The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B5. Beijing 2008 
distance. By the cosine-law this corresponds to a decrease of 
return energy of 2% (see Eq. (3)). 
For the angle series the minimum and maximum distance were 
within 14.89m and 15.15m 6 , originating in the rotation of the 
target frame. The largest angle of incidence for which the 
targets could be extracted from the measured data was 72.0°, 
which corresponds by the cosine-law to a decrease by 69%. The 
average intensities varied between 0.1156 and 0.1880. 
The Optech scanner operates at a wavelength of approximately 
1540nm. At this wavelength the reflectivity of the targets are 
0.986, 0.827, 0.653, 0.433, 0.233, and 0.080. For the distance 
series the minimum distance, at which measurements were 
recorded was 3.98m. No distances could be measured at the 3m 
step. The maximum incidence angle in this series is 4.5°, 
corresponding by the cosine-law to a decrease in return energy 
by 0.3%. The average intensities are between 0.0065 and 
0.2462. Originally, the values were up to 5 digit integers, but to 
make them comparable to the Riegl data they were scaled by 
setting the maximum single intensity in the entire data set to 1. 
For the series of measurements at different incidence angles, 
the ranges were between 14.84m and 15.09m. 7 Data from 
incidence angles up to 71.9° could be retrieved, corresponding 
by cosine to 0.311, i.e. a decrease in intensity by 69%. The 
averaged intensities for the individual targets are between 
0.0089 and 0.2138. 
Data of the range experiments is shown in Figure 3. 
Range [m] 
Figure 3. Mean intensities for the Riegl and the Optech 
laser scanner for three targets (99%, 40%, and 5% 
reflectivity) at different distances. 
5. RESULTS 
5.1 Modelling with the Lidar equation 
If the Lidar equation shall be used for modelling the intensity 
values, it is obvious that it can only apply to a selected range 
interval. For the Optech data this applies to the ranges above 
30m, thus for the experiments shown in the interval [30m,50m], 
This is also suggested by Larsson et al. (2006). However, this 
approach is not pursued because of the obvious limitations. 
6 From the analysis performed thereafter, this range difference 
has an influence of 3% on the intensities. 
7 The previous footnote does not only apply to the Riegl device 
but also to the Optech device. 
5.2 Separation approach 
The separation approach was investigated previously in Pfeifer 
et al. (2007). As is shown below, other approaches allow higher 
precision, and therefore the presentation of results will be very 
limited. It is interesting, however, to compare the results 
obtained now to previous investigations. 
In 2007 experiments with the same Riegl scanner as used in this 
study, i.e. the same physical device, were performed, using a 
similar setup of measurements. The main difference is, that the 
angle experiment was performed at a distance of 5m. For the 
data acquired now, the following functions were determined by 
optimization in the separation approach: 
r < 15m : g, (r) =- 0.0082r + 0.3338 
gi(r) = 1 (6) 
r > 18m : g! (r) = 0.0019r +0.1441 
g 2 ( P'cos(a)) = g 2 ( k) = k' a , with a = 0.22, g 3 =0 
The influence of the range in gi is described by a piecewise 
linear function, first decreasing and then increasing with range. 
A smooth transition between the two intervals is added. The 
function g 2 is an exponential function, and g 3 identical to zero 
as in the previous study. 
The distribution of the residuals is characterized by a r = 0.0114 
for all observations, separated for the range and the angle 
experiment into cr r = 0.0080 and a r = 0.0167, respectively. The 
mean values of the distribution are 0.0 and 0.0078. 
5.3 Nested approach 
The nested approach takes the form I + ej = g 4 ( k, g 5 ( r )). The 
function g 5 depends on the range only' but it determines one or 
several parameters of the function g 4 . These parameters are e.g. 
polynomial coefficients. In that case, g 5 is a vector valued 
function. Specifically, the following form was used in this 
study. g 4 is a cubic polynomial, and g 5 is a vector-valued 
function in 4D, determining the coefficients of g 4 . Precisely put, 
g 5 is a vector valued 3 rd order polynomial. This describes a Bi- 
Cubic Tensor-Product surface patch (Farin, 2002). 
For increasing flexibility, multiple patches can be laid out in the 
parameter domain ( r, k ), joining with suitable continuity 
conditions. For the data acquired with the Riegl and the Optech 
instrument, two patches were used, splitting the interval of r, 
similar as in the separation approach. Strict continuity was not 
enforced, but reached approximately. 
For the Riegl data the patch border is at >=15m. This split was 
obvious when looking at the curves of intensity vs. range 
(Figure 3). For estimating the functions, however, first the 
measurements to the six targets were analysed independently 
per distance step. 
For each distance step the data of mean target intensity I over k 
=p-cos(a) was analysed. As this approach is formally expressed 
as I = g 4 ( k, g 5 ( r )), this means that at fixed values of r the 
function g 4 could be fitted to the data. Different models were 
used for this, but for the clarity of the explanation only the 
cubic polynomial is considered. As six targets were available, 
this means that an overdetermination of two was available for 
the cubic fit. In Figure 4 the cubic polynomials together with 
the data are shown for all distance steps. The coefficients are 
shown in Figure 5.