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
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.
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.
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.
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.
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.
OUR PROBLEM OF INVERSION IS:
S = f()
= finv(S)
= 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
But, at its present state...
We propose a PCA inversion code based on the SIR performance
But, at its present state...
We propose a PCA inversion code based on the SIR performance
More work has to be done... and I hope to receive some suggestions!!
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
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
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
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
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
Modelling the solar atmosphere
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
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
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
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
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
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
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.
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.
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.
Testing the inversions
= 10-3 Ic
1.56 m
Testing the inversions
= 10-3 Ic
1.56 m
Testing the inversions
= 10-3 Ic
1.56 m
Testing the inversions
= 10-3 Ic
1.56 m
Testing the inversions
= 10-3 Ic
1.56 m
Testing the inversions
= 10-3 Ic
1.56 m
Testing the inversions
= 10-3 Ic
1.56 m
The errors are high but close to
the supposed error of the data base
Testing the inversions
= 10-4 Ic
630 nm + 1.56 m
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!!
- 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!!