Simulation of measured photospheric field components.
From the general analytical solutions, Eqs. (8)-(13)
one can obtain the
``photospheric" (i.e. at z = 0 ) field components
,
,
and
For illustration, these functions are represented in Fig. 1 (click here); the
``observation" domain, 
(z = 0) has the dimensions
=
= 5
(see Eq. (15 (click here))).
Figure 1: Simulated photospheric (z=0) observations,
(a), 
(b), 
(c) and 
(d)
as obtained from the analytical expressions
evaluated at z=0. For convenience, normalized quantities are used, namely
, 
and 
Simulation of photospheric-field components, after removal of
the 
ambiguity in the 
and 
components.
When suitable techniques are used to remove the 
ambiguity (see, e.g. Cuperman et al. 1993; Li et al.
1993), one obtains the field components
and 
shown in Figs. 2 (click here)a and b. The
corresponding contours of constant values are shown in Figs. 2 (click here)d and
e. For completness, the field component
is also represented in Figs. 2 (click here)c and f.
(In Figs. 2 (click here)d,e,c solid (dashed) curves indicate positive (negative)
contour values; the heavy solid curves represent the contour
value 
Figure 2: Simulated photospheric
(z=0) observations
(a) and
(b)
obtained after removal of the 
ambiguity;
(d), (e): contours of constant 
and 
-values; (c), (f) like (a), (d), for
Computation of the FFF-function 
.
Using the results illustrated in Fig. 2 (click here), by the aid of Eq.
(6 (click here)), one obtains the nonlinear FFF-function 
shown
in Fig. 3 (click here)a and the 
-function shown in
Fig. 3 (click here)d (recall the definition 
the corresponding relative error
(N-numerical, A-analytical)
as a function of 
is shown in Fig. 3 (click here)e. As can be seen, the computational accuracy
is exceptional: 
For completness, we show
in Figs. 3 (click here)b and c the spatial dependence of the functions
and
which enter the expression for 
Figure 3: (a), (d) Numerically computed non-linear FFF-function,
and
, respectively;
(e) relative error
as a function of 
;
(b), (c) spatial dependence of the functions
and
used for the calculation of the function 
Electrical currents.
From the 
-values shown in Fig. 3 (click here)a and the field
components 
, one can obtain the (normalized) FFF electrical current
density components
The computed quantities are shown in Figs. 4 (click here)a, b and c;
the corresponding contours of constant values are shown in
Figs. 4 (click here)d, e and c.
(Here, 
.
Figure 4: (a)-(c) Photospheric electrical current densities
and 
based on the results shown in Fig. 2 and Fig. 3;
(d)-(f) corresponding contours of constant current values
Simulation of the longitudinal component 
.
From the analytic expression, Eq. (10 (click here)) one obtains
discrete 
- values
in horizontal planes parallel to the observational
one, at vertical distances 
apart ( q=1,2,...);
these ``simulated" values are indicated by circles in Fig. 5 (click here).
Then, upon using a high-order interpolation method,
from these values one obtains
the much higher-density set of 
-values
in horizontal planes at distances
apart from each other, as shown by the continuous
curves in Fig. 5 (click here); actually,
, represents the vertical (z)
integration step.
The top (bottom) figure represents contours of constant
-values in the plane 
Figure 5: Circles: ``Simulated observations" of
- contours as obtained
from the analytical expression;
solid curves: interpolated values obtained upon using high order
interpolation techniques.
Top (bottom): y=0 (x=0) planes
Reconstruction of the magnetic field components,
& 
Upon using
Eqs. (1 (click here))-(6 (click here)) with (i)
``simulated" boundary conditions (at the
photosphere) represented by the functions
shown in Fig. 2 (click here), (ii) the non-linear FFF function
shown in Fig. 4 (click here)a, and (iii) the ``simulated"
longitudinal component 
shown in Fig. 5 (click here), by Eqs. (4 (click here))-(6 (click here)) and the high
order, corrective vertical extrapolation method described in Appendix B,
one obtains the final result - the three-dimensional
chromospheric magnetic field. Thus, Fig. 6 (click here) illustrates the
computed functions
and 
at the vertical distances
and 20 respectively; (
represents the
photosphere). Figure 7 (click here) shows contours of equal values of the
functions represented in Fig. 6 (click here).
For completness, the spatial structure of the ``measured"
- component is also indicated.
The ``stretching factors" indicated on the figures
(1, 1.2, 1.4 and 1.6) are used for the convenience of
graphical representation.
Figure 6: Reconstructed magnetic field components
(left column) and
(middle column)
at several height values 
, as indicated;
for comparison, the corresponding
- component
is shown in the right column.
(Actually, to emphasize the neutral line, 
we represent the quantity
,
rather than just
Figure 7: Contours of equal values of the functions represented in Fig. 6
Finally, in Fig. 8 (click here) we show contours of constant-value
magnetic field components 
and 
in the plane y=0 (top) and x = 0 (bottom);
and in Fig. 9 (click here) we show the reconstructed FFF electrical current
density components 
at 
( left: 
right: corresponding contours of constant values).
The average relative error
and maximum relative error
in the computation of 
and 
as a function of
the normalized height 
are shown in Fig. 10 (click here),
by solid 
and dotted 
curves.
As can be seen, the
computational relative accuracy is very good:
Figure 8: Contours of constant magnetic field values
and
in the plane y=0 (top) and x=0 (bottom)
Figure 9: Left: Reconstructed FFF electrical current densities
and
at several height values 
as indicated;
right: corresponding contours of constant current values