Neural network travel-time tomography

Travel-time tomography is a non-linear inverse problem. Monte Carlo methods are increasingly used to provide probabilistic solutions to tomographic problems, but these methods are computationally expensive. Neural networks can be used to solve some non-linear problems at a much lower computational cost. We show for the first time that a form of neural network called a mixture density network can perform fully non-linear, rapid and probabilistic tomographic inversion using travel-time data. We compare two methods to estimate the Bayesian posterior probability density functions: first a vector of networks are trained such that each estimates the marginal posterior probability distribution of wave speed in one grid cell; second, a single network estimates the entire posterior probability density function across all cells. While both methods provide estimates of the true structure in the means of their distributions, their uncertainty estimates differ: when separate networks are trained to solve for wave speeds at each location in the model the standard deviations exhibit uncertainty loops, as expected, whilst a network trained to solve for speeds on the whole model at once does not. The former method is therefore likely to be more robust.