# Vertical Seismic-Profile Data Local Full-Waveform Inversion Based on Marchenko Redatuming
## 3 Example
The wavefield \(P_{all}^{matr}\left(\text{x}_{\text{vpr}}^{\text{vsp}},\text{x}_{\text{vsp}}\right)\) of the retrieved Green's function \(G_{all}^{matr}(\text{x}_{\text{vpr}}^{\text{vsp}},\text{x}_{\text{vsp}})\) is shown in Figure 9a and can be compared with the wavefield signal (Figure 9b) obtained by waveform modeling with the true velocity model. Benefiting from the incorporation of VSP data, the errors between the redatumed wavefield and the full wavefield are very small. The laterally propagating waves caused by strike-slip faulting and cavities are accurately reconstructed. It should be noted that all the events before the arrival of the direct wave in redatumed signals need to be muted in further calculations. Since \(G_{all}^{matr}(\text{x}_{\text{vsp}},\text{x}_{\text{s}})\) and \(G_{all}^{matr}(\text{x}_{\text{vsp}},\text{x}_{\text{vsp}})\) are obtained in the same manner as \(G_{all}^{matr}(\text{x}_{\text{vpr}},\text{x}_{\text{s}})\) and \(G_{all}^{matr}(\text{x}_{\text{vpr}}^{\text{vsp}},\text{x}_{\text{vsp}})\), we will not repeat it here. In the standard Marchenko receiver redatuming-based local FWI, a total of 45 surface sources at x\({}_{\text{s}}\) = (X: 1200-4800 m; Z: 0 m) above the target area are used in the calculation, while the VSP-driven local FWI also contains 35 subsurface sources at x\({}_{\text{vsp}}\) = (X: 2500 m; Z: 6300-7000 m). Starting from the target region of the smooth velocity model in Figure 7b, the minimum frequency of the inversion is 8 Hz, the maximum frequency is 44 Hz, and the step size of the frequency is 2 Hz. To sharpen the inversion results, we use the minimum support regularization as a constraint. It should be noted that in this example, the weighting factor \(\alpha\) is 1 because the number of VSP virtual sources and surface sources in the inversion are similar and the energy is also balanced. The results of the local FWI based on the standard Marchenko redatuming are shown in Figure 10a. The result of the VSP and Surface data jointed local FWI is shown in Figure 10b. For reference, we also replaced the standard Marchenko retrieved signals with the corresponding measurement signals to perform local FWI, and the result is shown in Figure 10c. From Figure 10b, the cavity boundaries are well characterized to match the true model. Compared with Figure 10a, the inverted model in Figure 10b has a better lateral resolution, and the two cavities are clearly separated, while the strike-slip fault with a steep dip angle can be reconstructed. The result of VSP  
Figure 9: The common source gathers with a column of virtual receivers x\({}_{\text{vpr}}\) = (X: 2155 m, Z: 5555–6185 m) corresponding to a virtual source at x\({}_{\text{vsp}}\) = (X: 2500 m, Z: 6600 m) from (**a**) the Marchenko source-receiver redatuming with VSP data retrieved wavefield and (**b**) wavefield obtained by waveform modeling. and Surface data jointed local FWI still has certain errors for the characterization of the strike-slip fault compared to Figure 10c, and it is mainly due to the mismatches in the redatumed Green's function caused by the deep depth of the target area and the limited acquisition aperture. The comparison of the lateral model single trace at a depth of 5.85 km is shown in Figure 11, which can more clearly demonstrate the difference in lateral resolution and accuracy of the inversion results of different methods. Especially in cavity reservoir locations with low impedance, the results of the standard Marchenko method of local full waveform inversion cannot reconstruct accurate parameter values. The inversion result of the VSP and surface data jointed local FWI looks closer to the actual model. Figure 11: Comparison of velocity lateral profile from true model and inverted models using different methods. Figure 10: Comparison of local FWI results using different methods with (**a**) local FWI based on standard Marchenko receiver redatuming. (**b**) VSP and surface data jointed local FWI based on VSP data jointed Marchenko source-receiver redatuming.