(x cxt) 
(x -x! 2j 
io 
(y -y* ) 
1.50] 
coordi- 
groun- 
cients, 
, pitch 
ulation 
y affect 
values 
ice the 
the di- 
- in the 
derive 
which 
ssump- 
e been 
nearity 
Xf the 
f high 
| coor- 
y radar 
ich in- 
point 
**3/yi 
33*4/yi 
For single strips these equations at least have to be 
reduced for the unknown parameter for the 
terrainheight zi. For flat terrain the terms depending 
on the terrain height zi are zero. The principle idea of 
this method for the formulation of algorithms goes 
back to Baker ( 1975 ), who applied it on line 
scanner imagery, which has been modified for SLAR 
and SAR image geometry. These equations can al- 
ready be used without the knowledge of the content 
of the parameters AO to A4, respectively BO to B13, 
which are stated as follows: 
AGS -kx 7 x'oO/h -:k»i£$90 
Ais kx-k@/h 
A2- kx?-x'o1/h? 
AS- kxs445 - x'o2/h4*35 
A4= kx/h 
BO--ky-y'oO/h-*r'o-z'oO/h-ky-QO-r'o-dz/h*r'a-dky-dro' 
Bis-ky-z'oce/h?*-r'o-O0 /h*ky/n?-dz*(-ky/hr1/h) -dky 
B2= -kx-ky-y'oiíi/h* "kx -6'GO-2'01/h3 
B3m —kx?-ky-y'o2/hs*S9kx?-ro-z'o2/h«s«3 
B4- -kx-ky-z'oi/h**3 
BS= -ky-Q@/h? 
B6- -kx?-ky-y'o2/h**4 
B7= 2-kx**3-ky-z'o2-z'ol/h**4 
BB- kx**4-ky-z'o2?/h**5 
B9- -ky-2'o0/h 
BiO0- -kx-ky-z'ol/h*? 
Bii -kx?*-ky-2'o02/h**3 
B12-ky/h 
BiS3cky-(2i-2z'O0-kx-2'O01-xi/h-kx?-2'O1-xi?/h22)/th-y1) 
This approach can be extended by variation of the 
sensor behaviour, in particular by calculating for the 
"real flight path", as gained from housekeeping GPS 
and by additional parameters , including the 
introduction of the Doppler information of the Radar- 
image, to fit smoother to the ground control point 
field. As compared to heuristic approaches, which 
are still in use even for very advanced image 
processing devices, arbitrariness is subdued and the 
3rd dimension can be calculated. 
For recent radar campaigns the polynomial approach 
as derived from equivalent collinearity equations, 
showed an accuracy of approximately + 10m. 
To use the advantages of existing bundle block ad- 
justment software for conventional photography, an 
approximate transformation of the radar geometry 
into the conventional image geometry and vice versa 
also successfully has been carried out. The modified 
BINGO- program of the Institute for Photogrammetry 
of the University of Hannover according to Kruck et 
al. ( 1986 ), for 47 ground control points allowed to 
obtain an accuracy of about 22.5 pixels for a SIR -B 
image of the testsite Freiburg, as carried out by 
WIGGENHAGEN. 
2.3 Combined Approaches 
The following method gives an example for a 
combined approach between parametric and non- 
parametric solutions: Radar image coordinates (xi', 
709 
yi) for ground coordinates of particular points with 
known object coordinates (xi, yi, zi), including output 
pixels (e.g., anchor points), can be calculated from 
known ground control points (xp, yp, zp). Supposing, 
roll-, pitch- and yaw-values are neglectable and under 
the condition, the object coordinate system and the 
radar image coordinate system are in parallel, the 
following approach approximately is valid: 
x1'=xp'+kx((xi-xp)/h) 
yi—yp-ky((yi-DELTA Yi-yo'i)/(z-zo'i) 
-(yp-DELTAYp-yo'p)/(z-zo'p)) 
with DELTAYi-h (zi-z) / (yi-yoj). 
For the calculation of DELTAYp the index "i" has to 
be replaced by "p". The results improve by applying 
this approach onto surrounding control points. The 
deviations of the resulting image coordinate pairs for 
xi and yi' can be eliminated by the weighted mean 
approach. To apply this method, a great amount of 
ground control points is needed, which can be de- 
termined by radar block adjustment etc.. 
3. RADAR MAPS AND RADAR ORTHO- 
PHOTOS 
3.1 Quicklook results 
The proposed SAR image standard product is 
ground range imagery, because it is accepted by the 
user community. This output result for areas with flat 
terrain might even be considered as a final product, in 
particular for ocean regions and for coastal sites. 
3.2 Radar orthophoto 
For tasks, which require map accuracy(usually +1...3 
pixels, see DOYLE (1975) and KONECNY et 
al.(1984b)), like 
- GIS-input 
- mosaicing, 
- multisensor imagery and 
- change detection etc., 
for hilly and mountainous terrain, a digital geometric 
pixel by pixel image restitution, including terrain 
height effects is needed, preferable applying the indi- 
rect rectification method. To handle larger output 
blocks and to calculate for regional or even particular 
pixel wise terrain heights, according to suggestions of 
Egels and Massou d'Autume of the IGN (France) and 
Konecny (1985) the following method for digital im- 
age rectification has been established: 
Step 1 and 2: anchor point determination: 
1. For minimum and for maximum terrain height