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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B7, 2012 
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia 
2.2.3 Modelling of Atmospheric Quantities: Now 9, can be 
parametrized as a function of the observed atmospheric 
reflectance à, the view zenith angle 6,, the sun zenith angle 6;, 
the relative azimuth angle ¢ and a set of fixed parameters 
al...a4. 
Ô, = 0,(0,0, ---a4,0,,0,,P) (5) 
Since 6, is a nadir looking quantity there is no explicit azimuth 
angle dependence. However, the parametrization has to 
compensate the azimuth angle dependence of the observed 
atmospheric reflectance à at view zenith angle 0,. The azimuth 
angle dependence of à is caused by the path radiance Lo which 
is defined as the total radiance at ground reflectance 0. The total 
radiance is shown for several ground reflectances in Figure 3 
for a visibility of 3 km. For a satellite view the variation of Lo 
can be modelled with a cos(2*) dependence since it is caused 
mostly by Rayleigh scattering. For an airborne view at 1 km 
above ground a modelling with a function cos(1.4*@) is more 
adequate, since the predominant Mie scattering has a strong 
backscattering characteristic. 
Taowns Tup» and s do not have an azimuth angle dependence. 
  
  
  
  
  
  
  
  
  
  
5.60E-04 
5.40E-04 
y 520E-04 * 
“ = L(n=8) 
e IS j A = 
E 5.00E-04 —%——— paf a 
un p=0. 
t v pare / - = ‘cos(20) (p=0) 
9 480E-04 / 
2 A f — -tos(29) (p-0.2) 
& 460E-04 SC É en 
SA di 
4 ADE-04 We 
A 
4.20E-04 1 | 
0 50. 100150 
Az imuth Angle [*] 
2.90E-04 
2.70E-04 
= —L (p=0) 
N = 
> 2.50E-04 — | (20.2 
> L (p=04) 
2308-04 = = 'cos(1.49) (p=0) 
o 
S === «pOS(1.49) (p-0.2) 
3 9 10E-04 = <COS(I 49) (p=0.4) 
1.90E-04 
1.70E-04 
  
0 50 100 150 
Azimuth Angle ['] 
Figure 3. Variation of the at-sensor radiance as a function of 
the view azimuth angle for visibility 3 km and a sensor 
elevation of 100 km (upper image) and 1 km (lower image). A 
different ground reflectance p only adds a constant radiance 
offset. The variation can be modelled with a cosine function. 
The atmospheric reflectance 6, is the scaling factor for the other 
atmospheric quantities Ly, Tgown, Typ, and s. They are calculated 
as a function of the observed atmospheric reflectance 3, the 
view zenith angle 6,, the sun zenith angle 6;, the ground level H, 
the flight altitude over ground h and a set of fixed parameters 
(b;...b;, c,...cs, d;...d,, e;...eg). The parameters for the quantities 
in eqns (5)-(9) can be obtained using a multilinear regression 
from a sufficient number of model runs with all combinations of 
the input variables. 
L, = L,(b--5,,0,,0,,0,0,,h,H) (6) 
T mm T. miei c6, OS fi EDS (7) 
I, =T,(d\-d;.6,,0,) (8) 
$zs(e,--:e,, 6, ,h, H) (9) 
2.2.4 Broadband Sensors: The above calculation is strictly 
valid only for a single wavelength and the outputs being 
spectral densities. The evaluation of eqn (2) gives the 
contribution to the reflectance at this wavelength. For a 
broadband sensor the contributions of a spectral density x have 
to be integrated over the spectral response curve of the sensor 
using eqn (10) to give the band-averaged quantity. 
[x(A)R()aA 
ce 10 
de Eire a 
This would require to parametrize the quantities in eqns (5)-(9) 
for each wavelength separately which is not practical. So the 
parametrization is done for the band averaged quantities in eqns 
(5)-(9) and the reflectance is then calculated from the band- 
averaged quantities. (Richter, 2000) has found that the errors in 
the VISNIR range (400-1000 nm) are below 2% and therefore 
below the calibration accuracy. 
In the special case of a narrowband sensor with spectral 
sensitivities away from the gaseous absorption bands (like e.g. 
the ADS) the atmospheric quantities Lo, Tgown, Tup, and s change 
moderately with wavelength. Then the radiative transfer 
calculations can be made by using the effective bandwidth of 
the sensor and simply averaging the spectral quantities over the 
effective bandwidth without using the spectral response 
function as a weight. 
2.2.5 Reflectance Calibration for Images: Eqns (2) and (5)- 
(9) allow a fast image calibration to ground reflectance without 
any iteration. Since multiple scattering is a second order process 
p can be assumed constant and an average value of 0.15 for a 
midlatitude landscape is used. 
2.3 Bidirectional Reflectance 
2.3.1 Sampling and Model Inversion: As mentioned in 
(Beisl et al, 2004) the bidirectional reflectance process is 
influenced mainly by microscopic shadow casting and volume 
scattering processes with unknown influencing parameters. So 
the correction process also requires a model inversion. 
As suggested by the viewing geometry shown in sec. 2.1 the 
sampling has to be done in image columns for the line scan