# Vertical Seismic-Profile Data Local Full-Waveform Inversion Based on Marchenko Redatuming
## 2 Principles and Methods
### Local FWI Based on Standard Marchenko Redatuming
\begin{array}{ll}G_{all}(\mathbf{x}_{\text{\tiny{pr}}},\mathbf{x}_{\text{ \tiny{S}}})=\frac{1}{\omega}\oint_{\partial\text{D}_{b}}\frac{1}{\rho(\mathbf{ x}_{\text{\tiny{S}}})}[G_{in}(\mathbf{x}_{\text{\tiny{S}}},\mathbf{x}_{ \text{\tiny{pr}}})\frac{\partial}{\partial n_{\text{\tiny{S}}_{\text{\tiny{S} }}}}G_{all}(\mathbf{x}_{\text{\tiny{S}}_{\text{\tiny{S}}}},\mathbf{x}_{\text{ \tiny{S}}})-\frac{\partial}{\partial n_{\text{\tiny{S}}_{\text{\tiny{S}}}}}G_{ in}(\mathbf{x}_{\text{\tiny{S}}},\mathbf{x}_{\text{\tiny{pr}}})G_{all}( \mathbf{x}_{\text{\tiny{S}}_{\text{\tiny{S}}}},\mathbf{x}_{\text{\tiny{S}}})] |d\mathbf{x}_{\text{\tiny{S}}}\quad\mathbf{x}_{\text{\tiny{S}}}\in\partial \text{D}_{b}.\end{array}  \tag{9}\]  
where \(\mathbf{x}_{\text{\tiny{S}}}\) is the virtual source point along the target area boundary \(\partial\text{D}_{b}\), \(n_{\text{\tiny{S}}_{\text{\tiny{S}}}}\) is the outward unit normal to \(\mathbf{x}_{\text{\tiny{S}}}\) along the enclosed boundary \(\partial\text{D}_{b}\). According to the reciprocity theorem, \(G_{in}(\mathbf{x}_{\text{\tiny{S}}_{\text{\tiny{pr}}}},\mathbf{x}_{\text{ \tiny{pr}}},\mathbf{\omega})\) is equivalent to \(G_{in}(\mathbf{x}_{\text{\tiny{pr}}},\mathbf{x}_{\text{\tiny{S}}_{\text{\tiny{ pr}}}},\mathbf{\omega})\), while \(\mathbf{x}_{\text{\tiny{pr}}}\) can be any point in the target area \(\text{D}_{in}\). Our goal is to obtain the wavefield in the target region; therefore, we replace \(G_{all}(\mathbf{x}_{\text{\tiny{S}}_{\text{\tiny{S}}}},\mathbf{x}_{\text{ \tiny{S}}},\mathbf{\omega})\) with the wavefield represented as \(P_{all}(\mathbf{x}_{\text{\tiny{S}}_{\text{\tiny{S}}}},\mathbf{x}_{\text{ \tiny{S}}},\mathbf{\omega})\). We can then obtain the local forward modeling operator,  
\[ \begin{array}{ll}\text{F}(m_{in},\omega)=\frac{1}{\omega}\oint_{\partial\text {D}_{b}}\frac{1}{\rho(\mathbf{x}_{\text{\tiny{S}}_{\text{\tiny{S}}}})}[G_{in} (\mathbf{x},\mathbf{x}_{\text{\tiny{S}}_{\text{\tiny{S}}}},\mathbf{\omega}) \frac{\partial}{\partial n_{\text{\tiny{S}}_{\text{\tiny{S}}_{\text{\tiny{S}}}}}} P_{all}(\mathbf{x}_{\text{\tiny{S}}_{\text{\tiny{S}}}},\mathbf{x}_{\text{ \tiny{S}}},\mathbf{\omega})-\frac{\partial}{\partial n_{\text{\tiny{S}}_{\text{ \tiny{S}}_{\text{\tiny{S}}}}}}G_{in}(\mathbf{x},\mathbf{x}_{\text{\tiny{S}}_{ \text{\tiny{S}}}},\mathbf{\omega})P_{all}(\mathbf{x}_{\text{\tiny{S}}_{\text{ \tiny{S}}}},\mathbf{x}_{\text{\tiny{S}}},\mathbf{\omega})]dx_{\text{\tiny{S}}_{ \text{\tiny{S}}}}\;.\end{array}  \tag{10}\]  
where \(\mathbf{x}\) is an array representing all points within the target zone \(\text{D}_{in}\), and \(m_{in}\) is the model parameters in the target zone. The result obtained by the local forward modeling operator \(\text{F}(m_{in},\omega)\) is the local frequency-domain wavefield within the local model. According to the Marchenko redatuming introduced in the previous section, we only need the surface single-sided reflection data and the smoothed background velocity to obtain \(P_{all}(x_{\text{\tiny{S}}_{\text{\tiny{S}}}},\mathbf{x}_{\text{\tiny{S}}}, \mathbf{\omega})\). Meanwhile, \(G_{in}(\mathbf{x}_{in},\mathbf{x}_{\text{\tiny{S}}_{\text{\tiny{S}}}},\mathbf{ \omega})\) is obtained by using the wave equation frequency-domain finite-difference modeling with the local model parameters \(m_{in}\). When the model parameters are perturbed in the target zone, then \(G_{in}(\mathbf{x},\mathbf{x}_{\text{\tiny{S}}_{\text{\tiny{S}}}},\mathbf{\omega})\) can be calculated by finite-difference modeling. \(G_{all}(\mathbf{x}_{\text{\tiny{S}}_{\text{\tiny{S}}}},\mathbf{x}_{\text{ \tiny{S}}},\mathbf{\omega})\) is obtained by the standard Marchenko redatuming method. We will describe how to obtain the receiver redatumed signal \(P_{all}(\mathbf{x}_{\text{\tiny{S}}_{\text{\tiny{S}}}},\mathbf{x}_{\text{ \tiny{S}}},\mathbf{\omega})\) using the standard Marchenko method in Section 2.2. Referring to Equations (1)-(4) to obtain the objective function of the local FWI.