## 2 Problem Analysis and Theory
### Math Model of FWI  
The goal of an FWI is to determine model parameters that minimize the difference between the synthetic seismic response and actual observed data. Typically, this optimization problem can be expressed as follows [1; 2]:  
\[\mathbf{m}_{\mathrm{opt}}=\underset{\mathbf{m}}{\arg\min}\ J_{D}(\mathbf{m}), \tag{1}\]  
where \(\mathbf{m}\) represents the model parameter vector and \(J_{D}\) is the objective function that measures the difference between the observed data and synthetic data. To quantify this difference, the L\({}_{2}\) norm is typically used as follows:  
\[J_{D}=\sum_{s=1}^{N_{\mathrm{s}}}\sum_{t=1}^{T}\|\ \mathbf{d}_{\mathrm{obs}}( \mathbf{x}_{r},\mathbf{x}_{s},t;\mathbf{m})-\mathbf{d}_{\mathrm{cal}}( \mathbf{x}_{r},\mathbf{x}_{s},t;\mathbf{m}_{0})\|^{2}, \tag{2}\]  
where \(\mathbf{d}_{\mathrm{obs}}\) is the observed data, \(\mathbf{d}_{\mathrm{cal}}\) is the simulated data calculated using the initial model, m\({}_{0}\), x\({}_{r}\), and x\({}_{s}\) represent the receiver and source positions, respectively, N\({}_{\mathrm{s}}\) is the total number of sources, and T is the number of time steps. The \(\left|\left|\cdot\right|\right|^{2}\) operator denotes the squared L\({}_{2}\) norm, which calculates the sum of the squared differences between the elements of the enclosed data residual vector. The synthetic data are derived by solving the wave equation using numerical methods, and the observation system employed is vertical seismic profiling (VSP). VSP is an important geophysical exploration technique that requires placing receivers in a borehole to record seismic signals from the surface or borehole sources [28]. This method can produce high-resolution images of subsurface structures. Compared to conventional surface seismic exploration, VSP offers a higher resolution and signal-to-noise ratio, capturing more signals reflected from subsurface media and providing more accurate time-depth relationships and velocity information [29]. In FWI, VSP data can provide near-wellbore data or logging velocity as prior information, helping to reduce the dependence on the initial model [30].  
To minimize the objective function \(J_{D}\) (Equation (2)), it is necessary to compute its gradient with respect to the model parameter m. The gradient calculation formula is given as follows:  
\[\frac{\partial J_{D}}{\partial\mathbf{m}}=-2\sum_{s=1}^{N_{\mathrm{s}}}\sum_{t =1}^{T}\biggl{(}\frac{\partial\mathbf{d}_{\mathrm{cal}}(\mathbf{x}_{r}, \mathbf{x}_{s},t,\mathbf{m})}{\partial\mathbf{m}}\biggr{)}^{\top}(\mathbf{d} _{\mathrm{obs}}(\mathbf{x}_{r},\mathbf{x}_{s},t)-\mathbf{d}_{\mathrm{cal}}( \mathbf{x}_{r},\mathbf{x}_{s},t,\mathbf{m})), \tag{3}\]  
After computing the gradient, a suitable optimization algorithm was employed to update the model parameters based on the gradient. Commonly used optimization algorithms included the conjugate gradient and Newton's methods. Once the direction and step size for the update were determined according to the selected optimization algorithm, the model parameters were updated using the following rule:  
\[\mathbf{m}^{k+1}=\mathbf{m}^{k}-\alpha_{k}\frac{\partial J_{D}}{\partial \mathbf{m}}, \tag{4}\]  
where \(\alpha_{k}\) is the step size parameter computed by the optimization algorithm. The above process describes the traditional FWI workflow, in which the model parameters are iteratively updated to minimize the objective function until convergence is achieved, ultimately yielding the final FWI subsurface medium results.