## 2. Limitations of the KSB formalism for PSF anisotropy correctionWe first examine the KSB method, following our earlier tests in the context of analysis of Hubble Space Telescope images (Hoekstra et al. 1998, Appendix D). ## 2.1. The KSB methodThe technique of weak (or statistical) lensing involves measuring the systematic, gravitationally induced, distortion of background images behind a gravitational lens. In the weak lensing regime small background images are distorted by a shear and a convergence , whose combined effect is represented by the mapping where . For simplicity, in what
follows we neglect as it is small in
the weak lensing regime, and pretend we are deriving the true shear
instead of the reduced shear
KSB describe a method for recovering from images of distant galaxies. Essentially, they derive galaxy ellipticities from weighted second moments of the observed images, and then correct these for the effects of the weight function and of smearing by the point spread function (PSF). By averaging over many galaxies, which are assumed to be intrinsically randomly oriented, the effect of individual galaxy ellipticities should average out, leaving the systematic lensing signal. The KSB method has proved to be very effective, especially in the study of galaxy cluster potentials. KSB define various "polarizabilities", which express the ratio between an input distortion (gravitational shear or PSF anisotropy) and the measured polarization of an image . These polarizations are derived from weighted second moments of the image intensities, where Details of the method can be found in KSB, and in Hoekstra et al.
(1998, henceforth HFKS), where a few small errors in the formulae of
KSB were corrected. For the purposes of the present paper, it is
sufficient to know that in the KSB formalism, the "smear
polarizability" defines the ratio
between the PSF anisotropy ,
constructed from the unweighted second moments
of the (normalized) PSF, and the
resulting change in image polarization ## 2.2. How accurate is KSB?In the context of ground-based cluster weak lensing, the KSB method works well. Nevertheless, it does involve some approximations. Now that weaker and weaker signals are of interest, it is therefore important to understand the limitations of the method. As already discussed by HFKS, for strongly non-Gaussian PSF's the KSB method does not completely correct PSF anisotropy. This is particularly true when analyzing small galaxies in deep HST images, where it turns out that the choice of weight function in Eq. (3) is important. A simple PSF model can be used to illustrate why such residuals are, at some level, unavoidable. Consider the following PSF: where is a unit-integral Gaussian
of
The PSF of Eq. (4) has exactly zero anisotropy The polarization of the PSF plotted in Fig. 1 is plotted in Fig. 2. It has a value of 0.03 near , roughly the radius of maximum significance which should be used to minimize photon noise in the polarization measurement.
The KSB polarizabilities are derived assuming that the PSF can be
written as the convolution of a © European Southern Observatory (ESO) 1999 Online publication: December 2, 1999 |