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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 =
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