibbs–Metropolis Markov chain Monte Carlo (MCMC) methods to exploit the sparse structure of the sensitivity matrix. They demonstrated it on a realistic seismic tomography problem featuring about 11000 unknowns that had previously been solved by optimization (Sigloch, ).…”

“…For modelling the three‐dimensional structure of this velocity field, Zhang et al . () defined a neighbourhood within a fixed distance of the velocity parameters, but the number of neighbouring vertices within a fixed distance obviously depends on the geometry of the triangulation. Here we work in a more general setting, i.e.…”