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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B7, 2012 
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia 
SPACEBORNE SAR IMAGERY STEREO POSITIONING 
BASED ON RANGE-COPLANARITY EQUATION 
C.Q.Cheng® *, J.X Zhang *, G.M.Huang *,C.F.Luo* 
* Chinese Academy of Surveying and Mapping, Beijing 100039, China-cspring@casm.ac.cn 
Commission VII 
KEY WORDS: rigorous positioning, side looking radar, R-Cp equation, exterior orientation elements, SAR 
ABSTRACT: 
Stereo positioning is a important content for Spaceborne SAR imagery mapping, which is still lack of investigation depend on 
exterior orientation elements. In this paper, we firstly introduced the Range-Coplanarity (R-Cp) geometric imaging equation and its 
character for radar imagery, compared the difference and relationship between R-Cp and R-D model, and build stereo positioning 
rigorous model for spaceborne SAR imagery based on the R-Cp equation, satellite orbit, and pose refinement model. Then separate 
taking TerraSAR-X images in mountainous terrain observed from same and opposite side-looking direction as test data, we 
presented the initial attitude value acquirement method for TerraSAR-X image after moving compensation, analyzed stereo 
positioning precision according to amounts and spatial distributions of GCPs and lengths of baseline between SAR antennas, and 
also compared the precision with R-D model. the proposed stereo positioning model avoids SAR signal imaging parameters during 
the geometric processing, with one GCP it could improved the plan and height precision from (17.9m,29.2m) to (10.3m,6.1m) in the 
test of opposite side-looking direction images, and even had a better accuracy than R-D model when more GCPs were used, which 
indicate that R-Cp model have a potential application in spaceborne SAR image stereo positioning. 
1. INTRODUCTION 
The rigorous geometric imaging model of side-looking radar 
image mainly includes the range doppler equation and sensor 
exterior orientation elements. Leberl F model proposed by 
Leberl F (Leberl, 1978, 1990), according to range condition 
and zero doppler condition, complies with the radar imaging 
mechanism. The R-D model, with similar mechanism of Leberl 
F model, takes into account of the non-zero value situation of 
doppler frequency, and therefore has explicit geometric and 
physical meaning (Johnsen, et al., 1995). There have two kinds 
of model based on exterior orientation, the radar collinearity 
equation-based model(Konecny and Schuhr, 1988), is the 
extension of the application of collinearity equation in optical 
remote sensing image. The radar range coplanarity 
equation(Cheng et al, 2012),build according range condition 
and radar beam center coplanarity condition. 
Currently, the SAR image geometric correction under different 
conditions has been more extensively studied (Lauknes,2004, 
Johnsen, 1995; Gelautz,1998). The stereo positioning when the 
1960s was literature related to (La Prade,1963), However, due 
to the lack of suitable stereo image data, related research was 
unable to started, this situation changed when the RADARSAT 
launched in 1995,and most of SAR image positioning process 
adopted the R-D model. The stereo positioning research result 
used the elements of exterior orientation as positioning 
parameters still very scarce. As the Range-Coplanarity 
equation use the elements of exterior orientation as a 
directional parameters, the research of spaceborne SAR stereo 
  
* Corresponding author. 
positioning of sparse control point based on the R-Cp equation 
is significative. 
In 2005, China launched the 1:50,000 mapping project in 
western difficult region where without the topographic map 
ever before, for example the Hengduan Mountains, Yunnan 
and Guizhou Plateau are covered by snow or fog all the year 
round, optical remote sensing images are difficult to separately 
complete the mapping tasks, coupled with ruggedness terrain in 
these areas, field surveying of ground control points is very 
difficult, so mapping with high-resolution SAR images for 
these specific regions is an important part for the project, also 
the spaceborne SAR images stereo mapping under the 
conditions of sparse control point is a key technology for this 
project. In this case, the paper based R-Cp equation research 
the TerraSAR-X images stereo positioning models, techniques 
and methods. 
2. THESTEREO POSITIONING MODEL BASED ON 
R-CP EQUATION 
2.1 Introduction of R-Cp geometry imaging model 
R-Cp geometric imaging equation based on exterior elements is 
built through two conditions as following and its principle is 
shown as Fig 1 (Cheng et al, 2012): 
(DAI ground points corresponding to one line in image and 
the radar antenna at shoot time are within one center plane of 
radar beam, which is determined by state vector and attitude of