# Vertical Seismic-Profile Data Local Full-Waveform Inversion Based on Marchenko Redatuming
## 2 Principles and Methods
### Local FWI Based on Standard Marchenko Redatuming
For the two-dimensional isotropic acoustic medium, the Green's function in the frequency domain can be expressed by the second-order acoustic wave equation [28]:  
\[\frac{\partial}{\partial x}\big{[}\frac{1}{\rho(x)}\frac{\partial}{\partial x}G(x,x^{\prime},\omega)\big{]}+\frac{\partial}{\partial z}\big{[}\frac{1}{\rho(x)} \frac{\partial}{\partial z}G(x,x^{\prime},\omega)\big{]}+\omega^{2}m(x)\frac{1 }{\rho(x)}G(x,x^{\prime},\omega)=-\iota\omega\delta(x-x^{\prime}). \tag{5}\]  
where \(m(x)\) is the squared slowness at point x, \(\rho\) is the density and \(\iota\) is the imaginary unit. \(\omega\) is the angular frequency and \(G(x,x^{\prime},\omega)\) denotes the Green's function between the source x\({}^{\prime}\) and the receiver point x at \(\omega\) angular frequency. In the second computational state, only the truncated space D\({}_{in}\) exists. \(G_{all}(x,x_{s})\) can be represented by the source-free acoustic wave equation [14]:  
\[\frac{\partial}{\partial x}\big{[}\frac{1}{\rho(x)}\frac{\partial}{\partial x }G_{all}(x,x_{s},\omega)\big{]}+\frac{\partial}{\partial z}\big{[}\frac{1}{ \rho(x)}\frac{\partial}{\partial z}G_{all}(x,x_{s},\omega)\big{]}+\omega^{2}m( x)\frac{1}{\rho(x)}G_{all}(x,x_{s},\omega)=0\qquad x\in\mathrm{D}_{in}. \tag{6}\]  
and the Green's function of the point source x\({}_{vr}\) is noted as \(G_{in}(x,x_{vr})\) (red arrowed line in Figure 2), that is,  
\[\frac{\partial}{\partial x}\big{[}\frac{1}{\rho(x)}\frac{\partial}{\partial x }G_{in}(x,x_{vr},\omega)\big{]}+\frac{\partial}{\partial z}\big{[}\frac{1}{ \rho(x)}\frac{\partial}{\partial z}G_{in}(x,x_{vr},\omega)\big{]}+\omega^{2}m( x)\frac{1}{\rho(x)}G_{in}(x,x_{vr},\omega)=-\iota\omega\delta(x-x_{vr})\qquad x\in \mathrm{D}_{in}. \tag{7}\]  
Figure 1: Schematic of Green’s function in different computational states. Blue arrows mean the Green’s function between the surface source x\({}_{s}\) and a point in the target area x in the medium state of the full-domain model. Red arrows mean the Green’s function between x and a virtual receiver x\({}_{vr}\) in the truncated medium state. Figure 2: A simple model and the target zone (area enclosed by the solid white lines). VSP receivers are set along the black dotted line at 2500 m in the X direction. Combining Equation (6) multiplied by \(G_{in}(\mathbf{x},\mathbf{x}_{\text{\tiny{pr}}})\) and Equation (7) multiplied by \(G_{all}(\mathbf{x},\mathbf{x}_{\text{\tiny{S}}})\), and integrating both sides of the combined equation over \(D_{in}\), we have,  
\[ \begin{array}{ll}G_{all}(\mathbf{x}_{\text{\tiny{pr}}},\mathbf{x}_{\text{ \tiny{S}}})=\frac{1}{\omega}\int_{\text{D}_{in}}&G_{in}(\mathbf{x},\mathbf{x}_{ \text{\tiny{pr}}})\{\frac{\partial}{\partial x}[\frac{1}{\rho(\mathbf{x})} \frac{\partial}{\partial x}G_{all}(\mathbf{x},\mathbf{x}_{\text{\tiny{S}}})]+ \frac{\partial}{\partial x}[\frac{1}{\rho(\mathbf{x})}\frac{\partial}{ \partial x}G_{all}(\mathbf{x},\mathbf{x}_{\text{\tiny{S}}})]\}-\\ &G_{all}(\mathbf{x},\mathbf{x}_{\text{\tiny{S}}})\{\frac{\partial}{\partial x }[\frac{1}{\rho(\mathbf{x})}\frac{\partial}{\partial x}G_{in}(\mathbf{x}, \mathbf{x}_{\text{\tiny{pr}}})]+\frac{\partial}{\partial x}[\frac{1}{\rho( \mathbf{x})}\frac{\partial}{\partial x}G_{in}(\mathbf{x},\mathbf{x}_{\text{ \tiny{pr}}})]\}dx^{2}\qquad\mathbf{x}\in\text{D}_{in}.\end{array}  \tag{8}\]  
According to the Gaussian theorem, the volume integral is equal to the enclosed boundary surface integral; then Equation (8) can be written as the functional convolution-type representation theorem [12; 13; 14],  
\[