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The high cost of acquiring a sufficient amount of seismic data for training has limited the use of machine learning in seismic tomography. In addition, the inversion uncertainty due to the noisy data and data scarcity is less discussed in conventional seismic tomography literature. To mitigate the uncertainty effects and quantify their impacts in the prediction, the so-called Bayesian Physics-Informed Neural Networks (BPINNs) based on the eikonal equation are adopted to infer the velocity field and reconstruct the travel-time field. In BPINNs, two inference algorithms including Stein Variational Gradient Descent (SVGD) and Gaussian variational inference (VI) are investigated for the inference task. The numerical results of several benchmark problems demonstrate that the velocity field can be estimated accurately and the travel-time can be well approximated with reasonable uncertainty estimates by BPINNs. This suggests that the inferred velocity model provided by BPINNs may serve as a valid initial model for seismic inversion and migration.

Bayesian Physics-Informed Neural Networks for the Subsurface Tomography based on the Eikonal Equation