## 5 Results  
### Initial Model Dependency  
Figure 8a shows the initial constant-velocity model used in the Marmousi inversion, where the velocity at each point was 1800 m/s. Without incorporating any prior structural information as a constraint, this velocity model easily falls into local minima under conventional FWI, resulting in incorrect physical solutions. The corresponding true Marmousi velocity model is presented in Figure 8b. The inversion results under conventional FWI exhibited high uncertainty and deviated completely from the true velocity structure (Figure 8c). The inversion results obtained using our proposed method were compared against the conventional FWI (Figure 8d). Under the same initial model, the FAFWI method produced relatively accurate inversion results within the effective range of the VSP observation system. FAFWI reconstructed the velocity structures in the shallow and middle layers and successfully inverted the velocity structure and variation trends in deep regions. To quantitatively evaluate the velocity field characteristics in the vicinity of the wellbore, vertical velocity profiles at distances of \(\pm\)75 m from the wellbore center were extracted (Figure 9). The profile comparison analysis indicated that the inversion results showed high consistency with the true model in most regions, from the shallow to deep layers.  
The effective inversion area exhibited a distinct inverted triangular pattern based on the spatial distribution characteristics of the velocity field inversion results. This result is consistent with the physical principles; as in the VSP observation system, the geophones are arranged vertically along the wellbore, making it difficult to effectively receive seismic reflections from lateral areas far from the shot-well pairs. Based on this difficulty, we calculated the R\({}^{2}\) score, Structural Similarity Index (SSIM), and the normalized correlation coefficient between the inversion results and true model within this range to comprehensively evaluate the linear correlation and structural similarity of the results obtained by the FAFWI method when starting from the 1800 m/s constant-velocity initial model shown in Figure 8a for the Marmousi case. The results are summarized in Table 1. The model inverted by our proposed FAFWI method demonstrated a strong linear relationship and structural similarity with the true model (Table 1) as well as advantages in terms of the normalized correlation.  
Figure 8: Comparison of inversion results between conventional FWI and FAFWI under constant velocity model for the Marmousi case. (**a**) Initial model. (**b**) True model. (**c**) Conventional FWI. (**d**) FAFWI method.  
As the inversion process continues, the velocity parameters are updated in the correct direction, leading to an increasing number of seismic traces satisfying \(|\Delta\tau|<\Delta\text{threshold}\) in Equation (15) in each iteration. The total phase difference \(\tau=\sum_{\text{s}=1}^{\text{shots}}\Delta\tau^{\text{s}}\) gradually approaches 0, and the number of selected seismic traces shows an upward trend. These phenomena are reflected in the training metrics of the Marmousi model (Figure 10). This dynamic adjustment mechanism implements the dynamic objective function in Equation (16). As the inversion progresses, more seismic traces are included, resulting in more comprehensive and accurate inversion results.  
\begin{table}
\begin{tabular}{c c} \hline
**Parameters** & **Marmousi** \\ \hline R\({}^{2}\) Score & 0.5471 \\ SSIM & 0.8390 \\ Correlation Coefficient & 0.7858 \\ \hline \end{tabular}
\end{table}
Table 1: Quantitative comparison of inversion results using the FAFWI method on the Marmousi model.  
Figure 9: Velocity profiles produced using FAFWI method for the Marmousi model. (**a**) Velocity profile at 75 m left of the wellbore. (**b**) Velocity profile at 75 m right of the wellbore.