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range 0.3- 
2mrad typically (Baltsavias, 1999) what at the relative flying 
height of 1000m above ground results with a laser footprint of 
0.3 — 2 m in diameter. The object smaller of this footprint 
reflects a portion of laser radiance, but cannot shield other 
objects underneath it to be illuminated by laser beam, so they 
reflect laser radiance, too. Finally, the rest of laser radiance 
reaches the ground surface. So, we have multiple returns for 
one emitted laser pulse, depending on structure of illuminated 
object (fig. 2). 
1°" return 
CS 
ND 
2 
return 
  
3rd return 
  
last return 
  
  
Figure 2. Multiple reflections by laser scanning 
It is obvious that the first returned pulse belongs to point nearest 
to LIDAR sensor. If there are several returns in wooded areas, 
these points lie almost on canopy of trees. The last return could 
produce a terrain point, if laser beam reached the ground. If not, 
the off-terrain point is misclassified as terrain point. (fig. 3) 
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Figure 3. Last pulse over forested terrain; flight performed in 
October. (According Wever and Lindenberger 1999) 
This method is implemented in sensor, so there are sensors 
capable to register first or last returned pulse only. Recently 
developed sensors are capable to sample and register a shape of 
returned signal. Peterson et al. are shown that this is of great 
importance for forest inventory applications (Peterson et al. 
2001). 
101 
2.2 Iterative robust interpolation with linear prediction 
The discrepancies between off-terrain points, gathered by laser 
scanning, and real terrain surface can be divided into two parts. 
The first one consists of discrepancies induced by random 
errors only. They are relatively small in absolute amount, and 
normally distributed. Therefore, their impact on interpolated 
DTM can be minimised by using linear prediction as an 
interpolation method. The second one consists of discrepancies 
that originated from wrong classification of non-ground points. 
These discrepancies have almost the positive sign only, because 
they are induced by misclassified points belonging to objects 
above terrain surface. Their absolute value is relatively big and 
they should be treated as gross-errors and therefore removed 
from the dataset. 
There are well-known methods to filter out such outliers, but 
they all assumed that the discrepancies are normally distributed. 
So, Kraus propose a new approach of gross error detection, 
which are not normally distributed but skew (Kraus 1997). This 
approach is very well suited to filter laser scanner data, 
especially in wooded areas (Kraus and Pfeiffer, 1998) 
Classification and filtering is done iteratively with interpolation 
of DTM. For the interpolation is used an algorithm based on 
linear prediction (Kraus and Mikhail, 1972) with an individual 
accuracy for each measurement. As the first step, the rough 
approximation of the surface is interpolated. All points are 
included into interpolation under the same conditions, and at 
this point, there is no assumption about presence of gross errors 
in LIDAR dataset. In next step, the differences between 
interpolated surface and each individual measurement are 
computed and individual weight is determined on the basis of 
weight function adapted to skew distribution of errors. (fig. 4) 
  
20 4.0 5.0 80 19.0 
  
Figure 4. Residual distribution after the first interpolation step 
together with weight function p(r) superimposed 
(after Briese et al. 2000) 
Note that the origin g of the weight function is negative and that 
the left branch of weight function is identical to one. This 
function, expressed in analytical form is (Kraus & Pfeiffer 
1998): 
1 pss 
] 
(mc <Uv, <g+w (1) 
item es 75955 
0 g+w<y, 
where: 
p; — individual weight for each point, v; — individual residual, 
g; — shift of origin of the bell-curve, because of deviating the 
error distribution from the normal case. For laser scanner data, 
this shift is negative. 
w; — outlier limit value. If overrun, the point is removed from 
interpolation