Biases in, and corrections to, KSB shear measurements
  We analyse the KSB method to estimate gravitational shear from
surface-brightness moments of small and noisy galaxy images. We identify three
potentially problematic assumptions. These are: (1) While gravitational shear
must be estimated from averaged galaxy images, KSB derives a shear estimate
from each individual image and then takes the average. Since the two operations
do not commute, KSB gives biased results. (2) KSB implicitly assumes that
galaxy ellipticities are small, while weak gravitational lensing assures only
that the change in ellipticity due to the shear is small. (3) KSB does not
invert the convolution with the point-spread function, but gives an approximate
PSF correction which - even for a circular PSF - holds only in the limit of
circular sources. The effects of assumptions (2) and (3) partially counter-act
in a way dependent on the width of the weight function and of the PSF. We
quantitatively demonstrate the biases due to all assumptions, extend the KSB
approach consistently to third order in the shear and ellipticity and show that
this extension lowers the biases substantially. The issue of proper PSF
deconvolution will be addressed in a forthcoming paper.
