# Vertical Seismic-Profile Data Local Full-Waveform Inversion Based on Marchenko Redatuming
## 2 Principles and Methods  
### Local FWI Based on Standard Marchenko Redatuming  
According to waveform-matching criteria, the FWI method can be used to obtain parameters of underground media using amplitude, travel time, and phase information. The traditional frequency domain FWI objective function can be expressed as [6]:  
\[\min_{m}E(m,\omega)=\frac{1}{2}\left\|\mathbb{P}A(m,\omega)^{-1}f(x,z,\omega)-D_ {obs}(\omega)\right\|_{2}^{2}. \tag{1}\]  
where \(m\) is the model parameters, \(\omega\) is the angular frequency, \(A\) is the impedance matrix, and \(f\) is the source operator (to ensure the shot locations and amplitude energy). \(D_{obs}\) is the observed seismic data and \(\mathbb{P}\) is the receiver vector (to ensure the receiver location corresponds to the observed data). In the subsequent part, the spatial coordinates \(x\) and \(z\) will be implicitly included in the subsequent equations. In frequency domain FWI, the objective function is the functional under the L2 norm, which can be solved using the adjoint state method [27]:  
\[\delta E=\omega^{2}diag(u)^{*}v. \tag{2}\]  
where \(u\) is the forward wavefield, \(v\) is the adjoint wavefield and superscript symbol \({}^{*}\) means complex conjugate. This means that we have to solve the following forward modeling equations iteratively at each frequency or frequency group:  
\[A(m)u=f, \tag{3}\]  
and  
\[A(m)^{*}v=\mathbb{P}^{T}(\mathbb{P}u-D_{obs}). \tag{4}\]  
Under the conditions of a complete full-domain model, \(A\) is usually an enormously sparse matrix. The method, therefore, has high computational requirements and low computational efficiency, especially in high frequency because fine spatial sampling intervals are required to stabilize the wavefield simulations. When only a small area in the subsurface is the target, the inversion of the entire model could be computationally very wasteful. The finite difference wavefield injection (FD injection) method records the Green's functions of all points on the boundary of a target and injects the background wavefield of the full model into the local region. By applying the wavefield injection method, the seismic response of the entire model can be updated by simulating only the local wavefield when only the local model parameters are changed. A source field can be introduced into a localized finite difference grid by injecting an analytical source solution along a closed surface [15]. Similarly, the wavefile recorded along a closed surface can be used as a source wavefield to compute the wavefield within the injected grid region. The main idea of FD injection is to satisfy the principles of superposition and wavefield continuity across the injection surface. That is, we can simulate only [PERSON]'s function inside the injected volume and thus obtain the wavefield of the subsurface target range corresponding to the surface source [15]. The mathematical expression of this process can be given by the convolution-type representation theorem, which we will describe below. We follow this idea and derive the local forward modeling operator in the following sections. We define the model as two computational spaces. The two computational states can be illustrated as shown in Figure 1.  
The surface is represented as the black dashed line \(\partial\mathrm{D}_{s}\). The dashed rectangular box \(\partial\mathrm{D}_{b}\) is the boundary of the target. \(\mathrm{D}_{in}\) is the internal region of the target zone, \(\mathrm{D}_{out}\) is the external region. The first computational state is the complete model, containing \(\mathrm{D}_{in}\) and \(\mathrm{D}_{out}\). The second computational space is a truncated medium with \(\mathrm{D}_{in}\) only. Under the first computational state, assume a point \(\mathrm{x}_{s}\) at the surface as the source, then the Green's function received by a point \(\mathrm{x}\) inside \(\mathrm{D}_{in}\) is written as \(G_{all}(\mathrm{x},\mathrm{x}_{s})\) (blue arrowed line inFigure 1).