# Impact and mitigation of spectroscopic systematics on DESI DR1 clustering
measurements

A. Krolewski  J. Yu  0009-0001-7217-8006 A. J. Ross  0000-0002-7522-9083 S.
Penmetsa  W. J. Percival  0000-0002-0644-5727 R. Zhou  0000-0001-5381-4372 J.
Hou  J. Aguilar  S. Ahlen  0000-0001-6098-7247 D. Brooks  E. Chaussidon
0000-0001-8996-4874 T. Claybaugh  A. de la Macorra  0000-0002-1769-1640
Biprateep Dey  0000-0002-5665-7912 J. E. Forero-Romero  0000-0002-2890-3725 S.
Gontcho A Gontcho  0000-0003-3142-233X J. Guy  0000-0001-9822-6793 K.
Honscheid  S. Juneau  D. Kirkby  0000-0002-8828-5463 T. Kisner
0000-0003-3510-7134 A. Kremin  0000-0001-6356-7424 A. Lambert  L. Le Guillou
0000-0001-7178-8868 M. E. Levi  0000-0003-1887-1018 P. Martini
0000-0002-4279-4182 A. Meisner  0000-0002-1125-7384 R. Miquel  J. Moustakas
0000-0002-2733-4559 A. D. Myers  J. A. Newman  0000-0001-8684-2222 G. Niz
0000-0002-1544-8946 N. Palanque-Delabrouille  0000-0003-3188-784X G. Rossi  E.
Sanchez  0000-0002-9646-8198 E. F. Schlafly  0000-0002-3569-7421 D. Schlegel
M. Schubnell  H. Seo  0000-0002-6588-3508 D. Sprayberry  G. Tarlé
0000-0003-1704-0781 B. A. Weaver  C. Zhao  0000-0002-1991-7295

###### Abstract

The large scale structure catalogs within DESI Data Release 1 (DR1) use nearly
6 million galaxies and quasars as tracers of the large-scale structure of the
universe to measure the expansion history with baryon acoustic oscillations
and the growth of structure with redshift-space distortions. In order to take
advantage of DESI’s unprecedented statistical power, we must ensure that the
galaxy clustering measurements are unaffected by non-cosmological density
fluctuations. One source of spurious fluctuations comes from variation in
galaxy density with spectroscopic observing conditions, lowering the redshift
efficiency (and thus galaxy density) in certain areas of the sky. We measure
the uniformity of the redshift success rate for DESI luminous red galaxies
(LRG), bright galaxies (BGS) and quasars (QSO), complementing the detailed
discussion of emission line galaxy (ELG) systematics in a companion paper [1].
We find small but significant fluctuations of up to 3% in redshift success
rate with the effective spectroscopic signal-to-noise, and create and describe
weights that remove these fluctuations. We also describe the process to
identify and remove data from certain poorly performing fibers from DESI DR1,
and measure the stability of the redshift success rate with time. Finally, we
find small but significant correlations of redshift success rate with position
on the focal plane, survey speed, and number of exposures required, and show
the impact of weights correcting these trends on the power spectrum multipoles
and on cosmological parameters from BAO and RSD fits. These corrections change
the best-fit parameters by $<15\%$ of their statistical errors, and thus
contribute negligibly to the overall DESI error budget.

##  1 Introduction

The Dark Energy Spectroscopic Instrument [2, 3, 4, 5, 6] will obtain 40
million extragalactic spectra over its 5-year survey from 2021–2026, an order-
of-magnitude increase in cosmological constraining power and sample size over
previous samples from the Sloan Digital Sky Survey (SDSS). DESI has 5000
robotically-controlled fiber positioners [7] across a 7 deg2 focal plane [8]
at the 4-meter Mayall Telescope at Kitt Peak, Arizona, US [9], with a wide
field-of-view enabled by a new optical corrector [10]. DESI’s goal is to
obtain the tightest constraints to date on dark energy via baryon acoustic
oscillations and redshift-space distortions, in both cases using galaxies and
quasars as tracers of the dark matter field. In its main survey, DESi is
mapping the Universe using five tracers at $0.1<z<4.2$. During bright
telescope time, DESI observes the Bright Galaxy Sample (BGS), which is used
for cosmology at $0.1<z<0.4$ [11] (though the BGS targets span a slightly
wider redshift range). During dark time, DESI observes the Luminous Red
Galaxies (LRG) at $0.4<z<1.1$ [12], Emission Line Galaxies (ELG) at
$0.8<z<1.6$ [13], quasars (QSO) at $0.8<z<2.1$ [14]111The BAO and RSD analyses
only use quasars up to $z=2.1$, but higher-redshift quasars are used in cross-
correlation with the Ly$\alpha$ forest and to constrain primordial non-
Gaussianity, up to $z=3.5$ [15]., and the Ly$\alpha$ forest at $z>1.8$ [16].

The first year of DESI observations will be made available as DESI DR1 [17].
This sample already includes 6 million galaxies used in the cosmological
analysis (and another 4 million fainter BGS that were not used, since BGS is
far denser than necessary to achieve a cosmic-variance limited measurement)
[18]. The BAO measurements were presented in [19, 16] and their cosmological
interpretation given in [20]. The RSD measurements will come soon in [21], as
well as their cosmological interpretation [22] and primordial non-Gaussianity
constraints [15].

Making these percent-level measurements requires that the fluctuations in the
galaxy field are cosmological rather than driven by systematics. The largest
fluctuations are driven by inhomogeneities in the imaging used for targeting,
and such mitigations are extensively described in [18, 23, 24, 25, 26].

Figure 1: Flowchart describing the layout of the paper. A preliminary version
of the redshift success model must be fit to the data to define the low
success rate fibers; targets from these fibers are then removed from the final
data and the fit is re-run. The flow-chart ends once each test passes. The
impact on cosmological parameters is negligible for all tests; hence we show
the final line as dashed to indicate that all tests pass and we never reach
the final “yes” box.

This paper and the companion paper [1] characterize the impact of fluctuations
in spectroscopic observing conditions on large-scale structure measurements.
Specifically, we aim to ensure that the effective spectroscopic signal-to-
noise (or observing time) is as uniform as possible across all targets, and
that its variations do not cause variations in redshift success rate and thus
galaxy number density. This paper presents an overview of spectroscopic
systematics for all tracers, while [1] focuses specifically on ELG. ELG are
both the largest sample in DESI and challenging to redshift accurately, owing
to their faint fluxes and reliance on the narrow [OII] emission line, which at
high redshift can be confused with the abundant skylines at the red end of the
DESI wavelength range. Number density fluctuations driven by spectroscopic
conditions are typically quite small (roughly 1%, compared to the roughly 10%
fluctuations from imaging systematics), but can be detected at high
significance due to the large sample sizes in DESI. Obtaining a measurement of
the density field dominated by cosmological modes rather than systematics is
of paramount importance for DESI’s large-scale structure mission.

We focus on the discrete tracers (galaxies and quasars) since the Ly$\alpha$
forest has very different systematics. Nevertheless, redshift errors and
spectroscopic uniformity are also quite important for the Ly$\alpha$ analysis,
and the impact of these systematics are described in [27, 28] (see also [29]).

We provide a flow chart describing the methods and logic of this paper in Fig.
!1. We describe the DR1 DESI data processing in Sec. 2, as well as the mocks
used and the clustering measurements made. We describe the model for redshift
failures and their corresponding weight, $w_{\textrm{zfail}}$, in Section 3.
Then we describe the process to remove targets that were observed with low
success-rate fibers in Sec 4. Finally, we validate the redshift failure
weights in Section 5, showing that all trends of effective spectroscopic
signal-to-noise with hardware and observational variables are greatly reduced
and most are no longer significant after applying $w_{\textrm{zfail}}$. For
those trends that are significant, we measure their impact on galaxy and
quasar clustering in Section 6, and propagate their impact to cosmological
parameters for those that change the observed power spectrum noticeably. We
find $<20\%$ shifts in parameters inferred from BAO and RSD fits. We conclude
in Section 7.

We use the DESI fiducial cosmology in this work where appropriate, a flat
$\Lambda$CDM fiducial cosmology from the mean results of Planck [30]
base_plikHM_TTTEEE_lowl_lowE_lensing with
$\omega_{\mathrm{b}}=0.02237,\;\omega_{\mathrm{cdm}}=0.12,\;h=0.6736,\;A_{\mathrm{s}}=2.083\cdot
10^{-9},\;n_{\mathrm{s}}=0.9649,\;\sigma_{8}=0.8079,\;N_{\mathrm{eff}}=3.044,\;\sum
m_{\nu}=0.06\;\mathrm{eV}$ (single massive neutrino). This cosmology is used
to convert galaxy positions to Cartesian coordinates for computing two point
clustering statistics.

##  2 DESI Data

Figure 2: Variation of TSNR2 (top), modelled success rate (second row),
measured success rate (third row), redshift failure weight (fourth row), and
measured success rate weighted by the redshift failure weight (bottom row).
The model does not necessarily match the normalization of the measured
redshift success rate, especially for QSO, because it is designed to remove
fluctuations rather than match the normalization. Comparison of the third row
and the bottom row shows the model’s performance: the
$w_{\textrm{zfail}}$-weighted success rate is more uniform than the raw
measured success rate.

###  2.1 Observing strategy and catalog construction

DESI DR1 contains data from the first year of the DESI Main Survey observed
from May 2021 to June 2022 and covers over 7,400 deg2 [17, 18]. DESI targets
are selected from the DESI Legacy Survey Imaging, consisting of the
combination of the Beijing-Arizona Sky Survey (BASS) and the Mayall $z$-band
Legacy Survey (MzLS) at $\delta>32.375^{\circ}$ and the Dark Energy Camera
Legacy Survey (DECaLS) at $\delta<32.375^{\circ}$ [31, 32, 33]. Targets are
selected via multi-band color cuts, supplemented by infrared fluxes at 3.4 and
4.6 $\mu$m from the Wide-field Spectroscopic Explorer (WISE) for LRG and QSO,
and GAIA $G$ band for a morphological proxy to reject stars from BGS. We also
use flux cuts defined on fiber magnitudes, which measure the amount of
predicted flux within a DESI fiber (1.5” diameter) in 1” Gaussian
seeing.222See description here https://www.legacysurvey.org/dr9/catalogs/
These fluxes are the most relevant for this work, since redshift success rate
is connected to the total amount of light in a fiber, not the total amount of
light in the entire galaxy. The target selection pipeline is described in
[34]. Galaxies are subsequently assigned to fibers [35]. Fiber assignment has
a significant impact on small-scale galaxy clustering due to the 1.5 arcmin
patrol radius of each fiber. The impact of fiber assignment has been
extensively studied using both pairwise inverse probability weights [36] and
“fast fiberassign” [37] and its impact on two-point clustering is described in
[38]. Its impact on the RSD analysis is mitigated by removing all pairs with
separation less than the patrol radius [39].

BGS targets are split into “bright” and “faint” subsamples as described in
[11]. The cosmology sample uses “BGS_BRIGHT” with an additional restriction on
the $r$-band absolute magnitude, $M_{r}<-21.5$ [18]; this sample is called
“BGS_BRIGHT-21.5.” Hence there are three BGS samples for which large-scale
structure (LSS) catalogs are constructed, “BGS_ANY” (all BGS targets),
“BGS_BRIGHT” (without the absolute magnitude cut) and “BGS_BRIGHT-21.5” (used
for the cosmological analysis). In this paper we are primarily concerned with
“BGS_BRIGHT-21.5” but also briefly show the impact of spectroscopic
systematics on the other samples, which may be more relevant for other
applications requiring higher number densities. An important distinction
between the samples is that only “BGS_ANY” and “BGS_BRIGHT” exist for the BGS
targets; defining “BGS_BRIGHT-21.5” requires the galaxy’s absolute magnitude
(and thus its redshift), so “BGS_BRIGHT-21.5” can only be defined for
successful redshifts.

DESI survey operations are designed to ensure maximum uniformity in the data’s
spectroscopic quality [40]. The exposure time is dynamically adjusted based on
observed conditions [41]. Targets are observed from 5,000 fibers at once, with
4,326 available for the Y1 catalogs.333As explained in [42], the status of a
fiber positioner can change from night to night (e.g. due to transient
electronics issues). We therefore refer to the total number of fibers that
were usable at any point during DESI Y1, which is 4,326. The astrometric
performance and positioner accuracy are described in [43]. The fibers are
divided into 10 petals, labelled with zero-based indexing (see Fig. 2 in this
paper and Fig. 1 in [42] for a visualization of the focal plane). The fibers
are read out continuously in fiber number in the CCD, so CCD issues will
affect contiguous blocks of fibers. Amplifier issues are associated with half
of the fibers within a petal (i.e. fiber numbers 0 to 250 or 251 to 500).
Amplifier issues also cause flux offsets between halves of spectra in the
wavelength direction.

After nightly data is processed [44], QA is performed each morning, and tiles
with at least 85% of the target observing time can be approved, if the
redshift success rate, redshift distribution, and sky fiber residuals are
adequate (as described in [40]). The effective observing time for each target
is parameterized by the “template signal to noise ratio squared” (TSNR2), as
described in Section 4.14 of [44]. TSNR2, for a particular DESI target, is the
signal-to-noise that a fixed-brightness galaxy template would achieve on the
DESI spectrograph, if it were observed under exactly the same conditions as
the target under consideration. Hence despite having “signal-to-noise” in the
name, it is not related to the actual brightness of the target under
consideration. TSNR2 is defined for several templates corresponding to each
type of DESI target, including BGS, LRG, ELG, QSO, and Ly$\alpha$ forest. We
show the normalized distributions of TSNR2 for each target class in Fig. 3.

Figure 3: Normalized distribution of TSNR2 for targets in each tracer class.
For BGS we show the distribution of TSNR2 for BGS_BRIGHT targets; the
distribution is nearly the same for BGS_ANY targets.

TSNR2 is defined per fiber, and its average over an entire tile (specifically
using TSNR2_LRG for dark time and TSNR2_BGS for bright time) is used to
determine the tile’s effective exposure time. This is calculated via Eq. 22 in
[44], assuming a sky-limited observation, relating TSNR2 to spectroscopic
effective exposure time $T_{\textrm{spec}}$:

$T_{\rm spec}=(12.15\textrm{sec})\times TSNR^{2}_{\textrm{LRG}}$ (2.1)

TSNR2 for the other tracers is related to $T_{\textrm{spec}}$ in a similar way
but with different proportionality constants:

$T_{\rm spec}=(0.135\textrm{sec})\times TSNR^{2}_{\textrm{BGS}}$ (2.2) $T_{\rm
spec}=(8.60\textrm{sec})\times TSNR^{2}_{\textrm{ELG}}$ (2.3) $T_{\rm
spec}=(33.6\textrm{sec})\times TSNR^{2}_{\textrm{QSO}}$ (2.4)

If the effective exposure time is $<85\%$ of the goal effective exposure time
($0.85\times 1000=850$ s for dark time and $0.85\times 180=153$ s for bright
time), another exposure is required. Another relevant variable is the survey
speed, defined as the number of effective seconds observed per actual second
of observing time, as computed using the Exposure Time Calculator and
described in Section 5.3 of [40]. If the survey speed is low due to marginal
observing conditions, leading to long exposures, the exposure may be split to
facilitate cosmic-ray rejections. Thus tiles can be observed with up to 4
exposures in the Y1 sample, and these may be on different nights. A small
number of targets with low TSNR2 in the final data processing, TSNR2_ELG < 80
or TSNR2_BGS < 1000, are removed from the catalogs.

Once galaxies and quasars at $z<2.1$ are observed to the specified effective
exposure time, they are de-prioritized so that they can only be observed if no
other science target is available for a given fiber. Quasars at $z>2.1$ are
re-observed up to four times, to increase their signal-to-noise for the
Ly$\alpha$ forest measurements. Redshifts for the reduced and calibrated
spectra are measured using a homogeneous processing of the full dataset; for
Y1, this processing was internally referred to as “Iron.” Galaxies are
redshifted using the Redrock software [45], and quality is assessed by
redshift warning flags ZWARN and $\Delta\chi^{2}$, the difference between the
best-fit and the second best fit $\chi^{2}$. Quasars are redshifted using
updated DESI templates which separate low-redshift and high-redshift quasars
[46]; an additional fix to correctly account for the Ly$\alpha$ mean flux
evolution [27] was used for the Ly$\alpha$ analysis but not in the LSS
catalogs–we have verified that applying this fix makes little difference in
the LSS catalogs. The redshift success criterion follows that established for
the DESI SV3 LSS catalogs [5]

  * •

BGS: ZWARN==0, $\Delta\chi^{2}>40$

  * •

LRG: ZWARN==0, $\Delta\chi^{2}>15$

  * •

ELG: $\log_{10}(S_{\textrm{[OII]}})+0.2\log_{10}(\Delta\chi^{2})>0.9$

  * •

QSO: Passes QSO criterion from [14]

The ELG and QSO targets partially overlap, and thus some of the ELG targets
wind up as spectroscopically confirmed quasars. These targets are removed from
all of the ELG LSS catalogs; the ELG sample is therefore internally referred
to as “ELG_LOPnotqso.”

Both ELG and QSO require additional processing of the redrock outputs,
referred to as an “afterburner.” For ELG, this is to measure the [OII] flux
signal-to-noise ratio $S_{\textrm{[OII]}}$, and for QSO the QuasarNet and MgII
afterburners improve redshift accuracy and completeness. These success
criteria were extensively validated using the SV samples [11, 12, 13, 14] and
visually-inspected spectra from deep exposures [47, 48]. The cosmological
analysis also uses a volume-limited subset of BGS with absolute magnitude
$M_{r}<-21.5$, as determined using fastspecfit [49]. Since $M_{r}$ can only be
defined for galaxies with a successful redshift, we only apply this cut to
successful BGS targets, not all BGS targets.

Finally, LSS catalogs are created from the successful targets, with
appropriate weights. The process is described in detail in [18, 42]. In this
paper, we describe how the redshift failure weights are defined, which
upweight galaxies observed with less effective exposure time to account for
the lack of uniformity due to redshift failures.

###  2.2 Redshift success rate

The redshift failure weights are applied to ensure that the redshift success
rate (averaged over all redshift) is constant as a function of observing
conditions (i.e. TSNR2). Identically to [1], we always consider the normalized
redshift success rate, i.e. the spectroscopic success rate within a particular
tracer, observing condition, and/or redshift bin, divided by the overall
redshift success rate:

$\textrm{SSR}(x)=N_{\rm goodz}(x)/N_{\rm obs}(x)$ (2.5)
$f_{\textrm{goodz}}(x)=\frac{\textrm{SSR}(x)}{\sum_{x}N_{\rm
goodz}(x)/\sum_{x}N_{\rm obs}(x)},$ (2.6)

As a result of the normalization by overall redshift success rate,
$f_{\textrm{goodz}}$ always fluctuates around one; hence it does not
characterize the actual fraction of good redshifts, but rather the uniformity
of the good redshift fraction as a function of spectroscopic observing
condition.

Within Eqs. 2.5 and 2.6, “goodz” denotes targets that have obtained a
successful redshift following the classification in Sec. 2.1. Note that in
some cases, this means that the “failures” might actually have a robustly
classified redshift, but as the wrong type of target (e.g. QSO targets that
can be spectroscopically confirmed as stars or galaxies).

We differentiate here betweeen two defintions of “goodz.” In the first, used
to fit the redshift failure weights, a good redshift is merely any target that
passes the successful redshift criterion of Sec. 2.1, without any reference to
its redshift. In contrast, to validate the redshift failure weights for the
LSS catalogs, we are only concerned with the ipact of spectroscopic
systematics on tracers within the specified redshift ranges. Thus, in the
second definition of “goodz” used for validation, we also require that the
targets lie in the fiducial redshift range to be classified as “goodz”:
$0.1<z<0.4$ for BGS, $0.4<z<1.1$ for LRG, $0.8<z<1.6$ for ELG, and $0.8<z<2.1$
for QSO (and we also validate using wider redshift ranges in Appendix B). We
also consider narrower redshift ranges, where the successful redshifts must
lie in $\Delta z=0.1$ bins. For BGS, the validation definition requires that
“goodz” passes the $M_{r}<-21.5$ cut (but not necessarily the target), so the
overall redshift success rate passing this cut is quite low. The $M_{r}$ cut
is not applied to targets because the $M_{r}<-21.5$ cut is not well defined
for galaxies with poorly measured redshift, as determining $M_{r}$ relies on
the galaxy’s redshift. In contrast, the fitting definition of “goodz” for BGS
does not include the $M_{r}$ cut, i.e. the redshift failure weights are fit to
“BGS_BRIGHT” and not “BGS_BRIGHT-21.5.”

The overall “fitting” redshift success rates (with no extra redshift cuts) are
98.9% for LRG, 98.8% for BGS_BRIGHT (before the $M_{r}<-21.5$ cut), 72.4% for
ELG, and 66.2% for QSO. For LRG and BGS, failures are mainly faint galaxies
with no discernable emission features, with a small number of stellar
interlopers. For ELG, failures are mainly galaxies with insufficient [OII]
emission and minimal continuum detected, as well as ELGs at $z>1.6$ where
[OII] redshifts out of the DESI wavelength coverage.

For QSO, there are more possiblities. Based on visual inspection of deep
observations (deep VI), 71% of quasar targets are quasars, 16% are galaxies,
6% are stars, and 7% are inconclusive [14].444The fraction of quasars in deep
VI is higher than the DR1 pipeline classification fraction because the
pipeline misses some true quasars, and also due to sample variance from the
small size of the visually inspected dataset. This is a crude treatment of the
failures, since some of these spectra can be confidently classified,
particularly the stars. However, it is harder to devise a high-purity
classification for the quasar targets that are truly galaxies. A past study
aimed at constraining primordial non-Gaussianity from DESI quasar target and
Planck CMB lensing [50] created a spectroscopic success criterion for galaxies
targeted as quasars, in order to measure the redshift distribution of the
entire quasar targeting sample, including galaxies. This criterion is a hybrid
of the ELG and LRG spectroscopic success criterion but only achieves a 90%
purity, and the rest of the spectroscopically-classified galaxies are true
quasars. Moreover, the stellar classification has only 80% completeness (the
remaining 20% are “unclassified” by spectroscopic criteria), and, even worse,
we have no truth classification for the 7% of spectra that are unclassified
even in the deep VI exposures. To sidestep some of these difficulties, we will
also compare spectroscopic success rate trends in the full quasar sample to
those in a “high-purity” sample defined in [50], which removes much of the
contaminating stars and galaxies, though at the price of reducing the number
density by half. Overall, we continue to use the simple and crude
classification where all non-quasar quasar targets are classified as failures.
We discuss the impact of this choice on our results below and plan to revisit
this decision for future data releases.

###  2.3 Mocks

We use a suite of 25 AbacusSummit mocks [51, 52] to characterize the noise on
some of our redshift success statistics. These mocks have been passed through
the “fast fiberassign” [37] emulator to select DESI targets, and also to give
each mock target a per fiber and tile TSNR2 matched to the sky distribution in
the data. We refer to these mocks as the “Abacus FFA” mocks. The likelihood of
the redshift success rate within TSNR2 bins is complicated. Certain areas of
the survey preferentially have higher or lower TSNR2 (Fig. 4). As a result,
the probability of success for a single observation does not follow a binomial
distribution that would be expected if all galaxies were independent, but
contains a contribution from cosmic variance as well. The cosmic variance
contribution is akin to the super-sample covariance effect and is inversely
proportional to the volume of the patches (area of the TSNR2 patch times depth
of the redshift bin used). In other words, if a $\Delta z=0.1$ bin in TSNR2
just happened to contain a few more clusters than expected, its
$f_{\textrm{goodz}}$ would fluctuate higher than expected based on binomial
statistics alone. These fluctuations are possible for redshift bins of any
depth, but are suppressed as $\Delta z$ gets larger. Therefore, to assess the
significance of the fluctuations in redshift success rate in the narrow
redshift bins, we measure $\chi^{2}$ from the deviation between the observed
redshift success rate and unity, using binomial errors:

$\chi^{2}=\sum_{\textrm{TSNR2}}\frac{(f_{\textrm{goodz}}(\textrm{TSNR2})-1)^{2}}{\sigma_{\textrm{binomial}}^{2}}$
(2.7)

Rather than interpreting this statistic with the usual $\chi^{2}$
distribution, we directly compare the distribution of $\chi^{2}$ between data
and mocks with randomly-assigned redshift failures, but TSNR2 assigned
following the observed sky distribution. The mocks themselves do not have
redshift failures, so we randomly downsample the mocks by the observed mean
success rate, 99% for LRG, 97% for BGS_BRIGHT, and 66.5% for QSO.555While
these are slightly different from the success rate on data, the impact of the
difference on the TSNR2 trends in the mocks is negligible. This simple model
will under-estimate the scatter in success rate at low TSNR2, where the
success rate is lower than the mean (recalling that in the limit as the
failure rate $(1-p)$ goes to zero, the binomial error goes to $\sqrt{(1-p)}$).
This effect is most severe for the LRG and BGS, where the success rate is
close to 100%, and, at low flux, the failure rate may double at low TSNR2.
However, we primarily use the mocks to assess the significance of TSNR2 trends
in narrow redshift bins, where the “success rate” is typically low, due to the
redshift cut applied to the definition of a “success” (Eq. 2.5). As a result,
small changes in the success rate do not impact its error much.

Figure 4: Top: average TSNR2_LRG for the LRG sample, binned in nside=128
HEALPix pixels. Other dark time tracers show a similar pattern, since they
were observed in the same conditions in the same patches of sky. Bottom: Same
plot using TSNR2_BGS for the BGS_BRIGHT sample.

###  2.4 Clustering measurements

We test the impact of significant residual redshift success trends on the
observed power spectrum, quantifying their impact using
$\chi_{\textrm{sys}}^{2}$

$\chi_{\textrm{sys}}^{2}=(P_{\ell}^{\textrm{extra
wt.}}(k_{i})-P_{\ell}^{\textrm{std}}(k_{i}))(\boldsymbol{C}^{-1})^{ij}(P_{\ell}^{\textrm{extra
wt.}}(k_{j})-P_{\ell}^{\textrm{std}}(k_{j}))$ (2.8)

where $P_{\ell}^{\textrm{std}}$ are the power spectrum multipoles using the
standard weights (Eq. 2.12), $P_{\ell}^{\textrm{extra wt.}}$ are the
multipoles using the additional weight under consideration, and
$\boldsymbol{C}$ is the power spectrum covariance. Shifts in derived
parameters (i.e. cosmological, nuisance or BAO shift parameters
$\alpha_{\textrm{iso}}$ or $\alpha_{\textrm{AP}}$) from adding the additional
weights are at most $\sqrt{\chi_{\textrm{sys}}^{2}}$, if the impact of the
extra weight and the parameter under consideration are perfectly degenerate.
This sets the systematic uncertainty from residual inhomogeneities. If
$\chi_{\textrm{sys}}^{2}<<1$, we conclude that the systematic error does not
measurably affect parameters inferred from galaxy clustering, without actually
having to fit for those parameters. If $\chi_{\textrm{sys}}^{2}\gtrsim 1$, we
test the impact of weights by inferring cosmological parameters from both the
default power spectrum and the extra-weight power spectrum, to test how much
of the change projects onto cosmological parameters of interest, as opposed to
nuisance parameters.

We measure the power spectrum multipoles $P_{0}$, $P_{2}$, and $P_{4}$ using
pypower666https://github.com/cosmodesi/pypower [53], measured between $k=0$
and 0.4 $h$ Mpc-1, with bins of width $\Delta k=0.005$ $h$ Mpc-1. The
multipoles are calculated from the Fourier transform of the galaxy overdensity
field, using the Yamamoto estimator [54]

$P_{\ell}(k)=\frac{2\ell+1}{AN_{k}}\sum_{\boldsymbol{k}}F_{0}(\boldsymbol{k})F_{\ell}(-\boldsymbol{k})-\mathcal{SN}_{\ell}$
(2.9)

where $N_{k}$ is the number of modes in the $k$ bin, $A$ is a normalization
(defined below), and $F_{\ell}(k)$ is the Legendre-weighted Fourier transform
of the overdensity field:

$F(\boldsymbol{r})=n_{d}(\boldsymbol{r})-\alpha n_{r}(\boldsymbol{r}).$ (2.10)

$F_{\ell}(\boldsymbol{k})=\sum_{\boldsymbol{r}}F(\boldsymbol{r})\mathcal{L}_{\ell}(\mu)e^{i\boldsymbol{k}\cdot\boldsymbol{r}}.$
(2.11)

where $n_{d}$ and $n_{r}$ are the real-space densities of the data and
randoms, on a Cartesian grid with cell size 6 $h^{-1}$ Mpc, and $\mu$ is the
angle to the line of sight. The galaxies and randoms are weighted to correct
for incompleteness (due to the limited number of DESI passes in the Y1
dataset) $w_{\rm comp}$ and observational systematics ($w_{\rm sys}$ and
$w_{\rm zfail}$):

$w_{\rm tot}=w_{\rm comp}w_{\rm sys}w_{\rm zfail},$ (2.12)

In addition, the FKP weight [55] is applied to optimally combine clustering
measurements at different redshifts:

$w_{\rm FKP}=\frac{1}{1+n_{\mathrm{local}}(z,N_{\rm tile})P_{0}},$ (2.13)

where $n_{\mathrm{local}}(z,N_{\rm tile})$ is the local number density for a
given tracer, accounting for completeness variations as described in Section
7.3 of [42], and $P_{0}$ is the amplitude of the observed power spectrum at
$k\approx 0.15$ $h$ Mpc-1, as given in Section 8.2 of [18]. The systematics
weights include imaging systematics weights $w_{\rm sys}$, designed to null
the dependence of galaxy density on non-cosmological fluctuations (described
in Section 6 of [18]) and redshift failure weights $w_{\rm zfail}$, defined in
Section 3 below and designed to null the dependence of galaxy density on
spectroscopic observing conditions. Both data and randoms are weighted by
$w_{\rm tot}$.

The weighted field $F_{\vec{r}}$ is normalized to have mean zero:
$\alpha=\sum^{N_{d}}_{i=1}w_{\rm tot,i(d)}/\sum^{N_{r}}_{i=1}w_{\rm
tot,i(r)}$. The normalization of the power spectrum
$A=\alpha\sum^{k}n_{d,k}n_{r,k}$ is summed over a cell with 10 $h^{-1}$ Mpc in
size. The shot noise is only nonzero for the monopole:

$\mathcal{SN}_{0}=\frac{1}{A}\left[\sum^{N_{d}}_{i=1}w_{\rm
tot,i(d)}^{2}+\alpha^{2}\sum^{N_{r}}_{i=1}w_{\rm tot,i(r)}^{2}\right].$ (2.14)

We use analytic covariance matrices calculated by
THECOV777https://github.com/cosmodesi/thecov [56], based on the perturbative
covariance matrix code CovaPT888https://github.com/JayWadekar/CovaPT/ [57].
The covariance matrices are validated for DESI Y1 in [58].

##  3 Redshift failure model and weights

We define a set of redshift failure weights to remove density fluctuations due
to spectroscopic observing conditions for each of DESI’s four tracers.999This
is implemented in the DESI clustering code in
https://github.com/desihub/LSS/blob/v1.4-DR1/py/LSS/ssr_tools_new.py. The
process is very similar for LRG, BGS, and QSO, and quite different for ELG,
whose redshift failure weights are described in detail in Section 4.1 of [1].

For the LRG, BGS, and QSO weights, we start by defining a model for the
overall spectroscopic success rate (SSR), i.e. not normalized by the mean
spectroscopic success rate of each tracer. We fit a simple model to the
overall spectroscopic success rate as a function of TSNR2, allowing for an
exponentially rising failure rate at low TSNR2.

In the model, the spectroscopic success rate is defined by:

$\textrm{SSR}=1-\textrm{FR}_{\textrm{clip}}$ (3.1)

$\textrm{FR}_{\textrm{clip}}=\begin{cases}0&\textrm{FR}<0\\\
\textrm{FR}\equiv\exp{(-(\textrm{TSNR2}+a)/b)}+c&0<\textrm{FR}<1\\\
1&\textrm{FR}>1\\\ \end{cases}$ (3.2)

where the clipping ensures that the simple function does not go below zero or
above one. The model is always constrained to ensure that failure rate cannot
decrease with increasing TSNR2. While overall cuts to the sample are defined
in terms of TSNR2_LRG, the fits are done in terms of the tracer-specific
TSNR2.

We define 10 bins in TSNR2. For LRG, these are defined between 41.1 (=
500/12.15, corresponding to 500 “effective seconds”) and 139.9 (= 1700/12.15).
For QSO, the bins are defined between 13.3 (= 450/33.72, using the conversion
between effective time and TSNR2_QSO) and 53.4 (= 1800/33.72). For BGS, the
bins are defined between 889 (=120/0.135, using the conversion between
effective time and TSNR2_BGS) and 2222 (=300/0.135). In the regression we fit
against the median TSNR2 in each bin. When defining the weights and validating
their performance, we only use the first observation for high redshift quasars
with multiple observations, otherwise the high-TSNR2 bin for quasars would be
highly skewed towards these high redshift quasars.

This model is shown in Fig. 5 and the parameters $a$, $b$ and $c$ are given in
Table 1. The model fits the observed trends reasonably well in both
hemispheres, particularly at the low TSNR2 end. At moderate to high TSNR2, we
see difficulties in BGS_BRIGHT (South region) and QSO (North region). These
are further discussed below, but are believed to result from chance alignments
between TSNR2 and other properties controlling failure rate. For quasars, high
TSNR2 happens to lie at low galactic latitude, particularly in the North (see
Fig. 4), and for BGS, high TSNR2 happens to coincide with slightly fainter
targets, which have a much higher failure rate.

Figure 5: Overall failure rate model for tracers in the North (left column)
and South (right column), for BGS_BRIGHT (top), LRG (middle) and QSO (bottom).

Next, we account for the fiberflux dependence of the success rate. Faint
galaxies are considerably less likely to have successful redshift
measurements. This is particularly important because flux is correlated with
redshift, and having the correct flux dependence allows the redshift-failure
weights to properly account for spectroscopic observing fluctuations at all
redshifts. The fiberflux dependence of each tracer uses the band used for the
target selection fiber magnitude cut: $r$ for QSO and BGS, $g$ for ELG, and
$z$ for LRG. For BGS and QSO, we allow for a linear relationship between
success rate and fiberflux $F$, using two new parameters, $f$ and
$F_{\textrm{piv}}$. The weights are defined as follows:

$\textrm{SSR}_{\textrm{rel}}=\frac{\textrm{SSR}}{\textrm{max(SSR)}}$ (3.3)

$w_{\textrm{zfail}}=\begin{cases}1&w_{\textrm{raw}}<1\\\
w_{\textrm{raw}}\equiv\left[f\left(1-\frac{F}{F_{\textrm{piv}}}\right)+1\right]\left(\frac{1}{\textrm{SSR}_{\textrm{rel}}}-1\right)+1&w_{\textrm{raw}}>1\end{cases}$
(3.4)

The free parameters $f$ and $F_{\textrm{piv}}$ allow us to smoothly change
success rate with flux, and to change between the regime where success rate
strongly depends on flux and where it does not depend on flux. If $f=0$, then
the spectroscopic weight does not depend on flux, and is simply the inverse of
the global spectroscopic success rate fit. If $f=1$, flux maximally affects
success rate. If $F=0$ (i.e. a faint galaxy), then $w_{\textrm{zfail}}-1$ for
this galaxy is upweighted by $(f+1)$ relative to $w_{\textrm{zfail}}-1$ for a
galaxy with flux at $F_{\textrm{piv}}$. For LRG, we find that allowing for a
piecewise linear relationship between success rate and flux, with a third
parameter $C$, fits the data considerably better:

$w_{\textrm{zfail}}=\begin{cases}1&w_{\textrm{raw}}<1\\\
w_{\textrm{raw}}\equiv\left[G(F;F_{\textrm{piv}},f,C)\right]\left(\frac{1}{\textrm{SSR}_{\textrm{rel}}}-1\right)+1&w_{\textrm{raw}}>1\end{cases}$
(3.5)

where $G$ is a function of fiberflux $F$ with free parameters
$F_{\textrm{piv}}$, $f$, and $C$

$G=\begin{cases}f\left(1-\frac{F}{F_{\textrm{piv}}}\right)+1&F<F_{\rm
break}\\\ C+DF&F\geq F_{\rm break}\end{cases}$ (3.6)

where $D$ is defined to ensure contnuity of $G(F)$

$D=\left(f+1-\frac{fF_{\rm break}}{F_{\rm piv}-C}\right)\frac{1}{F_{\rm
break}}$ (3.7)

The break flux $F_{\rm break}$ is fixed to 3 nanomaggies;101010This is the
30th percentile in extinction-corrected $z$-band fiberflux. we do not find
significant improvements if we vary it.

The fit parameters are determined by applying the weights as a continuous
function to each galaxy as a function of its TSNR2 and fiberflux, and then
minimizing the difference between the weighted histogram and a uniform
distribution. The galaxies are binned in a histogram with the same 10 bins in
TSNR2 and 5 bins in flux, defined using the 20th, 40th, 60th, and 80th
percentiles. The fit parameters and $\chi^{2}$ are given in Table 1, both for
the overall TSNR2 fit (with 10 bins in TSNR2) and the flux-dependent fit.

The model success rate $f_{\textrm{suc}}$ is then directly related to
$w_{\textrm{zfail}}$

$f_{\textrm{suc}}=\textrm{SSRF}(F)/w_{\textrm{zfail}}$ (3.8)

where the function $\textrm{SSRF}(F)$ is a fit to the flux-dependent success
rate, with the following model

$\textrm{SSRF}(F)=a_{F}\erf{((b_{F}+F)/c_{F})}$ (3.9)

with $a_{F}$, $b_{F}$, and $c_{F}$ fit to the success rate in 20 equally
spaced flux bins. For this fit, the success rate is weighted by
$w_{\textrm{zfail}}$. The function $\textrm{SSRF}(F)$ accounts for the fact
that the success rate can vary strongly with flux, but this is normalized out
in the redshift failure weights, since we do not want to over-weight faint
galaxies with potentially very low success rates.

The ELG redshift success model is significantly different from what is
described above. Like the LRG, BGS, and QSO model, the ELG model is designed
to account for variation in redshift success rate with both spectroscopic
observing condition and redshift. However, for ELG the dominant galaxy
property controlling redshift success is not fiberflux, but rather [OII] line
emission strength. This is discussed extensively in Section 4.1 of [1].

Tracer | $a$ | $b$ | $c$ | TSNR2-fit $\chi^{2}$ | $f$ | $F_{\textrm{piv}}$ | C | Flux-fit $\chi^{2}$
---|---|---|---|---|---|---|---|---
|  |  |  |  |  | (nmgy) |  |
LRG N | -19.68 | 10.00 | 0.00668 | 4.92 | 14.72 | 2.94 | 1.50 | 42.76
LRG S | -18.89 | 10.42 | 0.00683 | 10.60 | 15.42 | 2.96 | 2.63 | 85.16
BGS_BRIGHT N | -238.15 | 181.71 | 0.0105 | 12.82 | 10.86 | 5.41 |  | 58.30
BGS_BRIGHT S | -469.99 | 141.72 | 0.00975 | 127.68 | 8.56 | 5.54 |  | 192.72
BGS_ANY N | -277.55 | 188.96 | 0.00999 | 15.86 | 3.52 | 5.42 |  | 94.50
BGS_ANY S | -464.11 | 151.49 | 0.00941 | 163.81 | 3.02 | 5.43 |  | 350.23
QSO N | -11.24 | 2.25 | 0.36 | 50.99 | 25.58 | 1.24 |  | 73.44
QSO S | -3.10 | 6.33 | 0.33 | 9.35 | 4.27 | 1.58 |  | 85.07
Table 1: Best-fit parameters for the overall TSNR2 fits (first set of columns;
$a$, $b$ and $c$ referring to parameters in Eq. 3.2) and the flux-dependent
fits (second set; $f$, $c$, and $F_{\textrm{piv}}$ referring to Eq. 3.4 and
3.5). We use 10 bins for the TSNR2 fits and 50 bins for the flux-dependent
fits.

##  4 Identifying and removing targets from low success rate fibers

We find that certain fibers have a systematically low success rate, perhaps
indicative of instrumental issues in the fiber or CCD, even beyond the success
rate expected from the average TSNR2 of observations in that fiber. We remove
the fibers with the worst success rates, rather than attempting to up-weight
these observations. Specifically, we remove any fiber that we deem to be more
than a 4$\sigma$ outlier, based on the observation that we expect less than 1
fiber out of the $\sim$4000 usable fibers to be a 4$\sigma$ outlier just by
random chance.

Determining whether to remove these fibers requires an accurate model for the
PDF of success rates. In this section, we neglect the cosmic variance
contribution to the success rate, since each fiber has observed galaxies
spread over the DESI Y1 footprint, and therefore has negligible cosmological
correlations. The success rate of each observation is a binomial distribution,
with probability mass function

$P(k)=\binom{n}{k}p^{k}(1-p)^{n-k}$ (4.1)

where $p$ is the success rate and $P(k)$ is the probability of having exactly
$k$ successes in $n$ independent Bernoulli trials. Many galaxies have been
observed with each fiber, at different times and locations on the sky. Each
observation has its own spectroscopic observing conditions, and thus its own
TSNR2. This means that the PDF of success rate for each fiber is the sum of
many binomial distributions with different success rates $p$. There exists an
analytic formula for the variance of $Z=\sum_{i=1}^{N}X_{i}$, the sum of $N$
binomial random variables $X_{i}$ with different success rates [59]

$\mathrm{Var}(Z)=N\bar{p}(1-\bar{p})-\sum_{i=1}^{N}(p_{i}-\bar{p})^{2}$ (4.2)

where $\bar{p}$ is the average success rate. As $N\rightarrow\infty$, the
central limit theorem guarantees that the PDF of $Z$ is Gaussian and fully
specified by its mean $\bar{p}$ and variance (Eq. 4.2).

The number of observations per fiber is typically large ($N\sim 600$ for LRG,
for instance). However, we want to measure probabilities deeply in the non-
Gaussian tail of the distribution to accurately determine whether a given
fiber is a 4$\sigma$ outlier. In the tail, the number of observations required
for the Gaussian approximation to hold is much larger than typically needed
for probabilities close to the core of the distribution. In particular, the
Gaussian approximation with Eq. 4.2 dramatically overstates the number of
4$\sigma$ outliers as compared to a Monte Carlo simulation (Fig. 6).

Figure 6: Comparison between Monte Carlo simulation and analytic (Gaussian)
approximation for the mean and variance of the average success rate per fiber.
The analytic approximation is a sum over binomial random variables with
different success rates for each galaxy observation with a different TSNR2. We
compute the likelihood of the observed success rate for each fiber, and
convert probabilites to a number of $\sigma$ offset (using a Gaussian PDF).
The line $y=x$ is shown in black. The probabilites shown are computed for the
LRG sample. The offset points to the upper right are all fibers with zero
failures; they are offset from all the fibers with exactly one failure, just
above.

As a result, we perform a Monte Carlo simulation to determine the likelihood
of the observed success rate for each fiber, within each tracer class,
following the Monte Carlo simulation for LRG in [12]. For each fiber, we
generated 3 million samples from the binomial PDF for each observation, using
the modelled success rate for each observation. We re-normalize the modelled
success rate to ensure that the mean success rate in the simulation matches
the observed mean success rate (note that in the model as defined in Sec. 3,
such a match is not guaranteed because we care about removing fluctuations
from the sample rather than accurately determining the true total number of
any given target). The large sample size is necessary to confidently determine
whether a given fiber is indeed a 4$\sigma$ outlier. We then directly
determined the probability of observing at least as many successes as were
seen in the actual data. This gives a $p$ value for each fiber, which was then
converted into a Gaussian “n-$\sigma$” deviation to put the numbers on a
familiar scale:

$n\sigma=-\sqrt{2}\times\text{erfc}^{-1}(2\times p\text{-value})$ (4.3)

where the complementary error function, erfc(x), is defined as:

$\text{erfc}(x)=1-\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}dt$ (4.4)

In Fig. 7, we show the observed distribution of success rates, compared to the
Monte Carlo simulation based on TSNR2-varying success rates and the analytic
approximation based on the central limit theorem. We see that there is a
significant excess of fibers with low success rate for LRG, BGS_BRIGHT, and
ELG. These fibers may have other issues impacting the success rate that are
not folded into the TSNR2, or the model linking TSNR2 to success rate may
break down for these fibers (if they preferentially sample regions in
fiberflux and TSNR2 where the model does not represent the data well). In
either case, the success rate for these fibers cannot be modelled well, and so
we remove them from further analysis.

While low success rate fibers are identified separately for each tracer, we
ultimately merge them into a single list of 55 unique low success rate fibers.
We remove all targets observed with these fibers from the LSS catalogs. We
also remove five additional fibers by hand, which are adjacent to low success
rate fibers already identified in petal 5. This leads to removing targets from
the contiguous range of fibers from 2675 to 2691.111111Further work has
identified a time-variable, non-linear, saturated feature in the 2D spectra
that leads to the poor redshift performance, as discussed here:
https://github.com/desihub/desispec/issues/2193#issuecomment-2004984304.
Overall, we remove 60 fibers from further analysis. This removes 1.2% of the
total redshifts from the DR1 analysis.

We show the position of these low success rate fibers on the focal plane in
Fig. 8. We also unroll these plots, to show the fiber-by-fiber success rate in
Figs. 9 and 10. The union of the per-tracer low success rate fibers are shown
in these plots, so the failure rate of the fibers is not necessarily higher
than average, if the fiber was identified as a low success rate fiber in a
different tracer.

Figure 7: Comparison of the success rate PDF between the observed data (red),
the Gaussian model using Eq. 4.2 (black) and the Monte Carlo simulations
(green), in both cases using the TSNR2 of each observation to model the
success rate. The dashed blue line indicates the cutoff for 4$\sigma$ outliers
in the data, as compared to the Monte Carlo simulations. The 4$\sigma$ cutoff
was chosen so that the expected number of outliers, just by chance, was $<1$
for the $\sim$4000 active DESI fibers. On this plot, the $x$ position of a
“4$\sigma$” outlier can change for each fiber, since it depends on the number
of galaxies contributing to that fiber. As a result, we cannot define a single
“4$\sigma$” cutoff line, but only show the maximum normalized success rate of
the fibers that we remove. This is why the blue line is at success rate
$\sim$15% for the quasars, since there is only one bad fiber identified with
success rate $\sim$15%.  Figure 8: Success rate for all tracers on the focal
plane, with low success rate fibers identified with red crosses. Targets on
these fibers are removed from large-scale structure catalogs.  Figure 9: Per-
fiber failure rate of LRG (blue) and BGS_BRIGHT (orange), with low success
rate fibers marked in dark red (LRG) and light red (BGS_BRIGHT). We use a
linear scale below failure rate of 0.025 and a logarithmic scale at higher
failure rate. Plots are arranged by petal from 0 to 9 (top to bottom).  Figure
10: Per-fiber failure rate of ELG (blue) and QSO (orange), with low success
rate fibers marked in dark red (ELG) and light red (QSO). Plots are arranged
by petal from 0 to 9 (top to bottom).

By visual inspection of the $n(z)$ per fiber,121212Plots and redshift
distributions are available in the Zenodo repository linked in the Data
Availability section. we find that several fibers in petal 2 have a
significant upward fluctuation in the ELG $n(z)$ at $z=1.5$: fibers 1261,
1269, 1295, 1296, and 1307. In these fibers, the spurious $z=1.5$ galaxies
were observed between Oct 1, 2021, and the end of DR1; they are not restricted
to observations at a particular time. This is roughly equivalent to a 0.1%
impact in the catastrophic redshift failure rate. The impact of the ELG
catastrophic rate, and its robustness to variations in the assumed rate, is
explored in Figs. 9 and 10 and Section 5.3 in [1].

We show further exploration of the interaction between the redshift
distribution and redshift success rate in Appendix A. We find weak evidence
for relationships between success rate deviation and mean redshift for some of
the tracers, but with differing signs. Due to the smallness of the observed
shifts, we do not correct for these trends in DR1, but flag them for further
study in future DESI data releases.

To robustly model the redshift success rate, we first remove targets from low
success rate fibers, and then fit the model described in Section 3. However,
identifying low success rate fibers requires a success rate model to determine
whether a given fiber is a 4$\sigma$ outlier. We therefore proceed
iteratively, first fitting a model with a fiducial “by-eye” low success rate
fiber list, then creating the systematic low success rate fiber list described
in this section, and then fitting the redshift success rate model and creating
$w_{\textrm{zfail}}$.

##  5 Validation of the redshift failure weights

We test that applying the redshift failure weights leads to a uniform
distribution of success rate within the redshift ranges of interest, as well
as in narrow redshift bins to make sure that we are not missing trends at any
particular redshift.

###  5.1 Redshift failure trends in the fiducial redshift ranges

In Fig. 11, we test the uniformity of the success rate against TSNR2, using
each tracer within the default redshift range: $0.4<z<1.1$ for LRG,
$0.1<z<0.4$ for BGS (and additionally applying the $M_{r}<-21.5$ cut, which
can only be applied to successful redshifts), $0.8<z<1.6$ for ELG, and
$0.8<z<2.1$ for QSO. This is in contrast to the fits, which use the redshift
success rate defined over the entire redshift range (as we make no reference
to redshift when defining the model and weights, since failed observations do
not have a good redshift by definition). We measure the significance of trends
using the $\chi^{2}$ difference from unity in all bins, using a binomial error
for each bin. As discussed in Sec. 2.3, this is not a good model for the
covariance in general, as it neglects the cosmic variance contribution, but in
the limit of a large redshift depth, the cosmic variance contribution is
suppressed.

We show several extensions to these plots in Appendix B: success rate trends
using the entire redshift range of each tracer (Fig. 35); success rate trends
for the extended quasar redshift range of $0.8<z<3.5$ (Fig. 35); and success
rate trends for BGS without the “BRIGHT” or absolute-magnitude cuts (Fig. 36).

Similar plots for ELG are shown in Fig. 4 of [1]. However, in those plots,
there is no redshift limit applied to the definition of a successful redshift
(not even the looser limits that we apply in Appendix B and Fig. 35); this
accounts for the slightly different trends and $\chi^{2}$ seen in [1].

For LRG, ELG, and QSO in the South, there is a significant relationship
between success rate and TSNR2 that is removed by applying the weights. Note
that the unweighted trends, while statistically significant due to the large
sample sizes and thus small errors, are quite small in magnitude, $<3\%$. For
BGS_BRIGHT-21.5, there is no significant relationship, and applying the
weights does not measurably change the $\chi^{2}$. In contrast, Fig. 5 shows a
clear increase in BGS_BRIGHT failure rate at low TSNR2. This difference is
driven by galaxies outside the fiducial redshift range, most likely at
$z>0.4$; referring to Fig. 36 in Appendix B shows that BGS_BRIGHT and
BGS_BRIGHT-21.5 have similarly mild success rate trends at low TSNR2.

####  5.1.1 Anomalous trends in the quasars

QSO N shows a minimal relationship between TSNR2_QSO and success rate at
TSNR2_QSO $<40$, and then a significant 8% drop in the success rate at high
TSNR2_QSO (Fig. 11). The $\chi^{2}$ indicates a significant trend that is not
mitigated by the weights, since they are unable to account for declining
success rate with increasing TSNR2_QSO. The drop in TSNR2_QSO contrasts with
our intuitive expectation that as we increase the signal-to-noise, the success
rate should increase. Using the classifications of [50], we find that at
TSNR2_QSO $>40$ and in the northern hemisphere, 62.9% of quasar targets are
classified as quasars, 24.0% are galaxies, 9.4% are bad, and 3.7% are stars.
At TSNR2_QSO $<40$, 65.1% of quasar targets are quasars, 21.6% are galaxies,
10.2% are bad, and 3.1% are stars.131313We point out that in the North, a
higher fraction of quasars targets are spectroscopic galaxies due to the worse
PSF in the BASS $g$ and $r$ bands [60], compared to the 16% galaxy in deep VI
tiles from SV [14]. The larger PSF size makes it more difficult to distinguish
stars and galaxies, leading to a higher galaxy contamination fraction. The
high TSNR2_QSO quasar targets lie at lower Galactic latitude, although the
only notable difference in their average imaging properties is a 20% increase
in Milky Way star density. It therefore appears that the drop in success rate
at high TSNR2_QSO is due to a targeting fluctuation, not a spectroscopic
fluctuation: these targets happen to lie at lower Galactic latitude where a
higher fraction of quasar targets are spectroscopic galaxies.

If we re-define the quasar success rate so that spectroscopic stars and
galaxies are counted as neither successes nor failures, the anomalous trend in
the North disappears (lower left panel of Fig. 11). While the relative success
rate is still $<1$ for the three highest TSNR2_QSO bins at TSNR2_QSO $>45$,
none of these deviations are statistically significant. This supports our
argument that the anomalous trend at high TSNR2_QSO is due to a chance
alignment between TSNR2_QSO and regions at low Galactic latitude with a higher
fraction of galaxies in the quasar targets.

This result suggests that we should use the classifications of [50] and
exclude successfully classified stars and galaxies from both the numerator and
the denominator of the quasar success rate. Redshift success rate should be
defined as the fraction of all true quasars that are successfully classified
as quasars by the DESI pipeline. While the stars and galaxies are quasar
targets, if they can be confidently classified as stars or galaxies, they are
clearly not true quasars. However, there is a problem with this proposal. The
galaxy classification of [50] is only 90% pure; the remaining 10% are almost
all true quasars (using the deep VI data as truth). Furthermore, it is not
clear that all of the “unclassified” spectra are true quasars, rather than
stars or galaxies. Of the 40% of the unclassified spectra with a VI
classification, 54% are quasars, 27% are stars, and 19% are galaxies. When
multiplying by the overall classified fraction (67% quasars, 16% galaxies, 12%
unclassified, 5% stars in the South region), this means that there are nearly
as many true quasars classified as galaxies (1.6% of the total) as those
classified as “unknown” (2.6%). Hence, without a higher-purity classification
of the galaxies, and a better understanding of the unclassified spectra (7% of
even the deep VI spectra cannot be confidently classified [14]), it is
difficult to properly modify the quasar success rate by removing stars and
galaxies from the calculation. As a result, we retain the simple
classification where all non-quasar targets are designated “failures” in the
success rate calculation, and defer re-considerations of this definition to
future DESI data releases. These choices only affect the tracer QSO clustering
and do not affect the Ly$\alpha$ forest, which depends on the observed
Ly$\alpha$ absorption along the line of sight to each quasar.

As a further validation of the quasar spectroscopic success rate definition,
we also measure the success rate for two subsets of the quasar target sample
with higher quasar purity, defined in [50]. The main quasar target sample has
relatively low purity, with only 60% of targets spectroscopically classified
as quasars, 18% unclassified, 16% galaxies, and 5% stars (using the [50]
classifications on the Y1 data rather than the SV or VI measurements). We also
consider a “no star” sample, where we use the photometric star probability of
[61], removing any quasar target with $P_{\rm star}>0.01$. This sample is 66%
quasars, 16% unclassified, 17% galaxies, and 1% stars. We also consider a
“high purity” sample, where we remove roughly the faintest half of the quasar
sample using cuts in W2 and $r$ magnitudes. This sample has half the number
density of the main quasar targets, but is 90% quasars, 4% unclassified, 4%
galaxies, and 2% stars.

We then measure the failure rate as a function of TSNR2_QSO for the main
quasar sample and the “no star” and “high purity” subsets. We find consistent
behavior between the “full,” “no star” and “high purity” subsamples. This
allows us to conclude that the measured quasar redshift success rate is
representative of the “true” redshift success rate, defined as the ratio of
spectroscopically-classified quasars to true quasars within the target sample,
and is robust to the presence of interlopers (stars and galaxies) within the
quasar sample. We provide more details in Appendix C.

Figure 11: Validation of the redshift failure weights for LRG, BGS, ELG, and
QSO in their fiducial redshift ranges. Background histogram shows the
distribution of TSNR2 for each tracer. Once redshift failure weights are
applied (points with errorbars), there are no significant trends between TSNR2
and success rate, removing the significant trends seen without the weights
(solid lines with shaded errorband). The $\chi^{2}$/dof are measured from the
deviation between the measured TSNR2 trend and a flat success rate at unity in
all TSNR2 bins. There remains a significant deviation from a flat success rate
trend in QSO N. This is because in the North imaging, high-TSNR2 regions of
the sky lie at lower Galactic latitude, where quasar targets are more likely
to be spectroscopically classified as galaxies. If we remove galaxies and
stars from the numerator and denominator of the quasar redshift success rate,
the dip at high TSNR2_QSO in the N is eliminated (light points). However,
removing stars and galaxies is complicated due to quasar contamination in the
galaxy classification and uncertainty in the classification of the residual
“unknown” spectra: see Section 5.1.1 for details.

###  5.2 Redshift failure trends in narrow redshift bins

We divide each sample into redshift bin subsets and test whether the redshift
failure weights successfully remove any redshift failure trends. We use bins
of $\Delta z=0.1$ for LRG and BGS, and $\Delta z=0.3$ for QSO (except for the
final QSO bin, which extends from $1.7<z<2.1$). The same analysis for ELG is
shown in Fig. 2 of [1]. That paper finds no significant redshift-dependent
trends for ELG once the redshift failure weights are applied.

####  5.2.1 LRG

The trends for LRG are shown in Fig. 12. The largest outlying trend is in the
South region at $0.8<z<0.9$, which shows a trend of increasing success rate as
TSNR2 increases. If the $\chi^{2}$ statistic follows a $\chi^{2}$ distribution
with the specified number of degrees of freedom, this trend is significant at
the equivalent of 4.1 Gaussian $\sigma$. The only other trend significant at
$>3\sigma$ is in the North region at $0.5<z<0.6$, which is significant at
3.1$\sigma$ and is largely driven by the point at highest TSNR2. The total
$\chi^{2}$ is 104.5 in N and 148.3 in S, across 70 bins each. This is
significant at a Gaussian equivalent of 2.6$\sigma$ in N and 5.1$\sigma$ in S.

The $\chi^{2}$ calculation uses binomial errors, but as discussed above, the
contribution from cosmic variance may be non-negligible, particularly in these
relatively narrow bins. As a result, we also compute $\chi^{2}$ for each
redshift bin and imaging region using the Abacus FFA mocks (Fig. 13). We find
that the mock mean and standard deviation (summing over all seven redshift
bins) $\chi^{2}=110\pm 23$ in the North and $96\pm 13$ in the South. Thus we
estimate that the North data is perfectly consistent with the mocks (with
$p=0.56$ since 14 of the 25 mocks have higher $\chi^{2}$ than data) while the
South is roughly a 4.0$\sigma$ outlier (none of the mocks have higher
$\chi^{2}$ than data). The most outlying bin in the South is the $0.8<z<0.9$
bin, which is a 3.2$\sigma$ outlier based on the mock mean and standard
deviation; none of the mocks have a higher $\chi^{2}$ than the data. The
combination of North and South has $p=0.08$, so it is possible that this is
simply a fluctuation in the $0.8<z<0.9$ bin driving the high $\chi^{2}$.
However, given that the shape of the trend matches what we expect from issues
in the redshift failure model, we test the application of an extra set of
weights to the South LRG at $0.8<z<0.9$ which explicitly nulls this trend. We
describe the impact of these weights on the observed power spectrum multipoles
in Section 6.

Figure 12: Relationship between success rate and TSNR2 in $\Delta z=0.1$ bins
for LRG, both without redshift failure weights (lines and shaded regions) and
with them (points with errorbars). The $\chi^{2}$ uses binomial errors in each
TSNR2 bin and requires comparison to mocks to interpret the significance, due
to the contribution of cosmic variance to the redshift failure errorbar within
each bin.  Figure 13: Distribution of $\chi^{2}$ within each redshift bin for
the 25 Abacus FFA mocks (gray lines) and the data (red) for LRG. The data lies
within the fluctuations of the mocks, except for the $0.8<z<0.9$ bin in the
South.

####  5.2.2 BGS

For BGS, we find significant trends in the South, especially in the
$0.2<z<0.3$ and $0.3<z<0.4$ bins (Fig. 14, where none of the mocks have higher
$\chi^{2}$ than observed in data (Fig. 15). We find $\chi^{2}$ in data is high
by 3.4$\sigma$ at $0.2<z<0.3$ and 2.2$\sigma$ at $0.3<z<0.4$, and the
combination in the South is high by 4.2$\sigma$ (all compared to mocks).
However, the shapes of the observed trends are puzzling, as the most
significant trend is the declining trend with increasing TSNR2_BGS in the
$0.2<z<0.3$ bin, rather than the more expected increasing trend.

This trend is likely due to slight differences in the $r_{\textrm{fiber}}$
distribution in the different TSNR2_BGS bins, i.e. again due to targeting
fluctuations that happen to align with regions with higher TSNR2_BGS. In Fig.
16, we show that targets in the higher TSNR2_BGS bins are generally slightly
fainter than those in the lower TSNR2_BGS bins. Combined with the steep drop
in spectroscopic success as a function of $r_{\textrm{fiber}}$, this leads to
a lower success rate in the higher TSNR2_BGS bins. We therefore divide the BGS
targets into 5 bins in $r_{\textrm{fiber}}$, matching the 5 bins used in the
flux-dependent redshift success fit. We then re-weight each TSNR2_BGS bin to
match the $r_{\textrm{fiber}}$ distribution of the first bin, and measure
$\chi^{2}$ for the re-weighted sample. We find that $\chi^{2}$ (with 10
degrees of fredom each) drops in the $0.1<z<0.2$ bin from 19.8 to 16.1, in the
$0.2<z<0.3$ bin from 34.4 to 20.2, and in the $0.3<z<0.4$ bin from 35.5 to
26.5. This drops the data from a 3.4$\sigma$ deviation to a 1.1$\sigma$
deviation at $0.2<z<0.3$, from 2.2$\sigma$ to 1.0$\sigma$ at $0.3<z<0.4$, and
from 4.2$\sigma$ to 1.8$\sigma$ considering the combination of all three bins.
This implies that the TSNR2_BGS trends are driven by the slight change in
$r_{\textrm{fiber}}$ distribution with TSNR2_BGS, and are not indicative of
underlying spectroscopic uniformity issues.

Figure 14: Same as Fig. 12, but for BGS_BRIGHT-21.5.  Figure 15: Same as Fig.
13, but for BGS_BRIGHT-21.5. The data lies within the fluctuations of the
mocks for North, but not for South. This is due to slightly fainter
$r_{\textrm{fiber}}$ targets in the high TSNR2_BGS bins. Re-weighting each
TSNR2_BGS bin to match the $r_{\textrm{fiber}}$ distribution in the first
TSNR2_BGS bin greatly reduces the discrepancy between data and mocks (dashed
red curve).  Figure 16: PDF of $r_{\textrm{fiber}}$ in bins of TSNR2_BGS,
normalized by the PDF in the lowest TSNR2_BGS bin. The right-hand $y$-axis
shows the success rate for BGS_BRIGHT (without the absolute magnitude cut);
the success rate is plotted in the thick red line as a function of
$r_{\textrm{fiber}}$.

####  5.2.3 QSO

We perform the same test for QSO as for LRG and BGS (Fig. 17). Due to the
fluctuation in the targeting sample leading to more galaxies at higher
TSNR2_QSO in the North (which has a particularly strong impact on the success
rates in the $1.7<z<2.1$ bin), we only use the first six points with TSNR2_QSO
$<40$ when computing $\chi^{2}$ in the North. Summing over the redshift bins,
we find a total significance of $\chi^{2}=21.6\pm 5.3$ in North and $37.8\pm
7.8$ in South. The data values are very consistent with the mocks in all
redshift bins (Fig. 18). Summing over all redshift bins in the data,
$\chi^{2}=18.4$ (24 dof) and 26.4 (40 dof) in North and South, with $p$ values
of 0.68 and 0.88, respectively.

Figure 17: Same as Fig. 12, but for QSO. While we show the success rate at
TSNR2_QSO $>40$ in the North, we do not include those points in the $\chi^{2}$
calculation, since they are affected by the targeting fluctuation leading to
more spectroscopic galaxies in regions of high TSNR2_QSO (lower right panel of
Fig. 11).  Figure 18: Same as Fig. 13, but for QSO.

###  5.3 Trends with focal plane radius

Even after applying the redshift failure weights, we find a residual
significant, though sub-percent, relationship between redshift success rate
and the target’s position on the focal plane. We show the relationship between
redshift success and focal plane radius on each of the ten petals for LRG
(where the trends are strongest) in Fig. 19. The trends are weaker for other
tracers, and we show them in Appendix D. In these figures, errorbars are
estimated by jackknife resampling, sequentially leaving out all galaxies
within each nside=32 HEALPix pixel.

Trends are generally much more significant when defined using the entire
redshift range of a tracer rather than the clustering redshift range. These
trends are only mildly reduced if we fit a linear relationship between focal
plane radius and success in each petal and apply it as a weight.

Using the entire redshift range, we find significant trends for LRG in all
petals except petal 7. Applying linear focal plane radius weights reduces the
significance of all trends, but the $\chi^{2}$ are still too high for all
petals except petal 2.

For LRG, we attempted to model these focal plane variations (and additional
angular variations apparent in Fig. 2) with Zernike polynomials defined on the
entire focal plane. However, we found that it took a very large number of
polynomials to even roughly fit the observed trends. Instead, we consider the
impact of these trends on clustering statistics by applying a set of weights
that fully nulls all fiber-to-fiber variation in the success rate,
$\eta_{\textrm{zfail}}$, defined similarly for ELG in [1]. These “focal plane
weights” are prone to overfitting, since they will null the Poisson
fluctuations to the per-fiber success rate as well as the true fluctuations.
We measure the impact of the focal plane weights on two-point clustering
statistics in Sec. 6 to provide an upper bound for the impact of these
residual variations in spectroscopic success rate on the focal plane.

Figure 19: Redshift success rate vs. focal plane radius for each of the ten
petals for LRG, using both the clustering redshift range $0.4<z<1.1$ (yellow)
and the entire redshift range, $0<z<1.6$ (green). Lines weight by the default
redshift failure weight $w_{\textrm{zfail}}$, while points apply an additional
linear focal plane radius weight to each petal (“linear wt”), which reduces
the significance of trends but does not eliminate them.

###  5.4 Time-dependent success rate

Figure 20: $z_{\textrm{fail}}$-weighted redshift success rate as a function of
time for LRG in the North (top) and South (bottom). The data is also split by
petal (abbreviated with P in the legend). We use a preliminary version of the
catalogs to make this figure (internally labelled “v0.6”) to show the dip in
success rate in petal 9 on December 12, 2021, which was fixed in the final
catalogs.

We split the data into 20 bins of approximately equal sample size using the
“LASTNIGHT” column indicating the last night that a particular target was
observed. As discussed above, some targets that are re-observed may be
observed on multiple nights. We show the dependence of LRG redshift failure
weighted success rate on petal and time in Fig. 20.

We find two significant dips in spectroscopic success. First, the petal 6
success rate drops by $\sim$1% between October 26, 2021 and November 6, 2021.
This is due to a known hardware issue: the $Z$6 CCD amplifiers were unstable
on these dates, but reverted to normal behavior after November 6. 0.95% of the
LRG sample is affected by this issue (observed on petal 6 in the specified
time period). We further measured the per-fiber redshift success rate during
this time period, and found that certain fibers were more affected than
others. The gray regions in Fig. 21 (identified as low success rate regions a
posteriori) have 96.4% success rate, compared to 98.6% for the other petal 6
fibers in this date range (white region), and 99.3% $z_{\textrm{fail}}$
success rate in the sample as a whole.141414This is higher than the unweighted
success rate of 98.9% because the redshift failure weights are always greater
than one.

Despite the lower success rate, we find no difference in redshift distribution
between the lowest-success rate fibers (gray region), the other petal 6 fibers
(white region), or the redshift distribution of the entire survey. We do not
find any evidence for spikes or pileups at any particular redshift, and the
mean redshifts are entirely consistent (Fig. 21). The mean redshift is
$0.7235\pm 0.0018$ in the good petal 6 regions between October 26 and November
6; $0.7222\pm 0.0022$ in the lowest success regions of petal 6; and $0.7204\pm
0.00013$ for the entire DR1. All errorbars are standard error of the mean. We
therefore see no significant offset in the mean redshift between the white and
gray regions of petal 6. We note that the $Z$6 amplifier issue was originally
identified by the redshift pileups that it created on the daily QA, but these
are entirely removed by the redshift success criterion.

We therefore test the application of a uniform weight to the affected
galaxies, the “Z6_ZFAIL” weight. We find that adding this weight makes no
practical difference to observed galaxy clustering. We therefore do not apply
the weight to the catalogs since it is somewhat arbitrary to determine which
fibers within petal 6 should be upweighted, and whether the upweighting should
be the same every day that was affected, or different.

We also identify an issue with petal 9 on December 12, 2021. Fig. 20
aggregates many nights together into the point labelled 2021-12-20, but we
find that the issue is isolated to a single night and leads to a 93% success
rate in petal 9 on that night. As described in Sec. 2.2 of [18], there was a
bug in the calibration of the ‘iron’ data processing affecting only December
12, 2021. This night was reprocessed and substituted for the original ‘iron’
reprocessing in the LSS catalog construction.

Other dips in the success rate are less significant, and we find that they are
associated with success trends with number of exposures and number of nights
over which a target was observed, as discussed below.

We also measure the time-dependence of the BGS, ELG, and QSO success rate. We
plot the BGS_BRIGHT success rate as a function of time in Fig. 22. We do not
see either the petal 6 or petal 9 outliers that are identified from the LRG
success rate. The overall redshift success rate is less stable than for LRG,
but fluctuates in a way that affects all petals equally.

We show the time-dependent ELG success rate in Fig. 23 and the time-dependent
QSO success rate in Fig. 24. Both ELG and QSO success rates fluctuate much
more than LRG or BGS_BRIGHT. This is because both samples are highly impacted
by inhomogeneities in the imaging; since DESI observes different areas of sky
at different times, the success rate will change depending on the imaging
properties of the regions that it is observing. Large variations are seen in
the ELG in the South because we observe the SGC in the fall and winter and the
NGC in the spring and summer; the deepest imaging is in DES in the SGC,
leading to higher success rates in fall and winter. The success rate is
generally consistent between petals at a given time, except for the $Z$6
amplifier issue. We plan to better model the imaging dependence of the success
rate in future releases to allow a better study of the spectroscopic success
rate stability, particularly for ELG.

Figure 21: Top: Per-fiber LRG success rate for petal 6 observed between
October 26, 2021 and November 6, 2021 (affected by the $Z$6 amplifier
instability). The two horizontal lines show the overall LRG success rate and
the average petal 6 success rate in this time period. The shaded regions
identify contiguous blocks of fibers with clearly lower success rates; the
success rate elsewhere is 98.6%, still slightly low compared to the other
(normal) time-dependent fluctuations. Bottom: Redshift distribution for both
the shaded (“bad”) and unshaded (“good”) petal 6 redshifts within the $Z$6
amplitifier instability time period, and also compared to the overall LRG
redshift distribution. There is no evidence for any difference in redshift
distribution or spikes at particular spurious redshifts, indicating that the
redshifts affected by the $Z$6 amplitifier instability are fine, just with a
reduced success rate.

Figure 22: $z_{\textrm{fail}}$-weighted redshift success rate as a function of
time for the BGS_BRIGHT in the North (top) and South (bottom).

Figure 23: $z_{\textrm{fail}}$-weighted redshift success rate as a function of
time for ELG in the South (top) and North (bottom).

Figure 24: $z_{\textrm{fail}}$-weighted redshift success rate as a function of
time for QSO in the South (top) and North (bottom).

We identified and investigated other fluctuations in the success rate, but
these were not traceable to known hardware or processing issues and instead
seemed to be the largest fluctuations from spectroscopic success rate trends
with overall properties such as the survey speed. In LRG we found that
galaxies coadded across multiple nights or exposures had a lower
$z_{\textrm{fail}}$-weighted success rate. We measured 99.35% redshift success
for galaxies observed in a single exposure; 99.15% success rate for galaxies
with two exposures (16.6% of the data); 98.95% for galaxies with three
exposures (3.2% of the data); and 99.22% success rate for galaxies with four
exposures (0.3% of the data). Galaxies split across multiple nights showed a
similar trend: 99.32% success for one night; 99.15% for two nights; and 98.67%
for three nights. We found a 0.5% drop in success rate between galaxies in the
lowest fifth percentile for survey speed (98.92%) compared to galaxies in the
top quarter of speed (99.4%); most of this drop is concentrated in the slowest
quarter of the sample, with a success rate of 99.26% for galaxies in the 20th
to 30th percentile in speed. These trends exist after applying the TSNR2-based
redshift failure weights, indicating that there are residual trends with low-
speed regions or observations requiring multiple exposures that are not
captured by TSNR2 or in the model linking TSNR2 to redshift success rate.

We see similar trends in the other tracers, with QSO showing by far the
largest impact for galaxies observed across multiple exposures or multiple
nights (3% drop between 1 night or exposure and 3 nights or exposures).
Success rate showed no clear trend with speed for QSO and BGS_BRIGHT, but the
slowest ELG had a $\sim$1% lower success rate than average. Due to the
fluctuations in success rate seen with these quantities, we define weights to
correct these trends and study their impact on galaxy clustering in Section 6.

| Nights | Exposures
---|---|---
Number | LRG | BGS_BRIGHT | ELG | QSO | LRG | BGS_BRIGHT | ELG | QSO
1 | 0.993 | 0.990 | 0.726 | 0.677 | 0.994 | 0.990 | 0.727 | 0.679
2 | 0.992 | 0.988 | 0.716 | 0.6453 | 0.992 | 0.989 | 0.721 | 0.659
3 | 0.987 | 0.984 | 0.727 | 0.644 | 0.990 | 0.988 | 0.712 | 0.650
4 |  |  |  |  | 0.992 | 0.995 | 0.734 | 0.673
Table 2: $z_{\textrm{fail}}$-weighted redshift success rates as a function of
number of nights or number of exposures required to observe a target for
sufficient effective time.

##  6 Impact of residual spectroscopic systematics on galaxy clustering

The overall impact of the redshift failure weights on clustering is small,
though not necessarily negligible, with $\chi^{2}_{\textrm{sys}}\sim 1$. When
turning off the redshift failure weights, we find
$\chi^{2}_{\textrm{sys}}=0.26$ and 0.63 for the two ELG bins at $0.8<z<1.1$
and $1.1<z<1.6$, and 0.24 for QSO. They have a much smaller impact on LRG and
BGS, with $\chi^{2}_{\textrm{sys}}<0.04$ for LRG and $<0.01$ for BGS. We show
the impact of turning off the redshift failure weights on the QSO in Fig. 25;
the impact of turning off these weights is roughly comparable to the largest
systematics arising from the additional weights.

Figure 25: Deviations between the QSO power spectrum multipoles measured with
the default weighting scheme, and without the redshift failure weights. We
show QSO as it shows the largest impact of the redshift failure weights on
clustering.

Next, we study the impact of the additional (optional) weights on galaxy
clustering. We find that the additional weights make a minimal difference in
the measured LRG clustering. First, we show the impact of the $Z$6 weights in
Fig. 26. We consider the clustering in both the NGC and SGC, as well as the
combined clustering (“GCComb”). We find a maximum
$\chi^{2}_{\textrm{sys}}=0.03$, and less than a 0.03$\sigma$ offset between
the default weights and the $Z$6 weights. This justifies our decision to not
include the $Z$6 weights by default, as excluding them has a completely
negligible impact on galaxy clustering.

In Fig. 27, we show the impact of adding extra weights to correct the
TSNR2_LRG trend in the South at $0.8<z<0.9$. Adding these weights makes up to
a 0.08$\sigma$ impact on the monopole and quadrupole at large scales. The
largest $\chi^{2}_{\textrm{sys}}=0.04$ for the SGC combination of monopole and
quadrupole, and 0.02 for the combined hemisphere monopole plus quadrupole.

Figure 26: Deviations between the LRG power spectrum multipoles measured with
the default weighting scheme, and adding an extra $Z$6 weight to fibers
affected by the $Z$6 amp instability issue. Deviations are presented in units
of $\sigma$, the diagonal errorbar from the analytic covariance matrix used in
the DESI Y1 analysis.  Figure 27: As in Fig. 26, but correcting the TSNR2_LRG
trend in the $0.8<z<0.9$ bin of LRG. We only show the $0.8<z<1.1$ bin since
this is the only one affected by the extra weights.

Finally, in Fig. 28, we show the impact of adding weights for survey speed on
the LRG clustering. The impact is very small at $k>0.01$ $h$ Mpc-1
($<0.01\sigma$) and $<0.05\sigma$ on large scales. The largest
$\chi^{2}_{\textrm{sys}}=0.02$.

Figure 28: As in Fig. 26, but adding weights correcting the survey speed trend
for LRG.

We also apply focal plane weights $\eta_{\textrm{zfail}}$ and weights
correcting trends with number of exposures or nights. We find that adding
these weights change $P(k)$ by $<0.01\sigma$ in all $k$ bins, and
$\chi^{2}_{\textrm{sys}}$ is less than 0.01.

We apply focal plane weights and number of night and exposure weights to
BGS_BRIGHT. We find $<0.02\sigma$ changes in all $k$ bins and
$\chi^{2}_{\textrm{sys}}<0.01$; i.e. negligible impacts on clustering.

We find ELG are more sensitive to applying these extra weights. The speed
weights have the largest impact on ELG clustering (Fig. 29), with up to a
$0.5\sigma$ impact on the large-scale monopole at $1.1<z<1.6$. These translate
into $\chi^{2}_{\textrm{sys}}$ of 0.004 and 0.47 in the two redshfit bins,
when combining hemispheres. $\chi^{2}_{\textrm{sys}}$ is similar between both
hemispheres and the combined case.

The weights for number of nights or number of exposures have a smaller impact
on ELG clustering. We show the number of exposures weights in Fig. 30. The
impact is small ($\chi^{2}_{\textrm{sys}}=0.05$ in both redshift bins) and
mainly concentrated on large scales. The impact of number of nights weight is
similar, but smaller, with $\chi^{2}_{\textrm{sys}}=0.05$, and a $0.05\sigma$
change in the low-$k$ monopole. The impact of focal plane weights is shown in
Figs. 5 and 6 and Table 1 of the companion paper [1]. The
$\chi^{2}_{\textrm{sys}}$ is at most 0.23 and individual $k$ bins shift by
$<0.1\sigma$ in an incoherent way, indicating that adding this weight has a
minimal impact on the clustering measurement.

Figure 29: As in Fig. 26, but adding weights correcting the survey speed trend
for ELG.  Figure 30: As in Fig. 26, but adding weights correcting the number
of exposures trend for ELG.  Figure 31: Impact on ELG clustering of removing
fibers 1261, 1269, 1295, 1296 and 1307, visually identified as having
anomalous spikes in the redshift distribution at $z=1.5$.

We show the impact of removing the fibers with anomalous spikes in the
redshift distribution at $z=1.5$ (1261, 1269, 1295, 1296 and 1307). Removing
these fibers leads to random scatter in the power spectrum multipoles with
amplitude $\sim 0.1\sigma$, leading to $\chi^{2}_{\textrm{sys}}=0.26$ and 0.49
in the two ELG redshift bins.

These additional weights have the largest impact on the QSO clustering. We
find $\chi^{2}_{\textrm{sys}}=1.9$ when applying the exposure weights (Fig.
32), an offset that is driven by the NGC ($\chi^{2}_{\textrm{sys}}=2.1$)
rather than the SGC ($\chi^{2}_{\textrm{sys}}=0.1$). The offset is largest at
large scales, reading 1.4$\sigma$ in the monopole and 0.6$\sigma$ in the
quadrupole. The impact of number of nights weights is similarly peaked towards
large scales, but much more modest overall, with
$\chi^{2}_{\textrm{sys}}=0.13$ and at most a 0.2$\sigma$ impact on the large-
scale monopole. Finally, we find a relatively large $\chi^{2}_{\textrm{sys}}$
impact from the focal plane weights, 1.2 in the combined hemisphere
measurement (Fig. 33). However, the pattern of the impact on each $k$ bin is
relatively random with a scatter of $\sim 0.3\sigma$; this may simply be due
to the scatter in the focal plane weights from the relatively small number of
quasar targets per fiber ($\sim$400).

Figure 32: As in Fig. 26, but adding weights correcting the number of
exposures trend for QSO.  Figure 33: As in Fig. 26, but adding weights
correcting the focal plane trend for QSO.

###  6.1 Impact on cosmological parameters

Sample | $\Delta\alpha_{\parallel}$ | $\Delta\alpha_{\perp}$ | $\Delta h$ | $\Delta\Omega_{m}$ | $\Delta\log(A_{s})$
---|---|---|---|---|---
QSO, no. of exposures | +0.001 (0.059) | -0.003 (0.030) | +0.001 (0.031) | -0.001 (0.020) | 0.008 (0.16)
QSO, focal plane | -0.003 (0.059) | +0.003 (0.030) | -0.004 (0.031) | 0.0 (0.020) | +0.005 (0.16)
ELG, $0.8<z<1.1$, speed | +0.002 (0.092) | +0.01 (0.1) | +0.002 (0.038) | -0.001 (0.034) | 0 (0.21)
ELG, $1.1<z<1.6$, speed | +0.002 (0.080) | 0 (0.073) | +0.001 (0.020) | -0.001 (0.020) | -0.014 (0.23)
ELG, $0.8<z<1.1$, mask $n(z)$ spike | -0.004 (0.092) | +0.01 (0.10) | +0.004 (0.038) | -0.003 (0.034) | -0.01 (0.21)
ELG, $1.1<z<1.6$, mask $n(z)$ spike | +0.002 (0.080) | -0.003 (0.073) | +0.002 (0.020) | 0.0 (0.020) | -0.036 (0.23)
Table 3: Change in parameters for BAO fits (left) and full-shape fits (right)
between the default clustering measurements and clustering measurements using
the weights and samples indicated in the left column. 1-$\sigma$ error for
each parameter is given in parenthesis. The parameter shifts are at most 30%
of the errorbar.

We propagate the impact of the trends with a significant $\Delta\chi^{2}$ onto
derived cosmological parameters (Table 3). We perform both BAO and full-shape
(RSD) fits. In both cases, we fit to the blinded data [62] and compare results
with and without the weights under consideration. For the BAO fits, we match
the default settings of [63, 19], fitting to the un-reconstructed power
spectrum. For the full-shape fits, we use the “velocileptors” LPT code [64,
65, 66] 151515https://github.com/sfschen/velocileptors with the “physically
motivated” prior bias (all bias parameters and counterterms are multiplied by
appropriate powers of $\sigma_{8}$) and “maximum freedom” in the bias
parameters (i.e. fixing them to the coevolution relations) as implemented in
the desilike code, fitting to the monopole and
quadrupole.161616https://github.com/cosmodesi/desilike/blob/41aec182cd0d5da0657a3bb11cda74b382cad4be/desilike/theories/galaxy_clustering/full_shape.py
We again fit to the blinded power spectrum data, without any small-scale
$\theta$-cut applied. For the full-shape fits, we vary the cosmological
parameters $h$, $\Omega_{m}$, $\log(A_{s})$, and $\omega_{b}$, with the
$\omega_{b}$ posterior dominanted by a Gaussian prior at
$\omega_{b}=0.02237\pm 0.00037$. We summarize the parameters $h$,
$\Omega_{m}$, and $\log(A_{s})$ for the full-shape fits in Table 3, and
$\alpha_{\parallel}$ and $\alpha_{\perp}$ for the template-based BAO fits.
While the settings in both the BAO and RSD cases do not exactly match the
fiducial choices for the DESI Y1 analysis, our primary concern is the
difference between the weighted and unweighted cases, not the actual values of
the parameters.

We find negligible changes on the cosmological parameters when applying the
extra weights. The largest change is a 0.15$\sigma$ shift in $\log{A_{s}}$
when masking the 5 fibers with a spike at $z=1.5$ in ELG at $0.8<z<1.1$. Apart
from this shift, every other parameter shift is $<10\%$ of the errorbar. These
fits all use $k_{\textrm{min}}=0.02$ $h$ Mpc-1, and are thus largely
unaffected by the low-$k$ uptick when applying the weights. However, we
caution that other parameter constraints which are more sensitive to large
scales (e.g. scale-dependent bias due to primordial non-Gaussianity) may be
more impacted by these additional weights.

In the end, due to the negligible impact of the extra weights, we do not
choose to use any of the extra weights in the LSS catalogs. We only report
$w_{\textrm{zfail}}$ in the LSS catalogs and in the large-scale BAO and RSD
analyses. However, we caution that other analyses of DESI data (for instance,
higher-point correlation functions) may want to make different choices about
what set of weights to use.

##  7 Conclusions

We develop and validate corrections for spectroscopic inhomogeneities in the
DESI DR1 galaxy clustering catalogs, which consist of luminous red galaxies
(LRG), bright galaxies (BGS), quasars (QSO), and emission line galaxies (ELG).
DESI tiles are observed for a uniform effective exposure time, but small
variations in the actual exposure times as well as fiber-to-fiber throughput
variations lead to variations in the effective observing time as parametrized
by the template signal-to-noise ratio squared, TSNR2, for different
observations. Out of 4326 fibers with valid spectra, we remove 60 fibers that
are $>4\sigma$ outliers in success rate, compared to expectations from a Monte
Carlo simulation using the modelled variation in success rate with TSNR2,
removing 1.2% of the total data. We then describe the creation of weights to
mitigate trends between redshift success rate and TSNR2 for LRG, BGS, and QSO.
The process for ELG is conceptually similar but different in detail, since ELG
redshift success depends on the flux of the [OII] line and is thus very
sensitive to fine variations in the sky background due to the forest of
atmospheric sky lines at the red end of DESI spectra. It is described in the
companion paper, [1].

We then validate the weights in Section 5. Without the redshift failure
weights, there are significant trends for LRG and QSO; after applying the
weights, their success rate is uniform. Even after applying the weights, we
find small but significant fluctuations in success rate as a function of
position on the focal plane, survey speed, or number of exposures required to
complete a galaxy observation. Finally, we study the time stability of the
redshift success rate and identify two periods of outlier performance. One of
these is related to a calibration error in the “iron” processing and was fixed
for production of the clustering catalogs. The other is related to a transient
hardware issue leading to instability in the $Z$-camera amplifier on petal 6
(lasting for 10 days). We find this issue leads to a uniform reduction in the
spectroscopic success rate, but no change in the redshift distribution.

We propagate the impact of the remaining residual statistically significant
trends on galaxy clustering in Section 6. We find that adding a weight for the
$Z$6 amplifier issue has no measurable impact on galaxy clustering, and so we
continue to use redshifts affected by this issue, without a weight to correct
them. The residual trends with the largest impact are survey speed trends for
the ELG, and number of exposure trends for QSO. Both of them have the largest
impact on very large scales ($k<0.03$ $h$ Mpc-1). We find that these trends
(along with other less significant trends) change cosmological parameters from
BAO and RSD by $<15\%$ of their errorbar: they are negligible compared to the
statistical uncertainty. However, we caution that other observables (small-
scale clustering, higher-order clustering, or very large scale observations)
may be affected differently by these residual trends, and their impact should
be tested to ensure that it is negligible.

We aim to further develop and test these corrections for future DESI data
releases. For instance, more detailed study of the $>4\sigma$ outlier fibers
may allow us to recover some of these fibers, slightly increasing the
available catalogs, or improve the modelling of success rate with TSNR2. While
the success rate is well-defined for the high-completeness LRG and BGS
samples, it is harder to define for the QSO, where a substantial fraction of
QSO targets result in spectroscopically confirmed galaxies or stars. Further
study of the properties of QSO targets that become galaxies or stars could
allow for a better and more consistent definition of spectroscopic success
rate for this sample. One possibility for an improvement is jointly modelling
the spectroscopic and imaging systematics, which will also allow us to better
distinguish chance alignments between targeting fluctuations and spectroscopic
properties (as discussed in Section 5.1.1). Another possibility is to use
machine learning techniques to perform a blind search for all possible
spectroscopic systematics affecting the redshift success rate–such an
extension is a possibility for future DESI data releases.

The DESI instrument and survey operations continue to improve. Most notably,
fiber positioning accuracy was substantially improved via a series of fiber
dither tests carried out throughout the first two years of the survey [67].
Fiber positioning inaccuracies are a substantial source of the focal plane
dependent redshift success trends, and the higher-order correction applied to
the positioning midway through the third year of operations will substantially
reduce their variability. Finally, we will continue to study whether
additional weights are required to mitigate spectroscopic systematics trends
that may be significant for other DESI analyses beyond the large-scale
clustering used for BAO and RSD fits.

## Data Availability

The data used in this work will be made public as part of DESI Data Release 1
(details at https://data.desi.lbl.gov/doc/releases/). The data points
corresponding to the figures are available on Zenodo at
https://doi.org/10.5281/zzenodo.11277146.

## Acknowledgments

AK was supported as a CITA National Fellow by the Natural Sciences and
Engineering Research Council of Canada (NSERC), funding reference
#DIS-2022-568580. WP acknowledges support from the Natural Sciences and
Engineering Research Council of Canada (NSERC), [funding reference number
RGPIN-2019-03908] and from the Canadian Space Agency. Research at Perimeter
Institute is supported in part by the Government of Canada through the
Department of Innovation, Science and Economic Development Canada and by the
Province of Ontario through the Ministry of Colleges and Universities. JY
acknowledges the support from the SNF 200020_175751 and 200020_207379
“Cosmology with 3D Maps of the Universe" research grant.

This material is based upon work supported by the U.S. Department of Energy
(DOE), Office of Science, Office of High-Energy Physics, under Contract No.
DE–AC02–05CH11231, and by the National Energy Research Scientific Computing
Center, a DOE Office of Science User Facility under the same contract.
Additional support for DESI was provided by the U.S. National Science
Foundation (NSF), Division of Astronomical Sciences under Contract No.
AST-0950945 to the NSF’s National Optical-Infrared Astronomy Research
Laboratory; the Science and Technology Facilities Council of the United
Kingdom; the Gordon and Betty Moore Foundation; the Heising-Simons Foundation;
the French Alternative Energies and Atomic Energy Commission (CEA); the
National Council of Humanities, Science and Technology of Mexico (CONAHCYT);
the Ministry of Science and Innovation of Spain (MICINN), and by the DESI
Member Institutions: https://www.desi.lbl.gov/collaborating-institutions. Any
opinions, findings, and conclusions or recommendations expressed in this
material are those of the author(s) and do not necessarily reflect the views
of the U. S. National Science Foundation, the U. S. Department of Energy, or
any of the listed funding agencies.

The authors are honored to be permitted to conduct scientific research on
Iolkam Du’ag (Kitt Peak), a mountain with particular significance to the
Tohono O’odham Nation.

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##  Appendix A Relationship between mean redshift and redshift success rate

We use the Monte Carlo simulations described in Section 4 to remove 55
4$\sigma$ outlier fibers from further analysis. In addition to removing
targets on low success rate fibers, this analysis also enables us to study the
uniformity of galaxy properties with success rate. In particular, the redshift
failure weights are intended to ensure that galaxies are homogeneously
distributed with respect to observing conditions. One important aspect of
homogeneity is the redshift distribution. We therefore measure the mean
redshift for all successful galaxies, averaged among fibers and then sorted by
each fiber’s “$n\sigma$” deviation from the expected success rate based on the
TSNR2 and fiberflux model. When computing mean redshift, galaxies are also
upweighted by their redshift failure weight (Fig. 34). We fit a line to each
trend, and find marginally significant relationships for LRG (2.1$\sigma$),
and larger significances for BGS_BRIGHT (4.7$\sigma$) and QSO (17.2$\sigma$).
The relationship is positive for LRG and QSO, and negative for BGS_BRIGHT. The
fluctuations in redshift are very small for LRG and BGS_BRIGHT, $\sim$0.002.
Quasars show a much larger variation, with $\Delta z$ $\sim$0.04 between the
highest bin (closest success rate to the mean) and the rest. However, this is
likely due to the re-observation of Ly$\alpha$ forest quasars, which were not
excluded when making this plot; they lie at $z>1.8$ and, since they are
observed multiple times, skew the distribution of quasar successes to higher
redshift. Given the small trends observed in the other tracers, we leave this
for future work in the next data release.

Figure 34: Relationship between success rate and redshift. On the $x$ axis, we
show the number of $\sigma$ deviation from the mean success rate, measured for
each fiber using Monte Carlo simulations. Fibers are then aggregated into bins
in $n\sigma$. The errorbar is from the dispersion of observed redshift values.
On the $y$-axis, we plot the redshift failure weighted mean redshift in each
bin. The 4$\sigma$ cut used to define low success rate fibers is given by the
dotted blue line.

##  Appendix B Validation of redshift failure weights in full redshift ranges
and for alternative BGS samples

We validate the performance of the redshift failure weights across the full
redshift ranges and for alternative BGS samples in Figs. 35 and 36. We find
that the LRG weights perform similarly well over the full redshift range as in
the clustering redshift range ($0.4<z<1.1$). Likewise, the weights work well
for the quasars in both the full redshift range and the extended range
$0.8<z<3.5$. These plots show the same dip at TSNR2_QSO $>40$ as Fig. 11, due
to the increased galaxy fraction in quasar targets at low Galactic latitude
(which aligns with the high TSNR2_QSO regions). ELG shows a $\sim 1.5\%$ drop
in success rate at low TSNR2_ELG, which is only partially corrected by the
weights, leading to a significant trend in the North and a marginally
significant trend in the South. As shown in [1], these trends are mitigated by
also applying the focal plane weights $\eta_{\textrm{zfail}}$.

We find generally worse performance for BGS, and extending these redshift
failure weights to alternative BGS samples or redshift ranges will require a
separate validation process. Considering BGS_BRIGHT-21.5 across the entire
redshift range, the trend in South is somewhat significant ($p=7.5\times
10^{-5}$, equivalent to 3.8$\sigma$ for a Gaussian). We find very large trends
for BGS_ANY and BGS_BRIGHT (i.e. without the absolute magnitude cut) across
the entire redshift range (Fig. 36). Some of these trends (e.g. at high
TSNR2_BGS) may be related to the change in $r_{\textrm{fiber}}$ distribution
with TSNR2_BGS (Fig. 16). The trends with TSNR2_BGS are much less significant
when considering the clustering range $0.1<z<0.4$ instead of the full redshift
range. However, we find somewhat significant drops in success rate at low
TSNR2_BGS, even after applying the redshift failure weights, indicating that
the redshift failure model may not adequately fit the data. In particular, the
redshift success rate is a very strong function of $r_{\textrm{fiber}}$, and
the simple model in Eq. 3.4 may not match the strong observed trends. We
emphasize again that for applications of the BGS sample (“BGS_BRIGHT-21.5”)
beyond those presented here, further validation of the redshift failure
weights is required.

Figure 35: As in Fig. 11, but using the entire redshift range of each tracer
rather than the clustering redshift range. We also show the quasars using the
extended redshift range ($0.8<z<3.5$) on the bottom.

Figure 36: As in Fig. 11, but for alternative definitons of the BGS sample:
BGS_BRIGHT (without the absolute magnitude cut) on top, and BGS_ANY on bottom.
We show both the clustering redshift range ($0.1<z<0.4$) on the left and the
full redshift range on the right.

##  Appendix C TSNR2_QSO–failure rate trends for “no star” and “high purity”
samples

We use the “no star” and “high purity” samples of [50] to assess whether the
TSNR2_QSO–failure rate trends are truly driven by fluctuations in the quasar
success rate, or by fluctuations in the success rate of quasar interlopers
(galaxies and stars). We find similar behavior of the TSNR2_QSO–failure rate
trend for the “full” and “no star” sample (Fig. 37). The “high purity” sample
shows a different behavior, with a more uniform rise in failure rate towards
smaller TSNR2_QSO, rather than a pronounced uptick at low TSNR2_QSO. However,
the “high purity” sample is quite different from “full” and “no star” since
many of the faintest quasar targets have been removed. This is important,
since the failure rate strongly depends on flux171717The quasar targets are
selected to be point sources, so flux and fiberflux are interchangeable. as
well as TSNR2_QSO. We compare the quasar success rate in fixed bins of
extinction-corrected $r$-band flux, defined as the quintiles of $r$ flux for
the “pure” sample.181818Since the “pure” sample is defined using a flux cut in
$r$ magnitude, the other samples will have a considerable number of targets
fainter than the faintest quintile and thus not shown in Fig. 38. We find that
within the quintiles of $r$ flux, the success rate–TSNR2_QSO trends are
indistinguishable within the errorbars, both in the North and the South (Fig.
38). We therefore conclude that the different behavior seen between the “high
purity” and “full” quasar samples in Fig. 37 is driven by the different $r$
flux distribution, rather than actual differences in the relative success
rate.

Thus we conclude that the measured quasar redshift success rate is
representative of the “true” redshift success rate, defined as the ratio of
spectroscopically-classified quasars to true quasars within the target sample,
and is robust to the presence of interlopers (stars and galaxies) within the
quasar sample.

Figure 37: Failure rate as a function of TSNR2_QSO for the full quasar sample
and two alternative samples with some quasars removed: a “no star” sample
where the star fraction is reduced from 5% to 1% using the photometric
classification of [61]; and a “high purity” sample where the quasar fraction
is increased from 60% to 90% by applying the “no star” cut and additionally
removing targets at the faint end in $r$ and W2 magnitudes. Since the failure
rate (i.e. the fraction of quasar targets that are not spectroscopically
confirmed quasars) is highest for full, the “no star” and “high purity”
failure rates are scaled by the indicated factors to allow better visual
comparison of the trends.

Figure 38: Failure rate as a function of TSNR2_QSO, dividing the quasar
targets into different subsets of extinction-corrected $r$ flux. The subsets
are defined using the quintiles of the “high purity” sample and are the same
flux cuts for all three samples. The faintest sample is at the top (blue
lines) and the samples increase in brightness moving down the plot. All
failure rates are normalized by the failure rate at TSNR2_QSO = 35, and an
arbitrary offset is added for clarity.

##  Appendix D Redshift success vs. focal plane radius for BGS, ELG, and QSO

We show trends of redshift success vs. focal plane radius for BGS_BRIGHT-21.5,
ELG, and QSO in Figs. 39, 40, 41. We only find a $>3\sigma$ trend in petals 1
and 6 for ELG using the clustering redshift range. We find significant trends
in petals 0, 2, 6 and 9 for ELG using the full redshift range. For petals 0
and 9, applying the linear focal plane radius weights improves the trends to
be statistically consistent with no trend. We do not find significant trends
for QSO and BGS when considering the entire redshift range.

Figure 39: As in Fig. 19, but for BGS_BRIGHT-21.5.  Figure 40: As in Fig. 19,
but for ELG.  Figure 41: As in Fig. 19, but for QSO.