Modern vector magnetographs provide the following magnetic field
components
:
(i) 
and
in a horizontal plane,
,
at the photosphere (z = 0); and (ii) the longitudinal (line
of sight) component 
(x, y, z) at a (finite) number of
horizontal planes 
(z) situated at distances 
(
) above the plane 
(0) and parallel to it.
These observational data need some theoretical treatment before they can
be used for reconstruction purposes: (a) the 
ambiguity in the
observed components 
(x, y, 0) and 
(x, y, 0) has to be
removed by the aid of some theoretical method (see, e.g., Cuperman et
al. 1991b and references therein); (b) high order interpolation
schemes (see, e.g., next section) have
to be applied in order to obtain 
(x, y, z) - values at a
relatively large number of intermediate horizontal planes parallel to
(0) and situated at distances 
apart,
where 
represents the vertical integration step.
With the magnetograph observations corrected and improved as indicated
above, the problem can be formulated as follows: given (i) the
transverse field components 
(x, y, 0) and 
(x, y, 0), as
well as the longitudinal field component 
(x, y, 0) in a plane
(z = 0) at the photosphere and (ii) the longitudinal field
component 
(x, y, z) at vertical distances 
apart, reconstruct the three-dimensional chromospheric magnetic
field in half the space above the domain 
(z = 0).
The steady state FFF equations are
and
Thus, Eq. (1) states that the electric current density
is proportional to the magnetic field 
.
Upon taking the divergence of Eq. (1 (click here)), by (2 (click here)),
one obtains
which indicates the constancy of 
along individual field lines.
In order of ascending complexity, Eqs. (1 (click here))-(3 (click here))
may describe the cases of
current-free configurations 
, linear force-free field
configurations 
and nonlinear force-free field
configurations 
. The components of Eqs. (1 (click here))
and (2 (click here)) are:
and
Thus, if the field components at the photosphere, 
(x, y; z = 0),
(x, y; z = 0) and 
(x, y; z = 0) are given (and, therefore
is determined) one can calculate numerically the horizontal
derivatives
and then, by Eqs.
(4 (click here))-(7 (click here)) one can obtain the vertical derivatives
,
and proceed to the vertical
integration to the height 
, and so on.
As a study case we consider a magnetic configuration generated by
electric currents satisfying the FFF condition
everywhere except along the line y = 0, z = -a (below
the photosphere), where an infinite straight line current 
is
flowing (Low 1982; Cuperman & Ditkowski 1990).
The solutions of Eqs. (1 (click here)) and (2 (click here)) for this case are
The free generating function 
is related to the quantity
by the equation
In this work we use the following generating function:
Thus, in the continuation we use the values of the field components 
and 
(and consequently, 
) at the
photospheric surface (z = 0) given by Eqs. (8 (click here))-(13 (click here))
as boundary
conditions and integrate the system of Eqs. (4 (click here))-(7 (click here))
to determine the
magnetic field components in the half space z > 0 . Comparison
of the results with the values given by Eqs. (8 (click here))-(12 (click here))
for z > 0
will indicate the reliability of the numerical integration procedure
proposed and implemented here.
For convenience, we use the normalizations (i = x, y, z):
The integration limits in the x, y-plane are:
