LAUSR

The eikonal equation plays an important role across multidisciplinary branches of science and engineering. In geophysics, the eikonal equation and its characteristics are used in addressing two fundamental questions pertaining to seismic waves: what paths do the seismic waves take (its spreading)? and how long do they take? There have been numerous attempts to solve the eikonal equation, which can be broadly categorized as finite-difference (FD)- and physics-informed neural network (PINN)-based approaches. While the former has been developed and optimized over the years, it still inherits some numerical inaccuracies and also the cost scales exponentially with the velocity model size. More importantly, it requires upwind calculations to satisfy the viscosity solution. PINNs, on the other hand, have shown great promise due to several features allowing for higher accuracy and scalability than conventional approaches. In this article, we demonstrate another unique feature of PINN solutions, specifically its flexibility resulting from the global nature of its NN functional optimization, allowing for functional gradients referred to as automatic differentiation. This feature allows us to overcome the inability of conventional methods to handle large areas of missing information (gap) in the velocity model. We find empirically that the PINNs interpolation–extrapolation inherent capability enables us to circumvent a scenario when traveltime modeling is performed on velocity models containing gaps. Such a capability is crucial when performing traveltime modeling using the global tomographic Earth velocity model.