  We investigate how the azimuthal ambiguity in solar vector magnetogram data
can be resolved by using the divergence-free property of magnetic fields. In a
previous article, by Crouch, Barnes, and Leka (Solar Phys. 260, 271, 2009),
error-free synthetic data were used to test several methods that each make a
different assumption about how the divergence-free property can be used to
resolve the ambiguity. In this paper this testing is continued with an
examination of the effects of Poisson photon noise and limited instrumental
spatial resolution. We find that all currently available methods based on the
divergence-free property can produce undesirable results when photon noise or
unresolved structure are present in the data. We perform a series of
experiments aimed at improving the performance of the global minimisation
method, which is the most promising of the methods. We present a two-step
approach that produces reasonable results in tests using synthetic data. The
first step of this approach involves the global minimisation of a combination
of the absolute value of the approximation for the divergence and a smoothness
constraint, which is designed to minimise the difference between the magnetic
field in neighbouring pixels. In the second step, pixels with measurements
known to be strongly affected by photon noise are revisited with a smoothing
algorithm that also seeks to minimise the difference between the magnetic field
in neighbouring pixels.
