Bayesian Variational Seismic Tomography using Normalizing Flows

Seismic travel time tomography is used widely to image the Earth's interior structure and to infer subsurface properties. Tomography is an inverse problem, and computationally expensive nonlinear inverse methods are often deployed in order to understand uncertainties in the tomographic results. Monte Carlo sampling methods estimate the posterior probability distribution which describes the solution to Bayesian tomographic problems, but they are computationally expensive and often intractable for high dimensional model spaces and large data sets due to the curse of dimensionality. We therefore introduce a new method of variational inference to solve Bayesian seismic tomography problems using optimization methods, while still providing fully nonlinear, probabilistic results. The new method, known as normalizing flows, warps a simple and known distribution (for example a Uniform or Gaussian distribution) into an optimal approximation to the posterior distribution through a chain of invertible transforms. These transforms are selected from a library of suitable functions, some of which invoke neural networks internally. We test the method using both synthetic and field data. The results show that normalizing flows can produce similar mean and uncertainty maps to those obtained from both Monte Carlo and another variational method (Stein varational gradient descent), at significantly decreased computational cost. In our tomographic tests, normalizing flows improves both accuracy and efficiency, producing maps of UK surface wave speeds and their uncertainties at the finest resolution and the lowest computational cost to-date, allowing results to be interrogated efficiently and quantitatively for subsurface structure.