*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

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