Full waveform inversion with quantified uncertainty in stratified media
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We consider the problem of determining inclusions in layered media from noisy measurement of wave fields on the surface. In a Bayesian inference framework to estimate uncertainties, we first determine the maximum a posteriori approximation (MAP) by optimizing error functionals which measure the deviation from the recorded data, regularized with a priori information. We devise an automatic Levenberg-Marquardt-Fletcher type optimization scheme based on the use of adaptive finite element meshes to solve wave equation constraints with changing discontinuities and algorithmic differentiation. Estimating uncertainty about the MAP point by a Laplace approximation, we expect to obtain the main mode of the a posteriory density. Markov Chain Monte Carlo studies provide a more detailed description, at a much higher computational cost. Adaptive meshes must be replaced by uniform meshes while sampling, which affects precision in the presence of interfaces between different materials. Caution should be exerted when interpreting secondary modes, since some of them may be related to the mesh quality and the numerical noise it creates.
