-71 Nov. 1999 
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laser profiling system 
n in (b) shows a shift 
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a laser profiling sys- 
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identical to the shift s 
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reconstruction. Only 
being constant is the 
other data acquisition 
tions. 
International Archives of Photogrammetry and Hemote Sensing, Vol. 32, Part 3W14, La Jolla, CA, 9-11 Nov. 1999 
  
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s 
Qx 
»- X 
  
Figure 6: The coordinate components qx, q, of the posi- 
tional error q are a function of the direction of the flight 
line, here expressed by the azimuth «. 
Profiler and Sloped Surfaces So far we only analyzed 
the effect of a positional error on horizontal surfaces. 
Let us now consider sloped surfaces, as illustrated in 
Fig. 7. As before, a positional error causes a shift of 
the reconstructed surface. This, in turn, introduces an 
elevation error that depends on the slope as follows 
ee TS 
  
  
  
  
  
  
  
  
  
Figure 7: Illustration of the reconstruction error on 
sloped surfaces. The elevation error Az depends on the 
slope angle y and the shift s' due to the positional error. 
Az s tany (9) 
Here, s’ is the shift component parallel to the maximum 
gradient of the slope. Let œ be the azimuth of the flight 
trajectory and f the azimuth of the slope gradient (see 
Fig. 8). Then s' can be expressed as a function of these 
two azimuths. We have 
s = gicos(f — a) (10) 
Az -— |qicos(f — o) tan(y) (11) 
An analysis of Eq. 1 1 reveals that the maximum absolute 
elevation error is reached if cos(B — «) — «1. This con- 
firms our intuition that the error is largest if the flight 
  
Figure 8: Relationship between the flight trajectory and 
the slope gradient. 
direction follows the maximum slope gradient. Conse- 
quently, the elevation error is zero if the flight path is 
perpendicular to the slope. Moreover, if the flight direc- 
tion « changes, then an angular error causes a distortion 
of the surface, even if the slope remains constant. 
3.2 Reconstruction Errors Caused by Angular Errors 
Let us now investigate the impact of a systematic angular 
error on the reconstruction of surfaces. An example of 
an angular error is the mounting bias, see, e.g. Filin et 
al. (1999). 
Profiler and Horizontal Surfaces Fig. 9(a) shows a 
profiling laser system acquiring data over a flat surface 
and an object, such as building. The reconstruction er- 
ror a depends on the angle ó and the range r. Assum- 
ing that 6 remains constant during data acquisition, the 
magnitude of a changes with the range; the orientation 
of the error changes with the flight direction. 
Because a, the angular displacement vector, depends on 
the range, the error is smaller for surface points closer 
to the laser system. Fig. 9(b) demonstrates the conse- 
quence; we realize that the reconstructed flight path is 
no longer a straight line. The angular error causes a 
change in the angular relationship of the surface. 
Profiler and Sloped Surfaces The next example, de- 
picted in Fig. 10, refers to a profiling system acquiring 
data on a slope. As the aircraft flies at a constant height 
in the direction of the maximum gradient, the range gets 
smaller and smaller. Since the angular displacement 
vector depends linearly on the range, its magnitude de- 
creases uphill. Consequently, the elevation error also 
decreases, causing a slope error Ay of the reconstructed 
surface. We find for the slope error the following equa- 
tion: