The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
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x _ f afX-X si ) + bfY-Y si ) + cfZ-Z si y 
a fX-X si ) + bfY-Y si ) + c 2 (Z-Z si ) 
0 f a 2 (X-XJ + b 2 (Y-Y si ) + C fZ-ZJ ' 
afX-X si ) + bfY-Y si ) + c 3 (Z-ZJ 
(1) 
Where a l ,b j ,c i (/ = 1,2,3) are elements of rotation matrix 
composed of (p,co,K, which are angle elements of exterior 
orientation. 
Two observation equations can be constructed with one ground 
control point (gcp). So at least 6 gcps and satellite orbit 
parameters are needed to achieve elements of exterior 
orientation of linear array push-broom imagery. 
2.2 F. Leberl Model 
The geometry relation between ground and image points in a 
SAR image is established by the Doppler and Range equations 
proposed by F. Leberl. (Xiao Guochao et al., 2001; He Yu, 
2005) 
because imaging time of each line is very short, the correlation 
of the exterior orientation elements of different lines is 
inevitable, which might induce no solution for collinearity 
equations. The ridge estimation has been used for solving this 
problem, but it is difficult to choose ridge estimation parameters 
in the classical ridge estimation algorithms. Zhang Yan et al. 
(Zhang Yan et al., 2004) proposed a robust combined ridge with 
shrunken estimator (RCRS), which could improve complex 
collinearity of coefficient matrix and detect singularity of 
observed value, making the estimator optimal and stable. Here 
we used this algorithm to orient SPOT image and achieved 
sub-pixel RMS error. 
In (3) for airborne SAR image, j^ is approximately zero for 
earth's rotation. But for spacebome SAR image, is not 
approximately zero but a linear function with time since flight 
eight is higher. Because y linearly varies with time, can 
be expressed as a n -order polynomial with y . (Wang 
Donghong et al., 2005; Chen Puhuai et al., 2001) Then (3) can 
be written as 
The range equation of slant range image is 
(X - Xs ) 2 + (Y - Ys ) 2 + (Z - Zs) 2 = (y s M y + Ds 0 ) 2 ( 2 ) 
Where Ds is the slant range delay, y is the across-track 
image coordinate of ground point P, M is the across-track 
pixel size, ( x,Y,Z ) are the object space coordinates of ground 
point P, ( Xs,Ys,Zs are 0 bject space coordinates of radar 
antenna center. 
Xv (X - Xs) + Yv(Y - Ys) + Zv (Z - Zs) ( 4 ) 
= a o + a \Y s + a iYs + a 3 yl + a *yt 
Where «.(/' = 0,1,2,3,4) are polynomial coefficients. 
When the exterior orientation elements of linear array 
push-broom imagery and SAR image have been achieved 
respectively, the stereoscopic pair is constructed. The space 
coordinate of ground point can be computed with homologous 
image points based on space intersection. 
The Doppler equation is 
Xv (X - Xs ) + Yv (7 - Ys ) + Zv (Z - Zs ) = - f DC ^ 
Where R is the slant range of ground point P, ^ is the 
radar wavelength, j- is the Doppler frequency. 
F. Leberl model is composed of formula (2) and (3). When 
f DC = o, they can be linearized to achieve elements of exterior 
orientation with gcps. 
3. THE CONSTRUCTION OF COMPOSITE STEREO 
MODEL 
3.2 Space Intersection 
A procedure in the composite stereo positioning is the 
determination of object space coordinates X,Y,Z from a pair 
of homologous image points ( x{ ,yf, x 2 >y 2 ) measured in both 
images of a stereo model. Using these measurements as input to 
the appropriate pair of mapping equations (i.e. equation (1) and 
(2) (4)) yields 4 equations to calculate 3 unknown entities. 
The overdetermined problem is solved by standard least squares 
adjustment using equation (1) and (2) (4) to calculate proper 
increments to the approximations given for the unknown 
coordinates. Using equation (1) and (2) (4), the corresponding 
position equation is as follows (in matrix notation): 
V = DA - L P (5) 
3.1 The Construction of Composite Stereoscopic Pair 
There usually are two ways to construct stereoscopic pair. One 
is first relative orientation with connection points and then 
absolute orientation with gcps. The other is direct exterior 
orientation of two images respectively and then combination of 
them. Since the equations of relative orientation are derived 
from collinearity equations, the former way is invalid for SAR 
image. In the later way, the orientation procedure of two images 
can be applied with two different groups of gcps respectively, 
which avoids the difficulty for collecting connection points. 
Here we choose the later way to construct the composite 
stereoscopic pair. 
The residual error vector is y - V2 Vj ^ y ; 
The coefficient matrix is 
D = (A D 2 D 3 D a ) t = 
^d\\ d n d X2 f 
d 2X d 22 d 2i 
^31 ^23 ^33 
V^41 ^24 ^43 J 
The increments vector of the unknown ground coordinates is 
A = {dX dY dZ) T \ 
As shown in (1), the exterior orientation elements of different 
line in linear array push-broom imagery are different. But