# Vertical Seismic-Profile Data Local Full-Waveform Inversion Based on Marchenko Redatuming
## 2 Principles and Methods
### Standard Marchenko Redatuming  
In order to clarify the VSP data joint source-receiver Marchenko redatuming, we use the standard Marchenko focusing function on relating the single-sided reflection signals received at the surface to the subsurface Green's function. The coupled Marchenko equations can be written as [19],  
\[f_{1}^{-}(\mathbf{x}_{\text{\tiny{S}}},\mathbf{x}_{i},\mathbf{\omega})+G^{-}( \mathbf{x}_{i},\mathbf{x}_{\text{\tiny{S}}},\mathbf{\omega})=\int_{ \partial\text{D}_{s}}R(\mathbf{x}_{\text{\tiny{S}}},\mathbf{x}_{\text{ \tiny{S}}}^{\prime},\mathbf{\omega})f_{1}^{+}(\mathbf{x}_{\text{\tiny{S}}}^{ \prime},\mathbf{x}_{i},\where \(\mathrm{x}_{\mathrm{s}}\) is a location point on the surface \(\partial D_{\mathrm{s}}\), \(\mathrm{x}_{\mathrm{s}}^{\prime}\) is also a location on the \(\partial D_{\mathrm{s}}\). In addition, \(\mathrm{x}_{\mathrm{f}}\) is a focal point within the medium. The superscript symbol \(*\) means complex conjugate. The superscripts \(+\) and \(-\) represent down-going and up-going wavefield components, respectively. \(R(\mathrm{x}_{\mathrm{s}},\mathrm{x}_{\mathrm{s}}^{\prime})\) is the single-sided surface reflection data with the source at \(\mathrm{x}_{\mathrm{s}}^{\prime}\) and the receiver at \(\mathrm{x}_{\mathrm{s}}\). \(G(\mathrm{x}_{\mathrm{i}},\mathrm{x}_{\mathrm{s}},\omega)\) is the redatumed Green's function to be obtained, while \(f_{1}{}^{\pm}(\mathrm{x}_{\mathrm{s}},\mathrm{x}_{\mathrm{i}},\omega)\) represents the down-going and up-going focusing functions. We can calculate the coupled Marchenko equations using an iterative approach. The details can be found in Reference [20]. The wavefield \(P(\mathrm{x}_{\mathrm{f}},\mathrm{x}_{\mathrm{s}},\omega)\) can be easily obtained by the convolution of \(G(\mathrm{x}_{\mathrm{i}},\mathrm{x}_{\mathrm{s}},\omega)\) with the source wavelet. The standard Marchenko method also has its limitations. That is, the standard single-sided reflection Marchenko redatuming method is insensitive to laterally propagating waves due to the limited aperture. Thus, the missing signal of the laterally propagating waves in the receiver redatumed Green's function retrieved by the standard Marchenko method would reduce the accuracy of the local FWI results for complex geological structures. Here, we will use the synthetic data of a simple model to illustrate the problem. This variable velocity, constant density, and meshing have 1601 grid blocks horizontally and 1201 grid blocks vertically, with a grid interval of 5 m (Figure 2). The solid white lines in Figure 2 correspond to \(\partial\mathrm{D}_{b}\) as shown in Section 2.1. The area enclosed by the solid white lines is the target zone. 121 sources are uniformly placed at the surface from 1 km to 7 km. A total of 401 receivers are fixed in the range of 0 to 8 km at the surface, with a receiver interval of 20 m. The source is a Ricker wavelet with a 25 Hz dominant frequency. We apply the standard Marchenko method to estimate the Green's functions of all points in the target area to obtain the local wavefield snapshots with a surface source at 4 km in the X direction (Figure 3a). As a benchmark, we also calculate the wavefield snapshots by using finite difference forward modeling in the full-model domain (Figure 3b). It can be clearly seen from Figure 3 that the laterally propagating wave components of the diffractions are missing in the result of the standard Marchenko method compared to the benchmark wavefield. Therefore, if the standard Marchenko method is used to estimate the redatumed signal \(G_{all}^{mar}(\mathrm{x}_{\mathrm{\pi}},\mathrm{x}_{\mathrm{s}},\omega)\) for local FWI, the virtual receivers \(\mathrm{x}_{\mathrm{\pi}}\) can only be set at the top of the target. In the next section, we will describe how to use the VSP data in the redatuming process so that the estimated redatumed signals contain the laterally propagating wave components.