4. PCA of polarised spectra
We present in this section the two approaches we have followed to assess the applicability of PCA techniques to Stokes profiles, first using a set of synthetic data, and then a set of real data.
4.1. Synthetic Stokes data
We consider the formation of the Fe I 5250.25 Å spectral line which is a normal Zeeman triplet commonly used in solar magnetic field measurements. For simplicity we assume a static plane-parallel Milne-Eddington model atmosphere with source function
(where the coefficient two is arbitrary, not influencing the results of the method) and with the following fixed line parameters: opacity ratio ; Doppler width = 29.4 mÅ; and damping constant = 0.001. We computed four databases of Stokes parameter profiles (2210 for each of at 300 wavelengths in steps of 2 mÅ) emerging normally from the surface of this model for a range of magnetic field parameters:
Integration of the radiative transfer equations for the Stokes parameters, including anomalous dispersion, was done using the DIAGONAL code (Lopez Ariste and Semel, 1999).
Fig. 6 shows the first 5 eigenvalues for each Stokes parameter. It is evident that the first three or four eigenprofiles contain most of the information needed to reconstruct any of the synthetic Stokes profiles to high precision. In Fig. 5 we show the first 4 eigenprofiles for and V. Note that because the azimuth varies uniformly through the whole possible range , the eigenprofiles for U are identical to those of Q. The reconstruction errors (here the r.m.s. errors normalised to the total flux of each parameter) are shown in Fig. 7: using 3 eigenprofiles for I and V give errors of order 1-2%; using 4 eigenprofiles for Q and U gives somewhat larger errors, probably because of the wide variation (positive and negative) of these linear polarisation profiles across the database. Obviously the errors could be reduced by using extra eigenprofiles.
The eigenfeatures of the Stokes parameters depend on 3 parameters, which makes them difficult to visualise. To simplify matters we restrict our attention to the subset of eigenfeatures with fixed azimuth at . Fig. 8 shows first 3 eigenfeatures for and V. These are surfaces which are not so simple as in previous sections, but they are single valued, smooth functions of the model parameters B and , indicating once again that interpolation can be used to estimate eigenfeatures at intermediate parameter values.
Fig. 9 shows the corresponding (smooth) model manifolds in 3-dimensional eigenfeature space. The I-manifold is actually drawn twice as the inclination varies over its whole range. In other words, points on this surface have an ambiguity of . The same ambiguity applies to the Q-manifold. The V-manifold consists of two anti-symmetric surfaces joining at a singular point. This singular point corresponds to a transverse magnetic field where , for which .
In the analysis of real Stokes data one can imagine using all the eigenfeature vectors to obtain an unambiguous determination of magnetic field intensity, inclination and azimuth, along with other parameters controlling line formation such as opacity ratio, Doppler width, line damping, source function slope, filling factor, etc (cf. del Toro Iniesta & Ruiz Cobo 1996).
4.2. Observed Stokes data
Decomposition of each Stokes parameter profile into a combination of just 3 or 4 eigenprofiles is not restricted to synthetic data from a simple model atmosphere. It also applies to observations, as we now demonstrate. One might say that here we are using the most complicated model available , the solar atmosphere itself. The data were taken by the High Altitude Observatory Advanced Stokes Polarimeter (ASP, Elmore et al. 1992,) on 17th June 1992. This is a standard ASP scan of a sunspot region using a spectral window around the 6301.5 and 6302.5 Å FeI lines. We selected 2000 observation points with V signal larger than 10% of the continuum. PCA was applied to these data; for each of the Stokes parameters we used all 2000 observed profiles as training data. Note that no further treatment or calibration has been done on the profiles: we have not separated the solar lines from the telluric ones or normalised the continuum or corrected noise, for instance.
The normalised eigenvalues shown in Fig. 10 indicate that 4 eigenprofiles are more than enough to reconstruct the observed data. Figs. 11 present the first 6 eigenprofiles for each of the Stokes aparameters There are several points to notice. The first eigenprofile of I is clearly an inverted mean solar profile of this spectral region. The V eigenprofiles (especially the second) are high asymmetric. We have included the eigenprofiles for both Q and U. This was not necessary with synthetic data: in that case they were identical, reflecting a uniform distribution of azimuth values in the dataset. For the observed data this was not the case. The first Q and U eigenprofiles are identical, but there are differences already evident in the second eigenprofiles, and these differences are more significant for the following ones. From this simple observation one can obtain two immediate conclusions: first, the selected profiles cannot be completely explained by a simple model of the type used in the previous subsection; second, there is an inhomogeneous distribution of azimuth of magnetic field in those selected profiles (as homogeneity was the cause of identical eigenprofiles in the synthetic data). Explaining the exact meaning of this inhomogeneity requires further analysis and is outside the scope of this paper. What is interesting, however, is the ease with which such conclusions can be obtained by the use of PCA.
Finally, Fig. 12 shows the reconstruction errors (here the r.m.s. errors normalised to the continuum) for the 2000 profiles using just 4 eigenprofiles for each of the Stokes parameters. In this PCA reconstruction the information discarded is consistent with estimated noise levels of the ASP (Lites et al., 1994). A more detailed study of PCA-based noise reduction (Hansen et al., 1992; Hansen, 1992; Hansen and Prost O'Leary, 1993) of Stokes profiles will be presented in another paper.
© European Southern Observatory (ESO) 2000
Online publication: March 9, 2000