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• Variation in the intrinsic ellipticity distribution as a function of galaxy magnitudes
• The residual PSF contamination for PSFs D and E
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From: Catherine Heymans, April 28, 2006

Hi all,
I think I may have found one of the sources of the magnitude/size dependence of the shear measurement accuracy in STEP2. Attached are two plots of the width of the intrinsic ellipticity distribution as a function of galaxy mag/size. Dashed lines are the mean from the full galaxy sample.
does vary with mag/size, and not in the way I was expecting. I thought as the gals got fainter/smaller they would become more irregular and would increase. Instead the fainter/smaller galaxies become intrinsically more circular. The exception is for PSFB, where the galaxies are all exponential and is fairly constant.
This will effect all methods who calculate a responsivity R~ (1-e^2) from the full galaxy ensemble to convert ellipticity to shear. I believe KSB will also be effected if Pgamma is fit over the full galaxy sample.
Richard, when the shapelet coefficients are taken from the COSMOS images, do they include a de-convolution by a rough ACS PSF? i.e is the isotropic smearing removed? Or are all these galaxies significantly larger than the ACS PSF size? Can we determine how realistic this trend is when it comes to real galaxies? It must exist at some level but is it as severe as we see here?
Ideas welcome from all,
Catherine
P.S fine print:
Ellipticity is estimated from the pre-PSF convolved images by measuring the quadrupole of the shapelet galaxies (correct me if I'm wrong there Richard). Intrinsic ellipticity is then calculated by removing the shear component. Mag is subject to a zeropoint correction (of ~9) Size is the shapelet size

Dear all,
The ellipticity measurements from the input catalogues are a little odd. The ellipticites are also defined in a different way for PSF B than for the others. The "input ellipticities" were really put in the input catalogues as column-fillers in a standard format, rather than as something intended for quantitative study. In the original simulation paper, we used SExtractor ellipticities of low noise objects to test the simulated images. My conclusion below is going to be that I would like to see SExtractor/KSB ellipticities for the STEP2 images. These will be more familiar to us all, and we will be able to interpret them with some intuition.
For now, let me explain what the "input ellipticities" actually are.

Exponential galaxies:
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These have concentric isophotes, with major and minor axes a and b. Their sizes are drawn from the distribution of galaxies in the COSMOS ACS images. Their ellipticities e=(a^2-b^2)/(a^2+b^2) are drawn from a random distribution with sigma_e=0.3, and are uncorrelated with the size or magnitude of the galaxy.
I store the major and minor axis lengths assuming that definition of ellipticity. Catherine then used these to calculate ellipticities e=(a-b)/(a+b).
I have an option in the code to let the width of the ellipticity distribution trace that in the shapelet galaxies. I didn't use that because the ellipticities of the shapelet galaxies are defined differently. However, now that we know all this is so important, it would have been good to turn on that flag...

Shapelet galaxies:
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The morhology of the shapelet galaxies really is expected to vary as a function of size and magnitude. The input galaxies are copies of those in the COSMOS ACS images. I didn't alter them in any way, other than reorienting them randomly and taking a mirror image of half of them. The morphologies of the faint ones do not influence the morphologies of the brighter ones, and vice versa. All of the galaxies in PSFs other than B are claculated this way; any variation between them is shot noise.
I first decomposed the COSMOS galaxies into shapelet space as they were seen (i.e. without deconvolving from the ACS PSF) but with a truncation in shapelet order n (which os close to a truncation in k-space). From the shapelet coefficients, I calculated the UNWEIGHTED quadrupole ellipticity of each galaxy - i.e. without the Gaussian weight function associated with KSB. There is a lot of weight given to flux at large radii, and this measure behaves in a peculiar fashion, against much of my intuition, at least.
Again using e=(a^2-b^2)/(a^2+b^2), I calculated appropriate values of a and b, storing these in the input catalogues. Catherine used these to work out e=(a-b)/(a+b). I also stored the unweighted rms size of the galaxies (i.e. the same integrals as the ellipticities, but with an r instead of a sine). These are also counterintuitive, and behave nothing like FWHMs.
The apparent variation in this quantity could arise from any of three sources:
* Reality. The unweighted ellipticities do not behave intuitively.
* ACS PSF residuals. The (very) small galaxies would have been made intrinsically (i.e. pre-shear) rounder by the ACS PSF in the original observation. I'm not entirely sure how to interpret the sizes on Catherine's x axes to decide if this is likely.
* Truncation effects. Since the unweighted ellipticites are inordinately sensitive to flux at large radii, they are affected by truncation even when measurements of FWHM are not.

Overall, I think it would probably be more instructive to see the variation of ellipticites measured by Konrad - or one of the KSB pipelines, since these are more familiar. Incidentally, that's why I would like to know the "KSB ellipticity" of the PSFs rather than my own peculiar statistic.
Cheers,
Richard

Hi all - a small comment or 2. For any method that measures a shear by calculating a responsivity, the responsivity should be determined from the subpopulation of galaxies that were used to measure the shear. So for example the STEP2 plots of size vs mag should definitely recalculate responsivity for each bin. In the real world, when one does tomographic cosmic shear, one will need a responsivity for each redshift bin, else you'll see this kind of bias crop up where the nearer bins have over/under-estimated shear calibrations.
As Catherine noted, this will be tough for the faintest galaxy bins, because the responsivities want to know the noise-free shape distribution but we only get the noisy one unless we have some deeper data to calibrate from. So we'll have the choice in the future of (a) obtaining the deeper data, or (b) trying to deconvolve the noisy shape distribution to get the intrinsic one.
I would think all of this would be true of the KSB polarizabilities also.
And yes, when I've examined the UDF image I have seen the shape distribution become much broader for fainter mags (which also means that shape noise is higher for future surveys than for today's). This seems to be something that several of you have detected in various kinds of data, which means it's probably true!
Cheers,
Gary

Gary, I quite agree about needing to calculate shear susceptibility/resposivity from each subsample in tomographic data. I am vaguely worried about what happens for galaxies with catastrophic redshift errors but, otherwise, that offers a very neat way out.
Unfortunately, the problem is still there - and possibly worse because it is concealed - in 2D surveys. Calculating only a global susceptibility will leave biases in shear measurement as a function of redshift, so that high redshift shears are overestimated, and low redshift ones underestimated (or vice versa). This'll change the effective redshift distribution of the source galaxies, with all the consequences that Ludo et al. wrote a paper about recently.
This has always been known in the KSB pipelines, of course. I think all of them either use shear susceptibilities from individual objects or fitted as a function of magnitude. So fitting your responsivity as a function of magnitude seems a perfectly viable too. Yes, ideally it would be as a function of redshift, but that's a lot closer.
Richard

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A quick update on STEP2:

i) Variation of intrinsic ellipticity distribution with galaxy mag/size is confirmed

ii) Updated improved m vs gal mag plots for BJ02-esque methods (RM,MJ,J2,RN) where the variation of is taken into account can be downloaded from here

iii)Should we also do something for the KSB-esque methods to account for this variation as we don't see such strong variation in real data?

iv) A new webpage with some of the interesting e-mail discussions about STEP2 can be found here
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Dear all,
We've investigated further the variation of the intrinsic ellipticity distribution as a function of galaxy magnitude/size by measuring galaxy ellipticity in several different ways on a PSF-free, noise-free, shear-free STEP2-esque image. We see the same trend that we saw with the 'input ellipticities', that STEP2 galaxies become intrinsically rounder at fainter magnitudes, using KSB, SExtractor and BJ02 ellipticity estimates (thanks to Reiko for the later).
I think we're all agreed that this trend is not what we see in real data, but is important to consider in future surveys. I think that it is likely a result, in the STEP2 simulations, of the fact that as the original template COSMOS galaxies get fainter they can only be fit with fewer number of shapelet coefficients. This is just something we have to live with, but we can do our best to reduce the impact on the final STEP2 mag/size dependent results;
For the BJ02-esque methods MJ,RN,RM, (Mikes, Reikos and Rachels), I have re-calculated the responsivity correction (where R~ 1-) for each magnitude and size bin, and then calculated new shear estimators with this magnitude/size dependent R. You can download the new plots of shear measurement accuracy vs galaxy mag and size for these methods from here
You'll see an improvement - but it's still not perfect!. This correction is also easy to perform for Konrads method (but Konrad I'll need the conversion factor as per my last e-mail).
Should we also do something for the KSB-esque methods? The equivalent of R is Pgamma. Henk fits this as a function of mag and rg, but might need to include a finner binning for the magnitude? Another alternative is to use raw Pgammas and a cut on |g|? KSBers - let me know if you have any ideas.
I think the main objective should be to show lensing in the best light. As this variation in STEP2 galaxy properties exaggerates the problem, we should be open to ways to correct for it in this STEP2 analysis, even though with in real data it is not going to be so easy.
Finally, there have been some interesting discussions on the e-mail. I've compiled a webpage of the most useful so everyone can read them. I'll continue to update this page during the project.
Many thanks
Catherine

The residual PSF contamination for PSFs D and E

From: Rachel Mandelbaum, May 3, 2006

Hi Catherine and Richard,
Chris and I met earlier today, and we may have an explanation for the strange trends seen with PSF contamination, namely that for the PSF with ellipticity in e_1 most groups see contamination in e_2, and vice-versa.
First, I should note that in the SDSS data, when we work in survey coordinates, the scan strategy almost always leads to a PSF that has some ellipticity along e_1 (the scan direction) but far less ellipticity along e_2. And when we examine the results of our PSF correction pipeline, they are consistent with slight PSF contamination along e_1, NOT e_2. This implies that there may be some differences in the PSFs between the real SDSS data and STEP that is leading to a very different result in STEP.
Second, Chris noted that the PSFs D and E are fairly skewed. Do they correspond to some real PSF of a real instrument? I know the non-elliptical ones correspond to Subaru but it's not clear where this one came from.
Third, looking both at the starfields and at the figure showing the PSFs in the paper draft, it seems clear to both of us that the PSFs are actually chiral; e.g., psf D is elliptical along e_1, but it's not symmetric about either the horizontal or vertical axis; likewise psf E is elliptical along e_2, but it's not symmetric about the 45 degree rotated axes. PSF D seems to have some strange bumps that may lead to a slight e_2 component, and vice versa for E, if the picture in the paper is really an accurate representation of it. So this could lead to some of the weird effects we see with the ellipticity seeming to contaminate the wrong component. I also think this problem could possibly have something to do with the methods of *measuring* the PSF. In other words, PSFs D and E have strong ellipticities in the e_1 and e_2 directions, respectively, and most people's PSF measurements are picking that up easily and removing it fairly successfully; but the small details that lead to chirality are a bit more subtle and perhaps not modeled (and therefore removed) as well. I'm in the process of investigating this with our pipeline now, but it may be worth seeing if there is some way of checking it for all the methods -- unfortunately everything I can think of is expensive in terms of computer time.
In any case, it seems clear from the comparison of how our method does on SDSS vs. on STEP that different PSFs can drastically change the PSF contamination, in component and likely in magnitude and sign as well. I suspect this is true for others, and should be pointed out in the paper so someone doesn't just look at it and say "oh, these methods all have weird PSF contamination, so clearly there's something wrong with all of them."
Feel free to pass this information along to other STEPpers if you feel it is useful.
Rachel

Firstly, don't worry about my saying unnecessarily bad things in the paper. I also want lensing to work! And, if STEP1 is anything to go by, draft copies are going to be bouncing around for a long time, and there'll be plenty of opportunities to tweak text. In fact, I'll incorporate some of your suggestions now.
Right! Back to frank discussion. I agree with your hypothesis that the e1/e2 contamination is likely to come from the PSF measurement. With perfect centering, and in the absence of pixellisation, the PSF really does have zero quadrupole moments, regardless of the radial weight function. The skewness and the blobs come from other structure that does not contribute to quadrupole moments. However, since you point out that the PSF is skewed, it's going to be difficult to guess the centroid. Any offset will certainly produce non-zero moments. Pixellisation will make centering even more difficult. I don't think that it has any more effect on the moments - after all, the PSF is measured from many stars, all at different pixel centres, and then (in effect) averaged. That is backed up by the almost identical results for e2 in PSF D as e1 in PSF E.
The PSF was originally from a Subaru data, but tweaked a little to make the ellipticity nice. I obviously rotated it. I also rotated individual parts of the quadrupole moments (m=2 shapelet coefficients) so that they all lined up with the first one. However, I didn't touch anything to do with the skewness (lower order shapelet coefficients) or substructure (higher order shapelet coefficients). That's real! In conclusion, all I can say is that the PSFs vary an enormous amount between telescopes. In fact, Subaru is proving very good for lensing. For the same FWHM, the Subaru PSF has *much* smaller wings than WHT, CFHT or even Keck.
Thanks for the ideas and comments. Please keep them coming!
Richard

Reiko,
I've thought again about Mike's comments during the telecon concerning m=6 and 10 shapelet coefficients. These don't contribute to quadrupole ellipticity (sin(2theta)sin(6theta) integrates to zero - that's why Fourier transforms work). So, with perfect centering, and in the absence of pixellisation, the PSF really does have zero quadrupole moments, regardless of the radial weight function. The blobs that you mention (and also some skewness) do not contribute to quadrupole moments.
However, this structure will make it more difficult to guess the centroid. Any offset will certainly produce non-zero moments. Pixellisation will make centering even more difficult. I don't think that pixellisation has any more effect on the moments - after all, the PSF is measured from many stars, all at different pixel centres, and then (in effect) averaged. That is backed up by the almost identical results for e2 in PSF D as e1 in PSF E. What do you think?
I'll look forward to your Airy ring test findings. Please let me know what you see - any explanations to go in the STEP paper are more than welcome.
Richard