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

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3. A new method

Here we present a new method, with which the PSF effects can be corrected for with greater accuracy. The essence of the method is not to work with the moments of the observed images; instead each image is fit directly as a PSF-convolved, sheared circular source of unknown radial profile.

Assume for the moment that we have managed to sum the images of many galaxies into an `average galaxy' image [FORMULA]. Analysing a stacked galaxy image is similar to the approach discussed by Lombardi & Bertin (1998), who average image second moments before corrections are applied. It differs from methods such as KSB or Bonnet & Mellier (1995) in which galaxies are individually corrected for PSF effects before they are combined to produce a shear estimate.

Intrinsically, [FORMULA] is circular if the galaxies are randomly oriented, but the image we observe has been distorted first by gravitational lensing shear, then by the atmospheric seeing, and finally by the camera optics. The observed [FORMULA] is therefore a sheared circular source, convolved with a (known) PSF. We fit [FORMULA] directly to such a model, with the minimum of further assumptions: in particular, the radial profile of [FORMULA] is left free. Note that the ellipticity of a sheared circular image is constant with radius, so after convolution with the PSF only a subset of ellipticity profiles is consistent with a shear.

If the PSF is known, e.g., from analysis of star images in the field, the model for [FORMULA] is specified by an unknown radial brightness profile, and by the shear parameters [FORMULA] that we are interested in. We model the radial profile as the superposition of several Gaussians of different fixed widths, and unknown amplitude. We have found that the following recipe for assigning the basis functions gives good results: (i) determine the best-fit circular Gaussian radii to the observed PSF and galaxy images, [FORMULA] and [FORMULA]. (ii) Take [FORMULA] as an estimate for the intrinsic radius of [FORMULA]. (iii) Use four components to describe the radial profile of [FORMULA], with Gaussian radii [FORMULA].

The algorithm is laid out in Fig. 3. Tests of its accuracy and its sensitivity to noise in the images are described next.

[FIGURE] Fig. 3. The schematic algorithm used to derive the shear from an observed mean galaxy and PSF image.

3.1. Simulations in the absence of noise

As our first test, we considered simulated images of intrinsically round sources (no shear) observed with PSF's of a range of shape and anisotropy. A weak lensing analysis with an accurate correction for the PSF should yield zero shear. On a large number of model images, described below, we compared the results of the algorithm of Fig. 3 with those from the KSB algorithm as described in HFKS (implying in particular that the same weight function is used in the derivation of polarizations and polarizabilities of galaxy and PSF images). KSB polarizations are converted to shear estimates by dividing by the "pre-seeing shear polarizability" [FORMULA], for which we use the expression given by Luppino & Kaiser (1997). Unless stated otherwise, in all our simulations KSB was implemented with a weight function given by the best-fit circular Gaussian to the post-seeing galaxy image.

3.1.1. Double-Gaussian images and PSF

In most of our simulations, we modeled the sources and PSF as double gaussians


where [FORMULA]. The parameter k is unity for a Gaussian profile, and is larger for more radially extended profiles. A reasonable, though admittedly crude, approximation to an exponential profile is given by setting [FORMULA], while [FORMULA] gives a reasonable approximation to a de Vaucouleurs profile (Fig. 4).

[FIGURE] Fig. 4. The radial profiles of the double-Gaussian models used in this paper. Left, the [FORMULA] profile is plotted logarithmically to show its similarity to an exponential profile; right the [FORMULA] profile is plotted logarithmically vs. [FORMULA] to show it is similar to a de Vaucouleurs model.

The [FORMULA] allow different radial variations of ellipticity to be prescribed. Before shearing, the average galaxy is intrinsically round, so we set the [FORMULA] equal to zero when modelling [FORMULA]. PSF shapes can be more complicated, and we considered three kinds of PSF ellipticity profile: [FORMULA] (constant ellipticity with radius), [FORMULA] (radially increasing ellipticity for [FORMULA]) and [FORMULA] (radially decreasing ellipticity). These three possibilities, though by no means exhaustive, form a representative set of PSF's.

The results of the simulations are presented in Figs. 5, 6 and 7. They show that the KSB method can suffer from systematic residuals around the 0.01 shear level once the PSF ellipticity exceeds 0.2 or so, whereas this is not so for the new method developed here. The KSB residuals are most important for small galaxies, for PSF profiles with long tails, and for radially increasing PSF ellipticity. The effect is clearly driven by the PSF shape, not by the galaxy brightness profile.

[FIGURE] Fig. 5. The result of correcting simulated unsheared images for PSF anisotropy, following the KSB method (solid lines) and the method presented here (dashed lines). [FORMULA] is in each case the shear that is deduced after the PSF correction, and should be zero for a perfect analysis. The k's are luminosity profile shape parameters for galaxy and PSF, and are explained in the text. In each panel the heavy line represents the case where the inner gaussian in the galaxy image is intrinsically of the same radius as the inner gaussian in the PSF (both have the same value of [FORMULA] in Eq. (6)), the lighter lines those where the galaxy is 0.5 (upper) and 1.5 (lower) times this size. The PSF ellipticity is constant with radius in these simulations, with axis ratio [FORMULA]. While for the [FORMULA] case (Gaussian PSF) the KSB method leaves a residual which is third-order in PSF ellipticity, other PSF luminosity profiles give rise to first-order residuals. The residuals of the new method in the upper panels disappear if more radial components are used in the fit for [FORMULA], highlighting that the dominant source of error in this method is the extent to which the radial profile is modeled correctly.

[FIGURE] Fig. 6. As Fig. 5, but only the outer component of the PSF is elliptical, with axis ratio [FORMULA].

[FIGURE] Fig. 7. As Fig. 5, but only the inner component of the PSF is elliptical, with axis ratio [FORMULA].

Notice that in the constant-ellipticity case (Fig. 5), with a Gaussian PSF ([FORMULA]) the residuals left by the KSB method are high order in PSF ellipticity, but that for non-Gaussian PSF's a low-order residual dominates. (We have verified this result analytically using symbolic mathematics.) This is a consequence of the fact that only the single elliptical Gaussian PSF can be written as a convolution of a compact anisotropic function with a round extended one, as assumed in the KSB derivation. It is clearly important to test algorithms not only for single-Gaussian PSF's!

3.1.2. A WFPC-2 PSF

In order to test whether our results are specific to the double-Gaussian formulation of the PSF, a test was also performed with a model PSF for the WFPC-2 camera on the Hubble Space Telescope. The model was generated with the TinyTIM software package, provided on-line at STScI by J. Krist. An oversampled PSF was calculated for a position near the corner of CCD#4, and convolved with a Gaussian circular galaxy of FWHM 0.25arcsec. This `galaxy' and the PSF (Fig. 8) were then binned to a resolution of half a WFPC-2 pixel to avoid under-resolving the PSF, and analyzed as above. The results are summarized in Table 1, and confirm the results obtained from the large number of double-Gaussian simulations described earlier.

[FIGURE] Fig. 8. PSF (jagged contours) and simulated galaxy image for an observation in the corner of one of the WFPC2 CCD's. Axis units are 0.05 arcsec.


Table 1. Results of a simulation based on a WFPC-2 PSF calculated using the TinyTIM software. In agreement with earlier results (HFKS), the KSB technique appears to over-correct for the anisotropic WFPC-2 PSF images slightly.

3.2. Noise properties

3.2.1. Analytic estimate

The error on the estimated shear due to photon noise can be estimated as follows. Let the 1-[FORMULA] error on each pixel of [FORMULA] be s (for simplicity we take this to be the same on every pixel, appropriate for background-limited work). Then the shear is obtained by minimizing


where [FORMULA] is the PSF, [FORMULA] is the intrinsic radial profile of the average galaxy, [FORMULA] denotes convolution, [FORMULA] is the position of the kth pixel, and [FORMULA] is the distortion matrix of Eq. (1). If the fit parameters [FORMULA] are uncorrelated with the radial profile, their inverse variances are given by [FORMULA]. For example, at the best fit


The right-hand side of Eq. (8) can be estimated assuming that the PSF and observed average galaxy are Gaussians with dispersions [FORMULA] and [FORMULA] pixels of integral 1 and F, respectively. Then the 1-[FORMULA] error on [FORMULA] evaluates to


where we have used the results that the PSF-fitting error on F for a Gaussian source is [FORMULA]. The error on [FORMULA] is the same. We have verified this formula by means of simulations, similar to those described below. Eq. (9) shows the expected increase in noise for small objects, as well as the lower limit, approached for fully resolved objects, of


3.2.2. Simulations with photon noise

We have checked the sensitivity to noise in the images by means of Monte Carlo simulation. Many realizations of random Gaussian noise superimposed on a PSF-smeared, intrinsically round galaxy image were analyzed with both algorithms, and the distributions of the resulting [FORMULA] estimates compared. Selected results are shown in Table 2. Interestingly, the effect of photon noise on the shear values derived with both methods is very similar, and there is once again no evidence for a bias in the results obtained with the new method.


Table 2. Results from representative noise simulations of the KSB method and the one presented in this paper. In each case, 100 noise realizations (noise per pixel of 0.001, with (see Eq. (6) [FORMULA]pixels, [FORMULA] and total flux 4) were analyzed with the standard KSB method, with the KSB method using a weight function double the radius of the best-fit Gaussian, and with the new method described in this paper. The first six simulations were of cases without PSF anisotropy, and in the last six the PSF has a constant axis ratio of 0.7. In all cases, the dispersions of the standard KSB method and the new one are very similar, but note the imperfect correction from the KSB method. The simulations of KSB with a wider weight function show better correction for PSF anisotropy than the standard KSB implementation, but at the cost of increased noise.

A possible way to avoid the systematic residuals of the KSB method is to increase the radius of the weight function W in Eq. (3), since the problems arise from the imperfect way in which the polarizabilities represent the effect of W. However, the primary function of W is to control the noise in the images. Doubling the Gaussian radius of W does in fact improve the anisotropy correction in the mean, but at the cost of almost doubling the noise on the result (see Table 2).

3.2.3. The effect of centroiding errors

The centroid of an image can be determined in different ways, each of them susceptible to errors due to photon noise. The effect of centroiding errors on the summed galaxy image will be a convolution with the distribution of centroid errors. Thus, the PSF needs to be convolved with this distribution before analysis of [FORMULA], so that the effect of the centroiding error can be compensated.

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© European Southern Observatory (ESO) 1999

Online publication: December 2, 1999