Up: Three-dimensional chromospheric magnetic

3. Computational algorithm

The computational algorithm used in this work for the reconstruction of the three dimensional magnetic field above the photosphere consists of the several steps.

a. Determination of the FFF-function

Using the observed photospheric field components (x, y, 0) (i = x, y, z), by the aid of Eq. (6 (click here)), one calculates the non-linear FFF function . Now, inspection of Eq. (6 (click here)) reveals the existence of mathematical singularities at points at which . While it can be shown analytically that the actual indeterminacy can be removed by techniques (see Cuperman et al. 1991b), for numerical computational purposes, some suitable methods have to be used. Thus, in our algorithm, is calculated at all points except in some finite width bands on both sides of the curves ; the missing are then obtained by efficient interpolation techniques. Thus, using 14-point derivative formulas (see Appendix A), is calculated with a maximum relative error of ! (This maximum error occurs at the largest r-values considered in this work, , where becomes very small).

b. Vertical extrapolation

We perform a progressive vertical (z) integration for the extrapolation of the photospheric magnetic fields within the FFF model, using as boundary conditions the field values at the photosphere. (As mentioned above, these values are simulated by the exact analytical solution of the FFF model, Eqs. (8)-(13 (click here))).

First, from the (known) (x, y, z) - components we compute the horizontal derivatives and using the high order derivative formulas given in Appendix A; this is achieved with a relative accuracy of about ;

Second, by the aid of Eqs. (4 (click here)) and (5 (click here)) we obtain the vertical derivatives and .

Third, using suitable 10-term extrapolation formulas developed in Appendix B, we obtain the sought for results - the three-component magnetic field in the half space z > 0. The somewhat less accurate results obtained at very low vertical height are corrected by a suitable iterative process; the maximum relative error at height is less than !

Up: Three-dimensional chromospheric magnetic

Copyright by the European Southern Observatory (ESO)
web@ed-phys.fr