The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B5. Beijing 2008 
1049 
Figure 4: Data of the Riegl distance experiment series. 
Each bullet represents the data of one target at a certain 
range and k (product of reflectivity and cosine of 
incidence angle). The vertical axis is the mean intensity 
per target. Targets are grouped by distance, and for each 
distance the least squares fit 3 rd order polynomial is 
shown. 
0 10 20 30 40 50 
Figure 5: Polynomial coefficients (fat black lines) of the 
functions of Figure 4, i.e. cubic polynomials in the 
monome form. These polygons are approximated by 
cubic polynomials each (thin gray lines), split for 
ranges below and above 15m. 
Experiment 
series 
g4- 
Model 
ct 0 abs., 
do rel. 
Distance 
Cubic 
0.00297, 1% 
0.00255 
Distance 
Log. 
0.00659, 2% 
0.00614 
Angle 
Cubic 
0.00241 
5% better 
Angle 
Log. 
0.00656 
7% worse 
Table 6: Precision values for the different data driven 
models (2 nd col.) of the Riegl data. “ct 0 abs.” is the a- 
posteriori precision (Eq. 5), and “ct 0 rel.” is the portion 
of ct 0 on the maximum mean target intensity. The 4 th col. 
shows the r.m.s. of the residuals, and the results can be 
compared between the distance experiment, used for 
parameter estimation, and the angle experiment (5 th 
col.). 
The coefficients of the cubic polynomials for each distance step 
can also be regarded as discrete observations of curves: one 
curve for the constant term, one for the linear, etc. The vertices 
of these curves can again be used for fitting a curve model. If a 
cubic polynomial is chosen, then the resulting patch is bi-cubic. 
For g 4 not only cubics, but the following functions were tested. 
• Logrithmic function, with constant offset and scale factor. 
This is suitable, if the conversion from optical to electronic 
power is a logarithmic amplification. 
• Linear scaling, suitable if received optical power and 
intensity value are directly proportional, termed “Scale” 
below. 
• Linear function: like “Scale”, but adding a constant offset, e.g. 
background noise. 
• Cubic polynomial, providing flexibility but still over 
determination in the estimation procedure. 
For g 5 always cubic polynomials were used, as the data does 
not follow the Lidar equation (Figure 3). However, some 
similarity of the Optech data to the curve shape of Figure 1 can 
be noted. 
Next, the results of the different models will be presented. Only 
those models describing the data reasonably well are presented. 
Also the Optech data had to be split into two patches for ranges 
above and below 18m. As it can be seen in Figure 5, the data 
cannot be approximated well for the last distance step (50m). 
These measurements were excluded from further analysis. 
The model is determined from the range series as explained. 
The angle series can then be used to verify the suitability of the 
model. It should be kept in mind, that the mean intensity values 
of the Riegl angle experiment data drop below the lowest mean 
intensities of the distance experiment (0.1156 vs. 0.1207). Thus, 
there is a certain amount of extrapolation with respect to k. For 
the Optech data this is not the case. The Riegl results are 
summarized in Table 6, the Optech results in Table 7. 
Experiment 
g4 
Co abs, (T 0 rel 
Distance 
Cubic 
0.00218, 1% 
0.00180 
Distance 
Linear 
0.00613,3% 
0.00508 
Distance 
Scale 
0.00681,3% 
0.00564 
Angle 
Cubic 
0.00814 
452% worse 
Angle 
Linear 
0.00716 
41% worse 
Angle 
Scale 
0.00765 
36% worse 
Table 7: Precision values for different data driven models 
for the Optech data. Columns as in Table 6. 
6. DISCUSSION 
With the data acquired in the previous experiment (Pfeifer et al., 
2007) with the Riegl scanner the precision a was 0.0108. The 
precision obtained now for the separation approach is 
comparable and approximately worse by 10%. The parameters 
also changed, which is attributed to the new data distribution in 
the experiments. Nonetheless, the behaviour of the device, 
namely decrease and increase of intensity depending on the 
range with a minimum at ~15m was the same. The precision 
obtained with the new data driven approaches are much higher 
and the separation method will not be discussed further. 
The two scanners show different, i.e. opposite, behaviour of 
intensity vs. range. Both deviate strongly from the pure Lidar 
equation, showing that it does not hold for the distances 
investigated. To reach a better agreement the model would have 
to be extended, depending on the system design. The effect of 
overlapping footprints, simulated in Sec. 3, appears to be 
similar to the function shown for the Optech scanner in Figure 3. 
For the Riegl scanner the situation is more complicated, and 
currently we do not have a good explanation. However, this 
behaviour was also found in Pfeifer et al. (2007) and the 
stability of the device could be confirmed.