# An Image Compensation-Based Range-Doppler Model for SAR High-Precision Positioning
## 1 Introduction  
### Geometric Models of SAR Images  
The synthetic aperture radar (SAR) geometric model plays a foundational role in high-precision geolocation processes, encompassing geometric correction, stereo positioning, block adjustment, and three-dimensional reconstruction of SAR imagery [1, 2, 3, 4]. Geometric models of SAR imagery are generally divided into two categories: rigorous and non-rigorous. Rigorous models rely on the mechanisms of SAR imaging, in which both the sensor's motion state and imaging parameters are incorporated. Meanwhile, non-rigorous models usually ignore the SAR imaging mechanism and disregard the sensor's geometric, physical, and imaging parameters, typically fitting the relationship between image and ground coordinates. A variety of rigorous SAR models have been developed, such as the range-zero Doppler model created by [PERSON], the three-dimensional model created by [PERSON] from the Canadian Remote Sensing Center, and the SAR collinear equation model created by [PERSON] from the Chinese Academy of Sciences [5, 6]. However, the most prevalent and effectivemodel is the range-Doppler (R-D) equations introduced by [PERSON] in 1982 [7]. From a geometric perspective, the Doppler equation confines the target of observation within an imaging Doppler cone, with the sensor's position serving as the vertex and the velocity vector as the axis. Meanwhile, the range equation bounds the target within a sphere centered on the sensor, with a radius equivalent to the SAR measurement distance. The intersection curve between the sphere corresponding to the range equation and the cone corresponding to the Doppler equation is a circle, representing the location range of the SAR observation target. In the course of image correction, the R-D model either takes elevation as a known parameter or as integrated with the Earth's ellipsoid model, which accounts for the elevation parameter [8]. The R-D model subsequently calculates the object's spatial coordinates (X, Y, and Z). In stereo positioning, two or multiple sets of R-D equations, derived from corresponding points in multi-view images, are used jointly to determine the object's spatial coordinates [9; 10]. At present, the R-D model is used as the basis for the majority of methods for the precise geometric processing of SAR images. [PERSON] et al. [11] developed an airborne SAR target positioning algorithm which refines the R-D model parameters through the use of corresponding points from airborne SAR and reference images, thereby enhancing the positioning accuracy. [PERSON] [10] presented a weighted spatial intersection algorithm that predicts algorithm accuracy based on prediction error analysis. [PERSON] et al. [12] achieved the high-precision geolocation of TerraSAR-X images using the R-D model without ground control points (GCPs), through the use of calibrated sensor parameters and correcting for atmospheric and tidal effects. Using the R-D and digital elevation models, accurate ground positioning within 2 m can be achieved for Tianhui-2 (TH-2) imagery [13]. Considering the scale differences between the range and Doppler equations, researchers have normalized the R-D model and improved the positioning accuracy through a weighting strategy [14]. The model's efficiency has been validated through experiments conducted on the Gaofen-3 C-band (GF3-C) and TerraSAR-X satellite data sets. To address the complexity of traditional geometric models, [PERSON] [15] introduced a fast algorithm based on local linear approximation to simplify the associated computations. The R-D model is also beneficial in other image processing tasks; for example, [PERSON] [9] employed the model to correct geometric distortions in SAR images, resulting in the improved accuracy of the SAR scale-invariant feature transform (SAR-SIFT) matching. Non-rigorous models are classified into true replacement models and correspondence models, as specified by the ISO 19130-1 standard [16].