3-D variational inference-based double-difference seismic tomography method and application to the SAFOD site, California

Seismic tomography is used to image subsurface structures at various scales, accomplished by solving a nonlinear and non-unique inverse problem. It is therefore important to quantify velocity model uncertainties for accurate earthquake locations and geological interpretations. Monte Carlo sampling techniques are usually used for this purpose, but those methods are computationally intensive, especially for large data sets or high-dimensional parameter spaces. In comparison, Bayesian variational inference provides a more efficient alternative by delivering probabilistic solutions through optimization. The method has been proven to be efficient in 2-D tomographic problems. In this study, we apply variational inference to solve 3-D double-difference (DD) seismic tomographic system using both absolute and differential traveltime data. Synthetic tests demonstrate that the new method can produce more accurate velocity models than the original DD tomography method by avoiding regularization constraints, and at the same time provides more reliable uncertainty estimates. Compared to traditional checkerboard resolution tests, the resulting uncertainty estimates provide a better measure for the reliability of the solution. We further apply the new method to data recorded by a local dense seismic array around the San Andreas Fault Observatory at Depth (SAFOD) site along the San Andreas Fault (SAF) at Parkfield. Similar to previous studies, the obtained velocity models show significant velocity contrasts across the fault. More importantly, the new method produces velocity uncertainties of less than 0.34 km s-1 for ${{{{V}}}_p}$ and 0.23 km s-1 for ${{{{V}}}_s}$. We therefore conclude that variational inference provides an effective tool for solving 3-D seismic tomographic problems and quantifying model uncertainties.