In: Wagner W., Székely. B. (eds.): ISPRS TC VII Symposium - 100 Years ISPRS, Vienna, Austria, July 5-7, 2010, IAPRS, Voi. XXXVIII, Part 7B 
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(3) Coplanarity equation derived by K-<p-G) system 
In the same way, the expansion of the coplanarity condition 
equation is derived as: 
{X - 3fr)(cos /c-cos<p) + (F-Xs)(sin vcos<p) + (Z-Zs)(- sin cp) = 0 (7) 
It can be concluded that the coplanarity condition is 
independent of co angle when the K-cp-co or cp-K-co rotation angle 
system is used. 
3. RANGE-COPLANARITY EQUATIONS IN 
DIFFERENT COORDINATE SYSTEMS 
3.1 The imaging equation in tangent orthogonal coordinate 
system 
It is suitable for airborne or small area spacebome radar 
images to orient in tangent orthogonal coordinate system. 
Thereinafter, imaging equations and image processing model 
will be derived based on the scanning plane confirmed by cp-K- 
co system. 
The distance equation based on range-coplanarity equation is 
the same as the one based on range-Doppler model. For slant 
range images, the equation is as follows: 
(X -Xs) 2 +(F- Ys) 2 + (Z - Zs) 2 = (y s M y + D s f (8) 
i x (X -Xs) + iy(Y-Ys) + i z (Z - Zs) = 0 
namely: 
W (Xo)x (Zo) x ~ 
cos<pcos/c 
'X-Xs 
( (X 0 )y (To)r (Zo)r 
sin K 
)• 
Y-Ys 
_(X 0 ) Z (Y 0 ) z (Z 0 ) z _ 
-sin^cos/c 
i 
N 
1 
where 
i x = cos (p cos ic(Xo) x +sin/r(Fo)^ - sin (p cos k(Zo) x 
\ i y = cos<pcos>c(Xo)y +sin/£:(Fo) y - sin (pcos k{Zo) y 
i z = cos (p cos ic(Xo) z + sin/r(Y0) z - sin cp cos k(Zo) z 
Thus, the range- coplanarity equation in GOCS is: 
j i x (X-Xs)+i y (Y-Ys)+i z (Z-Zs)=0 
j(X-Xs) 2 +(Y-Ysf HZ-Zsf ~{y s M y +Rff =0 
3.3 The Range-coplanarity Equation with Coordinates of 
Image Point as Explicit Function 
Thus, the range-coplanarity equations are obtained: 
f (X - XsXcos <p cos k) + ( Y — Ks) sin k — (Z - Zs)(sin (p cos k) = 0 (9) 
{X - Xs) 2 +(Y- Ys) 2 + (Z- Zsf = (.y s M y + R 0 f 
3.2 Imaging Equation in Geocentric Orthogonal 
Coordinate System (GOCS) 
The origin O of orbit coordinate system is the exposure 
station in the sensor orbit. The mathematic definitions of three 
axes in the coordinate system are as follows: 
'z o =[(Zo) x ,(Z 0 )y,(Z o ) z ] = P(t)/\\P(t)\\ ( 10) 
• F. =[(Y„) x ,(Y 0 ) r ,(Y 0 ) z ] = (Z B xF)/||(Z 0 x V)\\ 
X 0 =[(X 0 ) x ,(X 0 ) y ,(X 0 ) z ] = Y 0 xZ 0 
The normal vector of the scanning plane is: 
T 
i=K E K 
1 
o o 
L ^ 
- 
iy 
Jz _ 
And the coplanarity condition equation is: 
In order to express the above equation as a function of 
coordinates of the image point, the left-side of the coplanarity 
equation of the formula (9) is multiplied by a function F, with 
the value of coplanarity equation still being 0. Let F be: 
-J. (14) 
sin <p(X - Xs) + (Z - Zs) cos (p 
Where / is called equivalent focal length whose value does 
not affect the tenability of the equation, but affects the number 
of iterations while the equation is solved as well as the fixed 
weight of the observed values, f can be obtained by the 
following formula according to the geometric relation among 
pixel size, ground resolution and focal length: 
f Hs 'Fo (15) 
J GSDx 
where [J,q is the value of equivalent pixel, Hs is the height 
of the sensor, and GSDx is ground resolution of 
corresponding azimuth of the pixel. 
Meanwhile, the range equation of the formula (9) is also 
transformed into a function of y, that is, the range-coplanarity 
equation of SAR image positioning with coordinates of image 
point (x,y) as the explicit function can be obtained: