A multiline LTE inversion using PCA
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A multiline LTE inversion using PCA. Marian Martínez González. E. 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.

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A multiline LTE inversion using PCA

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A multiline lte inversion using pca

A multiline LTE inversion using PCA

Marian Martínez González


A multiline lte inversion using pca

E

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.

We model the physical mechanisms that takeplace in theline formation.

We model the Sun as a set of parameters contained in what we call amodel atmosphere.


A multiline lte inversion using pca

E

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.

We model the physical mechanisms that takeplace in theline formation.

STOKES VECTOR

We model the Sun as a set of parameters contained in what we call amodel atmosphere.


A multiline lte inversion using pca

E

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.

We model the physical mechanisms that takeplace in theline formation.

STOKES VECTOR

We model the Sun as a set of parameters contained in what we call amodel atmosphere.


A multiline lte inversion using pca

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()


A multiline lte inversion using pca

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.

OUR PROBLEM OF INVERSION IS:

S = f()

 = finv(S)


A multiline lte inversion using pca

 = finv(S)

sol

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

ini ± 

Forward modelling

ini

NO

Steor

Merit function

Converged?

Sobs

YES


A multiline lte inversion using pca

  • 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!


A multiline lte inversion using pca

  • 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 PROFILES

  • Asensio 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.


A multiline lte inversion using pca

  • 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


A multiline lte inversion using pca

  • 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 multiline lte inversion using pca

PCA inversion algorithm

DATA

BASE

Steor↔

Principal

Components

Pi i=0,..,N

Each observed profile can

be represented in the base

of eigenvectors:

Sobs=iPi

We compute the projection

of each one of the observed

profiles in the eigenvectors:

iobs= Sobs · Pi ; i=0,..,n<<N

PCA allows compression!!

SVDC

Sobs

iteor = Steor· Pi

Compute the

2

search

in

Find the minimum

of the 2


A multiline lte inversion using pca

PCA inversion algorithm

DATA

BASE

Steor↔

Principal

Components

Pi i=0,..,N

Each observed profile can

be represented in the base

of eigenvectors:

Sobs=iPi

We compute the projection

of each one of the observed

profiles in the eigenvectors:

iobs= Sobs · Pi ; i=0,..,n<<N

PCA allows compression!!

SVDC

Sobs

How do we construct a COMPLETE

data base???

iteor = Steor· Pi

This is the very key point

Compute the

2

search

in

Find the minimum

of the 2


A multiline lte inversion using pca

PCA inversion algorithm

DATA

BASE

Steor↔

Principal

Components

Pi i=0,..,N

Each observed profile can

be represented in the base

of eigenvectors:

Sobs=iPi

We compute the projection

of each one of the observed

profiles in the eigenvectors:

iobs= Sobs · Pi ; i=0,..,n<<N

PCA allows compression!!

SVDC

Sobs

How do we construct a COMPLETE

data base???

iteor = Steor· Pi

This is the very key point

How do we compute the errors of the

retrieved parameters??

Are they coupled to the non-completeness

of the data base ??

Compute the

2

search

in

Find the minimum

of the 2


A multiline lte inversion using pca

SIR

Montecarlo generation of the profiles of the data base

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

Is there any other

similar profile in the

data base ???

i

Siteor

2(Siteor, Sjteor) < ; j ≠ i

i=i+1

YES

NO

Save irej

i=i+1

Add it to the

data base


A multiline lte inversion using pca

SIR

Montecarlo generation of the profiles of the data base

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

Is there any other

similar profile in the

data base ???

i

Siteor

2(Siteor, Sjteor) < ; j ≠ i

Which are these parameters??

i=i+1

YES

NO

Save irej

i=i+1

Add it to the

data base


A multiline lte inversion using pca

Modelling the solar atmosphere

13 independent variables

  • - 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.


A multiline lte inversion using pca

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


A multiline lte inversion using pca

SIR

Montecarlo generation of the profiles of the data base

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

Is there any other

similar profile in the

data base ???

i

Siteor

2(Siteor, Sjteor) < ; j ≠ i

i=i+1

YES

NO

Save irej

i=i+1

Add it to the

data base


A multiline lte inversion using pca

SIR

Montecarlo generation of the profiles of the data base

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

Is there any other

similar profile in the

data base ???

i

Siteor

2(Siteor, Sjteor) < ; j ≠ i

We use the

noise level

as the reference

i=i+1

YES

NO

Save irej

i=i+1

Add it to the

data base


A multiline lte inversion using pca

SIR

Montecarlo generation of the profiles of the data base

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

Is there any other

similar profile in the

data base ???

i

Siteor

2(Siteor, Sjteor) < ; j ≠ i

i=i+1

YES

NO

Save irej

i=i+1

Add it to the

data base


A multiline lte inversion using pca

SIR

Montecarlo generation of the profiles of the data base

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

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

Is there any other

similar profile in the

data base ???

i

Siteor

2(Siteor, Sjteor) < ; j ≠ i

i=i+1

YES

NO

Save irej

i=i+1

Add it to the

data base


A multiline lte inversion using pca

SIR

Montecarlo generation of the profiles of the data base

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

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

The data base will never be complete..

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

Is there any other

similar profile in the

data base ???

i

Siteor

2(Siteor, Sjteor) < ; j ≠ i

i=i+1

YES

NO

Save irej

i=i+1

Add it to the

data base


A multiline lte inversion using pca

Degeneracies in the parameters

Studying the data base:

Degeneracies in the parameters

 = 10-3 Ic

1.56 m

~ 25 % of the proposed

profiles have been rejected.

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

For magnetic flux densities lower than ~50 Mx/cm2 the product of the three

magnitudes is the only observable.


A multiline lte inversion using pca

Degeneracies in the parameters

Studying the data base:

Degeneracies in the parameters

 = 10-4 Ic

1.56 m

~ 11 % of the proposed

profiles have been rejected.

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

For magnetic flux densities lower than ~8 Mx/cm2 the product of the three

magnitudes is the only observable.


A multiline lte inversion using pca

Degeneracies in the parameters

Studying the data base:

Degeneracies in the parameters

 = 10-4 Ic

630 m +1.56 m

~ 0.7 % of the proposed

profiles have been rejected!!

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

For magnetic flux densities lower than ~4 Mx/cm2 the product of the three

magnitudes is the only observable.


A multiline lte inversion using pca

Testing the inversions

 = 10-3 Ic

1.56 m


A multiline lte inversion using pca

Testing the inversions

 = 10-3 Ic

1.56 m


A multiline lte inversion using pca

Testing the inversions

 = 10-3 Ic

1.56 m


A multiline lte inversion using pca

Testing the inversions

 = 10-3 Ic

1.56 m


A multiline lte inversion using pca

Testing the inversions

 = 10-3 Ic

1.56 m


A multiline lte inversion using pca

Testing the inversions

 = 10-3 Ic

1.56 m


A multiline lte inversion using pca

Testing the inversions

 = 10-3 Ic

1.56 m

The errors are high but close to

the supposed error of the data base


A multiline lte inversion using pca

Testing the inversions

 = 10-4 Ic

630 nm + 1.56 m


A multiline lte inversion using pca

Testing the inversions

 = 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!!


A multiline lte inversion using pca

- 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.


A multiline lte inversion using pca

THANK YOU!!


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