Figure 2: Coordinate systems and definitions for radar projection equations; (a) planetocentric and antenna systems, 
K = intersection of range sphere and planetary surface (reference figure); (b) cone complement angle 7 and vector p 
p sA)p (3) 
where the rotation matrix A describes the rotation 
of vectors u, v, w parallel to x, y, z, SO that: 
À = Ju #2 (4) 
For a real aperture radar matrix A contains the 
orientation angles 0, o, K in their familiar form 
(see, for example, Leberl, 1972a). For synthetic 
aperture radar, however, vectors u, v and w are 
functions of the velocity vector 8 and position 
yector s: 
u = 3/5] 
+" tx Dis nd 5) 
Equations (1) or (3) can now be rewritten in sever- 
al ways, of which the most common (Konecny, 1970, 
1972a; Greve and Cooney, 1974: Gracie, et al., 
1970,:1972; Leberl, 1971c, 1972a, 1972b; DBA-Sys- 
tems, 1974) can be given in vector notation as: 
Ins 7 (6) 
2dQ-9-serqs|ip-a| (C? 
Equation (6) represents a sphere with its center 
at s and radius r ("range-sphere"). Equation (7) 
represents a cone with the axis along vector u 
(u defines the longitudinal axis of the real an- 
tenna; or it defines the direction of the velocity 
vector s, which in turn is the longitudinal axis 
of the synthetic antenna). If the radar operation 
is "normal", then r - 0 and the cone degenerates 
  
  
  
  
  
  
  
to a plane (the so-called "scanning plane"): 
8 -(p-8} =" (8) 
For completeness it should be mentioned that 
the previous equations neglect the small offset 
As between the sensor platform, whose position is 
defined by s, and the antenna. It might also be 
worthwhile to point out that the rotation matrix 
A for real aperture radar might be a composite of 
several rotations (antenna into antenna mount; 
antenna mount into sensor platform; sensor plat- 
form into reference system). 
5. SINGLE IMAGE RADAR MAPPING 
Mapping with single radar images has been 
carried out with the objective to derive object 
coordinates of imaged points either with or with- 
out the use of ground control points; to rectify 
the radar image for enhanced interpretability or 
for masaicking; and to produce a line plot as a 
cartographic product. The following will address 
first the mathematical methods of single image 
radar mapping, and then analyze the accuracy by 
reviewing the studies that have been carried out 
on this topic. 
5.1 Mapping Methods: 
The inputs to single image radargrammetry con- 
sist of (a) the measurements of slant range, r;, 
and time of imaging, t,, of a selected number ôf 
image points p; (i = 1.1. ip); (b) the measure- 
ments of the sensor-position s(t3), (j 13x. 3; 
ds) and sensor velocity s(t;) or attitude (tz), 
(ts), K(t3); (c) the position vectors gy of s 
number of ground control points g,, (k = |, - . 
ke); (d) perhaps a digital terrain model (DTM) of 
the imaged object. 
Computation of the position vector p. of a 
ground point can be based on either Equations (1) 
or (6) and (7). The three Equations (1) contain