# Bayesian Analysis for Remote Biosignature Identification on exoEarths
(BARBIE) II: Using Grid-Based Nested Sampling in Coronagraphy Observation
Simulations for O2 and O3

Natasha Latouf NSF Graduate Research Fellow, 2415 Eisenhower Ave, Alexandria,
VA 22314  Department of Physics and Astronomy, George Mason University, 4400
University Drive MS 3F3, Fairfax, VA, 22030, USA  NASA Goddard Space Flight
Center, 8800 Greenbelt Road, Greenbelt, MD 20771, USA  Sellers Exoplanents
Environment Collaboration, 8800 Greenbelt Road, Greenbelt, MD 20771, USA  Avi
M. Mandell NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt,
MD 20771, USA  Sellers Exoplanents Environment Collaboration, 8800 Greenbelt
Road, Greenbelt, MD 20771, USA  Geronimo L. Villanueva NASA Goddard Space
Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771, USA  Sellers
Exoplanents Environment Collaboration, 8800 Greenbelt Road, Greenbelt, MD
20771, USA  Michael D. Himes NASA Postdoctoral Program Fellow, NASA Goddard
Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771, USA  Michael
Dane Moore NASA Goddard Space Flight Center, Greenbelt, MD, USA.  Business
Integra, Inc., Bethesda, MD, USA.  Sellers Exoplanets Environment
Collaboration, 8800 Greenbelt Road, Greenbelt, MD 20771, USA  Nicholas
Susemiehl  Center for Research and Exploration in Space Science and
Technology, NASA Goddard Space Flight Center, Greenbelt, MD, USA.  NASA
Goddard Space Flight Center, Greenbelt, MD, USA.  Jaime Crouse NASA Goddard
Space Flight Center, Greenbelt, MD, 20771  NASA GSFC Sellers Exoplanet
Environments Collaboration  Shawn Domagal-Goldman NASA Goddard Space Flight
Center, Greenbelt, MD, 20771  Giada Arney NASA Goddard Space Flight Center,
Greenbelt, MD, 20771  NASA GSFC Sellers Exoplanet Environments Collaboration
Vincent Kofman NASA Goddard Space Flight Center, 8800 Greenbelt Road,
Greenbelt, MD 20771, USA  Sellers Exoplanents Environment Collaboration, 8800
Greenbelt Road, Greenbelt, MD 20771, USA  Integrated Space Science and
Technology Institute, Department of Physics, American University, Washington
DC  Amber V. Young NASA Goddard Space Flight Center, Greenbelt, MD, 20771
NASA GSFC Sellers Exoplanet Environments Collaboration Natasha Latouf
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present the results for the detectability of the O2 and O3 molecular
species in the atmosphere of an Earth-like planet using reflected light at
visible wavelengths. By quantifying the detectability as a function of signal-
to-noise ratio (SNR), we can constrain the best methods to detect these
biosignatures with next-generation telescopes designed for high-contrast
coronagraphy. Using 25 bandpasses between 0.515 and 1 m) and a pre-constructed
grid of geometric albedo spectra, we examined the spectral sensitivity needed
to detect these species for a range of molecular abundances. We first
replicate a modern-Earth twin atmosphere to study the detectability of current
O2 and O3 levels, and then expand to a wider range of literature-driven
abundances for each molecule. We constrain the optimal 20%, 30%, and 40%
bandpasses based on the effective SNR of the data, and define the requirements
for the possibility of simultaneous molecular detection. We present our
findings of O2 and O3 detectability as functions of SNR, wavelength, and
abundance, and discuss how to use these results for optimizing future
instrument designs. We find that O2 is detectable between 0.64 and 0.83 m with
moderate-SNR data for abundances near that of modern-Earth and greater, but
undetectable for lower abundances consistent with a Proterozoic Earth. O3 is
detectable only at very high SNR data in the case of modern-Earth abundances,
however it is detectable at low-SNR data for higher O3 abundances that can
occur from efficient abiotic O3 production mechanisms.

planetary atmospheres, telescopes, methods: numerical; techniques: nested
sampling, grids

##  1 Introduction

Over 5,000 exoplanets have been discovered and confirmed in the last three
decades. With the field of exoplanet exploration booming since the first
exoplanet atmosphere discovery (HD 209458; Charbonneau et al., 2002; Seager &
Deming, 2010; Kaltenegger, 2017; Madhusudhan, 2019), the next hurdle is the
detection and characterization of terrestrial exoplanet atmospheres. Current
space-based observatories (e.g. Spitzer, HST, JWST) are beginning to probe the
characteristics of potentially rocky planets with both transmission (de Wit et
al., 2016; Lim et al., 2023) and emission (Kreidberg et al., 2019; Zieba et
al., 2023) measurements, but potentially habitable planets will be remain
largely inaccessible except for 1-2 unique systems around the smallest stars
(e.g. TRAPPIST-1) and impossible to characterize for Sun-like stars. However,
with the advancements of high-contrast instrument technology and future
mission concepts (e.g., Roberge & Moustakas, 2018; The LUVOIR Team, 2019;
Gaudi et al., 2020), the ability to find a habitable Earth-twin around a Sun-
like star is now a realistic goal for the next flagship visible-light space
telescope, the Habitable Worlds Observatory (HWO).

Figure 1: Reflection spectra of a modern-Earth atmospheric composition but
with varied O3 and O2 abundances. We feature a high O3 abundance (3$\times
10^{-6}$), a 50% PAL O2 abundance (2.1$\times 10^{-2}$), and a modern Earth
(7$\times 10^{-7}$ O3, and 2.1$\times 10^{-1}$ O2), respectively. All spectra
are binned at R=140.

In the search for signs of planetary habitability, the detection of potential
biosignatures can hint at biological activity on the planetary surface. Since
Earth is currently our only example of a conclusively habitable (and
inhabited) planet, using model scenarios with conditions similar to those on
modern Earth or its distant past serves as a starting point for examining the
detectability of planetary characteristics (Arney et al., 2016; Rugheimer &
Kaltenegger, 2018). In particular, at different times in its history Earth
hosted varying concentrations of the atmospheric biomarkers O2 and O3, which
are primarily produced on Earth through photosynthesis and subsequent
photochemical reactions; therefore measurements of geochemical proxies for the
oxygenation of Earth’s surface and atmosphere over time provide examples of
possible global biogeochemical scenarios that we could encounter when
observing Earth-like exoplanets. It is also useful to consider planets that -
like Earth - have global liquid water oceans, but no biospheres. In some
conditions, such

The Phanerozoic (modern) epoch of Earth (541 million years ago – present),
which forms the basis for most habitability and biosignature studies, has
relatively high levels of both O2 (21%) and O3 (0.7 ppm) generated by abundant
plant life and subsequent photochemistry. However, in the Proterozoic epoch
(2.5 billion – 541 million years ago), photosynthetic life was less abundant
and measurements suggest oxygen accumulation in the atmosphere at
approximately 0.1% to 1% of modern Earth O2 levels but with potentially
detectable levels of O3 (Planavsky et al., 2014). In a planetary atmosphere
similar to the Archean epoch of Earth (4 – 2.5 billion years ago), we would
expect an oxygen-poor and ozone-poor atmosphere, with a large abundance of
greenhouse gases including CO2 and CH4, which may have in turn formed a
photochemical organic atmospheric haze (Zerkle et al., 2012; Arney et al.,
2016; Krissansen-Totton et al., 2018). In addition to past epochs of Earth’s
history, models of alternative atmospheric scenarios are capable of producing
of O2 and O3 dramatically higher than today’s values - even without the
presence of biology - due to a variety of mechanisms (Hu et al., 2012;
Domagal-Goldman et al., 2014; Gao et al., 2015; Tian, 2015; Harman et al.,
2018).

As shown in Figure 1, we can see how varying the abundances of O2 and O3 can
alter the resultant visible spectra and lead to differences in the
detectability of these gases. In this work we quantify the detectability of O2
and O3, using reflected-light measurements at visible wavelengths (0.515 – 1
m), for a range of abundance values and spectral bandpasses. This project is a
direct continuation from Latouf et al. (2023, hereafter BARBIE1) \- in
BARBIE1, we studied the detection of H2O as a function of SNR and abundance
throughout the visible spectral range, and in this paper we extend the same
methodology to the study of O2 and O3 and also examine the impact of the width
of spectral bands on the SNR required for detection.In $[\S]$ 2 we present the
methodology of our simulations, also providing a brief summary of BARBIE1. In
$[\S]$ 3 we present the results of our simulations for both the modern Earth-
like SNR study and the molecular abundance study. In $[\S]$ 4 we discuss the
presented results and analyze the impact for future observations of varying
Earth-twin epochs. In $[\S]$ 5 we present our conclusions and ideas for future
work.

##  2 Methodology

We follow a similar methodological approach to that of BARBIE1. Herein we
present a summary of the main steps in our analysis.

###  2.1 Inputs

####  2.1.1 Pre-Computed Spectral Grid

We use a geometric albedo spectral grid that was pre-computed by Susemiehl et
al. (2023, hereafter S23). This grid is housed in the Planetary Spectrum
Generator (PSG, Villanueva et al., 2018, 2022). The grid contains 1.4 million
geometric albedo spectra that have been pre-computed with the parameters,
minimum, and maximum values laid out in Table 1 of S23. The native resolution
of the grid is set to R=500 and binned to R=140 for our analysis. For more
information on the creation and verification of the grid, see S23 and BARBIE1.
There are three molecular species in the grid: H2O, O3, and O2, with N2 as the
assumed background gas. The minimum and maximum grid values for these
parameters are as follows: H2O in [$10^{-8}$, $10^{-1}$]; O3 in [$10^{-10}$,
$10^{-1}$]; O2 in [$10^{-10}$, $0.8$]; N2 = 1 - H2O - O3 - O2. The model is
comprised of 50% clear and 50% cloudy spectra, i.e. $\mathrm{C_{f}}$ = 50%.
There are three planetary parameters in the grid: surface pressure
($\mathrm{P_{0}}$), surface albedo ($\mathrm{A_{s}}$), and gravity (g). The
minimum and maximum values for these parameters are as follows:
$\mathrm{P_{0}}$ in [$10^{-3}$ Bar, 10 Bar]; $\mathrm{A_{s}}$ in [$10^{-2}$,
1]; gravity in [1 m/s2, 100 m/s2]; and $\mathrm{R_{p}}$ is fixed to 1
$R_{\earth}$. The grid covers a wavelength range from 0.515 – 1.0 m as this
range has been defined as the VIS channel in the exoplanet imaging instrument
concepts studied in the Astro2020 Decadal Survey that form the starting point
for HWO.

Log-Step O2 |  O2 (VMR)
---|---
0 | 2.1$\times 10^{-1}$†
-0.25 | 1.2$\times 10^{-1}$
-0.5 | 6.7$\times 10^{-2}$
-0.75 | 3.8$\times 10^{-2}$
-1 | 2.1$\times 10^{-2}$
-1.25 | 1.2$\times 10^{-2}$
-1.5 | 6.7$\times 10^{-3}$
-2 | 2.1$\times 10^{-3}$
-2.5 | 6.7$\times 10^{-4}$
-3 | 2.1$\times 10^{-4}$
Table 1: The above values were used in our simulations for O2, moving from
Earth-like values into different epochs of Earth’s history.
† Modern Earth-like value.

Log-Step O3 |  O3 (VMR)
---|---
+0.65 | 3$\times 10^{-6}$
+0.55 | 2.5$\times 10^{-6}$
+0.45 | 2$\times 10^{-6}$
+0.35 | 1.5$\times 10^{-6}$
+0.25 | 1.25$\times 10^{-6}$
+0.15 | 1$\times 10^{-6}$
+0 | 7$\times 10^{-7}$†
Table 2: The above values were used in our simulations for O3, moving from
Earth-like values up to a high-O3 build-up regime.
† Modern Earth-like value.

####  2.1.2 Mock Data and Retrieval Methodology

Our fiducial “data” spectrum is primarily set as a modern-Earth twin following
Feng et al. (2018), with constant volume mixing ratios (VMRs) H2O$=3\times
10^{-3}$, O3$=7\times 10^{-7}$, O2=$0.21$, a background gas of
N2=$1-$H2O$-$O3$-$O2, constant temperature profile at 250 K, $\mathrm{A_{s}}$
of 0.3, $\mathrm{P_{0}}$ of 1 Bar, and a planetary radius fixed at
$\mathrm{R_{p}}$ = 1 $\mathrm{R_{\earth}}$. We consider a resolving power of
140, binned from the native grid resolving power of 500. This fiducial
spectrum is given as data to the nested sampler in conjunction with the grid.
We use the modern-Earth twin fiducial spectrum for our initial SNR study,
wherein we focused on SNRs 3–16 moving in steps of 1. We then change the
fiducial spectrum for our abundance study, changing only the molecule of
interest one at a time, to the values listed in Table 2 and 2. All other
parameters were left to modern-Earth values, in order to specifically
constrain the molecule of interest. In BARBIE1, we focused on H2O, centered on
modern-Earth values and moving in increasing and decreasing log-steps. This
was due to the lack of constraint on H2O abundance throughout time. In this
study, we consider O2 and O3, and set our maximum O2 value as the modern-Earth
value of 0.21 VMR and move in decreasing log-steps of 0.25 and 0.5 down to
values of 1% to 0.1% to represent a mid-Proterozoic Earth epoch (Planavsky et
al., 2014). For O3 we wished to test the possibility that values higher than
on modern-day Earth might present stronger detectability. To study this
potential, we set the maximum value as 3 $\times 10^{-6}$ VMR, a value that is
$\sim 5$x the modern-Earth value of 7 $\times 10^{-7}$ VMR, and well within
the values created in models with high rates of abiotic O2 and O3 production
on planets around F-, K, or M-type stars (Hu et al., 2012; Domagal-Goldman et
al., 2014; Gao et al., 2015; Tian, 2015; Harman et al., 2018). and decrease to
the minimum value as the modern-Earth value.In log space, we decreased in
steps of 0.1, due to the small gap between the maximum and minimum values.

As in BARBIE1, to examine the detectability of the molecular species as a
function of the central wavelength and width of a bandpass, we chose 25 evenly
spaced values for the bandpass central wavelength. However, in this study we
also vary the total width of the spectral bandpass, examining values of 20%,
30%, and 40%. These ranges are consistent with the simultaneous bandpasses
that may be achieved with future high-performance coronagraphs (Ruane et al.,
2015; Por, 2020; Juanola-Parramon et al., 2022). Using these bandpass values
and the inputs laid out in 2.1.1 and 2.1.2, we run a series of Bayesian nested
sampling retrievals for each abundance and SNR combination using the PSGnest
application for the Planetary Spectrum Generator (PSG; Villanueva et al.,
2018, 2022).

PSG is a radiative transfer model and tool for synthesizing and retrieving
upon planetary spectra. This includes planetary atmospheres and surfaces
covering wavelengths from 50 nm to 100 mm (i.e. from UV to Radio) and a large
range of planetary properties. PSG includes aerosol, atomic, continuum, and
molecular scattering/radiative processes, implemented layer-by-layer. PSG also
includes the nested sampling routine PSGnest
111https://psg.gsfc.nasa.gov/apps/psgnest.php, which is an adaptation of the
algorithm used in Fortran MultiNest (Feroz et al., 2009). PSGnest is a
Bayesian retrieval tool based on the well-known MultiNest framework and
designed for exoplanetary observations; for more information on PSGnest and
our retrieval methodology, see S23 and BARBIE1.

###  2.2 Outputs

The output results file from PSGnest contains highest-likelihood values of
output parameters, the average value resulting from the posterior
distribution, uncertainties which are estimated from the standard deviation of
the posterior distribution, and the log evidence (logZ) (Villanueva et al.,
2022). It also includes the input data (wavelength, uncertainty, and input
spectrum, in respective columns) with the best-fit spectrum. We calculate the
median values, as well as the upper and lower limits of the 68% credible
region (Harrington et al., 2022) using the output results. We also extract the
posteriors and global log-evidence for use in our detectability calculations,
which is the numerical representation that quantifies the relative support per
each model given the input data (Feroz et al., 2009). Using these outputs, we
compute the Bayes factor. The Bayes factor is calculated by subtracting the
Bayesian log-evidence per retrieval using the fiducial gas abundances. The
resulting differences are referred to as the log-Bayes factor ($\mathrm{lnB}$;
Benneke & Seager, 2013). The log-Bayes factor determines which scenario is
most likely by examining the hypothesis where all parameters are present, and
systematically subtracting the evidences from scenarios where each parameter
in turn is nullified. In our studies, if $\mathrm{lnB}$ is less than 2.5, it
represents an unconstrained detection; if $\mathrm{lnB}$ is between 2.5 and 5,
it is a weak detection; if $\mathrm{lnB}$ is greater than 5, it is a strong
detection (reference Table 2 of Benneke & Seager, 2013). This differs slightly
from Table 2 of Benneke & Seager (2013), in which $\mathrm{lnB}$ represents a
weak, moderate, and strong detection respectively. We do not calculate the
log-Bayes factor for non-gaseous components, as those factors cannot be absent
and thus the calculation cannot represent a detection of those components.

Figure 2: Results of the fiducial case study for O2, but showing only a narrow
range of SNR values. The y-axis shows the abundance values for each molecule
in log scale, with the true value is shown with the black dashed line (in this
case, a modern Earth-like O2 value of 0.21 VMR). Each dot represents a
bandpass center, with the pink line portraying the median retrieved values of
O2 abundance for that bandpass, and the gray shaded region representing the
upper and lower limits for the 68% credible region. Increased retrieval
certainty can be seen where the gray regions narrow. Each point is colored to
indicate varying detection strength. Unconstrained regions ($lnB<2.5$) are
shown in light pink dots, weak detections ($2.5\leq lnB<5$) are shown in dark
pink diamonds, and strong detections ($lnB>5$) are shown in purple squares. We
present the range at which detectability materially changes for O2. Figure 3:
Results of the fiducial case study for O3 but focused on a specific narrow SNR
range. We show the SNR range at which detectability materially changes for O3.
All facets of the plot remain the same as in Figure 2. We notice that for
SNR=20, 3 bandpasses achieve a strong detection but none of these bandpasses
correctly constrain the abundance of O3 within the 68% credible region. We
present further investigation on this in Figure 4. Figure 4: Corner plots for
O3 at 0.64 m at SNR=20 to portray the lack of continuum and its impact on O3
abundance retrieval. We also show a corner plot for SNR = 200 with all
parameters set to exact grid points to portray the interpolation error in
$A_{s}$. The 68% credible region is shown as pink shading in the 1D
marginalized posterior distributions along the diagonal of the corner plot,
and the true values are represented by black lines in the diagonals of the
corner plot, and black stars within the 2D plots. It is clear that the error
in AS is due to a combination of parameter degeneracy and interpolation error
in the grid-derived forward models.

##  3 Results

###  3.1 Modern Earth Case Results

We begin by presenting the detectability of O2 and O3 as a function of SNR for
the fiducial modern-Earth case, as first examined by S23 and BARBIE1; all of
the O2 and O3 data and calculated log-Bayes factors across abundance, SNR, and
wavelength are available to the community on Zenodo22210.5281/zenodo.8349974.
We only present the narrow SNR range within which detectability strength
changes materially for O2 and O3 in Figures 2 and 3. We can see that based on
Figure 2, for SNR $\leq 7$ there is only a weak or unconstrained detection
possible for O2. However, beginning at SNR = 7.5, we can see strong detections
of O2 corresponding to bandpasses containing deep O2 spectral features, such
as at 0.74 m. A strong detection of O2 becomes possible from 0.68 - 0.84 m by
SNR = 8. At SNRs higher than 8, all bandpass locations yield a strong
detection of O2 with increasingly better constraints of the 68% credible
region.

Looking to Figure 3, it was required to significantly increase the SNR to
achieve a strong detection for O3. We look at an SNR range of 18 – 20, which
is quite high, however it is only at this point that we begin to see a strong
detection of O3 within the wavelength range of our simulations. At an SNR of
18, we can see a small area of weak detection between 0.6 and 0.67 m covering
six bandpasses. At an SNR of 19, strong detection becomes possible for two
bandpasses, and three bandpasses at SNR of 20. We can see it takes a very high
SNR to achieve a strong detection of O3, which would lead to an exceptionally
high integration time.

We can also see that although three bandpasses yield a strong detection at
SNR=20, none of those bandpasses correctly constrain the abundance of O3
within the 68% credible region. We present further investigation on this in
Figure 4a. In this corner plot, which is focused on 0.64 m at a 20% bandpass,
O3 is not retrieved within the 68% credible region, and there is a large
spread of the high probability region which does not center on the true value
of O3. This is largely due to the lack of continuum caused by the depth and
width of the ozone features; a similar problem for detecting H2O at long
wavelengths is discussed at length in BARBIE1. We present Figure 4b to show
that by increasing the bandpass width from 20% to 40% centered on 0.64 m, and
thus increasing the amount of spectral region covered in each bandpass, an
adequate continuum is constrained and the retrieval of O3 is firmly within the
68% credible region with little spread in the high probability regions.

However, we can see in the corner plots shown in Figure 4a and 4b that $A_{s}$
is also consistently retrieved at a value at least a factor of two away from
the true value. To investigate the source of this incorrect retrieved
parameter, we ran several retrievals with very high SNR (200) and with a data
spectrum using values centered on a grid point, and compared both sets of
results to our lower-SNR modern-Earth results. This allowed us to test whether
the result was due to degeneracies for lower-SNR data or due to the impact of
interpolation error in our grid-based retrieval scheme (as examined by S23).
In Figure 4c, where we use a very high SNR and all of the parameters are set
to exact values found in the S23 grid, we can see that all parameters are
retrieved within a 68% credible region except for $C_{f}$ which is known to be
degenerate. When $C_{f}$ is locked to its true value (0.5) as in Figure 4d, we
can see this issue disappears and the range in the highest likelihood region
decreases. There is higher interpolation error in $A_{s}$ likely due to the
scarce sampling in the grid points and the limited bandwidth of the sampled
spectra considered here. As there is only three grid points in this parameter
space, there is a higher likelihood for interpolation error as there is more
distance between the given true value and the nearest grid point. For a more
in-depth description of this interpolation error and its impact, see S23.

In Figure 5 we present the shortest wavelength at which a weak or strong
detection is achieved for O2 and O3; as described in BARBIE1, this is an
important metric since the number of planets amenable to high-contrast
coronagraphic imaging is higher when observing at shorter wavelengths due to
the smaller inner working angle. Figure 5a provides a summarial result of
Figures 2 and 3, covering the full range of SNR from 3-20 at a 20% bandpass.
In Figure 5b, we present the shortest wavelength for strong detection if we
assume a 30% bandpass, while in Figure 5c we present the shortest wavelength
for strong detection if we assume a 40% bandpass. We also present the range
for a strong detection of H2O as in BARBIE1, with additional SNRs to 20, to
provide context to the results and present the possibilities for dual or
triple molecule detection. To show the full range, we shade out to the longest
wavelength where detection is possible.

Figure 5: In all panels, the shortest bandpass center at which one can achieve
a strong detection for O2, or a strong or weak detection for O3 are shown in
purple triangles, dark purple diamonds, and light pink dots respectively. The
strong detection range for H2O is presented in pink squares, to provide
context for dual or triple molecule detection. The range between the shortest
and longest bandpass center at which detection is possible is filled in. We
present the range for a strong detection SNR is on the y-axis, and the
bandpass centers are on the x-axis. Note that the last bandpass center for
strong detection is shaded out to the long edge of the bandpass, to maintain
ability to directly compare the panels.

In Figure 5a we can see that O3 is only detectable, whether weakly or
strongly, with high SNR ($\geq 14$) data over a very narrow range. We can see
that O3 can be detected (albeit weakly) simultaneously with a strong detection
of H2O at high SNR ($\geq 17$) with careful bandpass selection. However, the
high SNR required for detection indicates that this would be costly in terms
of observing time. Conversely, O2 is strongly detectable at an SNR of 8, with
the range between shortest and longest wavelength encompassing all O2 features
in the visible wavelength range. This detectability range overlaps
significantly with the strong detection range of H2O. We can see that O2 and
H2O can be simultaneously observed with SNR = 10 with careful selection of
wavelength and corresponding bandpass. This also allows for a range of
possible SNR depending on the desired wavelength of detection - if a longer
wavelength such as 0.83 m is accessible, then SNR = 10 would allow for a dual
detection, but if a short wavelength such as 0.68 m is required, then a much
higher SNR is required. Next looking to Figure 5b, we see the detectability
change, with triple molecule detection possible from a shortest wavelength of
0.66 m out to 0.7 m at an SNR $\geq 13$. O2 remains strongly detectable
beginning at an SNR of 8 as in Figure 5a; however, while the shortest
wavelength of detectability for O2 starts at approximately 0.69 m with a
bandpass of 20%, the shortest wavelength of detectability for O2 with a
bandpass of 30% starts at approximately 0.66 m. When we look to Figure 5c, we
see that detectability changes drastically, with triple molecule strong
detection possible from a shortest wavelength of 0.625 m out to 0.725 m at an
SNR $\geq 11$. We can also see that once again, O2 remains strongly detectable
beginning at an SNR of 8 as in Figure 5a and 5b, the shortest wavelength of
detectability for O2 changes once more, starting at approximately 0.625 m with
a bandpass of 40%. Thus, although the SNR of strong detectability does not
change, the minimum possible wavelength of strong detectability significantly
decreases as a function of bandpass width.

###  3.2 Results for Varying Abundance Cases

Figure 6: Shown above is the lowest SNR values for strong detection of the
0.74 m O2 feature as a function of O2 abundance. We note that although the
0.1% and 1% modern O2 abundances (i.e. mid-Proterozoic abundances) are in our
study, they are not presented on the plot. We denote these abundances with a
pink arrow. The VMR values are on the x-axis, with SNR on the y-axis. Strong
detection is shown in purple squares, weak detection is shown in pink dots,
the modern abundance range is highlighted with a light pink strip.

At this point in our study, we shift to present our abundance case study,
wherein we vary the abundance of O2 below modern-Earth values and O3 above
modern-Earth values. To assess the trade-off between longer observations
(i.e., higher SNR) and different concentrations of O2 and O3, we varied the
SNR on the observations for the full range of different VMRs per molecule. In
Figure 6a, 6b, and 6c, we display the minimum SNR required to achieve a strong
detection for each abundance of O2 at 0.76 m at 20%, 30%, and 40% bandpasses
respectively. Looking first to Figure 6a, we notice that O2 quickly requires
mid- to high-SNR data to be strongly or weakly detected with even one order of
magnitude decrease in abundance. In fact, many of the abundances in our
simulation are not detectable at all in this SNR range. The Proterozoic
abundances of O2 (0.1% to 1% of modern Earth abundance) are not detectable at
any SNR $\leq 20$, and in fact will likely require an extremely high SNR to
become detectable, as these values are two to three magnitudes lower than
modern Earth abundance values. As seen in Figure 6b, varying the bandpass to
30% decreases the required SNR for detection across almost all abundances of
O2. For instance, where at a 20% bandpass it requires an SNR of 19 to strongly
detect O2 at 2.1$\times 10^{-2}$ VMR, with a 30% bandpass the required SNR
drops to 18. This is true for all detectable abundance cases except modern
Earth values, which consistently requires an SNR of 8 for strong detection. In
Figure 6c, we do not see a difference in required SNR for strong detection
with a 40% bandpass.

While bandpass width makes little difference to the detectability of O2, it
makes a significant difference when detecting O3. We can see in Figure 7 the
large impact of the change in bandpass width. With a 20% bandpass centered on
the 0.76 m O2, we capture the entirety of the O2 feature, along with the two
smaller H2O features at 0.74 and 0.84 m. When the bandpass is widened to 40%,
we capture all off the above, along with a portion of both the O3 feature that
peaks at approximately 0.63 m and the 0.9 m H2O feature. In Figure 8a, 8b, and
8c, we display the minimum SNR required to achieve a strong detection for each
abundance of O3 at 0.64 m with 20%, 30%, and 40% bandpasses respectively. In
Figure 8a, we can see that O3 is detectable, strongly and weakly, in the full
range of our simulation values. At modern Earth abundances, the required SNR
for a strong detection is high at 19, but it is possible to achieve a weak
detection at SNR = 14. When we increase to high O3 values, approximately 5x
higher in abundance than modern Earth, the required SNR drops drastically,
with strong or weak detection requiring SNRs of 7 or 5 respectively. The
largest decrease in required SNR occurs between modern Earth values (7$\times
10^{-7}$ VMR) and 1.25$\times 10^{-6}$ VMR, dropping steeply from a required
SNR of 19 to 11 for a strong detection, and 14 to 9 for a weak detection. When
we increase the bandpass width to 30% as in Figure 8b, we see the required SNR
for detection drop across all abundance values. For instance, at 1.5$\times
10^{-6}$ VMR, the required SNR for strong detection at a 20% bandpass is 10,
with a 30% bandpass the required SNR drops to 8. This is true for all values
across the abundances cases, to varying degrees of severity. This happens once
more when the bandpass is widened to 40% as in Figure 8c. Looking to the same
example case of 1.5$\times 10^{-6}$ VMR, the required SNR for strong detection
drops to 6 at a 40% bandpass. With each increase of bandpass width, strong
detectability at mid-low SNR data becomes more accessible, with a required SNR
as low as 4 for a strong detection at 3$\times 10^{-6}$ VMR.

Figure 7: Spectra with varied O3 abundances. We feature the highest abundance
(3$\times 10^{-6}$), the mid point abundance (1.5$\times 10^{-6}$), and lowest
abundance in the study (7$\times 10^{-7}$) (modern Earth), respectively. All
spectra are binned at R=140. We also present a 20% and 40% bandpass range
width, in light grey and light blue respectively. Figure 8: Shown above is the
lowest SNR values for strong detection of the 0.63 m O3 feature as a function
of O3 abundance. The VMR values are on the x-axis, with SNR on the y-axis.
Strong detection is shown in purple squares, weak detection is shown in pink
dots, and the modern abundance range is highlighted with a light pink strip.

###  3.3 Self-Consistent Chemistry Comparison

Parameter Symbol | Fiducial | New Values
---|---|---
$\mathrm{C_{f}}$ | 0.5 |
H2O | 3$\times 10^{-3}$ VMR |
O3 | 7$\times 10^{-7}$ VMR | 105%, 110%, 105% PAL
O2 | 0.21 VMR | 10%, 50%, 75% PAL
$\mathrm{P_{0}}$ | 1.0 Bar |
$\mathrm{g}$ | 9.8 m/s |
$\mathrm{A_{s}}$ | 0.3 |
Table 3: New input data spectrum values for the chemically consistent
retrievals, following the Atmos values found in Kozakis et al. (2022) for O2
and O3 Figure 9: The shortest bandpass center at which one can achieve a
strong detection for O2 or O3 at varying present atmospheric levels (PALs) for
a specific SNR are shown in purple triangles and dark purple diamonds
respectively. The range between the shortest and longest bandpass center at
which detection is possible is filled in. The strong detection range for H2O
is presented in pink squared to provide context to the results. Figure 10:
Shown above are the regions of single, dual, and triple molecular detection as
a function of wavelength. wavelength is presented on the x-axis, SNR is
presented on the y axis. The hatched regions symbolize regions with single
molecule detection, shaded regions symbolize only the overlapping areas
between dual molecular strong detection zones, and the black outlined shaded
regions symbolize the overlapping areas between triple molecular strong
detection. We present the 10% O2 PAL and 105% O3 PAL case in the first row,
the 50% O2 PAL and 110% O3 PAL case in the second row, and the 100% O2 PAL and
100% O3 PAL case in the last row. For all cases, we present a 20%, 30%, and
40% bandpass width.

As previously discussed, in our previous simulations we varied one molecular
abundance (for O2) while the other parameters were held fixed to modern Earth-
like values, thus we did not explore the relationship between varying both O2
and O3 and the resultant change in detectability. However, due to the
photochemical relationship between the O2 abundance and the production of O3,
the abundances of both molecules would actually be linked in any atmospheric
scenario. As found in Kozakis et al. (2022), the relationship between O2 and
O3 varies for model atmospheres depending on the type of host star, and the
linearity of the relationship also appears to vary. When looking to hotter
host stars (e.g., a G2 star), peak O3 abundance occurs at lower than modern
Earth O2 abundances due to the O3 layer shifting in the atmosphere downwards
to O2 levels due to O2 photochemical shielding.

This O2/O3 relationship means that we can constrain the overall molecular
oxygen abundance by detecting either of the species; O3 is in fact a highly
sensitive tracer of lower abundances of O2. In order to explore the impact of
this, we examined several scenarios where we varied our parameter values as
coupled parameters, i.e. 10% PAL O2/105% PAL O3, etc., consistent with the
results for a Sun-like star from Kozakis et al. (2022) (as seen in Table 3).
We present our detectability results for these abundance pairs in Figure 9. In
Figure 9a-c, we present the O2 results with H2O at 20%, 30%, and 40%
bandpasses at 10% PAL, 50% PAL, and 75% PAL of O2. We can see that at 10% PAL,
O2 detectability is unlikely, requiring a minimum SNR of 19 at a 20% bandpass,
and SNR of 18 at 30% and 40% bandpasses. However, between 50% PAL, 75% PAL,
and the previously presented 100% PAL of O2, there are little difference, with
slight variance in SNR (e.g. from requiring SNR of 8 for 100% PAL O2 to
requiring SNR of 10 for 50% PAL O2) and little change across bandpass width as
in Figure 5. In Figure 9d-f, we present the O3 results with H2O at 20%, 30%,
and 40% bandpasses at 105% PAL and 110% PAL of O3. We can see that
detectability does not shift significantly for O3, as our values do not vary
greatly from a modern-Earth like value. However, one notable difference is
that there is a larger range for strong O3 detection at both 105% and 110% O3
PAL with bandpass width 20% than the same bandpass width with 100% O3 PAL as
in Figure 5. We also see that O3 is detectable at an SNR of 17 rather than 19
as in the modern-Earth like results for a 20% bandpass. The detectability
range of O3 increases with increasing bandpass width, as expected following
Figure 5.

Following this, we analyze the limiting molecule in double or triple molecular
detection in Figure 10. We present 10% O2/105% O3 PAL in Figure 10a-c. We can
see that detection of O2 dually or triply with H2O or O3 is difficult. At a
20% bandpass, H2O and O2 are dual detectable only at an SNR $\geq 19$. At a
30% bandpass, triple H2O/O2/O3 detection is possible at an SNR $\geq 18$ in a
very narrow wavelength range (approximately 0.66 to 0.7 m), and dual H2O/O2
detection is also possible at an SNR $\geq 18$, while dual H2O/O3 detection
has a wider range of detectability, from SNR $\geq 14$, and from approximately
0.63 to 0.7 m. At a bandpass of 40%, the range for triple detection increases
in wavelength, to approximately 0.63 to 0.76 m. The range for dual H2O/O3
detection also increases, starting from SNR $\geq 11$ and from 0.63 to 0.75 m.
At all bandpasses, H2O has a large range in wavelength and SNR of single
detectability thus it is highly likely to strongly detect H2O. In all
bandpasses, O2 is the limiting molecule for dual or triple detection in both
SNR and wavelength. In terms of dual H2O/O3 detection, O3 is the limiting
molecule in wavelength, but H2O is the limiting molecule in SNR.

We present 50% O2/110% O3 PAL in Figure 10d-f. We can see that, as in the
modern Earth case, dual detection of H2O and O2 is always possible at all
bandpasses. The same is not true for H2O and O3 however, with dual strong
detection possible at only a single point in the 20% bandpass, and with a
wider range possible at 30%. At 40%, there is no dual detection of H2O and O3,
however there is a large range (approximately 0.12 m in width) wherein triple
detection is possible from SNR $\geq 11$. At 30% and 40% bandpasses there are
also areas where dual detection is possible with O2 and O3 at mid-high SNR and
short wavelengths. With a 20% bandpass, H2O is the limiting molecule for dual
H2O/O2 detection in both SNR and wavelength. At a 30% bandpass, H2O is still
the limiting molecule for dual H2O/O2 detection and triple H2O/O2/O3 detection
in SNR, however O3 is the limiting molecule in wavelength. When looking at
dual H2O/O3 detections, H2O is the limiting molecule in SNR and wavelength,
mimicking the dual H2O/O2 detection. At a 40% bandpass, O3 is the limiting
molecule in wavelength for a triple H2O/O2/O3 detection, and H2O is the
limiting molecule in SNR for a triple H2O/O2/O3 detection. We then have the
same H2O limiting for dual H2O/O2 detection in SNR, although there is no
limiting molecule for dual detection in wavelength since the final bandpasses
have detections for both H2O and O2.

We present 100% O2/100% O3 PAL in Figure 10g-i. In the 20% bandpass, we see
extremely similar results to Figure 10d, with the notable difference that O2
detection is possible down to an SNR of 8, compared to an SNR of 10. In the
30% bandpass, we again see almost identical results to the 50% O2/110% O3 PAL
case shown in Figure 10e. The differences here are once again that O2 is
detectable down to an SNR of 8, but we also notice that O3 has a smaller
region of single detectablity, which is also seen in the smaller region of
dual O2/O3 detection. As a result of the lower SNR requirements for O2
detectability, this results in a large region of dual H2O/O2 detection. In the
40% bandpass, once again the results mimic the 50% O2/110% O3 PAL counterpoint
case shown in Figure 10f. The differences shown are a smaller region of triple
H2O/O2/O3 detection and dual O2/O3, resulting in larger single O2 and dual
H2O/O2 detection regions. This also results in a smaller single H2O detection
region. In these cases, the limiting molecules are the same as above: At a 20%
bandpass, H2O is the limiting molecule for dual H2O/O2 detection in both SNR
and wavelength. At a 30% bandpass, H2O is the limiting molecule for dual
H2O/O2 detection and triple H2O/O2/O3 detection in SNR, however O3 is the
limiting molecule in wavelength. At a 40% bandpass, O3 is the limiting
molecule in wavelength and SNR for a triple H2O/O2/O3 detection.

##  4 Discussion

The primary utility of the retrieval results presented here is to help
understand how the impact of O2 and O3 abundance affects the optimal strategy
for detecting the presence of an atmosphere with water and oxygen with
spectroscopic bandpasses in the visible-light spectral region. Here we discuss
the major conclusions from this work, as well as some of the limitations to
the study that may impact these conclusions.

The most important result is the role that the width of the bandpass plays.
When examining Figures 6 and 8, we can see the notable difference in how a
widening bandpass changes the detectability of molecules across abundance and
type; where O2 detectability does not vary significantly as a function of
bandpass across abundance values, O3 changes significantly as the bandpass
widens. This is likely due to the fact that the O2 feature is narrow and deep,
thus making it easily detectable at high abundance values, but easily captured
in a smaller bandpass width, whereas O3 has a very broad feature thus the
widening of the bandpass allows for a stronger continuum and significant
change in detection as a function of bandpass. O3 also does not have any
strong features in the 0.515 – 1 m region, leading to inherent difficulty in
detection in this range. However, if we are able to observe the strong O3 band
at 0.36 m, detection would become stronger at lower SNR data. Recent works,
such as by Damiano et al. (2023), have investigated this principle and come to
the conclusion that even with impacts of possible confusion due to SO2 and
other species, O3 detection is indeed easier at lower SNR and lower resolution
data. Damiano et al. (2023) found that with spectral resolution of 7, and SNR
= 10, O3 can be detected in the UV at sub-PAL VMRs; they also note that
observing additional signatures of habitability in the NIR region is crucial
to interpreting O3 detections at UV and VIS wavelengths.We will examine these
questions further in future work addressing the complete UV, VIS and NIR
wavelength regions.

A wider bandpass also better enables detections of one or more molecules with
a single bandpass - in particular, the detection of O3 with other molecules.
In Figure 5, we can see that for modern Earth abundances a bandpass width of
$\geq 30\%$ would allow for a detection of H2O at slightly shorter wavelengths
(long-wavelength cutoff of $\sim$0.95 m versus 0.99 m for a 20% bandpass) and
also a joint measurement of a strong O2 detection and at least a weak O3
detection with SNR = 8-9 versus only O2 (which could help to limit false
positives and motivate additional observations). Furthermore, Figure 10 shows
that a 40% bandpass width enables a significant improvement in the ability to
detect all three species with a single bandpass at around 0.72 m; this could
either be used alone as a first reconnaissance bandpass or could be used as
the follow-up to a longer-wavelength search for H2O alone at low SNR. The
optimal choice between these two options depends significantly on whether a
planet is detectable at longer wavelengths (due to IWA) and whether the
exposure time required for longer-wavelength measurements is significantly
impacted by instrument sensitivity; we leave these instrument- and target-
dependent considerations to future work.

The second important result is that the abundance versus SNR results in
Figures 6 and 8 show a non-linear increase in the required SNR at smaller
abundances. Combined with the fact that SNR is linearly dependent on the
square of exposure time, this suggests that our exposure time will be
extremely sensitive to the limiting abundance that must be detectable. Figure
10 further demonstrates this, showing that when O2 drops from 50% PAL (12%
VMR) to 10% PAL (2% VMR), it essentially becomes undetectable. This motivates
a deeper analysis of whether detecting O2 at visible wavelength should be a
high priority in the progression of measurements if there is a high likelihood
of a non-detection for even a moderate abundance, compared with a measurement
of more sensitive markers of atmospheric abundance in the UV or NIR.

We note that high O3 abundances are challenging to model, in that it takes a
very small amount to completely overwhelm the spectrum. Spectra with high O3
have continua near zero, which also causes the error value to increase
substantially, thus high values of O3 must be handled with care. At high
abundances, the error value increase can lead to poorly constrained retrievals
using our grid. Susemiehl et al. (2023) found that this occurs at high O3
($log_{10}\ce{O3}>-5$) concurrently occurring with low $\mathrm{P_{0}}$
($log_{10}\mathrm{P_{0}}<-1.7$) using our grid. Our simulations go no higher
than $log_{10}\ce{O3}\leq-6$ and therefore avoid this degeneracy and resulting
poorly constrained retrievals.

As discussed in BARBIE1, there have been similar prior works that investigated
the relationship between SNR and detectability, specifically Feng et al.
(2018) and Damiano & Hu (2022). We explore the differences in retrieval
parameterization structure in BARBIE1, however even with the differences in
techniques and analysis, our results for O2 and O3 detectability are in
agreement. An SNR of 10 at R = 140 is sufficient to firmly constrain the
abundances for an Earth-twin atmosphere with O2 and O3. Our work also explores
the influence of varying bandpass width on molecular detection, which varies
from prior work, and thus presents a new analysis of detection possibilities.

##  5 Conclusions & Future Work

To summarize, by investigating bandpass width in tandem with SNR, we can see
that detectability is intrinsically linked to both factors as an additional
function of molecular type. The ability to properly prioritize and select the
best combination of parameters can drive efficient observing practices. By
understanding the SNR requirements for strong detection of O2 and O3 and also
understanding which bandpass width could result in simultaneous strong
detection of O2, O3, and H2O, we can properly prioritize the best combination
of bandpass and required SNR for detection, thereby informing the best options
for instrument design trades and observing procedure. O3 is most easily
observable using a 30 or 40% bandpass width at shorter wavelengths with mid-
low SNR data. With a 20% bandpass width, O3 is difficult to detect, requiring
SNR $\geq 19$ at modern-Earth values. O2 is not significantly affected by
bandpass width, and consistently requires an SNR of $\geq 8$ to be strongly
detected at modern-Earth values at 0.76 m and shorter. It is not detectable at
SNR $\leq 20$ at Proterozoic era abundances. Since the coronagraph and
instrument capabilities may dictate that the best observation occurs at
shorter wavelengths, O2 and O3 are well within the most optimized instrument
capabilities. O2 and O3 have strong geochemical markers that provide a depth
of knowledge to potential Earth observations, and a heightened ability to
constrain the observed Earth-like epoch.

By also investigating coupled atmospheric abundances of O2 and O3, we can
study how detectability of these molecules vary with the other parameter.
Allowing parameters to vary with each other following leading coupled
photochemistry and atmospheric models, we can prioritize the optimal bandpass
width for more realistic exo-Earth simulations.

In future work, we plan to build a new spectral model grid that includes a
wider range of molecular species, and we will extend to shorter and longer
wavelengths than the visible range. Specifically, we will cover the same
wavelengths as the proposed coronagraph instrument for the Habitable Worlds
Observatory (Juanola-Parramon et al., 2022), including UV, Optical, and NIR.
This will allow us to study more biosignatures and molecular species of
interest, and represent the physical chemistries more accurately. This will
allow us to expand our simulations of possible observations and establish best
observational practices for exoEarth observations using next generation
telescopes. We will also develop a PSG module to display all detectability
information in a publicly accessible format. We note that SNR results may be
subject to minor changes following grid reconstruction due to PSG radiative
transfer upgrades. We will also include a full exoplanet yield calculation
using the data contained within BARBIE1 and BARBIE2 to give an educated
baseline for detection of biomarkers. The interpolation metric used in this
work is a trilinear interpolation scheme which can be used on 3D grids as in
this work, but in future works we will investigate if another interpolation
metric, such as inverse distance weighted interpolation or multiplicative
weight interpolation, can minimize interpolation error in grid-based
retrievals.

We will also conduct photochemically self-consistent studies in order to
inform and broaden the results for future retrieval studies using grids. The
expansion of grid parameter space will allow for the possibility of chemically
consistent modeling, combined with input from updated photochemical models to
ensure self-consistent atmospheric gas compositions with retrievals. This will
be particularly important for considerations of gas detections for planets
around different star types, given the impact star type has on the
photochemistry of planetary atmospheres (Segura et al., 2005, 2010; France et
al., 2012).

N. L. gratefully acknowledges financial support from an NSF GRFP. N.L.
gratefully acknowledges Dr. Joesph Weingartner for his support and editing. N.
L. also gratefully acknowledges Greta Gerwig, Margot Robbie, Ryan Gosling,
Emma Mackey, and Mattel Inc.™ for Barbie (doll, movie, and concept), for which
this project is named after. This Barbie is an astrophysicist! The authors
also gratefully acknowledge conversation with Dr. Chris Stark regarding
exoEarth yields and instrument design. The authors would like to thank the
Sellers Exoplanet Environments Collaboration (SEEC) and ExoSpec teams at
NASA’s Goddard Space Flight Center for their consistent support. MDH was
supported by an appointment to the NASA Postdoctoral Program at the NASA
Goddard Space Flight Center, administered by Oak Ridge Associated Universities
under contract with NASA.

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