**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 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.

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* <sup>1</sup>, *f*2, *f* <sup>3</sup>, *f* <sup>4</sup>]= [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

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.

**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**).

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

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). Figure 9b (at *X* = 2200) and Figure 9c (at *X* = 3000) show a slightly better match than Figure 9a (*X* = 1900, inside the island) due to better illumination. The comparison shows the true and the estimated results match quite well, demonstrating the effectiveness of our method on complex models.

**Figure 8.** (**a**) A 2D transition zone velocity model; (**b**) true relative impedance perturbation after band-pass filtering; and (**c**) estimated relative impedance perturbation computed using the proposed method.

**Figure 9.** Comparison of filtered true impedance perturbation (red) and estimated results (blue) at three locations: (**a**) X = 1900; (**b**) X = 2200; and (**c**) X = 3000.

### **4. Discussion**

In Equation (11b), we can see that the summation of the time derivative and the spatial gradient images had an extra term of cos2*θ* compared to the time derivative image. When we designed the small-angle imaging condition in Equation (13), we mentioned that the exponential term approximated to *e*−*α*sin2*θ*. Since this exponential weighting was applied at each time step, we can consider the proposed imaging condition as the time derivative imaging condition weighted by the term cos2*θe*−*α*sin2*θ*. Figure 10 shows the analysis of this term by choosing different parameter *α*, compared to the term cos2*θ*. In the figure, we can see that a larger *α* attenuated the large reflections more rapidly than a small *α*; on the other hand, the attenuation curve was smoother for a small *α*, which generated less artifacts than a large *α*. In all our examples in the previous section, we empirically chose *α* = 5 based on a compromise between the preservation of accuracy, and artifacts reduction. Figure 11 shows the sensitivity analysis of the inversion results using different *α* (*α* = 0, 5, 10, 20) for the layered model (Figure 4). In the figure, we can see that for *α* = 0, the velocity and impedance perturbation cannot be distinguished, while for a larger *α* (*α* = 10 and *α* = 20), the shallow part is problematic. In future studies, we may consider the functions of different forms to attenuate large reflections while still preserving the imaging quality.

**Figure 10.** Analysis of different angle-dependent functions.

**Figure 11.** Estimated relative impedance perturbation for the layered model using different *α* in the weighting function in Equation (14): (**a**) *α* = 0; (**b**) *α* = 5; (**c**) *α* = 10; and (**d**) *α* = 20.

In all our numerical examples, broadband sources were used for synthetic data generation. Broad bandwidth played an important role in seismic inversion. Increasing the bandwidth at the high-frequency end improved seismic resolution, and adding more low frequencies in the data reduced the side lobes of the seismic wavelet [18]. Figure 12 demonstrates the impact of low frequencies on the inversion result. For the Sigsbee2b model, we applied a band-pass filter (*f*<sup>1</sup> = 3 Hz, *f*<sup>2</sup> = 5 Hz, *f*<sup>3</sup> = 56 Hz, *f*<sup>4</sup> = 60 Hz) to remove low frequencies in the input data, and then estimated the impedance perturbation using our proposed method. The estimated impedance perturbations with and without low frequencies are shown in Figures 12a and 12b, respectively. Figure 12a is the same as Figure 6c, and is repeated here for comparison. Figure 12c shows the trace comparison of Figure 12a,b at the horizontal location of 3500 m. The results demonstrate that low frequencies in the data affect the long-wavelength structures in the image. Low frequencies are crucial for both velocity and impedance inversions [8], and retaining low frequencies in the recorded data depends on broadband seismic acquisition and data processing.

**Figure 12.** Comparison of estimated impedance perturbation with (**a**) and without (**b**) low frequencies in input data. (**c**) Trace comparison of estimated impedance perturbation with (blue) and without (green) low frequencies at the horizontal location of 3500 m.

As in conventional RTM, the proposed method assumes primaries only in the data, so de-multiple is a necessary preprocessing step in the case of strong multiples in the data. Even though our method was derived from the framework of amplitude-preserving RTM, the inversion was based on the Born approximation (Equation (4)). Therefore, transmission losses due to the overburden effects were also a challenge for our proposed method. The proposed method could be extended to finite-frequency based inversion [19] or FWI approaches [20] in the future work.

### **5. Conclusions**

We proposed a weighted inverse scattering imaging condition for common-shot RTM, which outputs an estimation of the relative impedance perturbation. The construction of the weighting function was based on the two separate images from the conventional inverse scattering imaging condition. The proposed imaging condition was designed to select near-angle reflections during imaging, and therefore can separate the effect of impedance perturbation from velocity perturbation, for acoustic cases with variable velocities and densities. We demonstrated the effectiveness of the modified inverse scattering imaging condition using synthetic examples and showed that our method can produce reliable estimations of impedance perturbations.

**Author Contributions:** Conceptualization, H.L. (Hong Liang) and H.Z.; methodology, H.L. (Hong Liang); software, H.L. (Hong Liang), H.Z. and H.L. (Hongwei Liu); validation, H.L. (Hong Liang); formal analysis, H.L. (Hong Liang); investigation, H.L. (Hong Liang); resources, H.L. (Hongwei Liu); data curation, H.L. (Hong Liang); writing—original draft preparation, H.L. (Hong Liang); writing—review and editing, H.Z. and H.L. (Hongwei Liu); visualization, H.L. (Hong Liang); supervision, H.Z. and H.L. (Hongwei Liu); project administration, H.L. (Hongwei Liu); funding acquisition, H.Z. and H.L. (Hongwei Liu). All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Data Availability Statement:** Synthetic data used in this paper can be obtained from the corresponding author.

**Acknowledgments:** We use the Madagascar [21] for figure plotting in this paper.

**Conflicts of Interest:** The authors declare no conflict of interest.
