The Eikonal Solution for Attenuating VTI Media Using Physics-Informed Neural Networks

Traveltime computation in attenuating media is a challenging problem particularly when taking attenuation anisotropy into account. Body wave traveltimes consist of two parts when traveling in an attenuating medium, namely the real and imaginary parts. The real part corresponds to the phase of the waves while the imaginary part corresponds to the amplitude decay of the waves due to energy absorption. Analysis of this complex-valued traveltimes is important when amplitude of the waves is needed, e.g., seismic Q inversion and petrophysical properties analysis. Previous studies attempted to solve the complex eikonal equation using some sort of approximations. Here, we utilize a physicsinformed neural network (PINN) to solve for the complex-valued traveltimes in an attenuating transversely isotropic medium with a vertical symmetry axis (VTI). We incorporate the factored eikonal solution to deal with the point-source singularity as well as ensuring convergence. We impose the complex eikonal equation in the minimization of the loss function and compute the real and imaginary parts simultaneously. The result is remarkable accuracy of complex traveltimes in an attenuating VTI model with inhomogeneous velocity regardless of the strength of attenuation anisotropy. This demonstrates the potential of PINNs in solving challenging partial differential equations.