-11 Nov. 1999 
noticeable errors. 
  
ror of ó' introduces a 
rface is tilted by Ay, 
urface. 
| Surfaces Let us fi- 
Alar errors of a laser 
le, shown in Fig. 11, 
lown with a scanning 
0. This error is mea- 
range vector and the 
lane here. Fig. 12 de- 
It is obtained by pro- 
can plane. Thus, we 
r) (15) 
ror can be character- 
fset, and the azimuth 
The scan direction is 
e angular error is the 
he scan plane. 
ular error ó' causes a 
tude depends on the 
maller the error. The 
dicular to the range 
the following simple 
Scanning System and Sloped Surfaces The last ex- 
ample of a systematic angular error is related to a 
laser scanning system that flies across a tilted surface. 
Fig. 13(a) depicts this scenario; five individual footprints 
of one scanning sweep are shown. As in the previous 
example, the angle ¢’ indicates the effective angular off- 
set. 
  
  
  
Figure 13: A scanning laser system flies across a tilted 
surface. Five individual footprints of one scan sweep 
are shown. The reconstruction is affected by the effec- 
tive angular error ó'. It causes a displacement of the 
reconstructed points by an amount that is proportional 
to the range and 6°. Since the range changes on a tilted 
surface, the error also changes from point to point. As 
a consequence, the reconstructed surface has a wrong 
slope (a). If the slope changes within the swath, the 
angular relationship between the two slopes, in (b) ex- 
pressed by 4, also changes. 
The error a’ is a function of the range and the effective 
angular error 6’. We have 
a’ = |r|cos(e — T)tan(ô') (17) 
This error obviously reaches a maximum if € = T which 
would mean that the angular offset is exactly in the scan 
plane. On the other hand, the slope error vanishes if the 
offset is perpendicular to the scan plane. 
The magnitude of the error a' depends linearly on the 
range. Footprints that are closer to the laser system have 
a smaller error. With the error vectors at both ends of the 
scan sweep, we can determine the slope error as follows 
International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 3W14, La Jolla, CA, 9-11 Nov. 1999 
^y - tanó' tan? y (18) 
3.3 Summary 
The simple error analysis in this section demonstrated 
the effect of systematic positional and angular errors on 
the reconstructed surface. By and large, the resulting 
surface errors are a function of the topography, flight 
direction, and the systematic error. This functional re- 
lationship can be exploited to determine the errors if 
the true surface is known. In the interest of brevity, the 
error analysis was restricted to profiling and scanning 
systems. Non-planar scanning systems, such as nutat- 
ing mirror systems, will show similar effects, but closed- 
formed expressions are more difficult to derive. 
The simplest situation involves a profiling system and 
horizontal surfaces. Intuition tells that causes a horizon- 
tal shift that has no influence on the reconstructed sur- 
face. The positional error is a vector quantity and as such 
depends on the flight direction. This, in turn, causes 
shift errors of variable direction and magnitude. That is, 
reconstructed horizontal surfaces are distorted. We also 
recognized that horizontal surfaces, reconstructed from 
a scanning system with angular errors (mounting bias), 
have a slope error. 
Sloped surfaces present a more complex scenario. Here, 
the reconstruction error depends on several factors, 
such as slope gradiant, its spatial orientation, flight di- 
rection, and systematic positional and/or angular error. 
In most cases, the errors cause a deformation of the sur- 
face. That is, the relationship between the true and the 
reconstructed surface cannot be described by a simple 
similarity transformation. 
4 Processing and Analyzing Raw Surface Data 
Photogrammetry and ALR deliver discrete surface points, 
called raw surface data in this paper. The cloud of 3-D 
points is hardly a useful end result. Further processing, 
such as interpolating the irregularly distributed points to 
a grid, extracting useful surface properties, and fusing 
data sets. Moreover, as a quality control measure, the 
data need to be analyzed for consistency, correctness, 
and accuracy. We elaborate in the following subsections 
on some of the post-processing issues with an empha- 
sis on the similarity between data derived from the two 
methods. 
4.1 Calibration 
The computation of raw laser data has no internal redun- 
dancy. Whether a point is correct or not can only be in- 
ferred after processing. For example, systematic errors 
cannot be detected. The compelling conclusion from the 
error analysis in the previous section is that systematic 
errors cause not only elevation errors but also surface 
deformations. To take advantage of the high accuracy 
potential of ALR systems, it is imperative to detect and 
eliminate systematic errors—a process usually referred 
to as calibration. 
In the previous section we have derived some closed 
form expressions for elevation errors and surface defor- 
mations as a function of systematic errors and surface