Direct and inverse radiative transfer problems -...
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Direct and inverse radiativetransfer problems
How to go from stellar atmosphere modellingto aerosols problems ?
Olivier Titaud
Postdoctorant
Centro de Modelmatiento Matematico
Universidad de Chile
Doctor in Numerical Analysis
Universite of Saint Etienne, France
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1. The transfer equation
2. Numerical resolution of the di-
rect problem
3. Inverse problems
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The transfer theory
Studied physical process : Propagation of aparticles p in a diluted discrete medium;p[v] −→ p[v′] neutrons [speed].p[ν] −→ p[ν ′] photons [energy]p[m] −→ p[m′] aoerosols [mass] ?
The aim : describing the radiative field by itsproperties at each position and time,depending on the direction and frequency.
— Nov. 22 2004 – p.3
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The specific intensity
The main magnitude of the transfer theory is theSpecific Intensity I .
Measures the amount of energy carriedthroughout an elementary surface in aelementary direction cone and in anelementary frequency interval;
Depends on : the position, the direction, thefrequency and the time.
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A simplified model
The medium is modelized by a finite slab :stratified with plane-parallel homogeneous layers
static and in a steady state :
z = 0
z = z?
z
I−0 (µ), µ < 0
I+?
(µ), µ > 0
~n~k
θ
µ = cos θ
Examples : stellar atmospheres, clouds
.
— Nov. 22 2004 – p.5
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The transfer equation
It describes the propagation of energy carried bya radiation:
For all z ∈]0, z? [, µ ∈] − 1, 1[ , and ν > 0,
µ∂I
∂z(z, µ, ν) = −χ(z, ν)I(z, µ, ν)
+
∫ +∞
0
σ(z, ν, ν ′)J(z, ν ′) dν ′ + E(z, ν).
J(z, ν ′) =1
2
∫ 1
−1
I(z, µ, ν ′) dµ : mean intensity
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The transfer equation
It describes the propagation of energy carried bya radiation:
For all z ∈]0, z? [, µ ∈] − 1, 1[ , and ν > 0,
µ∂I
∂z(z, µ, ν) = −χ(z, ν)I(z, µ, ν)
+
∫ +∞
0
σ(z, ν, ν ′)J(z, ν ′) dν ′ + E(z, ν)
Contribution of emission
— Nov. 22 2004 – p.6
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The transfer equation
It describes the propagation of energy carried bya radiation:
For all z ∈]0, z? [, µ ∈] − 1, 1[ , and ν > 0,
µ∂I
∂z(z, µ, ν) = −χ(z, ν)I(z, µ, ν)
+
∫ +∞
0
σ(z, ν, ν ′)J(z, ν ′) dν ′ + E(z, ν)
Contribution of collisions
— Nov. 22 2004 – p.6
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The transfer equation
It describes the propagation of energy carried bya radiation:
For all z ∈]0, z? [, µ ∈] − 1, 1[ , and ν > 0,
µ∂I
∂z(z, µ, ν) = −χ(z, ν)I(z, µ, ν)
+
∫ +∞
0
σ(z, ν, ν ′)J(z, ν ′) dν ′ + E(z, ν).
Opacity = absorption + scattering = extinction >0
— Nov. 22 2004 – p.6
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The transfer equation
It describes the propagation of energy carried bya radiation:
For all z ∈]0, z? [, µ ∈] − 1, 1[ , and ν > 0,
µ∂I
∂z(z, µ, ν) = −χ(z, ν)I(z, µ, ν)
+
∫ +∞
0
σ(z, ν, ν ′)J(z, ν ′) dν ′ + E(z, ν).
Differential scattering coefficient.
— Nov. 22 2004 – p.6
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Connection with transport equation
Transport equation :
∂f
∂t(x, v, t)+v.∇f(x, v, t) = Coll(f, x, v, t)+E(x, v, t)
f : velocity distribution function of propagation particles of
energy1
2m|v|2.
Transfer equation : particles are photons of energy hν.
v → (ν, s := v/|v|)
f(x, s, ν, t) → (1/chν)I(x, s, ν, t)
aerosols[mass] → ?
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Changing of variable
τν(z) =
∫
z?
z
χν(z′) dz′ : optical depth
τν(0) =: τ ?ν
: optical thickness
The optical thickness represents the difficulty togo trough the medium:
Transition Lα : τ ?ν
= 2.146 × 1011
Transition Hα : τ ?
ν= 3.851 × 105
Continuum spectrum : τ ?ν
= 7.445
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Changing of variable
µ∂I
∂τ(τ, µ) = I(τ, µ) − $(τ)
∫ 1
−1
I(τ, µ) dµ − f(τ)
f(τ) =E(τ)
χ(τ): primary source function
$(τ) =σ(τ)
χ(τ): albedo : probability of scattering photons
Transition Lα : 0.99 ≤ $(τ) < 1
Transition Hα : 2 × 10−1 ≤ $(τ) < 1
Continuum spectrum : 2 × 10−4 ≤ $(τ) ≤ 1
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Boundary conditions
The specific intensity of the incident beam on theboundaries is known:
τ = τ ? τ = 0
interior vacuum~n
~k
θI+(µ)
I−(µ)
I(0, µ) = I−(µ) µ ∈ [−1, 0[
I(τ ?, µ) = I+(µ) µ ∈ ]0, 1]
— Nov. 22 2004 – p.9
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1. The transfer equation
2. Numerical resolution
3. Inverse problems
— Nov. 22 2004 – p.10
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Reduction of computation
Profile of matrix of the solved linear system
1
10
20
30
40
50
60
70
80
90
100
1 10 20 30 40 50 60 70 80 90 100
x : coefficient ofmodulus up than10−12 (more than50%)
Can us zeroing the small coefficients ?
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Iterative refinement
m = 1000
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Iterative refinement
Computation of anapproximation y(0)
on a coarse grid (n << m)
byone of the previous
approximationmethods
n = 20y(0)
m = 1000
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Iterative refinement
Computation of anapproximation y(0)
on a coarse grid (n << m)
byone of the previous
approximationmethods
n = 20y(0)
y(1)→y(2) � � � y(k) � � �
iterative refinementof y(0)
m = 1000
— Nov. 22 2004 – p.12
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Iterative refinement
Computation of anapproximation y(0)
on a coarse grid (n << m)
byone of the previous
approximationmethods
n = 20y(0)iterative refinement
of y(0)
y(1) (?)→ y(2)
(?)
� � � y(k)(?)
� � �
(?) Resolution of the samelinear system of rank n
m = 1000
— Nov. 22 2004 – p.12
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1. The transfer equation
2. Numerical resolution
3. Inverse problems
— Nov. 22 2004 – p.13
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Known and measured magnitudes
Momentums of I : Jn(τ) =
∫ 1
−1
µnI(τ, µ) dµ
Magnitude “solved star” (Sun) Other starsI(τ?, µ) known ∀µ known ∀µ
I(0, µ) measured for µ ≥ 0 unknownJ1(0) measured measuredJ1(τ?) known known
τ = τ? : interior the star - atmosphere boundary
τ = 0 : surface of the atmosphere - vaccuum boundary
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Inverse problems
First inverse problem:Get the function $ and the real τ? when
I(τ?, µ) and I(0, µ) are given for all µ;
f is given.
Second inverse problem:Get the function $ and the real τ? when
J1(0) is given;
I(τ?, µ) is given for all µ ∈ D
f is given.
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ReferencesS. Chandrasekhar, Radiative transfert, Oxford UP, 1960.
V. Kourganoff, Introduction a la Theorie Generale du
Transfer des Particules, Gordon and Breach, 1967.
H.C. Van de Hulst,Multiple light scattering, AcademicPress,1980.
D. Mihalas, Stellar atmospheres, Freeman and co, SanFrancisco, 1970, 1978.
M. Ahues, A. Largillier and B.V. Limaye, Spectral
Computations for Bounded Operators, Chapman and Hall/CRC,2001.
AHUES M., LARGILLIER A., TITAUD O., The roles of weaksingularity and the grid uniformity in relative error bounds,Numer. Funct. Anal. Optimz. 22(7-8) (2001), 789–814.
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ReferencesAhues M., D’Almeida F., Largillier A., Titaud O. andVasconcelos P., An L
1 refined projection approximate solutionof the radiation transfer equation in stellar atmospheres, J.
Comput. Appl. Math. 140(1-2) (2002), 13–26 (2002).
O. Titaud, Reduction of computation in the numerical resolutionof a second kind weakly singular Fredholm equation, Integral
Methods in Science and Engineering, 255–260, BirkhauserVerlag, 2004.
B. Rutily, Multiple scattering theory and integral equations,Integral Methods in Science and Engineering, Birkhauser,211-231, 2004.
L. Chevallier, Stellar atmosphere modelling, Integral Methods
in Science and Engineering, Birkhauser, 37-40, 2004.
— Nov. 22 2004 – p.16
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ReferencesN.J. McCormick, Inverse Radiativ Transfer Problems: a review,Nuclear Science and Engineering,112, 185–198 (1992).
G. Bal, Diffusion approximation of radiative transfer equations ina channel, Transport Theory Statist. Phys., 30(2-3), 269–293,(2001).
— Nov. 22 2004 – p.16