Elliptically weighted HOLICs for weak-lensing shear measurement. I. Definitions and isotropic point-spread function correction

We develop a new method of estimating gravitational shear by adopting an elliptical weight function to measure background galaxy images. In doing so, we introduce the new concept of "zero plane," which is an imaginary source plane where shapes of all sources are perfect circles, and regard the intrinsic shear as the result of an imaginary lensing distortion. This makes the relation between the observed shear, intrinsic shear, and lensing distortion much simpler, and thus higher-order calculations are easier. The elliptical weight function allows us to measure the multipole moments of the shapes of background galaxies more precisely by weighting brighter parts of the image highly, and to reduce systematic error due to insufficient expansion of the weight function in the original approach of Kaiser etal. (KSB). Point-spread function (PSF) correction in the elliptically weighted higher-order lensing image characteristics (E-HOLICs) method becomes more complicated than in the KSB method. In this paper, we study isotropic PSF correction in detail. By adopting the lensing distortion as the ellipticity of the weight function, we are able to show that the shear estimation in the E-HOLICs method reduces to solve a polynomial in the absolute magnitude of the distortion. We compare the systematic errors between our approach and that of KSB using the Shear Testing Programme2simulation. It is confirmed that the KSB method overestimates the input shear for images with large ellipticities, and E-HOLICs correctly estimates the input shear even for such images. Anisotropic PSF correction and analysis of real data will be presented in a forthcoming paper.