# An Image Compensation-Based Range-Doppler Model for SAR High-Precision Positioning
## 2 The Proposed Image Compensation-Based R-D Model
### Use of Image Compensation-Based R-D Model  
Applying the R-D model based on image compensation for the geometric processing of images mainly involves the mutual conversion of coordinates between image and object spaces, as well as related processing during image correction.  
Assuming that the image coordinates (\(c\), \(r\)) correspond to a ground elevation \(H\), the point coordinates \(P(X,Y,Z)\) in the geocentric Cartesian coordinate system with a ground height of \(H\) should satisfy the Earth ellipsoid model, as follows:  
\[\frac{X^{2}+Y^{2}}{(a+H)^{2}}+\frac{Z^{2}}{(b+H)^{2}}=1, \tag{10}\]  
where \(a\) and \(b\) represent the Earth's minor and major axes, respectively.  
The adjusted values (\(c_{a}\), \(r_{a}\)) corresponding to the original image coordinates (\(c\) and \(r\)) can be determined using the image compensation model. Based on the adjusted values (\(c_{a}\) = \(c+\Delta c\), \(r_{a}\) = \(r+\Delta r\)), the corresponding sensor position, velocity, and imaging Doppler parameters for the R-D model can be obtained. Therefore, the R-D model equation and the Earth ellipsoid model together form a system of three equations with three unknowns (\(X\), \(Y\), and \(Z\)), which can be solved.  
When transforming the coordinates of object point \(P(X,\,Y,\,Z)\) to the original image coordinates, the method described in Section 2.3 can be employed first, and the calculated values of the image points (\(c_{c}\), \(r_{c}\)) corresponding to object points can be obtained. Theoretically, this calculated value should match the adjusted values (\(c_{a}\), \(r_{a}\)) of the image compensation model from the original image point p(\(c\), \(r\)); that is, \(c_{c}=c_{a}\) and \(r_{c}=r_{a}\). For the image compensation model based on original image coordinates, these values are unknowns to be determined in this task. There are two solutions, as follows:  
1. The calculated coordinates of the image points (\(c_{c}\), \(r_{c}\)) obtained from object points can be used as the initial values for \(c\) and \(r\) in the compensation model, and the original image point coordinates and correction values can be refined through step-by-step iteration.
2. Considering that the difference between the calculated image point position (\(c_{c}\), \(r_{c}\)) and the real image point position (\(c\), \(r\)) corresponding to P is not significant (usually consisting of a few pixels) and the non-constant term coefficients of the compensation model are very small, the original image coordinate in the compensation model can be approximated as (\(c_{c}-a_{0}\), \(r_{c}-b_{0}\)). The increment values can be computed and the original image coordinate can be updated without iteration, typically resulting in an error that can be completely ignored.  
After obtaining the parameters for the image polynomial compensation function, high-precision ortho-rectification of SAR images can be achieved with the support of a digital elevation model (DEM), a process typically realized through an indirect method. Based on the geolocation coordinates of an image's four corner points, the corrected mapping image range can be determined. For each corrected image point, the object plane coordinates of the point can be calculated according to the range and the resolution of the corrected image, and the elevation value can be extracted from the DEM; thus, the three-dimensional object coordinates of the point can be obtained. Subsequently, the corresponding position in the original image can be calculated, and the pixel value can be resampled to match the corrected image.