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Astron. Astrophys. 345, 618-628 (1999)

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1. Introduction

Solar spectral inversion codes, which allow to recover information about the solar atmospheric plasma from line spectra, both in unpolarized and polarized light, have in the last years been designed and tested (Skumanich & Lites 1987; Keller et al. 1990; Ruiz Cobo & del Toro Iniesta 1992; Bellot Rubio et al. 1997; Sánchez Almeida 1997; López Ariste & Semel 1999; for an up-to-date review, see del Toro Iniesta & Ruiz Cobo 1996). Their application to solar data has permitted a better understanding of plasma conditions at photospheric levels.

However, although fast and reliable, all these inversion codes share the same particularity, namely the assumption that lines are formed under conditions of local thermodynamic equilibrium (LTE). On the contrary, in the upper photosphere and above, where LTE is no longer valid, one must consistently assume non-LTE, where the source function decouples completely from the Planck function, further complicating the radiative transfer problem.

Only in the last few years has the full non-LTE inversion problem begun to be tackled. Lites et al. (1988) derived an analytic solution to the transfer equation for polarized light in non-LTE, assuming an exponential form for the dependence of the source function on line center optical depth, and a depth-independent absorption matrix. Ruiz Cobo et al. (1997) introduced fixed non-LTE departure coefficients in order to invert KI 769.9 nm line spectra. Recently, Socas-Navarro et al. (1998) have developed a non-LTE inversion technique for strong chromospheric lines, based on the so-called "response functions", already used in LTE inversions, that allows for the recovery of the parameters describing the chromosphere. Response functions were first introduced by Mein (1971), who called them "weighting functions" and employed them to study the influence of temperature and velocity disturbances on line profiles.

A different idea was proposed by Mein et al. (1987). In their analysis, the observed profiles were decomposed into their Fourier components, and changes in the respective coefficients were connected with variations in the physical parameters of theoretical profiles. This method was applied to a time series of observed CaII 396.8 nm line spectra. Along the same line of work, Briand et al. (1996), who proposed the use of a grid of models to match directly observed profiles with theoretical ones, carried out non-LTE inversions of CaII 854.2 nm line spectra, in which the chromosphere was characterized by two temperatures corresponding to the bottom and the top of the atmosphere.

Although the implementation of inversion codes both in LTE and non-LTE has only lately become a very active area of research, synthetic non-LTE radiative transfer codes have on the other hand been available for a long time. Among them, and relevant to the present work, is the new multilevel non-LTE transfer code for isolated atmospheric structures implemented by Heinzel (1995), which is based on the multilevel accelerated lambda iteration (MALI) technique (Rybicki & Hummer 1991; Rybicki & Hummer 1992). This radiative transfer code is capable of treating the complex non-LTE hydrogen excitation and ionization equilibrium in a robust and accurate manner.

We are particularly interested in the formation of the H[FORMULA] Balmer line in chromospheric features, so-called chromospheric clouds, in which the observed emission is composed, firstly, of an H[FORMULA] background profile which has been partially absorbed in its passage through the cloud, and secondly, of the H[FORMULA] radiation emitted by the cloud itself. Traditionally, these absorption features have been modelled assuming that the cloud is optically thin enough such that the source function can be considered constant (Cram 1985). That assumption enables us to easily solve the radiative transfer equation (see Sect. 2 below), and thus we end up with a simple equation which describes the profile of the line as a function of optical depth at line center, Doppler shift, Doppler width and a constant source function (Beckers 1964; see also below). The success of this simple approach has led in the past to a better understanding of the physics of chromospheric mottles (Beckers 1964; Grossmann-Doerth & von Uexküll 1971; Tsiropoula et al. 1993), arch filament systems (Alissandrakis et al. 1990; Mein P. et al. 1996) and filaments (Schmieder et al. 1991; Mein et al. 1994).

Concerns about the validity of this approach have arisen, however, since the simple cloud model has failed to explain the H[FORMULA] profile of some dark features. This was already noticed early on by Cram (1975) and others (see references therein), and more complete model calculations were called for. Seeking to improve the cloud model, Mein & Mein (1988) introduced the concept of a "differential" cloud model where either the background intensity or the velocity gradient along the line of sight could vary. Later on, and based on results from an early version of Heinzel's code which showed that the source function inside the cloud has a quasi-parabolic dependence on the optical depth, Mein N. et al. (1996) presented an inversion strategy for cloud-like features which improved on earlier results. It was demonstrated that the velocity, optical thickness and Doppler width could be well recovered. More recently, Heinzel et al. (1999) studied the influence of non-zero cloud velocities on the magnitude and shape of the source function. They also suggested an iterative procedure for a fast determination of the true cloud velocities.

Although these studies are leading to better empirical inversions, it seems anyhow natural to attack the inversion problem by implementing a valid strategy which takes full advantage of the theoretical results provided by the MALI code. The approach that has been chosen for the present work comprises the construction of a grid of models which are then compared with the observations. Such an algorithm will be presented below, and its main characteristics will also be discussed.

The paper is organized in 6 sections. After this introduction, a short description of the non-LTE code is given. Then, a description of the grid, and of the numerical inversion procedure itself, are outlined in detailed. This is followed by an example application to real data. Finally, a discussion section and the conclusions will close the paper.

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© European Southern Observatory (ESO) 1999

Online publication: April 19, 1999