Eikonal Solution Using Physics-Informed Neural Networks for Global Seismic Travel Time Modelling

Abstract

Being able to determine how much time it takes for a seismic wave to travel from one point to another is essential in geophysics. One can achieve this goal under the asymptotic ray assumption and end up with the so-called Eikonal equation. The equation finds itself to be beneficial across science and engineering. In geophysics, especially the global seismology field, the solution of this equation is primarily used to perform travel time tomography and earthquake relocation application. In this research I propose a novel scheme to solve the Eikonal equation under two main objectives in mind: being able to compute more accurate first-arrival travel time using Three-dimensional (3-D) velocity model and also being as efficient as the standard procedure. The proposed method is using a physics-informed neural network (PINN). The forward problem is formulated such that the physical equation is the driving component of the minimization of the objective function. The velocity model used on this research is the second generation of the three-dimensional global adjoint tomographic model, GLAD-M25, to account for anelastic behaviour of the Earth.

From the numerical tests, I observed one unique feature in using PINNs to solve the Eikonal equation. I demonstrate that I can use a velocity model which has incomplete velocity information in it and still able to model accurately in some regions the travel time. The results show that the proposed method achieves a significant improvement on the velocity validation and more importantly, is able to calculate the first-arrival travel time using a full three-dimensional global tomographic model (GLAD-M25). The validation process is done by comparing the input velocity data with the recovered velocity from the modelled travel time. The residuals for all depth is below -1 to 1 % error and the recovered velocity and input data are align with a cosine similarity value around 0.999. The main limitation pertaining to the first iteration model proposed on this research is its training cost. For each epoch, given the large number of batches, the training takes around 52.383 minutes. However, once the model is trained, the inference process is comparable to a standard Eikonal solver.