SUMMARY: We test a fully non-linear method to solve Bayesian seismic tomographic problems using data consisting of observed traveltimes of first-arriving waves. Rather than using Monte Carlo methods to sample the posterior probability distribution that embodies the solution of the tomographic inverse problem, we use variational inference. Variational methods solve the Bayesian inference problem under an optimization framework by seeking the best approximation to the posterior distribution from a family of distributions, while still providing fully probabilistic results. We introduce a new variational method for geophysics—normalizing flows. The method models the posterior distribution by using a series of invertible and differentiable transforms—the flows. By optimizing the parameters of these transforms the flows are designed to convert a simple and analytically known probability distribution into a good approximation of the posterior distribution. Numerical examples show that normalizing flows can provide an accurate tomographic result including full uncertainty information while significantly decreasing the computational cost compared to Monte Carlo and other variational methods. In addition, this method provides analytic solutions for the posterior distribution rather than an ensemble of posterior samples. This opens the possibility that subsequent calculations that use the posterior distribution might be performed analytically.

Seismic tomography is a methodology to image the interior of solid or fluid media and is often used to map properties in the subsurface of the Earth. In order to better interpret the resulting images, it is important to assess imaging uncertainties. Since tomography is significantly nonlinear, Monte Carlo sampling methods are often used for this purpose, but they are generally computationally intractable for large data sets and high‐dimensional parameter spaces. To extend uncertainty analysis to larger systems, we use variational inference methods to conduct seismic tomography. In contrast to Monte Carlo sampling, variational methods solve the Bayesian inference problem as an optimization problem yet still provide fully nonlinear, probabilistic results. In this study, we applied two variational methods, automatic differential variational inference and Stein variational gradient descent, to 2‐D seismic tomography problems using both synthetic and real data, and we compare the results to those from two different Monte Carlo sampling methods. The results show that automatic differential variational inference provides a biased approximation because of its implicit transformed‐Gaussian approximation, and it cannot be used to find generally multimodal posteriors; Stein variational gradient descent produces more accurate approximations to the results of Monte Carlo sampling methods. Both methods estimate the posterior distribution at significantly lower computational cost, provided that gradients of parameters with respect to data can be calculated efficiently. We expect that the methods can be applied fruitfully to many other types of geophysical inverse problems.

SUMMARY: Seismic full-waveform inversion (FWI) provides high resolution images of the subsurface by exploiting information in the recorded seismic waveforms. This is achieved by solving a highly non-linear and non-unique inverse problem. Bayesian inference is therefore used to quantify uncertainties in the solution. Variational inference is a method that provides probabilistic, Bayesian solutions efficiently using optimization. The method has been applied to 2-D FWI problems to produce full Bayesian posterior distributions. However, due to higher dimensionality and more expensive computational cost, the performance of the method in 3-D FWI problems remains unknown. We apply three variational inference methods to 3-D FWI and analyse their performance. Specifically, we apply automatic differential variational inference (ADVI), Stein variational gradient descent (SVGD) and stochastic SVGD (sSVGD), to a 3-D FWI problem and compare their results and computational cost. The results show that ADVI is the most computationally efficient method but systematically underestimates the uncertainty. The method can therefore be used to provide relatively rapid but approximate insights into the subsurface together with a lower bound estimate of the uncertainty. SVGD demands the highest computational cost, and still produces biased results. In contrast, by including a randomized term in the SVGD dynamics, sSVGD becomes a Markov chain Monte Carlo method and provides the most accurate results at intermediate computational cost. We thus conclude that 3-D variational FWI is practically applicable, at least in small problems, and can be used to image the Earth’s interior and to provide reasonable uncertainty estimates on those images.

Seismic surface wave tomography is a tried and tested method to reveal the subsurface structure of the Earth. However, the conventional 2-step scheme of inverting first for 2-D maps of surface wave phase or group velocity and then inverting for the 3-D spatial velocity structure preserves little information about lateral spatial correlations, and introduces additional uncertainties and errors into the 3-D result. We introduce a 1-step 3-D non-linear surface wave tomography method that removes these effects by inverting for 3-D spatial structure directly from frequency-dependent traveltime measurements. We achieve this using the reversible jump Markov chain Monte Carlo (McMC) algorithm with a fully 3-D model parametrization. Synthetic tests show that the method estimates the velocity model and associated uncertainties significantly better than the conventional 2-step McMC method, and that the computational cost seems to be comparable with 2-step McMC methods. The resulting uncertainties are more intuitively reasonable than those from the 2-step method, and provide directly interpretable uncertainty on volumetrics of structures of interest.

SUMMARY: Seismic full-waveform inversion (FWI) can produce high-resolution images of the Earth’s subsurface. Since full-waveform modelling is significantly nonlinear with respect to velocities, Monte Carlo methods have been used to assess image uncertainties. However, because of the high computational cost of Monte Carlo sampling methods, uncertainty assessment remains intractable for larger data sets and 3-D applications. In this study, we propose a new method called variational FWI, which uses Stein variational gradient descent to solve FWI problems. We apply the method to a 2-D synthetic example and demonstrate that the method produces accurate approximations to those obtained by Hamiltonian Monte Carlo. Since variational inference solves the problem using optimization, the method can be applied to larger data sets and 3-D applications by using stochastic optimization and distributed optimization.

Constraining geophysical models with observed data usually involves solving nonlinear and nonunique inverse problems. Neural mixture density networks (MDNs) provide an efficient way to estimate Bayesian posterior marginal probability density functions (pdf's) that represent the nonunique solution. However, it is difficult to infer correlations between parameters using MDNs, and in turn to draw samples from the posterior pdf. We introduce an alternative to resolve these issues: invertible neural networks (INNs). These are simultaneously trained to represent uncertain forward functions and to solve Bayesian inverse problems. In its usual form, the method does not account for uncertainty caused by data noise and becomes less effective in high dimensionality. To overcome these issues, in this study, we include data uncertainties as additional model parameters, and train the network by maximizing the likelihood of the data used for training. We apply the method to two types of imaging problems: One‐dimensional surface wave dispersion inversion and two‐dimensional travel time tomography, and we compare the results to those obtained using Monte Carlo and MDNs. Results show that INNs provide comparable posterior pdfs to those obtained using Monte Carlo, including correlations between parameters, and provide more accurate marginal distributions than MDNs. After training, INNs estimate posterior pdfs in seconds on a typical desktop computer. Hence they can be used to provide efficient solutions for repeated inverse problems using different data sets. Also even accounting for training time, our results show that INNs can be more efficient than Monte Carlo methods for solving inverse problems only once.