next previous
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 tex2html_wrap_inline2535

Using the observed photospheric field components tex2html_wrap_inline2537 (x, y, 0) (i = x, y, z), by the aid of Eq. (6 (click here)), one calculates the non-linear FFF function tex2html_wrap_inline2549. Now, inspection of Eq. (6 (click here)) reveals the existence of mathematical singularities at points at which tex2html_wrap_inline2551. While it can be shown analytically that the actual indeterminacy can be removed by tex2html_wrap_inline2553 techniques (see Cuperman et al. 1991b), for numerical computational purposes, some suitable methods have to be used. Thus, in our algorithm, tex2html_wrap_inline2555 is calculated at all points except in some finite width bands on both sides of the curves tex2html_wrap_inline2557; the missing tex2html_wrap_inline2559 are then obtained by efficient interpolation techniques. Thus, using 14-point derivative formulas (see Appendix A), tex2html_wrap_inline2561 is calculated with a maximum relative error of tex2html_wrap_inline2563 ! (This maximum error occurs at the largest r-values considered in this work, tex2html_wrap_inline2567, where tex2html_wrap_inline2569 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) tex2html_wrap_inline2573 (x, y, z) - components we compute the horizontal derivatives tex2html_wrap_inline2581 and tex2html_wrap_inline2583 using the high order derivative formulas given in Appendix A; this is achieved with a relative accuracy of about tex2html_wrap_inline2585;

Second, by the aid of Eqs. (4 (click here)) and (5 (click here)) we obtain the vertical derivatives tex2html_wrap_inline2587 and tex2html_wrap_inline2589.

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 tex2html_wrap_inline2593 are corrected by a suitable iterative process; the maximum relative error at height tex2html_wrap_inline2595 is less than tex2html_wrap_inline2597 !

next previous
Up: Three-dimensional chromospheric magnetic

Copyright by the European Southern Observatory (ESO)