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. ATMOSPHERIC EFFECTS 
2.1 Viewing Geometry 
The viewing geometry of line sensors simplifies radiometric 
corrections considerably since for a single flight line the view 
azimuth p is constant for every pixel, assuming the airplane 
movements are compensated by a stabilized platform. The area 
of a constant view zenith angle 6, is an image column (cf. 
Figure 1). 
Column average = 
mean reflectance 
for constant view 
zenith angle 6, and 
constant relative 
azimuth angle ¢ 
  
Figure 1. Viewing geometry for a line scan sensor. 
For frame sensors the area of constant view zenith angle is a 
circle and still has a varying view azimuth (cf. Figure 2). 
constant view / 
zenith angle 0, and / 
varying relative 
azimuth angle @ 
  
   
  
Figure 2. Viewing geometry for a frame sensor. 
2.2 Reflectance Calibration of Aerial Images 
2.2.1 Radiative Transfer Equation: For the physical 
description of the nadir looking passive Earth observation the 
radiative transfer theory of Chandrasekhar is used (Fraser et al., 
1992). 
p ST, doy Lo 
L,=L, +2 (1) 
(1-sp) 
where Lm= measured at-sensor radiance 
Lo = path radiance for zero ground reflectance 
p = surface reflectance 
p = average surface reflectance of surrounding area 
S = mean solar spectral irradiance 
Town = total downward transmittance from top of the 
atmosphere (TOA) to the ground 
T, = total upward transmittance from ground to sensor 
s = spherical albedo of the atmosphere, i.e. the fraction 
of the upward radiance which is backscattered by the 
atmosphere 
For the case of atmospheric correction the surface reflectance p 
has to be calculated from the observed at sensor radiance L,, 
Solving eqn (1) for p gives: 
zl -L1,) = 
zu rl. 
ST.T (1- sg) 
wn up 
@) 
Suppose p is known then the quantities Lo, Taowns Tup> $, and S 
have to be calculated. 
The extraterrestrial irradiance S is given by a standard formula 
depending on the geographical latitude and the day of the year. 
2.2.2 Model Inversion: Photogrammetric image processing 
has to deal with huge amounts of data, so a very efficient 
algorithm is needed to calculate Ly, Tyown, Typ, and s. Therefore 
the modified Song-Lu-Wesely method described in (Beisl et al., 
2008) is used. 
The basic idea is the following: Standard atmospheric models 
calculate the at-sensor radiance, irradiance and transmittances 
based on a given ground reflectance, atmosphere and aerosol 
load. Since the ground reflectance is the unknown quantity, the 
model has to be inverted for a given at-sensor radiance. 
Unfortunately, none of the atmospheric parameters is known in 
case of aerial images. So standard values are assumed for all 
atmospheric and aerosol parameters except the horizontal 
visibility (which is related to the aerosol concentration) and, 
which has the biggest influence on the radiative transfer. 
A series of forward model runs is performed with all 
combinations of the input variables (ground elevation, flying 
height, sun zenith angle, view zenith angle, relative azimuth 
angle, and visibility), giving a multi-dimensional look-up table. 
The total transmittances are calculated indirectly from runs at 
three ground reflectances, because radiative transfer programs 
will typically only output the direct transmittance between sun 
and ground and between ground and sensor. 
The aim is to replace the non-observable quantity visibility by 
an observable quantity. The only available choice is the 
atmospheric reflectance à, in nadir view which is determined 
from the observed atmospheric reflectance à of a dark pixel 
viewed from a certain view zenith and azimuth angle. 
The atmospheric reflectance à is defined here as the difference 
of the observed top-of-atmosphere (TOA) reflectance a and the 
ground reflectance p of the dark pixel (eqn 3). The TOA 
reflectance is calculated from the at-sensor radiance L,, and the 
TOA irradiance S (eqn 4). 
ózo-p | (3) 
zl, 
S 
  
(4) 
For the dark pixel we assume an average reflectance p of 2 %. If 
the true reflectance spectrum of the dark pixel were known, a 
more accurate modelling would be possible. But if a wrong 
spectrum is taken then the whole image will be calibrated with 
this spectral error. Therefore a spectrally constant dark pixel 
reflectance is chosen.