# Whither or wither the Sulfur Anomaly in Planetary Nebulae?

Shuyu Tan  The Laboratory for Space Research, Faculty of Science, The
University of Hong Kong, Cyberport 4, Hong Kong  Quentin A. Parker  The
Laboratory for Space Research, Faculty of Science, The University of Hong
Kong, Cyberport 4, Hong Kong

###### Abstract

We present a thorough investigation of the long standing sulfur anomaly
enigma. Our analysis uses chemical abundances from the most extensive dataset
available for 126 planetary nebulae (PNe) with improved accuracy and reduced
uncertainties from a $10\times 10$ degree Galactic bulge region. By using
argon as a superior PNe metallicity indicator, the anomaly is significantly
reduced and better constrained. For the first time in PNe we show sulfur
$\alpha$-element lock-step with both oxygen and argon. We dispel hypotheses
that the anomaly originates from underestimation of higher sulfur ionization
stages. Using a machine learning approach, we show that earlier ionization
correction factor (ICFs) schemes contributed significantly to the anomaly. We
find a correlation between the sulfur anomaly and the age/mass of PNe
progenitors, with the anomaly either absent or significantly reduced in PNe
with young progenitors. Despite inherent challenges and uncertainties, we link
this to PNe dust chemistry, noting those with carbon-dust chemistry show a
more pronounced anomaly. By integrating these findings, we provide a plausible
explanation for the residual, reduced sulfur anomaly and propose its potential
as an indicator of relative galaxy age compositions based on PNe.

ISM: abundances — planetary nebulae: general — stars: evolution

##  1 Introduction

Sulfur, as an $\alpha$ element, should be produced in lockstep with others
like oxygen, neon, argon, and chlorine in more massive stars, so its cosmic
abundance should also be proportional. Strong correlations between sulfur and
oxygen abundances are seen in H ii regions and blue compact galaxies. However,
historically, Planetary Nebula (PNe hereafter) sulfur abundances, which arise
from low-to-intermediate mass progenitors, have consistently been lower,
giving rise to the so-called “sulfur anomaly” first identified in PNe by Henry
et al. (2004).

Extensive efforts to explain the PNe sulfur anomaly through observational and
theoretical approaches has so far proved elusive. One suggestion is past
failure to properly account for the abundances of unseen ionization stages
above S2+. Another factor is the difference in evolutionary stages in H ii
regions, blue compact galaxies and PNe. H ii regions are the current sites of
chemical enrichment, from which the next generation of stars emerge. In
contrast, PNe form a highly diverse and complex family, with progenitor masses
spanning from $\sim$8 $M_{\odot}$ to 1 $M_{\odot}$ and so exhibiting a wide
range of ages, from tens of millions of years to over 10 Gigayears.

Shingles & Karakas (2013) looked at whether the s-process is responsible for
destroying sulfur during late stage stellar evolution but their study did not
support this. Another idea is dust depletion of sulfur (Henry et al., 2012;
García-Hernández & Górny, 2014). Delgado-Inglada et al. (2015) suggested
oxygen and neon may be enriched during PN progenitor evolution which could
give more scatter, depending on progenitor mass (and so effectively age). As a
result they consider argon and chlorine more reliable metalicity indicators.
Given difficulties in measuring weak chlorine lines, argon is by far the
easier to work with and is adopted here.

###  1.1 The context of the sulphur anomaly

Our Galaxy is an ideal testing ground for refining chemical evolutionary
models. Galactic H ii regions represent zero-age systems while PNe represent
older to considerably older populations of intermediate- to low-mass stars for
study. For PNe the observed abundances can deviate to sub- and super-solar
depending on progenitor age. PNe allow accurate determination of abundances
among various chemical tracers through their emission lines, including precise
abundance measurements of alpha elements. These elements, created by massive
stars, remain unaltered by their progenitors and reflect molecular cloud
composition when the stars were formed. If $\alpha$ elements like oxygen,
neon, sulfur, and argon are produced in lockstep in massive stars, their
abundances should exhibit equivalent correlations. H ii region and PNe
observations have revealed strong correlations between oxygen and neon. They
are considered indicators of their formation environments. Similar lockstep
behaviour has been noted for argon. However, previous sulfur abundance studies
in Galactic disk PNe, e.g. (Henry et al., 2004) and (Henry et al., 2012), and
Galactic Bulge PNe by (Cavichia et al., 2010), report data points that
considerably deviate from the narrow, linear tracks for H ii regions. Instead
they exhibit a broad spread in sulfur abundances that generally fall lower
than for H ii regions. This discrepancy is the sulphur anomaly. It has been
discussed in detail by e.g. Henry et al. (2004) and Henry et al. (2012) and
has been based on O/H values as the metalicity indicator.

We examined steps from previous studies that derived PNe sulfur abundances
from their spectra. These directly determine the ionic abundances of S+ and
S2+, when observable, and then multiplied by an “ionisation correction factor”
(ICF) e.g. (Mathis, 1985; Delgado-Inglada et al., 2014), calculated to account
for unobserved ionisation stages. Possible reasons given for the sulfur
anomaly include underestimation of higher ionisation stages when using these
ICFs (see later). Henry et al. (2012) tested the sulfur abundance deficit
across a range of nebular properties but found no strong correlations.
Recently, by studying the born-again nebula V4334 Sgr and using ESO VLT
spectroscopic observations from 2006 to 2022, Reichel et al. (2022) confirmed
an [S iii] level, contrary to expectation of a decrease, because the S2+
stage, if it dominates, will rapidly recombine to S+. A possible explanation
is recharge from underestimated fractions of higher ionisation stages. They
propose this could explain the anomaly but this object is itself anomalous and
only a single data point.

##  2 ESO VLT FORS2 Spectroscopy

Our new catalogue of abundances for Galactic bulge PNe from quality VLT/FORS2
spectra, Tan et al. (2024) (Paper III) provides the most comprehensive and
reliable PNe abundances available to re-examine the sulfur anomaly. We found
the first clear example of a lock-step relation for argon and neon with oxygen
unlike previous studies, reflecting a reduction in our data’s error
dispersion. We re-examined this lockstep behaviour and the relationship
between sulfur and oxygen abundances using our high S/N data from the same
telescope and instrument and for a well-defined PNe sample. We use these new
data to investigate the sulfur anomaly. Our PNe selection was based on likely
Galactic bulge membership, as detailed in Paper I (Tan et al., 2023).

To derive abundances, we used the Fitzpatrick (1999) extinction law to account
for interstellar reddening. We used the ICF scheme in Delgado-Inglada et al.
(2014), hereafter DMS14, to correct for unobserved sulfur ionisation stages.
This is a key difference to previous work, e.g. Kingsburgh & Barlow (1994) and
Kwitter & Henry (2001), hereafter KW01, as used by HSK12. We derive S/H and
O/H values for 124 objects on the standard 12 + $\log$(X/H) scale, yielding
high-quality results and $<0.2$ dex uncertainties.

##  3 The sulfur anomaly and a comparison with literature studies

Fig. 1 shows the relationship between S/H and O/H for 162 Galactic disk PNe
from Henry et al. (2012), hereafter HSK12, and the 124 PNe in our ESO VLT
bulge sample. HSK12 incorporates data from Henry et al. (2004), Milingo et al.
(2010), and Henry et al. (2010), which adopt the Kwitter & Henry (2001) ICF
scheme.

As proposed by Delgado-Inglada et al. (2015), AGB star evolution may modify
nebular oxygen abundances compared to their progenitors. Their analysis
suggested using Ar/H or Cl/H as a more reliable metalicity indicator. Given
the large uncertainties associated with Cl/H derivation, we adopt Ar/H. In
Fig. 1 we plot S/H against Ar/H. We include data for H ii regions in the
nearby spiral galaxy M101 from Kennicutt Jr et al. (2003) and metal-poor blue
compact galaxies from Izotov & Thuan (1999) (collectively hereafter H2BCG),
where the $\alpha$-element lockstep behaviour is demonstrated. We exclude data
from H ii regions with an uncertainty in $\log$(S/O) $>0.1$ dex, as they
exhibit a noticeable deviation from the main H2BCG data. Table 1 presents
best-fit parameters for S/H versus O/H and Ar/H of each sample from linear
least-squares regression using scipy.stats.linregress, together with Pearson’s
correlation coefficients ($r$).

The left-most panel of Fig. 1 presents the non-lockstep behaviour of S/H
versus O/H for Galactic disk PNe from HSK12 (as blue and purple data points)
as two trends: a sparsely populated (13%) upper trend that aligns with the
H2BCG data (green points), and a well-populated (87%) locus below the H2BCG
trend encapsulating the anomaly. The slope of the best fit line (in blue)
significantly deviates from unity with a formal $R^{2}=0.26$, providing the
main anomaly locus.

We calculate the sulfur deficit, $d$(S), as the discrepancy between the
predicted $12+\log$(S/H) values (from the best-fit line for $12+\log$(S/H)
versus $12+\log$(O/H) of H2BCG) and the actual observed PNe values, following
HSK12. Using only results in HSK12 with reported measurement uncertainty
$<0.2$ dex (denoted as HSK12+), the median $d$(S) is $-0.40^{+0.31}_{-0.29}$
dex. The sulfur deficit seen in the S/H versus O/H plot is at least partially
due to the ICF scheme used as we show in the next section.

The middle-left plot of Fig. 1 is the equivalent S/H versus Ar/H plot. Here,
there is only a single, broad locus of PNe data points falling below but
closer to the H2BCG line. This shows that using the O/H metallicity indicator
is a key factor in the sulfur anomaly in HSK12. The data scatter in the left
panels may reflect uncertainties in determining reliable PNe chemical
abundances using low-resolution spectroscopic observations on 2-m telescopes.
Our Paper III shows that literature results based on low-resolution spectra
from 2-m telescopes generally show a deviation $>0.1$ dex from those from the
VLT observations for the $\alpha$ elements.

In the right panels of Fig. 1, we show Galactic bulge PNe data from Paper III
that displays a broader metalicity range, with many objects having super-solar
O/H and Ar/H. Compared to HSK12, our high-quality VLT/FORS2 abundances show
less scatter with most points on a narrower track below the H2BCG trend. The
bulge PNe show clear lockstep behaviour for S/H versus O/H with a strong
correlation coefficient of $r=0.84$. The best-fit line has a near unity slope.
This is the first time lockstep behaviour between S/H and O/H has been shown
for any PNe sample. The S deficit is a smaller systematic offset of $\sim 0.2$
dex, with a median $d$(S) of $0.24_{-0.18}^{+0.27}$ dex. The scatter at the
sub-solar O/H remains significant. The sulfur lockstep behaviour is further
tightened in S/H versus Ar/H, as in the right-most panel of Fig. 1. These are
our major new findings.

We reanalysed the HSK12 chemical abundances using their de-reddened line
intensities following methods described in Paper III111 Due to modest blue
resolution of the HSK12 spectra ($\sim 1.4$ Å), the [Ar IV] $\lambda$4711 and
He I $\lambda$4713 lines, and the [O II] $\lambda\lambda$3726+28 doublet,
cannot be resolved. Hence, no electron densities could be determined for [Ar
IV] or [O II]. We also excluded [Ar IV] $\lambda$4711 from calculating the
Ar3+ abundances.. Our recalculated values show median differences compared to
HSK12 for S/H and O/H of 0.1 and $-0.07$ dex, respectively, while agreeing
within the measurement uncertainty for Ar/H. These discrepancies could be
attributed to the differences in atomic datasets and ICF schemes used. As a
result of our recalculations, the median $d$(S) reported in HSK12 and shown in
Figure 2 would be halved to $0.19_{-0.27}^{+0.24}$ dex.

Figure 1: Comparison of S/H versus O/H and Ar/H among PNe samples c.f. H2BCG data (green crosses). The two left panels show Galactic disk PNe from HKS12, with blue and pink triangles triangles for high and lower quality measurements with uncertainties $<0.2$ dex and $>0.2$ dex. The right two panels give our Galactic bulge PNe results from Paper III as orange squares. In all panels, best-fit lines are the same color as the data. Only HSK12 data points with uncertainties $<0.2$ dex (HSK12+) were fitted. Our VLT results for O/H and Ar/H display best-fit lines parallel to the H2BCG trends with clear lock-step behaviour. The red asterisk is solar abundance. Color-coded boxplots at the top and right of each panel depict distributions of O/H, Ar/H and S/H. Red lines indicate median values with boxes extending from the 25th to 75th percentiles and whiskers from 1% to 99% with outliers as crosses.  2*Sample |  2*X/H vs. S/H | S/H from ICF scheme |  | S/H from ML
---|---|---|---|---
|  | slope |  $-$intercept | $R^{2}$ | $r$ |  | slope |  $-$intercept | $R^{2}$ | $r$
VLT [124] | O | 1.05$\pm$0.06 |  $-$2.22$\pm$0.53 | 0.71 | 0.84 |  | 0.99$\pm$0.06 |  $-$1.70$\pm$0.51$-$ | 0.70 | 0.84
| Ar | 0.98$\pm$0.06 |  $-$0.58$\pm$0.38 | 0.69 | 0.83 |  | 0.95$\pm$0.06 | 0.75$\pm$0.38 | 0.70 | 0.83
HSK12+ [140] | O | 0.66$\pm$0.10 |  $-$0.96$\pm$0.82 | 0.26 | 0.51 |  | 0.78$\pm$0.11 |  $-$0.02$\pm$0.93$-$ | 0.29 | 0.54
| Ar | 0.63$\pm$0.06 |  $-$2.69$\pm$0.38 | 0.44 | 0.67 |  | 0.70$\pm$0.07 | 2.31$\pm$0.44 | 0.44 | 0.67
H2BCG [61] | O | 1.00$\pm$0.03 |  $-$1.56$\pm$0.21 | 0.96 | 0.98 |  | - | - | - | -
| Ar | 0.95$\pm$0.04 |  $-$1.00$\pm$0.21 | 0.92 | 0.96 |  | - | - | - | -
Table 1: Slope, intercept , corresponding errors, and the coefficient of
determination ($R^{2}$) from the least-squares regression for the
$12+\log$(S/H) versus both $12+\log$(O/H) and $12+\log$(Ar/H) data, along with
correlation coefficients ($r$) for Galactic disk PNe from HSK12+ and bulge PNe
from Paper III, as shown in Fig. 1. The best-fit parameters derived with S/H
from the literature ICF schemes are presented in cols. 3-6 while those
obtained with S/H from machine learning-derived icf(S) are presented in cols.
7-10. The final two rows presents results for H2BCG.

##  4 Understanding the sulfur anomaly

We examine the sulfur anomaly in Galactic disk and bulge PNe samples. Despite
a notable reduction in the S/H deficit found in our high-quality bulge PNe
data, the sulfur anomaly persists for some PNe. We focus on potential
artificial causes of this anomaly, including underestimation of sulfur higher
ionization stages beyond S2+ in photoionization models. We implement a machine
learning approach to derive an ad hoc sulfur ICF for each object to examine
the ICF scheme proposed in the literature. We compare our optical observations
with infrared literature results where higher ionisation stages could be
directly observed, to examine possible issues associated with use of optical
observations.

###  4.1 Validation of ICF scheme via machine learning

The KW01 and DMS14 ICF literature formulae were obtained through analytical
fitting to photoionization models. The sulfur anomaly has been attributed to
failure to properly account for high ionisation stages through use of an
incorrect ICF for sulfur in HSK12, something our re-calculations of their data
confirm. Sabin et al. (2022) first introduced machine learning (ML) techniques
to determine ICFs tailored to individual PNe. Here, we apply ML to a large
sample. To minimise uncertainties with the ICF formulae, we adopt a random
forest (RF) regressor (Breiman, 2001) (a learning method based on decision
tree regression), to predict ICF values (S/(S++S2+) or S/S+). RF input
features include He2+/He+, O2+/O+, S2+/S+, Ar3+/Ar2+, and Cl3+/Cl2+. We used
the RandomForestRegressor in the Python library Scikit-Learn (Pedregosa et
al., 2011). Objects with unobserved O2+ were excluded due to large O/H
uncertainties and challenges in accurately determining the O2+/O+ ratio. Where
other ionic stages in input features were not seen, such as Ar3+ or Cl3+, a
separate RF regressor predicted the ionic fraction ratios (e.g., Ar3+/Ar2+ and
Cl3+/Cl2+) using He2+/He+ and O2+/O+ as input features.

An extensive photoionization model grid of PNe was computed as in Delgado-
Inglada et al. (2014). We selected PNe photoionization models from an extended
grid version with a larger effective temperature range for PNe central stars.
These models were run with version c17.01 of Cloudy (Ferland et al., 2017) and
hold in the 3MdB_17 database (Morisset et al., 2015) under the reference
PNe_2020. From the $>700,000$ models we chose $\sim 120,000$ that met
realistic PN model selection criteria in Delgado-Inglada et al. (2014) to
train the RF regressor. The full sample was divided into a training set (75%
of the sample), a validation set (12.5%), and a test set (12.5%). The training
set trained the RF regressor and the validation set tuned the hyperparameter
$\alpha$ of the RF regressor to enhance efficiency and accuracy achieved via a
grid-search cross-validation (CV) of $\alpha$ values (see Appendix B).

Figure 2 displays the distribution of differences between the $12+\log$(S/H)
values obtained using the icf(S) from literature-based ICF schemes and our ML
approach for the PNe sample in HSK12+ and Paper III. The median differences
for these samples are $-0.04_{-0.24}^{+0.13}$ and $-0.0006^{+0.15}_{-0.15}$
dex. The ICF scheme in DMS14, applied in Paper III, shows no systematic
difference and smaller discrepancies to the ML values, unlike the KW01 used in
HSK12. The ML comparison for HSK12+ (right panel of Figure 2) has a left-
skewed distribution, suggesting underestimation of S/H and, consequently, a
slightly larger average sulfur deficit of $\sim$0.07 dex. The best-fit line
parameters for ML S/H values are presented in Table 1 for both HSK12+ and
Paper III PNe samples. The higher $R^{2}$ and $r$ values indicate a better
linear relationship, while the best-fit line for HSK12+ is closer to unity.
Considering the sulfur anomaly typically exceeds 0.1 dex, the inaccuracies
linked to the ICF schemes are likely not the only cause of the sulfur anomaly.
The increased consistency between the ICF scheme in DMS14 and the ML results
shows that the DMS14 scheme is more reliable, supporting the argument that
their ICF scheme reduces the sulfur anomaly, as shown here.

Figure 2: Differences between $12+\log$(S/H) values from literature ICF
formulae and our machine learning approach. The HSK12+ sample in the left
panel adopted the KW01 ICF scheme, while the right panel presents PNe from
Paper III using the DMS14 ICF scheme. The left-skewed distribution in the
HSK12+ panel suggests a contribution to the sulfur anomaly from the KW01 ICF
scheme.

###  4.2 Mid-infrared observations of PNe

The S/H deficiency seen in Galactic PNe is suggested as due to underestimation
of higher ionization stages of sulfur in photoionization models, e.g. Henry et
al. (2012); Reichel et al. (2022). S/H values from mid-infrared _Spitzer_
spectra in Pottasch & Bernard-Salas (2015), PBS15 hereafter, enables
observation of the higher ionization stage S3+ and are systematically higher
than our results, with a median difference of 0.17 dex. The left panel of Fig.
3 shows the VLT S/H versus O/H plot for eight bulge PNe in common with PBS15.
The VLT results show the sulphur anomaly in terms of the H2BCG linear fit with
an $R^{2}$ of 0.34. The PBS15 O/H values are from optical observations but our
VLT spectra are superior. The right panel shows good agreement between PBS15
sulfur abundances from mid-infrared spectra and oxygen abundances from our VLT
spectra. We find $R^{2}=0.71$, consistent with the H2BCG results. A comparison
of our sulfur ionic abundances and PBS15 shows that the S2+/H+ mid-infrared
values from [S iii] $\lambda$18.7 $\mu m$ and $\lambda$36.5 $\mu m$ in PBS15
are systematically higher than from the equivalent optical [S iii]
$\lambda$6312 line, with a median difference of 0.16 dex. The two outliers in
the right hand plot in Fig. 3 have no lower S2+/H+ values from mid-infrared
spectra. The dashed green line is the H2BCG fit and the red asterisk is solar
abundance. For 6/8 PNe in common the mid-infrared S/H values follow the
expected trend line. Comparison of our icf(S) and S/(S++S2+) or S/S+ in PBS15
shows a median difference of $\sim$4%, with underestimation $>$10% in only two
PNe.

Hence, the sulfur anomaly is unlikely to arise from underestimation of higher
sulfur ionization stages in the photoionization model and their ICFs. Optical
measurement of [S iii] lines gives lower S2+ abundances compared to their mid-
infrared counterparts, if the values of S2+ inferred from the mid-infrared are
more accurate.

Figure 3: Comparison of S/H versus O/H for 8 bulge PNe in common with Pottasch
& Bernard-Salas (2015). The left panel shows S/H values from optical VLT
observations in Paper III. The right panel gives S/H values inferred from
PBS15 mid-infrared data. The dashed green line is the fit to the H2BCG data;
the red asterisk denotes the solar abundance. For 6 out of 8 PNe in common the
mid-infrared S/H values essentially follow the expected trend line.

We highlight some issues associated with mixing the optical and mid-infrared
data for chemical abundance derivation. First, Appendix.A gives the same plots
for neon and argon as for sulfur in Fig. 3. Our Paper III VLT results show
good agreement with the H2BCG data, while the mid-infrared data yields
systematically higher results by 0.29 and 0.33 dex for neon and argon. A
similar issue with neon abundance was reported in Dors Jr et al. (2013) when
studying star-forming regions. This was partly attributed to abundance
variations across the nebulae. It is possible mid-infrared and optical data
intrinsically provide different abundance estimates, with the mid-infrared
data yielding higher estimates. This is supported by detailed PNe
photoionization modelling in Bandyopadhyay et al. (2021) and dusty PNe
photoionization modelling in Otsuka et al. (2017), where best-fit
photoionization models for the optical emission lines and PNe physical
conditions yielded lower fluxes of [S iii] $\lambda$18.7 $\mu$m and
$\lambda$36.5 $\mu$m than the observed mid-infrared line fluxes.

###  4.3 Underestimation of S2+/H+from optical spectra

To address the potential underestimation of S2+ ionic abundances using optical
spectra that could lead to a discrepancy between optical and mid-infrared
observations, we re-examined our Paper III S/H measurements. We considered
several factors, including errors in interstellar correction at the red [S
iii] lines from the use of a single, standard extinction law that may not be
valid in the Galactic bulge (Gould et al., 2001; Udalski, 2003), uncertainties
in [S iii] line flux measurements and the varying spatial distribution of
sulfur and oxygen in PNe.

To quantify the impact of interstellar extinction correction uncertainties, we
recalculated S/H using emission line lists from Paper III and compared two
extinction laws introduced in Howarth (1983) and Cardelli et al. (1989), with
that of Fitzpatrick (1999) used in Paper III. We showed good agreement between
the S2+/H+ and S/O abundance ratios derived using the Howarth (1983) and
Fitzpatrick (1999) extinction laws, with a median difference of
$-0.001^{+0.004}_{-0.006}$ dex and $-0.00\pm 0.007$ dex. The extinction law of
Cardelli et al. (1989) yielded slightly lower values, indicating even lower
results compared to the mid-infrared, with median differences in S2+/H+ and
$\log$(S/O) of $-0.07^{+0.04}_{-0.06}$ and $-0.03\pm 0.03$ dex. We find the
extinction correction impact on the sulfur anomaly is insignificant compared
to the discrepancy between optical and mid-infrared S2+ results and conclude
that the extinction correction is not a contributing factor to the anomaly.

Our Paper III VLT data covered a wavelength range up to 8500 Å, which only
allowed detection of the [S iii] $\lambda$6312 line. We investigated whether
lower signal-to-noise ratios for the [S iii] $\lambda$6312 line might lead to
underestimation of its line fluxes. Our analysis showed that the sulfur
anomaly size had no correlation with measured raw fluxes of the [S iii]
$\lambda$6312 line, indicating that the sulfur deficiency is not due to weak
line fluxes.

The near-IR [S iii] $\lambda\lambda$9069, 9531 lines were not observed in our
VLT spectra. The ionization structure of the PNe may introduce uncertainties
in the S2+ abundance determinations due the absence of electron temperatures
derived from [S iii] line ratios, as S2+ (22.34–34.79 eV) probes lower
ionization zones than O2+ (35.12-54.94 eV). While the analysis of H ii regions
utilising deep spectra in Freeman et al. (2013) suggests no such systematic
difference between Te([S iii]) and Te([O iii]), the ionising sources and the
gas and dust components in H ii regions are considerably different from PNe. A
comprehensive exploration of high-quality PNe data with the near-IR [S iii]
line observations, where the telluric contamination can be satisfactorily
removed, is imperative to examine how the ionization structures of PNe affect
the sulfur deficit.

Abundances from long-slit spectroscopy only sample a portion of a PN often
weighted towards high-ionization regions near the central star (Ali et al.,
2016). Hence, we also consider literature on PNe integral field unit (IFU)
spectroscopy. Such 2-D spectroscopy can provide global abundance data. Using
ionic abundances of O and S derived from IFU observations, the sulfur anomaly
persists for four PNe in Ali et al. (2016) and three PNe in García-Rojas et
al. (2022). For most objects, the sulfur deficit reduced by 0.10 to 0.34 dex;
however, two PNe show a greater $d$(S) by $\sim 0.12$ dex using IFU
observations. Thus, ionization stratification and spatial variation in sulfur
and oxygen abundances within a PN may partly account for the observed sulfur
anomaly.

To conclude, firstly HKS12 and Paper III employ a non-biased ICF for sulfur,
as shown by our ML icf(S) values. Secondly, possible intrinsic differences
between optical and mid-infrared spectroscopy results may lead to higher
abundances derived from mid-infrared spectra. Thirdly, among the application
of different extinction laws, uncertainties in measuring weak [S iii]
$\lambda$6312 lines and any abundance inhomogeneities within the PNe, none
appear to have a significant impact that could lead to underestimation of S2+
ionic abundances. The PNe sulfur anomaly, though reduced in our work, remains
a genuine effect but is unlikely to arise from underestimating sulfur
ionization stages above S2+.

##  5 The sulfur deficit across different PNe samples and properties

After confirming in Section 4 that the sulfur anomaly is a real, albeit much
reduced in PNe, it is likely that the varying PNe S/H deficiencies in
different Galactic regions, as discussed in Section 3, could indicate a
progenitor population effect. Below we present an analysis of the sulfur
deficit considering links to PN progenitor age/mass, dust chemistry and
Galactic location. The results are summarised in Fig. 4 and explored in terms
of PNe progenitor age and sulfur deficit.

###  5.1 PNe progenitor ages/masses

We developed a basic PNe age estimator based on our high-quality VLT
abundances as detailed in Appendix C. We differentiate between young
($t_{\star}<1$ Gyr) and old ($t_{\star}>7.5$ Gyr) PN populations by combining
the Peimbert classes with an extant scheme in Stanghellini & Haywood (2018)
that establish a link between the effectiveness of nitrogen production in PN
progenitors and their ages/masses. We used this criteria to divide the HSK12+
disk sample and our VLT bulge sample into crude young and old progenitor star
PNe populations.

In Fig. 4, we give a combined plot for sulfur deficit, $d$(S), as a function
of Galactic height in kpc; 12 + $\log$(O/H) and histograms of $d$(S) values
with bins of 0.11 dex for the HSK12+ disk PNe sample (top row) and VLT bulge
PNe sample (bottom row). We color-coded the data according to these crude
ages.

In the upper left panel of Fig. 4, the HSK12+ disk PNe show a broad spread in
$d$(S) with a peak at $\sim 0.4$. The old PNe population is restricted to a
narrower range of higher $d$(S), while the younger population have a broader
distribution from $1.0$ to $-0.6$ dex in $d$(S). Objects with no sulfur
deficit primarily have young progenitors. In the lower left panel of Fig. 4,
the bulge PNe sample has a narrower uni-modal $d$(S) distribution peaking at
$\sim 0.3$ dex. The old population is restricted to a narrower range of higher
$d$(S) while the younger population dominates at lower $d$(S) values up to
$-0.2$ dex.

A two-sample Kolmogorov–Smirnov (KS) test (Smirnov, 1939) using
scipy.stats.ks_2samp compared the $d$(S) distribution of the young and old
populations. The KS p-values are 8E-4 for disk PNe and only 1E-5 for bulge
PNe. Adopting a stringent significance level $\alpha$ of 0.001, we reject the
null hypothesis that the $d$(S) of different age groups come from the same
population.

Given the crudeness of the age classification these results are indicative,
but they imply the sulfur deficit is PN age related and hence to the initial
progenitor mass.

###  5.2 Galactic height from the mid-plane

We use identifications of central stars of Galactic PNe from _Gaia_ EDR3 in
Chornay & Walton (2021) (CW21) and González-Santamaría et al. (2021) (GSM21)
and their parallax distances, to determine Galactic heights from the mid-
plane. For Galactic disk PNe in HSK12+ and the VLT Galactic bulge PNe sample
in Paper III, there are 94 and 47 objects with _Gaia_ parallax distances.
Preliminary analysis by Parker et al. (2022) and Tan et al. (2023), indicated
that the CW21 identifications are more reliable though up to 12% of wrongly
identified CSPN remain. We use the CW21 results when _Gaia_ distances were
available in both studies. We calculated the Galactic Plane vertical distance
($z$) using _Gaia_ distances and Galactic ($l$, $b$) coordinates.

The right panel of Fig. 4 displays $d$(S) against $|z|$ in kpc. In the young
disk PNe progenitor population, 62% have $|z|<0.25$ kpc while the old
population is more evenly split. Indeed, the stellar population associated
with the Galactic thin disk is expected to having younger members at lower
Galactic scale heights (e.g. Li & Zhao, 2017). For the bulge sample the
Galactic height issue is more complex probably due to the greater mixing of
stellar populations. No correlation was found between $d$(S) and $|z|$ for the
old PNe populations.

###  5.3 Metallicity indicator O/H

The middle panel of Fig. 4 depicts $d$(S) as a function of $12+\log$(O/H). For
both the disk and bulge PNe the sulfur deficit exhibits a weak association
with oxygen abundance. PNe with extreme sulfur deficits ($d$(S)$>0.5$ dex)
exist mainly in the disk PNe with sub-solar to solar O/H values. Bulge PNe
showing low or no sulfur deficit ($0.1<d$(S) $<-0.1$ dex) have slightly sub-
solar to super-solar O/H values and are from the young progenitor population.
Some disk PNe with low or no sulfur deficit display sub-solar to super-solar
O/H values. Several young disk objects have an overabundance of sulfur with
sub-solar to extremely sub-solar O/H.

At lower O/H values, $d$(S) increases as O/H increases, while at higher O/H,
$d$(S) decreases as the O/H increases. The overall point colour distribution,
particularly evident in the bulge sample suggests distinct mechanisms may
drive the sulfur deficit in PNe, dominating in oxygen-poor and oxygen-rich
nebulae.

###  5.4 PN morphology

Morphological classifications of disk and bulge PNe are from HASH and Paper I.
Most PNe were classified as elliptical, bipolar or round. For both disk and
bulge PNe, the morphological classes, based on their average sulfur deficit
values, could be arranged in decreasing deficit as round, elliptical and
bipolar. There is no major difference between $d$(S) values for the
morphological classes, and their averages agree to 1$\sigma$.

Figure 4: A combined figure for sulfur deficit $d$(S) on the vertical axis for
the HSK12+ sample (top row) and VLT sample (bottom row) against i) histograms
of $d$(S) values with a bin size of 0.11 dex, ii) 12+$\log$(O/H) and iii)
absolute height above the Galactic plane ‘$|z|$’ in kpc. The Galactic disk and
bulge PNe samples are color-coded to their basic age group from criteria
outlined in Appendix C. Green indicates young and orange indicates old
progenitor stars according to the adopted dating scheme. The grey dashed line
in the left panel outlines the overall distribution. The middle panel red
vertical line indicates solar oxygen abundance.

###  5.5 PNe dust chemistry

AGB stars produce different dust grains based on their C/O abundance ratio.
Possible gas phase sulfur depletion due to dust or molecule formation was
investigated in Henry et al. (2012). They concluded it is unlikely to explain
the sulfur anomaly as no correlation between $d$(S) and C/O ratios was seen.
The formation of key sulfur-bearing compounds, MgS and FeS, depends on a
C-rich (C/O $>1$) environment in AGB stars (e.g. Nuth et al., 1985; Lodders &
Fegley Jr, 1995). However, to classify PNe as C-rich or O-rich (C/O $<1$),
observing the dust features in their infrared spectra is a more reliable way
than measuring the C/O abundance ratios from optical emission lines. The
latter could be affected by uncertainties and depletion of carbon or oxygen
into dust grains (Delgado-Inglada et al., 2015).

PNe dust features from _Spitzer_ (García-Hernández & Górny, 2014) show that
33% of Galactic disk PNe and 8% of bulge PNe have C-rich dust features. Such
C-rich PNe generally have sub-solar O/H and a larger sulfur deficit (median
$d$(S)$=$0.51 dex) while those with O-rich or C-rich and O-rich dust features
(dual chemistry) scatter above and below the BCG data, with median $d$(S)
values of 0.34 and 0.07 dex (see their Fig. 8).

This work suggests missing gas-phase sulfur in C-rich PNe could be due to
efficient dust formation, while reduced sulfur deficit in O-rich objects may
be due to sulfur depletion into molecules (e.g. SO, SO2, H2S, and CS Omont et
al., 1993; Bujarrabal et al., 1994) in C-rich and O-rich environments.
Furthermore, Delgado-Inglada et al. (2015) found that C-rich PNe with sub-
solar O/H were enriched in oxygen by $\sim 0.3$ dex. Hence, the sulfur deficit
in C-rich nebulae may be due to both dust and/or molecule depletion and oxygen
enrichment, while gas-phase sulfur depletion into molecules leads to a smaller
sulfur deficit in oxygen rich PNe.

AGB star evolution can lead to transformation of low to intermediate mass
stars (1.5-4$M_{\odot}$) from O-rich to C-rich stars through the efficient
third dredge-up process (Karakas & Lattanzio, 2014). However, low-mass stars
($\leq 1.5$$M_{\odot}$) and more massive stars ($>$ 3-4$M_{\odot}$) which are
classified as young PNe in this study, are expected to remain O-rich
throughout their evolution.

##  6 Summary

Based on our robust abundances for a significant sample of Galactic bulge PNe,
we report important new insights into the PNe sulfur anomaly while reducing
its significance. We rule out some previous suggestions of where it might
originate and validate others that contribute to its reduced form. We reveal
that it is the young and more massive progenitors of the current PNe
population and their associated dust chemistry that show low or absent sulfur
deficit. Our key findings are:

* we show lockstep behaviour for sulfur versus both oxygen and argon abundances for the first time, emphasising the role high-quality data plays

* adopting the DMS14 ICF scheme reduces the sulfur anomaly as our machine learning approach shows

* based on mid-infrared data for a small, overlapping PNe sample, we find the sulfur anomaly is unlikely due to underestimation of higher sulfur ionization stages in the photoionization models and their derived ICFs

* the impact of different extinction laws on the sulfur anomaly is insignificant

* the sulfur deficit is not due to weak sulfur line fluxes in the optical range. While IFU observations show sampling whole PNe is important for an overall sulfur deficit determination it is also reduced for some PNe

* correlations of lower sulfur deficit with Galactic height above/below the mid-plane and their close relation to PNe progenitor mass and chemistry are shown

* the sulfur deficit is almost absent for PNe from higher mass, younger progenitors and those with oxygen or dual dust chemistry related to progenitor mass. Larger deficits for intermediate progenitor mass PNe that are carbon rich are seen, the solution is likely there

Acknowledgments
We thank the anonymous referee for their prompt review and constructive
comments that improved this manuscript. ST thanks HKU and QAP for an MPhil
scholarship and a research assistantship. QAP thanks the Hong Kong Research
Grants Council for GRF research grants 17326116, 17300417 and 17304520. We
made use of NASA’s Astrophysics Data System; the SIMBAD database, operated at
CDS, Strasbourg, France; Astropy, a community-developed core Python package
for Astronomy (Robitaille et al., 2013); HASH, an online database at the
Laboratory for Space Research at HKU federates available multi-wavelength
imaging, spectroscopic and other data for all known Galactic PNe and is
available at: http://www.hashpn.space.

## References

  * Ali et al. (2016) Ali, A., Dopita, M., Basurah, H., et al. 2016, Monthly Notices of the Royal Astronomical Society, 462, 1393
  * Bandyopadhyay et al. (2021) Bandyopadhyay, R., Das, R., & Mondal, S. 2021, Monthly Notices of the Royal Astronomical Society, 504, 816
  * Breiman (2001) Breiman, L. 2001, Machine learning, 45, 5
  * Bujarrabal et al. (1994) Bujarrabal, V., Fuente, A., & Omont, A. 1994, Astronomy and Astrophysics, Vol. 285, p. 247-271 (1994), 285, 247
  * Calvet & Peimbert (1983) Calvet, N., & Peimbert, M. 1983, Revista Mexicana de Astronomia y Astrofisica, 5, 319
  * Cardelli et al. (1989) Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, Astrophysical Journal, Part 1 (ISSN 0004-637X), vol. 345, Oct. 1, 1989, p. 245-256., 345, 245
  * Cavichia et al. (2010) Cavichia, O., Costa, R., & Maciel, W. 2010, Revista mexicana de astronomía y astrofísica, 46, 159
  * Chornay & Walton (2021) Chornay, N., & Walton, N. 2021, Astronomy & Astrophysics, 656, A110
  * Corradi & Schwarz (1995) Corradi, R. L., & Schwarz, H. E. 1995, Astronomy and Astrophysics, Vol. 293, p. 871-888 (1995), 293, 871
  * Delgado-Inglada et al. (2014) Delgado-Inglada, G., Morisset, C., & Stasińska, G. 2014, MNRAS, 440, 536, doi: 10.1093/mnras/stu341
  * Delgado-Inglada et al. (2014) Delgado-Inglada, G., Morisset, C., & Stasińska, G. 2014, Monthly Notices of the Royal Astronomical Society, 440, 536
  * Delgado-Inglada et al. (2015) Delgado-Inglada, G., Rodríguez, M., Peimbert, M., Stasińska, G., & Morisset, C. 2015, Monthly Notices of the Royal Astronomical Society, 449, 1797
  * Dors Jr et al. (2013) Dors Jr, O. L., Hägele, G. F., Cardaci, M. V., et al. 2013, Monthly Notices of the Royal Astronomical Society, 432, 2512
  * Ferland et al. (2017) Ferland, G., Chatzikos, M., Guzmán, F., et al. 2017, Revista mexicana de astronomía y astrofísica, 53
  * Fitzpatrick (1999) Fitzpatrick, E. L. 1999, Publications of the Astronomical Society of the Pacific, 111, 63
  * Freeman et al. (2013) Freeman, K., Ness, M., Wylie-de-Boer, E., et al. 2013, MNRAS, 428, 3660, doi: 10.1093/mnras/sts305
  * García-Hernández & Górny (2014) García-Hernández, D., & Górny, S. 2014, Astronomy & Astrophysics, 567, A12
  * García-Hernández & Górny (2014) García-Hernández, D. A., & Górny, S. K. 2014, A&A, 567, A12, doi: 10.1051/0004-6361/201423620
  * García-Rojas et al. (2022) García-Rojas, J., Morisset, C., Jones, D., et al. 2022, Monthly Notices of the Royal Astronomical Society, 510, 5444
  * González-Santamaría et al. (2021) González-Santamaría, I., Manteiga, M., Manchado, A., et al. 2021, Astronomy & Astrophysics, 656, A51
  * Gould et al. (2001) Gould, A., Stutz, A., & Frogel, J. A. 2001, The Astrophysical Journal, 547, 590
  * Henry et al. (2010) Henry, R., Kwitter, K. B., Jaskot, A. E., et al. 2010, The Astrophysical Journal, 724, 748
  * Henry et al. (2004) Henry, R. B., Kwitter, K., & Balick, B. 2004, The Astronomical Journal, 127, 2284
  * Henry et al. (2012) Henry, R. B., Speck, A., Karakas, A. I., Ferland, G. J., & Maguire, M. 2012, The Astrophysical Journal, 749, 61
  * Howarth (1983) Howarth, I. D. 1983, Monthly Notices of the Royal Astronomical Society, 203, 301
  * Izotov & Thuan (1999) Izotov, Y. I., & Thuan, T. X. 1999, The Astrophysical Journal, 511, 639
  * Karakas & Lattanzio (2014) Karakas, A. I., & Lattanzio, J. C. 2014, Publications of the Astronomical Society of Australia, 31, e030
  * Kennicutt Jr et al. (2003) Kennicutt Jr, R. C., Bresolin, F., & Garnett, D. R. 2003, The Astrophysical Journal, 591, 801
  * Kingsburgh & Barlow (1994) Kingsburgh, R. L., & Barlow, M. 1994, Monthly Notices of the Royal Astronomical Society, 271, 257
  * Kwitter & Henry (2001) Kwitter, K., & Henry, R. 2001, The Astrophysical Journal, 562, 804
  * Li & Zhao (2017) Li, C., & Zhao, G. 2017, The Astrophysical Journal, 850, 25
  * Lodders & Fegley Jr (1995) Lodders, K., & Fegley Jr, B. 1995, Meteoritics, 30, 661
  * Mathis (1985) Mathis, J. S. 1985, ApJ, 291, 247, doi: 10.1086/163063
  * Milingo et al. (2010) Milingo, J., Kwitter, K., Henry, R., & Souza, S. 2010, The Astrophysical Journal, 711, 619
  * Morisset et al. (2015) Morisset, C., Delgado-Inglada, G., & Flores-Fajardo, N. 2015, Revista mexicana de astronomía y astrofísica, 51, 101
  * Nuth et al. (1985) Nuth, J. A., Moseley, S. H., Silverberg, R. F., Goebel, J. H., & Moore, W. J. 1985, Astrophysical Journal, Part 2-Letters to the Editor (ISSN 0004-637X), vol. 290, March 1, 1985, p. L41-L43., 290, L41
  * Omont et al. (1993) Omont, A., Lucas, R., Morris, M., & Guilloteau, S. 1993, Astronomy and Astrophysics (ISSN 0004-6361), vol. 267, no. 2, p. 490-514., 267, 490
  * Otsuka et al. (2017) Otsuka, M., Ueta, T., Van Hoof, P. A., et al. 2017, The Astrophysical Journal Supplement Series, 231, 22
  * Parker et al. (2022) Parker, Q. A., Xiang, Z., & Ritter, A. 2022, Galaxies, 10, 32
  * Pedregosa et al. (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., et al. 2011, the Journal of machine Learning research, 12, 2825
  * Peimbert (1978) Peimbert, M. 1978, in Symposium-International Astronomical Union, Vol. 76, Cambridge University Press, 215–224
  * Peimbert & Serrano (1980) Peimbert, M., & Serrano, A. 1980, Revista Mexicana de Astronomia y Astrofisica, vol. 5, Apr. 1980, p. 9-18., 5, 9
  * Peimbert & Torres-Peimbert (1983) Peimbert, M., & Torres-Peimbert, S. 1983, in Symposium-International Astronomical Union, Vol. 103, Cambridge University Press, 233–242
  * Phillips (2005) Phillips, J. 2005, Monthly Notices of the Royal Astronomical Society, 361, 283
  * Pottasch & Bernard-Salas (2015) Pottasch, S., & Bernard-Salas, J. 2015, Astronomy & Astrophysics, 583, A71
  * Reichel et al. (2022) Reichel, M., Kimeswenger, S., van Hoof, P. A., et al. 2022, The Astrophysical Journal, 939, 103
  * Robitaille et al. (2013) Robitaille, T. P., Tollerud, E. J., Greenfield, P., et al. 2013, Astronomy & Astrophysics, 558, A33
  * Sabin et al. (2022) Sabin, L., Gómez-Llanos, V., Morisset, C., et al. 2022, MNRAS, 511, 1, doi: 10.1093/mnras/stab3649
  * Shingles & Karakas (2013) Shingles, L. J., & Karakas, A. I. 2013, MNRAS, 431, 2861, doi: 10.1093/mnras/stt386
  * Smirnov (1939) Smirnov, N. V. 1939, Bull. Math. Univ. Moscou, 2, 3
  * Soderblom (2010) Soderblom, D. R. 2010, Annual Review of Astronomy and Astrophysics, 48, 581
  * Stanghellini & Haywood (2018) Stanghellini, L., & Haywood, M. 2018, The Astrophysical Journal, 862, 45
  * Stanway & Eldridge (2018) Stanway, E. R., & Eldridge, J. 2018, Monthly Notices of the Royal Astronomical Society, 479, 75
  * Tan et al. (2023) Tan, S., Parker, Q. A., Zijlstra, A., & Ritter, A. 2023, Monthly Notices of the Royal Astronomical Society, 519, 1049
  * Tan et al. (2024) Tan, S., Parker, Q. A., Zijlstra, A. A., & Rees, B. 2024, MNRAS, 527, 6363, doi: 10.1093/mnras/stad3496
  * Udalski (2003) Udalski, A. 2003, The Astrophysical Journal, 590, 284
  * Ventura et al. (2016) Ventura, P., Stanghellini, L., Di Criscienzo, M., García-Hernández, D., & Dell’Agli, F. 2016, Monthly Notices of the Royal Astronomical Society, 460, 3940

##  Appendix A Comparison of Ne/H and Ar/H versus O/H relationships using
optical and mid- infrared observation

We present additional figures that give a comparison of Ne/H and Ar/H with O/H
and with Ne/H and Ar/H derived from both the optical VLT data from Paper III
and the mid-infrared observations presented in PBS15.

Figure A1: Comparison of Ne/H and Ar/H versus O/H relationships using optical
VLT observations and mid-infrared observations in PBS15, with symbols defined
in Fig. 3.

##  Appendix B Selection of hyperparameters and performance evaluation for the
random forest regressor

The RF regressors, used to predict either the ionic fraction or the icf(S),
all attained an $R^{2}$ of 0.99, indicating a reliable regression. For ICF
values for sulfur, icf(S), used when S2+ is observed, i.e.
icf(S)$=$S/(S++S2+), the fractional differences between the predicted and
actual test sample values are generally $<5\%$. Only 0.9% of cases exceed 15%
fractional differences (primarily for icf(S) values from 2.5 to 15). The
median fractional difference is near zero (0.0002). This suggests no
systematic bias between the predicted and true icf(S) values of the test set.
For the icf(S) values used when only S+ is observed, icf(S)$=$S/S+, predicted
values exhibit slightly larger deviations from test sample actual values.
Nonetheless, the fractional difference is $<15\%$ for $\sim$92% of cases, with
a median fractional difference of 0.005. This indicates no significant
systematic differences between the predicted and true icf(S) values of the
test set.

To enhance the performance of the RF regressor, which was used for predicting
either icf(S) or unobserved ionic fractions, we compared different sets of
hyperparameters using the k-fold cross-validation (CV) on the validation set.
The hyperparameters tuned for RF include the number of trees, (n_estimators),
the maximum depth of each tree, (max_depth), the number of features allowed to
make the best split while building the tree, (max_features), minimum number of
samples to split an internal node (min_samples_split), minimum number of
samples to form a leaf node (min_samples_leaf) and whether bootstrap samples
are used when building trees (bootstrap). The default values were used for the
remaining hyperparameters.

###  B.1 RF regressor for icf(S)

For icf(S), the RF model generates highly accurate predictions for both
S/(S2++S+) and S/S+, with $R^{2}$ values exceeds 0.99. We presents the
fractional uncertainties in predicted icf(S) with respect to the true values
of the test set as a function of true icf(S) values on a log scale in Fig.B1
for icf(S)$=$S/(S2++S+) (top panel) and icf(S)$=$S/S+ (bottom panel). Both RF
regressors exhibit no systematical discrepancies from the actual values, with
most of the predicted values having a fractional uncertainty less than 5% and
10%, respectively. Although, the upper panel of Fig.B1 reveals slightly higher
predicted values than the actual values for intermediate S/(S2+ \+ S+) values
($\sim 1.26$ to 3.16), the median fractional uncertainty within this range is
0.01 dex. Thus, no systematical bias in predicted values presents for this
range.

2*Parameter |  2*Grid | Optimal parameter
---|---|---
|  | S/(S2++S+) | S/S+
n_estimators | [100, 600] | 500 | 100
min_samples_split | [2, 7] | 3 | 3
min_samples_leaf | [1, 10] | 2 | 1
max_depth | [5, 50] | 13 | 50
max_features | [sqrt, auto] | sqrt | sqrt
bootstrap | [True, False] | False | True
Table B1: Hyperparameter adjustments for icf(S) RF regressors.

Each model’s performance was evaluated with a 5-fold CV and quantified by the
mean squared error (MSE) between the predictions and true values of the
validation set. We used RandomizedSearchCV from scikit-learn to randomly
select a subset of 50 among all combinations of $\alpha$ values. The best
parameter selections are presented in Table B1. The optimal set of $\alpha$
values, which minimised the MSE, was chosen to train the final model. This
model was then trained on the training set and applied to the test set.

Figure B1: Fractional uncertainty in predicted icf(S) from the RF regressor as
a function of log-scale true icf(S) for the test set. The contours illustrate
the number density based on kernel density estimation. The black solid line
denotes zero fractional uncertainty, with a high concentration of data points
along this line indicating no systematic discrepancy between the predicted
icf(S) from the RF regressor and the true values from the test set.

###  B.2 RF regressor for Ar and Cl ionic fractions

We used two separated RF regressors to predict the unobserved ionic fraction,
Ar3+/Ar2+, Cl3+/Cl2+, which are then used as input features to predict icf(S).
Similarly, the hyperparameters were tuned through a 5-fold CV. We used
GridSearchCV from scikit-learn to test all possible combinations of $\alpha$
values. The parameter grids and best parameter selections are presented in
Table.B2. Using He2+/He+ and O2+/O+ as the input features, the RF models to
predict Ar3+/Ar2+ and Cl3+/Cl2+ achieved $R^{2+}$ values of 0.99.

2*Parameter |  2*Grid | Optimal parameter
---|---|---
|  | Ar3+/Ar2+ | Cl3+/Cl2+
n_estimators | [500, 2E3, 2E3] | 2E3 | 1E3
min_samples_split | [2, 4, 6] | 2 | 6
min_samples_leaf | [1, 5] | 1 | 1
max_depth | [50, 75, 100] | 75 | 100
max_features | [sqrt, auto] | auto | auto
bootstrap | [True, False] | True | True
Table B2: Hyperparameter adjustments for Ar3+/Ar2+ and Cl3+/Cl2+ RF
regressors.

The fractional uncertainties in Ar3+/Ar2+ and Cl3+/Cl2+ as a function of the
ionic fractions on a log scale is given in Fig.B2. The predicted values show
larger deviation when Ar2+ or Cl2+ is the dominant ionization stage of Ar or
Cl. Still, the predicted values agree with the true values of the test set
within 50% for 89% and 90% of the cases.

Figure B2: Fractional uncertainty in predicted Ar3+/Ar2+ (left) and Cl3+/Cl2+
(right) from the RF regressor as a function of log-scale true ionic fractions
of the test set. The plots were constructed in the same manner as Fig.B1.

##  Appendix C Classifying Planetary Nebulae into basic Young and Old Groups

Based on previous studies on PNe abundances, Peimbert (1978) divided PNe into
four distinct types, each with unique general properties. Type I PNe are those
enriched in He and N abundances (He/H$>0.125$ and $\log$(N/O)$>-0.3$, as
defined in Peimbert & Torres-Peimbert, 1983), and they typically exhibit
bipolar morphologies, high central star temperatures, and lower mean scale-
heights above the Galactic plane (Corradi & Schwarz, 1995; Peimbert & Torres-
Peimbert, 1983). These properties are consistent with more massive progenitors
compared to other PNe nuclei, as younger and more massive stars are likely to
form in a more enriched ISM and undergo more efficient ‘dredge-up’ processes
(Phillips, 2005), leading to higher He and N abundances. The progenitor masses
of Type I PNe are suggested to be greater than $2.4M_{\odot}$ (Peimbert &
Serrano, 1980; Calvet & Peimbert, 1983), with small variations in the
literature, corresponding to an stellar age of about 1 Gyr.

Using AGB evolutionary models that incorporate dust formation in circumstellar
envelopes, Ventura et al. (2016) investigated the relationship between
progenitor masses and chemical composition of the resulting PNe. They found
that carbon stars produce C-rich PNe by ejecting their shells, while stars
undergoing hot-bottom burning, a phase during which most of their C is
converted into N, do not go through the carbon star phase and instead eject a
N-rich shell. These findings were used to classify PNe into young and old
populations according to their C/H and N/H ratios in Stanghellini & Haywood
(2018), hereafter referred to as SH18. A similar classification scheme using
O/H instead of C/H was also derived in SH18 as follows: for the old PNe
population with a stellar age greater than 7.5 Gyr,
$12+\log$(N/H)$<0.8\times[12+\log$(O/H)$]+1.4$; and for the young PNe
population with the central stars younger than 1 Gyr,
$12+\log$(N/H)$>0.6\times[12+\log$(O/H)]$+3.3$.

We consider both PN dating methods as equally reliable. We select a group of
young PNe where the two schemes yield non-contradictory results, meaning
objects are either classified as young based on the SH18 criteria and are Type
I, or are Type I PNe without an age group determined by the SH18 scheme.
Similarly, we choose a group of old PNe that are classified as old using the
SH18 criteria and are not Type I objects. Despite the large uncertainties,
which may exceed 3-4 Gyr due to the abundance determination and the dating
methods themselves, the young PNe group’s progenitor ages are estimated to be
less than 1 Gyr, while the old PNe population is expected to descend from
those older than 7.5 Gyr. Despite this crude age demarcation, a clear effect
in the size and distribution of the sulfur anomaly is shown in Fig. 4 in the
main body of the text, between the young and old PNe populations.

It is important to note that this age classification relies on the single-star
assumption for CSPNe. The presence of companion stars, particularly in close-
binary systems, can affect their evolutionary pathways and introduce
uncertainty in age estimation (see e.g. Soderblom, 2010; Stanway & Eldridge,
2018).


# Critique of optical coherence tomography in epistemological metrology

Yoshiaki Yasuno  Computational Optics Group, University of Tsukuba, Tsukuba,
Ibaraki 305-8573, Japan<EMAIL_ADDRESS>
###### Abstract

This treaties aims to introduce a speculative framework to comprehend modern
metrology. In a classical sense, measurement measures physically existing
quantities, such as length, mass, and time. By contrast, modern measurement
measures a wide spectrum of objects, for example, an imaging biomarker, which
is a user-defined pattern. Artificial-intelligence (AI)-based diagnosis is
another example. If we regard an optical coherence tomography (OCT) device
equipped with diagnostic AI as a comprehensive “measurement system,” the
object being measured is a “disease,” which is not a physical quantity but a
concept.

To comprehend the wide range of modern measurements, we introduce a
speculative, i.e., philosophical and theoretical, model called the
“epistemological-metrology model.” In this model, we describe the act of
measurement as cascading encoding-and-decoding processes. In the encoding
processes, three types of objects-to-be measured are considered, which include
substance, existence, and concept. Then we classify acts of measurement into
“sensing,” “understanding,” and “reasoning,” which measure the substance,
existence, and concept, respectively.

We note that the measurements in the understanding and reasoning classes are
constructive. Namely, they proactively define the quantity-to-be-measured by
the measurement modalities themselves. A speculative method to warrant the
relevance of such constructive measurements is presented.

We investigate several modern OCT-related measurements, including AI-based
diagnosis, types of polarization sensitive OCT, and attenuation coefficient
imaging, using our theoretical framework.

††journal: opticajournal††articletype: Research Article

https://optics.bk.tsukuba.ac.jp/COG/

##  1 Background: Modern conception of measurement

The modern conception of measurement includes far broader fields than that in
the classical sense. In classical measurement, the quantities to be measured
are quantities that are physical substances, such as mass, length, and time.
In the case of conventional, or classical, optical coherence tomography (OCT)
[1], the quantity to be measured is back-scattering light intensity. However,
modern measurement systems measure a far broader spectrum of quantities.

An example is a clinical imaging device equipped with an automatic diagnostic
function, such as an OCT device with glaucoma diagnostic software, for example
Refs. [2, 3]. The final measurement target of such a device is “glaucoma,”
that is, a “disease.” A disease is not a physical entity but a concept.

A similar example is an “imaging biomarker,” which is a characteristic pattern
in an image that correlates with a specific disease, for example Ref. [4].
When a measurement system acquires an image and computationally obtains the
imaging biomarker, the object being measured by such a system is a biomarker.
This is not a physical entity but a user-defined artificial quantity.

A more specific example can be found in polarization-sensitive (PS)-OCT [5,
6]. Some PS-OCT measures a quantity called the “degree-of-polarization-
uniformity” (DOPU) [7]. Although the DOPU is highly related to a physically
defined quantity; degree of polarization or depolarization, DOPU and these
quantities are not the same. The DOPU is defined as a variance of Stokes
vectors on a Poincaré sphere measured within a small spatial region in an
image. DOPU is not defined based on physical models but is a quantity
arbitrarily defined by PS-OCT developers. As metrology and measurement science
progress, the spectrum of the object being measured will be further broadened
and further exceeds the domain of physical entities.

##  2 Complication in modern measurement

Modern measurement systems are based on a variety of technologies, theories,
and conceptions. They cover, for example, physical phenomena (e.g.,
interference), physics-based signal processing (e.g., OCT reconstruction),
image processing (e.g., tissue segmentation and imaging biomarker extraction),
and image recognition (e.g., artificial-intelligence (AI)-based computer
diagnosis). Although these technological concepts are used together, their
mutual and theoretical relations are not well understood. As we discuss in
later sections, the lack of understanding has occasionally caused irrelevant
uses of some of these concepts. Additionally, this irrelevance will become
more severe as modern measurement begins to cover wider conceptions.

##  3 Our approach to comprehend “measurement”

This treaties aims to present a speculative (philosophical) framework to
comprehend modern measurement. In this framework, conceptions of modern
measurement are organized as a theoretical and speculative model called
“epistemological metrology.” To induce this model, we start from a readily
acceptable generalized model of measurement: the “cascading encoding-decoding
model.” Then we identify the similarity between this model and the
epistemology of Immanuel Kant [8]. By exploiting this similarity, we induce
the “epistemological-metrology model,” which comprehends the conceptions of
classical and modern measurements.

In epistemological metrology, we classify measurements into three classes:
“classical,” “semi-classical” (or “semi-cognitive”), and “cognitive.”
Cognitive and semi-cognitive measurements are characterized by their
constructive nature; that is, cognitive measurement is defined as a
measurement methodology that defines, or constructs, the object-to-be-measured
(OTBM) using the measurement methodology itself. By contrast, classical
measurement is measurement in a well-accepted sense, i.e., it measures
quantities that were defined before the measurement methodology was
established.

To further understand the conceptions of classical, semi-cognitive, and
cognitive measurements, we introduce three types of OTBM including (1)
substance, (2) existence, and (3) concept. They are measured in the classical,
semi-cognitive, and cognitive measurements, respectively.

We also show that some currently well-accepted measurements, including PS-OCT
and AI-based diagnosis, are not classical but semi-cognitive or cognitive;
that is, they construct (i.e., define) the quantities to be measured
themselves. We also note that cognitive measurement must be constrained by
some means to maintain its relevance. For this constraint, we introduce the
idea of “inevitability.” We show that the idea of inevitability not only
warrants the relevance of the measurement but also clarifies the difference
between signals and artifacts in a speculative sense.

##  4 Cascading encoding-decoding model

Figure 1: Sequence of OCT measurement. The first half, from tissue to the
interference field, can be regarded as an encoding process, whereas the second
half, from the digitized field to the contextualized OCT image, is regarded as
a decoding process. The first and second halves show a symmetricity centered
at photo-detection. See the first two paragraphs of Section 4 for details.

Measurement can be regarded as a sequential process of encoding and decoding.
We denote this view of measurement as a “cascading encoder-decoder model.”
Figure 1 exemplifies the measurement process of OCT. The first object is
biological tissue or the human body. It interacts with the probe beam, which
results in a modulated (and backscattered) probe beam. This modulated probe
beam interferes with a reference beam and forms an interference field. This
interference field is detected and digitized. The digitized interference field
is processed, and a raw OCT signal is yielded. The raw OCT signal is further
processed and a contextual image of tissue or the body is obtained. The
contextual image is exemplified by tissue-segmented OCT images, an attenuation
coefficient (AC) image (e.g., Ref. [9]), a DOPU image of PS-OCT [7], optical
coherence refraction tomography images [10, 11], etc.

We find that this sequential process is symmetric and centered at the
inference-field-detection step (the dashed line in Fig. 1). We also find that
the left-hand side of the symmetric axis is a cascading encoding process from
a physical entity (i.e., tissue), to a phenomenon, that is, the interference
signal. The right-hand side is a corresponding decoding process that starts at
the digitized interference field and reaches the image of the measurement
target.

Figure 2: Measurement in its wide sense is depicted as a cascading encoding-
and-decoding process. The objects in the upper half of the diagram are the
objects-to-be-measured, whereas those in the lower half are the measured
object/quantity. Objects at the same horizontal position correspond to each
other.

This model of an imaging process becomes more comprehensible in the depiction
in Fig. 2. The objects in the encoding process (upper half of the diagram) and
corresponding objects in the decoding process (lower half of the diagram) are
written at the same horizontal locations, which are referred to as “levels.”
Additionally, we added more objects at the beginning and end of the process,
that is, diseases and diagnosis. A disease abnormalizes tissue and the
diagnosis software of the measurement system provides a diagnosis using the
abnormality of the contextualized image.

It is noteworthy that the distance between the objects at the same level
indicates the similarity between the objects. For example, the similarity
between the interference signal and its digitized version is higher than that
between the human body and its contextualized image.

##  5 Epistemology and metrology

In this section, we note that the cascading-encoding-decoding model is similar
to the epistemological model of Kant [8]. The Kant’s model explains the act of
human recognition as a sequence of three processes: sensing, understanding,
and reasoning.

In the sensing process, the human cognitive system captures the phenomenon as
it appears and passes it to the understanding process. In this process, the
sensed object is not yet contextualized. This process is similar to the
digitization of the interference field, in which the interference electric
field is converted into a digital signal, but its meaning, that is, context,
is not yet known.

The understanding process binds the sensed object to a particular meaning,
such as the particular name of the phenomenon. This process is similar to OCT
image reconstruction and tissue segmentation. The meaningless interference
signal is properly processed, and a meaningful, on the other word,
“contextualized,” OCT image is generated.

The final process, that is, reasoning, leads us to the “concept behind the
contextualized phenomenon.” This concept is purely conceptional and its
existence or nonexistence cannot be proven. The “god” is an example of such a
concept in Kant’s philosophy. The reasoning process is similar to “diagnosis”
in our measurement. In the diagnosis process, we measure the disease, that is,
the reason for tissue abnormality. The disease is assumed to be the cause of
abnormality and is purely conceptional. It is also noteworthy that, in
reality, the disease did not abnormalize the tissue, but we defined a concept
of the disease to explain the cause of abnormality.

Considering the similarity between Kant’s model and our measurement model, we
find that the OTBM are more physical on the left-hand side of the diagram
(Fig. 2) and become more conceptual, or metaphysical, on the right-hand side.
It is also noteworthy that, on the right-hand side, the object in the upper
half of the diagram is more fundamental, whereas on the left-hand side, the
object in the lower half is more fundamental; that is, the interference
electric field is more fundamental than the digitized signal, that is, the
electric field “ rules” the digitized signal. By contrast, the diagnosis, that
is, the recognized disease, is more fundamental and important than the
conceptual “disease.” We provide details regarding this point in the next
section.

##  6 Classical and cognitive measurements

Figure 3: Model of “epistemological metrology.” The objects-to-be-measured
(OTBM) are classified into substance, existence, and concept according to the
ruling relationship between the OTBM and the measured object (MO). Acts of
measurement are classified into sensing, understanding, and reasoning, and
they correspond to classical, semi-cognitive, and cognitive measurements,
respectively. Table 1: The key conceptions appeared in the epistemological
metrology model. The object-to-be-measured (OTBM) is classified into the
substance, existence, and concept (first row), based on the idea of ruling
(second row). The measurement acts are then classified as sensing,
understanding, and reasoning, respectively for the substance, existence, and
concept. These measurement acts are recognized as classical, semi-cognitive
(semi-constructive), and cognitive (constructive), respectively.

By exploiting the similarity between the epistemology and the measurement, we
can define a “epistemological metrology model,” as depicted in Fig. 3. Table 1
also summarizes the key conceptions appeared in this model. Here the first
point is that the OTBM and the measured object/quantity (MO) are explicitly
distinguished. In the diagram, the OTBM are appeared in the upper side, while
the MOs are appeared in the lower side. The second point is that the OTBM are
classified into three types: (1) substance, (2) existence, and (3) concept.

Substance is a physical quantity or entity, such as an electric field.

Existence is an object that is definitively analogized from the measurement
result of existence. Contextualized OCT images, such as AC, DOPU, and tissue-
segmented images, are the measurement results of existence. A raw OCT image is
the measurement result of substance in one perspective, or existence in
another perspective. Namely, if we consider that the OCT image represents the
electric field distribution in a sample, OCT measures the substance. On the
other hand, if we consider the OCT image represents “the scattering ability”
of the tissue, not the scattered electric field itself, then OCT measures the
existence.

The concept is an object that is arbitrarily defined to explain the measured
existence. Disease is an example of a concept.

Based on the classification of the OTBM, we can define three levels of
measurements. The most fundamental level of measurement is sensing, which
measures physical existence (rightmost of the diagram in Fig. 3). The second
level of measurement is understanding, which measures existence. Existence is
more contextualized than substance (middle of the diagram). The final level of
measurement is reasoning, which measures a concept to explain the reason
behind the measured existence (leftmost). For example, an interference field
(substance) is sensed by a photo-detector and is understood as a
contextualized OCT image (existence) and is finally reasoned as a disease
(concept). The further to the left the quantity is in the diagram, the more
contextualized.

To further understand the levels of measurement, we introduce the idea of
“classical” and “cognitive” measurements. A cognitive measurement is
characterized by its constructive nature; that is, cognitive measurement
proactively defines the OTBM. So, we also denote the cognitive measurement as
“constructive measurement.” Reasoning is a cognitive (constructive)
measurement. In our example, the disease is “defined,” i.e., the concept of
the disease is constructed, by being diagnosed; that is, the observation
(diagnosis) “rules ” the object (disease), as shown by the red arrow in the
diagram.

By contrast, classical measurement measures a predefined quantity, such as an
electric field. In this case, the predefined physical quantity (i.e., OTBM)
rules the measured quantity (MO). Sensing is a classical measurement.

The understanding process is classified as “semi-classical” or equivalently
“semi-cognitive (semi-constructive).” The measured quantity is highly related
to the predefined physical object (or phenomenon), but the measured quantity
is contextualized by the measurement. Hence, the OTBM and the MO mutually rule
each other.

##  7 “Inevitability” constrains cognitive measurements

An immediate question that arises about cognitive and semi-cognitive
measurements is “if the OTBM is defined by the act of measurement itself, how
can the measured quantity be meaningful?” In cognitive measurement, anything
could, in principle, be arbitrarily defined as an OTBM. Hence, it should be
constrained in some manner. To constrain cognitive and semi-cognitive
measurements and make them meaningful, we introduce the idea of the
“inevitability of the measurement.” Inevitability explains why the quantity
“should be measured.” We define two types of inevitabilities: pragmatic and
transcendental, as summarized in Table 2.

Table 2:  Two types of inevitabilities warrant the cognitive and semi-
cognitive (constructive and semi-constructive) measurements.

The terminology of “pragmatic inevitability” was inspired by the philosophical
school of “pragmatism” [12]. In this philosophy, the proposition is thought to
be meaningful only if it is effectual. Based on a similar idea, we regard the
OTBM to be “pragmatically inevitable” if the measured object is meaningful in
any context; that is, pragmatic inevitability is defined based on the utility
of the measurement. For example, a “disease” is pragmatically inevitable if it
can be used to make a clinical decision or develop an effective therapy.
Similarly, a segmentation result is pragmatically inevitable if it can be used
for the next step of the measurement, that is, diagnosis.

The second type of inevitability is “transcendental inevitability,” which is
inspired by Kant’s transcendental, or metaphysical, philosophy. In this
philosophy, some transcendental objects, such as the soul or a god, are
thought to be inevitably recognized. However, this recognition is not based on
the existence of the object but purely based on the recognition system of the
human being. By adopting this concept, we define a transcendentally inevitable
OTBM as follows: what appears in the measurement results but is originated
from the measurement system itself instead of the measured sample. Namely, the
transcendentally inevitable object is the “artifact,” which inevitably appears
in the measurement results because of the property of the measurement system.

One important characteristic of the transcendentally inevitable object, that
is, the artifact, is that it can be distinguished from the signal only if we
have a priori knowledge about the sample and/or measurement system. For
example, “fixed pattern noise” and/or the “mirror image” of OCT can be
recognized as an artifact only if we have known that the sample does not have
such a structure and the principle of OCT can create such pseudo-signals.

##  8 Critique of OCT in epistemological metrology

By exploiting the epistemological metrology model, we can preclude some
irrelevant directions of OCT research. We provide examples from AI-based
diagnosis, PS-OCT, and AC measurement.

###  8.1 AI-based diagnosis

AI-based diagnosis is “reasoning,” and hence, cognitive (i.e., constructive).
Some diagnostic AIs were designed to mimic the established classification
systems of clinical findings, such as the classification of the optic nerve
head. If the training target of AI is human-classified labels, then the
attempt of AI-based classification could, in principle, be cyclic. This is
because both the training target, that is, the classification results of human
experts, and the AI are in the lower half of the diagram in Fig. 3. In
practice, this type of AI can be meaningful as long as the accuracy of the
AI’s inference is lower than the inter-human expert agreement. However, if we
attempt to increase accuracy more than the agreement, the AI does not lead us
closer to the real disease, which is in the upper half of the diagram. Namely,
the pragmatic inevitability of such over-learning is low.

We can consider actively defining the disease based on the AI-based diagnosis.
For example, we can train the AI to provide the best responding treatment
using clinical observations as input. In this case, the concept of the disease
is defined by the AI because it can maximize clinical intervention.

###  8.2 Degree-of-polarization-uniformity (DOPU)

Another example can be found in DOPU-related research. Low DOPU values are
believed to be correlated with melanin [13]; hence, a low DOPU region is
collocated with retinal pigment epithelium (RPE) [7]. However, DOPU
measurement is semi-cognitive.

Historically, DOPU was “defined” to highlight RPE [7] after it was found that
RPE exhibits random phase retardation patterns [14]. Hence, there is no
logical inevitability that relates DOPU to melanin.

This logical speculation implies that “proving low DOPU is inevitably
associated with melanin” is not really a reasonable research direction, but
defining modified versions of DOPU or similar polarization randomness metrics
[15, 16, 17] that are sensitive and specific to the tissue- or disease-of-
interest can be more reasonable. This point also implies a fundamental
similarity between DOPU and the idea of an imaging biomarker.

###  8.3 Attenuation coefficient (AC) imaging

AC imaging is another example of semi-cognitive measurement. AC is the
attenuation rate of the OCT signal or the back-scattering of the probe beam
[9, 18]. Several attempts have been made to correct the confocal effect to
achieve a more “accurate” AC measurement [19, 20]. We may find that if AC is
defined as described above, the confocal effect is a part of the attenuation
source, and hence should not be corrected.

This contradiction can be explained by considering the semi-cognitive nature
of AC measurement; that is, AC measurement, in reality, is not intended to
measure the signal attenuation rate but to measure “some quantity being
sensitive to tissue abnormalities;” that is, AC measurement constructs the
quantity-to-be-measured to meet the purpose of the measurement. Hence, despite
AC measurement seemingly being a measurement of physical quantity, it is semi-
cognitive and its inevitability is pragmatically warranted.

AC is meaningful as far as it can be used to detect tissue abnormalities;
hence, the definition of AC can be modified, for example, corrected for the
confocal effect, to be more suitable for this purpose.

##  9 Summary

Figure 4:  The summary of the logic structure on this treaties. Three primary
concepts are highlighted by red. And other concepts are defined from these
primary concepts.

The logic structure of this treaties are summarized in a logic diagram shown
in Fig. 4. In this treaties, we introduced a speculative model of wide-sense
measurement called epistemological metrology to comprehend classical and
modern measurements. This model based on three key concepts: (1) the
distinction between object-to-be-measured (OTBM) and measured-object (MO), (2)
the ruling direction between the OTBM and the MO, and (3) inevitability of
measurement.

Based on the ruling direction, we classified the OTBM into three types:
substance, existence, and concept.

We classified the measurement measuring these objects as classical, semi-
cognitive, and cognitive, respectively. The latter two are characterized by
their constructive nature, that is, they define the measured quantity as a
part of the measurement. Although the semi-cognitive and cognitive (i.e.,
constructive) measurements are not well constrained in a classical sense, the
relevance of them can be warranted by the inevitability. We found that several
well-accepted OCT-based measurements are cognitive or semi-cognitive.
Additionally, we showed that this model is useful for evaluating the relevance
of research directions in measurement science.

Although this treaties discussed only OCT and its related measurement
modalities, the epistemological metrology model could be used to comprehend
other measurement modalities.

## Funding

Core Research for Evolutional Science and Technology (JPMJCR2105); Japan
Society for the Promotion of Science (21H01836);

## Disclosures

Yasuno: Yokogawa Electric Corp. (F), Sky Technology (F), Nikon (F), Kao Corp.
(F), Topcon (F), Tomey Corp (F).

## References

  * [1] D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. H. a. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
  * [2] Z. Burgansky-Eliash, G. Wollstein, T. Chu, J. D. Ramsey, C. Glymour, R. J. Noecker, H. Ishikawa, and J. S. Schuman, “Optical Coherence Tomography Machine Learning Classifiers for Glaucoma Detection: A Preliminary Study,” Invest. Ophth. Vis. Sci. 46, 4147–4152 (2005).
  * [3] J.-C. Mwanza and D. L. Budenz, “Optical coherence tomography platforms and parameters for glaucoma diagnosis and progression,” Curr. Opin. Ophthalmol. 27, 102–110 (2016).
  * [4] D. C. Sullivan, N. A. Obuchowski, L. G. Kessler, D. L. Raunig, C. Gatsonis, E. P. Huang, M. Kondratovich, L. M. McShane, A. P. Reeves, D. P. Barboriak, A. R. Guimaraes, and R. L. Wahl, “Metrology standards for quantitative imaging biomarkers,” Radiology 277, 813–825 (2015). Publisher: Radiological Society of North America.
  * [5] J. F. de Boer, C. K. Hitzenberger, and Y. Yasuno, “Polarization sensitive optical coherence tomography – a review [Invited],” Biomed. Opt. Express 8, 1838–1873 (2017).
  * [6] B. Baumann, “Polarization sensitive optical coherence tomography: A review of technology and applications,” Appl. Sci. 7 (2017).
  * [7] E. Götzinger, M. Pircher, W. Geitzenauer, C. Ahlers, B. Baumann, S. Michels, U. Schmidt-Erfurth, and C. K. Hitzenberger, “Retinal pigment epithelium segmentation by polarization sensitive optical coherencetomography,” Opt. Express 16, 16410–16422 (2008).
  * [8] I. Kant, _Critique of Pure Reason (Kritik der reinen Vernunft)_ (1787), 2nd ed.
  * [9] K. A. Vermeer, J. Mo, J. J. A. Weda, H. G. Lemij, and J. F. de Boer, “Depth-resolved model-based reconstruction of attenuation coefficients in optical coherence tomography,” Biomed. Opt. Express 5, 322–337 (2014).
  * [10] K. C. Zhou, R. Qian, S. Degan, S. Farsiu, and J. A. Izatt, “Optical coherence refraction tomography,” Nat. Photon. 13, 794–802 (2019). Number: 11 Publisher: Nature Publishing Group.
  * [11] K. C. Zhou, R. P. McNabb, R. Qian, S. Degan, A.-H. Dhalla, S. Farsiu, and J. A. Izatt, “Computational 3D microscopy with optical coherence refraction tomography,” Optica 9, 593 (2022).
  * [12] W. James, _Pragmatism: A New Name for Some Old Ways of Thinking_ (1907).
  * [13] B. Baumann, S. O. Baumann, T. Konegger, M. Pircher, E. Götzinger, F. Schlanitz, C. Schütze, H. Sattmann, M. Litschauer, U. Schmidt-Erfurth, and C. K. Hitzenberger, “Polarization sensitive optical coherence tomography of melanin provides intrinsic contrast based on depolarization,” Biomed. Opt. Express 3, 1670–1683 (2012).
  * [14] M. Pircher, E. Götzinger, R. Leitgeb, H. Sattmann, O. Findl, and C. Hitzenberger, “Imaging of polarization properties of human retina in vivo with phase resolved transversal PS-OCT,” Opt. Express 12, 5940–5951 (2004).
  * [15] S. Makita, Y.-J. Hong, M. Miura, and Y. Yasuno, “Degree of polarization uniformity with high noise immunity using polarization-sensitive optical coherence tomography,” Opt. Lett. 39, 6783–6786 (2014).
  * [16] N. Lippok, M. Villiger, and B. E. Bouma, “Degree of polarization (uniformity) and depolarization index: unambiguous depolarization contrast for optical coherence tomography,” Opt. Lett. 40, 3954–3957 (2015).
  * [17] M. Yamanari, S. Tsuda, T. Kokubun, Y. Shiga, K. Omodaka, N. Aizawa, Y. Yokoyama, N. Himori, S. Kunimatsu-Sanuki, K. Maruyama, H. Kunikata, and T. Nakazawa, “Estimation of Jones matrix, birefringence and entropy using Cloude-Pottier decomposition in polarization-sensitive optical coherence tomography,” Biomed. Opt. Express 7, 3551–3573 (2016). Publisher: Optica Publishing Group.
  * [18] K. A. Vermeer, J. v. d. Schoot, H. G. Lemij, and J. F. d. Boer, “RPE-normalized RNFL attenuation coefficient maps derived from volumetric OCT imaging for glaucoma assessment,” Invest. Ophth. Vis. Sci. (2012).
  * [19] G. Smith, N. Dwork, D. O’Connor, U. Sikora, K. Lurie, J. Pauly, and A. Ellerbee, “Automated, depth resolved estimation of the attenuation coefficient from optical coherence tomography data,” IEEE Trans. Med. Imag. PP, 1–1 (2015).
  * [20] J. Kübler, V. S. Zoutenbier, A. Amelink, J. Fischer, and J. F. de Boer, “Investigation of methods to extract confocal function parameters for the depth resolved determination of attenuation coefficients using OCT in intralipid samples, titanium oxide phantoms, and in vivo human retinas,” Biomed. Opt. Express 12, 6814 (2021).