## 2 Materials and Methods  
### Finite-Difference Acoustic Wave Equation  
The numerical simulation of wave equation forward modeling is fundamental to inversion. The time-space domain finite difference algorithm is widely used in geophysical forward modeling due to its simplicity and high computational efficiency, and it is employed to simulate the propagation of seismic waves. In the process of solving partial differential wave equations using the finite difference method, the computation area is first discretized into a grid, then grid differences are utilized to approximate the derivative operators, and the wave equation is converted into a difference equation. In a 2D isotropic acoustic medium, the time-domain wave equation is  
\[\
abla^{2}\mathrm{u}(\mathrm{r},\mathrm{t})=\frac{1}{\mathrm{v}^{2}(\mathrm{ r})}\cdot\frac{\partial\mathrm{u}^{2}(\mathrm{r},\mathrm{t})}{\partial\mathrm{t}^{2}}+ \mathrm{s}(\mathrm{r},\mathrm{t})\delta(\mathrm{r}-\mathrm{r}_{\mathrm{s}}) \tag{1}\]  
where \(\
abla\mathrm{u}^{2}(\mathrm{r},\mathrm{t})\) represents the pressure or displacement at coordinate \(\mathrm{r}\) at time \(\mathrm{t}\), \(\mathrm{V}^{2}(\mathrm{r})\) represents the speed of sound at coordinate \(\mathrm{r}\), \(\mathrm{s}(\mathrm{r},\mathrm{t})\) represents the source function at coordinate \(\mathrm{r}\) at time \(\mathrm{t}\), and \(\delta(\mathrm{r}-\mathrm{r}_{\mathrm{s}})\) is a Dirac (Delta) function, indicating that energy is released only at the source position. For the 2D medium, spatial discretization is undertaken by converting coordinates (x, y) into grid points (i, j), where i represents the index in the x direction and j representsthe index in the y direction. Uniform grid division is usually adopted, dividing the space into a series of equal-sized cells. Time t is discretized into time steps \(\Delta\)t, allowing time to progress in fixed intervals. Based on discrete spatial grids and time steps, difference approximations are used to replace derivative terms. A centered difference approximation is used to approximate the second-order spatial derivative \(\
abla\mathrm{u}^{2}(\mathrm{r},\mathrm{t})\):  
\[\
abla\mathrm{u}^{2}(\mathrm{i},\mathrm{j},\mathrm{t})\approx\frac{\mathrm{u} (\mathrm{i}+1,\mathrm{j},\mathrm{t})-2\mathrm{u}(\mathrm{i},\mathrm{j},\mathrm{ t})+\mathrm{u}(\mathrm{i}-1,\mathrm{j},\mathrm{t})}{\Delta\mathrm{x}^{2}}+ \frac{\mathrm{u}(\mathrm{i},\mathrm{j}+1,\mathrm{t})-2\mathrm{u}(\mathrm{i}, \mathrm{j},\mathrm{t})+\mathrm{u}(\mathrm{i},\mathrm{j}-1,\mathrm{t})}{\Delta \mathrm{y}^{2}} \tag{2}\]  
where \(\Delta\mathrm{x}\) and \(\Delta\mathrm{y}\) represent the grid spacing in the x and y directions, respectively. A second-order centered difference approximation is used to approximate the second-order time derivative \(\partial\mathrm{u}^{2}(\mathrm{r},\mathrm{t})/\partial\mathrm{t}^{2}\):  
\[\frac{\partial\mathrm{u}^{2}(\mathrm{r},\mathrm{t})}{\partial\mathrm{t}^{2}} \approx\frac{\mathrm{u}(\mathrm{i},\mathrm{j},\mathrm{t}+\Delta\mathrm{t})-2 \mathrm{u}(\mathrm{i},\mathrm{j},\mathrm{t})+\mathrm{u}(\mathrm{i},\mathrm{j},\mathrm{t}-\Delta\mathrm{t})}{\Delta\mathrm{t}^{2}} \tag{3}\]  
Substituting the difference approximations into the original wave equation yields the 2D time-domain finite difference acoustic wave equation:  
\[\mathrm{u}(\mathrm{i},\mathrm{j},\mathrm{t}+\Delta\mathrm{t})=\mathrm{V}^{2}( \mathrm{i},\mathrm{j})\Delta\mathrm{t}^{2}\Big{[}\
abla\mathrm{u}^{2}( \mathrm{i},\mathrm{j},\mathrm{t})-\mathrm{s}(\mathrm{i},\mathrm{j},\mathrm{t })\delta(\mathrm{i},\mathrm{j}-\mathrm{r}_{\mathrm{s}})\Big{]}+2\mathrm{u}( \mathrm{i},\mathrm{j},\mathrm{t})-\mathrm{u}(\mathrm{i},\mathrm{j},\mathrm{t}- \Delta\mathrm{t}) \tag{4}\]  
Through the difference acoustic wave equation, the wavefield at the current time can be calculated using the wavefields of the past two time steps, the source, and the velocity model.