Abstract
Travel-time tomography for the velocity structure of a medium is a highly nonlinear and nonunique inverse problem. Monte Carlo methods are becoming increasingly common choices to provide probabilistic solutions to tomographic problems but those methods are computationally expensive. Neural networks can often be used to solve highly nonlinear problems at a much lower computational cost when multiple inversions are needed from similar data types. We present the first method to perform fully nonlinear, rapid and probabilistic Bayesian inversion of travel-time data for 2D velocity maps using a mixture density network. We compare multiple methods to estimate probability density functions that represent the tomographic solution, using different sets of prior information and different training methodologies. We demonstrate the importance of prior information in such high-dimensional inverse problems due to the curse of dimensionality: unrealistically informative prior probability distributions may result in better estimates of the mean velocity structure; however, the uncertainties represented in the posterior probability density functions then contain less information than is obtained when using a less informative prior. This is illustrated by the emergence of uncertainty loops in posterior standard deviation maps when inverting travel-time data using a less informative prior, which are not observed when using networks trained on prior information that includes (unrealistic) a priori smoothness constraints in the velocity models. We show that after an expensive program of network training, repeated high-dimensional, probabilistic tomography is possible on timescales of the order of a second on a standard desktop computer.
Highlights
Seismic travel-time tomography is often used to reconstruct images of the interior of the Earth [1,2,3], but is a significantly nonlinear and nonunique inverse problem
We show for the first time that Mixture density networks (MDNs) can perform fully nonlinear, rapid and probabilistic 2D tomography from travel-time data
A clearly advantageous strategy for the future of neural network tomography is to invest effort in finding and using more sophisticated and correct prior information [40]. Recent efforts in this direction include [41] who use expert elicitation to constrain prior multipoint geostatistics, Mosser et al [42] who use neural networks to parametrise geological prior information and Nawaz and Curtis [30,31,32] who use Markovian models and variational methods with embedded neural and mixture density networks to combine geological and geophysical information; these various directions appear to be strategically important for the future of this field
Summary
Seismic travel-time tomography is often used to reconstruct images of the interior of the Earth [1,2,3], but is a significantly nonlinear and nonunique inverse problem. To find solutions with minimal computation, the physics relating local wave speed to measured travel times is usually simplified by linearisation [4], but this creates large differences between linearised and true probabilistic solutions [5, 9]. An alternative approach to estimate the posterior pdf is to use prior sampling [11, 12]. In this case, samples are created before inference using only available prior knowledge. The set of samples can be interrogated for examples that are consistent with any particular data set (a method called resampling [13]) or used to parametrise a function that relates data to models which can be used to solve the inverse or inference problem [14].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
