The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B5. Beijing 2008 
not be disclosed by the device manufacturers. Also estimating 
these parameters requires knowledge of the system design. 
So far, the relation of emitted to received optical power was 
discussed. The received signal is usually converted to an 
electrical signal and amplified by an APD (avalanche photo 
diode) and then digitized. This amplification is not necessarily 
linear, and frequently a logarithmic behaviour is assumed. 
Furthermore, for round-trip-time systems the emitted pulse is 
not a Dirac impulse but has a duration in the order of a few ns, 
e.g. 5ns, corresponding to a pulse length of 1.5m. The 
“intensity” may refer to the maximum, the average below the 
pulse shape, or the energy level after a certain amount of time. 
It does stand to reason to evaluate alternative approaches, 
relating intensity values, ranges, and target reflectivity to each 
other without considering the physical foundation. 
3.2 Data driven approach 
In the previous section it has been shown that a number of 
unknown system parameters may exist. This complicates using 
a model driven approach, because the system parameters need 
to be known for estimating object properties, especially object 
cross section a or object reflectivity p-cos(a) for Lambertian 
targets, from the observed range r and intensity I. 
In a data driven approach the function / is determined. In a 
general case, but restricted to Lambertian targets, it can be 
written as f[rj,a,p)=0. From a set of observations and 
corresponding known object properties the parameters of/ can 
be estimated. For typical data from laser scanners r is 
determined more precisely, with a coefficient of variation 3 
below 1%, whereas the intensity value has a coefficient of 
variation of 3% to 10% (Pfeifer et al., 2007, Hofle and Pfeifer, 
2007). It therefore stands to reason to choose r, p, and a as 
function parameters, because they are determined more 
precisely, than the function value I. This allows formulating 
residuals in an observed quantity. 
I + e, = g(r, p, a) (4) 
The function g should fulfil the following requirements. Firstly, 
it should explain the data well. The / should have small 
residuals: ejj = g (/}, py, aj) - Ij , for sets of data quadruples with 
y'=l,... n, (n the number of points). The standard deviation ct 0 of 
the residuals 4 , obtained from a function g with u parameters can 
be used to judge the explanatory power of g. 
a 0 = ( £ ejf /( n-u )) 0 5 (5) 
Secondly, the function g should not contradict basic physical 
principles. As details of the device model are unknown, this is 
limited. However, the parameters p and a of the object have 
similar effects on the backscattered power, namely decreasing it 
for smaller values of p and larger values of a. The beam 
diameter is small, in the order of a few mm, whereas the pulse 
length for a round-trip-time measurement system is in the order 
of meter. Thus, the stretching of the pulse may be neglected, 
too 5 . The effects of p and a may therefore be bundled to k = 
P’cos(a), which is in line with the basic physical principles. 
3 The coefficient of variation is standard deviation / mean. 
4 N.B.: a now refers to a quality measure, and not to the 
scattering target cross section, introduced in section 2. 
5 Assuming a pulse with constant distribution of power across 
and along the beam, a 5mm spot size, a target inclination of 45°, 
and a 4ns pulse duration (1.2m), this stretch is below 1%. 
Thirdly, the function should avoid effects not observed in the 
data. Properties like monotonic behaviour or smoothness, which 
are visually apparent in the data, should also be properties of 
the function g. A decreasing series of values strictly 
interpolated by a polynomial may give rise to an oscillatory 
function, which cannot be justified by the data. With an 
increasing number of parameters и the function g may also start 
modelling the noise. The noise would be different in a 
repetition of the experiment and should not be represented by g. 
Finally, the function should be invertible. This is necessary for 
uniquely finding the reflectivity given observations of r and I. 
The following types of functions are possible, of which 
especially the nested approach will be investigated. 
• Separation approach: 1 = gfr )*g 2 ( к) + g3 
•Nested approach: I = g 4 ( k, g 5 ( r)) 
• Surface fitting approach: 1 = g 6 (r, k) 
Next to ct 0 also a r will be used to judge the distribution of 
residuals. The term a r refers to the root mean square value of 
the residuals, and is different from a 0 by the denominator under 
the root. 
4. EXPERIMENT 
Reference targets with known reflectivity behaviour were used. 
These targets are made of Spectralon®, which reflects 
according to a Lambertian scatterer (cosine-law). The targets 
are quadratic with an edge length of approximately 13cm, and 
mounted onto a metal frame, holding all six targets with 
reflectivities of about 5%, 20%, 40%, 60%, 80%, and 99%. The 
frame was placed in different distances to the laser scanners and 
at different aspects. Points measured on one target were 
selected, and for the point set of each target at each range and 
aspect, a number of parameters was determined. Also, a plane 
was fitted to the points minimizing the orthogonal distances. 
This allows computing further parameters. Finally, for each 
target the following parameters were used (Table 2). 
p[] 
Target reflectivity 
on 
Mean angle of incidence (to the plane) 
r [m] 
Mean range 
JU 
Mean observed intensity value 
Table 2. Parameters determined for each target in the 
experiments. Symbols are given with their units. 
Scanning was performed with a Riegl LMS-Z420i (termed 
Riegl) and an Optech ILRIS 3D (termed Optech). For each 
device, two series of measurements were performed. In the first 
series the distance of the target frame to the scanner was 
changed: up to 15m in lm-steps, thereafter in 5m steps, up to 
the maximum length of the laboratory (50m). In the second 
series the target frame was placed at a distance of 
approximately 15m to the scanner, and rotated in 9° steps from 
0° to 72°. 
The Riegl scanner operates at a wavelength of approximately 
1550nm. At this wavelength the reflectivity of the targets are 
0.986, 0.828, 0.653, 0.433, 0.233, and 0.081. 
For the distance series the minimum and the maximum mean 
range were 2.02m and 50.04m, and mean intensities varied 
from 0.1207 to 0.3085. The maximum angle of incidence was 
11.4° for the target on the outer end of the frame at the shortest