A multiline LTE inversion using PCA Marian Martínez González.

A multiline LTE inversion using PCA

Marian Martínez González

In astrophysics we can not directly measure the physical properties of the objects, we do always retrieve them.

We are always dealing with inversion problems.

E

We model the Sun as a set of parameters

contained in what we call a model atmosphere.

We model the physical mechanisms that take place

in the line formation.

In astrophysics we can not directly measure the physical properties of the objects, we do always retrieve them.

We are always dealing with inversion problems.

E

We model the Sun as a set of parameters

contained in what we call a model atmosphere.

We model the physical mechanisms that take place

in the line formation. STOKES VECTOR

Model atmosphere: - Temperature (pressure, density) profile along the optical depth.

- Bulk velocity profile.

- Magnetic field vector variation with depth.

- Microturbulent velocity profile.

- Macroturbulent velocity.

Let’s define the vector containing all the variables:

= [T,v,vmic,vmac,B,...]

Mechanism of line formation Local Thermodynamic Equilibrium.

Population of the atomic levels Saha-Boltzmann

Energy transport The radiative transport is the most efficient.

Radiative transfer equation.

S = f()

Model atmosphere: - Temperature (pressure, density) profile along the optical depth.

- Bulk velocity profile.

- Magnetic field vector variation with depth.

- Microturbulent velocity profile.

- Macroturbulent velocity.

Let’s define the vector containing all the variables:

= [T,v,vmic,vmac,B,...]

Mechanism of line formation Local Thermodynamic Equilibrium.

Population of the atomic levels Saha.

Energy transport The radiative transport is the most efficient.

Radiative transfer equation.

S = f()OUR PROBLEM OF INVERSION IS:

= finv(S)

= finv(S)

The information of the atmospheric parameters is encoded in the Stokes

profiles in a non-linear way.

Iterative methods (find the maximal of a given merit function)

Sobs

iniForward modelling

Steor

Sobs

Merit function Converged?

YES

NO

sol

ini ±

The noise in the observational profiles induce that:

-Several maximals with similar amplitudes are possible in the merit

function.

This introduces degeneracies in the parameters.

We are not able to detect these errors!

The noise in the observational profiles induce that:

-Several maximals with similar amplitudes are possible in the merit

function.

This introduces degeneracies in the parameters.

We are not able to detect these errors!

BAYESIAN INVERSION OF STOKES PROFILESAsensio Ramos et al. 2007, A&A, in press

- Samples the Likelihood but is very slow.

PCA INVERSION BASED ON THE MILNE-EDDINGTON APPROX.López Ariste, A.

- Finds the global minima of a 2 but the number of parameters increases in a multiline

analysis.

We propose a PCA inversion code based on the SIR performance

- A single model atmosphere is needed to reproduce as spectral lines

as wanted.

- Does not get stuck in local minima.

- We can give statistically significative errors.

- It is faster than the SIR code.

- It can be a very good initialitation for the SIR code.

But, at its present state...

- It is limited to a given inversion scheme (namely, the number of

nodes)

- The data base seems to be not complete enough.

We propose a PCA inversion code based on the SIR performance

More work has to be done... and I hope to receive some suggestions!!

- A single model atmosphere is needed to reproduce as spectral lines

as wanted.

- Does not get stuck in local minima.

- We can give statistically significative errors.

- It is faster than the SIR code.

- It can be a very good initialitation for the SIR code.

But, at its present state...

- It is limited to a given inversion scheme (namely, the number of

nodes)

- The data base seems to be not complete enough.

PCA inversion algorithm

DATABASE

Steor ↔

Principal Components

Pi i=0,..,NSVDC

Sobs

Each observed profile canbe represented in the base of eigenvectors:

Sobs=iPi

We compute the projectionof each one of the observedprofiles in the eigenvectors:

iobs

= Sobs · Pi ; i=0,..,n<<N

PCA allows compression!!

iteor = Steor · Pi

Compute the 2

Find the minimumof the 2

searchin


DATABASE

Steor ↔


Pi i=0,..,NSVDC

Sobs


Sobs=iPi


iobs

= Sobs · Pi ; i=0,..,n<<N


iteor = Steor · Pi

Compute the 2


searchinHow

do

we co

nstru

ct a

COMPLE

TE

data

bas

e???

This

is th

e ve

ry k

ey p

oint


DATABASE

Steor ↔


Pi i=0,..,NSVDC

Sobs


Sobs=iPi


iobs

= Sobs · Pi ; i=0,..,n<<N


iteor = Steor · Pi

Compute the 2


searchinHow

do

we co

nstru

ct a

COMPLE

TE

data

bas

e???

This

is th

e ve

ry k

ey p

oint

How do we compute the errors of the

retrieved parameters??

Are they coupled to the non-completeness

of the data base ??

Montecarlo generation of the profiles of the data base

i

i=0,.... ?? from a random uniform distribution

SiteorSIR

Is there any othersimilar profile in the

data base ???

YES NO

Add it to the data base

i=i+1

i=i+1Save irej

2(Siteor, Sj

teor) < ; j ≠ i


i


SiteorSIR


data base ???

YES NO


i=i+1

i=i+1Save irej

Which are these parameters??

2(Siteor, Sj

teor) < ; j ≠ i

Modelling the solar atmosphere

- A field free atmosphere (occupying a fraction 1-f):

Temperature: 2 nodes linear perturbations.

Bulk velocity: constant with height.

Microturbulent velocity: constant with height.

- A magnetic atmosphere (f):

Temperature: 2 nodes.

Bulk velocity: constant.

Microturbulent velocity: constant.

Magnetic field strength: constant.

Inclination of the field vector with respect to the LOS: constant.

Azimuth of the field vector: constant.

- A single macroturbulent velocity has been used to convolve the

Stokes vector.

13 independent variables

Synthesis of spectral lines

The idea is to perform the synthesis as many lines as are considered of interest to study the solar atmosphere.

In order to make the numerical tests we use the following ones:

Fe I lines at 630 nm

Fe I lines at 1.56 m

Spectral synthesis We use the SIR code.Ruiz Cobo, B. et al. 1992, ApJ, 398, 375

Reference model atmosphere HSRA (semiempirical)

Gingerich, O. et al. 1971, SoPh, 18, 347


i


SiteorSIR


data base ???2(Si

teor, Sjteor) < ; j ≠ i

YES NO


i=i+1

i=i+1Save irej


i


SiteorSIR


data base ???2(Si


YES NO


i=i+1

i=i+1Save irej

We use thenoise level

as the reference


i


SiteorSIR


data base ???2(Si


YES NO


i=i+1

i=i+1Save irej


i


SiteorSIR


data base ???

YES NO


i=i+1

i=i+1

How many do we need in order the base to be “complete” ??

Save irej

2(Siteor, Sj

teor) < ; j ≠ i


i


SiteorSIR


data base ???

YES NO


i=i+1

i=i+1

How many do we need in order the base to be “complete” ??

Save irej

2(Siteor, Sj

teor) < ; j ≠ iThe data base will never be complete..

We have created a data base with ~65000 Stokes vectors.

Degeneracies in the parameters

The noise has made the B, f, parameters not to be.

For magnetic flux densities lower than ~50 Mx/cm2 the product of the threemagnitudes is the only observable.

Studying the data base:


= 10-3 Ic 1.56 m

~ 25 % of the proposedprofiles have been rejected.






= 10-4 Ic 1.56 m

~ 11 % of the proposedprofiles have been rejected.






= 10-4 Ic 630 m +1.56 m

~ 0.7 % of the proposedprofiles have been rejected!!

Testing the inversions

= 10-3 Ic 1.56 m

= 10-3 Ic 1.56 m


= 10-3 Ic 1.56 m

The errors are high but close tothe supposed error of the data base


= 10-4 Ic 630 nm + 1.56 m


= 10-4 Ic 630 nm + 1.56 m


Apart from some nice fits, it is

impossible to retrieve any of the

parameters with a data base of 65000

profiles!!

- The inversions should work for two spectral lines with ~105

profiles in the data base for a polarimetric accuracy of

10-3-10-4 Ic.

- The inversion of a lot of spectral lines proves to be very

complicated using PCA inversion techniques.

- IT IS MANDATORY TO REDUCE THE NUMBER OF

PARAMETERS.

- The model atmospheres would be represented by some other

parameters that are not physical quantities (we would not

depend on the distribution of nodes) but that reduce the

dimensionality of the problem and correctly describes it.

THANK YOU!!

A multiline LTE inversion using PCA Marian Martínez González.

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