Estimating the Redshift Distribution of Faint Galaxy Samples

Abstract

We present an empirical method for estimating the underlying redshift distribution N(z) of galaxy photometric samples from photometric observables. The method does not rely on photometric redshift (photo-z) estimates for individual galaxies, which typically suffer from biases. Instead, it assigns weights to galaxies in a spectroscopic subsample such that the weighted distributions of photometric observables (e.g., multi-band magnitudes) match the corresponding distributions for the photometric sample. The weights are estimated using a nearest-neighbor technique that ensures stability in sparsely populated regions of color-magnitude space. The derived weights are then summed in redshift bins to create the redshift distribution. We apply this weighting technique to data from the Sloan Digital Sky Survey as well as to mock catalogs for the Dark Energy Survey, and compare the results to those from the estimation of photo-z's derived by a neural network algorithm. We find that the weighting method accurately recovers the underlying redshift distribution, typically better than the photo-z reconstruction, provided the spectroscopic subsample spans the range of photometric observables covered by the photometric sample.

Figures (9)

• Figure 1: Distributions of: $i$ magnitude ( left panel); redshift $z$ given $i$-magnitude for $i=$ 20, 22, 24 ( middle panel); and galaxy type $t$ ( right panel) for the fiducial DES mock catalog. Lower (higher) values of $t$ correspond to early (late) spectral types, and the $t$ distribution shows evidence of bimodality.
• Figure 2: Distributions of $r$ magnitude ( left panels), $r-i$ color ( middle panels), and spectroscopic redshift $z_{\rm spec}$ ( right panels) for each spectroscopic catalog used with SDSS photometry. Also shown in the left panels are the total numbers of galaxies in each spectroscopic sample, counting repeated objects.
• Figure 3: Idealized magnitude-redshift hypersurface for $N_m=2$ magnitudes. Without degeneracies, the hypervolume in magnitude space surrounding a galaxy, $V_{m} \propto d_m^{N_m}$, corresponds to an approximate redshift interval $\Delta z$. Whereas empirical photo-z methods usually make this implicit assumption, the weighting method works under more general conditions.
• Figure 4: Distributions of magnitudes $grizY$ and colors $g-r$, $r-i$, $i-z$, $z-Y$, for the DES mock photometric catalog and for the first spectroscopic training set. Grey regions indicate the distributions in the photometric sample, horizontal hatched regions indicate those for the spectroscopic training set, and the solid black histograms are those for the weighted training set.
• Figure 5: Left panel: Photometric redshift $z_{\rm phot}$ vs. spectroscopic redshift $z_{\rm spec}$ for a random sampling of galaxies in the DES mock catalog. Photo-z's were computed using the neural network algorithm described in Appendix A, using the first spectroscopic training set described in the text. The dashed and dotted curves are the contours containing $68\%$ and $95\%$ of the galaxies in narrow bins of $z_{\rm spec}$. Also indicated are the overall rms photo-z scatter $\sigma$ and $68\%$ confidence region $\sigma_{68}$ (see their definition in the text). Right panel: Redshift distributions. The shaded grey region shows the redshift distribution of the photometric sample that we are aiming to reconstruct. The horizontal hatched distribution shows the redshift distribution of the spectroscopic training set corresponding to the magnitude and color distributions shown in Fig. \ref{['fig:all.mag.col.dist.DESmock.tflat.igfdf']}. The solid black histogram shows the reconstructed redshift distribution using the weighting method. The dotted lines show the neural network photo-z distribution of the photometric set, showing peaks due to photo-z biases. Also indicated are the $\chi^2$ and KS statistics for both the weighting method and the photo-z distribution.