# Vertical Seismic-Profile Data Local Full-Waveform Inversion Based on Marchenko Redatuming
## 2 Principles and Methods
### Marchenko Source-Receiver Redatuming with VSP Data  
Based on the one-way reciprocity theorems of convolution and correlation [29], [PERSON] and [PERSON] present the formula for the source-receiver Marchenko redatuming method [25]:  
\[G^{-}(\mathrm{x}_{i},\mathrm{x}_{j},\omega)=\int_{\partial D_{s}}2(\iota\omega p (\mathrm{x}_{s}))^{-1}\partial_{z}G(\mathrm{x}_{s},\mathrm{x}_{j},\omega)f^{+} (\mathrm{x}_{s},\mathrm{x}_{i},\omega)dx_{\mathrm{s}}\;, \tag{14}\]  
and  
\[G^{+}(\mathrm{x}_{i},\mathrm{x}_{j},\omega)=-\int_{\partial D_{s}}2(\iota\omega p (\mathrm{x}_{s}))^{-1}\partial_{z}G(\mathrm{x}_{s},\mathrm{x}_{j},\omega)f^{-*} (\mathrm{x}_{s},\mathrm{x}_{i},\omega)dx_{\mathrm{s}}\;. \tag{15}\]  
where \(\iota\) is the imaginary unit. \(G(\mathrm{x}_{i},\mathrm{x}_{j},\omega)\) is the Green's function between the subsurface virtual source \(\mathrm{x}_{j}\) and the subsurface virtual receiver \(\mathrm{x}_{i}\). \(G(\mathrm{x}_{s},\mathrm{x}_{j},\omega)\) and \(f^{\pm}(\mathrm{x}_{s},\mathrm{x}_{i},\omega)\) on the right-hand side of Equations (14) and (15) can be calculated by the standard Marchenko Equations (12) and (13). However, as shown in Section 2.2, the Green's function \(G(\mathrm{x}_{s},\mathrm{x}_{j},\omega)\) obtained by the standard Marchenko method may lack the laterally propagating waves. VSP data \(R_{VSP}(\mathrm{x}_{j},\mathrm{x}_{s})\) is the signal at the receivers in the borehole. Compared to the redatumed Green's function \(G(\mathrm{x}_{s},\mathrm{x}_{j})\) retrieved by the standard Marchenko method, the VSP data contain accurate laterally propagating wave information. The reciprocity theorem suggests that \(R_{VSP}(\mathrm{x}_{j},\mathrm{x}_{s})\) and \(R_{VSP}(\mathrm{x}_{s},\mathrm{x}_{j})\) are equivalent. This means that if we replace \(G(\mathrm{x}_{s},\mathrm{x}_{j},\omega)\) Equations (14) and (15) with \(R_{VSP}(\mathrm{x}_{s},\mathrm{x}_{j},\omega)\), we can obtain more accurate [PERSON]'s functions. Therefore, we can derive the formula of the Marchenko source-receiver redatuming method with VSP data [26]:  
\[G(\mathrm{x}_{i},\mathrm{x}_{j},\omega)=\int_{\partial D_{s}}2(\iota\omega p( \mathrm{x}_{s}))^{-1}\partial_{z}R_{VSP}(\mathrm{x}_{s},\mathrm{x}_{j},\omega )(f^{+}(\mathrm{x}_{i},\mathrm{x}_{s},\omega)-f^{+*}(\mathrm{x}_{i},\mathrm{x }_{s},\omega))dx_{s}\;. \tag{16}\]  
It is important to note that Equations (14)-(16) need to satisfy the condition that \(\mathrm{x}_{i}\) is above \(\mathrm{x}_{j}\) in depth. In addition, due to the acquisition system of the VSP data, the positions of \(\mathrm{x}_{j}\) are fixed (the positions of the receivers in the well). In other words, with Equation (16), we can obtain the accurate Green's function between the virtual source at the locations of the receivers in the borehole and the virtual receivers at any points above the depth of that virtual source. Similarly, we use the simple model in Section 2.2 as an example. Keeping the original acquisition system unchanged, the VSP receivers are set at depth intervals of 20 m in the well and at 2500 m in the X-direction. We estimate the Green's function \(G(\mathrm{x}_{j},\mathrm{x}_{s})\) with an array of surface sources and a single subsurface virtual receiver at coordinates (X: 2500 m, Z: 5040 m) using the standard Marchenko receiver redatuming. Its comparison with the measured VSP signal \(R_{VSP}(\mathrm{x}_{j},\mathrm{x}_{s})\) is shown in Figure 4.  
It can be clearly noticed from Figure 4 that the Green's function \(G(\mathrm{x}_{j},\mathrm{x}_{s})\) obtained by the standard Marchenko receiver redatuming method lacks the lateral propagating wave components from diffractions compared to the actual measured VSP signal.