  Vector magnetogram data are often used as photospheric boundary conditions
for force-free coronal magnetic field extrapolations. In general, however,
vector magnetogram data are not consistent with the force-free assumption. In
this article, we demonstrate a way to deal with inconsistent boundary data, by
generalizing the "self-consistency procedure" of Wheatland & Regnier (2009). In
that procedure, the inconsistency is resolved by an iterative process of
constructing two solutions based on the values of the force-free parameter
alpha on the two polarities of the field in the boundary (the P and N
polarities), and taking uncertainty-weighted averages of the boundary alpha
values in the P and N solutions. When the alpha values in the P and N regions
are very different, the self-consistent solution may lose high alpha values
from the boundary conditions. We show how, by altering the weighting of the
uncertainties in the P or N boundary conditions, we can preserve high alpha
values in the self-consistent solution. The weighted self-consistent
extrapolation method is demonstrated on an analytic bipole field and applied to
vector magnetogram data taken by the Helioseismic and Magnetic Imager (HMI)
instrument for NOAA active region AR 12017 on 2014 March 29.
