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Astron. Astrophys. 355, 759-768 (2000)

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1. Introduction

In this paper we suggest a new approach to solving inverse problems in solar and stellar atmospheric physics using spectral line observations. The primary motivation for the work is the need to develop fast methods of analysis to handle the huge amount of spectral data that will be obtained by the French-Italian THEMIS solar telescope (Mein and Rayrole, 1985; Rayrole and Mein, 1993). The techniques described are also relevant to analysis of data from other solar instruments, eg. the Advanced Stokes Polarimeter (Elmore et al., 1992), and to the broad problem of stellar chromospheric modelling using high resolution spectra (eg. Mihalas 1978).

Our approach is similar in spirit to the work of Briand et al.(1996) and Molowny-Horas et al.(1999). Compute a database of spectral profiles using a large number of models. Inversion of an observed profile to obtain an atmospheric model is then equivalent to a problem in pattern recognition, ie. finding the nearest profile in the model profile database. Treating inversion as pattern recognition is an attractive alternative to the trial-and-error method often used in chromospheric modelling, and, in particular, the computationally intensive non-linear least squares fitting method used in Stokes parameter profile inversion (for a review see del Toro Iniesta & Ruiz Cobo 1996b). To make database search efficient it is important to reduce the dimensionality of the search space. For the CaII infrared triplet line at 8542 Å formed in a grid of solar chromospheric models, Briand et al. expressed the spectra as a linear combination of a small number of Fourier components (typically 6). In this example the database search was in a 6-dimensional space, rather than the higher dimensional space of the number of wavelengths sampled in the spectral line. They showed that one could interpolate in the Fourier coefficients to obtain estimates of the atmospheric parameters.

The classical statistical method of principal component analysis or PCA (eg. Murtagh & Heck 1987) is a preferable method of dimensionality reduction to Fourier decomposition. The reason for this is that PCA provides a more natural, signal specific, orthogonal decomposition of the spectral profile. PCA exploits the fact that spectra have similar profile shapes and magnitudes, and thus they tend to cluster tightly in the signal space. PCA is now widely used to analyse astrophysical spectra, eg. classification of stellar, galactic and QSO spectra (Ibata and Irwin, 1997; Lahav et al., 1996; Francis et al., 1992), automated galactic redshift determination (Glazebrook et al., 1998), and even spectral data calibration (Lopez Ariste et al., 1999). Typically in these applications observed spectra classified by human experts are used as training data.

The fundamental idea in this paper is to use synthetic spectra associated with known model parameters as training data. This method was used by Gulati et al. (1997) to estimate the effective temperatures of G-K dwarfs. Their training data set was a library of synthetic spectra based on Kurucz's (1992) line blanketed stellar atmospheric models. Using simple analytic models of photospheric and chromospheric line formation, Rees et al. (1998) showed that unpolarised spectra could be reconstructed to high accuracy with as few as two PCA components. Here we extend their work to more complex models, including multi-level line formation in non-Local Thermodynamic Equilibrium (non-LTE) and polarisation. Recently Rohen et al. (1999) applied PCA to synthetic galaxy spectra and discussed how the principal components could be used to estimate model parameters.

Inspiration for our approach to spectral line analysis, which we call PCA inversion , came from an unexpected non-astrophysical source, research on pattern recognition in computer vision. Can we estimate the pose (angle of orientation) of an object from an image of the object? This is an inverse problem: we must derive a continuous physical parameter (pose angle) from a high dimensional signal (the image). Murase & Nayar (1995) solved this problem by applying PCA to a training set of images of the object at known poses. This is analogous to applying PCA to a database of model spectra. Estimating the pose of the object from an image not in the database is analogous to estimating the atmospheric model parameters for an observed spectral line.

Our purpose here is to establish the principles of PCA inversion. Application to inversion of real data is described in a separate paper. The basic theory is presented in Sect. 2. In Sect. 3 we illustrate the main ideas using two examples of unpolarised spectra: intensity profiles computed with the classical analytic Milne-Eddington model (Sect. 3.1); and H[FORMULA] flux profiles formed in non-LTE in models of K-dwarf chromospheres (Sect. 3.2). In Sect. 4 we apply PCA to polarised spectra. First we consider synthetic data, the Stokes parameter profiles of a line formed in the presence of a magnetic field, again computed using the Milne-Eddington atmospheric model (Sect. 4.1). This is the model most often used for Stokes inversion by non-linear least squares. Then, in Sect. 4.2 we illustrate how PCA analysis of observational data can itself provide useful insights into the inverse problem, in the absence of detailed modelling. We use Stokes profiles observed with the High Altitude Observatory Advanced Stokes Polarimeter (Elmore et al., 1992). In the conclusion in Sect. 5 we summarise the main points, and speculate on the use of neural networks for real-time inversion.

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© European Southern Observatory (ESO) 2000

Online publication: March 9, 2000