a multiline lte inversion using pca
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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
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
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
searchinHow
do
we co
nstru
ct a
COMPLE
TE
data
bas
e???
This
is th
e ve
ry k
ey p
oint
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
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
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
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
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 ???2(Si
teor, Sjteor) < ; j ≠ i
YES NO
Add it to the data base
i=i+1
i=i+1Save irej
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 ???2(Si
teor, Sjteor) < ; j ≠ i
YES NO
Add it to the data base
i=i+1
i=i+1Save irej
We use thenoise level
as the reference
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 ???2(Si
teor, Sjteor) < ; j ≠ i
YES NO
Add it to the data base
i=i+1
i=i+1Save irej
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+1
How many do we need in order the base to be “complete” ??
Save irej
2(Siteor, Sj
teor) < ; j ≠ i
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+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:
Degeneracies in the parameters
= 10-3 Ic 1.56 m
~ 25 % of the proposedprofiles have been rejected.
Degeneracies in the parameters
The noise has made the B, f, parameters not to be.
For magnetic flux densities lower than ~8 Mx/cm2 the product of the threemagnitudes is the only observable.
Studying the data base:
Degeneracies in the parameters
= 10-4 Ic 1.56 m
~ 11 % of the proposedprofiles have been rejected.
Degeneracies in the parameters
The noise has made the B, f, parameters not to be.
For magnetic flux densities lower than ~4 Mx/cm2 the product of the threemagnitudes is the only observable.
Studying the data base:
Degeneracies in the parameters
= 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
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 tothe 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
Testing the inversions
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!!
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