the sensor; (2) The spatial range between sensor and ground 
points is equal to the range measured by radar wave. The 
Coplanarity equation is build according to the fact that radar 
beam plan is perpendicular to x' axes of satellite body 
coordinate system which is changing with attitude angles. 
Range-coplanarity condition is established by the following 
equations: 
R -|OP -OS | (1) 
i -(OP-OS)=0 
i is the normal vector of the plane of radar beam center (the 
same as x' axis which is converted from x axis in the sensor 
coordinate system by rotating attitude angles), op.osare the 
position vectors of ground points and the radar antenna, 
respectively. R is the slant distance. 
  
  
P à 
x terrain 
Fig 1. Range-coplanarity imaging geometry 
for side-looking radar 
When the rotation of attitude angles of 9-x-« system is used, 
the expansion equation of imaging equation for radar image of 
planar scanning mode is correspondingly obtained as follow: 
(X — Xs)(cosp cos x) + (Y — Ys)sinx — (Z — Zs)(sin pcosx) = 0 (2) 
(X - Xsy! «(Y -Ysy! €(Z - Zsy. - (M, Ry 
In formula(2), (Xs, Ys,Zs) is sensor location coordinate, 
(X, Y,Z) is ground point coordinate, M, is the across-track 
resolution; Ro is nearest slant distance; y is the column 
coordinate of the pixel in the image. 
For satellite radar image, paper (Cheng,2012) has given two 
kinds of R-Cp model in ECR coordinate system, one is: 
i (X X) i (Y -3) -(Z - Z5) 20 (3) 
(X-X)Y «(Y -Ysy «(Z- 45 - GM, -Ry) -0 
The other kind is explicit function of image coordinate (x,y) in 
ECR: 
pee AT f (4) 
y S[JCX — Xsy. (Y - Ysy +(Z —Zs} -RJ/ M, 
where , 
i =[i, iy ey =R"R,° [1 0 ol 2 RE R^ (o, Kk) are 
translate matrix special from orbit coordinate system to ECR 
coordinate system and from satellite body coordinate system to 
    
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 
    
orbit coordinate system. formula(4) is suitable for combined 
adjustment for multi-source data than formula(3). 
2.2 The connection and difference between R-Cp model and 
R-D model 
The Range-Doppler (R-D) model is the most widely used 
physical sensor model for spaceborne SAR remote sensing 
system(Curlander and McDonough, 1991). In the R-D model, 
the geometric characteristics of SAR imaging procedure are 
rigorously described by two equations in azimuth and range 
(Curlander,1982): 
| ROT OS (5) 
fa = UVs -Vp)-(OP-OS)/(AR) 
where fp is the Doppler frequency, À is the radar wavelength, 
Vs and Vp represents velocity vectors of SAR sensor and 
target respectively. 
We can see from the range equation that it is the spatial point 
set, which the distance is R from ground points to sensor, and 
the geometric shape is range sphere; The doppler equation is a 
cone over the sensor; The coplanarity equation shows the 
ground points and the sensor in the same plane. On a sampling 
point, the geometry figure of R-D equation and R-Cp equation 
are shown in the left part of Figure (2). 
For any radar image pixel we can calculate the doppler 
frequency of corresponding ground point, the distance between 
ground point and sensor antenna of photography time and the 
attitude values, then the R-D and R-Cp model can be built, 
which together with Earth ellipsoid model and the ground 
elevation model could process the radar image direct 
positioning. The intersection of the Earth ellipsoid equation 
with range equation, coplanar equation, or doppler equation 
separately presented on ground are a circle, an arc/line, and a 
hyperbolic. Therefore, the hyperbolic and circle intersection 
point on earth surface is the solution of the R-D model 
positioning, the arc/line and circle intersection point is the 
solution of positioning point under the R-Cp model, the 
principles of two models shown in the right part of Figure (2). 
Figure 2 also shows the link and difference when the R-Cp 
model and R-D model positioning, and also concluded that 
different pixel points at the same image line for non side- 
looking radar image with one group attitude values but 
different doppler frequency values. For the real aperture radar, 
when the precision sensor attitudes and the doppler frequency 
are known, the positioning result of R-D and R-Cp model is 
same. But for SAR images, because of its variety of imaging 
modalities, yet not one rigorous positioning model is common 
to all the images. If SAR imagery has the same features of 
geometric deformation as that of the image shoot by the real 
aperture radar under the ideal motion state referenced by the 
motion compensation process, the real aperture radar-based R- 
Cp model established can still be useful to the SAR images. If