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The Rational Function Model (RFM) has been widely used as an alternative to rigorous sensor models of highresolution optical imagery in photogrammetry and remote sensing geometric processing. However, not much work has been done to evaluate the applicability of the RF model for Synthetic Aperture Radar (SAR) image processing. This paper investigates how to generate a Rational Polynomial Coefficient (RPC) for highresolution TerraSARX imagery using an independent approach. The experimental results demonstrate that the RFM obtained using the independent approach fits the RangeDoppler physical sensor model with an accuracy of greater than 10^{−3} pixel. Because independent RPCs indicate absolute errors in geolocation, two methods can be used to improve the geometric accuracy of the RFM. In the first method, Ground Control Points (GCPs) are used to update SAR sensor orientation parameters, and the RPCs are calculated using the updated parameters. Our experiment demonstrates that by using three control points in the corners of the image, an accuracy of 0.69 pixels in range and 0.88 pixels in the azimuth direction is achieved. For the second method, we tested the use of an affine model for refining RPCs. In this case, by applying four GCPs in the corners of the image, the accuracy reached 0.75 pixels in range and 0.82 pixels in the azimuth direction.
Sensor models are required to represent the functional relationship between 2D image space and 3D object space. In general, sensor models are classified into two categories: physical and generic models. The choice of a sensor model depends on a variety of factors, including the performance and accuracy required, the physical information of the acquisition system and the available control information [
In physical sensor models, the imaging process is described by parameters defining the position and orientation of a sensor with respect to an objectspace coordinate system [
Accurate geolocation of Synthetic Aperture Radar (SAR) images depends on the quality and errors in SAR orbital data. Raggam
The focus of our study is different from the previous studies mentioned above in that we focus on SAR sensor parameter correction before RPC generation. In our procedure, four sensor parameters are first updated by using a different number, and distributed Ground Control Points (GCPs) are calculated, followed by RPCs. We test our methodology for producing RPCs for TerraSARX spotlight images by independent methods, and we calculate refined RPCs by using an independent approach. The sections of this article are described below. First, the basic concepts of physical and generic sensor models for SAR images are described. Then, in Section 2, the methodology of using an RF model for SAR images is explained, followed by an introduction to RPC adjustment models. Section 3 is devoted to the experiments and the results obtained by calculating RPCs in a terraindependent fashion and by applying two methods to calculate RPCs. A discussion is provided in Section 4.
The Range–Doppler (RD) model is the most widely used physical sensor model for spaceborne SAR remote sensing systems [
The Rational Function Model (RFM) is defined as the ratio of two bicubic polynomials involving 39 parameters in any direction in a 3D form:
In this equation,
Zhang
The general workflow of developing the RF model used in this paper is shown in
Because RangeDoppler equations and SAR sensor parameters are used for RPC generation, any error in the orbital data directly affects the coefficients. Therefore, it is important to analyse errors in the SAR sensor parameters.
The normal way to correct errors in the RPC is to use accurate Ground Control Points (GCPs) together with a suitable model, such as an affine transformation, to refine the RPCs [
In this method, physical sensor parameters that are involved in the generation RPCs are first updated, and then RPCs are calculated. The azimuth time for the first line, the Pulse Repetition Frequency (PRF), the range time for the first pixel and the Range Sampling Rate (RSR) in the azimuth and range directions are the four parameters that are updated in this approach [
Because the RF model is calculated from the physical imaging model without the aid of ground control points, errors in the direct measurement of sensor orientation can cause biases in the RPC mapping. The biases can be taken into account by using a biascorrected RPC model, expressed as follows:
In these equations,
Case one.
Case two.
Case three.
For more information about the physical meanings of the parameters, please see the relevant references [
In this study, we use a TerraSARX SingleLook SlantRange Complex (SSC) image as a test case for RF modelling. The image was acquired over the city of Jam, southern Iran, in spotlight mode. It was acquired on May 17 2011, in a descending orbit, covering an area of approximately 42 km^{2}. The ground elevation in the study area is between 590 and 800 m. From the complex SSC product, an amplitude image was generated for use in this paper.
In the experiment, a total of 11 distinct ground control points were measured by GPS. The accuracy of these points is less than 10 cm. The image coordinates of these 11 points were carefully measured up to a nominal accuracy of 1 pixel. All of these points were used as GCPs and check points (CKPs) for different configurations in the experiment.
With these datasets, we conducted three experiments to provide a comprehensive evaluation of the RF model and to compare the performances of different bias correction methods. The experiments had the following aims:
To examine the RF model in an independent approach.
To examine the effects of GCP number and distribution on sensor orientation parameters and RPC generation with parameter adjustment.
To examine the use of an affine model for RPC refinement with different GCP numbers and distributions.
In the first experiment, the method described in Section 3 was applied to the SAR image to solve the RF model in an independent way. We took our lead from previous studies [
In this section, we assess the accuracy of the original RPCs that were computed using the independent method by calculating the image coordinates for all GCPs using original RPCs. The results show a mean difference of 4.2 pixels in range and 3.7 pixels in azimuth between coordinates calculated by using original RPCs and true coordinates. These results demonstrate that there is a timedependent drift error in the image orientation, as illustrated in
To account for the drift error, we adjust the physical sensor parameters to improve the accuracy of the RPCs. We use four physical parameters: azimuth time for the first line (ta_{0}), Pulse Repetition Frequency (PRF), range time for first pixel (tr_{0}) and Range Sampling Rate (RSR). These parameters are updated by ground control points. The initial values of these parameters are available in the SAR metadata. However, due to atmospheric effects and other disturbing factors, these parameters change with time. Therefore, using the initial values creates errors in imaging geometry and disturbs the conditions of the physical equations. The relationships between image coordinates and these four parameters are described by the following equations [
In these equations,
Note that the first columns in
As is clear from the results in
In this section, we use an affine model for RPC refinement. In this case, the coefficients of the affine model (A_{0}–A_{2} and B_{0}–B_{2} in
As indicated in
In this article, we applied the RF model to a spotlight TerraSARX image using an independent approach, with the aim of replacing the physical model with an appropriate generic sensor model. We showed that using the RF model with the independent method provides a good fit to the rigorous Range–Doppler model, with an accuracy of greater than 10^{−3} pixels in image space.
Two techniques were also tested for improving the accuracy of the RPCs. In the first method, SAR sensor orientation parameters are corrected by using a number of GCPs, and these are used to generate RPCs. In the second method, the RPCs are generated by using the initial values of the SAR sensor orientation parameters, and an affine model is used to refine the RPCs.
Experiments involving the first method indicate that, by using three GCPs in the image (GCPs 6, 8 and 11 in
For the second method, in which an affine model is applied to refine the RPCs, selecting four GCPs in the corners of the image achieves the best accuracy. Using GCPs at the edges or centre of the image increases cost but does not improve accuracy.
Our implementations of the two methodologies for RPC adjustment show that increasing the number of GCPs from four to eight does not improve RPC adjustment accuracy, and it even causes a decrease in accuracy in the range direction. This is because GCPs have measurement errors of approximately 1 pixel in image space, and increasing the number of GCPs causes measurement errors to accumulate at the check points. Additionally, tests of different GCP scenarios show that it is not necessary to have GCPs in the centre of the image. By comparing the two methods, we find that they both produce similar results, with the first method requiring fewer control points. The main advantage of the first method, in which SAR sensor orientation parameters are corrected by using GCPs, is that the corrected parameters have interpretable physical meanings. In the second approach, in which affine coefficients are used to mitigate the accumulation of errors, the parameters are difficult to interpret from physical point of view.
Zhang
This study demonstrates the use of a Rational Function Model for TerraSARX imagery. Although the Rational Function Model (RFM) has been widely used as an alternative to rigorous sensor models of high resolution optical imagery in photogrammetry and remote sensing geometric processing, not much has been done to assess its applicability for Synthetic Aperture Radar (SAR) image processing. We applied the RFM to a TerraSARX SingleLook SlantRange Complex (SSC) image acquired over the city of Jam, southern Iran, which was acquired in spotlight mode with 11 control points that were measured using GPS. The accuracy of the control points was better than 10 cm. The RF model that we obtained with an independent approach fit the RangeDoppler physical sensor model with accuracy better than 10^{−3} pixels. However, errors in the physical SAR sensor parameters impair the absolute accuracy of the RPCs, and Ground Control Points (GCPs) should be used for RPC adjustment. Two methods were used for RPC adjustment. One uses updated physical SAR sensor orientation parameters, and the other uses an affine model for RPC refinement. The results show that, for both methods, using 3–4 GCPs is a good choice in terms of accuracy and cost for calculating RPCs with high accuracy. Additionally, the experiments indicate that control points at the corners of the image provide better accuracy than other placements. Planimetric accuracy reaches 1.12 pixels when we update the SAR sensor orientation parameters with three GCPs, and it reaches 1.11 pixels with the affine model method when four GCPs are used.
We would like to thank Farazamin Consulting Engineers for providing GPS Data for the GCPs. We also acknowledge the constructive comments of two anonymous reviewers that helped improve the quality of the manuscript.
We declare that we have no conflict of interest.
General workflow of developing a rational functional model [
Distributions of ground control points (GCPs) and check points (CKPs) in the study area for a scenario involving five GCPs. Note: triangles represent the GCPs, and circles represent the CKPs.
An example of a ground control point (GCP) selected in (
Plot of rational function modell (RFM) fitting errors versus number of elevation layers.
Discrepancies in the image space between the calculated rational polynomial coefficient (RPCs) and the true ground control points (GPCs).
Results for rational function modelling by an independent method. RMSE: root mean square error, CNP: control point, CKP: check point.

 

LCurve  0.031  0.035  0.047  0.042  0.049  0.065 
Changes in physical sensor parameters for different ground control points (GCP) scenarios. The locations of points listed in column 1 are shown in
6,8  0.0309  8.602  2.27 × 10^{−5}  218.43 
6,11  0.2329  2.457  9.81 × 10^{−5}  192.14 
3,9  0.0811  5.582  3.88 ×10^{−5}  273.11 
6,8,11  0.3273  3.607  1.29 × 10^{−5}  124.53 
3,6,11  0.2497  2.252  1.20 × 10^{−5}  114.87 
2,6,8,11  0.3298  2.571  1.28 × 10^{−5}  122.15 
1,5,8,10  0.3138  1.711  1.22 × 10^{−5}  117.68 
1,5,8,9,10  0.2897  1.627  1.21 × 10^{−5}  115.78 
2,3,6,8,11  0.3286  2.614  1.356 × 10^{−5}  130.12 
1,2,3,5,6,8,10,11  0.2912  1.723  1.225 × 10^{−5}  117.51 
The results of rational polynomial coefficient (RPC) adjustments by using synthetic aperture radar (SAR) sensor orientation correction in different ground control point (GCP) scenarios. The locations of points listed in column 1 are shown in



 

6,8  1.9 × 10^{−4}  9.1 × 10^{−5}  4.1 × 10^{−4}  −4 × 10^{−4}  1.2 × 10^{−4}  4.1 × 10^{−4}  0.54  2.13  2.19  −0.8  2.85  2.71 
6,11  4.2 × 10^{−5}  4.7 × 10^{−5}  5.3 × 10^{−5}  −4.2 × 10^{−4}  2.3 × 10^{−5}  4.3 × 10^{−4}  −1.62  1.02  1.89  0.7  0.81  1.03 
3,9  −3.8 × 10^{−6}  1.7 × 10^{−5}  1.2 × 10^{−5}  −4.1 × 10^{−4}  1.8 × 10^{−5}  4.1 × 10^{−4}  1.51  2.14  2.75  0.031  1.61  1.55 
6,8,11  6.1 × 10^{−5}  1.09  1.16  −3.9 × 10^{−4}  0.74  0.61  −0.74  0.82  0.69  0.27  0.85  0.88 
3,6,11  4.7 × 10^{−6}  1.38  1.42  −4.4 × 10^{−4}  0.75  0.76  −0.68  1.02  1.18  0.49  0.86  0.94 
2,6,8,11  7.4 × 10^{−5}  1.31  1.13  −4 × 10^{−4}  0.88  0.76  −0.45  0.97  0.73  −0.35  0.65  0.80 
1,5,8,10  1.8 × 10^{−5}  0.81  0.69  −4.1 × 10^{−4}  0.94  0.82  0.27  1.25  1.07  −0.57  0.61  0.82 
1,5,8,9,10  −1.8 × 10^{−4}  0.72  0.65  −4 × 10^{−4}  0.9  0.81  0.16  1.37  1.26  −0.29  0.77  0.77 
2,3,6,8,11  5 × 10^{−5}  1.28  1.15  −4.1 × 10^{−4}  0.77  0.68  0.12  1.05  0.97  −0.29  0.85  0.83 
1,2,3,5,6,8,10,11  −1.8 × 10^{−5}  1.01  0.94  −4.1 × 10^{−4}  0.81  0.76  −0.39  1.48  1.23  −0.62  0.32  0.68 
The results of rational polynomial coefficient (RPC) refinement using the affine model in different ground control point (GCP) scenarios. The locations of the points in column 1 are shown in



 

6,8,11  −5.2 × 10^{−4}  6.8 × 10^{−4}  5.3 × 10^{−4}  6.1 × 10^{−4}  7.1 × 10^{−4}  4.5 × 10^{−4}  −0.87  2.00  2.17  0.86  1.17  1.38 
3,6,11  −1.7 × 10^{−4}  0.0017  0.0021  2.7 × 10^{−4}  2.6 × 10^{−4}  3.4 × 10^{−4}  0.19  2.02  1.93  0.21  1.25  1.18 
2,6,8,11  −2.5 × 10^{−4}  1.30  1.02  4.7 × 10^{−4}  0.75  0.31  −0.37  1.02  0.75  −0.48  0.74  0.84 
1,5,8,10  −0.0012  0.56  0.48  −4.9 × 10^{−4}  0.35  0.65  0.17  0.8  1.03  −0.15  0.99  0.96 
1,5,8,9,10  −0.0011  0.52  0.47  −5.5 × 10^{−4}  0.45  0.41  0.03  1.36  1.03  0.11  0.94  0.93 
2 ,6,8,9,11  −1.5 × 10^{−4}  1.28  1.14  4.8 × 10^{−4}  0.66  0.59  0.11  1.05  0.96  −0.43  0.81  0.82 
1,2,3,5,6,8, 10,11  −3.3 × 10^{−4}  1.00  0.94  1.3 × 10^{−4}  0.69  0.65  −0.19  1.48  1.22  −0.64  0.32  0.69 