Rapid Bayesian Seismic Tomography using Graph Mixture Density Networks

A wide range of academic and practical applications require that we interrogate the Earth’s subsurface for answers to scientific questions. A common approach is to image the subsurface properties using data recorded at or above the Earth’s surface, and to interpret those images to address questions of interest. Seismic tomograph is one such method which has been used widely to generate those images. In order to obtain robust and well-justified answers, it is important to assess uncertainties in property estimates.

To solve seismic tomographic problems efficiently, mixture density networks (MDNs) have been used to estimate Bayesian posterior probability density functions (pdfs) which describe the uncertainty of tomographic images. However, the method can only be applied in cases where the number of data is fixed, and consequently cannot be used in a large number of practical applications that have variable sizes of data. To resolve this issue, we introduce graph neural networks (GNNs) to solve seismic tomographic problems. Graphs are data structures that provide flexible representation of complex, variable systems. GNNs are neural networks that manipulate graphs. GNNs can be combined with MDNs (called graph MDNs) to provide estimates of posterior pdfs for graph data. In this study we use graph MDNs to solve seismic tomographic problems by representing seismic travel time data using a graph. We demonstrate the method using both synthetic and real data, and compare the results with those obtained using Monte Carlo sampling methods. The results show that graph MDNs can provide comparable solutions to those obtained using Monte Carlo methods for problems with variable number of data. After training, graph MDNs estimate posterior pdfs in seconds on a typical desktop computer. Hence the method can be used to provide rapid solutions for similar problems with variable sizes of data. We therefore conclude that graph MDNs can be an important tool to solve many practical tomographic problems.