Probabilistic neural network-based 2D travel-time tomography
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ORIGINAL ARTICLE
Probabilistic neural network-based 2D travel-time tomography Stephanie Earp1
•
Andrew Curtis1,2
Received: 27 June 2019 / Accepted: 6 April 2020 Ó The Author(s) 2020
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. Keywords Neural networks Mixture density networks Uncertainty estimation Seismic tomography
1 Introduction Seismic travel-time tomography is often used to reconstruct images of the interior of the Earth [1–3], 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]. Increases in compute power now allow fully
& Stephanie Earp [email protected] Andrew Curtis [email protected] 1
School of Geosciences, University of Edinburgh, Edinburgh, UK
2
Institute of Geophysics, ETH Zu¨rich, 8092 Zurich, Switzerland
nonlinear Monte Carlo sampling solutions to be found without linearisation, to solve problems in 2D [5, 6] and 3D [7–10]. Using Bayesian methods, such solutions provide samples (example tomographic models) that fit the data to within their measurement uncertainties, are consistent with available pri
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