Direct and inverse radiativetransfer problems

How to go from stellar atmosphere modellingto aerosols problems ?

Olivier Titaud

titaud@dim.uchile.cl

Postdoctorant

Centro de Modelmatiento Matematico

Universidad de Chile

Doctor in Numerical Analysis

Universite of Saint Etienne, France

— Nov. 22 2004 – p.1

1. The transfer equation

2. Numerical resolution of the di-

rect problem

3. Inverse problems

— Nov. 22 2004 – p.2

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

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.

— Nov. 22 2004 – p.4

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

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

— Nov. 22 2004 – p.6

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

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

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

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

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] → ?

— Nov. 22 2004 – p.7

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

— Nov. 22 2004 – p.8

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

— Nov. 22 2004 – p.8

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

1. The transfer equation

2. Numerical resolution

3. Inverse problems

— Nov. 22 2004 – p.10

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 ?

— Nov. 22 2004 – p.11

Iterative refinement

m = 1000

— Nov. 22 2004 – p.12

Iterative refinement

Computation of anapproximation y(0)

on a coarse grid (n << m)

byone of the previous

approximationmethods

n = 20y(0)

m = 1000

— Nov. 22 2004 – p.12

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

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

1. The transfer equation

2. Numerical resolution

3. Inverse problems

— Nov. 22 2004 – p.13

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

— Nov. 22 2004 – p.14

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.

— Nov. 22 2004 – p.15

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.

— Nov. 22 2004 – p.16

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

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