Measuring the Reduced Shear
  Neglecting the second order corrections in weak lensing measurements can lead
to a few percent uncertainties on cosmic shears, and becomes more important for
cluster lensing mass reconstructions. Existing methods which claim to measure
the reduced shears are not necessarily accurate to the second order when a
point spread function (PSF) is present. We show that the method of Zhang (2008)
exactly measures the reduced shears at the second order level in the presence
of PSF. A simple theorem is provided for further confirming our calculation,
and for judging the accuracy of any shear measurement method at the second
order based on its properties at the first order. The method of Zhang (2008) is
well defined mathematically. It does not require assumptions on the
morphologies of galaxies and the PSF. To reach a sub-percent level accuracy,
the CCD pixel size is required to be not larger than 1/3 of the Full Width at
Half Maximum (FWHM) of the PSF. Using a large ensemble (> 10^7) of mock
galaxies of unrestricted morphologies, we find that contaminations to the shear
signals from the noise of background photons can be removed in a well defined
way because they are not correlated with the source shapes. The residual shear
measurement errors due to background noise are consistent with zero at the
sub-percent level even when the amplitude of such noise reaches about 1/10 of
the source flux within the half-light radius of the source. This limit can in
principle be extended further with a larger galaxy ensemble in our simulations.
On the other hand, the source Poisson noise remains to be a cause of systematic
errors. For a sub-percent level accuracy, our method requires the amplitude of
the source Poisson noise to be less than 1/80 ~ 1/100 of the source flux within
the half-light radius of the source, corresponding to collecting roughly 10^4
source photons.
