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6. Summary

We have developed and implemented a computational algorithm for the reconstruction of the vector - chromospheric magnetic fields within the framework of the non-linear FFF theoretical model. This algorithm represents the extension, modification and improvement of previously proposed progressive vertical extrapolations (see, e.g. Wu et al. 1985; Cuperman et al. 1990) in the following sense:

1. In the calculation of the non-linear FFF-function tex2html_wrap_inline2863 (by the aid of Eq. (6 (click here))), the derivatives tex2html_wrap_inline2865 and tex2html_wrap_inline2867 are computed by the aid of 14-term formulas, rather than by 2-3 term formulas as previously used. Moreover, 14-versions of the 14-term formulas are used at various points, in order to ensure about equal minimum computational error. The maximum relative error in the computation of these derivatives is smaller than tex2html_wrap_inline2869. As a consequence, tex2html_wrap_inline2871 is computed with a maximum relative error smaller than tex2html_wrap_inline2873 (see Fig. 3 (click here)).

2. The same 14-term formula (and its 14 versions) is applied for the calculation of the derivatives tex2html_wrap_inline2875 and tex2html_wrap_inline2877 which enter the expressions of tex2html_wrap_inline2879 and tex2html_wrap_inline2881, Eqs. (4 (click here)) and (5 (click here)), respectively. Consequently, the same high computational precision is obtained also for this purpose.

3. At points where tex2html_wrap_inline2883, to avoid inherent ``mathematical" discontinuities, rather than calculating tex2html_wrap_inline2885 by the aid of Eq. (6 (click here)) in conjunction with some smoothing techniques, a suitable, different approach is used. The same holds at points where tex2html_wrap_inline2887 or tex2html_wrap_inline2889, including the case in which tex2html_wrap_inline2891. Thus, at all points in the range tex2html_wrap_inline2893, the maximum relative error in the computed function tex2html_wrap_inline2895 is smaller than tex2html_wrap_inline2897 (See Fig. 3 (click here)).

4. The progressive vertical (z) extrapolation is based on a ``moving" 10-term formula, including information from ten consecutive grid points, q=0,1,2,...,9. Thus, the extrapolated value tex2html_wrap_inline2903 is expressed in terms of its derivatives at grid points q=0,1,2,...,9 as well as the value of the function itself at the point q=0; the various derivatives are calculated according to Eqs. (4 (click here)) and (5 (click here)), with tex2html_wrap_inline2909, tex2html_wrap_inline2911 and tex2html_wrap_inline2913 obtained as indicated above.

5. A special correction is applied to the extrapolation results obtained for the first few grid points above the photosphere (especially at the first point, tex2html_wrap_inline2915 because in these cases the information required by the ``ten-term formula" is not available.

6. As a result ot the computational algorithm summarized above, an exceptional good extrapolation accuracy is obtained: at tex2html_wrap_inline2917, the maximum relative error in the extrapolated vector magnetic field component tex2html_wrap_inline2919 is smaller than tex2html_wrap_inline2921tex2html_wrap_inline2923

Figure 10: Average relative error tex2html_wrap_inline2925 (solid curve) and maximum relative error (dotted curve) in the reconstruction of the magnetic field components tex2html_wrap_inline2927 and tex2html_wrap_inline2929 as a function of the normalized height, tex2html_wrap_inline2931


The authors are grateful to G. Ai, J.J. Aly, P. Demoulin, C. Fang, J.C. Henoux, B. Leroy, J. Rayrole, T. Sakurai, M. Semel and H. Zhang for valuable discussions or/and for a critically reading of the manuscript.

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