# Vertical Seismic-Profile Data Local Full-Waveform Inversion Based on Marchenko Redatuming
## 2 Principles and Methods
### VSP-Driven Local FWI Method  
Section 2.1 stated that when using only the standard Marchenko receiver redatuming method, we can only use the estimated signal between the surface sources and the virtual  
Figure 4: The common receiver gathers corresponding to the virtual receiver (VSP receiver) at x\({}_{j}\) (X: 2500 m; Z: 5040 m). From (**a**), the standard [PERSON] receiver redatuming method retrieved wavefield of Green’s function \(G(\mathrm{x}_{j},\mathrm{x}_{s})\); from (**b**), the measurement VSP data from direct modeling. Figure 5: A comparison of wavefields of the retrieved Green’s functions between a virtual source (X: 2500 m; Z: 5040 m) and virtual receivers (X: 3500 m; Z: 4170–5020 m) based on different methods. (**a**) The estimated wavefield using the standard Marchenko source-receiver redatuming. (**b**) The estimated wavefield using the Marchenko source-receiver redatuming with VSP data. (**c**) The waveform modeling result based on the true model. receivers at the top position of the target area as the observed data due to the inaccuracy of the lateral propagation wave redatuming. However, with the VSP Marchenko redatuming method, [PERSON]'s function can be obtained more accurately for the virtual receivers on the lateral side of the target, provided that the location of the virtual receivers is above the VSP receivers. Therefore, the VSP-driven local FWI requires only a simple substitution of the Green's functions between VSP receivers and virtual receivers into the local FWI method. The array of VSP receiver points is denoted as \(\mathsf{x}_{vsp}\). We calculate the Green's functions \(G^{marr}_{all}(\mathsf{x}_{vss},\mathsf{x}_{vsp},\omega)\) and \(G^{marr}_{all}(\mathsf{x}_{vpr}^{sp},\mathsf{x}_{vsp},\omega)\) by the VSP Marchenko method. \(P^{marr}_{all}(\mathsf{x}_{vss},\mathsf{x}_{vsp},\omega)\) and \(P^{marr}_{all}(\mathsf{x}_{vpr}^{sp},\mathsf{x}_{vsp},\omega)\) are the wavefields of \(G^{marr}_{all}(\mathsf{x}_{vss},\mathsf{x}_{vsp},\omega)\) and \(G^{marr}_{all}(\mathsf{x}_{vpr}^{spsp},\mathsf{x}_{vsp},\omega)\), then Equation (10) can be modified as follows,  
\[\mathrm{F}^{vsp}(m_{int},\omega)=\frac{1}{\iota\omega}\oint_{\partial\mathrm{ D}_{b}}\frac{1}{\rho(\mathsf{x}_{vgs})}[G_{int}(\mathsf{x},\mathsf{x}_{vgs} )\frac{\partial}{\partial n_{\mathsf{x}_{vgs}}}P^{marr}_{all}(\mathsf{x}_{vgs },\mathsf{x}_{vsp},\omega)-\frac{\partial}{\partial n_{\mathsf{x}_{vgs}}}G_{ in}(\mathsf{x},\mathsf{x}_{vgs},\omega)P^{marr}_{all}(\mathsf{x}_{vgs}, \mathsf{x}_{vsp},\omega)]dx_{\mathrm{vgs}}\;. \tag{17}\]  
and Equation (11) can be written as,  
\[\min_{m_{in}}E(m_{in},\omega)=\frac{1}{2}\sum_{\mathsf{x}_{vsp}}\left\| \mathbb{P}_{vr}^{vsp}\mathrm{F}^{vsp}(m_{in},\omega)-P^{marr}_{all}(\mathsf{x }_{vpr}^{vsp},\mathsf{x}_{vsp},\omega)\right\|_{2}^{2}. \tag{18}\]  
It should be noted that \(\mathbb{P}_{vr}^{vsp}\) in Equation (18) is different from \(\mathbb{P}_{vr}\) in Equation (11). The differences are shown in Figure 6. The capability of including the laterally propagating waves from the VSP Marchenko source-receiver redatuming method allows the improvement of the lateral resolution for the inversion. Therefore, \(\mathbb{P}_{vr}^{vsp}\) is set mainly vertically on the lateral side of the target. By combining Equations (11) and (18), the objective function of the VSP and Surface data jointed local FWI method can be written as:  
\[\min_{m_{in}}E(m_{in},\omega)=\frac{1}{2}\sum_{\mathsf{x}_{s}}\left\|\mathbb{ P}_{vv}\mathrm{F}(m_{int},\omega)-P^{marr}_{all}(\mathsf{x}_{vpr},\mathsf{x}_{s },\omega)\right\|_{2}^{2}+\frac{1}{2}\alpha\sum_{\mathsf{x}_{vsp}}\left\| \mathbb{P}_{vr}^{vsp}\mathrm{F}^{vsp}(m_{in},\omega)-P^{marr}_{all}(\mathsf{x }_{vpr}^{vsp},\mathsf{x}_{vsp},\omega)\right\|_{2}^{2}. \tag{19}\]  
where \(\alpha\) is a weighting factor to balance the residuals and gradients of the two terms on the right side of the equation.