  Given the basic parameters of a cosmic shear weak lensing survey, how well
can systematic errors due to anisotropy in the point spread function (PSF) be
corrected? The largest source of error in this correction to date has been the
interpolation of the PSF to the locations of the galaxies. To address this
error, we separate the PSF patterns into components that recur in multiple
exposures/pointings and those that vary randomly between different exposures
(such as those due to the atmosphere). In an earlier study we developed a
principal component approach to correct the recurring PSF patterns (Jarvis and
Jain 2004). In this paper we show how randomly varying PSF patterns can also be
circumvented in the measurement of shear correlations. For the two-point
correlation function this is done by simply using pairs of galaxy shapes
measured in different exposures. Combining the two techniques allows us to
tackle generic combinations of PSF anisotropy patterns. The second goal of this
paper is to give a formalism for quantifying residual systematic errors due to
PSF patterns. We show how the main PSF corrections improve with increasing
survey area (and thus can stay below the reduced statistical errors), and we
identify the residual errors which do not scale with survey area. Our formalism
can be applied both to planned lensing surveys to optimize instrumental and
survey parameters and to actual lensing data to quantify residual errors.
