## 4. PCA of polarised spectraWe 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 dataWe 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: -
Intensity G. -
Azimuth . -
Inclination to line of sight .
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
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
Fig. 9 shows the corresponding (smooth) model manifolds in
3-dimensional eigenfeature space. The
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 dataDecomposition 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 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
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 |