Biases in, and corrections to, KSB shear measurements

We analyse the Kaiser-Squires-Broadhurst (KSB) method to estimate gravitational shear from surface-brightness moments of small and noisy galaxy images. We identify three potentially problematic assumptions. These are as follows. (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 only assures that the change in ellipticity due to the shear is small. (3) KSB does not invert the convolution with the point spread function (PSF), 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 counteract 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 Melchior et al.