# Bayesian Analysis for Remote Biosignature Identification on exoEarths
(BARBIE) I: Using Grid-Based Nested Sampling in Coronagraphy Observation
Simulations for H2O

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 Dane Moore NASA Goddard Space Flight Center, Greenbelt,
MD, USA.  Business Integra, Inc., Bethesda, MD, USA.  Sellers Exoplanents
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.  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  Michael D. Himes NASA
Postdoctoral Program Fellow, NASA Goddard Space Flight Center, 8800 Greenbelt
Road, Greenbelt, MD 20771, USA Natasha Latouf<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Detecting H2O in exoplanet atmospheres is the first step on the path to
determining planet habitability. Coronagraphic design currently limits the
observing strategy used to detect H2O, requiring the choice of specific
bandpasses to optimize abundance constraints. In order to examine the optimal
observing strategy for initial characterization of habitable planets using
coronagraph-based direct imaging, we quantify the detectability of H2O as a
function of signal-to-noise ratio (SNR) and molecular abundance across 25
bandpasses in the visible wavelength range (0.5-1 m). We use a pre-constructed
grid consisting of 1.4 million geometric albedo spectra across a range of
abundance and pressure, and interpolate to produce forward models for an
efficient nested sampling routine, PSGnest. We first test the detectability of
H2O in atmospheres that mimic a modern-Earth twin, and then expand to examine
a wider range of H2O abundances; for each abundance value, we constrain the
optimal 20% bandpasses based on the effective signal-to-noise ratio (SNR) of
the data. We present our findings of H2O detectability as functions of SNR,
wavelength, and abundance, and discuss how to use these results for optimizing
future coronographic instrument design. We find that there are specific points
in wavelength where H2O can be detected down to 0.74 m with moderate-SNR data
for abundances at the upper end of Earth’s presumed historical values, while
at 0.9 m, detectability is possible with low-SNR data at modern Earth
abundances of H2O.

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

##  1 Introduction

Figure 1: The full modern-Earth high-resolution spectrum from the spectral
grid in Susemiehl et al. (accepted), with a zero cloud fraction. We show the
full resolution (R) of 500 (shown in pink, top panel), the spectrum binned to
R=140 that are used in the nested sampling simulations conducted in this work,
(shown in purple, middle panel), and a spectrum at R=70 for reference (shown
in dark purple, bottom panel). We also present the bandpass centers used in
this study as dots (only in the top panel), and the first and last 20%
bandpasses shaded in grey. We note that bandpasses overlap heavily with each
other, so we only show the first and last for clarity.

We have seen the discovery and confirmation of thousands of exoplanets since
the first planet found orbiting a Sun-like star almost 30 years ago (Mayor &
Queloz, 1995), and we are now on the verge of entering an exciting new era of
exoplanetary exploration: detection and characterization of terrestrial
exoplanet atmospheres. Characterization of an exoplanet’s atmosphere can
provide vital information on the formation, history, and physical composition
of the planet; for potentially habitable planets, we can also search their
atmospheres for biosignatures such as H2O, O2, and O3 that can hint at the
likelihood of clement conditions and biological activity (Kaltenegger, 2017;
Schwieterman et al., 2018). With current advances in instrumentation and
proposed future missions that are set to prioritize terrestrial exoplanet
atmosphere characterization (e.g., Roberge & Moustakas, 2018; The LUVOIR Team,
2019), the ability to find a habitable Earth-twin is becoming a realistic
prospect within the next few decades. The looming goal is to discover,
characterize, and analyze an Earth-twin terrestrial planet which could provide
insight into our own Earth’s history and formation and potentially evidence of
biological activity.

A true Earth-twin analog would occupy the habitable zone (HZ) of its host star
at such a distance that liquid water can exist at the surface (Kasting et al.,
1993; Kopparapu et al., 2013); hence, detecting H2O generally stands as the
first step in the search for habitability. Terrestrial planets in the HZ with
strong water detections are the prime habitable planet candidates and thus
will be priority targets for re-observation. However, observed Earth twins
could exhibit a range of H2O abundances; at different times in the Earth’s
history, the planet hosted varying concentrations of different atmospheric
biomarkers. Interestingly, little geologic evidence of the atmospheric H2O
abundance through Earth’s history is available. Even though other biomarkers
such as O2 and O3 have geochemical proxies that allow for better estimation of
varying epoch atmospheric abundances (Planavsky et al., 2014), the presence of
H2O in geochemical records does not constrain atmospheric abundance. This is
due to the fact that water is present throughout the Earth’s history and,
unlike O2 and O3, present in solid, liquid as well as gaseous phases. However,
general circulation models (GCMs) have been able to provide constraints on
Earth’s water vapor abundance through time using constrained parameter sets.

Parameter Symbol | Fiducial | Minimum | Maximum
---|---|---|---
$\mathrm{C_{f}}$ | 0.5 | N/A | N/A
H2O | 3$\times 10^{-3}$ VMR |  $10^{-8}$ VMR |  $10^{-1}$ VMR
O3 | 7$\times 10^{-7}$ VMR |  $10^{-10}$ VMR |  $10^{-1}$ VMR
O2 | 0.21 VMR |  $10^{-8}$ VMR | 0.8 VMR
$\mathrm{P_{0}}$ | 1.0 Bar |  $10^{-3}$ Bar | 10 Bar
$\mathrm{g}$ | 9.8 m/s | 1 m/s | 100 m/s
$\mathrm{A_{s}}$ | 0.3 | $10^{-2}$ | 1
Table 1: Modern Earth-like fiducial parameter values used in these
simulations, and in S23, along with minimum and maximum values.

Looking forward to next-generation instrument design, such as the
instrumentation for the upcoming Habitable Worlds Observatory (HWO),
efficiency is key to maximizing the science output from limited observation
time. With this in mind, the optimal wavelength for the spectral bandpass used
for measuring the reflected flux and atmospheric absorption is a crucial
factor to consider. Coronagraph designs for future exoplanet direct imaging
mission concepts generally feature two deformable mirrors (DMs) to control the
wavefront of light, along with a suite of focal plane masks (FPM) such as the
vortex coronagraph and the apodized pupil Lyot designs, both of which were
included for consideration in the LUVOIR and HabEx studies (Ruane et al.,
2015; The LUVOIR Team, 2019; Por, 2020; Juanola-Parramon et al., 2022). These
designs have an out-of-band wavefront sensor that takes reflected light from
the FPMs to sense pointing errors and wavefront drifts, but the accuracy of
the wavelength control degrades as the observed wavelength diverges from the
central wavelength. It is for this reason that when using a coronagraph, the
light must pass through a bandpass filter, which filters out light beyond a
defined range on either side of the central wavelength; light outside of this
range may not be well corrected by the optical system and thus degrade overall
performance. Currently, the width of bandpasses (i.e. 10, 20 or 30% of the
central wavelength) is an area of active technology development, but we chose
to adopt an intermediate value of 20% in our simulations.

There are multiple factors which affect the efficiency of atmospheric
characterization at different wavelengths. First, when we observe at shorter
wavelengths, we can observe the HZs of more distant and lower mass stars, as
the HZs for these targets are smaller in angular separation and the inner
working angle (IWA) of coronagraphs is proportional to wavelength. Second,
prioritizing shorter wavelengths also enables us to decrease astrophysical
noise terms, since the telescope point spread function (PSF) area is also
proportional to $\lambda$, and background noise scales with PSF solid angle,
thus allowing for less noisy observations. Third, the stellar SED peaks at 0.5
- 0.7 m for FGK-type stars; thus, positioning our bandpass effectively allows
for the observation of a higher photon flux, thus increasing the achievable
SNR with the same exposure time. However, H2O has an increasing number of deep
spectral features at longer wavelengths; thus, observing at longer wavelengths
would allow us to minimize the SNR needed to detect these features but would
decrease the photon flux and HZ visibility, while observing at shorter
wavelengths with shorter features would optimize observing efficiency but
would require a higher SNR. Quantifying the required SNR for H2O detection in
each wavelength region is therefore a crucial component to determining the
optimal observing strategy.

In order to quantify the SNR required to detect H2O for different spectral
bandpasses and different molecular abundances, we utilized a Bayesian spectral
retrieval methodology. Spectral retrieval studies have been used to explore
the detectability of atmospheric compositions for the direct imaging of
exoplanets using both the Roman Space Telescope (e.g., Lupu et al., 2016;
Nayak et al., 2017) as well as a future Earth-twin imaging mission similar to
HWO (Feng et al., 2018; Smith et al., 2020; Robinson & Salvador, 2022; Damiano
& Hu, 2022). However, traditional Bayesian retrievals using real-time
radiative transfer calculations require large amounts of computational runtime
to effectively explore multiple parameters and mission capabilities - and the
ongoing and future addition of more complex model features such as high-
accuracy aerosol calculations, surface albedo models, photochemical hazes,
etc. will only increase runtimes even more. With this in mind, more recent
methods to accelerate retrievals have been explored, such as ultra-efficient
radiative transfer schemes (Robinson & Salvador, 2022) and machine learning
(e.g., Zingales & Waldmann, 2018; Márquez-Neila et al., 2018; Cobb et al.,
2019; Fisher et al., 2020; Himes et al., 2022). In this study, we achieve an
extremely short computational time for Bayesian retrievals using a spectral
grid-based method, which was first described in Susemiehl et al. (accepted,
hereafter S23). In this framework, a grid of model spectra at defined
parameter values is pre-generated using a full high-fidelity radiative
transfer scheme, and forward models for the Bayesian inference algorithm are
then interpolated from the grid.

In this work, we constrain the detectability of H2O absorption in the spectra
of Earth-like exoplanets as a function of observational SNR and central
bandpass wavelength, in order to lay the groundwork for optimizing exoplanet
characterization observations with the Habitable Worlds Observatory. By using
the pre-computed grid by S23, we reduce runtime significantly and explore a
large span of parameter space (i.e. SNR by increasing signal and varying
molecular abundance) at little computational cost. We present the results of
this study and showcase the benefits to grid-based retrieval studies. In
$[\S]$ 2 we present the methodology in three parts; a summary of the grid-
building work done by S23, a summary of the Planetary Spectrum Generator and
the application PSGnest, and a thorough description of our simulations and
model parameters. 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

###  2.1 Spectral Grid

Figure 2: Comparison of spectra with varying H2O abundance. Shown are the
lowest and highest abundance in our simulations (3$\times 10^{-4}$ VMR and
3$\times 10^{-2}$ VMR) as well as the modern Earth value (3$\times 10^{-3}$
VMR). All spectra are binned at R=140. At the highest abundances, H2O spectral
features begin to overlap.

Our ability to perform thousands of nested sampling runs to retrieve
exoplanetary atmospheric parameters in a reasonable computational timeframe
relies heavily on the efficiency inherent in grid-based retrievals. Using a
pre-built grid allows for each retrieval to take on the order of seconds to
minutes.

For this study, we used the grid of rocky planet reflectance spectra built by
S23 using the the Planetary Spectrum Generator (PSG, Villanueva et al., 2018,
2022). PSG is a radiative transfer model and a tool for synthesizing and
retrieving planetary spectra, including atmospheres and surfaces for a wide
range of planetary properties and covering wavelengths from 50 nm to 100 mm
(i.e. UV/Visible/NIR/IR/FIR/THz/sub-mm/Radio). PSG includes aerosol, atomic,
continuum (i.e. CIAs, Rayleigh, Raman), and molecular scattering and radiative
processes; these are implemented with a layer-by-layer radiative transfer
framework. For this study and the building of the grid, the latest HITRAN-2020
(Gordon et al., 2022) molecular absorption parameters were used, implemented
in PSG with the correlated-k methodology. The HITRAN database also includes
collision-induced absorption (CIA) bands, and we characterize the MT_CKD water
continuum as H2O-H2O and H2O-N2 CIAs (Kofman & Villanueva, 2021)

S23 present a novel approach to grid-building that uses an adaptive grid point
approach. The development of the grid featured an algorithm which adds new
grid points iteratively at the location of highest interpolation error to
reach a specified error contribution based on the expected uncertainty of the
data that will be analyzed; the interpolation error is thus reduced at each
step by balancing between the peak computational complexity and peak
interpolation accuracy. The grids can then be used to produce forward models
which are input to the nested sampling routine
PSGnest111https://psg.gsfc.nasa.gov/apps/psgnest.php. PSGnest is a Bayesian
retrieval tool adapted from the Fortran Multinest retrieval algorithm (Feroz
et al., 2009), housed in the PSG code base and designed for application to
exoplanetary observations. Computation is accelerated via memory-mapping coded
in C, enabling ultra-efficient retrievals. The output contains highest-
likelihood values of output parameters, the average value resulting from the
posterior distribution, uncertainties (also estimated from posterior
distribution), and the log evidence (logZ) (Villanueva et al., 2022).

The current grid of geometric albedo spectra covers a wavelength range from
0.515–1.0 m, as this range has been defined as the VIS channel in the
conceptual exoplanet imaging instruments for future missions such as LUVOIR
and HabEx, and was designed to analyze data up to SNR=20 for a spectral
sampling of R=140. The grid uses a simpified set of planetary parameters
defining the atmospheric composition, with the range of values centered on
those of modern Earth as defined in Feng et al. (2018). These values consist
of constant volume mixing ratios (VMRs) of 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, and surface albedo ($\mathrm{A_{s}}$) of 0.3.
For our simulations, the pressure model-top was set to $10^{-4}$ bars with
$\mathrm{P_{0}}$ as surface pressure, and the planetary radius was fixed at
$\mathrm{R_{p}}$ = 1 $R_{\earth}$. The radius was fixed due to the intrinsic
degeneracy between the planetary radius $\mathrm{R_{p}}$ and a wavelength-
independent surface albedo for reflected-light clear sky models; this
degenerate effect on the total reflected light of the planet would mean that a
specific continuum geometric albedo spectrum could be interpreted as a large
planet with low overall albedo or small planet with a high albedo (an issue
also addressed in Feng et al., 2018). Specific complementary observations at
very short or very long wavelengths or (in rare cases) planetary transit
measurements may be able to mitigate this degeneracy and permit the
measurement of the absolute radius of the planet; however, for now we have
kept the radius fixed in our investigation of on an Earth-like twin, and
allowed for a varying surface albedo.

Cloudy versions of each spectrum were also created to complement the clear
versions of the same spectrum. The clouds were assumed to be wavelength-
independent with a 0.23509 ppm mass mixing ratio and a peak of 1m for particle
size distribution (corresponding to a surface optical depth of 10). We adopt a
generic isotropic scattering cloud model appropriate for an Earth-twin
observed at a distance, with a cloudiness fraction ($\mathrm{C_{f}}$) computed
by linearly combining a clear ($\mathrm{s_{clear}}$) and cloudy
($\mathrm{s_{cloudy}}$) spectrum:
$\mathrm{C_{f}}\times\mathrm{s_{clear}}+[1-\mathrm{C_{f}}]\times\mathrm{s_{clear}}$.

The total grid consists of 1.4 million spectra with parameter points for the
parameters laid out in the left column of Table 1, along with the minimum and
maximum grid values as in Table 1 of S23; the full set of data files for the
grid are currently available for use within the PSG nested retrievals
application222https://psg.gsfc.nasa.gov/apps/psgnest.php. The minimum and
maximum grid values are also used as the priors in our simulations. The grid
was generated using API calls to a distributed configuration of PSG called
GridRunner housed on Goddard Space Flight Center’s (GSFC) local Explore cloud
computing cluster. In future work, the grid will be rebuilt introducing an
increased number of parameters and a wider wavelength range; however, the
current pre-built grid allowed for a rapid analysis and is currently available
for use on PSG.

###  2.2 Simulated Retrievals

To explore the optimal bandpass for H2O detection as function of SNR, we chose
25 evenly-spaced bandpass centers across the full grid wavelength range of
0.515–1.0 m. We chose the fiducial parameter set shown in Table 1 with a
bandwidth of 20% and R$=$140 to be a baseline for comparison. In practice, the
bandpass widths are approximately 19% due to rounding to fully include or
exclude the discretized points in the R=140 spectra; however, coronagraphy
design for the future HWO observatory is not yet decided and could vary in
bandpass width as discussed above. The clear versions of the two spectra are
presented in Figure 1 to directly compare the feature smoothing between
spectra, along with the resolving power of 70 for reference. One can see that
there is loss of smaller features and smoothing of minutia in deeper features,
such as the 0.9 m H2O feature as the resolving power decreases. The bandpass
centers are presented in the top panel to show the method of visualizing the
data, as well as the first and last 20% bandpasses. It is evident that the
bandpasses overlap significantly with each other, resulting in a very thorough
study of coronagraphy bandpass possibilities.

Figure 3: Results of the analysis for the fiducial abundance (3$\times
10^{-3}$) for H2O, as an example case. Each dot represents a bandpass center,
with the pink line portraying the median retrieved values and the gray shaded
region representing the upper and lower limits for the 68% credible region;
the true value is shown with the black dashed line. Increased statistical
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.

There were 400 live points used in PSGnest, and an evidence tolerance of 0.1.
For each retrieval performed, the median values as well as the upper and lower
limits of the 68% credible region were calculated by inferring the 1D
marginalized posterior probability density function (PDF) using a kernel
density estimation (KDE), and then integrating the PDF top-down until summing
68% of it (Harrington et al., 2022). We also calculate the Bayes factor for
each bandpass retrieval to confirm constraints and detection strength for each
retrieved atmospheric constituent. The Bayes factor is calculated by
subtracting the Bayesian log-evidence per retrieval using the previously-
assumed gas abundances (Trotta, 2008); the differences in log-evidences is the
log-Bayes Factor, $\mathrm{lnB}$ (Benneke & Seager, 2013). If $\mathrm{lnB}$
is less than 2.5, it represents an unconstrained detection for that data
simulation; if $\mathrm{lnB}$ is between 2.5 and 5 a weak detection is
represented, and if $\mathrm{lnB}$ is greater than 5 a strong detection is
represented (reference Table 2 of Benneke & Seager, 2013). For our
simulations, the comparison occurs between a retrieval with all parameters
included and retrieved versus a retrieval with one abundance parameter (the
parameter of interest) removed from the retrieval and the abundance set to
zero. The 68% credible region contains the true value 68% of the time, whether
or not the molecule is present, whereas $\mathrm{lnB}$ directly examines if an
improved fit occurs with the molecule included. Thus, we prioritize
$\mathrm{lnB}$ as it directly addresses the likelihood of the presence of the
molecular species and its detectability. The log-Bayes Factor is not
calculated for non-gaseous parameters as the values are not associated with a
presence or absence of evidence.

Figure 4: Results of the analysis for the fiducial abundance for H2O, but for
a narrower SNR range. All facets of the plot remain the same as in Figure 3.
Detectability at wavelengths below 0.84m occurs for SNR$>$11.

We also examined a range of assumed signal-to-noise ratios (SNRs) for the data
set. Since exoplanets are so much fainter than their host stars (i.e. 10
orders of magnitude for the Earth-Sun system), observations are going to be
extremely challenging and likely require long observing times. Signal
increases linearly with time, whereas noise increases with the square root of
the time (in the source-noise-dominated regime). The combined atmospheric
signal across all wavelengths will have to be much stronger than the standard
deviation of the noise in order be able to derive any meaningful constraints
on the planet’s atmosphere. Thus, we delve deeply into the study of SNR and
the derivable constraints and detections as signal increases. The noise term
for the SNR here refers to the standard deviation of the noise at all pixels
and is added as Gaussian noise after the simulations of the spectra to simply
mimic SNR in real observations. Similar to Feng et al. (2018), who studied an
SNR range of 5, 10, 15, 20, we studied a preliminary broad range (3 - 16 in
steps of 1), and then added smaller SNR steps (8.5, 9.5, 10.5, 11.5). We
explore a broader range of SNR to thoroughly investigate the change in H2O
detectability and the associated detection strength, specifically between SNRs
of 10 and 12.

To add another dimension to our study, we explored the effect of varying
molecular abundances on detectability as a function of SNR. Figure 2 portrays
how varying the abundance of H2O influences the spectrum. One can see that as
the abundance of H2O increases, the feature depth also increases, as expected.
The lowest abundance of 3$\times 10^{-4}$ VMR has very shallow features, with
increasing depth moving up to 3$\times 10^{-2}$ VMR, the highest abundance. To
finely explore the influence of each molecule, we only vary the abundance of
the molecular species of interest and maintain the other components at the
modern Earth-like values as presented in Table 1. We vary our molecular
abundances by stepping through log-space, increasing or decreasing by 0.25
and, at larger values, 0.5. We center around the modern Earth-like values for
each molecule, to provide a familiar starting point. This also means that
although the physical chemistry of each model is not accurate to any specific
epoch of Earth, we can match the individual abundances as closely as possible
to various epochs of interest. Our abundance values range from 3$\times
10^{-4}$ to 3$\times 10^{-2}$ VMR.

For the purposes of this study, we will be focusing on our results for the H2O
abundance study, with a mention of the preliminary results found for O2. We
note that the absolute SNR results are subject to change with newer grid
construction, due to upgrades in PSG and the addition of more parameters and
molecules; the results of this study are therefore most valuable for
understanding the relationships between bandpass wavelength, SNR and H2O
abundance with regard to detectability. Future work will explore the impact of
more physically realistic atmospheric compositions and the retrieval of
additional atmospheric parameters, and we will produce updated spectral grids
accordingly.

Each case study (i.e. every SNR in the previous range applied to each
abundance study) took approximately 5 minutes to run on the GridRunner
application of PSG; comparatively, each case study on a local machine (MacBook
Air 2017, 1.8GHz dual-core Intel Core i5) took approximately 1.5 to 2 hours.
Each case study consists of thousands of spectra to be retrieved upon, thus
GridRunner massively accelerated our progress and saved days of time.

##  3 Results

###  3.1 Modern Earth Case Results

We begin by presenting the detectability of H2O as a function of SNR for the
fiducial modern Earth case, as first examined in S23. We present a portion of
the SNR range in Figure 3; all of the H2O data and calculated log-Bayes Factor
across abundance, SNR, and wavelength are available to the community on
Zenodo33310.5281/zenodo.7897198. As SNR increases, the detectability of
smaller H2O spectral features at shorter wavelengths (such as the 0.74 m
feature) improves; this is evident in the narrowing of the 68% credible region
for the strong detection points in Figure 3. There is a significant change in
detectability at shorter wavelengths between an SNR of 8 and an SNR of 10 — at
SNR = 8 we achieve a strong detection at 0.9 m but it is unconstrained at all
other points, while for SNR = 10 we find weak detections down to 0.71 m. There
is also a tipping point of diminishing returns at approximately SNR = 14,
after which there is no change in the bandpasses where H2O is detectable and
very slight changes in the sizes of the credible regions. There is a notable
shift in detectability between SNR = 11 and SNR = 12, wherein the 0.74 m water
feature goes from a firmly weak detection to a firmly strong detection with a
narrower credible region. Thus we investigated further with smaller SNR steps
in this transition region to fully understand the shift in detectability,
portrayed in Figure 4; for example, at SNR = 11.5 the 0.74 m bandpass center
begins to yield a strong detection of H2O where previously it was a weak
detection.

Figure 5: Summary of the bandpasses where H2O and O2 are detectable for our
fiducial Modern Earth scenario, based on results as shown in Figures 3 and 4.
The shortest bandpass center at which one can achieve a strong or weak
detection for H2O, or a strong detection for O2, are shown in dark purple
squares, pink dots, and light purple triangles, respectively. SNR is on the
y-axis, and the bandpass centers are on the x-axis. We also show the range out
to the longest wavelength at which the same detection is achieved as shaded
regions. We note that a strong detection of both H2O and O2 can be achieved
with bandpasses at or below 0.83 m and SNR$\geq$10.

As we discussed in §1 and is demonstrated in Figures 3 and 4, the minimum
wavelength for detection is directly related to the SNR of the data, but
longer wavelengths also yield detections due to the increasing depth of the
H2O spectral features. We summarize this result in Figure 5, where we plot the
positions of the minimum bandpass where either a strong or weak detection of
H2O can be achieved for a specified SNR; the results show an almost linear
relationship between the shortest detectable bandpass center (at either weak
or strong detection) and the SNR value required. We note that there are no H2O
detections for wavelengths shorter than 0.66 m since there are no water vapor
features available shortwards of this bandpass, and for SNR $>$ 14 we are
strongly detecting H2O at all wavelengths longer than 0.65 m. To show the full
range of detectability, we shade the region out to the longest wavelength at
which a detection is achieved. H2O is always strongly detectable at the 0.9 m
feature, leading to larger likelihood of H2O detectability as we move to
higher SNRs. The weak detection of H2O shows a similar result, with 0.9 m
weakly detectable to the lowest SNR in our study. For comparison, we also show
a similar analysis for O2, which only has one strong spectral feature at 0.76
m in our spectral range; therefore above SNR = 8, the longest and shortest
wavelengths at which O2 is detectable are constant.

Figure 6: Three corner plots, one for each notable H2O feature, with one
corner plot at very high SNR to portray the intrinsic parameter degeneracies.
Each has the same parameters with SNR of 12, shown in Table 1, with the
exception of locking $\mathrm{C_{f}}$ and an SNR of 200 for corner plot d. In
all corner plots, the 68% credible regions are 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.

We note that as we approach the longest wavelengths in the visible range, we
notice that the true value of H2O is not retrieved, but rather finds a higher
abundance of H2O than the true value. This is due to a degeneracy with
pressure in the last three bandpasses driven by a lack of continuum due to the
depth of the water features; in other words, the water absorption region is
saturated and wide enough that the continuum cannot be correctly retrieved.
This is confirmed by investigating the corner plots made for bandpasses
covering different H2O spectral features.

In Figure 6a, which shows the corner plot for the retrieval results for the
0.74 m bandpass, the high-probability region is perfectly aligned with the
true values for $\mathrm{P_{0}}$ and H2O; in contrast, we see in Figure 6b
that the 0.86 m bandpass fares slightly less well compared to the 0.74 m
bandpass, with the high-probability regions not as well constrained, but still
aligned in part to the true values. However, the high-probability region here
is much more elongated, still leading to uncertainty and inaccurately
retrieved values, with the true values found at the edges of the 1D 68%
credible regions. Comparing these figures to the corner plot for the 0.9 m
bandpass in Figure 6c, the difference is noticeable. The high-probability
region shown between surface pressure $\mathrm{P_{0}}$ and H2O is distanced
away from the true value, which is found to be on the tail of the high
probability region and again at the edges of the 1D 68% credible regions; this
leads to a poorer constraint on H2O in selected bandpasses with longer
wavelengths, despite the fact that they cover the largest water feature in the
visible wavelength range, with a much better constraint at shorter
wavelengths. Interestingly, the lack of continuum and related degeneracy does
not affect the ability to detect H2O at any wavelength, nor does it affect the
strength of the detection, as we can see from Figures 3 and 4. It is also
important to note that there is also a degeneracy with albedo
($\mathrm{A_{s}}$) throughout the full spectrum, which leads to a three-way
degeneracy at longer wavelengths.

Another way to investigate this degeneracy is to effectively remove the impact
of spectral noise in order to investigate the impact of parameter degeneracies
alone. We increased the SNR to 200, thus minimizing posterior broadening due
to SNR, and locked $\mathrm{C_{f}}$ to examine whether that is a factor in
driving H2O from the true value. We present the corresponding corner plot in
Figure 6d (note that the axis ranges have changed in order to visualize the
degeneracies). Without the noise introduced by low SNR, we can vividly see the
degenerate relationships between variables, such as $\mathrm{P_{0}}$ and H2O;
although both are well within 1-$\sigma$ of the true values, there are still
elongated probability regions. O2 is poorly constrained, as expected due to
the absence of the molecule in this region, while gravity becomes very well
constrained at high SNR without $\mathrm{C_{f}}$ interference. The impact of
variations in cloud coverage on H2O detectability was not investigated in
detail in this work, but earlier work such as Kawashima & Rugheimer (2019) has
shown that clouds can increase the detectability of atmospheric species due to
the increase in overall planet brightness; however, for H2O this is only
relevant for extreme cases of high cloud coverage at high altitudes, as the
0.9 m H2O feature has significant depth regardless of cloud coverage.

Figure 7: Default spectrum used for simulations, plotted with the high-
probability values as drawn from the corner plots and posterior values of
$\mathrm{P_{0}}$ and H2O. The top panel shows a zoomed-in view of the longer
wavelength ranges, and the bottom panel is identical but zooms in to the
shorter wavelengths. Each portray the three bandpasses that are most centered
with the features of interest. The presence of O2 in the short-wavelength
region helps to break the degeneracies between multiple parameters.

To further illustrate this degeneracy, Figures 7a and 7b show the full-
wavelength spectrum with all the input modern-Earth values, plotted with a
spectrum with $\mathrm{P_{0}}$ and H2O replaced with the maximum probability
values shown in Figure 6c (approximately -0.4 for $\mathrm{P_{0}}$ and -1.6
for H2O in log-space). Each figure also presents the three bandpasses most
central to the wavelength of interest. One can clearly see that the maximum
probability spectrum mirrors the spectrum in the 0.9 m section almost
perfectly, which is what one would expect following the corner plot. However
in the 0.74 m region the spectrum fits poorly, specifically in the regions
with O2. This contribution of O2 allows the code to differentiate and break
the $\mathrm{P_{0}}$ and $\mathrm{A_{s}}$ degeneracies. Thus we see the
results presented previously, wherein abundances are well constrained in
shorter wavelengths (with more molecular species present) in contrast to
longer wavelengths (where there is only water present). This occurs at all
SNR, however the credible region changes as SNR increases, and we see that the
true value of H2O shifts to the edge of the credible region for longer
wavelengths, most notably at an SNR of 14, and at higher SNRs is shown to
still be retrieved at the edge of the 1D 68% credible region in Figure 6d. As
such, although we can strongly detect H2O at longer wavelengths with low SNRs,
the trade-off is accuracy in abundance measurement. However, the insufficient
credible region could also be affected by the known deficiencies in Multinest
(Buchner, 2016; Ardévol Martínez et al., 2022; Himes, 2022); we will explore
this possibility in more depth in future work.

###  3.2 Results for Varying Abundance Cases

At this point in our study, we shift to present our abundance case study,
wherein we vary the abundance of H2O above and below modern-Earth values. To
assess the trade-off between longer observations (i.e. stronger signals and
lower noise) and different concentrations of H2O, we varied the SNR on the
observations for the full range of different H2O VMRs. As examples, in Figures
8a and 8b we present two heat maps summarizing the H2O detectability for the
0.74 m bandpass and the 0.9 m bandpass. Both color bars cover the full range
of results between 0 and 5 log-Bayes factor, to represent the change in
detection strength from weak to strong. Focusing first on Figure 8b, we can
see that for almost all abundances of H2O it is possible to achieve a strong
detection. At a VMR of 3$\times 10^{-4}$, it requires an SNR = 12 to achieve a
strong detection. There is a significant improvement beyond VMR = 1$\times
10^{-3}$, thus resulting in a strong detection at SNR of 7. Thereafter, the
SNR required for strong detection decreases linearly as abundance rises. We
can see that as a general pattern, detection using the 0.9 m bandpass is
achievable for a wide range of abundances at relatively low SNR values.
Looking now to Figure 8a, we see a far different story — strong detection for
the 0.74 m bandpass does not occur until a VMR of 1.7$\times 10^{-3}$, and
that occurs at SNR = 16. It is at this point that we see the same linear
progression as in the top panel, with decreasing SNR and increasing abundance,
until we once again study the highest abundance tested in the study (3$\times
10^{-2}$ VMR) and we see a strong detection occur at SNR of 5.

Figure 8: Heatmap plots illustrating detection strength as a function of SNR
and varying H2O abundance. SNR is on the y-axis, H2O abundance is on the
x-axis, and the colorbar shows the range of log-Bayes Factor ($\mathrm{lnB}$)
from 0 to 5 to describe detection strength. $lnB<2.5$ are unconstrained,
$2.5\leq lnB<5$ are weak, and $lnB>5$ are strong, as in prior figures.

In Figure 9 we display the minimum SNR required to achieve a strong detection
for each abundance of H2O for four representative bandpasses across our
wavelength range - 0.9, 0.85, 0.83, and 0.74 m. We notice that at very low
abundances of H2O, it requires a high SNR even for the 0.9 m bandpass for a
strong detection, and it is not strongly detectable at any wavelengths lower
than 0.85 m. Conversely, at very high abundances of H2O, every selected
bandpass with H2O is strongly detectable at low SNRs, with a highest required
SNR of 5 for H2O at 0.74 m. To provide context to the results, we present
ranges for the zonally averaged abundance of H2O for several epochs of Earth’s
history derived from global circulation models (GCMs), including a lower-CO2
Neoproterozoic Slushball scenario (minimum VMR of 3.1$\times 10^{-5}$, maximum
VMR of 1.6$\times 10^{-3}$), a warmer Mid-Cretaceous Greenhouse period
(minimum VMR of 1.6$\times 10^{-3}$, maximum VMR of 1.6$\times 10^{-2}$), and
the range of values for Modern Earth (minimum VMR of 3.1$\times 10^{-5}$,
maximum VMR of 7.6$\times 10^{-3}$). The Mid-Cretaceous, Neoproterozoic, and
Modern values were taken from simulations of Earth epochs performed using the
ROCKE-3D GCM and incorporating the SOCRATES radiation scheme (Way et al.,
2017), with all values other than CO2 and solar insolation left to modern
Earth values (L. Sohl, priv. comm.). The particulars for each run are provided
in Table 2.

Figure 9: Plot of the lowest SNR values at which a strong detection is achieved for notable H2O feature bandpasses, as a function of H2O abundance. The VMR values are on the x-axis, with SNR on the y-axis. The 0.9, 0.83, 0.85, and 0.74 m bandpasses are shown in light pink stars, pink dots, dark pink diamonds, and purple dots, respectively. We also mark the rough H2O abundance ranges for several epochs through Earth’s history, including a Neoproterozoic Slushball period (in dark pink), a Mid-Cretaceous Greenhouse warm period (pink) and the measured range for Modern Earth values (light pink). We note that for display purposes, we have left off the lowest abundance values for the Neoproterozoic Slushball and Modern epochs from the plot range. Earth Epoch | Age | Solar Insolation |  CO2 | Continent
---|---|---|---|---
Neoproterozoic Slushball | ca. 715 Ma | 94% modern solar insolation | 40 ppm | Sturtian Supercontinent
Mid-Cretaceous Greenhouse | ca. 100 Ma | 99% modern solar insolation | 2282 ppm | Mid-Cretaceous Continental
Modern | pre-industrial (1850) | 100% solar insolation (1361 W$\mathrm{m}^{-2}$) | 285 ppm | Modern Continental
Table 2: Details and parameter values for the ROCKE-3D simulations examining
several epochs of Earth’s history; values were provided by L. Sohl through
private communication.

##  4 Discussion

Understanding the impact of H2O detectability across multiple selected
bandpasses at varying wavelengths is illustrated in Figure 5. This is crucial
when looking into optimizing observations — since next-generation telescope
designs might prioritize shorter wavelengths based on photon flux or
instrument efficiency, we can use these results to optimize observations by
selecting a shorter bandpass that could meet different observational goals. As
we can see in Figure 5, O2 becomes strongly detectable at an SNR of 8. We can
see that by carefully selecting the best bandpass, we can detect both O2 and
H2O at varying strengths, depending on the SNR needed. For example, if we wish
to strongly detect both O2 and H2O in the same bandpass, a slightly higher SNR
is needed with the 0.74 m bandpass (SNR = 12) versus the 0.84 m bandpass (SNR
= 11).

There is also a trade-off between pure detectability versus an accurate
constraint on abundance. If we purely want to confirm the presence of H2O, we
would prioritize the longer wavelengths close to 0.9 m, which are certain to
yield a strong detection at mid to low SNRs for most abundance values;
however, these bandpasses do not accurately retrieve the abundance values due
to parameter degeneracies driven by a lack of spectral continuum. If we
prioritize detecting an accurate abundance level for H2O with a single
bandpass measurement, we would need to utilize a shorter bandpass that
requires higher SNRs and observation time. By prioritizing detectability above
all else, we can confirm H2O presence within more exo-atmospheres, and earmark
more planets for follow-up observation at lower initial observation cost,
which can then be observed for longer times and at higher SNRs (this will be
explored further in a companion paper by Stark et al. (submitted)). However,
by prioritizing shorter wavelengths, the instrument capabilities are optimized
and accurate abundance values retrieved, leading to better knowledge of the
exoplanet composition (i.e. slushball or water worlds) and more effectively
designating interesting planets for follow-up observations of biosignatures.
We note that a more thorough integration-time optimization can provide a firm
single metric for evaluating these trades, but is immensely complex and model-
dependent due to the combination of telescope and detector inputs, noise,
resolution, etc., and thus is generally treated on a case by case basis for
different mission designs. In this way, one can study the impact of specific
parameters on the necessary integration time, such as in Checlair et al.
(2021), Kopparapu et al. (2021) and Villanueva et al. (2022).

In Figure 9 we examine the detectability of H2O for several Earth epochs. We
can see that detecting H2O during a Neoproterozoic Slushball period would be
difficult, requiring SNR = 7 in order to detect H2O at 0.9 m or SNR = 9 at
0.85 m, with detection at shorter wavelengths unconstrained or improbable.
Detecting a Mid-Cretaceous Greenhouse abundance would be relatively easy, with
H2O at multiple wavelengths detectable with mid to low SNRs; the caveat is
that this is applicable only for the mid-to-high abundance values believed to
be present for the Mid-Cretaceous Greenhouse. We can also see that the
abundance values for Modern Earth cover a large range, encompassing all the
same values of a Neoproterozoic Slushball and sharing a large portion of the
Mid-Cretaceous Greenhouse. The Modern era values, however, do not climb to the
same high abundances as the Mid-Cretaceous Greenhouse, and thus require an SNR
of 10 to detect H2O at all wavelengths of interest. H2O at longer wavelengths,
however, is accessible to be detected at an SNR of 4 at the larger abundance
values. We also note that the ROCKE-3D simulations find a mean of 3.5$\times
10^{-3}$ VMR (or 3500 ppm) for the Modern Earth Epoch, landing very close to
our fiducial value of 3$\times 10^{-3}$ VMR. All of the states discussed
above, however, are to be used as examples of VMRs that may be possible for
atmospheric conditions matching epochs of Earth’s history, but we note that
since there is little constraint on specific H2O values over geologic time,
one could not determine a specific epoch when a H2O abundance is observed -
the uncertainty and potential range of H2O in these periods is large, thus at
certain points a given abundance of H2O could correspond to any of the three
epochs we present. We present these values as a statement to the detectability
of water vapor as a function of time using the best estimates from high-
fidelity atmospheric models.

As stated above, there have been similar prior works that investigated the
relationship between SNR and detectability, in particular Feng et al. (2018)
and Damiano & Hu (2022). Our overall retrieval parameterization structure is
quite similar to that of F18, and in S23 we performed an apples-to-apples
comparison with F18 to show that our results are largely equivalent; however,
since F18 used the full visible wavelength range and we only use limited
bandpasses in this work, we did not conduct a similar comparison. Damiano & Hu
(2022) also used much different wavelength ranges for their analyses, and also
established the ability for their retrieval algorithm EXOREL to determine the
background gas of an atmosphere, whereas we define our background gas to be
N2. Even with these differences, our results for H2O detectability agree with
both Damiano & Hu (2022) and Feng et al. (2018) that by an SNR of 10, at R =
140, firmly constraining the abundance for an Earth-twin atmosphere is very
likely.

##  5 Conclusions & Future Works

To summarize, SNR and detectability are intrinsically linked, and knowledge of
the connection can drive efficient observing practices. By understanding the
SNR requirements for detecting molecules of interest such as H2O, and properly
prioritizing the spectral bandpasses to optimize detectability of different
atmospheric constituents, we can inform the best instrument designs and
observing procedure. H2O is most easily observable at longer wavelengths such
as 0.9 m, however the stellar SED and instrument capabilities dictate that
best observation occurs at shorter wavelengths, such as 0.74 m. Although the
absorption is less deep in this wavelength range, one can also capture
multiple molecular species such as O2. Due to the lack of constraint on H2O
through time, observations of water vapor alone could not inform the epochs of
an Earth-twin. However, other molecular species with geochemical markers could
provide more depth of knowledge to potential Earth observations (Planavsky et
al., 2014).

In follow-up analyses, we will present a similar molecular abundance study for
O2, O3 and other potential molecular species that could be present in Earth-
twin atmospheres, and expand on the trade space in choosing bandpasses for
some or all of these species. We also note that there are a number of
limitations to our model framework that we plan to improve on in subsequent
work. We do not vary noise within the bandpasses as in real coronagraph
observations (i.e. prioritizing the bandpass center and increasing noise
towards the bandpass edges), but rather assume a consistent noise throughout
the bandpass. We use isotropic abundance profiles for the molecular species
and cloudiness fraction, we have a limited selection of molecular species
(i.e. only H2O, O2, and O3), and we also do not include any dependence on
orbital phase. We also use units of geometric albedo for our spectral models;
thus we cannot retrieve on radius due to the degeneracy described above in §2.
We intend to address these concerns in future works using new grids of forward
models. We once again note that SNR results are subject to change with a new
grid construction, due to upgrades in PSG and the addition of more parameters
and molecules. This will allow us to represent the physical chemistry more
accurately and study further molecular detections such as CH4 and CO2. By
improving and expanding the spectral grid, we will be able to expand our
simulations of possible observations and establish best practices for exoEarth
observations using next generation telescopes such as HWO.

We thank the referee for their thoughtful review. N. L. gratefully
acknowledges financial support from an NSF GRFP. N. L. gratefully acknowledges
Dr. Joesph Weingartner for his support and editing, as well as Dr. Ferah
Munshi and Jordan Van Nest for their aid in plot coding. 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 gratefully
acknowledge conversations with Dr. Giada Arney, Dr. Michael Way, and Dr. Linda
Sohl on water vapor abundances in Earth’s atmosphere through time. 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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