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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
    
  
3.4 Radar Equation 
The intensity of the received laser pulse can be determined from 
the range equation, which describes the influences of sensor, 
target, and atmosphere (Jelalian, 1992): 
pp 
2 m : i 2 : Toys ai GE Ch 
4zR' ff; 
where P, — received signal power (watt) 
P, — transmitted signal power (watt) 
D, — diameter of receiver aperture (meter) 
R — range from sensor to target (meter) 
f= laser beamwidth (radian) 
Mss = System transmission factor 
Num = atmospheric transmission factor 
o = target cross section (square meters) 
The radar equation shows that the received power is a function 
of transmitter power, laser beamwidth, aperture size of the 
receiver, system losses, and atmospheric transmission. The 
properties of the target are described by a single quantity, the 
backscattering cross section. 
The backscattering cross section is, as its name suggest, the 
effective area for collision of the laser beam and the target, 
taking into account the directionality and strength of the 
reflection (Jelalian, 1992). Therefore it has a unit of square 
meters. In the case of airborne laser scanning the wavelength is 
always much smaller than the size of the scattering elements 
(e.g. leaf, roof). Therefore, the effective area for collision is 
simply the size of the projected area of the scatterer. The 
magnitude and directionality of the reflected energy depends 
strongly on the surface properties of the target and its 
orientation with respect to the incoming beam. 
3.2 Pulse Form 
Models which are capable of simulating the waveform from 
different targets are already available. For example, Sun and 
Ranson (2000) present a model for forest canopies which 
assumes that laser scanners are working on a hot spot condition 
(because the light source and detector are at the same point). 
Here we build upon the radar equation, noting that in the way as 
it is written in equation (1) it only applies for point scatterers or 
non-tilted surfaces. In case of distributed targets, where 
scattering of the incoming laser beam takes place within the 
range interval [R;, R,], the return-signal strength is the 
superposition of echoes from portions of the target at different 
ranges (Ulaby et al, 1981). Mathematically, this can be 
expressed by an integral: 
R 2 
nbi 2R 
P()= |— 177 ssn" B| 1 —— | o(R)dR ( 
AzR' fi y 
t g 
N 
) 
where f is the time, v, is the group velocity of the laser pulse, 
and o(R)dR is the differential backscattering cross section do in 
the interval dR (Wagner et al., 2004). The group velocity v, for 
optical and near-infrared radiation in dry air differs from the 
speed of light in vacuum by at most 0.03 % (Rees, 2001), so 
  
here we use v, « 3-105 ms. The term 2R/v, in the brackets is 
the round trip time. 
It must be considered that at range R some scattering elements 
may be shaded by objects situated above this range, i.e. in the 
interval [R,, R]. Since these shaded areas do not contribute to 
the return signal, do represents an "effective" or “apparent” 
cross section that represents only illuminated scatterers in dR. 
As an example, lets us consider the ground surface beneath a 
tree. If 90 % of the ground surface is shaded by leaves and 
branches, then the “effective” cross section of the ground is 
10 % of its actual cross section. 
In the next section we will study the impact of different pulse 
detection methods on the distance measurements. For this 
purpose it is sufficient, in a first step, to focus solely on the 
waveform of the return pulse. In more advanced studies also the 
different parameters which modulate the signal strength should 
be considered. 
4 IMPACTS OF PULSE DETECTION 
4.1 Detection Methods 
Pulse detection is applied on the backscattered waveform. The 
task of the detector is to derive from the continuous waveform 
discrete, time-stamped /rigger-pulses, which encode the 
position of the individual targets, thus allowing to compute the 
distance between the scanner system and the generator of the 
return pulses, i.e., the illuminated objects. Since the details of 
detection methods applied by commercial laser scanner systems 
are currently not known, we will consider here a number of 
standard detection methods: threshold, centre of gravity, 
maximum, zero crossing of the second derivative, and constant 
fraction. 
The most basic technique for pulse detection is to trigger a 
pulse whenever the rising edge of the signal exceeds a given 
threshold (Figure 1); although conceptually simple and easy to 
implement, this approach suffers from a serious drawback: the 
position of the triggered pulse (and thus the accuracy of any 
distance measurements derived from it) is rather sensitive to the 
amplitude and width of the signal. The same holds for the 
centre of gravity when computed over all points above a fixed 
threshold. More sophisticated schemes are based on finite 
differences respectively numerical derivatives — e.g. the 
detection of local maxima or the zero crossings of the second 
derivative — or, more generally, the zero-crossings of a linear 
combination of time-shifted versions of the signal. An example 
of the latter approach is the constant fraction discriminator, 
which determines the zero crossings of the difference between 
an attenuated and a time delayed version of the signal. 
Maximum, zero crossing, and constant fraction are invariant 
with respect to amplitude variations and, to some degree, also 
changes in pulse width. In practice, these detectors should only 
trigger for signal amplitudes above a given threshold (this 
"internal" threshold is not to be confused with the threshold 
detector) in order to suppress false positives, i.e. spurious 
trigger-pulses due to noise. This is especially true for operators 
based on higher order derivatives like zero crossing, which are 
known to be rather noise-sensitive. 
Figure 2 illustrates the trigger-pulses generated by the five 
detectors discussed above for the case of a single mode return 
signal from rough terrain, assuming a Gaussian differential