12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Ronald Prinn
3. Scattering of Radiation by Molecules and Particles
a. Introduction
Here, we’ll deal with wave aspects of light rather than quantum aspects.
Consider components of electric field in 2 mutually perpendicular directions,
parallel and perpendicular to the plane of propagation and propagating in the z
direction:
Er = ar ei( ωt −kz)
ω = circular frequency
E = a ei( ωt −kz)
k =
(1)
2π
λ
The intensity is given by:
I = Er Er* + E E* = ar2 + a2
(2)
Let us first consider Single Scattering. We may consider a single particle or a
small volume of particles such that scattering events will all be single scattering
events.
Fig. 1
p ′′ = phase function
p = phase matrix
I = k sca P I0
dV
4πR 2
or
I = hsca
P11 I0 dV
4πR 2
k sca = scattering cross-section per unit volume
k sca =
σ sca
1
=
dV
dV
N
∑σ
i =1
sca
[k sca ] = length−1
,i
[σsca ] = length2
N = total no. of particles
Qsca,i
σ ,i
= sca
Gi
where G i = geometrical cross sec tion
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Ronald Prinn
Lecture
Page 1 of 17
Qsca =
σsca
=
G
∑σ , i
∑G
sca
Qsca = Scattering efficiency
i
P is the phase matrix and provides the angular distribution and polarization of
the scattered light. For our purpose here, lets consider the total intensity of the
radiation whether polarized or not. Then the term, p11 , is the phase function or
scattering diagram which defines the probability for scattering of unpolarized
incident light in any direction. p11 is normalized such that:
p11 dΩ
∫ 4π = 1
where dΩ = element of solid angle
Let us define <cos α> =
∫ cos α
(3)
p11 dΩ
where α is the scattering angle (see Fig.
4π
1).
<cos α> = anisotropy parameter (or asymmetric parameter) and can vary
between +1 and -1. <cos α> = 0 for isotropic scattering.
We define analogous terms for particle absorption and we have:
k ext = k sca + k abs
σ = π r 2Q
σext = σsca + σabs
Qext = Qsca + Qabs
(4)
Then, we define:
ω = single scattering albedo =
k sca σsca Qsca
=
=
k ext
σext Qext
For practical applications, k sca and ω can be taken as constants and P11 is a
function only of the scattering angle, α.
P
This special case is valid for:
(1) randomly oriented particles, each of which has a plane of symmetry
(2) randomly oriented asymmetric particles, if half the particles are mirror
images of the others.
(3) Rayleigh scattering and Mie scattering.
Another important definition: χ =
2πa
λ
where a = particle radius.
m = nr − ini where nr = real part of the index of refraction and ni = imaginary part.
ni is responsible for absorption and nr is responsible for scattering.
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Ronald Prinn
Lecture
Page 2 of 17
For water, nr = 1.33 across the visible and near infrared. And ni will depend on
the kinds of materials dissolved in the water drops. It will therefore be much
more of a function of wavelength.
Returning to the analytical expression for the electric field as a beam of radiation
passing through a single particle:
E(z, t) = E0e i(ωt −kz)
The intensity of radiation varies as the square of the E-field. So, we have:
I = E20 e 2 i( ωt −kz)
k =
So, we have : E(z, t) = E0 e
2π
and
λ
and I = E02 e
⎛ −4 πzni ⎞
⎜⎜ λ
⎟⎟
0
⎝
⎠
e
λ0
m
⎡ −2 πz
⎤
i⎢
(nr −ini ) + ωt ⎥
⎣ λ0
⎦
−2 πzni
= E0 e
λ =
λ0
e
⎛ −2 πznr
⎞
+ ωt ⎟⎟
i ⎜⎜
⎝ λ0
⎠
⎛ 4 πznr
⎞
+ 2 ωt ⎟⎟
i ⎜⎜ −
λ0
⎝
⎠
and – using the definition of size parameter χ =
2 πa
where we will take z = 2a =
λ
diameter of drop, we have:
I = E02 e −4 χ ni e i( −4 χ nr + z ω t)
(5)
absorption
b. Mie scattering – still single scattering.
⎧⎪Ers ⎫⎪ exp(−ikR + ikz)
⎨ s⎬ =
ikR
⎪⎩E ⎪⎭
i
⎧⎪S1 (α, θ) S 4 (α, θ)⎫⎪ ⎧⎪E r ⎪⎫
⎨
⎬⎨ ⎬
⎪⎩S3 (α, θ) S2 (α, θ)⎪⎭ ⎪⎩E i ⎪⎭
(6)
at distance R in the
far field
If we consider isotropic, homogenous, spheres, we have:
⎧⎪S1 (α)
S=⎨
⎪⎩ 0
0
⎫⎪
⎬
S2 (α)⎪⎭
I =
1
F I0
k R2
2
And we have the transformation matrix:
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Ronald Prinn
The S values are in general
complex number and functions of
scattering angle.
Lecture
Page 3 of 17
⎧1
*
*
⎪ 2 S1S1 + S2 S2
⎪
⎪
⎪1
⎪ S1S1* − S2S*2
⎪2
⎪
F = ⎨
⎪
⎪
0
⎪
⎪
⎪
⎪
0
⎪⎩
(
)
1
S1S1* − S2S*2
2
)
0
(
)
1
S1S1* + S2S*2
2
)
0
(
(
0
1
S1S2* + S2 S1*
2
0
−
(
)
i
S1S2* − S2S1*
2
(
⎫
⎪
⎪
⎪
⎪
⎪
0
⎪
⎪
⎬
⎪
i
S1S*2 − S2 S1* ⎪
2
⎪
⎪
⎪
1
*
* ⎪
S1S2 + S2 S1 ⎪
2
⎭
0
(
)
(
which is proportional to the phase matrix: F = C P
The normalization condition on P leads to: C =
)
(8)
F11 dΩ
∫ 4π
4π
and since σsca = effective cross section, we have:
σ sca =
IR 2 d Ω
∫ I0
4π
and
C=
F11 d Ω k 2 σ sca
∫ 4π = 4π ,
4π
where we’ve used I= F11 k 2 R 2
So – we have:
F11 =
k 2 σsca p11 1
= (S1S1* + S2S2* )
4π
2
F21 =
k2 σsca p21 1
= (S1S1* − S2S2* )
4π
2
F33 =
k 2 σsca p21 1
= (S1S2* + S2S1* )
4π
2
defines phase function
(9)
F 43 =
k2 σsca p43
i
= − (S1S2* − S2S1* )
4π
2
For a single sphere, we have:
∞
2n + 1
S1 =
∑ n(n + 1) ⎡⎣a π
S2 =
∑ n(n + 1) ⎡⎣b π
n
n =1
∞
n=1
n
2n + 1
n n
+ bn τn ⎤⎦
+ an τn ⎤⎦
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Ronald Prinn
(10)
Lecture
Page 4 of 17
)
πn and τn are function only of α and relate to Legendre Polynomials.
2πa
and m = nr − ini and involve Spherical Bessel
λ
an and bn are functions of x =
functions.
We also have:
Qsca =
2
x2
Qext =
2
x2
∞
∑ (2n + 1)(a a
*
n n
n =1
∞
∑ (2n + 1) R
n=1
< cos α > =
4
x Qsca
2
e
+ bnbn* )
(an + bn )
(11)
n(n + 2)
2n + 1
R e (anan*+1 + bnbn*+1 ) +
R e (anbn* )
n
+
1
n(n
+
1)
n =1
∞
∑
All above is for Single Scattering from a Single Sphere. In general, if optical
thickness is not too large, single scattering can be applied to a distribution of
particles assumed to be independent. We then have:
k sca =
r2
∫σ
sca
(r)n (r) dr =
r1
k ext =
r2
∫ πr
Qsca (r)n (r) dr
r1
r2
r2
r1
r1
∫ σext (r)n(r) dr =
2
∫ πr
2
Qext (r)n(r) dr
(12)
where n(r) = size distribution, describing the number of the particles having radii
between r and r+dr over the range r1 to r2.
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Ronald Prinn
Lecture
Page 5 of 17
c. Geometric Optics:
When r >> λ, we can use ray theory of light due to Fresnel (see Van de Hulst, Ch.
3, Light Scattering by Small Particles). Terminology is:
0 – diffraction
1 – external reflection
2 – double refraction
3 – first rainbow
4 – second rainbow
______________________________________________________________
and if P(θ) =
1
∞
2π π
∑ P (θ), then 4π ∫ ∫ P (θ) sin θ dθ dφ , is for non-absorbing spheres
=0
0 0
from Fresnel theory.
real = 1.33
= 2.00
0
.500
.500
1
.033
.081
2
.442
.364
3
.020
.043
4
.033
.008
5
.002
.004
always true in geometric optics
often sufficient to consider just
these
For non-absorbing particles, diffraction = 1\2 of scattered light. Thus, the
geometric optics limiting value of Qext = 2.0.
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Ronald Prinn
Lecture
Page 6 of 17
d. Rayleigh Scattering:
r
λ
and
r
λ
m
where m = nr − ini
Radiation penetrated particle
quickly. ∴ particle own field is
negligible in the process.
Particle can be considered
to be in homogenous
external electric field
Esca =
k 2 αpEincident
R
where β = angle between dipole
moment and direction of scatter
sin β e −ikn
⎧ 3 (1 + cos2 α)
⎪ ψ
⎪ −3
2
⎪⎪ ψ sin α
p(α) = ⎨
⎪
0
⎪
⎪
0
⎪⎩
−3
ψ
sin2 α
3 (1 + cos2 α)
ψ
0
0
0
0
3 cos α
2
0
(13)
⎫
⎪
⎪
0
⎪⎪
⎬
⎪
0
⎪
3 cos α ⎪
⎭⎪
2
0
(14)
p11 = 3 (1 + cos2 α)
ψ
(15)
which gives angular distribution of intensity scattered by small particles – the
Rayleigh scattering.
We also obtain:
Qsca =
8 4 m2 − 1
x
3
m2 + 2
⎧ m2 − 1 ⎫
Qabs = −4x Im ⎨ 2
⎬
⎩m + 2 ⎭
(16)
note that Qabs > Qsca as x → 0
note 4th power of x or λ-4 dependence
This result is the same as Mie scattering as limiting case as x → 0 (r<<λ).
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Ronald Prinn
Lecture
Page 7 of 17
12.815 Lecture Notes (Atmospheric Radiation)
Multiple Scattering
Refer back to Eq. 22 from the first set of Atmospheric Radiation lecture notes where
we discussed Case III which arises due to the following two conditions:
Fν >> B ν (T)
(1)
I(θ’,φ’,τν)>>Bν(τν)
(2)
The resulting equation of transfer is:
μ
dIν
(θ, φ, τν ) = Iν (θ, φ, τν ) − Jν (θ, φ, τν )
dτ ν
where Jν (θ, φ, τν ) =
(3)
ω
ω
P(θ, φ, θ ', φ ') Iν (θ ', φ ', τν ) sin θ ' dθ ' dφ ' + e−τν / μo P(θ0 , φ0 )Fν
4π ∫
4
and the formal solution is:
τ
I(τν , μ, μ0 , φ0 , φ) ↑ = ∫ J(τν ', μ, μ0 , φ, φ0 ) e(τν −τν ') / μ
0
dτν '
μ
μ>0
(4)
I(τν , μ, μ0 , φ, φ0 ) ↓ = −
τνs
∫ J(τ
ν
', μ, μ0 , φ, φ0 ) e−( τν '−τν ) / μ
τν
dτ ν '
μ
μ<0
Due to the complexities of evaluating the integrals in Eq. 4, a number of techniques
have been used to generate numerical results:
1.
2.
3.
4.
5.
6.
7.
8.
9.
Discrete Ordinates
Doubling or Adding Method
Successive Orders of Scattering
Iteration of Formal Solution
Invariant Embedding
Method of X and Y Functions
Spherical Harmonics Method
Expansion in Eigenfunctions
Monte Carlo Method
We will focus some attention on the Discrete Ordinates Method and apply an
available computer program to some exercises.
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Ronald Prinn
Lecture
Page 8 of 17
Radiative Transfer in a Scattering Atmosphere
1. Coordinate system in a “plane parallel” atmosphere
Here position defined by z (or τ) only. Recall that optical depth τ related to
altitude z by dτ = -αdz where α is the extinction coefficient.
cos θ = μ ; θ = inclination to
Notation
upward
outward
cos θo = μ o ; θo = inclination to
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Ronald Prinn
normal
upward
outward
normal
Lecture
Page 9 of 17
cos θ‘ = μ‘ ; θ‘ = inclination to
upward
outward
normal
From spherical geometry, the cosine of the scattering angle, α can be
expressed in terms of the incoming and outgoing directions in the form:
(
cos α = μμ '+ 1 − μ2
) (1 − μ ' )
1
2
2
1
2
cos ( φ '− φ )
(5)
Let us now digress for a moment and examine the properties of Legendre
polynomials (which come to play in a variety of ways in radiative transfer
problems).
We may consider writing the phase function in terms of Legendre polynomials
in the form:
P(cos α) =
N
∑C
P (cos α)
(6)
=0
Legendre polynomials have the following form, and orthogonal and recurrence
properties:
So,
P0 (μ) = 1
Pn (μ) =
1
dn
(μ2 − 1)n
2n n! dμn
P1 (μ) = μ
P2 (μ) =
3 2 1
μ −
..........
2
2
⎧0
⎪
∫−1 P (μ)Pk (μ) dμ = ⎨ 2
⎪⎩ 2 + 1
1
μ P (h) =
(n = 1, 2.....)
≠h
(7)
=h
+1
P (μ)+
P (μ)
2 + 1 +1 2 + 1 +1
Using Eq. 5, the Phase Function defined above may be written as follows:
P(μ, θ, μ′, φ′) =
N
∑C P
=0
⎡μ μ′ + (1 − μ2 )12 (1 − μ′)12 cos (θ′ − θ)⎤
⎣⎢
⎦⎥
(8)
From the orthogonality condition, the expansion coefficients are given by:
1
C =
2 +1
P (μ) P (μ) dμ
2 −∫1
= 0,1......N
where we note that the phase function is normalized to unity:
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Ronald Prinn
Lecture
Page 10 of 17
1
4π
2π 1
∫ ∫ P (μ) dμ dφ ≡ 1
0 −1
There is an addition theorem for Legendre polynomials which allows us to
write the Phase Function as follows:
P(μ, φ, μ′, φ′) =
N
N
∑ ∑C
m
Pm (μ)Pm (μ′) cos m(φ′ − φ)
(9)
m= 0 =0
where Pm denotes the Associated Legendre polynomials and:
(
(
C m (2 − δ0,m ) C
− m) !
+ m) !
,
⎧1,
δ0, m = ⎨
⎩0,
= m,......N
0≤m≤n
(10)
m=0
otherwise
In view of the expansion of the phase function, the diffuse intensity may also
be expanded in a cosine series in the form:
I(τ, μ, φ) =
N
∑I
m
m= 0
(τ, μ) cos m(φ0 − φ)
(11)
Substituting Eqs. 9 and 11 into Eq. 3, and using the orthogonality of the
associated Legendre polynomials, the equation of transfer splits into (N+1)
independent equations, and may be written as:
μ
dIm (τ, μ)
ω
= Im (τ, μ) − (1 + δ0, m )
dτ
4
1
x
∫P
m
N
∑C
m
Pm (μ)
(12)
=m
(μ′) Im (τ, μ′) dμ′
−1
−
−τ
ω N m m
C P (μ)Pm (−μ0 )F e μ0
∑
4π =0
m = 0, 1, …… N
Let us rewrite these equations as follows:
μ
dIm (τ, μ)
= Im (τ, μ) − J m (τ, μ)
dτ
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Ronald Prinn
(13)
Lecture
Page 11 of 17
with the source function given by:
ω
4
Jm (τ, μ) = (1 + δ0,m )
ω
+
4π
N
∑C
m
=m
N
∑ Cm Pm (μ)
=m
m
1
∫P
m
(μ′) Im (τ, μ ′) dμ′
(14)
−1
m
P (μ)P (−μ0 )F e
− τ
μ0
To proceed with the solution of Eq. 13, we first discretize the equation by
replacing μ with μi (i= -n,…., n, with n= 1, 2,….) and the integral with a sum
with the weights, aj
1
∫
f(μ) dμ =
n
∑ f(μ ) a
j
j=−n
−1
(15)
j
The homogeneous solution for the set of first-order differential equations may
be written:
Im (τ, μ i ) =
n
∑L
j=−n
m
j
ψ jm (μi ) e
−k m
j τ
(16)
where ψ jm (μi ) and k jm denote the eigenvectors and eigenvalues, respectively,
and L m
j are coefficients to be determined from appropriate boundary
conditions. On substituting Eq. 16 into the homogeneous part of Eq. 13, the
eigenvectors may be expressed by
ψ jm (μi ) =
(1 + δ0,m ) ω
m
j
4(1 + μ j k )
N
∑C
m
=m
n
Pm (μi ) ∑ aq Pm (μq ) ψm
j (μ q )
(17)
q=−n
The particular solution may be written in the form
Ipm (τ, μi ) = Zm (μi ) e
−τ
μ0
(18)
From Eq. 13, we have
Z m (μi ) =
ω
μ
⎛
⎞
4 ⎜1 + j ⎟
μ
0⎠
⎝
N
∑C
=m
m
Pm (μi )
(19)
F ⎞
⎛ n
x ⎜ ∑ aq Pm (μq ) Zm (μq ) + Pm (−μ0 ) ⎟
π ⎠
⎝ q=−n
m
Equations 17 and 19 are linear equations in Ψ m
and may be solved
j and z
numerically. The complete solution for Eq. 13 is the sum of the general
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Ronald Prinn
Lecture
Page 12 of 17
solution for the associated homogeneous system of the differential equations
and the particular solution. Thus,
Im (τ, μi ) =
n
∑L
j=−n
m
j
ψm
j (μi ) e
−k m
j τ
+ Zm (μi ) e
−τ
μ0
(20)
i = -n, ………. +n
In order to determine the unknown coefficients, Lmj , aq, boundary conditions
must be imposed.
In the discrete-ordinates method for radiative transfer, analytical solutions for
the diffuse intensity are explicitly given for any optical depth. Thus the
internal radiation field can be evaluated without additional computational
effort. And furthermore, useful approximations can be developed from this
method for flux calculations.
Advantages of Discrete Ordinate Method
a) In principle - numerical computations can be done for any order of
approximation.
b) The internal radiation field is determined - not just the Reflection &
Transmission.
c) Accurate results (to about 1%) are achievable with only a few streams
(3-4) for most cases.
We will utilize the Discrete Ordinate computer program to do a few
excercises.
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Ronald Prinn
Lecture
Page 13 of 17
Multiple Scattering Computational Techniques
1. Discrete Ordinates (We’ll discuss in detail in a few minutes.)
2. Doubling or Adding
Principle: If reflection and transmission is known for each of two layers, the
reflection and transmission from the combined layer can be obtained by
computing the successive reflections back and forth between the two layers.
If the two layers are chosen to be identical, the results for a thick
homogenous layer can be built up rapidly in a geometric (doubling) manner.
3. Successive Orders of Scattering
Principle: Intensity is computed individually for photons scattered once, twice,
three times, etc. with the total intensity obtained as the sum over all orders.
If the intensity is expanded in a Fourier series, the high frequency terms arise
from photons scattered a small number of times. Therefore, most Fourier
terms can be obtained with some accuracy by computing a few orders of
scattering.
4. Iteration of Formal Solution
Direct solution of integral over source function by dividing atmosphere into
layers with small optical thickness.
5. Invariant Imbedding
Differential Equations are developed which give the change of reflection and
transmission matrices when an optically thin layer is added to the atmosphere.
It is a special case of the Doubling or Adding technique.
6. Method of X and Y Functions
Involves the determination of integral equations for functions which depend
upon only one angle and are directly related to Reflection and Transmission
matrices. The integral equations need to be solved numerically. The integral
equations are completely specified by a character function depending on the
particular phase function. This method is due to Chandrasekhar.
7. Spherical Harmonic Method
Intensity is immediately expanded into a finite number of spherical harmonics
and then the Phase Function is expanded in Legendre polynomials similar to
the Discrete Ordinate method.
8. Expansion in Eigenfunctions
Standard technique for solving differential equations. Find homogenous
solution and particular solution. Apply boundary condition. Direct application
to complete RTE is ponderous. Discrete Ordinates technique depends on this
approach for solving discretized set of equations.
9. Monte Carlo Method
Scattering of an individual photon can be considered to be a stochastic
process, with the Phase Function being the probability density function for
scattering at a given angle. Photons are allowed to play a game of chance in a
computer and by recording the history of a sufficient number of photons, the
radiation field can in principle be determined to an arbitrary accuracy. The
basic simplicity of this method allows great flexibility, and hence it can be
applied to complicated problems which would be virtually insoluble by other
methods.
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Ronald Prinn
Lecture
Page 15 of 17
Isotropic Scattering and Discrete Ordinates
Pertinent RTE:
μ dI(μ, φ, τ) / dτ = I(μ, φ, τ) −
ω
P (μ, φ, μ ′, φ′) I(μ ′, φ′, τ) dμ ′ dφ′
4π ∫∫
ω − τ μ0
e
P (μ0 , φ0 )F
4
+
(1)
For isotropic scattering, we have:
P (μ, φ, μ′, φ′) = 1 and I (μ, τ) =
1
2π
2π
∫ I(μ, φ, τ) dφ
(2)
0
i.e. – Intensity is azimuthally independent.
1
μ
−τ
dI(τ, μ)
ω
ω
= I(τ, μ) − ∫ I(τ, μ′) dμ′ −
F e μ0
dτ
2 −1
4π
(3)
Applying Gaussian Quadrature, and setting Ii = I (τ, μi), we have:
μi
−τ
dIi
ω +n
ω
= I i − ∑ Ijaj − F e μ0
dτ
2 j=−n
4π
(4)
i = −n,........., +n
Since this is linear differential equation, we need to seek the general solution
(sometimes called the homogenous solution) and then the particular solution.
Homogenous solution:
Try (guess) I i = gi e−kτ where g i and k are constants.
μi
dIi
ω
= Ii − ∑ Ij aj
dτ
2 j
∴ gi (1 + μi k) =
ω
2
(5)
∑a
j
j
gi
So, g i must be of the form L
(1 + μi k )
where L is a constant.
Substituting this back into Eq. 5, we get the characteristic equation for eigenvalue k
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Ronald Prinn
Lecture
Page 16 of 17
aj
ω +n
=ω
∑
2 j=−n (1 + μ j k)
aj
n
∑ (1 − μ
j=1
2
j
k2 )
=1
Note difference in summation
(6)
This Eq. has 2n roots, ±k α α = 1……..n which when ω = 1 includes 2K α values of
zero.
General Solution is:
Ii =
L ± α e ∓k α τ
α =1 1 ± μik α
n
∑
i = −n,......, +n
(7)
Particular Solution:
Try:
Ii =
− τ
ω
F hie μ0
4π
−μ i
We have:
i = −n,......, +n
hi = hi −
μ0
ω +n
μ
hi ⎛⎜ 1 + i ⎞⎟ = ∑ ajhj + 1
μ0 ⎠ 2 j=−n
⎝
or
hi must be of the form =
with
⎛
γ = ⎜1 − ω
⎜
⎝
1 +n
ω ∑ ajhj − 1
2 j=−n
1+
γ
μi
⎞
⎟
∑
2 ⎟
j=1 [1 − μ j 4μ 0 ⎠
aj
n
(8)
(9)
μ0
−1
(10)
Adding the homogenous and particular solutions, we obtain:
Ii =
n
Lj e
−k j τ
∑1+ μ k
j=1
i
+
j
ωF γ e
− τ
μ0
(11)
μ
4π ⎛⎜ 1 + i ⎞⎟
μ0 ⎠
⎝
i = -n, …….. +n
The L j are determined from boundary conditions.
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Ronald Prinn
Lecture
Page 17 of 17