Bayesian spatial modelling for high dimensional seismic inverse problems

We study the application of Bayesian spatial modelling to seismic tomography, a geophysical, high dimensional, linearized inverse problem that infers the three-dimensional structure of the Earth's interior. We develop a spatial dependence model of seismic wave velocity variations in the Earth's mantle based on a Gaussian Matern field approximation. Using the theory of stochastic partial differential equations, this model quantifies the uncertainties in the parameter space by means of the integrated nested Laplace approximation. In resolution tests using simulated data and in inversions using real data, our model matches the performance of conventional deterministic optimization approaches in retrieving three-dimensional structure of the Earth's mantle. In addition it delivers estimates of the full parameter covariance matrix. Our model substantially improves on previous work relying on Markov chain Monte Carlo methods in terms of statistical misfits and computing time.