i 
EE ER 
  
en a EEE 
meme re 
MEC 
  
for limited areas of approximately 40x40 km? of flat 
terrain and stable flight conditions already allow to 
obtain accuracies of about +1 pixel for ground con- 
trol points. 
The advantages of 2 dimensional polynomial 
equations beside others are 
- the didactic value for introducing into digital geo- 
metric image processing, 
- suitability for quick programming 
- satisfying for limited areas of flat terrain 
- support for approximate value determination 
- support for blunder detection 
Some disadvantages of 2 dimensional polynomials 
are 
- arbitrariness 
- limited area of validation 
- a blockadjustment based on arbitrary polynomial 
equations of higher than first order shows an 
extremely bad error propagation. 
- restriction for 2 dimensions. 
2.2 PHYSICAL PARAMETRIC APPROACH 
In order to formulate strict geometric algorithms for 
SAR- and SLAR radar imagery, a physical 
parametric solution is envisaged, which allows to 
calculate 
- the global and local behaviour of the sensor posi- 
tion and attitudes, 
- 3dimensional ground control point coordinates and 
- computation of imagecoordinates for 3 dimensional 
output raster data (resampling). 
Extended collinearity equations as derived by the 
authors (see Konecny, Schuhr, 1986) fulfil this geo- 
metric requirements for radar images. For the 
abscisses values x' of the image coordinates the fol- 
lowing constraint is valid: 
a up DELTA fo ai is DELTAY - x (z - 2! 
j 
0 =-kx 
) 
137 oj 
  
a (x -DELTAX-x' )+a (y.7 DELTAY — y! )+a (2.-—:2' 
3134 oj 323 À 
For the groundrange ordinate values y'gr follows: 
a (x -DELTAX-x' )+a . (y - DELTAY - y' 3a (z - 2° 
13 i oj 22) =i oj 
oj 33] oj 
23j oj 
  
J 
LL po -DELTAX-x' )+a (y - DEUTAY = y' )ia (2 - z' 
31} i oj 32j i 
Notice, the z value carries no index, because it repre- 
sents the (unknown) (constant) terrainheight for the 
groundrange image calculation, while DELTAX and 
DELTAY depend on the individual terrainheight zi. 
To achieve the measured y' value in the ground range 
image, the near range distance r'o has to be 
subtracted from the computed groundrange ordinate 
value y'gr 
Y-y gro 
oj 333 oj 
708 
While for slant range ordinates follows 
With sufficient approximation is valid: 
DELTAX = 
  
  
  
2 2 2 2 2 2 
(xx) Hy!) ICS 721 )+(y_-y' )+(z -2' )-h 
+ i. oj i.ej J|: i o) i oj i oj 
  
(x ex") 
-------------------- i oj 
| xt) ay yt) 
je Por y y oj 
= F X (x. -x' ) 
1-0 
DELTAY = F x (Y -y* ) 
io} 
As usual 
X1,y1,Z1 =3dimensional object point coordi- 
nates 
XO0j,yoj,Zo)  -instantaneous sensor position 
z =terrain height chosen for groun- 
drange calculation 
h =70'] - Z 
kx, ky = equivalent focal length 
al lj until a33j = instantaneous rotation coefficients, 
which ,as sinusoidal functions, depend on roll, pitch 
and yaw values as a function of x' ("= time"): 
roll = omega = Qj = Qe + Qi - x'i +» Q2 - x'i? 
pitch = phi = &j = $0 + $1.7 x'1 t $2 - x'i? 
yaw = kappa= kj = ke + ki - x'i + k2 - x'i? 
xo'j = x'900 t x'oi - x'i + »'g2 - x'j* 
yo‘; = y'00 + y'oi - x'i + v'02 - x'i? 
Zo) = 2'Q0 + 2'01 - x'i * z'o2 - x'i*? 
This expressions are valid for the general formulation 
of SLAR- and SAR- image geometry. 
While deviations in the sensor position directly affect 
the SAR image geometry, general attitude values 
only, but not changes of attitudes do influence the 
SAR image geometry, which is in opposite to the di- 
rect influence of the unstable sensor behaviour in the 
SLAR image geometry. 
Linearized collinearity equations allow to derive 
polynomial equations of equivalent content, which 
are valid under particular flight behaviour assump- 
tions. The following observation equations have been 
derived from linearized equivalent radar collinearity 
equations ‚for second order variations of the 
orientation elements after the elimination of high 
correlated terms and after transition to ground coor- 
dinates. For a block consisting of overlapping radar 
strips, the complete observation equations, which in- 
clude the calculation of 3dimensional point 
coordinates are 
vx'iz AO + Al - yi + AZ - xi t AS - xi? 
+ A4 - dxi - x'i measured 
vy'l= Bo + Bl - vi + BZ2 - xi + B3 - xi? 
TO B4-xi-yi * BO^yi* € B6-xi*-yi + B/-xi*<3/yi 
*t BB-xi**4/yi 
4 B9 2i/yi t B10-xi-z2i/yi *B11-xi?-zi/yi 
* Bi2-dyi + B13-dzi - y'i measured 
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