ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002 
  
Using this relation (2) one gets the height-error of an unknown 
point N as follows: 
AH y - (r-1) (Hy - Hacp) (3) 
So we see, that during the orientation step of indirect 
georeferencing of aerial images with fixed IOR (according to a 
valid laboratory calibration) in the NPS the PRCs get a vertical 
shift of AH proportional to the flight height above ground 
(more exactly: above the level of the GCPs). Since Hr > Hgep 
and T > 1 (for Gauss-Krueger), the determined flight height will 
always be higher than in reality. In the restitution step the 
heights of unknown points in the level of the GCPs will be 
determined correctly, whereas points above resp. below this 
level Hacp get an error AHy which is proportional to the height 
difference (Hy — Hcp). 
Numerical example: Principal distance ¢ = 150 mm, image 
scale 1:10.000 — (Hr — Hgep) = 1.5 km — AHg = 40 cm (at the 
end of the Gauss-Krueger overlap A = 2.0°). With (Hx — Hgep) 
= 200 m we get AHy = 6 cm, which is slightly smaller than the 
best achievable height accuracy at this flight height of 
0.06%0o(Hy — Hace) = 9 cm [Kraus 1996]. 
This specified problem of indirect georeferencing in the NPS is 
well known in Photogrammetry for a long time; e.g. [Rinner 
1959]. In the work of Wang [Wang 1980] it is addressed 
thoroughly. He determines the introduced error in unknown 
points empirically with simulation computations depending on 
the kind of projection (Gauss-Krueger, Lambert, stereographic — 
all three being conformal; true ordinate — being non conformal), 
the size of the block of images, the position of the block relative 
to the central meridian, the flight direction, the image scale, the 
number of planar and height control points and the number of 
TPs. The main outcome of his investigations is that the impact 
of the length distortion (ie. P2 and P3) on the determined 
points in conformal map projections using Earth curvature 
corrected images is negligible. Although it should be mentioned 
that in Wang’s investigations horizontal terrain was always 
assumed, therefore errors in the determined heights due to the 
height difference to the mean level of the GCPs are not 
documented. 
Note: This height problem is only relevant as long as the 
principal distance c is not allowed to be corrected during the 
orientation step. If c (common for all images) is free, then 
problem P2 is solved in the middle of the area of interest. 
Problem P3, however, remains uncorrected. The equations (2) 
and (3) still hold but there T needs to be replaced by T/Telobal 
where Tgloba IS a mean value for the area of interest (see end of 
section 4). 
4. DIRECT GEOREFERENCING IN CONFORMAL 
MAP PROJECTIONS 
In this case the following quantities are given: the elements of 
the images’ XOR referred to the NPS, the image measurements 
of unknown object points and the IOR according to a valid 
calibration protocol. The coordinates of the unknown points are 
to be determined. Problem P1 is solved using the Earth 
curvature correction. What about P2 and P3? 
In this case none of the images’ XOR elements are free (they 
are already measured directly; i.e. the orientation step does not 
exist) and therefore the flight height can not adjust to the local 
planar scale — caused by T and realized in the known planar 
coordinates of the PRCs. In Figure 3 this situation is depicted 
(two images and one unknown point N, which shall be 
determined during image restitution). 
Cartesian: NPS: 
ge Gon Olpe A 
ho | 
! AH \ N 
Hy | | "n, 
Figure 3: Direct georeferencing 
The pencil of projection rays is congruent for both systems (the 
Cartesian one and the distorted one, i.e. the NPS), since in each 
the same value for the principal distance c is used. Therefore it 
holds: 
AHy -(r—1) (Hy - Hy.) (4) 
We see, that during the restitution step of direct georeferencing 
the heights of the unknown points will be determined with 
errors proportional to the imaging distance and since Hy»Hy, 
and t 71 (for Gauss-Krueger) all determined points will always 
lie below their real level. 
Numerical example continued: With the imaging distance 
(Hr — Hy) = 1.5km we get AHy = 40 cm and this value lies 
clearly above the achievable height accuracy of 9 cm. 
So we see, that in contrary to indirect georeferencing the height 
errors induced in the NPS during direct georeferencing are not 
negligible. If one compares equations (4) with (3) and 
exchanges Hr by Hac», one sees the equivalence of these two 
equations; ie. the height errors of the determined points 
increase for both — direct and indirect georeferencing — with the 
difference to the level of the height control points. During 
indirect georeferencing the unknown points lie approximately in 
the level of the (ground) control points, whereas during direct 
georeferencing they do not — they lie below by the amount of 
the flight height. This clearly shows the interpolating behavior 
of indirect georeferencing and the extrapolating behavior of 
direct georeferencing. 
Now the question arises, how to remove these errors. Three 
possibilities can be offered: 
MI) computation in a Cartesian tangential system 
M2) correction of the heights 
M3) correction of the principal distance 
The advantages and disadvantages of M1 were already 
discussed in section 2. The methods M2 and M3 are alternatives 
for solving the two problems P2 and P3 when performing direct 
georeferencing in the NPS and will be discussed in the 
following. 
In M2 the principal distance remains unchanged but all heights 
that are used in the AT are corrected by the respective planar 
scale (Hoor ^ Hgrt). If the area of interest is not too large, it 
should be sufficient to compute one representative value for t 
(— Tgzpoval) In the center of the area of interest and to correct all 
heights with T,,5,. In this case only P2 is solved and P3 is 
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