  Routine measurements of the solar magnetic field are mainly carried out in
the photosphere. Therefore, one has to infer the field strength in the higher
layers of the solar atmosphere from the measured photospheric field based on
the assumption that the corona is force-free. Meanwhile, those measured data
are inconsistent with the above force-free assumption. Therefore, one has to
apply some transformations to these data before nonlinear force-free
extrapolation codes can be applied. Extrapolation codes in cartesian geometry
for modelling the magnetic field in the corona do not take the curvature of the
Sun's surface into account and can only be applied to relatively small areas,
e.g., a single active region. Here we apply a method for nonlinear force-free
coronal magnetic field modelling and preprocessing of photospheric vector
magnetograms in spherical geometry using the optimization procedure.We solve
the nonlinear force-free field equations by minimizing a functional in
spherical coordinates over a restricted area of the Sun. We extend the
functional by an additional term, which allows to incorporate measurement error
and treat regions with lacking observational data. We use vector magnetograph
data from the Synoptic Optical Long-term Investigations of the Sun survey
(SOLIS) to model the coronal magnetic field. We study two neighbouring
magnetically connected active regions observed on May 15 2009. For vector
magnetograms with variable measurement precision and randomly scattered data
gaps (e.g., SOLIS/VSM) the new code yields field models which satisfy the
solenoidal and force-free condition significantly better as it allows
deviations between the extrapolated boundary field and observed boundary data
within measurement errors. Data gaps are assigned to an infinite error. We
extend this new scheme to spherical geometry and apply it for the first time to
real data.
