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Some Shear Results From SDSS Runs 752 & 756 | |
| Some Shear Results From SDSS Runs 752 & 756 | ||
| Some Shear Results From SDSS Runs 752 & 756 | ||
| Some Shear Results From SDSS Runs 752 & 756 |
For our analysis of shear in SDSS we have used shear estimators computed with an algorithm developed by Bernstein, Smith, Jarvis, and Fischer (in preparation) and a lot of the ancillary work is done by PHOTO. It is interesting to compare with other shear estimator algorithms which are commonly in use. Another method is the imcat suite of routines developed by Nick Kaiser. Here we present a comparison of our method with an IMCAT derived algorithm on the same set of SDSS data. For short we denote our method by FB (Fischer-Bernstein), these estimators were computed at U. Michigan (see their SDSS Weak Lensing Homepage). A description of this data product is given here. These are the same shear estimators used in the recent SDSS Galaxy-Galaxy Lensing publication (Paper 34). The comparison method is a modification of IMCAT by Ludovic van Waerbeke which include SExtractor components, which was in turn modified by our own Istvan Szapudi for use with the SDSS data.
The data used was from SDSS imaging runs 752 and 756, using columns 2 and 3; and fields 294-303 for run 752 and field 480-489 for run 756. Actually field 481 for column 2 run 756 was missing so there are a total of 39 frames used. In both case only the r' images were used. Good FB objects were gleaned using the criteria and all IMCAT objects were used. I then matched objects with the requirement that the object centers be within one pixel of each other. This yielded 8208 matches, leaving 774 unmatched FB objects and 20112 unmatched IMCAT objects. With a 1 pixel cut there is no confusion due to crowding. All objects are indicated in the following table.|
Table of the # of objects in each field. On top is the # of matched objects and below the # of unmatched FB/IMCAT objects. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
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In order to better understand which objects we are considering let us first consider the relationship of the magnitude system of the two catalogs. Note that the FB catalog magnitudes are based on PHOTO and are calibrated, while I don't think the IMCAT catalog magnitudes are calibrated at all. Below we plot the two magnitudes for the matched objects.
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Scatter plot of the FB and IMCAT magnitudes for the matched objects. No extinction correction has been applied. |
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Adjusting the magnitude systems we now plot the distribution of matched and unmatched objects.
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Histograms of the (adjusted) magnitudes of the matched FB (black), matched IMCAT (magenta), unmatched FB (cyan), and unmatched IMCAT (green) objects. |
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The large number of unmatched IMCAT objects is partly a result of the limited magnitude range considered by the FB objects (the IMCAT catalog goes much deeper) and (I believe) because the IMCAT catalog includes stars. Note that the unmatched IMCAT objects dominate at the dim and bright end and are never small compared to the matched IMCAT objects. Also note the spurious number of r'fb>24 objects apparent in both matched and unmatched objects. For r'fb<24 the unmatched objects are a small fraction of the total. The greater number of IMCAT objects at all magnitudes raises the question of whether we loose shear signal because FB has a higher standard for detecting objects and hence detects fewer galaxies - but if the excess are all stars or has poor shear estimators then we would not be loosing signal.
To compare the shear estimators by the two algorithms we can compare the relative magnitudes of the two shear estimator for the same (matched) objects as well as comparing the orientation of the estimators. It is also interesting to see how the estimators vary with the brightness of the objects. To do this we have divided the matched sample in half, cutting at the median r'fb, 21.1. Objects w/ r'fb>21.1 are called bright and objects with r'fb>21.1 are called dim. Let us begin by comparing the magnitude of the shear estimators in the following figure.
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Scatter plot of the magnitude of the shear estimator for matched objects from the two catalogs. The blue dots are for bright objects and the red dots are for dim objects. |
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From this figure we see that the two estimators are rather different, showing qualitative differences as well as statistical variation. Some points
While these points do not tell us which is a better estimator, the long tail in the distribution of IMCAT estimators suggest that the "noise" induced by these estimators will be larger than that of the FB estimators. In any case given the qualitative difference in these estimators it is is clear that the differences in the shear estimators in "shear space" (i.e. including orientation) will be quite different as well.Next consider the orientation the the estimators. This is given by a position angle on the sky relative to the row direction, which we denote by PA. Since rotating shear by 180° yields the same shear, we restrict PA to the range [-90°,+90°].
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Scatter plot of the position angle (PA) of the shear estimator for matched objects from the two catalogs. The blue dots are for bright objects and the red dots are for dim objects. |
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The PAs of both estimators seem uniformly distributed in angle, showing no visible preference for alignment wrt the pixel direction, and the two PAs tend to be close to each other. The PAs agree much better than the magnitudes and from this plot there is visible difference (to my eye) between the bright and dim galaxies. We may take a closer look with a histogram of the angle between the two PAs.
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Histogram of the relative position angle of the two shear estimators for from the two catalogs. The black line is for the entire matched sample, the blue line is for the matched bright objects, and the red line is for the matched dim objects. |
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We see that the difference in PAs between the two estimators is sharply peaked near zero (good) although the correlation indicated by the cosines is far from perfect (unity). Also we see that the PAs agree significantly better for the bright galaxies than the dim ones (not surprising) although the correlation is not that different, dim decreased by 23% relative to bright.
So far nothing we have looked at addresses the issues of bias, accuracy, or systematics errors in the shear estimators. Since we are not using a field with known shear it is difficult to address the 1st two issues; and in any case this is too small a sample to measure the level of shear we are interested in for SDSS. However we can get some handle on systematic effects if they are large enough; since we can look for preferential alignment of the shear estimator toward the direction of the ellipticity of the PSF. FB provides us with an estimator of the ellipticity of the PSF (see here) and we have used this to compute the position angle of the PSF: PAPSF. This PA should be uncorrelated with the true shear and hence uncorrelated with the PA of the shear estimator.
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Scatter plot of the PA of the two shear estimators relative to the PA of the PSF. The blue dots are for for bright objects, and the red dots are for dim objects. |
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The above plot shows no visible preference for alignment toward or away from the PSF (good) but we can look more closely with a histogram.
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Histograms to the PAs of the two shear estimators relative to the PA of the PSF. The black line is for the matched FB PAs and the magenta line is for matched IMCAT PAs. |
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Noting the vertical range of the plot we see that the PAs relative to the PSF are nearly uniformly distributed. However the correlation given by the averaged cosine relative to it's error show that there is a significant tendency for the PA to be directed in the same direction as the PSF ellipticity; indicating an under-correcting for the anisotropic PSF. The magnitude of this correlation is about the same for the FB and IMCAT estimators when considering all matched objects, but in the FB case it appears there is no significant difference in alignment between dim and bright objects; while for the IMCAT case the alignment seem stronger for bright objects than for dim objects, which is surprising. In fact the dim objects show only a barely significant, 1.3-sigma, correlation with the PSF. The fact that these correlations are inconsistent with zero indicates the presence of some systematic errors.
Note that one can always adjust the smear polarizability (or similar quantity) to make this correlation zero. This might mess up other properties of the estimator such as error or bias.
One way of estimating the magnitude of the systematic error in the shear is to weight the cosine by the shear estimator magnitude. If the PSF anisotropy has constant orientation over the region and one used an unweighted sum to estimate the average shear, then this average provides and indicator of the bias in this estimate of the average shear. This average for the FB estimator is 0.023±0.006 and for the IMCAT estimator is 0.017±0.36. While this bias is not significantly the different between the two estimators, the errors in for the IMCAT estimator is so large that it is consistent with zero. The reason for this is that the average in the IMCAT case is dominated by a relatively small number of objects with large shear estimator size, while the FB estimators never have large size. This is one manifestation of the larger noise of the IMCAT estimators discussed above. The fact that the average is not itself much larger for the IMCAT estimator indicates that the objects with larger estimators are not as correlated with the PSF as the objects with smaller estimators. We might find from a larger sample that the IMCAT estimators do introduce a smaller bias than the FB estimators.
From the previous paragraph one might worry that the estimators as they stand would always be dominated by systematics if the shear is smaller than 0.02, which would be rather discouraging. However this would only be true if the PSF maintained the same orientation over the region in which one wants to measure the shear. Fortunately this is not the case for SDSS data and the systematics error examined here falls off as one goes to larger and larger areas as the PSF bias averages out, just as the shape noise averages out. Still these results suggest that some improvement can be made in the estimators.
From this first comparison of the FB and IMCAT estimators we do not have much indication that one is better than the other, however we do see that they are significantly different in some qualitative features.