# An Image Compensation-Based Range-Doppler Model for SAR High-Precision Positioning
## 2 The Proposed Image Compensation-Based R-D Model
### Solving Polynomial Coefficients of Image Compensation Functions  
The determination of parameters involves two main steps, carried out with the support of GCPs. The first step is to calculate the image coordinates using the ground coordinates of GCPs; in other words, the image coordinates (\(c_{ci}\), \(r_{ci}\)) corresponding to the ground point (\(X_{i}\), \(Y_{i}\), \(Z_{i}\)) are calculated utilizing the original orientation parameters. The second step is to determine the parameters of the image compensation polynomial. The least squares fitting approach is employed to compute the differences between the calculated image coordinates and those from the GCPs. The two steps are detailed below. 2. _Translating Ground Coordinates into Image Calculation Coordinates_ The iterative method used to determine the image coordinates (\(c_{ci}\), \(r_{ci}\)) from the ground coordinates (\(X_{i}\), \(Y_{i}\), \(Z_{i}\)) via the R-D model includes the following steps: 1. [label=(0)]
3. Begin with temporary values of the image coordinates (\(c_{0}\) and \(r_{0}\)); 2. Extract the Doppler value and the SAR range measurement value corresponding to the temporary image coordinates (\(c_{0}\), \(r_{0}\)); 3. Determine the image row time and interpolate to determine the sensor's position and velocity at that time; 4. Calculate the Doppler value and the range distance based on the sensor motion information and the ground target position, and then compare these with the values from Step (2) to obtain the differences \(df_{D}\) and \(dR\); 5. Calculate the increments of the image coordinates using the formulas \(dc_{0}=df_{D}/Vy\) and \(dr_{0}=dR/Mr\); 6. If \(\mid dc\mid<0.001\) and \(\mid dr_{0}\mid<0.001\), terminate the iteration, setting (\(c_{ci}\), \(r_{ci}\)) equal to (\(c_{0}\), \(r_{0}\)). Otherwise, update the temporary coordinate values using \(c_{0}=c_{0}+dc_{0}\), \(r_{0}=r_{0}+dr_{0}\), and return to Step (2). 3. _Determining the Parameters of the Polynomial Coefficients in the Image Compensation Function_ For each reference point or GCP (\(X_{i}\), \(Y_{i}\), \(Z_{i}\), \(c_{i}\), and \(r_{i}\)), the calculated values of the image point--denoted as \(c_{ci}\) and \(r_{ci}\)--can be determined based on the ground point coordinates (\(X_{i}\), \(Y_{i}\), \(Z_{i}\)) using the R-D model. Meanwhile, the image coordinates of the GCP are denoted as (\(c_{i}\), \(r_{i}\)) and the corresponding difference values (\(\Delta c_{i}\), \(\Delta r_{i}\)) for this GCP can be computed. Taking the parameters \(a_{0}\)-\(a_{5}\) and \(b_{0}\)-\(b_{5}\) in Equation (2) as unknowns, we can establish two error equations for each GCP, resulting in a total of \(2n\) error equations for \(n\) GCPs. When one-, three-, four-, and six-parameter models for image refinement are employed, the minimum number of GCPs required is one, three, four, and six, respectively. Once the minimum requirement is met or exceeded, the \(n\) parameters can be solved using the least squares method. Based on Equation (2), error equations for row and column coordinates can be created separately and solved. The error equations for the column coordinate of the GCP, in the form of a vector, can be expressed as follows:  
\[V_{\text{C}}=AX_{a}-L_{\text{C}}, \tag{4}\]  
where \(V_{\text{C}}=[v_{\text{c1}},v_{\text{c2}},\ldots\ldots,v_{\text{cn}}]^{T}\) is the error vector of image column coordinates, \(L_{\text{C}}=[\Delta c_{1},\Delta c_{2},\ldots\ldots\Delta c_{n}]^{T}\) is the constant vector of image column coordinates, and \(n\) is the number of GCPs. The coefficient matrix \(A\) for the six-parameter model is shown below:  
\[A= \begin{bmatrix}1&c_{1}&r_{1}&c_{1}r_{1}&c_{2}^{2}&r_{2}^{2}\\ 1&c_{2}&r_{2}&c_{2}r_{2}&c_{2}^{2}&r_{2}^{2}\\.&.&.&.&.&.\\ 1&c_{n}&r_{n}&c_{n}r_{n}&c_{n}^{2}&r_{n}^{2}\end{bmatrix} .