## 3 Characteristics of Wavefield Response in Layer Models
### Analysis of Inversion Results  
To validate the effectiveness of the proposed propagation matrix inversion method, we also conducted inversions using a two-dimensional model. The real model is shown in Figure 17, where the P-wave model is taken from the overthrust model, and the S-wave and density models were calculated using empirical formulae. The actual model, shown in Figure 17, was smoothed to obtain the initial model, as shown in Figure 18. This study utilised the theoretical computation of AVA reflectivity spectra to evaluate the pre-stack PP-wave seismic responses. The incident angle ranged from 0\({}^{\circ}\) to 40\({}^{\circ}\), and a Ricker wavelet with a dominant frequency of 45 Hz was used. Figure 19 shows the inversion results of the seismic data with a signal-to-noise ratio (SNR) of two. Figure 19 shows that the new propagation matrix method can effectively invert the stratigraphic parameters. The comparison in Figure 20 shows the results of the inversion based on the Zoeppritz equation and the new propagation matrix method, where the black curve represents the true model parameters, the green curve represents the initial inverted model, and the red curve represents the inversion result. Figure 20a-c show the inversion results of various parameters using different methods at an SNR of two, where the green curve represents the initial model, the red curve represents the inversion result using the new propagation matrix method, and the blue curve represents the inversion result obtained using the Zoeppritz equation. Figure 20d-f show the error values of various parameters inverted by different methods, with the red curve indicating the absolute normalised error of the inversion results using the new propagation matrix method and the blue curve representing the absolute normalised error of the inversion results using the Zoeppritz equation. Figure 20a-c show that the inversion results based on the modified propagation matrix method are more accurate than those based on the Zoeppritz equation. Moreover, Figure 20d-f show that the AVA inversion based on the Zoeppritz equation exhibits reduced accuracy with increasing depth. This can be attributed to the fact that the Zoeppritz equation does not consider the impact of transmission loss in the geological layers. As the depth of the layers increased, the influence of transmission loss increased, leading to greater errors in the AVA seismic data simulated using the Zoeppritz equation. Figure 21 shows the field data used for the inversion in this study. Figure 22 shows an inversion of the field data for a construction area in eastern China. The incidence angles were \(7^{\circ}\), \(13^{\circ}\), and \(19^{\circ}\) when the Ricker wavelet was used, which was the same as the peak frequency of the actual data, and the peak frequencies of the wavelets were 28.16 Hz, 26.34 Hz, and 23.61 Hz, respectively. Figure 23 shows the inversion results of the offset data from the well in the actual data from Figure 21 obtained using different methods. As seen in Figure 23, the accuracy and stability of the pre-stack inversion of the propagation matrix were better than those of the Zoeppritz equation inversion. Figure 19: Two-dimensional inversion results (SNR = 2). (**a**) The P-wave velocity model, (**b**) S-wave velocity model, and (**c**) density model. Figure 21: Field data. (**a**-**c**) represent the field seismic data for incidence angles of \(7^{\circ}\), \(13^{\circ}\), and \(19^{\circ}\), respectively. Figure 20: One-dimensional inversion results. (**a**-**c**) show the model data. (**d**–**f**) show the absolute error value between the different methods and the real logging data. The black curve represents the logging data. The green curve represents the initial model. The blue curve denotes the inversion result obtained using the Zoeppritz equation. The red curve is the inversion result obtained using the new propagation matrix method. Figure 23: Field data inversion results. (**a**) \(V_{\rm P}\); (**b**) \(V_{\rm S^{\prime}}\); (**c**) density.