  We apply a linear Bayesian model to seismic tomography, a high-dimensional
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.
