# An Image Compensation-Based Range-Doppler Model for SAR High-Precision Positioning
## 1 Introduction
### Refinement of Geometric Model Parameters  
The geometric model describes the relationship between the object's spatial coordinates and the SAR image coordinates. However, the achievement of high-precision positioning is contingent on calibrating the sensor parameters and refining the model parameters through the use of GCPs. The positioning accuracy of the R-D model is affected by various factors, such as initial range errors, synchronization discrepancies between the SAR sensor and the position and orientation system (POS), sensor motion parameter errors, and errors in the transmission of electromagnetic waves. Several studies have conducted thorough analyses on the effects of parameter errors on the accuracy of SAR image positioning [9; 24; 25]. Some of these affecting factors, such as initial slant range, slant range rate, and time synchronization, can be calibrated in a laboratory or testing field, while others, such as atmospheric delay and tidal correction, can be obtained through relevant rigorous models [12]. The optimization of sensor system parameters is contingent on the availability of a high-precision calibration field, and the timeliness of calibration is pivotal to ensuring the accuracy of image positioning. However, some sensors may encounter challenges regarding the prompt calibration of parameters, potentially leading to suboptimal geolocation accuracy when only the initial parameters are used. To further enhance the positioning accuracy of SAR images, it is often necessary to employ high-precision reference points for the comprehensive optimization of the sensor position, velocity, slant range, time synchronization, and Doppler parameters within the R-D model [26; 27]. Although conventional methods for enhancing the accuracy of the R-D model are theoretically rigorous, they have several limitations with regard to the practical use of reference points for comprehensive optimization. These limitations include the difficulty of dealing with a large number of model parameters and the necessity of using additional models to express variations throughout an entire image. For example, in terms of additional models, positional and velocity corrections typically require six distinct sets of polynomial parameters [26; 27]. Strong parameter correlations in the R-D model present challenges related to decorrelation when parameters are solved under conditions of sparse GCPs [27]. The process of obtaining refined R-D model parameters using GCPs is complex and requires a high level of expertise, significantly hindering advanced users from efficiently performing high-precision SAR image geolocation processing independently. These limitations restrict the application of the R-D model in certain scenarios. Therefore, it is necessary to develop a simpler and more user-friendly technique for SAR positioning based on the R-D model. In contrast, the RFM can relatively easily achieve high-precision geolocation through introducing an error compensation model for the image space with the support of GCPs. The most common approach involves using a linear affine transformation model to correct image errors caused by various factors. Studies have suggested that, with the support of adequate GCPs, the positioning accuracy of RFMs enhanced with affine transformations is comparable to that of R-D models refined with model parameters [20; 28; 29]. For instance, [PERSON] et al. [23] demonstrated that RFMs with affine transformations can achieve sub-pixel accuracy. This methodology has been gradually applied to multi-scene imagery, as demonstrated in the studies conducted by [PERSON] et al. [30] and [PERSON] et al. [31], who employed RFM-based block adjustment techniques to accomplish high-precision positioning for GF3-C and Yaogan-5 (YG-5) multi-scene images.