In: Wagner W., Székely, B. (eds.): ISPRS TC VII Symposium - 100 Years ISPRS, Vienna, Austria, July 5-7, 2010, IAPRS, Vol. XXXVIII, Part 7B 
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adapted formulation of the radar equation (see equation 1), which 
considers all influencing factors: the receiver aperture diameter 
D r , the range between sensor and target R, the laser beam di 
vergence fit and the backscatter cross section, as well as losses 
occurring due to the atmosphere or in the laser scanner system 
itself, i.e. a system and atmospheric transmission factor r} sys and 
rjatm respectively. The backscatter cross section combines all tar 
get parameters such as the size of the area illuminated by the laser 
beam Ai, the reflectivity p and the directionality of the scattering 
of the surface Q (Wagner et al., 2006, Briese et al., 2008, Jelalian, 
1992): 
Pr 
PtD 2 r 
4TrR*fi? ' ' VsysVatrn 
with 
(1) 
Parameters which are unknown but can be assumed to be constant 
during one ALS campaign can be combined to one constant, the 
so-called calibration constant C ca i. Due to the fact that, in case 
of ALS systems with Gaussian system waveform, the received 
power is proportional to the product of the amplitude Pi and the 
echo width s Pl i, it can be replaced by the term PiS p> i (Wagner 
et al., 2006, Hofle et al., 2008). This yields the following form 
of the calibration equation for calculating the backscatter cross 
section a: 
local angle of incidence di (Lutz, 2003): 
ttR 2 fi 2 A R 2 fit n 
Aif 
resp. Ai 
¿if 
4 cosOi cosOi 
(3) 
Once the calibration constant is derived the calibrated backscatter 
cross sections of the individual echoes for the whole data set can 
be determined. 
Due to different flight heights or beam divergence, the illumi 
nated area Ai and therefore also the backscatter cross section a 
can vary a lot. Therefore, Wagner et al. (2008a) introduce area- 
normalized values, so-called backscattering coefficients, which 
have the advantage that measurements with different resolution 
can be compared more easily. The backscatter cross section can 
be related to the illuminated area Ai, which leads to the cross 
section per unit-illuminated area <7° [m 2 m~ 2 ] (Wagner et al., 
2008b): 
(4) 
Since the incidence angles change, it might be more convenient 
to normalize the backscatter cross section to the illuminated area 
at zero angle of incidence, i.e. the cross section of the incoming 
beam, which results in the so-called bistatic scattering coefficient 
7 [m 2 m -2 ] (Wagner et al., 2008b): 
CcalaitR PiS P ,i 
datm 
with C ca l 
fiì 
PtDrTjsys 
(2) 
(5) 
The range, the amplitude and the echo width in equation 2 are re 
sults of the Gaussian decomposition of the full-waveform data, 
while the atmospheric transmission factor r/ aim can be deter 
mined by meteorological data and radiative transfer models such 
as MODTRAN (Berk et al., 1998, Briese et al., 2008). In order to 
estimate the calibration constant in equation 2 only the backscat 
ter cross section of a reference surface is necessary. This can be 
achieved by the second formula of equation 1, the assumption of 
a Lambertian scatterer, which means that the scattering solid an 
gle Q is 7r steradians, and the knowledge of the reflectivity p of 
the reference surface. For a fast estimation the illuminated area 
Ai in equation 2 can be replaced by the laser footprint area at the 
scattering object Ai/ (see figure 1 and equation 3). 
Figure 1: Laser footprint area at the scattering object Aif, i.e. the 
circular area perpendicular to the laser beam at distance R (green 
area); area illuminated by the laser beam Ai at distance R and di 
angle of incidence (red area). 
The laser footprint area at the scattering object (see green area in 
figure 1) can easily be calculated by the range R and the beam 
divergence fit, while the area illuminated by the laser beam (see 
red area in figure 1) can be approximated by an ellipse whose 
calculation of the area additionally requires an estimation of the 
The backscatter cross section a as well as the backscattering coef 
ficients <r° and 7 are not free from influences caused by the angle 
of incidence. In case of ideal Lambertian scatterers incidence an 
gle corrected values can be achieved by division with the cosine 
of the local angle of incidence: 
ere 
cr 
cosdi 
resp. 76i 
7 
cosdi 
(6) 
Although (To seems to be the most suitable value at first sight, it 
amplifies the effect of the angle of incidence (for an Lambertian 
scatterer by the square of the cosine law). The computational ad 
vantages of 7 are obvious since no time-consuming local plane 
fits are necessary in order to estimate the local surface normal, 
which is required for the calculation of the local incidence an 
gle. However, in case homogeneous values are aimed at for a 
homogeneous surface, incidence angle corrected values such as 
<70 have to be computed. Since the estimation of the local surface 
normal can be uncertain or even impossible for some echoes, e.g. 
in vegetated areas, incidence angle corrected values cannot be 
guaranteed for the whole data set. Therefore, it depends on the 
subject of interest which calibration value to choose for further 
processing. 
2.2 Practical Method 
The practical method for radiometric calibration based on natural 
surfaces as reference targets is already presented in Briese et al. 
(2008). Therefore it will be mentioned only briefly. It consists of 
three parts (see figure 2). 
Prior to the ALS flight natural reference targets should be se 
lected in order to be able to measure the reflectivity by using the 
RIEGL reflectometer and Spectralon® diffuse reflectance stan 
dards (Labsphere Inc., 2010) at approximately the same time as 
the ALS data is acquired (see figure 2(a)) (Briese et al., 2008). 
Meteorological data such as the visibility at the time of the flight 
either have to be observed during data acquisition or gained from