# An Image Compensation-Based Range-Doppler Model for SAR High-Precision Positioning
## 2 The Proposed Image Compensation-Based R-D Model
### The Proposed Image Compensation Model  
The image compensation model was developed by adding correction values (\(\Delta c,\Delta r\)) to the original image coordinates. These values were derived from two-dimensional low-order polynomial functions of the original image row and column coordinates (\(c\), \(r\)). When a second-order polynomial is used, it can be expressed as follows:  
\[\begin{cases}\Delta c=a_{0}+a_{1}c+a_{2}r+a_{3}c^{2}+a_{4}cr+a_{5}r^{2}\\ \Delta r=b_{0}+b_{1}c+b_{2}r+b_{3}c^{2}+b_{4}cr+b_{5}r^{2},\end{cases} \tag{2}\]  
where \(a_{i}\) and \(b_{i}\) (\(i\) = 0, 1, \(\ldots\), 5) represent the coefficients of polynomial functions for image compensation. According to the minimum number of GCPs required and the application requirements, the polynomial order can also be chosen as the 0 th order (with parameters \(a_{0}\) and \(b_{0}\)) or 1 st order (with parameters \(a_{0}\), \(a_{1}\), \(a_{2}\), \(b_{0}\), \(b_{1}\), and \(b_{2}\)). In the case of order 0, the equations only contain constant terms, while, in the case of order 1, they represent the affine transformation model.  
SAR image row data, obtained by moving the sensor along the azimuth direction, primarily involve parallel projections. Image column data depend on the sensor sampling-reflected electromagnetic waves at equal time intervals, constituting range projections. From the perspective of the R-D model form, the main factors affecting SAR imaging or positioning errors include SAR sensor time and POS time synchronization errors, as well as the initial slant range error, which are the main factors influencing SAR imaging errors and are mainly characterized by translation [37]. These imaging errors can be absorbed through the two parameters \(a_{0}\) and \(b_{0}\) in the image compensation model. Within approximately 15 min, the sensor position and velocity errors measured via GNSS are primarily characterized by translation and drift, which can be expressed by a linear model [39]. Although there is a non-linear functional relationship between the image column coordinates and sensorposition error, the position measurement accuracy of GNSS is easily better than the 1 m level and even reaches 0.01 m [32]. Imaging errors caused by sensor motion errors, as well as the effects of other low-frequency errors, can be corrected using linear or second-order polynomial models. Considering the parallel projection relationship between image lines, a simplified quadratic polynomial model was constructed, as follows:  
\[\begin{cases}\Delta c=a_{0}+a_{1}c+a_{2}r+a_{3}c^{2}\\ \Delta r=b_{0}+b_{1}c+b_{2}r+b_{5}r^{2}\end{cases} \tag{3}\]  
The above polynomial refinement model was used to correct the comprehensive impact of all parameter errors on image imaging and positioning. For ease of expression, image-compensated R-D models were categorized based on the number of polynomial parameters in the compensation functions. The functions containing only parameters consisting of the constant terms \(a_{0}\) and \(b_{0}\) are referred to as the one-parameter model. Polynomials with parameters \(a_{0}\), \(a_{1}\), \(a_{2}\), \(b_{0}\), \(b_{1}\), and \(b_{2}\) are called the three-parameter model, while those with parameters \(a_{0}\), \(a_{1}\), \(a_{2}\), \(a_{3}\), \(b_{0}\), \(b_{1}\), \(b_{2}\), and \(b_{5}\) are termed the four-parameter model. Similarly, Equation (2) is labeled as the six-parameter model, and the R-D model refined in the classical way is denoted as the classical model. Thus, in this article, a total of five types of R-D refinement methods are used. In theory, different polynomial forms yield varying processing accuracies, a notion that will be experimentally evaluated later.