The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008 
9 
immense computation effort, so (Richter and Schlapfer, 2002) 
use a value of 0.15 for p . 
In our case, we will use a resampled image of minification level 
32 to calculate a low resolution image with eqn. (2), thereby 
neglecting the adjacency effect for the already large pixels. 
Interpolation then gives an estimate of p for each high 
resolution pixel. 
A: A =j3-r a (9) 
The wavelength exponent a is a measure for the aerosol size 
distribution. The Angstrom turbidity coefficient p describes the 
aerosol concentration. The values range from a = 1.3 for small 
particles to a = 0.1 for large particles and p = 0.1 for clear 
atmosphere to p = 0.4 for hazy atmosphere. 
The mean solar irradiance S at solar zenith angle 0j is calculated 
from the solar constant S 0 and the ratio of Earth-Sun distance a 
to mean Earth-Sun distance r. 
S = S 0 (a/r) 2 cosO j 
(7) 
The diffuse forward transmission T dif contains the parameter co 0 
which is set to 0.9 for rural and to 0.6 for urban aerosols. 
T = T T T 
1 dif 1 o 1 g 1 w 
(l — T s )~- + F c <a„(\ - T A (r, j (10) 
The measured at-sensor radiances L m can be obtained from the 
ADS40 data as described in (Beisl, 2006b). Now we need a 
method to calculate the unknown quantities L 0 , T down , T up , and s. 
Angstrom Method: This method provides a way of calculating 
the quantities L 0 , T down , T up , and s using the following 
assumptions: clear atmosphere with rural aerosols and a 
horizontal visibility above 11 km. The approximations follow 
(Iqbal, 1983) and have three free parameters: aerosol size a, 
aerosol concentration p, and single scattering albedo co 0 . For 
brevity only an outline is given here. 
*■' « 
direct diffuse propagated forward 
To-T w -T A -T R -T a T 0 -T* -T 0 -{ 0.5*(1-T„)'T a + F t tM 1 -TJT*} 
Figure 1. Direct and scattered components of the radiation 
(Iqbal, 1983) 
For calculating the downward transmittance T down the direct T dir 
and the diffuse T djf contribution have to be considered 
according to Figure 1: 
L,r=T R T 0 T 0 T w T A (8) 
The Rayleigh contribution T R only depends on wavelength X 
and solar zenith angle and is described by the formula from 
(Bucholtz, 1995). The gaseous absorption T G in the ADS40 
channels can be neglected except the ozone absorption T 0 , 
where the values of Leckner quoted by (Iqbal, 1983) are used. 
The water absorption T w is only relevant in the near Infrared 
(NIR), where we use again (Iqbal, 1983). The aerosol 
contribution T A is described by the Angstrom turbidity formula 
and introduces two parameters a and p, which determine the 
extinction coefficient k A . 
Turning Figure 1 upside down we can calculate the upward 
transmittance T up for an illumination coming from the ground. 
The spherical albedo s is the diffuse reflected part of this 
process. Since the multiple scattering is only a second order 
effect, we approximate the incidence zenith angle for 
calculating T w , T R , T 0 , T A , and F c by 60° and use a relative air 
mass of 1.9 . 
s = T 0 T G T W (0.5(l - T R )T A + 
(l-F c K(l-r,)T s ) 
For calculating the path radiance for zero ground reflectance L 0 
we also consider two contributions. The direct part originates 
from the diffuse reflected part between ground and sensor of the 
direct irradiance at sensor level. The diffuse part comes from 
the diffuse reflected part between ground and sensor of the 
diffuse forward scattered irradiance at sensor level. Again the 
incidence zenith angle of the diffuse irradiance is approximated 
by 60°. 
Modified Song-Lu-Wesely Method: For the case of a satellite 
sensor (AVHRR) (Song et al., 2003) use a linear 
parametrisation in atmospheric reflectance 5 0 , view zenith angle 
0 r and sun zenith angle 0; to calculate L 0 , T down , T up , and s. 
Since we are dealing with an airborne sensor we have the 
additional variables ground level H and flight altitude over 
ground h. The atmospheric reflectance for nadir sun and nadir 
view zenith angle 8 0 itself is parametrised (a!...a 4 ) and 
calculated from the measurement of the atmospheric reflectance 
5 above a dark pixel. 
8 Q =d Q {S,a r -a^0 r ,6 i ) (12) 
The atmospheric reflectance 8 is defined here as the difference 
of a pseudo reflectance a and the ground reflectance p. 
n L m 
< 13 ) 
S = a-p (14) 
For a dark water pixel we can model the reflectance p with the 
Fresnel reflectance of water = 2 %. It turned out that it is a good 
approximation to use this value also for dark dense vegetation 
or a grey shadow pixel. Then 8 0 can be calculated for this pixel, 
which is observed at a view zenith angle 0 r and a sun zenith