Hessian-based approximate solution of large-scale statistical inverse seismic wave propagation
Friday, November 18, 3PM – 4PM
George Stadler, ICES
Hessian-based approximate solution of large-scale statistical inverse seismic wave propagation. We are interested in inferring local wave speeds in acoustic and elastic media from waveforms, as well as in estimating the uncertainty in the inferred solution. This inverse problem is cast as a large-scale statistical inverse problem in the framework of Bayesian inference. The complicating factors are the high-dimensional parameter space (due to discretization of the infinite-dimensional parameter fields for the local wave speeds) and the expensive forward problems given by the time-dependent elastic-acoustic wave equation.
We exploit that at the maximum likelihood point, the covariance matrix for the Gaussian approximation of the posterior probability density is the inverse of the Hessian. A low rank approximation of the Hessian that exploits the compact nature of the data misfit operator for problems with limited measurements is constructed by the Lanczos method. The covariance operator can then be computed using the Woodbury identity. This allows, for instance, to compute the pointwise variance field, or samples for the inferred local wave speed field.
We use a first-order system formulation of the wave equation, which allows to treat the elastic and the acoustic equation in the same framework. This system is discretized using a high-order spectral discontinuous Galerkin (dG) method, which uses an upwind numerical flux. To obtain accurate gradients and Hessians, derivatives of the misfit are computed for the dG-discretized equation. The resulting discretization for the adjoint wave equation is a dG scheme with a downwind numerical flux. The method is applied to infer global earth models from synthetic measurements.
This work is joint with: Tan Bui-Thanh, Carsten Burstedde, Omar Ghattas, James Martin and Lucas C. Wilcox
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