An efficient pseudoelastic pure P-mode wave equation and the implementation of the free surface boundary condition

Based on the elastic wave equation, a pseudoelastic pure P-mode wave equation has been recently derived by projecting the wavefield along the wavefront normal direction. This pseudoelastic pure P-mode wave equation offers an accurate simulation of P-wave fields with accurate elastic phase and amplitude characteristics. Moreover, considering no S-waves are involved, it is computationally more efficient than the elastic wave equation, making it an excellent choice as a forward simulation engine for P-wave exploration. Here, we propose a new pseudoelastic pure P-mode wave equation and apply the stress image method to it to implement the free surface boundary condition. The new pseudoelastic wave equation offers significantly improved computational efficiency compared to the previous pseudoelastic wave equation. Additionally, the wavefields simulated by this new pseudoelastic wave equation exhibit clear physical interpretations. We evaluate the accuracy of the new wave equation in simulating elastic P-waves by employing a model with high-velocity contrasts. We find that this new equation, which purely admits P-waves, though having exact amplitude and phase behaviour as the elastic waves for transmission components, the amplitudes slightly suffer in the scattering scenario. The difference in amplitude between the elastic and our pseudoelastic increases as the contrast in velocity at the interface (interlayer velocity ratio) increases, especially the S-wave velocities. This has negative implications on scattering from the free surface boundary condition or the sea bottom interface, especially if the shear wave velocity below the surface or the sea bottom is high. However, in cases where, like for land data in the Middle East, the transition to a free surface is smoother, the accuracy of the pseudoelastic equation is high. In all cases, regardless of the interlayer velocity ratio, the accuracy of the pseudoelastic wave equation in simulating the elastic case, for scattered waves, exceeds that of the acoustic wave equation in phase and amplitude.