The solar chromospheric and coronal magnetic fields play a major role in the physics of the solar atmosphere. So far, these magnetic fields were determined by the extrapolation of observed photospheric magnetic fields upon assuming either (1) a potential, current-free model and using (only) the line of sight component or (ii) a force-free field (FFF) model (, ), and using all three components of the photospheric field. Various analytical or/and numerical methods for the determination of the magnetic field in half the space above the photosphere have been suggested or/and implemented. (See, e.g., the reviews by Sakurai 1989 and Gary 1990; also, the papers by Schmidt 1964; Semel 1967, 1988; Sturrock & Woodburg 1967; Molodensky 1969; Nakagawa & Raadu 1972; Nakagawa 1974; Levine 1975, 1976; Chiu & Hilton 1977; Seehafer 1978; Sakurai 1981, 1982; Alissandrakis 1981; Low 1982, 1985; Aly 1984, 1987, 1989, 1992; Schmal et al. 1982; Wu et al. 1985; Wu et al. 1990; Gary et al. 1987; Cuperman et al. 1989a, 1989b, 1990, 1991a, 1991b, 1993; Low & Lou 1991; Amari & Demoulin 1992; Faubert-Sholl et al. 1992; Bruma & Cuperman 1993).
Obviously, in both the potential and FFF cases, calculated chromospheric and coronal magnetic field configurations suffer from the rather limited information used for their computation: only at the bottom of the three-dimensional integration domain are boundary conditions (i.e., photospheric observations) used.
Referring to the more general FFF-case (also considered in this paper), the pertaining progressive vertical extrapolation method used in the past can be summarized as follows: (a) Starting from the vector magnetograph data at the photosphere (say, z=0), use the z-component of the FFF-equation in order to calculate the function ; (b) next, by the aid of the x- and y-components of the FFF equation, as well as of the divergence equation and the obtained at step (a), calculate the derivatives (i = x, y, z); (c) finally, utilizing a Taylor series expansion with the first two (or three)terms retained, obtain the values at a vertical (height) position , etc. Various aspects related to the ill-posed character (in the sense of Hadamard 1923) of the progressive vertical method for the extrapolation of the photospheric magnetic field have been addressed by several authors and will not be repeated here (see, e.g., Morse & Feshbach 1955; Tikhonov 1963; Antokhim 1968; Bakushinsky 1968; Kryanev 1973; Tikhonov & Arsenin 1977; Low 1982, 1985; Cuperman et al. 1990). Based on the fact that many ill-posed problems do occur in science and technology these authors recommend the development and use of suitable ``regularization" methods leading to satisfactory solutions.
Recently, it has been pointed out (Zhang 1993, 1994) that physical situations may exist in which, besides measurements of three-component photospheric magnetic fields, data on longitudinal (line of sight) chromospheric magnetic fields are also available; and that these two types of information can be combined in order to determine the full three-component chromospheric magnetic field. Thus, when NLTE effects are relatively small (e.g., in the case of quiet and plage regions), longitudinal chromospheric magnetograms at different wavelenghts in the blue wing of line are indicative of the longitudinal magnetic fields at different chromospheric altitudes. For example, using a set of eight magnetograms in the blue wind of the line at wavelengths , (monochromatic images of Stoke's parameter V of the line), and considering that the core of the line forms at about 1900 km in the solar atmosphere (Allen 1973) Zhang (1993) estimated the formation height of the wavelength to be about 1200 km. It is anticipated that in the near future, many more (40-50) multi-level longitudinal magnetic field measurements will be possible (Ai 1994a, 1994b; Fang 1994).
In this work we develop - within the FFF framework - a high order precision method for the reconstruction of chromospheric magnetic configurations. The method is based on the utilization of both types of existing magnetic field observations: (i) vector (three-component) photospheric measurements and (ii)longitudinal (along the line of sight) chromospheric measurements. The goal is to compute the full three-component magnetic field configuration at all points above active photospheric regions. A high order computational algorithm was developed and utilized for this purpose. As it will be shown, this algorithm enables one the reconstruction of the chromospheric vector magnetic field with a maximum relative error of .
The paper is organized as follows: Sect. 2 presents the general formulation of the problem, including a discussion of the observations, the basic equations utilized and analytical solutions of FFF model equations used as a test case; Sect. 3 describes the general computational algorithm developed and its implementation; Sect. 4 presents the results of the calculation demonstrating the high performances of the algorithm; Sect. 5 elaborates on the elimination/minimization of measurements errors as well as on the fitting of the corrected data to a FFF-state; A summary is given in Sect. 6.