Joint P- and S-Wave VSP Traveltime Tomography via Well Log-Guided Physics-Informed Neural Networks

Traveltime tomography is widely used in seismology to construct accurate long-wavelength subsurface velocity models and to investigate the Earth’s internal structure and dynamic processes. The joint inversion of vertical seismic profile (VSP) P- and S-wave velocities presents added complexity due to the limitations of conventional numerical solvers, which are generally constrained to first-order accuracy and perform poorly on irregular computational grids. To overcome these limitations, physics-informed neural networks (PINNs) have been introduced, enabling the PINN-based tomography framework (PINNtomo) to utilize the representational strength of neural networks to solve the eikonal equation, reconstruct velocity models in a data-consistent and physically constrained method, and address these challenges more effectively. We propose a physics-informed neural network framework guided by well log information for the joint P-wave and S-wave VSP traveltime tomography (W-PINNPStomo). Specifically,W-PINNPStomo performs joint multi-parameter inversion of P-wave and S-wave velocities, with model updates constrained by P- and S-wave log data serving as soft constraints. The W-PINNPStomo framework incorporates two neural networks: a travel time network dedicated to predicting P- and S-wave traveltimes, and a velocity network tasked with estimating the corresponding wave velocities. By minimizing the misfit between initial P- and S-wave traveltimes derived from the eikonal equation and those predicted by the model, the associated velocity fields can be iteratively updated. Since the original PINNtomo loss function relies on multiplicative-factor eikonal equations that depend on background traveltimes based on uniform velocities at the source location, we introduce a source-independent multiplicative decomposition of the eikonal equation to eliminate this velocity dependence. Numerical experiments conducted on the Marmousi, Fault and Salt velocity models under VSP acquisition geometries confirm the effectiveness of the proposed improvements. By incorporating well log information as soft constraints, the W-PINNPStomo framework achieves faster convergence and improved multi-parameter inversion accuracy compared to PINNPStomo.