# Sensitivity Analysis of P-Wave Polarization Direction and Velocity Gradient Inversion
## 5 Tests and Discussion
### The Effect of Improving Inversion Results  
In this section, we conducted a numerical experiment on a flat-layer model with dimensions of \(200\times 400\) m\({}^{2}\). Within this model, two horizontal layers with velocities of 5250 m/s and 5300 m/s are embedded in the background medium with a velocity of 5000 m/s, as shown in Figure 3. We discretized the model, with 16 grid points in the horizontal direction and 32 grid points in the vertical direction. A total of 32 sources were placed at equal intervals between deep 2 m and 392 m on the left side of the model. Additionally, 32 receivers were placed at equal intervals between deep 4 m and 389 m on the right side of the model. Figure 2: Partial derivative values. (**a**) The partial derivative values at the eighth row for eight parallel rays generated by the homogeneous velocity model; (**b**) The partial derivative values at the eighth row for eight parallel rays generated by the velocity model with a 50 m/s velocity difference; (**c**) The partial derivative values at the eighth row for eight parallel rays generated by the velocity model with a 100 m/s velocity difference; (**d**) The partial derivative values at the eighth row for eight parallel rays generated by the velocity model with a 150 m/s velocity difference; (**e**) The partial derivative values at the eighth row for eight parallel rays generated by the velocity model with a 200 m/s velocity difference; (**f**) The partial derivative values at the eighth row for eight parallel rays generated by the velocity model with a 250 m/s velocity difference. The actual projection data are obtained through numerical simulation by solving Equations (1) and (2). As shown in Figure 4, a total of 1022 rays were generated from left to right, which were used for the inversion of the relative velocity gradient. Based on the flat-layer model, we used both the ART method and our proposed method to invert for the relative velocity gradient. The estimated projection data used in both methods are based on Equation (6). ART methods employ the iterative scheme of Equation (21), whereas our proposed approach adopts the iterative scheme of Equation (23). Figure 5 shows the relative velocity gradient maps obtained through two methods. To provide a clearer visualization of the inversion results near the source, we enlarged the six red dashed boxes marked in Figure 5, as shown in Figure 6. Specifically, Figure 6a through Figure 6f correspond to the dashed boxes labeled 1 through 6, respectively, in Figure 5. From Figures 5 and 6, it can be observed that the relative velocity gradients obtained using the ART inversion method perform poorly near the source location. This is attributed to the limited number of projections and the insensitivity of P-wave polarization direction data to relative velocity gradients near the source, resulting in inadequate updates of the relative  
Figure 4: The whole ray paths of the flat-layer model. Figure 3: Flat-layer velocity model. velocity gradients in that vicinity. In contrast, the relative velocity gradients obtained by our proposed method show significant improvement near the source location. The comparison chart of the vertical slices of the inversion results at 25 m in Figure 5 is shown in Figure 7. As depicted in Figure 7, it can be observed that the relative velocity gradient values obtained from our proposed method are closer to the true values compared to those from the ART method, especially near the source. This demonstrates the significant accuracy advantage of our inversion method, particularly in regions closer to the source. Figure 5: Relative velocity gradient maps obtained through two methods. (**a**) The relative velocity gradient map obtained through the ART inversion method; (**b**) The relative velocity gradient map obtained through the proposed method.