Summary We test a fully non-linear method to solve Bayesian seismic tomographic problems using data consisting of observed travel times 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, while still providing fully probabilistic results. We introduce a new variational method for geophysics – normalizing flows. The method models the posterior distribution by employing 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.

The ultimate goal of a scientific investigation is usually to find answers to specific, often low‐dimensional questions: what is the size of a subsurface body? Does a hypothesized subsurface feature exist? Existing information is reviewed, an experiment is designed and performed to acquire new data, and the most likely answer is estimated. Typically the answer is interpreted from geological and geophysical data or models, but is biased because only one particular forward function is considered, one inversion method is applied, and because human interpretation is a biased process. Interrogation theory provides a systematic way to answer specific questions by combining forward, design, inverse, and decision theories. The optimal answer is made more robust since it balances multiple possible forward models, inverse algorithms and model parametrizations, probabilistically. In a synthetic test, we evaluate the area of a low‐velocity anomaly by interrogating Bayesian tomographic results. By combining the effect of four inversion algorithms, the optimal answer is very close to the true answer, even on a coarsely gridded parametrization. In a field data test, we evaluate the volume of the East Irish Sea basins using probabilistic 3D shear wave speed depth inversion results. This example shows that interrogation theory provides a useful way to answer realistic questions about the Earth. A key revelation is that while the majority of computation may be spent solving inverse problems, much of the skill and effort involved in answering questions may be spent defining and calculating target function values in a clear and unbiased manner.

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The spatial derivatives in decoupled fractional Laplacian (DFL) viscoacoustic and viscoelastic wave equations are the mixed-domain Laplacian operators. Using the approximation of the mixed-domain operators, the spatial derivatives can be calculated by using the Fourier pseudospectral (PS) method with barely spatial numerical dispersions, whereas the time derivative is often computed with the finite-difference (FD) method in second-order accuracy (referred to as the FD-PS scheme). The time-stepping errors caused by the FD discretization inevitably introduce the accumulative temporal dispersion during the wavefield extrapolation, especially for a long-time simulation. To eliminate the time-stepping errors, here, we adopted the [Formula: see text]-space concept in the numerical discretization of the DFL viscoacoustic wave equation. Different from existing [Formula: see text]-space methods, our [Formula: see text]-space method for DFL viscoacoustic wave equation contains two correction terms, which were designed to compensate for the time-stepping errors in the dispersion-dominated operator and loss-dominated operator, respectively. Using theoretical analyses and numerical experiments, we determine that our [Formula: see text]-space approach is superior to the traditional FD-PS scheme mainly in three aspects. First, our approach can effectively compensate for the time-stepping errors. Second, the stability condition is more relaxed, which makes the selection of sampling intervals more flexible. Finally, the [Formula: see text]-space approach allows us to conduct high-accuracy wavefield extrapolation with larger time steps. These features make our scheme suitable for seismic modeling and imaging problems.