## 1 Introduction  
Seismic pre-stack inversion is an important tool for predicting reservoirs. It has always been a focus of geophysical research. Amplitude variation with angle (AVA) three-parameter inversion can provide models for P-wave and S-wave velocity and density at the same time. These models are crucial for reservoir prediction [1, 2]. The conventional seismic pre-stack AVA inversion method is based primarily on the Zoeppritz equations and their approximations for the inversion of subsurface parameters [3, 4, 5, 6], and is widely used. Nevertheless, it is limited by assumptions such as the small-angle incidence and weak reflectivity, which restrict its applicability to incidence angles [7]. Pre-stack inversion based on a precise Zoeppritz equation can overcome these limitations. However, it only considers the primary reflection from P-wave incidence and ignores the impacts of transmission loss and multiple reflections on seismic responses [8; 9; 10]. These factors ultimately constrain the precision of pre-stack inversion [11]. The finite difference wave equation forward modelling can accurately simulate the seismic wave response in complex media. However, the widespread application of full-waveform inversion to practical data is limited by its substantial computational burden and vulnerability to local minima problems [12]. The layered media propagation matrix theory provides a theoretical framework for the spectral calculation of P- and S-wave reflection coefficients in multilayered media at arbitrary incidence angles [13; 14; 15]. The reflection coefficient spectrum theory considers transmission loss and the effects of multiple reflections between layers. It uses the analytical solutions of the wave equation and provides a higher resolution than seismic response simulations based on the Zoeppritz equation and its approximations. The recursive method initially introduced by [PERSON] forms the basis of most propagation matrix theories. [PERSON] [16] further utilised [PERSON]'s matrix formulation to derive a phase-velocity dispersion equation for upgoing waves in multilayered solid media. Seismic response simulations based on recurrence matrices can be categorised into the reflectivity method (RM) [13] and the propagation matrix method (PM) [17]. The most discussed reflectivity method is the Kennett recurrence matrix [18]. [PERSON] et al. [19] described a vectorisation process that uses the innermost recurrent reorganisation to calculate the recurrence matrix, effectively accelerating the modelling process. [PERSON] and [PERSON] [20] derived second-order approximations of the reflection and transmission coefficients, which improved the simulation accuracy. [PERSON] et al. [21] investigated the relationship between the P-wave reflection coefficient and the converted wave reflection coefficient in azimuthally anisotropic elastic media. Their study examined how these coefficients vary with respect to incidence and azimuth angles. By integrating the analysis of the P-wave and converted wave reflection coefficients, they achieved robust estimates for several useful anisotropic parameters. [PERSON] et al. [22] employed the Hanning window method to suppress truncation effects in the temporospatial domains of seismic records caused by integration truncation. [PERSON] and [PERSON] [23] proposed a generalised reflection and transmission matrix algorithm for calculating wave propagation in stratified fluid-saturated porous media in half-space. [PERSON] et al. [24] proposed AVA inversion based on the reflectivity method. [PERSON] [25; 26] estimated the reflection coefficients of PP- and PS-waves at individual interfaces in elastic and viscoelastic isotropic media. Similar to the reflection method, [PERSON] [27] used the propagation matrix method in conjunction with previous equations to derive a generalised propagation matrix method for wavefield simulations in viscoelastic anisotropic media. [PERSON] et al.