1. A multiline LTE inversion using PCA Marian Martínez González

2. 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.

3. 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

4. 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

5. 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(  )

6. 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:  = f inv (S)

7. 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) S obs  ini Forward modelling S teor S obs Merit functionConverged? YES NO  sol  ini ± 

8. 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!

9. 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.

10. 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.

11. 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.

12. PCA inversion algorithm DATA BASE S teor ↔  Principal Components P i i=0,..,N SVDC S obs Each observed profile can be represented in the base of eigenvectors: S obs =  i P i We compute the projection of each one of the observed profiles in the eigenvectors:  i obs = S obs · P i ; i=0,..,n<<N PCA allows compression!!  i teor = S teor · P i Compute the  2 Find the minimum of the  2 search in

13. PCA inversion algorithm DATA BASE S teor ↔  Principal Components P i i=0,..,N SVDC S obs Each observed profile can be represented in the base of eigenvectors: S obs =  i P i We compute the projection of each one of the observed profiles in the eigenvectors:  i obs = S obs · P i ; i=0,..,n<<N PCA allows compression!!  i teor = S teor · P i Compute the  2 Find the minimum of the  2 search in How do we construct a COMPLETE data base??? This is the very key point

14. PCA inversion algorithm DATA BASE S teor ↔  Principal Components P i i=0,..,N SVDC S obs Each observed profile can be represented in the base of eigenvectors: S obs =  i P i We compute the projection of each one of the observed profiles in the eigenvectors:  i obs = S obs · P i ; i=0,..,n<<N PCA allows compression!!  i teor = S teor · P i Compute the  2 Find the minimum of the  2 search in How do we construct a COMPLETE data base??? 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 ??

15. Montecarlo generation of the profiles of the data base ii i=0,.... ?? from a random uniform distribution S i teor SIR Is there any other similar profile in the data base ??? YES NO Add it to the data base i=i+1 Save  i rej  2 (S i teor, S j teor ) <  ; j ≠ i

16. Montecarlo generation of the profiles of the data base ii i=0,.... ?? from a random uniform distribution S i teor SIR Is there any other similar profile in the data base ??? YES NO Add it to the data base i=i+1 Save  i rej Which are these parameters??  2 (S i teor, S j teor ) <  ; j ≠ i

17. 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

18. 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

19. Montecarlo generation of the profiles of the data base ii i=0,.... ?? from a random uniform distribution S i teor SIR Is there any other similar profile in the data base ???  2 (S i teor, S j teor ) <  ; j ≠ i YES NO Add it to the data base i=i+1 Save  i rej

20. Montecarlo generation of the profiles of the data base ii i=0,.... ?? from a random uniform distribution S i teor SIR Is there any other similar profile in the data base ???  2 (S i teor, S j teor ) <  ; j ≠ i YES NO Add it to the data base i=i+1 Save  i rej We use the noise level as the reference

21. Montecarlo generation of the profiles of the data base ii i=0,.... ?? from a random uniform distribution S i teor SIR Is there any other similar profile in the data base ???  2 (S i teor, S j teor ) <  ; j ≠ i YES NO Add it to the data base i=i+1 Save  i rej

22. Montecarlo generation of the profiles of the data base ii i=0,.... ?? from a random uniform distribution S i teor SIR Is there any other similar profile in the data base ??? YES NO Add it to the data base i=i+1 How many do we need in order the base to be “complete” ?? Save  i rej  2 (S i teor, S j teor ) <  ; j ≠ i

23. Montecarlo generation of the profiles of the data base ii i=0,.... ?? from a random uniform distribution S i teor SIR Is there any other similar profile in the data base ??? YES NO Add it to the data base i=i+1 How many do we need in order the base to be “complete” ?? Save  i rej  2 (S i teor, S j teor ) <  ; j ≠ i The data base will never be complete.. We have created a data base with ~65000 Stokes vectors.

24. Studying the data base: Degeneracies in the parameters  = 10 -3 I c 1.56  m

25. Degeneracies in the parameters The noise has made the B, f,  parameters not to be. For magnetic flux densities lower than ~50 Mx/cm 2 the product of the three magnitudes is the only observable. Studying the data base: Degeneracies in the parameters  = 10 -3 I c 1.56  m ~ 25 % of the proposed profiles have been rejected.

26. Studying the data base: Degeneracies in the parameters  = 10 -4 I c 1.56  m

27. Degeneracies in the parameters The noise has made the B, f,  parameters not to be. For magnetic flux densities lower than ~8 Mx/cm 2 the product of the three magnitudes is the only observable. Studying the data base: Degeneracies in the parameters  = 10 -4 I c 1.56  m ~ 11 % of the proposed profiles have been rejected.

28. Degeneracies in the parameters The noise has made the B, f,  parameters not to be. For magnetic flux densities lower than ~4 Mx/cm 2 the product of the three magnitudes is the only observable. Studying the data base: Degeneracies in the parameters  = 10 -4 I c 630 m +1.56  m ~ 0.7 % of the proposed profiles have been rejected!!

29. Testing the inversions  = 10 -3 I c 1.56  m

30.  = 10 -3 I c 1.56  m Testing the inversions

31.  = 10 -3 I c 1.56  m Testing the inversions

32.  = 10 -3 I c 1.56  m Testing the inversions

33.  = 10 -3 I c 1.56  m Testing the inversions

34.  = 10 -3 I c 1.56  m Testing the inversions

35.  = 10 -3 I c 1.56  m The errors are high but close to the supposed error of the data base Testing the inversions

36.  = 10 -4 I c 630 nm + 1.56  m Testing the inversions

37.  = 10 -4 I c 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!!

38. - The inversions should work for two spectral lines with ~10 5 profiles in the data base for a polarimetric accuracy of 10 -3 -10 -4 I c. - 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.

39. THANK YOU!!