Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption Radiative transfer 1 Radiation What is radiation? Radiance I and irradiance E Blackbody radiation 2 Radiative transfer equation Derivation Direct-diffuse splitting of radiation field Horizontally homogeneous atmosphere 3 Discrete ordinate method Solution of RTE using the DOM DOM - Impact of number of streams DOM - Deltascaling and intensity correction 4 Single scattering properties Single scattering theory Size distribution Examples 5 Molecular absorption Introduction Line-by-line calculations Broad-band calculations Radiative transfer 10. February 2010 1 / 52
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Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Radiative transfer1 Radiation
What is radiation?Radiance I and irradiance EBlackbody radiation
2 Radiative transfer equationDerivationDirect-diffuse splitting of radiation fieldHorizontally homogeneous atmosphere
3 Discrete ordinate methodSolution of RTE using the DOMDOM - Impact of number of streamsDOM - Deltascaling and intensity correction
4 Single scattering propertiesSingle scattering theorySize distributionExamples
Figure: An opaque container at absolutetemperature T encloses a “gas” of photonsemitted by its walls. At equilibrium, thedistribution of photon energies is determinedsolely by this temperature. The distributionfunction is called Planck (distribution) function(Figure from Bohren and Clothiaux, 2006)
Radiative transfer 10. February 2010 6 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Planck radiation
10-2 10-1 100 101 102 103
wavelength [µm]
10-1
100
101
102
103
104
105
106
107
108
109
irra
dia
nce
[W
/(m
2 µ
m]
6000 K300 K
Figure: Planck functions for surface temperature of sun (≈ 6000 K, blueline), surface temperature of earth (≈ 300 K) and solar irradiance at top ofatmosphere (dotted green line).
Radiative transfer 10. February 2010 7 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Radiative transfer1 Radiation
What is radiation?Radiance I and irradiance EBlackbody radiation
2 Radiative transfer equationDerivationDirect-diffuse splitting of radiation fieldHorizontally homogeneous atmosphere
3 Discrete ordinate methodSolution of RTE using the DOMDOM - Impact of number of streamsDOM - Deltascaling and intensity correction
4 Single scattering propertiesSingle scattering theorySize distributionExamples
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Radiative transfer equation
~n∇Iν = −kext,ν Iν +ksca,ν
4π
∫4π
Pν(~n′ → ~n)Iν(~n′)dω + kabs,νBν
(integro-differential equation for radiance for specific direction ~n)
RTE includes the following processes:Exchange of photons with surrounding of volume element∆V ∆ω∆ν
ExtinctionAbsorptionOutscattering: Scattering of photons from ~n into ~n′
Inscattering: Scattering of photons from ~n′ into ~nEmission of photons into ~n
Stationary form of RTE because time dependence can be neglectedin Earth’s atmosphere
Radiative transfer 10. February 2010 9 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Direct-diffuse splitting of radiation field
total solar radiation field = diffuse solar radiation + direct solar beam
Iν = Id,ν + Sνδ(~n − ~n0)
Direct radiation Sν can be separated and calculated usingLambert-Beer’s law:
dSνds
= −kext,νSν , ~n = ~n0
RTE for diffuse solar radiation must be further simplified
Radiative transfer 10. February 2010 10 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Horizontally homogeneous atmosphere
plane-parallel approximation:curvature of Earth’s atmosphere is neglectedall optical properties are independent of horizontal positionsolar beam independent on horizontal position
only one spatial coordinate required, altitude z oroptical thickness τ =
∫ z0 kext(z′)dz′
approximation not valid for e.g. inhomogeneous clouds or verylow sun
Figure from Mayer 2009
Radiative transfer 10. February 2010 11 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Radiative transfer1 Radiation
What is radiation?Radiance I and irradiance EBlackbody radiation
2 Radiative transfer equationDerivationDirect-diffuse splitting of radiation fieldHorizontally homogeneous atmosphere
3 Discrete ordinate methodSolution of RTE using the DOMDOM - Impact of number of streamsDOM - Deltascaling and intensity correction
4 Single scattering propertiesSingle scattering theorySize distributionExamples
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Separation of µ and φ
Assumption: Phase function is rotationally symmetric alongdirection of incident light, correct for spherical and randomlyoriented particlesPhase function expansion in Legendre series
P(cos Θ) =∞∑l=0
plPl (cos Θ)
p0 =12
∫ 1
−1P(cos Θ)d cos Θ = 1 (normalization of P)
p1 =32
∫ 1
−1cos ΘP(cos Θ)d cos Θ = g (asymmetry parameter)
Phase function with µ = cos θ and φ separated using additiontheorem of associated Legendre polynomials:
P(cos Θ) =∞∑
m=0
(2− δ0m)∞∑
l=m
pml Pm
l (µ)Pml (µ′) cos m(φ− φ′)
Radiative transfer 10. February 2010 13 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
System of differential equations for each Fouriermode of radiance field
Fourier expansion of the radiance field:
I(τ, µ, φ) =∞∑
m=0
(2− δ0m)Im(τ, µ) cosφ
DE for each Fourier mode of radiance field, depends only on 2variables τ and µ:
µd
dτIm(τ, µ) = Im(τ, µ)− Jm(τ, µ) m = 0,1, ...,Λ
Radiative transfer 10. February 2010 14 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Scattering integral – Gaussian quadrature
Gaussian quadrature: method to approximate integral offunctions which can well be approximated by a polynomialfunctionSeparate differential equation (DE) for each quadrature point(also called stream):
µidIm(τ, µi )
dτ=Im(τ, µ)− ω0
2
r∑j=1
wj Im(τ, µj )∞∑
l=m
pml Pm
l (µi )Pml (µj )
− ω0
4πS0 exp
(− τ
µ0
)P(µi , µ0)− (1− ω0)B(τ)δ0m
Inhomogeneous DE⇒ solution= particular solution forinhomogeneous DE + general solution for homogeneous DE
Radiative transfer 10. February 2010 15 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Figure: Cloudy radiance field, TOA.DISORT, nstr=16 with delta-scaling(exercise 8).
Radiative transfer 10. February 2010 23 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Calculation for water cloud - intensity correction
DISORT2 includesintensity correctionmethod by Nakakjimaand Tanaka (1988),which calculates the firstand second orders ofscattering using thecorrect phase function
Figure: Cloudy radiance field, TOA.DISORT2, nstr=16 with intensity correction(exercise 8).
Radiative transfer 10. February 2010 24 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Radiative transfer1 Radiation
What is radiation?Radiance I and irradiance EBlackbody radiation
2 Radiative transfer equationDerivationDirect-diffuse splitting of radiation fieldHorizontally homogeneous atmosphere
3 Discrete ordinate methodSolution of RTE using the DOMDOM - Impact of number of streamsDOM - Deltascaling and intensity correction
4 Single scattering propertiesSingle scattering theorySize distributionExamples
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Single scattering theory
Scattering calculations in planetary atmospheres:1 single scattering by small volume element
(Mie theory, geometrical optics ...)2 multiple scattering by entire atmosphere
(solution of RTE, e.g. DOM)
Assumption: scattering particles are sufficiently separated sothat they can be treated as independent scatterers (nointerference of radiation scattered by independent particles)
Scattered radiance at distanceR in far field:
~Isca = kscaPdV
4πR2
Figure from Hansen and Travis, 1974
Radiative transfer 10. February 2010 26 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Geometrical optics method
Geometrical optics method can be applied for particles that arelarge compared to the wavelength, e.g. cloud droplets in UV/Vissize parameter x = 2πr
λ � 1Trace individual rays through particleSnell’s law: direction of refracted rays
n1 sinα = n2 sinβ
Fresnel equations: Intensity and polarization of radiationreflected and refracted by particle surface
Radiative transfer 10. February 2010 27 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Geometrical optics
Figure from Hansen and Travis, 1974
Radiative transfer 10. February 2010 28 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Rayleigh scattering
Figure from Hansen and Travis, 1974
Radiative transfer 10. February 2010 29 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Mie theory
Calculation of optical properties (P,qsca,qabs) of sphericalparticles (Mie, 2008)Solution of Maxwell equations (Input: refractive index, sizeparameter)physical explanation: multipole expansion of scattered radiation
Radiative transfer 10. February 2010 30 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Size distributions
A cloud consists of droplets of various sizes following a sizedistribution n(r):
N =
∫ rmax
rmin
n(r)dr
optical properties are averaged over size distribution
ksca =
∫ rmax
rmin
σscan(r)dr
kext =
∫ rmax
rmin
σextn(r)dr
P(cos Θ) =4π
k2ksca
∫ rmax
rmin
P′(cos Θ, r)n(r)dr
Radiative transfer 10. February 2010 31 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Effective radius
A “mean radius” for scattering may be defined as follows (scatteringcross section σsca = πr 2Qsca):
rsca =
∫ rmax
rminrπr 2Qsca(r)n(r)dr∫ rmax
rminπr 2Qsca(r)n(r)dr
In the UV/VIS water cloud droplets fulfill x � 1 and ω0 ≈ 1 , thenQsca ≈ 2
reff =1G
∫ rmax
rmin
rπr 2n(r)dr
Generalization for non-spherical particles (e.g. ice crystals or aerosols)
reff =
∫ rmax
rminV (r)n(r)dr∫ rmax
rminA(r)n(r)dr
r – equivalent sphere radius; A – geometrical cross section averagedover all possible orientationsEffective variance of a size distribution:
veff =1
Gr 2eff
∫ rmax
rmin
(r − reff)2A(r)n(r)dr
Radiative transfer 10. February 2010 32 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Extinction efficiency
100 101 102 103
size parameter x
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
exti
nct
ion e
ffic
iency
Qext
single particlegamma size distribution
Figure from exercise 14
major maxima and minimacaused by interference ofdiffracted radiation (l=0) andtransmitted radiation (l=2)phase shift for ray passingthrough sphere ρ = 2x(nr − 1)
superimposed “ripple” structurelast few sigificant terms in Mieseriesexplanation: surface wavesvanish by integration over sizedistribution
geometrical optics limit of 2 forlarge x
Radiative transfer 10. February 2010 33 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Asymmetry parameter
100 101 102 103
size parameter x
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
asy
mm
etr
y p
ara
mete
r g
single particlegamma size distribution
Figure from exercise 14
geometrical optics limit of 0.87 forlarge xRayleigh limit of 0 for small x
Radiative transfer 10. February 2010 34 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Size distributions
Mie calculations for size distributionswith the same reff=10 µm anddifferent veff (exercise 15)
Optical properties in UV/Vis/NIR forall size distibutions very similar, butlarger differences in thermal spectralregion
0 5 10 15 20r [micrometer]
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.045
n(r
)
veff=0.15veff=0.10veff=0.05veff=0.01
103 104100
120
140
160
180
200
220
240
q ext
[1
m·g/m
3]
Gamma distribution, reff=10µm
103 1040.0
0.2
0.4
0.6
0.8
1.0
ω0
veff = 0.15
veff = 0.1
veff = 0.05
veff = 0.01
103 104
λ [nm]
0.70
0.75
0.80
0.85
0.90
0.95
1.00g
Radiative transfer 10. February 2010 35 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Dependence on effective radius
Mie calculations for size distributionswith different reff and the sameveff=0.1 (exercise 16)
Optical properties in UV/Vis/NIR forall size distibutions very similar, butlarger differences in thermal spectralregion
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Absorption coefficients in atmospheric window(8–14µm)
8 9 10 11 12 13 14wavelength [µm]
10−15
10−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
abso
rptio
nco
effic
ient
[1/m
]
H2OCO2O3N2OCH4HNO3
altitude: 0.5 km, ARTS calculation
Radiative transfer 10. February 2010 43 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Molecular physics
Molecules have 3 forms of internal energy
Eint = Erot + Evib + Eel
According to quantum mechanics energy states are quantized:Erot - rotational energy (microwave)Evib - vibrational energy (IR)Eel - electronic energy (NIR/Vis/UV)
Erot < Evib < Eel
absorption: transition from lower to higher energy stateemission: transition from higher to lower energy stateabsorption/emission lines characteristic for particular molecule
Radiative transfer 10. February 2010 44 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Line broadening
1 Natural broadeningHeisenberg’s uncertainty priciple ∆E∆t & hlifetime of molecule in excited state is finiteemitted energy is distributed over finite frequency interval ∆νnegligible in Earth’s atmosphere
2 Collision / Pressure broadeningduring emission molecule collides with other moleculeslifetime is shortenedinteraction causes line-broadening (larger than natural broadeningbecause lifetime of molecule much longer than time betweencollisions)dominant below 20 km in Earth’s atmosphere
3 Doppler broadeningrandom thermal motion of moleculesdifferent relative velocities between molecules and radiation sourcecauses Doppler broadening of emission linesdominant above 50 km in Earth’s atmosphere
Radiative transfer 10. February 2010 45 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Line-shapes
Figure from Zdunkowski et al.
Radiative transfer 10. February 2010 46 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
k-distribution method
aim: obtain average transmission in a particular spectral bandresort frequency grid according to absorption coefficient k andreplace wavenumber integration by integration over k:
Tν̄ =
∫∆ν
e−k(ν)ds dν∆s
=
∫ ∞0
e−kdsh(k)dk
h(k) - probability density function (pdf) for occurence of k
integration over cumulative pdf g(k) =∫ k
0 h(k)dk :
Tν̄ =
∫ 1
0e−k(g)dsdg
g(k) is a smooth monotonically increasing function between 0and 1 and the integral can be approximated by very few gridpoints (e.g. using Gaussian quadrature)
Radiative transfer 10. February 2010 47 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
k-distribution method
Illustration of k-distribution method.Figures from Zdunkowski et al.
Radiative transfer 10. February 2010 48 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
correlated-k-distribution method
k-distribution method exact only for homogeneous layerfor inhomogeneous atmosphere correlated-k method may beusedTransmission for 2 trace gases:
Tν̄)(1,2) =
∫∆ν
Tν(1)Tν(2)dν∆s
Approach results in integration over two cumulative PDFsapproximate method, accuracy investigated in e.g. Fu and Liao(1992)
Radiative transfer 10. February 2010 49 / 52
Radiation Radiative transfer equation Discrete ordinate method Single scattering properties Molecular absorption
Radiative transfer1 Radiation
What is radiation?Radiance I and irradiance EBlackbody radiation
2 Radiative transfer equationDerivationDirect-diffuse splitting of radiation fieldHorizontally homogeneous atmosphere
3 Discrete ordinate methodSolution of RTE using the DOMDOM - Impact of number of streamsDOM - Deltascaling and intensity correction
4 Single scattering propertiesSingle scattering theorySize distributionExamples