# An Image Compensation-Based Range-Doppler Model for SAR High-Precision Positioning
## 3 Experiments  
The purposes of the conducted experiments were to test the precision of the four aforementioned image compensation models, namely the one-parameter, two-parameter, four-parameter, and six-parameter model, under different GCPs, compare the accuracy of the four models with that of the classical R-D refinement model, and evaluate the effectiveness of the proposed method across different satellite and airborne SAR images.  
In these experiments, we adopted SAR data sets from three platforms and four sensors--namely, the spaceborne X-band TerraSAR-X, spaceborne C-band GF3-C, manned airborne X-band SAR-X, and low-altitude unmanned aerial vehicle (UAV)-mounted Ku-band miniSAR-Ku image data sets.  
The classical R-D refinement model, developed by the Chinese Academy of Surveying and Mapping and integrated into the SARplore software (Version 3.3) product, was used for comparative studies. The unknowns of the model comprised sensor position, velocity, initial slant range, and Doppler parameters. Sensor position and velocity errors were represented using a linear refinement model, and a combined adjustment approach was employed to solve all unknowns simultaneously.  
Under the given conditions of initial orientation parameters, refinement parameters, and ground elevation, there exists a strict transformation between object space coordinates and image coordinates. Accuracy assessment can be conducted using either the ground coordinates of reference points (for assessing geolocation accuracy) or the image coordinates of reference points (for assessing orientation accuracy). In the former method, the image coordinates and elevation of the reference point are treated as known parameters, and the refined model is used to calculate their geolocation plane coordinates, which are then compared with the plane coordinates of the reference point to calculate the accuracy. In the latter method, the refined R-D model and the object coordinates of reference points are used to calculate image coordinates, which are then compared with the image coordinates of the reference points. That means that, for the geolocation accuracy values in Tables 1-6, which follow, we first calculated the planar error for each CP based on the positioning errors \(dX_{i}\) and \(dY_{i}\) using the formula \(dp_{i}=\sqrt{dX_{i}^{2}+dY_{i}^{2}}\). Subsequently, the geolocation accuracy was computed with all \(n\) CPs using the formula \(m_{p}=\sqrt{\sum d{p_{i}^{2}}/n}\) (unit: meters). The calculation of image orientation accuracy was similar to that of geolocation accuracy, obtained through calculating the errors in the image coordinates of the reference points and then computing the mean error (unit: pixels). When different models are compared under the same conditions, the same set of GCPs and check points (CPs) is used to prevent their error differences and distribution differences from affecting the accuracy. When some experiments required multiple GCPs but the number of reference points was insufficient, we employed the leave-one-out cross-validation (LOOCV) method for accuracy validation [40]; namely, one reference point was selected as a CP in turn, and the remaining reference points were used as GCPs for testing. Through tallying the errors for all the CPs, the accuracy was obtained. In the following experiments, when the number of CPs was equal to the total number of reference points, LOOCV was applied.