## 2 Theory  
### FWI Math Model  
As mentioned earlier, FWI is a high-resolution imaging technique that uses the complete seismic waveform information to invert subsurface medium parameters. The core principle of FWI is to estimate these parameters through an iterative optimization process that minimizes the difference between observed and simulated seismic data. This process is fundamentally based on the wave equation, often represented in matrix form as follows [23]:  
\[M(x)\frac{d^{2}u(x,t)}{dt^{2}}=A(x)u(x,t)+s(x,t), \tag{1}\]  
where \(M\) and \(A\) represent the mass and stiffness matrices, respectively. The source term is denoted as \(s(r,t)\), while the seismic wavefield is represented by \(u(x,t)\), where \(x\) denotes the spatial position and \(t\) represents time. For a 2D point source, the acoustic wave equation can be expressed as:  
\[\
abla^{2}u(x,t)=\frac{1}{v^{2}(r)}\cdot\frac{\partial^{2}u(x,t)}{\partial t^ {2}}+s(x,t)\delta(x-x_{s}), \tag{2}\]  
where \(\
abla^{2}u(x,t)\) represents the pressure at coordinate \(r\) at \(t\), \(v^{2}(r)\) denotes the acoustic velocity at coordinate \(r\), \(s(r,t)\) is the source function at coordinate \(r\) at time \(t\), and \(\delta(x-x_{s})\) is the Dirac delta function, indicating that the energy of the source function is released exclusively at the source location. Forward-wave equation modeling is performed using numerical methods to generate simulated data records in the time-space domain. The objective function is defined as the discrepancy between the observed and simulated data. Typically, the L2 norm is employed to quantify this discrepancy, as shown in (3) [24]:  
\[J_{D}=\left\|d_{\text{obs}}(x_{r},x_{s},t,m)-d_{\text{cal}}(x_{r},x_{s},t,m_{0} )\right\|^{2}, \tag{3}\]  
where \(m_{0}\) is the model parameter vector, \(d_{obs}\) represents the observed data obtained from field survey data or simulated data from a true model, \(d_{cal}\) denotes the simulated data obtained by solving the wave equation, and \(x_{r}\) and \(x_{s}\) represent the receiver and source positions, respectively. The method of acquiring observational data \(d_{obs}\) depends on the observation system. For instance, surface wave observation systems deploy multiple receivers on the ground surface to record seismic waves [25], enhancing FWI resolution in shallow layers. This study employs VSP, a specialized seismic exploration technique where seismic receivers are positioned in boreholes, and the seismic source is typically located on the surface [26]. This configuration allows for the direct measurement of seismic waves propagating from the surface to the borehole, providing high-resolution data on subsurface medium properties. In FWI, VSP data offer several advantages, including high signal-to-noise ratio (SNR) near-well measurements, which can serve as initial models or prior constraints. Moreover, VSP data are rich in low-frequency information, which facilitates FWI convergence and enhances inversion accuracy and stability [27; 28]. Constructing an appropriate objective function is crucial for enhancing the resolution of FWI, and the integration of additional seismic information into the objective function can significantly accelerate convergence. By defining the objective function \(J_{D}\) for full-waveform inversion, the FWI process is reformulated as an optimization problem:  
\[m_{\text{opt}}=\arg\min_{m}J_{D} \tag{4}\]where \(J_{D}\) represents the objective function defined under the L2 norm. FWI is a highly nonlinear and ill-posed inverse problem that typically requires an iterative solution. The gradient of the objective function with respect to the model parameters is generally computed using the adjoint-state method [29]. Subsequently, the model is updated based on the gradient by selecting an appropriate optimization algorithm. Common optimization algorithms include the steepest descent method and the conjugate gradient method, both of which enhance the robustness of FWI [30].