Analytical Non-Decoupled Solution and Dispersion Characteristics of Rayleigh Waves in Multi-Layered Vertical Transverse Isotropic Media

Seismic wavefield simulation is the primary technique used to study the effects of vertical transverse isotropy (VTI) on the propagation of Rayleigh waves. However, conventional Rayleigh wave dispersion equations are based on isotropic assumptions and cannot be applied to the dispersion characteristics of multi-layered VTI media. Based on the Rayleigh wave potential functions in VTI media, this study derives inhomogeneous wave equations governing the Rayleigh wave potentials. These equations exhibit a distinctive duality; the particular solution associated with the inhomogeneous term in the P-wave equation coincides exactly with the solution of the homogeneous SV-wave equation. Compared to existing methods, the solution to the wave equations does not require decoupling. Using conventional exponential-form potential function solutions, this study realizes the analytical computation of Rayleigh wave inhomogeneous wave equations in VTI media and establishes a dispersion equation for multi-layered VTI media. The reliability of the method is verified through mathematical back substitution and numerical validation. To further explore the dispersion characteristics of Rayleigh waves in VTI media, a three-layered model is designed, and the dispersion response features under different VTI parameters are computed, indicating the high sensitivity of the dispersion curves to changes in any of the five VTI parameters. This paper presents a non-decoupled recursive analytical method for computing Rayleigh wave wavefields and dispersion curves in VTI media. The approach requires solving only a second-order inhomogeneous boundary-value differential equation and adopts the standard exponential potential representation used for isotropic media. This makes the method more practical and yields a fast, convenient algorithm for seismic parameter inversion and data processing in VTI media.