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Technical Commission VII

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
2.1 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.,
p ST, doy Lo
L,=L, +2 (1)
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
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)

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.