Yp- Similarly, erroneous slant ranges ( Ar) create
errors in Yor One finds:
Ay, = (r Ar -h Ah)/y, (13)
so that for standard deviations et and UE
2 2
2 2 2 = 2 er n à
og = 73 g + T3 Th = 2 + 2 (1 )
y r y sin ^9 tan“
Equation (14) clearly shows how the across-track
error reduces for increasing look angles but
increases with the errors of range and height.
The results listed in Table 2 are thus not com-
parable if they are obtained from imagery taken
under different look angles, of terrain with differ-
ent relief, using different resolutions, or ground
points of different identifiability. The orbital
radar (Tiernan, 1976; Leberl, 1975e) and the JPL
L-band aircraft radar (Leberl, Farr et al.,1976)
both employ very steep look angles (elevation
angles in orbital radar: 0? to 229; in JPL air-
craft radar: 09 to 559) so that errors of y are
much larger than with systems using larger eleva-
tion angles. Also the identifiability of surface
features is far inferior on the Moon, or in the
Alaskan tundra, to that of man-made features in
well-developed areas, where all other results were
obtained.
Two accuracy analyses concern specific cases;
they are therefore marked by an asterisk (*). The
study by DBA-Systems (1974) reports on the plani-
metric accuracy employing a single radar image and
radar interferometer data (for an explanation of
the radar interferometer, see: Manual of Photogram-
metry (1966)). However, use of the interferometer
provides the elevation angle €) to an object point.
FLIGHT CONTROL ACCURACY
Data per (Meters)
100 km
along across height
lc lc lc
HIRAN 0.0 66.9 23.3 23.3
HIRAN 0.5 57.8 23.3 19.7
None 0.5 51.8 26.6 19.7
Table 3: Mapping accuracy from a single radar image and an
interferometer measuring depression angles; AN-ASQ 142 ra-
dar system with 3-m resolution used (See DBA-Systems, 1974)
This permits computation of topographic heights
even from single images.
For this case, DBA-Systems (1974) obtains re-
sults shown in Table 3. It should be mentioned
that these results concern geometrically probably
the best side-looking radar system ("All Weather
Mapping System" with AN/ASQ-142) with well-defined
control and check points.
The other study is by Raytheon (1973 ,see also
Greve and Cooney ,1974) and employs a digital
terrain model (DTM) for digital monoplotting. Con-
sideration of topographic heights should thus re-
duce the mapping errors.
As a general the achieved accura-
cies vary from better than the radar resolution to
several times the range resolution. However, to
achieve the sub-resolution accuracy, a very high
control point density is required (see Gracie et
al., 1970), with, at the same time, the absence of
any topographic relief.
conclusion,
6. MAPPING FROM A SINGLE RADAR STEREO MODEL
Far less work has been performed on radar
stereo mapping than on single image radargrammetry.
However, the methods of stereo mapping and the
order of magnitude of what can be expected have
been rather well-established, essentially since
the 1972 ISP-Congress.
6.1 Mapping Methods - Mathematical Formulations
Radar stereo mapping has limited itself to the
purely analytical approach. Only for PPI-radar |
(non-side-looking) has there ever been the concept
for a plotting instrument proposed (Levine, 1963). |
Derenyi (1970) and Konecny (1970, 1971) have
established that the formation of a stereo model
(relative orientation) is not determined for strip
imagery. This leads to the conclusion that radar €
stereo models can only be formed if the elements
of exterior orientation are known (measured).
Model formation consists, thus, of the solu-
tion of two pairs of equations (Equations (6) and
(7)); each pair is obtained from one image of the
stereo set, denoted by (') and (") (see Leberl,
1972b, 1975e): |
ps} =r'; up > 5!) > sinrju'tip - s'i
(15)
Ip. = s"| = re”; op = s") = sinr[|u"| |p. = s"
This is a set of four non-linear equations with
three unknowns (x , Z ). Linearization leads
to a set of RS "walt tone
CV «C^ DAD tf, - 0 (16)
OR E] Fol A —1
where vector v,, contains the conreccion to the
14 observations (s, s" , $', S", I” ), Ap
is the vector of unknown, s of diete Miet in
C and D are coefficient matrices. Solution of this e
set of equations follows the rules of the method of
least squares.
Instead of directly introducing s, $ as obser-
vations into the projection equation, Gracie et
al., (1970, 1972) first computed spline polynomi-
als "ehrough all observed values of s, $. The co-
efficients of the splines were assumed to be con-
stants. Only the time is observed which serves as
the parameter of the spline polynomials. This
approach thus reduces the number of observations to
four (rt! , t” r',r"). A completely rigorous
approach would however require also the s, S to be -
considered as observations. Such a formulation is | >
used by DBA-Systems (1974) and is also described
by Leberl (1976b). |
Instead of Equation (15), one can also equate
the two sets of Equations (3) (Leberl, 1972b,
1975e):
(p^) = 0 (17)
s' + A' (p*) 1 - s" + A"
known look angles Q', MO". Linearization leads
to:
This is a set of three equations with the two un- €
EAN + EE, y (18)
2 =2
This approach minimizes the distance between the
homologue projection circles.
