Faint object optimization

After introducing the basic functionality of the main science pipeline components in the last section, we will now turn to a discussion of the effects of the implemented reduction optimizations. Figure 7.6 illustrates the different optimization schemes for 10_×10 _co-added image cutouts centered on the calibration cluster RX J0018.2+1617 at z= 0.55, displayed with high image contrast. The upper panels show the sum images after the first reduction loop, whereas the lower panels have the results after the iterated background subtraction. Going from left to right adds theoptimal weightingscheme and the fractional pixel offsets. Hence the upper left panel displays the reduction quality achievable with a single loop (quicklook) reduction, and the lower right panel shows the same data with all optimizations incorporated requiring about the threefold CPU time.

We will first consider the effects of theiterated background subtraction,i.e.the difference between the upper and lower panels of Fig. 7.6. By masking out all objects and their halos during the second reduction loop, the sky background determination is unbiased compared to the systematically slightly higher median levels when source fluxes are included. This lower background value at the object positions results in additional object flux when the sky background is subtracted. The iterated background subtraction hence recovers object flux and consequently makes the sources brighter. This flux recovery is expected to be

weighting & fractional pixel stacking it erat e d ba c kgr ound s ubtr acti on

Figure 7.6: Faint object optimization. The upper left panel shows a high-contrast 10_×10 _{zoom on} the calibration cluster RX J0018.2+1617 at z= 0.55 as seen in the stacked sum image after the first reduction pass (see panel 5 in Fig. 7.5). Going to the right panels adds fractional pixel offsets and optimal weighting. The lower panels show the image results after the second iterated sky subtraction. The final best image including all optimizations is thus placed in the lower right panel. While the iterated sky subtraction recovers a significant amount of flux of faint and extended objects, the optimal weighting scheme increases the limiting magnitude by improving the signal-to-noise ratio of the faint sources (see Fig. 7.7 for quantitative results).

more significant for extended low surface brightness objects compared to stellar point sources. A useful approximation to convert measured magnitude offsets into flux differences ∆f=f2−f1 for small variations (∆m¿1) is given by the relationship

∆m=−2.5·log à f2 f1 ! =−2.5·log à 1 +∆f f1 ! ≈ − 2.5 ln(10) · ∆f f1 ≈ −1.1· ∆f f1 , (7.4) implying that small magnitude differences reflect approximately the fractional flux differ- ence.

Figure 7.7: Effects of advanced reduction schemes. Top panels: Difference photometry in the H-band for various reduction stages as shown in Fig. 7.6 using the same deep detection image. Left: Total magnitude difference between the sum image with iterated background subtraction (here H1, lower left panel in Fig. 7.6) and the sum image after a single reduction loop (here H2, top left panel in Fig. 7.6). The dashed red line represents a zero offset (i.e. the same measured magnitudes), black crosses show all objects in the FoV, big blue circles represent the galaxies seen in Fig. 7.6 (i.e.cluster members of RX J0018.2+1617), and the green solid lines follows the median offset taken in 0.5 mag bins. At H>16, the recovered flux from the iterated background subtraction becomes noticeable and increases to a median difference of 0.1–0.15 mag (about 10-15% in flux) between H magnitudes of 18–21. At H>21 the difference drops back to zero which reflects the depth of the object mask. Note that the recovered flux is drastically increased for the faint extended cluster galaxies (blue circles) compared to the median difference. Right: Same plot for the difference between the final weighted image (H1, lower right panel in Fig. 7.6) and the iterated background subtracted image without weighting (H2). The total magnitudes are now consistent, since weighting cannot change the flux, but only the signal-to-noise ratio of the sources. Left bottom panel: logN–logS of the final reduced image (black solid line) and the first sum image (red solid line) in a field with significant seeing and transparency variations during the observations. The systematic vertical offset is due to the iterated background subtraction. The difference in total number of detected sources on the other hand can be attributed to the optimal weighting scheme. The effect is separated in the right panel, which shows the results of the best final image (black) and the non-weighted image (red). The limiting magnitude is significantly improved by the weighting and the total number of detected sources can be boosted by as much as 30–40% in fields with strongly varying conditions.

The effect of the iterated background subtraction is quantified in the upper left panel of Fig. 7.7 where the difference photometry with and without the second reduction loop is shown against the total object magnitudes. For both images the measurements were performed in the exact same apertures by using a common deep multi-band detection image (see Sect. 7.4.1), i.e. any offsets are due to object flux differences. No measurable effects would thus imply zero offsets (dashed null line), and more object flux after the iterated loop will manifest itself in a positive magnitude offset (the brighter, i.e. smaller, magnitudes are subtracted). The measurements for all objects are plotted as black dots, for the cluster members in Fig. 7.6 as blue circles, and the median averaged differences in 0.5 mag bins are traced by the green solid line. Following this line shows practically matching magnitudes out to H∼15, which is expected due to the photometric calibration procedure using bright 2MASS stars as discussed in Sect. 7.4.2. At H>16, the recovered flux from the iterated background subtraction becomes noticeable, i.e.the magnitudes are systematically brighter, which increases to a median difference of 0.1–0.15 mag (about 10– 15% in flux) between H magnitudes of 18–21. At H>21 the averaged difference drops back to zero reflecting the depth of the object mask which does not cover the faintest sources anymore. Note that the recovered flux is drastically increased for the cluster galaxies (blue circles) compared to the median difference. The difference for the bright cD galaxy with its extended halo is 0.1 mag and in the important regime between 18< H <21 the effect grows to 0.2–0.4 magnitudes or equivalently to approximately 20–40% in flux. This pronounced effect for the cluster environment, i.e. for the main science applications, has two reasons. (i) Most cluster galaxies are faint extended low surface brightness objects and (ii) the large halo of the central cD galaxy biases the background modelling in its vicinity, which is the main contribution for this particular environment.

The effects of fractional pixel offsets and the optimal weighting are investigated in combination, since both optimizations only improve the signal-to-noise ratio of faint sources but do not change the flux of the objects. This can be seen in the right panel of Fig. 7.7 where now the difference photometry is applied to the fully optimized image (lower right panel of Fig. 7.6) and the non-weighted version (lower left panel of Fig. 7.6). The median net flux differences is now consistent with zero and the increased scatter at the faint end can primarily be attributed to photometric measurement errors dominated by the image with lower SNR.

The main contribution of the stacking procedure using fractional pixel offsetsis the im- proved reconstruction of the natural seeing-limited point-spread function (PSF). In Fig. 7.6 this is best seen for the stellar source at the right edge center of the panels and the fainter galaxies which change from box-like object cores in the left panels to symmetrically round PSF shapes (for the star) on the right. The recovered natural PSF shape is also the most compact flux configuration with an expected improved FWHM of approximately the av- erage displacement error of O(0.100_{). On the other hand, the implemented fractional pixel} offset scheme introduces correlated noise, since the flux from a single independent pixel is now distributed onto four pixels of the master grid. A main concern would be spuri- ous detections introduced from correlated single pixel noise, e.g. an uncorrected cosmic or flickering pixel, that passed the object detection threshold of at least four adjacent pixels

more than one standard deviation above the background (see Sect. 7.4.1). However, for a stacked sum of only ten images (typically 20–80) this would already require a single image pixel-noise outlier of σsingle>∼4

√

10≈12 above the background, which implies that it would have been easily detected and removed by the cosmics removal procedure. Even a constantly hot pixel not included in the bad pixel mask would most likely not lead to a spurious source since (i) it would show up in the sky model and be subtracted with it, and (ii) these outliers would again be removed from thecosmics routine since they do not align in world coordinates. In conclusion, the fractional pixel offsets procedure is not expected to cause any significant number of spurious source detections through correlated noise.

The last step towards the maximally achievable image depth is the implementedoptimal weighting scheme. The measurable effects of this optimization strongly depend on the variability of the observing conditions according to Equ. 7.3. However, since the execution time per field and filter is typically 30–80 min long, changes of the external conditions are likely, leading to significant improvements if weighting is applied. Variations that modulate the weighting factors include (i) pronounced background changes at the beginning or end of the night or due to the moon setting or rising, (ii) transparency drops induced by partial cloud coverage, and most sensitively (iii) seeing variations in the atmosphere or the telescope dome,e.g. if the primary mirror is warmer than the ambient air.

The optimization of the signal-to-noise ratio at the faint source end has two important effects. Firstly, objects can be characterized with higher photometric accuracy, i.e. the measurement errors decrease. Secondly, very faint formerly undetected sources are now pushed over the detection limit requiring a minimum fixed SNR. This latter effect is of particular importance since the total number of sources increases rapidly with the flux limit as Ntot ∝flim−3/2 (Equ. 3.26) and allows to probe the galaxy luminosity function in

clusters to fainter levels. The number counts for a field observed under strongly varying conditions in the lower right panel of Fig. 7.7 illustrates the achievable gain. The black curve shows the detection results for the image with the applied optimal weightingscheme, the red line gives the outcome on the non-weighted image. Up to H<_∼20 the number counts are basically identical but at H∼20.5 the red non-weighted curve turns over and reaches its plateau. The black weighted line on the other hand exhibits a significantly improved limiting magnitude, which in turn boosts the total number of detected sources for this field by almost 40% at no extra observational cost. The depth optimization via fractional pixel offsets and optimal weighting thus takes effect in the magnitude range 20_∼<H_∼<22, which includes about half of all detectable sources.

For comparison, the lower left panel of Fig. 7.7 shows the number counts in the same field but with the first loop non-weighted (quicklook) image used for the red line. The num- ber counts now exhibit a systematic vertical offset starting at H>16 due to the discussed iterated background subtraction, which is the consequence of the objects being systemati- cally brighter, hence more sources exist at a given fixed magnitude.

In summary, the implemented optimization schemes17 _{for the science-grade reduction}

17_{The faint object optimization effects in this section have been discussed using H-band data, since this}

pipeline have an important impact on the targeted science applications. The iterated background subtraction corrects the systematic biasing of the total object magnitudes, whereas the combined effects of thefractional pixel offsetsand theoptimal weightingscheme improve the limiting depth of the data. With respect to the intended distant galaxy cluster identifications the achieved improvements are of prime interest. (i) The measured object magnitudes need to be accurate in the full range 16<_∼H<_∼21 in order to derive accurate colors and thus allow a red-sequence identification and model comparisons. (ii) The increased number of detected cluster galaxies enhances the cluster signature, eases the evaluation, and increases the redshift grasp.