Simultaneous P- and S-wave seismic traveltime tomography using physics-informed neural networks

Seismic tomography has long been an effective tool for constructing reliable subsurface structures. However, simultaneous inversion of P- and S-wave velocities presents a significant challenge for conventional seismic tomography methods, which depend on numerical algorithms to calculate traveltimes. A physics-informed neural network—based seismic tomography method (PINNtomo) has been proposed to solve the eikonal equation and construct the velocity model. We propose extending PINNtomo to perform multiparameter inversion of P- and S-wave velocities jointly, which we refer to as PINNPStomo. In PINNPStomo, we employ two neural networks: one for the P- and S-wave traveltimes and another for the P- and S-wave velocities. By optimizing the misfits of P- and S-wave first-arrival traveltimes calculated from the eikonal equations, we can obtain the predicted P- and S-wave velocities that determine these traveltimes. Recognizing that the original PINNtomo utilizes a multiplicative factored eikonal equation, which depends on background traveltimes corresponding to a homogeneous velocity at the source location, we propose to use an effective-slowness-based factored eikonal equation for PINNPStomo to eliminate this dependency. The proposed PINNPStomo, incorporating the effective-slowness-based factored eikonal equation, demonstrates superior convergence speed and multiparameter inversion accuracy. We validate these improvements using two-dimensional Marmousi, two-dimensional Overthrust and three-dimensional foothill elastic velocity models across three different seismic data acquisition geometries.