Radiative Transfer Radiative transfer equation (RTE) - IAP ...

Radiative Transfer 

Radiative transfer equation (RTE) 

without scattering 

N. Kämpfer 

RTE 

Beer’s law 

Abs. cross section 

Opacity & 

Transmission 

Mean free path 

Scattering 

Schwarzschild eq. 

Institute of Applied Physics 

University of Bern 

12. Oct. 2010

Outline 

Beer’s law in nonhomogeneous medium 

Absorption cross section 

Opacity & Transmission 

Photon mean free path 


Beer’s law 


Opacity & 

Transmission 


Scattering 


A first look at scattering 

Schwarzschild equation

Absorption by a non homogeneous medium 

Example for absorption of solar radiation in the atmosphere 

Radiation is passing through 

layer dz under zenith-angle θ 

Radiation is attenuated by 

amount dI along path ds 

ds = sec θdz = 1 

cos θ dz = 1 µ dz 

1/µ is called air mass factor 

Law of Beer-Lambert 

to consider the 

ase of scattering by a spherical particle of radius r, 

rr rvhich the scattering, absorption, or extinction 

fficiency K1 in (4.16) can be prescribed on the 

(- d( 

ig. 4.10 Extinction of incident parallel beam solar radiacn 

as it passes through an infinitesimally thin atmospheric 

Lver concaining absorbing gases and/or aerosols. 

dI (λ) = −k a (λ)I (λ)ds 

particle. Figu 

eters for va 

phere and ra 

For the scatt 

the spectrum 

air molecule 

)>1 for rain 

Particles w 

scattering rad 

scattering reg 


Beer’s law 


Opacity & 

Transmission 


efficiency is o 

Scattering 

and the scat 

forward and b 

Fig.4.12a. For 

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Figure 4.13 s 

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◮ k a (λ) is called absorption coefficient. Dimension is m −1 

◮ k a (λ) is proportional to number density: k a (λ) = σ a (λ)n 

◮ σ a (λ) is called absorption cross section 

◮ k a (λ) = ρk m (λ): k m is mass abs. coefficient, [m 2 /kg]

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Absorption cross sections of some molecules 

from the visible to the microwave 


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Beer’s law 


Opacity & 

Transmission 


Scattering 

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copied from Bohren, Fundamnetals of atmospheric radiation

Opacity and transmittance 

Atmosphere consists of many different species 

→ every constitutent contributes in its own way 

k a = ∑ k a,i = ∑ n i σ a,i = ∑ ρ i k m,i 

i 

i 

i 

Absorption along a path element ds is given by Beer Lambert 

dI (λ) = −k a (λ)I (λ)ds 


Beer’s law 


Opacity & 

Transmission 


Scattering 


which after integration over the whole atmosphere yields 

I (λ) = I 0 (λ)e −τa(λ,s) 

Opacity τ: τ a (λ, s) = ∫ s σ a(λ)n(s)ds = ∫ s k a(λ, s)ds 

Penetration depth: δ = 1/k a 

Transmittance: t(λ, s) = e −τa(λ,s) ⇒ t ≈ 1 − τ a for τ a ≪ 1 

Absorptivity: a(λ, s) = 1 − t(λ, s) ⇒ a ≈ τ a for τ a ≪ 1 

CAVE! absorptivity is not the same as absorption coefficient!

Transmission of Earth atmosphere 

Contribution of different species in the UV to infrared 

Transmittance 

Transmittance 

Transmittance 

Transmittance 

1 

0 

1 

0 

1 

0 

1 

ZENITH ATMOSPHERIC TRANSMITTANCE 

UV VIS Near IR Thermal IR 

CO 2 

0 

0 

0.3 0.4 0.5 0.6 0.8 1 1.2 1.5 2 2.5 3 4 5 6 7 8 910 12 15 20 25 30 40 50 

Wavelength [µm] 

copied from Petty, Atmospheric radiation 

CO 

N 2 

O 

CH 4 

O 2 

0 

O 3 

H 2 

O 

Total 

1 

0 

1 

1 

1 

0 

Transmittance 

Transmittance 

Transmittance 

Transmittance 


Beer’s law 


Opacity & 

Transmission 


Scattering 

Schwarzschild eq.


Microwave regime 

1 

Zenith Microwave Transmittance 


Transmittance 

0.9 

0.8 

0.7 

0.6 

0.5 

0.4 

Oxygen 

Water Vapor (21.3 kg m -2 ) 

Cloud Liquid Water (0.2 kg m -2 ) 

Total 

Beer’s law 


Opacity & 

Transmission 


Scattering 


0.3 

0.2 

0.1 

0 

0 50 100 150 200 250 300 

Frequency [GHz] 

copied from Petty, Atmospheric radiation


Microwave regime 

1 

Zenith Microwave Transmittance 


Transmittance 

0.9 

0.8 

0.7 

0.6 

0.5 

0.4 

H 2 O O 2 

O 2 

H 2 O 

Dry (0 kg/m 2 ) 

Polar (3.1 kg/m 2 ) 

Midlatitude (21.3 kg/m 2 ) 

Tropical (53.6 kg/m 2 ) 

Beer’s law 


Opacity & 

Transmission 


Scattering 


0.3 

0.2 

0.1 

0 

0 50 100 150 200 250 300 

Frequency [GHz] 

copied from Petty, Atmospheric radiation

Mean free path of a photon 

What happens to a photon launched from a particular point 

in a medium? 

→ We cannot say what happens to a particular photon 

→ We can determine what happens in a statistical sense 

What is the probability that a photon propagating along the 

x-axis is absorbed between x and x + ∆x? 

Given by the integral of a probability density function p(x) 


Beer’s law 


Opacity & 

Transmission 


Scattering 


∫ x+∆x 

x 

p(x) dx 

where 

∫ x+∞ 

x 

p(x) dx = 1 

Assume exponential attenuation according Beers law 

→ probability photon is not absorb over distance x is e −kax 

→ probability photon is not absorb over distance x + ∆x is 

e −ka(x+∆x) 

→ probability photon is absorb over interval ∆x is difference 

∫ x+∆x 

x 

p(x)dx = e −kax − e −ka(x+∆x)

Mean free path of a photon 

probability photon is absorb over interval ∆x 

∫ x+∆x 

x 

p(x) dx = e −kax − e −ka(x+∆x) 

According mean value theorem 

p(¯x)∆x = e −kax −e −ka(x+∆x) where ¯x lies between x + ∆x 

Divide both sides by ∆x and take limit ∆x → 0 


Beer’s law 


Opacity & 

Transmission 


Scattering 


p(x) = − d dx e−kax = k a e −kax 

Mean distance 〈x〉 photon travels before being absorbed 

〈x〉 = 

∫ ∞ 

0 

xk a e −kax dx = 1 k a 

= l a 

... but this is the penetration depth ☺

Scattering of radiation 

◮ In addition to absorption, light may also be scattered by 

air molecules, cloud droplets and aerosols 

◮ Scattering is a redistribution of radiation in different 

directions → phase function 

◮ In analogy to absorption define a scattering cross section 

◮ Effect of absorption plus scattering is called extinction 

σ e = σ s + σ a 

◮ Accordingly an extinction coefficient, k e is defined 


Beer’s law 


Opacity & 

Transmission 


Scattering 


k e = k s + k a = σ e n 

◮ Relative importance of scattering versus absorption is 

given by the single scatter albedo, ˜ω 

˜ω = k s 

k e 

= 

k s 

k s + k a 

= σ s 

σ e 

◮ For ˜ω = 0 → absorption prevails 

◮ For ˜ω = 1 → scattering prevails 

1 − ˜ω = k a 

k e

Scattering of radiation 

◮ The scattering cross section is a kind of shadow 

However this ’shadow’ can be bigger than the actual 

geometrical cross section 

◮ The ratio of the scattering cross section to the 

geometrical area A is called scattering efficiency: 

Q s = σ s 

A 

◮ Q s in general will be a strong function of λ and ε 

◮ For visible wavelength σ s of air molecules is approx. 

proportional to 1/λ 4 → Rayleigh scattering 

◮ Rayleigh scattering explains blue sky and red sunsets 

and sunrises 

◮ The loss of radiation along a line-of-sight due to 

scattering is always associated with a gain in radiation 

along other lines-of-sight 

⇒ source term in radiative transfer 


Beer’s law 


Opacity & 

Transmission 


Scattering 

Schwarzschild eq.

Example of scattering efficiency 


Beer’s law 


Opacity & 

Transmission 


Scattering 


copied from K.N.Liou, Atmospheric radiation 

Q s as function of size parameter x = 2πr/λ 

m i = imag part of N. m r = 1.5

Extinction of solar radiation 


Transmittance 

I I 

('l 

O) 

Beer’s law 


Opacity & 

Transmission 


Scattering 


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Schwarzschild equation 

The atmosphere not only absorbs but it also emits radiation 

according Kirchhoff’s law 

→ we have to combine extinction and thermal emission! 

For the infrared part we will neglect scattering 

Consider radiation at λ propagating through a layer ds 

→ initial intensity I will be attenuated due to absorption 

dI abs = −k a Ids 


Beer’s law 


Opacity & 

Transmission 


Scattering 


Fraction lost of intensity due to absorption is k a ds 

→ This is the absorptivity of the slice of the medium 

According Kirchhoff this absorptivity is equal to the 

emissivity ε of the slice 

→ The slice will thus emit radiation 

dI emit = εB(T ) = k a B(T )ds 

where B(T) is the Planck function

Radiance of cloudy sky 

Radiance [W/(m 2 .sr.µm)] 

10 

9 

8 

7 

6 

5 

4 

3 

2 

Clear sky 

9 km alt. cloud layer 

5 km alt. cloud layer 

0.7 km alt. cloud layer 

Planck curves 


Beer’s law 


Opacity & 

Transmission 


Scattering 


1 

0 

4 6 8 10 12 14 16 18 20 

Wavelength [µm] 

copied from E. Brocard, Ph.D. thesis 

Planck curves for 215K, 230K, 250K, 270, 294K superposed


The net change in intensity is 

dI = dI abs + dI emit = k a (B − I )ds 

⇒ 


dI 

ds = k a(B − I ) = k m ρ(B − I ) 

This can be rewritten with the help of the opacity dτ = k a ds 

dI 

dτ = B − I 

This is nice and compact, BUT what does it mean? Solve 

the equation, i.e. integrate it. 

Assume a sensor on a satellite looking down on the Earth 

Use opacity as vertical coordinate 


Beer’s law 


Opacity & 

Transmission 


Scattering 


⇒ I (0) = I (τ)e −τ + 

∫ τ 

0 

Be −τ ′ dτ ′

Radiative transfer equation 


Interpretation: 

I (0) = I (τ)e −τ + 

∫ τ 

0 

Be −τ ′ dτ ′ 

◮ I (τ) emission at level τ, e.g. by surface of the Earth, or 

any other level where opacity is τ when looking down 

from sensor 

◮ but reduced by the absorption along the path to the 

sensor by e −τ , i.e. the transmissivity 

◮ plus emission of atmospheric layers Bdτ ′ 

◮ Note: Planck function depends on T , B = B(T (τ)) 

◮ taking into account attenuation on the way to the 

sensor e −τ ′ 

☞ Almost all radiative transfer problems involving 

emission and absorption, without scattering, can be solved 

with this equation 


Beer’s law 


Opacity & 

Transmission 


Scattering 

Schwarzschild eq.

Radiative transfer equation 

Line by line calculations for atmospheric radiative transfer 

The total flux F of radiation could in principle be obtained 

by integrating over all wavelengths, or frequencies, i.e. over 

thousands and thousands of transitions, and in all directions 

F = 

∫ ∞ ∫ 2π ∫ π/2 

0 

0 

0 

I ν (θ, ϕ) cos θ sin θdθdϕdν 

This is done by so called line by line calculations based on 

spectral data sets, such as HITRAN☼ or MODTRAN 

Band transmission models are sometimes used where flux 

transmissions ˆT , averaged over a spectral band, are used 


Beer’s law 


Opacity & 

Transmission 


Scattering 


∫ 1 

F ↑ (z) = σTS 4 ˆT (z s , z) + 

F ↓ (z) = 

∫ 1 

ˆT (z,∞) 

ˆT (z S ,z) 

σT 4 (z ′ )d ˆT (z ′ , z) 

Leading to a net flux of F (z) = F ↑ (z) − F ↓ (z) 

σT 4 (z ′ )d ˆT (z ′ , z)

Radiative Transfer Radiative transfer equation (RTE) - IAP ...

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