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Astron. Astrophys. 352, 355-362 (1999)

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2. Limitations of the KSB formalism for PSF anisotropy correction

We 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 method

The 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 [FORMULA] and a convergence [FORMULA], whose combined effect is represented by the mapping


where [FORMULA]. For simplicity, in what follows we neglect [FORMULA] as it is small in the weak lensing regime, and pretend we are deriving the true shear [FORMULA] instead of the reduced shear g. Thus, our results on shape measurements are valid, but their interpretation as a lensing signal may require consideration of the [FORMULA] factor. Our analysis makes no assumptions on the smallness of [FORMULA], though.

KSB describe a method for recovering [FORMULA] 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 [FORMULA]. These polarizations are derived from weighted second moments


of the image intensities, where W is a weight function which goes to zero at large radii. The weight function is required as otherwise the sky noise in the outer parts of the image dominates the measured moment. The significance of the measurement is optimized by taking the weight function to be relatively compact, of a size comparable to the image itself.

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" [FORMULA] defines the ratio between the PSF anisotropy [FORMULA], constructed from the unweighted second moments [FORMULA] of the (normalized) PSF, and the resulting change in image polarization e. The "shear polarizability" [FORMULA] is the ratio between the applied shear [FORMULA] and the resulting change in the image polarization. KSB show how the polarizabilities can be derived from higher weighted moments of the observed PSF and galaxy images.

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 [FORMULA] is a unit-integral Gaussian of x- and y-dispersions a and b. [FORMULA] is a small parameter. The case [FORMULA] is plotted in Fig. 1.

[FIGURE] Fig. 1. An example of a PSF with radially varying ellipticity, but zero overall polarization. (Eq. (4) with [FORMULA]). Contours differ by a factor of [FORMULA]. The dotted curves are circles, shown for comparison.

The PSF of Eq. (4) has exactly zero anisotropy p: the second moments in x and in y are equal. However, the ellipticity of the PSF varies with radius, which means that the weighted second moments are not equal: weighting the central parts more will enhance the x-moment preferentially. In fact, it is easy to show that the polarization constructed with weighted moments is [FORMULA]. The precise result for a Gaussian weight function [FORMULA] is


The polarization of the [FORMULA] PSF plotted in Fig. 1 is plotted in Fig. 2. It has a value of 0.03 near [FORMULA], roughly the radius of maximum significance which should be used to minimize photon noise in the polarization measurement.

[FIGURE] Fig. 2. The polarization of the PSF of Fig. 1 as a function of the weight function's Gaussian radius w. The plot shows that with compact weight functions such as those that are required to suppress photon noise, even mildly elongated PSF's may erroneously yield a polarization of several percent.

The KSB polarizabilities are derived assuming that the PSF can be written as the convolution of a compact anisotropic part with an extended circular part (KSB Eq. A1). This assumption allows the anisotropy to be characterized in terms of p only. However, our example shows that this assumption may be too restrictive: it effectively couples the radial intensity profile of the PSF with its ellipticity profile. For example, a single Gaussian with constant ellipticity can be written as such a convolution, but a sum of two elliptical Gaussian such as the PSF of Eq. (4) cannot. The systematic errors that arise are the result of this.

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© European Southern Observatory (ESO) 1999

Online publication: December 2, 1999