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Scattering
In-scattering
Absorption
Figure 2.1. Types of interaction of light with particles.
2.4.1 Physical Model of the Air
Sunlight interacts with the particles distributed in the air as it propagates through
the atmosphere. Two types of interaction are important: scattering (Figure 2.1,
left), which changes the light direction, and absorption (Figure 2.1, right), which
transforms the energy into other forms. Scattering in the view direction is called
in-scattering (Figure 2.1, center). The amount of light energy per unit length
scattered at point x is expressed by the total scattering coecient β s ( x ). The
angular distribution of the scattered light is described by the phase function p ( θ )
where θ is the angle between incident and outgoing directions. 1 The total losses of
energy per unit length caused by absorption and both absorption and scattering
are given by the absorption and extinction coe cients β a ( x )and β e ( x )= β s ( x )+
β a ( x ), respectively.
Air is usually modeled as a mix of two types of particles: air molecules and
aerosols. Scattering by air molecules is accurately described by the Rayleigh
theory. It is considerably wavelength-dependent, and almost isotropic with the
following phase function: p R ( θ )= 3
16 π (1+cos 2 ( θ )). Precise derivation of scatter-
ing coecients is irrelevant for this paper and can be found for instance in [Nishita
et al. 93,Preetham et al. 99]. As in previous works [Riley et al. 04,Bruneton and
Neyret 08], we use the following values for Rayleigh scattering coecient at sea
level: β R .rgb =(5 . 8 , 13 . 5 , 33 . 1)10 6 m 1 .
Scattering by aerosols is described by the Mie theory. Cornette-Shanks func-
tion is commonly used as an approximation to the phase function of Mie parti-
cles [Nishita et al. 93,Riley et al. 04]:
g 2 )
2(2 + g 2 )
(1+cos 2 ( θ ))
(1 + g 2
p M ( θ )= 1
4 π
3(1
2 g cos( θ )) 3 / 2 .
As in [Bruneton and Neyret 08], we use g =0 . 76 and β s M .rgb =(2 , 2 , 2)10 5 m 1 .
For aerosols we also assume slight absorption with β a M =0 . 1 β s M .
In the model of the atmosphere it is assumed that the particle density de-
creases exponentially with the altitude h : ρ = ρ 0 e −h/H ,where ρ 0 is the density at
sea level and H is the particle scale height (which is the height of the atmosphere
if the density was uniform). We use the following scale heights for Rayleigh and
1 We assume that the phase function is normalized to unity such that Ω p ( θ ) =1where
integration is performed over the whole set of directions Ω.
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