## 4 Conclusions  
The primary objective of this study was to enhance the simulation of AVA seismic responses by introducing modifications to the existing propagation matrix theory. To validate the stability and reliability of the novel algorithm designed for computing seismic AVA responses, this study derived theoretical solutions for AVA reflection coefficients based on propagation matrix theory. These theoretical solutions were subsequently utilised to accurately compute the AVA seismic responses for the specific model being investigated. The results obtained from the forward modelling of various models were meticulously compared with the theoretical solutions of the AVA reflection coefficients. The conclusive findings demonstrated that the newly proposed propagation matrix method outperforms the Zoeppritz equation and conventional propagation matrix theory in delivering precise AVA seismic responses. Moreover, extensive seismic pre-stack multiparameter inversion was performed to comprehensively evaluate the accuracy and stability of the new propagation matrix method for pre-stack inversion. The numerical inversion outcomes unambiguously revealed that the inversion accuracy achieved using the new method surpassed that obtained using the Zoeppritz equation. Nevertheless, the new propagation matrix method does possess certain limitations, including being marked by a greater computational intricacy than that of traditional propagation matrix methodologies. Although we have meticulously evaluated several distinct models and successfully demonstrated that this innovative approach exhibits enhanced forward precision, it must be acknowledged that the underlying premises of the Zoeppritz equations are comparatively less intricate. This inherent simplicity may afford the Zoeppritz equations a broader spectrum of applicability, thereby potentially eclipsing the propagation matrix method in certain contexts. Future research directions for the propagation matrix method may encompass investigations into its application within anisotropic media and the integration of the stratigraphic attenuation factor Q into the propagation matrix framework. Data curation, Z.H. ; Methodology, Q.D. ; Supervision, S.C. and J.S. ; Writing--original draft, Q.D. ; Writing--review and editing, G.W. All authors have read and agreed to the published version of the manuscript. This research was funded by the Science and Technology Project of China National Petroleum Corporation, grant number 2023 ZZ05. Funding was also provided by The National Natural Science Foundation of China, grant number 42074127, and The Science and Technology Project of China National Petroleum Corporation, grant number 2023 ZZ15 YJ02. Not applicable. The original contributions presented in this study are included in the article; further inquiries can be directed to the corresponding author. The authors would like to thank [PERSON] and [PERSON] for their guidance during the research. We would also like to thank the reviewers for their feedback and [PERSON] for his assistance. Finally, we are grateful for the support from family members. Mr. [PERSON] is employed by BGP, CNPC China. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. The Frechet derivative of the function \(G(\mathbf{m})\) is redefined as  
\[\frac{\partial G}{\partial m}=W\frac{\partial r}{\partial m}. \tag{12}\]where \(r\) represents the reflectivity in the time-angle domain; \(W\) is the wavelet matrix; m represents the inversion parameters, and, in the three-parameter pre-stack inversion, it can be represented as  
\[\mathbf{m}=\Big{[}V_{p}^{1},\cdots,V_{p}^{nz},V_{s}^{1},\cdots,V_{s}^{nz},\rho^{1}, \cdots,\rho^{nz}\Big{]}^{T}. \tag{10}\]  
where \(3 nz\) represents the number of inversion parameters.