Three-dimensional chromospheric magnetic field configurations based on photospheric-vector and chromospheric-multi-level longitudinal-magnetic field observations

The three-dimensional (3D) reconstruction of magnetic configurations above the photosphere is considered within the framework of the nonlinear force-free-field (FFF) model. The physical- computational algorithm proposed and tested incorporates, for the first time, the following basic features: 1) Both photospheric vector field, $ {\vec B} (x,y,0) $ and chromospheric line of sight field component, $ B_{z} (x,y,z) $ data are utilized; this reduces significantly the degree of ill-posedness characterizing the Cauchy problem corresponding to the case when only $ {\vec B} (x,y,0) $ - values are used as boundary conditions. 2) A high-order, very efficient computational algorithm is developed and used: horizontal derivatives are evaluated by 14 - terms formulas in 14 different forms, selected such as to provide optimal computational accuracy; the vertical integration is achieved by the use of “moving" 10 - term formulas expressed in terms of 10 derivatives and the first $ B_{i} (x,y,z) $ values $(i=x,y,z)$. 3) At neutral points, where inherent computational singularities in the values of the FFF-function α arise, rather than using smoothing techniques based on four-neighbouring- values averages, suitable procedures ensuring continuity are developed and used. The overall result of the incorporation of these novel features is an improvement by orders of magnitude of the accuracy with which the chromospheric fields are reconstructed in the case in which one uses (i) only $ {\vec B} (x,y,0) $ - values as boundary conditions and (ii) relative simple computational formulas and smoothing techniques; at $ \bar{z} = 20 $, $\Delta B_{i} / B_{i} < 10^{-3} $ ! The elimination/minimization of measurement errors as well as the fitting of the corrected date to FFF-model-states is also discussed.