What was wrong in the previous version of data?

Fig.1 compares the photometry in our old (v1) and new (v2) catalogs (gray dots). Faint objects in the old catalog appear to be 0.1 mag fainter than they should be. This effect only has weak aperture dependence. It also exists on galaxy photometry derived from SExtractor auto aperture, as well as photometry of stars. This is noticed by several groups when they compare our photometry with photometry independently derived from other instruments. Our new reduction resolves this issue. Here we explain the nature of this systematic effect, and warn the community that such effect can exist in other deep imaging data reduced with a similar method.

Fig.1 -- Systematic effect in the old WIRCam Ks-band photometry. The gray dots show the comparison of galaxy photometry in the old and new reduction. The photometry here is measured with 2" apertures and the systematics is weakly dependent on the aperture size. The diamonds show the simulated systematic effect (see text).

Difference in Old and New Reduction

In our reduction, we flattened a sequence of dithered exposures with sky flat. In the old reduction, a flat field image is created by median-combining all images in the dither sequence after the images are normalized to 1. We call this "single flat," since all images share the same flat field image.

In the new reduction, each image in a dither sequence has its own flat field image. The flat field image is created by median-combining all other images in the dither sequence (i.e., not including the image to be flattened). We call this "multi flat," since all images have their own associated flat field images.

Bias in Single Flat

We believe the systematic effect shown in Fig.1 is caused by a bias in the single-flat method. In single flat, any objects existing in the images upward bias the flat at their own locations, and therefore bias their own fluxes. (This is avoided in the multi flat.) To understand this effect, we carried out Monte Carlo simulations.

In our WIRCam observation and reduction, several to less than 15 dithered exposures can be grouped together and processed in one reduction run. We thus choose to simulate 11 dithered exposures as a representative case, and see what is the bias in a flat field median-combined from the 11 exposures. In one Monte Carlo realization, 11 pixels are assigned Gaussian random numbers according to the typical background brightness (240 uJy per pixel) and the typical background fluctuation (4.5 uJy per pixel) in our Ks-band observations. Then a celestial object of a specific magnitude is assigned to one of the 11 pixels. A median value of the 11 pixels is calculated and compared with the median without such an object. Because of the bias caused by the celestial object, the median is systematically larger than the case where there are no objects. In other words, flat field generated by median-combining images is always overestimated at the locations where there are objects.

Fig.2 -- Upward bias in a median-combined flat field caused by objects in the image.

Fig.2 shows the above simulated upward bias. This bias is fairly small. For objects brighter than the background fluctuation, the upward bias is only 0.0023 mag. For objects close to our detection limit (25 mag), the bias is an extremely small 0.0002 mag. From the point of view of bias in flat field, this effect should not have any noticeable impact to the photometry. Furthermore, this effect should be larger at the bright end, as bright objects bias the flat field more strongly. The picture changes entirely if we take into account the very bright sky background in the near-IR.

Bias in the Single-Flat Flux

In the actual reduction or photometry, there is always a step of subtracting background. The background region around an object is not affected by the above bias, and thus would be slightly brighter than the background underneath the object after the flat fielding. When such an over-estimated background is subtracted from the object, it takes away some flux from the object. Because the background in the near-IR is very high, even though the bias in flat is small, it can result in a significant downward bias in the measured flux.

Fig.3 -- Bias in flux after taking into account background subtraction.

Fig.3 shows the result of such biased background subtraction. At the bright end, because the object brightness is comparable to the background, so the downward bias (positive mag in Fig.3) in flux is comparable to the upward bias in flat, which is less than 0.01 mag. On the other hand, at the faint end, the background is much brighter than the object. A small error in the background estimate can lead to a huge error in the object flux. The simulation in Fig.3 shows that such a bias is approximately 0.15 mag under the assumed conditions.

We compare the simulated result (Fig.3) with the actual data, which is shown by diamonds in Fig.1. The bias is reverted because of the definition of the y-axis. In addition, the flux bias is shifted to exactly zero at the bright end of Fig.1, instead of having a small non-zero value in Fig.3. This is because in both the old and new reduction, the bright-end fluxes are calibrated with 2MASS objects and any systematic effects would not show up at the bright end.

In Fig.1, the behavior of the simulations (diamonds) and data (gray dots) is very similar. There is a noticeable difference in where the systematic effect starts to appear (Ks ~ 18 for the data vs. Ks ~ 16 for the simulations). This has something to do with the assumed morphology of objects and crowdness of the field. In the simulation, we assume one object just occupies one pixel. If we assume a more realistic surface brightness profile, the diamonds would shift horizontally to the left. We also assume there is only one object in the field, so only one pixel out of the 11 contains an object and flat is only biased at the location of the object. In reality, multiple objects can exist both at the locations of objects and background region. In addition, in real observations, the number of dithering, the sky background, and the background fluctuation are all variable. Simulating these effects is out of the scope of this study.

Conclusion

The broad consistency between the simulations and data implies that the systematic effect is caused by bias in single flat. This is corrected in our new reduction with the multi-flat method. The newly released data should be free from such an effect.