## 3 Methodology
### Multi-GPU Cooperative Acceleration Strategy for SAR Image Matching in Large Area
In summary, the optimal image pixel size for each GPU can be set based on the actual graphics memory of the GPU, which is \(M_{s}\times N_{s}\). For image pairs with overlapping areas exceeding this size, block operations are performed in advance. For overlapping area image pairs smaller than this size, multiple regions can be combined together for full utilization of the GPU devices. Through the above-standardized processing of the image data to be matched, the GPU node can process without considering the changing imaging parameters of the SAR images to be matched. At the same time, the single GPU can achieve the optimal load, thus significantly improving the processing efficiency. 2. Dynamic adjustment of optimal size of image to be matched considering ground object type Due to the geometric characteristics of side view imaging of SAR image, its geometric positioning model has a higher correlation with elevation. As shown in Figure 3, according to the analysis of the SAR imaging geometric positioning model, the two SAR images to be matched were obtained from satellite SAR sensors at different incidence angles \(\theta_{1}\) and  
Figure 2: SAR image matching differences of different ground object types (red dots are matching point pairs). (**Left**): SAR image matching in flat areas makes it easier to find corresponding points. (**Right**): SAR image matching in undulating areas is difficult to match to corresponding points. \(\theta_{2}\), the relationship between relative positioning error of \(\Delta r=\Delta r_{1}-\Delta r_{2}\) and error with elevation \(\Delta h\) and the incident angle is:  
\[\Delta r=\Delta r_{1}-\Delta r_{2}=\frac{\Delta h}{\theta_{1}}-\frac{\Delta h}{ \theta_{2}} \tag{12}\]  
From the above formula, when the elevation error is fixed (the same terrain), the relative positioning error decreases with the increase in the incidence angle. When the incident angle is fixed, the relative positioning error increases with the increase in elevation error, so the relative positioning error in flat areas (small elevation error) is small, and that in undulating areas (large elevation error) is large. For SAR image matching, the incidence angle is fixed, so SAR image matching is less difficult to process in flat areas, and it is easy to obtain more homonymous points. The process of image matching search and model fitting is also faster. However, in undulating areas, the processing is difficult, there are fewer homonymous points to be matched, the image matching search is complex, and the simulation fitting is not easy to converge. The above theoretical analysis is consistent with the experimental results shown in Figure 2.  
In view of the above analysis, this study introduces the elevation change trend to dynamically adjust the optimal processing size of the SAR image to be matched. In the digital elevation model (DEM), slope and aspect are important parameters to describe surface morphology. The slope is a measure of slope gradient at a point on the surface, usually expressed in percentage or decimal form, and its calculation formula is as follows:  
\[Slope=\frac{\Delta Z}{\Delta D}\times 100\% \tag{13}\]  
Among them, \(\Delta Z\) is the elevation difference between the point and its adjacent points and \(\Delta D\) is the distance between the point and its adjacent points. Different size windows can be used for slope calculation, such as \(3\times 3\) or \(5\times 5\) windows. The size of the window will affect the accuracy and details of slope calculation. Take the \(3\times 3\) window as an example, where the elevation of the center point is \(Z(i,j)\) and the elevations of \(8\) adjacent points around it are \(Z(i-1,j),Z(i+1,j),Z(i,j-1),Z(i,j+1),Z(i-1,j-1),Z(i-1,j+1),\)\(Z(i+1,j-1),\)\(Z(i+1,j-1),\)\(Z(i+1,j+1)\). The slope calculation formula at the center point is:  
\[Slope=[(Z(i+1,j)-Z(i-1,j)+Z(i,j+1)-Z(i,j-1))/(2d)]\times 100\% \tag{14}\]  
where d is the side length of the grid. The slope aspect is the inclination direction of a slope facing due north at a point on the surface.