Target Selection and Validation of DESI Luminous Red Galaxies

The LRG targets are selected from the DESI Legacy Imaging Surveys Data Release 9 (Dey et al. 2019; D. Schlegel et al. 2023, in preparation, hereafter LS DR9), specifically the g, r, and z optical bands (4000–10000 Å, without the i band at 7800 Å) and forced photometry of the Wide-field Infrared Survey Explorer (WISE, Wright et al. 2010) W1 band in the infrared (3.4 μm). The imaging footprint is shown in Figure 2. Table 1 lists the sky areas and other summary information about the DESI LRG sample.

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The optical grz imaging consists of two regions separated at decl. δ ≃ 32°, with each region observed by different telescopes (with similar grz filter sets). The southern (δ ≲ 32°) part of the imaging footprint (hereafter "the South") is observed by DECam on the 4 m Blanco Telescope at the Cerro Tololo Inter-American Observatory (CTIO). Most of the observation is done by the DECam Legacy Survey (DECaLS, Dey et al. 2019), and data from other observations, most importantly the Dark Energy Survey (DES, The Dark Energy Survey Collaboration 2005), are also used. The northern (δ ≳ 32°) part of the imaging footprint (hereafter "the North"), which is inaccessible from CTIO, is observed by two telescopes at the Kitt Peak National Observatory in two surveys: the Beijing–Arizona Sky Survey (BASS, Zou et al. 2017) observed in the g and r bands using the 90Prime Camera on the 2.3 m Bok Telescope, and the Mayall z-band Legacy Survey (MzLS) observed in the z band using the Mosaic-3 Camera on the 4 m Mayall Telescope. The Mayall Telescope has since been repurposed for DESI.

For the same photometric band, small differences in the filter sets, detectors, observing conditions, and image processing between the North and the South cause their photometry to differ slightly. For the LRGs, these differences are often nonlinear functions of magnitude and color, and cannot be quantified by a constant offset. Thus we find it necessary to implement slightly different color cuts for each region. The cuts are optimized using both photometric and spectroscopic data to achieve uniform density and redshift distributions across the footprint. The specific color cuts are described in the next subsection.

The LRG targets are selected using optical photometry in the grz bands and near-infrared photometry in the WISE W1. The LRG selection cuts for the South, shown in Figure 3, are

$$\begin{eqnarray}&&{z}_{\mathrm{fiber}}\lt 21.60\end{eqnarray} \tag{ 1a }$$

$$\begin{eqnarray}&&z-W1\gt 0.8\times (r-z)-0.6\end{eqnarray} \tag{ 1b }$$

$$\begin{eqnarray}&&(g-W1\gt 2.9)\ \mathrm{OR}\ (r-W1\gt 1.8)\end{eqnarray} \tag{ 1c }$$

$$\begin{eqnarray}&&\begin{array}{l}((r-W1\gt 1.8\times (W1-17.14))\ \mathrm{AND}\\ (r-W1\gt W1-16.33))\ \mathrm{OR}\ (r-W1\gt 3.3)\end{array}\end{eqnarray} \tag{ 1d }$$

where g, r, z, and W1 are magnitudes and z_fiber is the z-band fiber magnitude, i.e., the magnitude corresponding to the expected flux within a DESI fiber. ³⁸ Throughout this paper, unless otherwise specified, all the magnitudes are in the AB system and are corrected for Galactic extinction ³⁹ based on Schlegel et al. (1998) with correction from Schlafly & Finkbeiner (2011).

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Equation (1b) utilizes the 1.6 μm (rest-frame) "bump" (John 1988; Sawicki 2002) to efficiently remove stars from the sample (as shown in the upper left panel of Figure 3), similar to the stellar rejection cut in Prakash et al. (2015), resulting in a low stellar contamination rate of ∼0.5% (see Section 4.2). Equation (1c) removes galaxies at lower redshifts while retaining high completeness of massive galaxies at z > 0.4. The sliding color–magnitude cuts in Equation (1d) function as luminosity cuts: as shown in the lower left panel of Figure 3, the r − W1 color is a good proxy for redshift, and the W1 magnitude limit that shifts with r − W1 effectively selects the most luminous (in the observed W1 band) galaxies at any redshift. For objects with r − W1 ≳ 3.3, which are the faintest LRGs in our sample and are near the faint limit (Equation (1a)), the r − W1 versus W1 sliding cut is dropped to boost the number density of the highest-redshift LRGs. The cuts are tuned so that the comoving number density of the selected sample is close to constant at 0.4 < z < 0.8.

Finally, Equation (1a) sets the faint limit for the sample to ensure a high redshift success rate for DESI spectroscopic observations (see Sections 4.3 and 4.4). We choose the fiber magnitude over the total magnitude as the faint limit because the former is much more strongly correlated with the spectroscopic S/N. Note that the fiber magnitude cut, which is effectively a morphology cut, could introduce systematics in both sample selection (e.g., systematics dependence on seeing) and theoretical modeling (as galaxy orientation could be aligned with the tidal field; e.g., see Lamman et al. 2022), and such effects will need to be carefully studied in the clustering analysis.

The photometry in the North is slightly different from that in the South, and the selection cuts are tuned to match the number density and the redshift distribution in the South. The cuts for the North are

$$\begin{eqnarray}&&{z}_{\mathrm{fiber}}\lt 21.61\end{eqnarray} \tag{ 2a }$$

$$\begin{eqnarray}&&z-W1\gt 0.8\times (r-z)-0.6\end{eqnarray} \tag{ 2b }$$

$$\begin{eqnarray}&&(g-W1\gt 2.97)\ \mathrm{OR}\ (r-W1\gt 1.8)\end{eqnarray} \tag{ 2c }$$

$$\begin{eqnarray}&&\begin{array}{c}((r-W1\gt 1.83\times (W1-17.13))\,{\rm{A}}{\rm{N}}{\rm{D}}\\ (r-W1\gt W1-16.31))\,{\rm{O}}{\rm{R}}\,(r-W1\gt 3.4).\end{array}\end{eqnarray} \tag{ 2d }$$

In addition to the aforementioned cuts, we apply the following quality cuts (e.g., to remove objects with bad photometry) in the target selection pipeline (Myers et al. 2022). We require that each object be observed at least once in all of the three optical bands. We also require that the inverse-variance values for r, z, and W1 fluxes be positive; this rejects problematic imaging data. A small number of stars are not removed by the aforementioned selection cuts due to saturation in the imaging. We remove them by requiring z_fibertot > 17.5, and if Gaia (Gaia Collaboration et al. 2018) photometry is available, we also require G_Gaia > 18.

The DESI target selection pipeline also removes objects close to bright stars (in Gaia and Tycho-2), star clusters, and large galaxies, which are flagged by the LS DR9 MASKBITS ⁴⁰ 1, 12, and 13, respectively. These masks are very minimal and only remove regions with the worst contamination, and additional masking is needed for the clustering analysis. We describe these additional masks in Section 2.4.

The selection cuts for the Survey Validation and the 1% Survey samples are described in Appendices A and B, respectively.

In order to accurately model galaxy clustering (e.g., Zu & Mandelbaum 2015; Zhou et al. 2021) and study galaxy–galaxy lensing (e.g., Alam et al. 2017; Jullo et al. 2019), halo occupation distribution (e.g., Rodríguez-Torres et al. 2016), and evolution of the most massive galaxies (e.g., Bundy et al. 2017), it is desirable to have a large spectroscopic sample of strongly clustered galaxies with well defined stellar populations. The target selection cuts for the DESI LRG sample have therefore been optimized to select the most massive galaxies with a high degree of completeness. We define completeness as the ratio of the number of selected LRGs to the total number of galaxies brighter than the LRG magnitude limit (defined by Equations (1a) and (2b)). In this section we refer to objects that satisfy our stellar rejection cut (defined by Equation (1b)) as "galaxies."

The cuts defined by Equations (1c) and (2b) reject objects with low redshifts while retaining the most massive galaxies for redshifts over 0.4. The design of these cuts was guided by estimates of stellar masses of galaxies obtained using a random forest algorithm (Breiman 2001) trained on DESI Legacy Imaging Survey photometry and stellar masses of galaxies from Bundy et al. (2015). A detailed description of the method used to obtain the stellar masses can be found in Appendix C.

Figure 4 shows the stellar mass completeness of the DESI LRG sample as a function of both stellar mass and redshift. As spectroscopic redshifts are only available for some of the selected LRGs and not the magnitude-limited sample, we use the photometric redshifts in LS DR9 ⁴¹ (Zhou et al. 2021) for this analysis. The selection cuts result in a sample that is highly complete for the most massive galaxies (i.e., ${\mathrm{log}}_{10}({M}_{\star }[{M}_{\odot }])\gt 11.5$) in the redshift range 0.4–1.0. The completeness decreases significantly for redshifts lower than 0.4 but the decrease is less severe for redshifts above 1.0. This high mass completeness is one of the defining characteristics of the DESI LRG sample and will aid a multitude of scientific studies.

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Here we describe the additional veto masks specifically optimized for the LRG targets to create a clean sample for the DESI clustering analysis. They are mostly masks of bright stars, although they also include masking for large galaxies and star clusters, etc. The veto masks comprise four separate sets of masks:

• 1.  
  unWISE (Meisner et al. 2019) pixel-level bitmask: we use all but bit five ("AllWISE-like circular halo") of the collapsed mask bits as listed in Table A4 of Meisner et al. (2019). We exclude bit five because these circular masks are not optimal (either too large or too small, depending on the magnitude of the star) for the LRG targets, and they are replaced by
• 2.  
  WISE circular geometric masks: these masks replace the "AllWISE-like circular halo" masks in unWISE. The relation between radius and W1 magnitude is optimized for the LRGs, so that the excess or deficit of LRG targets at the edge of the mask is less than ∼10%.
• 3.  
  Gaia/Tycho-2 circular masks: similarly, we obtain the relation between radius and magnitude for Gaia stars. We use stars in Gaia Early Data Release 3 (EDR3, Gaia Collaboration et al. 2021) supplemented at the bright end (where Gaia photometry is unreliable) by Tycho-2 and Two Micron All Sky Survey (2MASS) photometry.
• 4.  
  Custom masks: these are masks for large galaxies, star clusters, and planetary nebulae that were not masked by the LS DR9 MASKBITS, and regions with other imaging artifacts (identified from visual inspection of regions with high LRG densities). The total area of the custom masks is much smaller than the area of the other masks.

We describe sets 2–4 in more detail in Appendix D. The combined veto masks remove 8.5% of the DESI footprint. Within the masked (contaminated) area, the LRG target density is 1100 deg⁻² and the stellar contamination rate (based on spectroscopic classification) is ∼10%, compared to the 605 deg⁻² density and ∼0.5% stellar contamination in the unmasked ("clean") area. The stellar contamination rate is much higher in the masked region because the photometry (especially in the W1 band) used in the stellar rejection cut (Equation (1b)) is contaminated by the nearby bright stars.

While the full LRG spectroscopic and target catalogs include objects flagged by the LRG veto masks, we recommend that those objects be removed for analyses that require a clean sample with uniform selection. In the rest of the paper, we only use the "clean" LRG sample (with the LRG veto masks applied) instead of the full target/spectroscopic sample.

Finally, we note that the LRG veto masks presented here are not definitive, and they may see further improvements, e.g., for the DESI Year 1 science analyses.

The variation of imaging properties (such as depth and seeing) over the footprint and the presence of astrophysical foregrounds (such as Galactic dust) can imprint on the density of galaxy targets. Here we examine the impact of these systematics on the LRG target density. The systematics properties that we examine here include depth (galaxy depth in grz and point-spread function (PSF) depth in W1), seeing (in grz), Gaia stellar density, and Galactic extinction E(B − V) (from Schlegel et al. 1998).

While the photometry used in target selection has been corrected for Galactic extinction, we include E(B − V) here because imperfections in the correction, e.g., due to errors in the dust map, can still bias the photometry and affect the target density.

We use the STARDENS values from Myers et al. (2022) for Gaia stellar density, and we use values from the imaging catalog ⁴² for all other systematics properties (GALDEPTH, PSFDEPTH, PSFSIZE, and EBV).

Figure 5 shows the dependence of LRG target density on the imaging and foreground systematics. The LRG sample is much brighter than the imaging detection limit (with the faintest LRG targets being at least ∼2 mag brighter than the median z-band 5σ detection limit), and the stellar rejection cut and the LRG veto masks efficiently remove the contamination caused by stars. Therefore the LRG sample is relatively robust against these imaging/foreground systematics. The density deviations caused by these systematics are almost all within ±5%.

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The systematics trends in Figure 5 can be almost completely removed via linear regression of the systematics properties:

$$\begin{eqnarray}&&{N}_{{\rm{predict}},k}={c}_{0}+\displaystyle \sum _{i=1}^{8}\displaystyle \sum _{l=1}^{{N}_{{\rm{rand}},k}}{c}_{i}{S}_{i,l},\end{eqnarray} \tag{ 3 }$$

where N_predict,k is the "predicted" number of LRG targets in the kth HEALPix pixel, S_i,l is the value of the ith systematics property of the lth random point, N_rand,k is the number of random points in the kth pixel, and c_i are the coefficients. We use the random catalogs in LS DR9. The "corrected" systematics trends are shown in Figure 6. For the linear regression, we include all systematics properties except stellar density, because (1) the stellar contamination rate is already very low and (2) the stellar density maps are inherently noisy and the stars in the Gaia catalog (on which the stellar density map is based) is much brighter than the stellar contamination in the LRG targets. Indeed we find little correlation with stellar densities after applying the systematics weights. The coefficients for the North and the South are different, but DECaLS and DES are treated as one sample in the linear regression and have the same coefficients. The LRG density in the DES region is ∼3% lower than the average density due to its deeper photometry, and the linear regression weights accurately predict this density difference.

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There is an unexpected dependence on E(B − V) that remains after correcting for depth and seeing dependence: the LRG density at E(B − V) ≲ 0.015 is more than 5% lower than average. We also see similar trends at very low E(B − V) in the other DESI tracers (which have very different selections and redshifts), so it is unlikely to be a statistical fluke. While we are not certain what causes this drop in target density at very low E(B − V), we speculate that it is caused by systematics in the Galactic extinction map. Specifically, the E(B − V) map from Schlegel et al. (1998) is based on dust emission in the far infrared (FIR), which may have been contaminated by FIR emissions from background galaxies. The remaining trends (at higher E(B − V)) might be due to effects dependent on the spectral energy distribution (SED): ideally, we would calculate the extinction coefficients for each galaxy based on its SED, but in practice we use a single stellar spectrum for calculating the extinction coefficients, and this could lead to systematic errors in the dereddened fluxes. We will examine the Galactic dust-related systematics in future investigations.

Note that while the aforementioned linear weights work well for the projected density of the full LRG sample, for subsets of the sample (e.g., in redshift bins) the coefficients and weights should be recomputed for each subset, because different subsamples are affected by the selection cuts differently (e.g., brighter subsamples might be more sensitive to the sliding cut in r − W1 versus W1 while fainter subsamples might be more sensitive to the z_fiber faint limit) and have different sensitivities to the different systematics.

There were no formal requirements on DESI targeting with respect to these systematic trends in target density. Instead, this analysis was used as a diagnostic to ensure no trends existed beyond expectations, e.g., from previous surveys. One can observe that the trends are generally more moderate than those found in BOSS and eBOSS (Ross et al. 2017, 2020). Further, we have demonstrated that a simple linear regression, similar again to that applied to BOSS/eBOSS, successfully models the trends. The exception is a relationship with E(B − V) that should be studied in more detail in future DESI work. The details and modeling here thus represent a first step in the process of determining the selection function for the DESI LRG sample, a vital component for producing 3D clustering measurements once the redshifts for the LRG sample have been measured, and a reference that will aid future steps.

Another source of imaging systematics is the zero-point (ZP) uncertainty—the uncertainty of the photometric zero-point for each exposure of the imaging data. While depth and seeing can be accurately measured and their effects on targeting can in principle be modeled, the ZP uncertainty is a systematic uncertainty that is mostly due to imperfect modeling of the observing conditions. Thus, the imprint of ZP uncertainties on the sky is difficult, if not impossible, to correct for. We design the LRG target selection to be as insensitive to the ZP uncertainties as possible.

One way to quantify the effects of ZP uncertainties on the LRG targets is to estimate the level of fluctuation in target density caused by the ZP uncertainties. The estimated ZP uncertainties in g, r, z, and W1 are 3, 3, 6, and 1 mmag, respectively (D. Schlegel et al. 2023, in preparation). For the LRG selection, a net change of +10 mmag in g, r, z, and W1 causes changes of +0.11%, +1.40%, −1.23%, and −2.89% in target density, respectively. If we assume that the ZP errors in the four bands are all uncorrelated with each other, we can treat the combined effect on the target density as sums of independent Gaussian random variables, and the resulting rms of the target density is 0.9%. During Survey Validation, we considered an alternative LRG selection that implements the luminosity cut using z-band fiber magnitude and r − z color (see Appendix A), and this selection has a much larger rms of ∼4% due to the large ZP uncertainty in the z band. (The z-band photometric calibration has large uncertainties mainly because the effective z-band filter transmission can vary significantly due to absorption by telluric water vapor.) Its insensitivity to zero-point errors is the main reason why we chose to adopt the WISE-based luminosity cut.

We use spectroscopic data from SV1, SV3, and the first two months of the Main Survey. We only select LRG+QSO tiles in SV1 and dark tiles in SV3 and the Main Survey. The sky coverage of the spectroscopic LRGs is shown in Figure 7. Figure 8 shows some example spectra of LRGs observed during the Main Survey and the image cutouts (from the Legacy Surveys Viewer ⁴³ ).

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SV1 has several flavors of coadds, and we use the single-exposure coadds, the nominal (1×) depth coadds, and the cumulative (deep) coadds. LRG targets are assigned the target bit 2⁰ in all three programs. A significant number of brighter LRGs are also observed in the BGS program under very different observing conditions, and we do not include them here. We remove objects affected by instrument issues by requiring that the COADD_FIBERSTATUS value in the catalogs is equal to 0. We apply the veto mask (Section 2.4) to create a clean sample.

The redshift fitting is done using Redrock (S. Bailey et al. 2023, in preparation). It uses 1D spectra produced by the DESI spectroscopic pipeline (Guy et al. 2022) as input. For each spectrum, it computes the best-fit redshift and Δχ², which is the difference in χ² between the best-fit model and the second best-fit model and is an indication of the reliability of the best-fit redshift.

We use a sample of "true" redshifts as the reference sample for measuring the catastrophic failure rates. While the redshifts from visual inspection (Lan et al. 2022) could also be used as true redshifts, they are only available for a few thousand SV LRGs and a few hundred Main Survey LRGs. Therefore we use the much larger sample of redshifts from deep coadded SV1 spectra as the true redshifts. We require a minimal effective exposure time (t_eff, see its definition in Guy et al. 2022) of 3000 s, which is three times the DESI nominal depth of t_eff = 1000 s. We validate these deep redshifts by comparing them with the redshifts from visual inspection, and we find that the redshifts disagree for less than 0.5% of the Main Survey LRGs.

For assessing the spectroscopic performance at nominal depth, we require t_eff > 800 s. And we also require t_eff < 1200 s for the SV1 coadds (which have a wider range of t_eff than the Main Survey). To assess whether a redshift (e.g., obtained at nominal depth) is correct, we compare it with the redshift of the same object from the deep coadded spectra. If it differs from the deep redshift by more than 1000 km s⁻¹, that redshift is considered a "catastrophic redshift failure." (A small number of deep redshifts have ZWARN ≠0 or z_redrock > 1.5, and are likely not reliable. We treat the corresponding nominal-depth redshifts as catastrophic failures regardless of their redshift values.)

Figure 1 shows the redshift distribution of LRGs observed in the first two months of the Main Survey, which covers roughly 2000 deg⁻². The sample has a roughly constant comoving density of 5 × 10⁻⁴ h³ Mpc⁻³ in 0.4 < z < 0.8 and a high-z tail that extends beyond z = 1.0. Figure 1 excludes a small (1%) fraction of the observed LRG targets that are rejected by the quality cut (see Section 4.3); these objects are the faintest LRG targets and are mostly at the high-redshift end of the sample.

Only 0.5% of the LRG targets are spectroscopically classified by Redrock as stars and 1.7% as QSOs. The rest are classified as galaxies. Almost all the stellar contaminants are cool stars such as M dwarfs whose strong infrared emissions allow them to pass the stellar rejection cut. About 0.6% of the LRG targets are also QSO targets, and 61% of these are spectroscopically classified as QSOs and 2% as stars. If we exclude QSO targets, the stellar fraction is 0.5% and the QSO fraction is 1.2%. A large fraction of the area observed in the first two months of the Main Survey is at relatively low Galactic latitude, and the full DESI footprint will include a larger fraction of high-latitude area where the stellar density is lower. Therefore we expect the 0.5% stellar contamination rate to be an upper bound for the full DESI sample.

Note that in the first two months of the Main Survey observations, the observed LRG targets include a larger fraction (by a factor of ∼2) of objects that are also QSO targets than in the full survey. This is because the overall fiber-assignment completeness is lower at the beginning of the survey and the QSO targets have a higher fiber-assignment priority than the LRGs (see Raichoor et al. 2022). We correct for this by downweighting the QSO targets so that the fractions described above are estimates for the final Main Survey sample.

The catastrophic redshift failure rate of LRGs observed at nominal depth is 0.7% (110/15,379) from comparison with the deep redshifts. This is consistent with what we find from repeat observations: in the overlap between SV3 and the Main Survey: we find that 1.2% (84/7233) of the repeats have different redshifts, which means that the per-object catastrophic failure rate is 0.6% if we assume that the redshift efficiency is the same in SV3 and the Main Survey.

To reject incorrect redshifts, we apply the following redshift quality cut (shown in Figure 9): we require Δχ² > 15, z_redrock < 1.5, and the ZWARNING flag ZWARN = 0. The z_redrock < 1.5 cut removes the pileup of catastrophic failures at z ∼ 1.6. (While it is not entirely clear what causes this pileup, based on the fact that these objects have mostly noisy and featureless spectra, we speculate that to best match the observed featureless spectra, Redrock finds the best fit at z ≳ 1.5 so that the 4000 Å break feature of the principal component analysis templates is redshifted outside the DESI spectral coverage.) We note that the redshift quality cut is preliminary, and it may change, e.g., as the spectroscopic pipeline and Redrock evolve.

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The redshift quality cut removes 1.1% of the LRGs, 43% (82/191) of which are catastrophic failures. The catastrophic failure rate in the accepted (confident) redshifts is 0.2% (28/15,188). From visual inspection of the spectra and images, we find that a significant fraction of the catastrophic failures that pass the redshift quality cut are blends (i.e., there is more than one object within the fiber diameter), and both redshift solutions produce reasonable fits to the observed spectra.

Hereafter, we refer to the fraction of objects that meets the quality cut as the "redshift success rate" and the fraction that fails the cut as the "failure/rejection rate," and we refer to the incorrect redshifts (as determined by comparing with deep or repeat spectra) as "catastrophic failures."

In the above assessments, we have excluded QSO targets and objects spectroscopically classified as QSOs, because they have much higher redshift failure rates and are a very different population than the "normal" LRGs. The objects that are targeted or classified as QSOs have approximately 10 times higher catastrophic failure rates than the rest of the LRG targets, and they are also about 10 times more likely to fail the redshift quality cut. It should be noted that the distinction between a "galaxy" and "QSO" is not always sharply defined (e.g., Redrock may classify some "galaxies" with features of active galactic nuclei as "QSOs" but not others), and more careful consideration may be needed for the selection of the DESI clustering sample.

The spectroscopic redshifts of the LRGs are primarily based on absorption lines and the 4000 Å break, and sufficient spectroscopic S/N is critical for confident redshift estimation. The S/N mainly depends on two factors: the source brightness and the spectroscopic depth (t_eff). Here we investigate how the two factors affect the LRG redshift failure rate. (The redshift failure rate also depends on other factors such as the strength of the absorption and emission lines and prominence of the 4000 Å break, but they do not vary significantly within the LRG sample and are thus less important factors for the redshift determination.)

The z-band fiber magnitude (z_fiber) is very well correlated with Δχ² and catastrophic redshift failures, as shown in Figure 10, and the correlation is much stronger than the fiber magnitudes in the g and r bands. Therefore we use z_fiber as the parameter for source brightness. In Figure 11 we show the catastrophic redshift failure rate and rejection rate as a function of z_fiber at nominal depth. The error bars show the uncertainty for a binomial distribution: ${\sigma }_{p}=\sqrt{{Np}(1-p)}/N$ where N is the total number of objects and p is the failure/rejection rate. At the z_fiber limit, the LRGs have the highest catastrophic failure rate of ∼2% and rejection rate of ∼4%. The catastrophic rate after applying the redshift quality cut is much less than 1% at the z_fiber limit.

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Figure 12 shows the catastrophic redshift failure rate and the rejection rate as a function of t_eff. The rejection rate flattens out above t_eff ≃ 1000 s. And while the catastrophic failure rate and rejection rate increase significantly at t_eff < 800 s, the catastrophic failure rate of the accepted redshifts remains well below 1%.

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For clustering measurements, it is important to correct for redshift incompleteness caused by targeting and observational factors. Here we use the following function of the effective exposure time and z-band fiber flux to predict the LRG redshift failure/rejection probability:

$$\begin{eqnarray}&&\begin{array}{l}{P}_{\mathrm{fail}}({t}_{\mathrm{eff}},{f}_{z,\mathrm{fiber}})=\exp ({a}_{0}S+{a}_{1})\\ \,+\,{a}_{2}/({f}_{z,\mathrm{fiber}}/1\,\mathrm{nanomaggy}),\end{array}\end{eqnarray} \tag{ 4 }$$

where nanomaggy is a linear flux unit used in LS DR9 catalogs (with a flux of 1 nanomaggy corresponding to an AB magnitude of 22.5), $S\equiv ({f}_{z,\mathrm{fiber}}/1\,\mathrm{nanomaggy})\sqrt{{t}_{\mathrm{eff}}/1\,{\rm{s}}}$ is (approximately) proportional to the spectroscopic S/N when the sky is much brighter than the source (which is true for the fainter LRGs), f_z,fiber is the z-band fiber flux, and a₀, a₁, and a₂ are constant coefficients.

The exponential term is motivated by the observation that the redshift failure rate decreases exponentially with S. At brighter magnitudes and in deeper exposures (where the per-object failure rate becomes less than ∼1%) the exponential term approaches zero faster than the observed failure rate, so we add the second term a₂/f_z,fiber to account for the higher observed redshift failure rate. From visual inspection, we find that many of the redshift failures at brighter magnitudes are blends or have problematic spectra due to instrument issues.

The best-fit coefficients are found by minimizing ${\sum }_{i}{({P}_{\mathrm{fail},i}-{Q}_{i})}^{2}$, where P_fail,i is the predicted failure probability of the ith object, and Q_i = 0 if the object passes the quality cut and Q_i = 1 if it fails the cut. For the fitting we use Main LRGs observed in SV1 and the first two months of the Main Survey, and SV3 LRGs (which are ∼0.1 magnitude fainter than the Main LRGs). To best match the redshift failure rates of the Main Survey, we only use LRGs with 500 s < t_eff < 2000 s; this prevents the large number of redshift failures at very low t_eff (mainly from SV1) from dominating the fit. The best-fit coefficients are a₀ = −0.0911, a₁ = 3.34, and a₂ = 0.0228.

The gray crosses in Figures 11 and 12 are the predicted failure rates, and they match the observed failure rates (green error bars) very well. Figure 13 shows the observed and predicted dependence of the LRG redshift failure rate on both z_fiber and t_eff. The residuals are negligible in the range of z_fiber and t_eff relevant for LRGs in the Main Survey.

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For the DESI clustering analysis, it is important that variations in the spectroscopic efficiency of each fiber do not imprint on the measured galaxy densities. For example, Ross et al. (2012) found that the redshift efficiency of the BOSS CMASS sample varies with the fiber location on the focal plane. To quantify the per-fiber efficiency, we compute the average LRG redshift failure rate for each fiber using observations during the Main Survey. The per-fiber LRG failure rate is shown in Figure 14 (left panel), and we do not see clear patterns in the fiber efficiency given the statistical uncertainties. To more rigorously assess whether the observed failure rates are consistent with uniform fiber efficiency, we perform Monte Carlo simulations with the per-object failure probability given by the model described in Section 4.4. We find that the observed distribution of the measured per-fiber failure rates is mostly consistent with the simulated distributions for all except a few fibers, as shown in Figure 14 (right panel), indicating very uniform spectroscopic efficiencies for almost all the fibers. We will revisit the fiber-to-fiber uniformity when more data become available.

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