## 3 Numerical Examples  
In this section, we demonstrate the effectiveness of the proposed imaging condition for the impedance perturbation estimation (Equation (14)) on synthetic data. The first example was designed to demonstrate the effectiveness of the proposed method on a laterally invariant model with uncorrelated velocity and density. Figure 4a,b show the velocity and density model, respectively. Figure 4c shows the true relative impedance perturbation. We used the velocity and density in the first layer as the background velocity and density, respectively. We first subtracted the background impedance from the exact impedance and then analytically calculated the relative impedance perturbation. Figure 5a,b show the filtered true velocity and impedance perturbation, respectively. Figure 5c illustrates the impedance perturbation estimated by our proposed method, which matches well with the true results. Note that the depth errors in the estimated acoustic impedance are due to the fact that we used a constant migration velocity. Figure 3: Comparison of imaging results of a single trace (offset = 0 m) (a) using Equation (13) and (b) using Equation (14). Next, we demonstrated the acoustic impedance estimation on the Sigsbee2b model [17], and focused on the sediment areas with fine impedance structure. We performed acoustic finite difference modeling for synthetic data generation using a broadband source wavelet (\([f_{1},f_{2},f_{3},f_{4}]\)= [0, 2, 56, 60] Hz). The exact velocity model is shown in Figure 6a, from which the density model was generated using a predefined relationship of the two parameters. A \(V(z)\) velocity model and a constant density model were used for migration. We analytically calculated the exact impedance perturbation and applied the band-pass filtering to the exact result according to the bandwidth of input data. The filtered true impedance perturbation is shown in Figure 6b. In comparison, Figure 6c illustrates the  
Figure 4: (**a**) Velocity model; (**b**) density model; and (**c**) true relative impedance perturbation. Figure 5: (**a**) Filtered true relative velocity perturbation; (**b**) filtered true relative impedance perturbation; and (**c**) estimated relative impedance perturbation. impedance perturbation results estimated by our proposed method. Figure 6d displays the overlay of the two images (Figure 6b,c). The detailed comparison between Figure 6b,c at the three different locations are shown in Figure 7. An amplitude calibration was performed using reflections from the water bottom. In Figure 7, the filtered true impedance perturbation is shown in red and the estimated impedance perturbation is shown in blue. The numerical results demonstrate the overall good match between the estimated and the true impedance perturbation. In the third example, we demonstrated the proposed method on a 2D transition zone model. The exact velocity model (labels show grid numbers) is illustrated in Figure 8a, and the density model had similar structural characteristics (not shown here). As in the  
Figure 6: (**a**) Velocity model for synthetic data generation; (**b**) filtered true impedance perturbation; (**c**) impedance perturbation estimated by the proposed method; and (**d**) overlay plot of the two images (red: true, blue: estimated). Figure 7: Comparison of filtered true impedance perturbation (red) and estimated results (blue) at three horizontal locations: 2000 m (**a**); 3500 m (**b**); and 5000 m (**c**). previous example, synthetic data were generated using a broadband source wavelet. A smoothed version of the true velocity was used for migration, with the assumption of a constant density. Figure 8b shows the filtered true impedance perturbation, while Figure 8c illustrates the estimated results using the proposed method. Detailed trace comparisons between the two are shown in Figure 9 at the three different horizontal locations (\(X=1900\), \(2200\), \(3000\)).