## 2 Materials and Methods
### RPC Model Reconstruction and Model Optimization  
Reconstructing the RPC model and optimizing its parameters are crucial for the atmospheric correction of SAR imagery. Converting atmospheric delay into geometric model parameters is a key aspect of RPC model reconstruction. Slant-range measurement errors caused by atmospheric delay are typically manifested as positional displacements of ground objects along the range direction within the image. This primarily results in translational shifts in the column direction, while the row direction remains largely unaffected [53]. Therefore, the transformation of atmospheric delay into pixel correction parameters can be expressed as follows:  
\[\Delta C=\frac{\Delta R_{atm}}{m_{r}} \tag{18}\]  
where \(\Delta C\) is the correction of the column coordinates, and \(m_{r}\) is the range-oriented resolution of the image. The reconstruction of the RPC model based on pixel correction parameters can be expressed as follows:  
\[ \begin{split} R&=R_{scale}R_{n}+R_{off}\\ C+\Delta C&=C_{scale}C_{n}+C_{off}\end{split}  \tag{19}\]  
Next, in the subsequent phase of the analysis, the parameters of the RPC model are recalibrated. The methodology for determining these parameters can generally be categorized based on the availability and reliance on terrain data. There are primarily two classifications: terrain-dependent methods, which utilize topographic data, and terrain-independent methods, which do not. For this investigation, the terrain-independent approach was selected, a method that obviates the necessity for data from ground control points. Instead, this approach utilizes solely the range of the maximum and minimum elevation values that are encapsulated within the extent of the imagery's coverage area. Virtual control points are generated by constructing a three-dimensional spatial grid, where the elevation direction is stratified at specified intervals, and regular grids are established in the planar directions with uniform spacing. Assume the image is evenly divided into \(m\cdot n\) regular grids in the horizontal directions, and the elevation range is uniformly stratified into \(k\) elevation layers. To avoid ill conditioning of the design matrix, the number of elevation layers is generally required to exceed 2. Consequently, a total of \((m+1)\cdot(n+1)\cdot(k+1)\) virtual control points can be generated. By sequentially taking the image coordinates of each grid center and the midpoint elevation values of adjacent elevation layers as checkpoints, a total of \(m\cdot n\cdot k\) virtual checkpoints can be generated. The number of control points required depends on the structure of the RPC model. Among the nine different forms, when the denominator terms vary and a third-order polynomial is used, the RPC model achieves the highest fitting accuracy [30]. In this case, there are a total of 78 RPC parameters, requiring a minimum of 39 known control points. Subsequently, with sufficient control points, the RPC parameters can be solved using the least squares method. Firstly, transform Equations (1) and (2) into the following standard forms:  
\[ \begin{array}{l}F_{1}=N_{1}(B_{n},L_{n},H_{n})-R_{n}\cdot N_{2}(B_{n},L_{n},H_{ n})=0\\ F_{2}=N_{3}(B_{n},L_{n},H_{n})-C_{n}\cdot N_{4}(B_{n},L_{n},H_{n})=0\end{array}  \tag{20}\]  
The system of error equations is constructed as follows:  
\[V=A_{2}x-l_{2} \tag{21}\]  
Among them, for \(i=1,\cdots,20\); \(j=2,\cdots,20\),  
\[A_{2}=\left[ \begin{array}{cc}\frac{\partial F_{1}}{\partial a_{i}}\frac{ \partial F_{1}}{\partial b_{j}}\frac{\partial F_{1}}{\partial c_{i}}\frac{ \partial F_{1}}{\partial d_{j}}\\ \frac{\partial F_{2}}{\partial d_{i}}\frac{\partial F_{2}}{\partial b_{j}} \frac{\partial F_{2}}{\partial c_{i}}\frac{\partial F_{2}}{\partial d_{j}} \end{array} \right],x=\left[a_{i}\;b_{j}\;c_{i}\;d_{j}\right]^{T},l_{2}=\left[-F_ {1}^{0}\;-F_{2}^{0}\right]^{T}\]  
Then, the error equation can be solved using the least squares method.