The reliable determination of magnetic fields and the thermodynamic structures of the solar atmosphere is still crucial for our understanding of many complex processes in and outside magnetic active regions. The inference of atmospheric parameters from Stokes profiles is a nonlinear inverse problem. Typically, those problems are solved by linearizing an appropriate forward model, computing the sensitivities and then iteratively solving a regularized optimization problem. In this sense most of the inversion procedures for Stokes profiles are based on a non-linear least-square minimization (i.e. Levenberg-Marquardt algorithm) (Auer et al. 1977; Landolfi et al. 1984; Skumanich & Lites 1987). All the latter are using a simplified forward model (Milne-Eddington model) in order to yield an analytical solution for the fast evaluation of the requiered derivatives. The most advanced Stokes profile inversion so far was introduced by Ruiz Cobo & del Toro Iniesta (1992). This inversion code is based on response functions and does not rely on the analytical Milne-Eddington solution and therefore it is able to retrieve height dependent information within a reasonable time.
But the improving techniques in spectro-polarimetry and the improving spatial resolution of especially future projects like GREGOR (von der Luehe et al. 1999), SOLIS (Keller 1998) or SolarB (Lites 2000) provide us with huge amounts of polarimetric data. This emphasizes the need for a fast and stable automated analysis and interpretation of Stokes profiles. In this sense the new inversion method described by Socas-Navarro et al. (2001) which operates in the low-dimensional eigenfeauture space, produced by a principle component analysis (PCA) of Stokes profiles, bypass the iterative behaviour and seems to be a very promising alternative and supplement to existing inversion methods. In this paper we introduce another alternative to an iterative solution, a direct inversion, based on the approximation of the nonlinear inverse mapping between the Stokes parameter and some of the underlying atmospheric parameters. The inversion based on this approximate inverse mapping proved to be stable, accurate and extremely fast.
Artificial neural networks were already succesfully applied in many different astrophysical fields such as the classification of stellar spectra (Bailor-Jones et al. 1998; Weaver & Torres-Dodgen 1997; Gulati et al. 1994), the retrieval of stellar parameters from low resolution spectra (Bailor-Jones 2000), the time series analysis of solar active regions (Calvo et al. 1995) or the sunspot index prediction (Fessant et al. 1996).
We have used one of the most popular types of ANNs, the Multi-Layer Perceptron (MLP),
which is a general model for approximating nonlinear multivariate functions. To adapt
(i.e. train) our MLP to the inverse problem we take advantage of the fact that we have
a good knowledge of the forward problem (i.e. the polarized radiative transfer) in
the case of magnetic sensitive absorption lines formed under the conditions of local
thermodynamical equilibrium (LTE). Thus we are able to generate a sufficient large database
of synthetic Stokes profiles for the training process of the MLP.
We have applied this model in the following work to simulated observation of the
infrared line (IR) Fe I 
15648.
Once the MLP is trained and has found a good approximation of the
inverse mapping it can recover the complete magnetic field vector, the line of sight
velocity, the microturbulence, the macroturbulence and the filling factor as well as an
estimate of the temperature stratification with exceptional speed.
This paper is organized as follows: in Sect. 2 we describe the general structure and properties of a Multi-Layer Perceptron. In Sect. 3 we investigate the capabilities of the MLP to retrieve some of the temperature information encoded in the Stokes profiles. We consider the problem if an MLP can recognize a specific signature in a given Stokes profile to distinguish between the temperature structure of different semi-empirical model atmospheres. Extensive statistical tests are made to evaluate the performance of the MLP. Section 4 deals with the inference of various atmospherical parameters from the Stokes parameters. The training of the MLPs to approximate the inverse relation and the statistical tests of the trained MLPs are described. Because our estimates are single valued we consider, in Sect. 5, the influence of magnetic field strength and LOS velocity gradients on the MLP calculations. To assess the results we use the concept of heights of formation (HOFs) defined by Sanchez Almeida et al. (1996), to demonstrate that the calculations of the MLP do represent very good averaged values of the stratified atmosphere. In Sect. 6 we employed the MLP model to retrieve estimates for the filling or stray light factor. We simulated a simple two component atmosphere to demonstrate that a trained MLP can yield reliable estimates about the fraction of the magnetic component. In Sect. 7 we give a short comment on the remarkable speed of the inversion. Finally, in Sect. 8, we summerize the main conclusions of this work and give an outlook to some future applications.
Copyright ESO 2001