four unknowns (p;=(xp» Ypi> Z pi) and Q,:) the 
alternative formulation of Equitions (6) and (7) 
consists of = equations with three unknown 
(“pi » Ypi» Obviously, one of the unknowns 
has to be ie measured or given so that the 
equations can be solved. 
In single image radargrammetry, the unknown to 
be given (or to be assumed to be known) is the 
height of the object. Two distinct cases can be 
differentiated: (a) the simplified case, with a 
straight, level flight parallel to the x-coordin- 
ate axis; (b) the general case, with measured 
flight parameters, in a geocentric coordinate sys- 
tem. 
The following will discuss these two cases and 
then present a method of using ground control 
points; it will treat the rectification of imagery 
and the merging of the radar image with digital 
height data in a process of "digital mono-plotting." 
Simplified Case of Coordinate Computation: 
  
Often a simplified flight configuration is 
assumed for single image radargrammetry (e.g. 
Rydstrom, 1968; Moore, 1969; Bosman et al., 1971, 
1972a, b,; Koopmans, 1974; Hirsch et al., 1976) 
With a straight, level, and undisturbed flight 
parallel to the x-coordinate axis the inputs to 
Equations (1) or (3) simplify to: 
$m (x.20, h) 3* (6,0, 0) 
d= K=O r=20 
so that 
x = x 
p S 
2 2 15 
= (he 9 
Yo (r ( 2.07) (9) 
2 -7 
P P 
General Case of Coordinate Computation: 
  
The position of the sensor is measured by an 
inertial or similar guidance system, by SHORAN or 
HIRAN- tracking, or by satellite orbit determination. 
In all these cases it is convenient to work in a 
geo-(planeto-)centric coordinate system. The 
height of the object is in this case expressed as 
the radius R of the reference spheroid or sphere. 
According to Leberl (1975 e) one obtains from 
Figure 2: 
BF hy + h, + h, (10) 
where (note that R = Ip]: 
2 2 2 
h, = (|e] + |s| - r )/{(2 Ils)» 
h, = h, s/ |S|] 
|e 
* 
Il 
< 
xs/ivx s| (for v, see Equ. 5) 
cose c u*- S/]8| 
h, = ut fs] - hj) tane * r sin "| 
Ground Control Points: 
  
Ground control points can be used in one of 
two ways: (a) interpolatively (b) parametrically. 
In the interpolative method, the radargrammetrical- 
ly computed position vectors p, and the given 
ground points By define difference vectors p 
(k 9 1,5... k.. These can be, used to. intefpo- 
late correction in radargrammetric points. 
Leberl (1971c, 1972a) evaluated a series of differ- 
ent interpolation algorithms with the result that 
piece-wise polynomials performed best; and Derenyi 
(1974a) experimentally searched for the optimum poly- 
nomial. 
An elegant parametric method of using ground 
control points was formulated by Gracie et al., 
(1970, 1972) and used by Greve and Cooney, (1974) 
and Leberl (1975e). This method applies to synthe- 
tic aperture radar. According to Figure 3, a plane 
through the ground point g, but normal to the 
flight line, is intersected by the flight line. 
This is represented as a polygon of linear pieces 
carrying an index £(L= 1,5 . DE 
e, 7 En (85 7 8) (11) 
5 $? * 8) e, / 1354] (12) 
From position vector Sg» the slant range rg and 
time of imaging tg can easily be determined. The 
radar image also provides an independent measure- 
ment of slant range and time, namely ry, tp. The 
discrepancies Ar = Ip -,fg and At = tp - te 
can define the coefficients of a correction poly- 
nomial for the range and time measurements in the 
radar image. 
  
ZERO-DOPPLER 
PLANE 
  
  
  
  
Figure 3 
Computation of the radar sensor's orbit posi- 
tion from where ground point g was imaged (sin- 
gle image radar resection) 
Rectification: 
The along track and across track coordinates 
of radar images are generated independently of each 
other. There is, therefore, the possibility of in- 
consistencies in along- and across-track scales. 
In the case of synthetic aperture radar images 
these inconsistencies can, if known, be removed in 
the process of converting the raw sensor output 
(the "signal-histories") into the "map film." 
The method employs a variable Scale setting in the