A Bayesian approach to linear inverse problems in seismic tomography

Friday, 19 December 2014
Yuan Tian1, Ying Zhou1,2, Julianne Chung2, Matthias Chung2 and Jieyuan Ning1, (1)School of Earth and Space Sciences, Peking University, Beijing, China, (2)Virginia Tech, Blacksburg, VA, United States
Seismic tomography is often an ill-posed linear inverse problem and regularization such as damping and smoothing has been widely applied to find an approximate solution to the inverse problem. The “optimal” solution is chosen based on the tradeoff between model norm (or model roughness) and data misfit. The main difficulty associated with this deterministic approach is in determining a balance between model uncertainty and data fit. This can make interpretation of tomographic structures subjective because models at the “corner” of the tradeoff curve often show large variability. In this study, we investigate a Bayesian approach to the linear inverse problem by minimizing an empirical Bayes risk function based on training dataset generated for the tomographic problem. We show that sample average approximation can be used to find optimal spectral filters to solve the linear tomographic problem based on singular value decomposition (SVD). We compare optimal truncated SVD, optimal Tikhonov filtering as well as independent optimal spectral filtering in finite-frequency tomography and ray theoretical tomography using a global dataset of surface-wave dispersion measurements.