## 3. PCA of unpolarised spectraIn this section we illustrate these ideas using unpolarised intensity and flux profiles formed in models of increasing complexity. ## 3.1. Intensity profilesConsider a spectral line formed in static Milne-Eddington model of a plane-parallel atmosphere with source function equal to the Planck function written as a linear function of , the continuum optical depth, as The radiative transfer equation can be solved analytically in this case to obtain the emergent intensity normal to the surface, where First we fix the line broadening parameter at , and generate a database of 19 profiles using the opacity ratio values , i.e. from 1 to 10 in steps of 0.5. The profiles computed at 61 wavelengths are shown in Fig. 2a. Only the first two PCA eigenvalues in Fig. 2b are significant; note that in this and later figures we plot the normalised eigenvalues . Thus the profiles can be reconstructed to high accuracy using the mean profile and the two eigenprofiles in Fig. 2c. The absolute percentage errors between the reconstructed and the original data for all wavelengths are plotted in Fig. 2d. Note that the eigenfeatures in Fig. 2e are smooth functions of . Clearly eigenfeatures at parameter values not used originally can be estimated accurately by interpolation. This is a recurring theme in this paper. As shown in Fig. 2f the model manifold in two-dimensional eigenfeature space is a smooth curve parameterised by .
Now we allow both parameters to vary, generating a database of 209 profiles with and for 81 wavelengths . Profiles for and 10 are shown in Fig. 3a. Eigenvalues and eigenprofiles are shown in Fig. 3b and Fig. 3c respectively. The reconstruction error (Fig. 3d) using only 5 eigenprofiles is less than 2% for the entire database. As we cannot visualise the model manifold in 5 dimensions, we consider instead its projection on the 3-dimensional space spanned by the first 3 eigenprofiles. The first 3 eigenfeatures shown in Figs. 3d-f are again smooth functions of the model parameters, and the model manifold in Fig. 3g is a smooth surface.
## 3.2. H flux profilesThe efficacy of PCA analysis is not restricted to such simple
models. To emphasise this fact we consider the strong line
H formed in non-LTE in the
chromosphere of a cool star. The stellar atmosphere model adopted here
is typical of that for a K0 dwarf star with log g = 4.478 (eg. Kelch
1978, Simon et al. 1980), and follows the solar paradigm, having a
temperature distribution consisting of a negative photospheric
temperature gradient with height, a temperature minimum region, a
gradual positive gradient (the chromospheric temperature rise),
followed by a temperature plateau and a steep transition zone. We use
a model with only two parameters: The H line exhibits strong
variations. As a photo-ionisation controlled line in quiet
atmospheres, the line is in absorption, but as atmospheric parameters
are changed, source function control moves to collisional, resulting
in core filling and wing emissions which are readily modelled. It is
well known that varying the parameter For our computations, we used the non-LTE computer code "Multi" (Carlsson, 1986) to solve the coupled statistical equilibrium and radiative transfer equations in a semi-infinite, plane parallel atmosphere under the constraint of hydrostatic equilibrium. The adopted hydrogen atom is the five bound level plus continuum model of Lites et al. (1987). All bound-bound transitions and the Lyman, Balmer, Paschen and Brackett continua are calculated in detail. Scattering in frequency is modelled by complete redistribution which is a good approximation for H (Thomas, 1957; Shine et al., 1975). To approximate the strong partial redistribution effects in the hydrogen Lyman lines, we restricted the radiative transfer calculations in these lines to the Doppler core (Milkey and Mihalas, 1973; Lites et al., 1987). This has a significant effect on the results, causing as much as a 5% variation in H and H core depths and a factor of 10 change in the departure coefficients when compared with results where the transfer calculations are not restricted to the core. All computations and atomic models are discussed in more detail in Thatcher (1994) and Thatcher et al. (1991). Flux profiles normalised to the background continuum for 81
wavelengths -4(0.1)4 Å measured from
H line centre were used in the PCA
analysis. Examples are shown in Fig. 4a for
and 8500, and all values of
© European Southern Observatory (ESO) 2000 Online publication: March 9, 2000 |