# The Impending Opacity Challenge in Exoplanet Atmospheric Characterization

Prajwal Niraula1,∗ Julien de Wit1,∗ Iouli E. Gordon2 Robert J. Hargreaves2
Clara Sousa-Silva2 Roman V. Kochanov2,3,4

###### Abstract

With a new generation of observatories coming online this decade, the process
of characterizing exoplanet atmospheres will need to be reinvented. Currently
mostly on the instrumental side, characterization bottlenecks will soon stand
by the models used to translate spectra into atmospheric properties.
Limitations stemming from our stellar[2] and atmospheric[3] models have
already been highlighted. Here, we show that the current limitations of the
opacity models used to decode exoplanet spectra propagate into an accuracy
wall at $\sim$0.5-1.0 dex (i.e., 3 to 10$\times$) on the atmospheric
properties, which is an order of magnitude above the precision targeted by
JWST Cycle 1 programs and needed for, e.g., meaningful C/O-ratio constraints
and biosignatures identification. We perform a sensitivity analysis using nine
different opacity models and find that most of the retrievals produce
harmonious fits owing to compensations in the form of $>$5$\sigma$ biases on
the derived atmospheric parameters translating in the aforementioned accuracy
wall. We suggest a two-tier approach to alleviate this problem involving a new
retrieval procedure and guided improvements in opacity data, their
standardization and optimal dissemination.

∗These authors contributed equally to this work.

Department of Earth, Atmospheric and Planetary Sciences, Massachusetts
Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts
02139, USA;

Harvard-Smithsonian Center for Astrophysics, Atomic and Molecular Physics
Division, Cambridge, MA, USA;

V.E. Zuev Institute of Atmospheric Optics, Laboratory of Theoretical
Spectroscopy, Russian Academy of Sciences, 1 Akademichesky Avenue, 634055
Tomsk, Russia;

QUAMER laboratory, Tomsk State University, 36 Lenin Avenue, 634050 Tomsk,
Russia.

##  1 Introduction

Robust data interpretation requires building models that encompass the
ensemble of physical processes possibly at play and accounting for the
uncertainty associated with our understanding of these processes. For
transmission spectroscopy, the four building blocks of such models relate to
(1) the host star as a source of photons, (2) the planetary atmosphere
attenuating the host star light, (3) the opacity describing the wavelength-
and condition-dependence of the attenuation process, and (4) the instruments
collecting the photons and converting them into electronic data. The
associated uncertainty budget, long dominated by instruments through photon
noise and systematics, needs a re-evaluation at the dawn of JWST and the
Extremely Large Telescopes (ELTs).

A closer look at the current state of our models has already revealed
challenges associated with the interpretation of the next-generation data[4].
Stellar inhomogeneities can mimic or mute a planetary signal and our limited
understanding of them may prove a bottleneck[2]. The complex 3D structure of
planetary atmospheres may play a similar role[5]; simple 1D models are unable
to capture the day-night temperature asymmetry[3] or highlight the role of
clouds and hazes in muting the spectral features[6, 7]. Uncertainties (and
biases) associated with atmospheric models are also expected to stem from our
understanding–or lack thereof–of the underlying photochemical processes[8]. In
addition, degeneracies amongst model parameters and dependencies to the model
formulations will contribute to the challenges of robust high-precision
retrievals[9, 10].

The impact of uncertainties and inaccuracies in opacity models is a common
hurdle. Atmospheric studies within the Solar System faced limitations
associated with unavailable pressure broadening parameters[11]. More recently,
studies[12, 13, 14, 7, 15, 16] explored the possibilities for similar hurdles
in the field of exoplanetary sciences and highlighted the large discrepancies
between synthetic spectra generated from different pre-computed cross-sections
supporting the need for additional experimental works[17]. In 2021, a study
detected atmospheric constituents of a hot-Jupiter via the cross-correlation
technique using different sets of cross-sections, each yielding a different
detection significance[18].

We expand the scope of these studies by assessing and quantifying the main
sources of biases and uncertainties in opacity models considering their impact
on the retrieved atmospheric parameters (scientific insights, i.e., final data
product) rather than on the transmission spectra (observable, i.e.,
intermediate data product). It is a fundamental nuance as large differences
between opacity models may not always translate into equivalent differences in
the retrieved atmospheric parameters (see Methods). A transmission spectrum
does not encode the atmospheric information uniformly across wavelengths[19].
Its information density is a non-monotonic function of wavelength therefore
mapping discrepancies in opacity models onto atmospheric parameters is non-
trivial. To understand the impacts of imperfections in opacity models, we
generate nine different sets of opacity cross-sections representative of
standard assumptions and perform a perturbation/sensitivity analysis via self-
and cross-retrievals (see Methods).

CS-DFLT is our nominal cross-section generated using standard assumptions on
the default calculation options of hapi[20] (v 1.0). CS-1SUP and CS-1SDN are
designed to assess the effect of measurement uncertainties of line parameters
from line lists to retrievals. CS-SELF, CS-MAXB, and CS-MINB are designed to
assess the effect of incomplete knowledge regarding line broadening. CS-500W
is designed to assess the effect of incomplete knowledge regarding far-wings
of line profiles. While the aforementioned cross-sections use HITRAN2016[21],
CS-EXML and CS-HTMP use respectively ExoMol[22] and HITEMP2010[23] to assess
the effects of completeness and line-list imperfections through “cross-
database” retrieval.

Figures 1 & 2 shows the subtle differences emerging at the level of an opacity
cross-section and a transmission spectrum, respectively, for warm-Jupiter
observations simulated using each of the nine cross-sections. Although CS-DFLT
(HITRAN) and CS-HTMP (HITEMP) are more similar to each other than to CS-EXML
(ExoMol) at the level of cross-section for 0.1 atm due to their accounting of
pressure shift (Figure 1.a), CS-HTMP and CS-EXML are more alike at the level
of transmission spectra which probes mostly lower pressure levels where the
effect of the pressure shifts is not as substantial but that of line-list
completeness is (Figure 2). We present the transmission spectrum both in terms
of wavelength-dependent variations of the transit depth and effective
atmospheric height[24], $\rm h_{\rm eff}$, in scale height. Such a dual
presentation of a transmission spectrum highlights the intermediate nature of
this data product and connects it to the original signal and its noise (light-
curve) and the final data product (atmospheric parameters).

##  2 Results

###  2.1 Blind to opacity model perturbations.

We find that despite the perturbations to the opacity models, most cross-
retrievals lead to fits with reduced $\chi^{2}$ close to 1. Meaning that
despite the opacity perturbations, it is possible to bias other parameters of
the global model to compensate and yield a good fit (example in Figure 3). We
find for example that perturbations to the opacity model through the
broadening coefficient for our super-Earth scenario lead to biases on the
atmospheric temperature, abundances of water, carbon dioxide, methane, and
ozone of -4.02$\sigma$, 3.31$\sigma$, 6.91$\sigma$, 7.14$\sigma$, and
6.11$\sigma$, respectively. We highlight that these biases affect each
parameter individually. The abundances of some elements are overestimated
while others are underestimated, which results in compounding biases on
important elemental ratios, such as C/O for formation modeling. Additionally,
we find that a cross-retrieval using the ExoMol and HITEMP2010 databases over
HITRAN2016 for the same super-Earth scenario leads to similarly strong biases
on the atmospheric temperature, abundances of water, carbon dioxide, methane,
and ozone of -18.18/-19.37$\sigma$, -8.32/-20.64$\sigma$,
-12.84/-16.42$\sigma$, -17.99/-65.95$\sigma$, and -8.61/-17.58$\sigma$ (see
Methods), respectively.

We find that the significance of these biases is in some cases amplified by a
shrinking of the posterior probability distributions (PPDs) of the derived
parameters. For example, we find that the 1-D PPDs are 3 to 6 times narrower
than in the nominal (i.e., self-retrieval) case. Such shrinking relates to a
change in the $\chi^{2}$ distribution over the parameter space associated with
a change in mapping from atmospheric parameters to transmission spectrum with
different opacity models. We discuss further the complex topology of the
$\chi^{2}$ distribution, the related relationship between the atmospheric
parameter estimates, and the underlying dependency to the opacity models in
Methods to provide further insights into the intricacies of this problem. This
highlights that imperfections in opacity models may not only lead to biased
parameter estimates, they may also affect the estimated uncertainty associated
with them. Figure 4 highlights both effects.

###  2.2 Opacity-driven accuracy wall straight ahead.

These biases are statistically significant and are also, and most importantly,
physically substantial–often larger than 0.5 dex (equivalent to a factor of
3). While the former implies that our findings regarding opacity-induced
biases are relevant within the context of observations reaching the signal-to-
noise ratio necessary to highlight such biases, the latter is independent of
such considerations and can be directly related to scientific insights that
may thus be out of reach. Figure 5 highlights this difference between a
statistically- and physically-significant bias while contextualizing the
latter with the accuracy targeted by upcoming observation programs.

Figure 5.a shows the positive correlation between the Spectral Statistical
Distance (SSD) and the Parametric Statistical Distance (PSD). The SSD is
defined as
$\sum\sigma_{\lambda}^{-2}\times(\rm{Base~{}Model}_{\lambda}-Model_{\lambda})^{2}$,
where Base model is the synthetic spectrum calculated using the default cross-
section (i.e. CS-DFLT), Model is the synthetic spectrum generated using
another cross-section based on the same atmospheric parameters, and
$\sigma_{\lambda}$ is the noise model associated with these observations (see
Methods). The PSD is defined as the Bhattacharya distance[25] that measures
the statistical distance between two PPDs while accounting for the covariance
between the parameters. While SSD appears to correlate positively with PSD in
log-log space, the large spread in value seen around their mean relationship
highlights that perturbations at the level of a transmission spectrum cannot
readily be translated into perturbations at the level of atmospheric
parameters. Indeed, cross-sections initially yielding a large SSD may still
enable a good fit (reduced $\chi^{2}\sim 1$) owing to efficient compensation
mechanisms (see Methods).

Figure 5.b introduces the same information as Figure 5.a, but without the
context of a specific observation, defining the statistical nature of the PSD.
Instead, Figure 5.b uses the Parametric Physical Distance, defined as
$|1-\theta_{ret}/\theta_{true}|$, where $\theta_{ret}/\theta_{true}$ is the
ratio of the retrieved versus true atmospheric parameter. Figure 5.b shows
that, for most cross-retrievals, parameters are biased by $\sim$0.5 to
$\sim$1.0 dex due to the opacity model. This corresponds to a 3 to 10$\times$
drop in accuracy compared to the accuracy expected from instrument models only
for our synthetic observations. This 0.5-1.0 dex accuracy wall is best
contextualized by the ambition of upcoming JWST observations. GO 1633, for
example, aims to constrain the metallicity of the hot Jupiter HD 189733 b to
the 0.025-dex level to determine whether it is consistent with the Solar
System’s relation. Such precision level is over one order of magnitude below
the accuracy wall highlighted here. Our analysis reveals that for planetary
systems similar to the synthetic systems used here (H-Mag 10, see Methods),
this accuracy wall can be reached within 10 to 20 transits, while it would be
reached within only a few transits for systems with bright hosts such as HD
189733 b (H-Mag 5.6).

##  3 Discussion

We assess the extent to which complementary datasets such as emission spectra
could help mitigate the observed biases by performing the same retrievals
while introducing relevant priors on the p-T profile. We find that the biases
were only marginally reduced (by up to 20%). The current state of opacity
models is thus yielding a hard accuracy wall around 0.5 to 1 dex. Such an
accuracy wall will have severe consequences. For example, not being able to
distinguish via transmission spectroscopy alone between a temperature of 300 K
or 600 K, a “surface” pressure of 1 bar vs 2 bar, and/or an abundance of gases
of 5% vs 20% will drastically impact our future capability to address key
scientific questions spanning from formation mechanisms (e.g., via constraints
on metallicity trends and elemental ratios) to the search for Life via the
vetting of potential biosignature gases.

###  3.1 A two-tier mitigation strategy

In order to support robust data interpretations and scientific claims while
aiming to maximize the scientific output of JWST and the ELTs, we suggest the
following two-tier approach. First, develop community standards for opacity
models such as tierraCrossSection introduced in this work, to adequately
estimate the sensitivity of derived inferences to opacity models. This first
step is aimed at propagating the biases and uncertainties from opacity models
(line lists, cross-sections, and thermodynamic data) to planetary parameters.
Doing so will provide robustness to scientific inferences but will also
substantially limit the scientific outputs of the aforementioned observatories
(marked as “Accuracy Drop” in Figure 5.b). For that reason, the second step
suggested aims at investigating the origins of the biases and uncertainties in
current opacity models that dominate biases and uncertainties on scientific
inferences to guide the improvement of these models and match the accuracy
enhancement within reach with JWST and the ELTs (marked as dotted line in
Figure 5.b).

###  3.2 Identifying the dominant bottlenecks of opacity models

We find that the effect of measurement uncertainties (CS-1SUP and CS-1SDN)
leads to the least significant biases (see Figure 5). The biases found are
contained below 0.15-dex and 4$\sigma$. We find that this effect is currently
dominated by uncertainties on the line intensity and to a lesser extent the
line precision, which will be a bottleneck for future missions (see Methods).
Accounting for the fact that these two cross-sections were generated to
represent the maximal compounding effect of the line list uncertainties, we
thus find that the current precision at which line parameters are reported is
adequate for a robust interpretation of JWST data–barring the absence of
systematic errors and the completeness of the list (see CS-EXML and CS-HTMP).

We find that the effect of incomplete knowledge regarding line broadening (CS-
SELF, CS-MAXB, and CS-MINB) leads to strong biases reaching values of up to
1-dex and 7$\sigma$. We expect that the additional consideration of
collisional broadening would only further compound the underlying problem.

We find that the effect of incomplete knowledge regarding far-wings of line
profiles (CS-500W) is larger for the super-Earth synthetic case than for the
warm-Jupiter one (see Figure 5). While for the latter the biases are confined
below 0.5-dex and for all but one parameter, for the former the biases are
around 1-dex for half of them. This difference is explained by the high
sensitivity of the retrieved abundances of molecules with strong parse
absorption features such as methane to their line profiles, combined with
their connection to the atmospheric mean molecular mass and thus scale
height–which is independently constrained by the data[24]–for which all the
other atmospheric parameters (incl. abundances and temperature) can try to
compensate (see Methods).

We find that “cross-database” retrievals (CS-EXML and CS-HTMP) produce the
strongest biases due to completeness issue, as expected (Figure 2). Molecules
known to have discrepant line lists such as methane[26] understandably have
the strongest biases -17.99/-65.95$\sigma$ and -11.2/-20.9$\sigma$ for super-
Earth and warm-Jupiter, respectively. Although the significance of these
biases is increased by reducing PPDs (by a factor up to 6, Figure 4), they
still correspond to biases up to 2 dex. Fortunately, we find that the
residuals of such retrievals had reduced $\chi^{2}$ significantly above one
(see Methods), which indicates a strong mismatch between data and model. This
is however the only case where the model parameters could not allow for total
compensation of the cross-section perturbation.

###  3.3 Steps towards alleviating the dominant bottlenecks of opacity models

Following these findings, we suggest that the accuracy wall stemming from
opacity models would be best addressed via a refined understanding and
parametrization of (1) line broadening effects and (2) far-wing behaviors
together with (3) a refined tracking of differences between line lists and
underlying data sources. (1) and (2) would benefit from additional laboratory
measurements and theoretical calculations.

Regarding (1) the line broadening effect specifically, measurements and
calculations of collisional broadening for any relevant species, as well as an
analysis of the temperature dependence of these parameters are needed.
Currently, there are significant gaps in the knowledge of broadening behavior
of many molecules of interest for atmospheric characterization. Traditionally,
the majority of the efforts in this area have focused primarily on spectral
lines for terrestrial atmospheric molecules broadened by air. Recently, the
HITRAN database started to add broadening parameters due to the pressure of
other ambient gases dominating planetary atmospheres, including H2, He, CO2
(see for instance ref.[27]) and H2O[28]. As described in ref.[27] there was a
severe lack of laboratory measurements or calculations for some of these
molecules, and crude estimates had to be used. The call for more measurements
has been heard for some of the systems. For instance, only one line of SO2
broadened by CO2 had been measured[29] prior to the publication of ref[27].
Since then, however, several experimental and theoretical works have
appeared[30, 31, 32]. Nevertheless, the situation has not improved for many
other molecules, for instance, dozens of articles are devoted to measurements
of the CO2 lines broadened by CO2, O2 and N2[33], but almost none for CO2
broadened by H2.

It is worth noting that experiments that measure broadening parameters are not
trivial to perform. For instance, ref[34] shows examples when a single
parameter was measured in different laboratories but the reported values often
did not agree within their uncertainties. However, with the advent of modern
experimental techniques and fitting softwares, the situation is improving. The
quality of the theoretical calculations of the broadening parameters has also
improved recently[35, 36], although these are still quite limited since the
calculations are computationally expensive, especially when carried out for a
wide range of temperatures. Nevertheless, theoretical calculations do provide
a promising avenue to complement experimental measurements.

Regarding (2) the far-wing effect specifically, this relates to one of the
least studied and understood areas of molecular spectroscopy. For different
molecules and their possible collisional broadeners (or perturbers), the
difference in far-wing behavior can be drastic[37]. As an example, the far
wings of water vapor lines, which are major contributors to the water
continuum[38], are parametrized differently than those of CO2 lines that are
assumed to be sub-Lorentzian[39].

Regarding (3) the refined tracking of differences between line lists and
underlying data sources, the community would benefit from evaluations between
theoretical line lists and available laboratory data[26], and
atmospheric/planetary data[40, 41]. For several molecules, there are multiple
sources of opacity data, each with different formats, advantages, and
limitations. A centralized database, with standardized formats and informative
metadata, is a crucial step in responsibly providing reliable and up-to-date
spectral data to the astronomical community.

Each of the aforementioned actions are key to alleviating the opacity-driven
accuracy wall revealed here, which is likely to similarly influence other
remote sensing techniques and applications. Therefore, such additional work
should be viewed as foundational to the success of JWST and the ELTs, and a
priority for our community.

###  3.4 Framework

We map the key components to a transmission spectrum to guide the development
of our framework aimed at assessing the effect of imperfections in opacity
models on retrieval capabilities. Supplementary Figure 1 shows these
components, and our process to identify accuracy bottlenecks. The framework
uses minimal complexity for stellar, atmospheric, and instrumental models to
focus on the role of opacity models. We notably assume long observations to
reach a regime that allows us to reveal from the observation noise the
limitations stemming from opacity models. Doing so allows us to study how
biases and uncertainties propagate from line lists to transmission spectra
and, ultimately, retrieved atmospheric parameters via a
perturbation/sensitivity analysis. We perform such sensitivity analysis via
self- and cross-atmospheric injection retrievals based on nine opacity cross-
sections generated with different sets of assumptions to be representative of
underlying model uncertainties.

###  3.5 Opacity Database & Cross-Sections

We use molecular data from the HITRAN2016[21], HITEMP2010[23] and ExoMol[22]
databases. The line-by-line section of the HITRAN database contains important
spectroscopic parameters for individual transitions. These parameters include
(but are not limited to) line position, line intensity (and Einstein-A
coefficient), half-widths (and their temperature dependences) associated with
the pressure broadening by different ambient gases, quantum identifications of
the transition, and degeneracy factors of the energy states involved in the
transition, along with the energy value of the lower state (see ref.[44] for
the description of some of the main parameters). The HITRAN database focuses
on spectra at the terrestrial atmosphere temperatures and pressures. Therefore
for many molecules, it can not be accurately used at elevated temperatures,
primarily due to the paucity of transitions that become stronger when the
temperature increases. The HITEMP database[23] is similar to HITRAN in its
parametrizations, however, it contains substantially more transitions for the
eight molecules that are currently provided, which is referred to as the
completeness issue. This allows for the modeling of spectra at elevated
temperatures. The ExoMol database[22] provides line positions, Einstein-A
coefficients, and rotational quanta assignments and has thus also been
addressing the completeness of line lists. The molecular data from the ExoMol
database contains many orders of magnitude more transitions than those derived
from experiments, due to its theoretical approach to calculating line lists.
In many cases the parameters in ExoMol and HITRAN agree very well, in fact,
many parameters in HITRAN are from ExoMol and vice versa. Nevertheless, in
some cases, significant discrepancies between the two databases have been
reported (e.g., for methane[26]).

For atmospheric retrieval, the real-time generation of opacity cross-sections
at present is computationally infeasible. Therefore, the field standard is to
pre-compute the cross-sections. The generation of an opacity cross-section
requires a series of assumptions. We, therefore, generated several cross-
sections representing such assumptions to explore their impacts on the
retrieved parameters. For this study, we generated nine equivalent cross-
sections for seven different molecules. Supplementary Table 1 introduces the
parameters used. The cross-sections are generated using TierraCrossSection,
which re-purposes functions from HAPI[20] to enable the generation of cross-
sections under a wider range of assumptions. Such cross-sections are relevant
for sensitivity analysis similar to the one introduced here. To generate
cross-sections from the ExoMol database, we use exocross[43] for producing
files with line intensities and line positions, which can be used by
TierraCrossSection. We introduce below the nine different cross-sections and
their purpose in our sensitivity analysis.

  1. 1.

CS-DFLT: This cross-section represents the standard for our study and is based
on the default calculation options of hapi (v 1.0). We use the Voigt profile
for lines, air broadening parameters for pressure broadening, and calculate
the extent of the line profile to 50 Voigt half widths at half maximum (HWHM).

  2. 2.

CS-1SUP: This cross-section aims at assessing the effect of uncertainties on
line parameters on retrieval capabilities. We thus modify line intensity,
pressure broadening, line position, and temperature dependence of air
broadening by +1$\sigma$ error. The uncertainties for the aforementioned
parameters are reported in HITRAN. When the values are not available (i.e.,
corresponding to uncertainty index 0), we assume an error of 20 percent,
although in some cases it maybe too conservative as, for instance, some of the
excellent ozone data in HITRAN2016 is from the times where uncertainties were
not provided in the database. Supplementary Figure 2 shows the distribution of
uncertainties on the line parameters for four different molecules. It
highlights the wide range of uncertainties each parameters span and their
correlation to the line intensity. Using these cross-sections, we can
understand the impact of the current accuracy of these lines.

  3. 3.

CS-1SDN: Similarly to CS-1SUP, the CS-1SDN cross-section aims at assessing the
effect of uncertainties on line parameters on retrieval capabilities. Here the
aforementioned line parameters are perturbed by -1$\sigma$.

  4. 4.

CS-SELF: This cross-section aims at assessing the effects of (1) unknown
background broadeners and (2) unknown broadening parameters for the dominant
broadener. To that end, instead of using air as the perturber (broadener), we
use self-broadening parameters. We note that the processes behind line
broadening are complex and lead to broadening parameters that are themselves a
function of the rotational and vibrational quanta of the transition
represented (i.e., non-monotonic dependence to wavelength). The difference
between air- and self-broadening is molecule dependent, for example, some
molecules such as nitrogen have a similar range of broadening coefficients,
whereas for water, the broadening coefficients (self and air) range between
values spanning over 0.5 dex (from 0.15 to 0.55 and from 0 to 0.15,
respectively).

  5. 5.

CS-500W: This cross-section aims to assess the effect of imperfections in line
profiles on retrieval capabilities. We generate CS-500W by modeling the line
wings up to 500 HWHM otherwise truncated to the default value of 50. The far-
wing behavior is affected by the wind patterns and turbulence in the
atmosphere, phenomena not considered in current retrievals. Yet, the far wings
for the strong lines can determine the overall spectral bands and opacity
continuum, particularly at higher pressures.

  6. 6.

CS-MAXB: This cross-section aims to further explore how the pressure
broadening factor can impact retrieval. We take the broadening parameter as
double of the larger value between the air and the self-broadening parameter
for each transition (i.e., +0.3 dex perturbation). This cross-section
represents the case when the pressure broadening parameters are over-
estimated. Higher pressure broadening does not change the overall opacity of
the molecule, but simply redistributes the opacity from the line center into
the wings.

  7. 7.

CS-MINB: Similarly to CS-MAXB, the CS-MINB cross-section aims to further
explore how the pressure broadening factor can impact retrieval. Here, we use
half of the smaller value of each line of the broadening parameters between
self-broadening and the air-broadening (i.e., -0.3 dex perturbation). This
cross-section aims to reveal how underestimating the pressure broadening
parameter can impact the retrieval.

  8. 8.

CS-EXML: This cross-section aims at assessing the effect of differences in
opacity databases used to perform a retrieval. We generate this cross-section
using the ExoMol line list. To ensure the same level of completeness, we use
the intensity cut-off used in HITRAN. As for the pressure broadening, we
assumed a universal value of 0.07 cm-1 with the temperature dependence
exponent of 0.5 across all the molecules. ExoMol suggests a more accurate
$J$-dependence for a few of the molecules, which however was not implemented
to ensure the same treatment across the molecules. Molecules such as nitrogen
and ozone, for which no equivalent cross-section could be created, were
replaced with their equivalents from CS-DFLT.

  9. 9.

CS-HTMP: This cross-section aims at further exploring the effect of
(in)completeness of line lists comparing HITEMP2010 to HITRAN2016 as it
extends the HITRAN line lists to higher temperature [23] primarily to address
the completeness issue that arises from using cutoff for the weaker lines.
Molecules such as methane, and water, significant contribution towards opacity
come from weaker lines at these higher temperatures. We note that both
HITEMP2010 and the HITRAN2016 data are regularly updated when better
experiments or theoretical calculations are performed. Hydrogen, nitrogen and
ozone whose line lists are not part of HITEMP2010, their cross-sections were
taken from CS-DFLT.

###  3.6 Additional Opacity

We added two attenuation processes to our opacity models, Rayleigh scattering
and collision-induced absorption (CIA). We accounted for Rayleigh scattering
for all molecules except ozone (refs.[45, 46]), for which such data is
unavailable and which does not occur at concentrations high enough to impact
Rayleigh scattering. We accounted for CIA using the data from HITRAN CIA[47]
for $\rm H_{2}-H_{2}$[48], $\rm H_{2}-He$[49], and $\rm N_{2}-N_{2}$[50, 51,
52].

###  3.7 Molecule Selection

Interpreting a transmission spectrum requires an inventory of molecular
opacities, often complex molecular networks that contribute to a final
spectrum[53]. For this exploratory work, we consider seven molecules that have
traditionally been important in the context of planetary and exoplanetary
sciences: water, methane, carbon dioxide, carbon monoxide, ozone, hydrogen,
and nitrogen. Our work with these seven molecules should be representative of
the typical systematic problems with any other molecules, and represent the
prominent molecules for hydrogen, carbon, and oxygen chemistry. Among the
seven molecules, hydrogen (H2) and nitrogen (N2) do not have strong
spectroscopic features. Similarly, we add helium as a background gas at a
constant volume mixing ratio of 15 to 85 with respect to hydrogen[54, 55].

###  3.8 Spectral Resolution

It is necessary to simulate the planetary spectrum at a spectral resolution
that is higher than the one offered by the observing instrument and then bin
down to a lower resolution. Observations at R$\sim$10 with Hubble require
model evaluation at R$\sim$1,000 for example[56]. For our applications
focusing on JWST observations, we tested the retrieval performances using
resolutions of 10,000, 100,000, and 2,000,000. We find that the R$\sim$10,000
model leads to deviations in the simulated spectra of up to 50 p.p.m. (part
per million) when compared to the R$\sim$2,000,000 model, whereas for
R$\sim$100,000, the mean difference is below 10 p.p.m. Most significantly, we
find that cross-retrievals using the two later resolutions lead to consistent
results (while using R$\sim$10,000 lead to deviations above the 3$\sigma$
level), implying equivalent performance for R$\sim$100,000 and
R$\sim$2,000,000 for the intended application. We thus choose to use
R$\sim$100,000 to mitigate the contribution of the spectral resolution choice
to the uncertainty budget while keeping our models computationally effective,
as in ref.[24].

We note that there is an intrinsic maximum resolution attainable with a given
line list due to the uncertainty on the line position (see Supplementary
Figure 2). The line lists used here report the position of most dominant lines
with uncertainties below or within the 0.01 to 0.001 per cm range. This
translates into a resolution between 70,000 and 1.6 million over the
wavelength range covered (0.6 to 15 microns), which is comparable to the
resolution selected above based on an instrumental argument (data quality and
detector resolution). This means that the precision on the line positions in
current opacity line lists will not affect upcoming atmospheric retrieval with
JWST.

###  3.9 Grid Optimization

We choose our temperature-pressure grids to minimize interpolation errors.
Line profiles are more sensitive to the changes in pressure at a high-pressure
regime than at a low-pressure regime. Conversely, they are more sensitive to
small changes in temperature at a lower temperature compared to the higher
temperature regime. We minimize the effect of interpolation error by using a
non-homogeneous p-T grid (see Supplementary Table 1), similarly to ref.[24].

###  3.10 Transmission Spectroscopy Code and Benchmarking

We develop a 1D code for transmission spectra, tierra (TransmIssion
spEctRoscopy of tRansiting plAnets), following the code used in ref[24]. We
benchmark tierra against petitRadtrans[57] for different atmospheres.
petitRadtrans has itself been benchmarked against other codes[58, 59] using
the recommended cross-section. We find that the two transmission spectra are
in excellent agreement (median absolute deviation of 1.9 p.p.m., $<$0.005
scale height) as shown in Supplementary Figure 3. This difference ultimately
found with our sets of models is thus marginal in comparison to the effects
intended to be studied here (see Figure LABEL:fig:ModelComparison) implying
that our transmission code implementation will not contribute significantly to
the uncertainty budget.

###  3.11 Planet, Star, and Instrument Models

For this exploratory work, we use simple models for the planet and for the
star to focus on the limitation stemming from current opacity models. We
assume the star does not present heterogeneities which could induce
contamination at the level of the planetary spectrum[2]. We assume that the
planetary atmosphere can be modeled via a 1D atmosphere with elemental
abundances that are independent of the altitude and a temperature profile
following $T(z)=T_{\infty}+(T_{0}-T_{\infty})e^{-H_{T}/z}$, where $H_{T}$ is
the temperature scale height. We note that such models would be too simplistic
to perform a robust atmospheric retrieval of JWST data[2, 5, 3]. Yet the
related limitations are not present in our approach using
injection-(cross)retrieval analysis, which uses the same planet and star
models to generate the synthetic spectra and analyze them.

For the instrument model, we use a model as realistic as possible to best
compare the contributions of the opacity model to the total noise budget. We,
therefore, use pandexo[60] for our instrument and noise models. We consider
two observational cases: (1) an Earth-sized planet around a Hmag 10 M-dwarf
star and (2) a Jupiter-sized planet around a late Hmag 10 K-dwarf star (see
Supplementary Table 2). We assume in both cases combined observations with
Near Infrared Spectrograph (NIRSpec) and Mid-Infrared Instrument (MIRI). In
order to isolate the effect of opacity models, we assume a total of 100
transits both in NIRSpec and MIRI with a noise floor at 10 p.p.m.[61]. We
chose this large number of transits to ensure that if there is an accuracy
wall relevant in the context of the JWST and ELTs era it will be revealed. And
in fact, we find that for the synthetic systems chosen around H-Mag 10 stars
the accuracy wall is hit within 10 to 20 transits (see Figure 4), meaning that
systems such as HD 189733 b with a H-Mag 5.6 host only requires a few
transits.

###  3.12 Analysis Technique

We study the effect of perturbations to our opacity models on two data
products: (1) the transmission spectrum and (2) the retrieved atmospheric
parameters. We quantify the deviations between transmission spectra via the
“Spectral Statistical Distance” (SSD) defined as
$\sum\frac{(\rm{Base~{}Model}_{\lambda}-Model_{\lambda})^{2}}{\sigma_{\lambda}^{2}}$,
where Base model is the synthetic spectrum calculated using the default cross-
section (i.e., CS-DFLT), Model is the synthetic spectrum generated using
another cross-section based on the same atmospheric parameters, and
$\sigma_{\lambda}$ is the noise model associated with these observation. The
“Spectral Statistical Distance” corresponds to the Deviation Quotient used as
detection metrics in previous studies[62, 63]. We later show how these metrics
come with limitations.

In order to quantify the deviation between the retrieved parameters we need a
framework to derive their best estimates and uncertainties from a given
spectrum and introduce the “Parametric Statistical Distance”. We use Monte
Carlo Markov Chain (MCMC) technique to assess the Posterior Probability
Distributions (PPDs) of the atmospheric parameters associated with a spectrum.
For these retrievals, we fixed planetary mass and radius as well as stellar
ones to focus on the effect of opacity models. Our jump parameter set consists
of three parameters for the temperature profile and seven molecular number
densities taken at the reference radius following the formulation introduced
in ref[24].

To expedite the fitting process and minimize the burn-in, we initiate the
MCMCs using the original values used to generate the synthetic model with CS-
DFLT. We use affine invariant MCMC as implemented in emcee[42] for exploring
the parameter space, and ran our models with 20 walkers for a minimum of
25,000 steps. Our walkers are in log-space for the number densities and the
temperature lapse rate. We compute the base pressure from the number densities
of the molecules and allow it to vary up to 100 atmospheres. We ensured the
convergence of the MCMCs by checking for the good mixing of the walkers, and
the evolution of the log probability.

Using the PPDs we can derive a “Parametric Statistical Distance” (PSD) to
quantify the deviation in the retrieved parameters introduced by the
perturbations in the opacity models. To do so, we used the Bhattacharya
distance[25]. By measuring PSD, we can quantify biases between PPDs in a
multi-dimensional fashion (see Figure 4 and Supplementary Figure 4, and 6).
The metric increases proportionally as the mean deviates from one another and
accounts for the covariance among the parameters. If the posterior
distributions are identical, the distance between them would be zero.

We highlight the need for assessing model-driven accuracy limit at the level
of final data products (here, retrieved atmospheric parameters) rather than
intermediate data product (here, transmission spectrum) via the non-monotonic
relationship between PSD and SSD. While SSD appears to correlate positively
with PSD in log-log space, the large spread in value seen around their mean
relationship highlights that perturbations at the level of a transmission
spectrum cannot readily be translated into perturbations at the level of
atmospheric parameters. Cross-sections initially yielding a large SSD may
still enable a good fit (reduced $\chi^{2}\sim 1$) owing to efficient
compensation mechanisms. For example in the super-Earth scenario, while
CS-500W and CS-MAXB have SSDs over three times those of CS-1SUP and CS-1SDN
they similarly yield reduced $\chi^{2}$ around 1 owing to larger compensations
(ie., biases) on the atmospheric parameters resulting in larger PSDs
(Supplementary Table 3).

###  3.13 Super-Earth Retrievals

In using CS-1SUP, we find that most values are retrieved within less than
2.5$\sigma$, except for carbon dioxide which deviate by 3.18$\sigma$ (see
Supplementary Table 3). For CS-1SDN, we find four parameters: $T_{0}$, along
with molecular abundance of methane, ozone and hydrogen biased by
2.87$\sigma$, 4.02$\sigma$, 2.87$\sigma$ and -4.51$\sigma$ respectively. While
no significant precision change was seen for CS-1SUP, retrievals from CS-1SDN
lead to a significant precision change for $T_{0}$, lapse rate, and
$T_{\infty}$ by a factor of 0.35, 2.47, and 14.49 respectively. CS-SELF
retrievals show strong biases among temperature parameters $T_{0}$
(-4.02$\sigma$), and similarly are the abundances of water (3.31$\sigma$),
carbon dioxide (6.91$\sigma$), methane (7.14$\sigma$), and ozone
(6.11$\sigma$). All three temperature parameters exhibit strong biases for
CS-500W: $T_{0}$ (-6.62$\sigma$), lapse rate (2.67$\sigma$), and $T_{\infty}$
(5.48$\sigma$). Similarly, molecular abundances of carbon monoxide
(-3.23$\sigma$), water (8.09$\sigma$), carbon dioxide (8.08$\sigma$), methane
(10.11$\sigma$), and ozone (9.83$\sigma$). Such biases are accompanied by
greater than 2$\times$ change in the precision for lapse rate (0.34),
$T_{\infty}$ (0.44), and the abundances of nitrogen (39.13) and hydrogen
(73.55). Retrievals using CS-MAXB show significant biases in $T_{0}$
(-6.61$\sigma$), lapse rate (3.0$\sigma$), molecular abundance of water
(6.13$\sigma$), carbon dioxide (9.19$\sigma$), methane (11.26$\sigma$), and
ozone (10.23$\sigma$) as shown in Supplementary Figure 4, while retrievals
using CS-MINB show significant biases in $T_{0}$ (6.19$\sigma$), and molecular
abundance of ozone (-4.41$\sigma$). CS-MAXB led to change in the precision of
posterior distribution for abundances of lapse rate (0.38), nitrogen (2.89),
and hydrogen (2.9), and similarly for CS-MINB led to change in precision for
$T_{0}$ (0.29), lapse rate (2.78), T∞ (15.61), and molecular abundances of
hydrogen (0.45). CS-EXML shows biases greater than 2$\sigma$ for all of the
parameters with the largest bias seen for the retrieved abundance of the
methane (-17.99$\sigma$): $T_{0}$ (-18.18$\sigma$, and precision factor of
0.31), lapse rate (precision factor of 4.66), $T_{\infty}$ (-19.37$\sigma$,
and precision factor of 0.18), molecular abundance of nitrogen (precision
factor of 0.21), carbon monoxide (2.73$\sigma$), water (-8.32$\sigma$), carbon
dioxide (-12.84$\sigma$), ozone (-6.61$\sigma$) and hydrogen (precision factor
0.4). Similarly to CS-EXML, CS-HTMP shows the strongest biases, due to a
difference in completeness with CS-DFLT. We find that the cross-retrieval with
CS-HTMP leads to PPDs often narrower than those of CS-EXML as well as an
atypical PPD for CO (see last subsection for discussion).

###  3.14 Warm-Jupiter Retrievals

Using CS-1SUP, most parameters are retrieved within less 2.5$\sigma$, except
for significant lapse rate (a factor of 2.4), see Supplementary Table 4. For
CS-1SDN, we similarly observe lapse rate a significant precision change of
lapse rate (2.32). Similar to the case of super-Earth, CS-SELF comparatively
exhibit stronger biases among numerous parameters as shown in Supplementary
Figure 6: $T_{0}$ (-7.54$\sigma$), lapse rate (2.87$\sigma$), molecular
abundances of carbon dioxide (3.37$\sigma$), and ozone (2.54$\sigma$). CS-500W
retrievals exhibited strong biases for $T_{0}$ (-7.89$\sigma$), lapse rate
(precision change factor of 2.12), and abundances of carbon monoxide
(-5.77$\sigma$). Retrievals using CS-MAXB showed significant biases in $T_{0}$
(-25.68$\sigma$), lapse rate (3.85$\sigma$), molecular abundance of carbon
monoxide (3.73$\sigma$), and ozone (2.73$\sigma$), while retrievals using CS-
MINB show significant biases just for molecular abundance of methane
(-2.60$\sigma$) and hydrogen (-11.1$\sigma$). The change in precision when
using CS-MAXB is for $T_{0}$ (0.2) whereas for CS-MINB it is observed for
lapse rate (11.85), $T_{\infty}$ (2.54). and hydrogen (0.25). Using CS-EXML
yield biases for the warm-Jupiter for methane (-11.2$\sigma$), and significant
change in the precision for lapse rate (10.85), and $T_{\infty}(11.36)$. As
for the super-Earth case, CS-HTMP and CS-EXML similarly show the strongest
biases, due to a primary difference CS-DFLT in the form of increased
completeness.

###  3.15 Information Content and Bias Propagation in Transmission Spectra

A planetary transmission spectrum encodes information on the pressure,
temperature, and the number densities at the pressure levels probed, which
thus also includes their dependence with altitude via quantities such as the
pressure scale height[24]. This complex information encoding is guided by the
opacity model, resulting in a complex $\chi^{2}$ topology that is strongly
sensitive to the assumptions and uncertainties of opacity models (see e.g.,
Figure 3 and Supplementary Figure 5). In this subsection, we focus on two
specific aspects of our sensitivity analysis to provide the reader with an
increased understanding of the opacity challenge. First, we look at the
uniquely sharp PPD of CO for the CS-HTMP retrieval of the super-Earth case
(see Figure 3, top right panel). Then, we turn to the correlation between
atmospheric parameters highlighted notably in Supplementary Figure 4.

The top right panel of Figure 3 highlights that CS-HTMP yields a uniquely
tight constraint on the base-pressure number density of CO (Log10NCO). We
leverage this unique characteristic to provide insights into why CS-HTMP
yields a topology of the $\chi^{2}$ distribution for the super-Earth case
resulting in such a different constraint on CO. For the case of the super-
Earth, CS-HTMP and CS-EXML lead to the strongest shift (and bias) towards
lower temperatures and lower number densities for all molecules but N2, H2,
and CO. This is due to the increased level of completeness of these cross-
sections yielding higher opacity in the valleys between molecular features and
resulting in biases in perceived number densities and base pressure. Indeed,
the same level of absorption in these valleys requires an overall lower base
pressure and number density for the related absorber. We note that this effect
dominates the perturbations induced in the cross-retrievals with CS-HTMP and
CS-EXML for the super-Earth case but not for the warm-Jupiter case owing to
the absence of scattering and CIA for the former, which exacerbates its
sensitivity to deeper inter-feature valleys in the VIS-NIR region of the
spectra (see Figure 3 vs Supplementary Figure 7). Second, CS-HTMP yields
smaller number densities than CS-EXML for all but H2 and CO, resulting in CO
bearing more of the constraints associated to the atmospheric scale height
(via the mean molecular weight) and the base pressure. Supplementary Figure 7
highlights this aspect by presenting the effect of perturbing Log10NCO from
the best-fit value for both CS-DFLT and CS-HTMP by -0.25, which correspond to
the 1$\sigma$ confidence interval for Log10NCO with CS-DFLT (Supplementary
Table 1). We show in the lower panel that doing so results in a worsening of
the $\chi^{2}$ over three and a half times more significant for CS-HTMP than
for CS-DFLT, echoing the differences seen in the top right panel of Figure 3
(namely a tighter constraint on CO). In addition, the bottom panel of
Supplementary Figure 7 highlights that in both cases the primary effect of
perturbing Log10NCO for this super-Earth case leads to amplifying/shrinking
all molecular features more so than only changing the molecular features of
CO, meaning that the primary effect of such a perturbation is indeed on the
mean molecular weight (and to a lesser extent the base pressure).

In Supplementary Figure 4, we report the PPDs of the retrieved parameters for
the case of the super-Earth for self-retrieval (CS-DFLT, red) and cross-
retrieval (CS-MAXB, green). The only difference between the two cross-sections
relates to broadening; CS-DFLT assumes air-broadening (geocentric) while CS-
MAXB assumes twice the maximum between air- and self-broadening. This
perturbation is only impacting pressure levels at which pressure-related
broadening effect dominate temperature-related broadening effect. Therefore,
above $\sim$0.1 mbar the p-T profiles retrieved are consistent, and
$T_{\infty}$ is not biased (see top right panel). At deeper levels, the
increased broadening for CS-MAXB implies–all other things being constant–a
decrease of line intensities and an increase in line width. The primary
compensation mechanism for this perturbation is an increase in the number
densities for molecules with strong molecular features and a decrease in
temperature. This explains the trend seen between $T_{0}$ and CO2, CH4, H2O
and O3 within and between the PPDs of CS-DFLT and CS-MAXB.

##  4 Captions of Figures

Figure 1. Ensemble of opacity-model perturbations at the level of an opacity
cross-section. a: Calculated cross-sections showing the 1.3663 micron water
line at 450 K, 0.1 atm, and 2-million resolution. Each cross-section is
identified by its color and label on the right and is generated following
specific assumptions on parameters such as broadening coefficients, far-wing
behavior, and data source (see section Opacity Database & Cross-Sections in
Methods). The 1$\sigma$ envelope is the bound between CS-1SUP and CS-1SDN,
which respectively account for the +1 and -1$\sigma$ perturbations on line
parameters stemming from lab measurements (incl., pressure broadening and line
intensity and position). CS-EXML based on ExoMol[22] differs from the other
models at pressure above $\sim$1 mbar due to the absence of pressure shift on
the line positions. b: Calculated cross-sections showing 1.4 micron water
feature at 450 K and 100 atm.

Figure 2. Propagation of the ensemble of opacity-model perturbations to the
level of a transmission spectrum. Comparison of synthetic warm-Jupiter
transmission spectra for a combined broadband NIRSpec and MIRI observation for
the nine different opacity models used in this perturbation/sensitivity
analysis (see section Opacity Database & Cross-Sections in Methods) (Top).
Non-random differences with amplitudes ranging from 20 to 150 part per
millions (p.p.m.) are observed compared to the nominal (i.e., unperturbed) CS-
DFLT model (Bottom), which is substantial in the era of JWST.

Figure 3. Transmission-spectrum fit quality unaffected by opacity-model
perturbations. Best-fit model for the cross-retrieval of a warm-Jupiter from
its synthetic transmission spectrum simulated with nominal cross-section CS-
DFLT and retrieved with cross-section CS-SELF (see Methods) (Top).
Spectroscopic features/bands caused by absorption have been indicated. Despite
the significant differences among the opacity models (Figures 1 & 2), the
absence of structure in the residuals with $\chi^{2}_{\nu}$=1.068 (Bottom)
implies that a good fit is enabled by compensation errors on the retrieved
atmospheric parameters (i.e., biases). This implies that the limitations of
current opacity models may remain undetected in upcoming atmospheric retrieval
efforts.

Figure 4. Propagation of the ensemble of opacity-model perturbations to the
level of retrieved atmospheric properties. Posterior probability distributions
(PPDs) of the retrieved atmospheric parameters (i.e., final data product) for
the super-Earth scenario highlighting the biases induced by perturbations to
our opacity model. Each cross-section is identified by its color and label on
the right. The dotted black vertical lines represent the true values used in
generating the synthetic spectrum. Deviations with a statistical significance
of up to $\sim$20$\sigma$ and physical significance of over 1 dex are
reported. This suggests the significant sensitivity of retrieved atmospheric
properties to opacity models in upcoming atmospheric retrieval efforts.

Figure 5. The 0.5-1.0 dex opacity-driven accuracy wall for exoplanet
atmospheric characterization. a: Relationship between the Spectral Statistical
Distance (SSD) and the Parametric Statistical Distance (PSD) relating the
statistical deviation between the transmission spectrum generated with a
cross-section and the one generated with the nominal cross-section to the
statistical deviation between the PPDs retrieved through cross- and self-
retrieval. b: Relationship between the SSD and the Parametric Physical
Distance revealing biases between 0.5 to 1 dex on the retrieved parameters,
labelled “Opacity-driven Accuracy Wall”. This translates into an accuracy drop
from what the instrument alone could deliver as highlighted by the shaded
green area for our observation scenario and the dotted line for Cycle 1
Program. The filled markers show the maximum distance amongst the retrieved
atmospheric parameters, the open markers their median distance.

P.N. acknowledges the support of the Grayce B. Kerr Fellowship Fund and Elliot
Fellowship at MIT. The authors would like to thank Dr. B. V. Rackham, Dr. E.
K. Conway, and Prof. S. Seager for discussions on various topics presented in
this paper. P. N. would like to thank Dr. P. Molliere for benchmark testing
with petitRadtrans. The authors acknowledge the MIT SuperCloud and Lincoln
Laboratory Supercomputing Center for providing (HPC, database, consultation)
resources that have contributed to the research results reported within this
paper. The authors express gratitude to the reviewers for their comments and
suggestions that resulted in a substantial improvement of the manuscript.

J.d.W. designed the study. P.N. developed the computational framework for the
study with the support of J.d.W., I.E.G., R.J.H., C.S.-S., and R.V.K., P.N.
and J.d.W. led the analysis and interpretation with the support of I.E.G.,
R.J.H., and C.S.-S. I.E.G., R.J.H. and C.S.-S. lead the discussions regarding
future avenues for improving opacity models. Every author contributed to
writing the manuscript.

The authors declare that they have no competing financial interests.

This paper makes use of the opacity data from the HITRAN2016[21],
HITEMP2010[23] and ExoMol[22] databases. The perturbed opacity cross-sections
and the synthetic exoplanet transmission spectra are available from the
corresponding author upon request and can be (re)generated using the
tierraCrossSection and tierra codes (see Code Availability), respectively.

This work makes use of the following publicly available codes: emcee[42],
ExoCross[43], hapi[20]. Additionally, it is based on tierra which we is now
publicly available at https://github.com/disruptiveplanets /tierra, and
tierraCrossSection which is also available at
https://github.com/disruptiveplanets/tierraCrossSection.

Correspondence and requests for materials should be addressed to JdW (email:
jdewit@mit.edu).

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##  5 Tables and Figures for Methods

Supplementary figure 1:  Framework for sensitivity analysis of retrieved
atmospheric properties to opacity model. The four building blocks of the
transmission spectra are shown in green, analysis techniques are marked in
blue, and the orange parallelogram highlights the remote sensing technique at
the center of this perturbation/sensitivity analysis. The arrows are shown to
indicate the direction of the information flow. We perform self- and cross-
retrieval for two planetary cases (a super-Earth around M-dwarf and a Jupiter-
sized planet around K-dwarf, see Supplementary Table 2) with nine distinct
sets of cross-sections (see Supplementary Table 1) to access their impacts.
Table 1: Parameters and line list sources used in the generation of our set of
nine different cross-sections

Profile: | Voigt
---|---
Resolution: | 100,000
Wavelength Range: | 0.6 $\mu$m to 30 $\mu$m
Molecules: | H2O, CO, CO2, O3, CH4, N2, H2 |
Line List Source |
HITRAN: | H2O [G17], CO [L15, G17], CO2 [G17], O3 [G17], CH4 [G17], N2 [G17], H2 [W98, G17]
ExoMol: | H2O [P18], CO [L15], CO2 [Y20], CH4 [Y14, Y17], H2 [R19]
HITEMP: | CH4 [H20], H2O [R10], CO2 [R10], CO [L15]
Rayleigh: | H2O [SU05], CO [SU05], CO2 [SU05], CH4 [SU05], N2 [SU05], H2 [D65] |
CIA: | H2-H2 [A11], H2-He [A12], N2-N2 [L96, K15, H17] |
Grid |
Temperature (K): | [100, 110, 120, 130, 140, 160, 180, 200, 230, 260, 290, 330, 370, 410, 460, 510, 580, 650, 730, 810]
Log10 Pressure (atm): | [ -5.00, -4.20, -3.80, -3.50, -3.25, -3.05, -2.95, -2.85, -2.75, -2.65, -2.55, -2.45, -2.30, -2.15, -2.00, -1.85,
| -1.70, -1.55, -1.4, -1.25, -1.1, -0.95, -0.80, -0.70, -0.60, -0.50, -0.40,
-0.30, -0.20, 0.10, 0.00, 0.10, 0.20,
| 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, 0.90, 1.00, 1.10, 1.20, 1.30, 1.40,
1.50, 1.60, 1.70, 1.80, 1.90, 2.00]
Individual Parameters
| CS-DFLT | CS-1SUP | CS-1SDN | CS-SELF | CS-500W | CS-MAXB | CS-MINB | CS-EXML | CS-HTMP
Values | Center | +1$\sigma^{\dagger}$ | -1$\sigma^{\dagger}$ | Center | Center | Center | Center | Center | Center
Broadening ($\gamma$) | Air ($\gamma_{A}$) | $\gamma_{A}$ | $\gamma_{A}$ | Self ($\gamma_{s}$) | $\gamma_{A}$ | 2$\cdot$max($\gamma_{A},\gamma_{S}$) |  $\frac{1}{2}\cdot$ min($\gamma_{A},\gamma_{S}$) |  $\gamma$=0.07, n‡=0.5 | $\gamma_{A}$
Linewing (HWHM) | 50 | 50 | 50 | 50 | 500 | 50 | 50 | 50 | 50
Database | HITRAN | HITRAN | HITRAN | HITRAN | HITRAN | HITRAN | HITRAN | ExoMol | HITEMP

References: G17[21], L15[64], W98[65], P18[66], Y20[67], Y14[68], Y17[69],
R19[70], SU05[46], D65[45], H20[26], R10[23], A11[48], A12[49], L96[50],
K15[51], H17[52].
$\dagger$Perturbed line position, line intensity, pressure broadening, and
temperature exponent of the pressure broadening.

$\ddagger$Temperature exponent dependence of the pressure broadening.

Supplementary figure 2:  Measurement uncertainties on absorption line
parameters reported in HITRAN. Uncertainties on the line parameters vs line
intensities as reported in HITRAN for methane (red), water (yellow), ozone
(blue), and carbon dioxide (brown). For each uncertainty range reported (y
axis), the mean and median line intensity values are shown as empty and full
diamonds, with the 1 and 2$\sigma$ intensity ranges. The number of lines in
each uncertainty range is reported on the right side of its 2$\sigma$
interval. These uncertainties are used to perturb the four different
parameters in the generation of CS-1SUP and CS-1SDN. The parameters of the
strongest lines are reported with the smallest uncertainties. a: Uncertainty
on the line position. b: Uncertainty on the line intensity. c: Uncertainty on
the air broadening coefficient. d: Uncertainty on the temperature dependency
of the air broadening coefficient. The uncertainty codes (average, default,
unreported) are introduced in Ref.[44]. Supplementary figure 3:  Benchmarking
of tierra against petitRadtrans. Benchmarking of tierra (R=100,000) against
petitRadtrans for a Jupiter-sized planet around a K dwarf (0.55 R⊙) for 365 K
isothermal temperature with base pressure of 1 atm containing water at mixing
ratio of $10^{-5}$ for a combined NIRSpec and MIRI observation. We use the
cross-sections from petitRadtrans, which present a lower resolution on their
temperature and pressure grids, to focus this benchmarking on tierra rather
than possible difference on underlying cross-sections. The median absolute
deviation between two models is 1.95 p.p.m. and RMS is 4.4 ppm which is
marginal in comparison to the deviations seen in the model comparisons (Figure
LABEL:fig:ModelComparison).

Table 2: Parameters used for the generation of synthetic model  super-Earth
---
$\rm M_{Pl}$ | 1 M⊕ | T0 | 300 K
$\rm R_{Pl}$ | 1.1 R⊕ | T∞ | 400 K
$\rm M_{*}$ | 0.1 M⊙ | HT | 10-2 km-1
$\rm R_{*}$ | 0.1 M⊙ | H-Mag | 10.0
MR${}_{\rm N_{2}}$ | 3.0$\times 10^{-1}$ | MR${}_{\rm CH_{4}}$ | 1.0$\times 10^{-1}$
MR${}_{\rm CO_{2}}$ | 1.0$\times 10^{-3}$ | MR${}_{\rm O_{3}}$ | 1.0$\times 10^{-4}$
MRCO | 1.0$\times 10^{-1}$ | MR${}_{\rm H_{2}}$ | 3.391$\times 10^{-1}$
MR${}_{\rm H_{2}O}$ | 1.0$\times 10^{-1}$ | P0 | 1.0 atm
Warm-Jupiter
$\rm M_{Pl}$ | 1.0 MJup | T0 | 500 K
$\rm R_{Pl}$ | 1.0 RJup | T∞ | 600 K
$\rm M_{*}$ | 0.55 M⊙ | HT | 10-2 km-1
$\rm R_{*}$ | 0.55 M⊙ | H-Mag | 10.0
MR${}_{\rm N_{2}}$ | 1.0$\times 10^{-5}$ | MR${}_{\rm CH_{4}}$ | 2.0$\times 10^{-5}$
MR${}_{\rm CO_{2}}$ | 2.0$\times 10^{-5}$ | MR${}_{\rm O_{3}}$ | 1.0$\times 10^{-6}$
MRCO | 1.0$\times 10^{-5}$ | MR${}_{\rm H_{2}}$ | 8.499$\times 10^{-1}$
MR${}_{\rm H_{2}O}$ | 2.0$\times 10^{-5}$ | P0 | 1.0 atm
Table 3: Retrieved parameters$\dagger$ and their observed biases$\ddagger$ for
the case of super-Earth

| CS-DFLT | CS-1SUP | CS-1SDN | CS-SELF | CS-500W | CS-MAXB | CS-MINB | CS-EXML | CS-HTMP
---|---|---|---|---|---|---|---|---|---
T0 [K] | $286.5^{+24.63}_{-45.66}$ | $206.29^{+50.94}_{-56.13}$ | $337.84^{+7.47}_{-13.2}$ | $186.53^{+28.22}_{-29.98}$ | $147.26^{+23.07}_{-20.22}$ | $130.66^{+25.6}_{-19.49}$ | $357.93^{+7.53}_{-9.36}$ | $110.21^{+10.44}_{-7.67}$ | $107.67^{+9.93}_{-5.39}$
| -0.55$\sigma$ | -1.84$\sigma$ $|$ 1.82 |  2.87$\mathbf{\sigma}$ $|$ 0.35 |  -4.02$\mathbf{\sigma}$ $|$ 0.99 |  -6.62$\mathbf{\sigma}$ $|$ 0.73 |  -6.61$\mathbf{\sigma}$ $|$ 0.77 |  6.19$\mathbf{\sigma}$ $|$ 0.29 |  -18.18$\mathbf{\sigma}$ $|$ 0.31 |  -19.37$\mathbf{\sigma}$ $|$ 0.22
Log10HT [m-1] | $2.05^{+0.24}_{-0.21}$ | $-1.7^{+0.26}_{-0.18}$ | $-2.94^{+0.49}_{-0.3}$ | $-1.84^{+0.09}_{-0.09}$ | $-1.84^{+0.05}_{-0.06}$ | $-1.79^{+0.05}_{-0.07}$ | $-3.01^{+0.43}_{-0.46}$ | $0.11^{+0.63}_{-0.86}$ | $0.29^{+0.47}_{-0.49}$
| -0.21$\sigma$ | 1.67$\sigma$ $|$ 1.38 | -1.92$\sigma$ $|$ 2.47 | 1.78$\sigma$ $|$ 0.56 |  2.67$\mathbf{\sigma}$ $|$ 0.34 |  3.0$\mathbf{\sigma}$ $|$ 0.38 | -2.35$\sigma$ $|$ 2.78 | 2.45$\sigma$ $|$ 4.66 |  4.67$\mathbf{\sigma}$ $|$ 2.13
T∞ [K] | $412.21^{+16.83}_{-10.86}$ | $399.21^{+9.94}_{-6.14}$ | $553.88^{+166.78}_{-112.72}$ | $405.76^{+8.22}_{-7.18}$ | $421.64^{+4.48}_{-3.95}$ | $408.89^{+7.25}_{-5.63}$ | $531.03^{+210.67}_{-90.37}$ | $370.75^{+1.51}_{-1.89}$ | $371.36^{+1.88}_{-1.87}$
| 1.12$\sigma$ | -0.08$\sigma$ $|$ 0.83 | 1.37$\sigma$ $|$ 14.49 | 0.8$\sigma$ $|$ 0.8 |  5.48$\mathbf{\sigma}$ $|$ 0.44 | 1.58$\sigma$ $|$ 0.67 | 1.45$\sigma$ $|$ 15.61 |  -19.37$\mathbf{\sigma}$ $|$ 0.18 | -15.23$\mathbf{\sigma}$ $|$ 0.14
Log10N${}_{\rm N_{2}}$ [m-3] | $23.24^{+0.13}_{-0.13}$ | $23.26^{+0.28}_{-0.35}$ | $23.08^{+0.08}_{-0.11}$ | $23.19^{+0.29}_{-0.33}$ | $11.74^{+7.1}_{-7.77}$ | $23.12^{+0.38}_{-0.72}$ | $23.17^{+0.08}_{-0.13}$ | $23.24^{+0.04}_{-0.04}$ | $23.27^{+0.06}_{-0.07}$
| 0.18$\sigma$ | 0.13$\sigma$ $|$ 1.66 | -1.7$\sigma$ $|$ 0.5 | -0.09$\sigma$ $|$ 1.63 | -1.62$\sigma$ $|$ 39.13 | -0.25$\sigma$ $|$ 2.89 | -0.58$\sigma$ $|$ 0.55 | 0.59$\sigma$ $|$ 0.21 | 0.77$\sigma$ $|$ 0.5
Log10NCO [m-3] | $22.63^{+0.24}_{-0.31}$ | $23.05^{+0.23}_{-0.27}$ | $22.05^{+0.34}_{-0.44}$ | $23.08^{+0.25}_{-0.39}$ | $21.9^{+0.26}_{-0.27}$ | $23.12^{+0.33}_{-0.4}$ | $22.58^{+0.33}_{-0.37}$ | $22.03^{+0.26}_{-0.32}$ | $22.96^{+0.04}_{-0.04}$
| -0.45$\sigma$ | 1.15$\sigma$ $|$ 1.11 | -2.03$\sigma$ $|$ 1.73 | 0.87$\sigma$ $|$ 1.42 |  -3.23$\mathbf{\sigma}$ $|$ 1.18 | 0.95$\sigma$ $|$ 1.62 | -0.48$\sigma$ $|$ 1.56 |  -2.73$\mathbf{\sigma}$ $|$ 1.29 | 5.52$\mathbf{\sigma}$ $|$ 0.15
Log10N${}_{\rm H_{2}O}$ [m-3] | $22.9^{+0.08}_{-0.09}$ | $22.93^{+0.09}_{-0.06}$ | $22.83^{+0.06}_{-0.06}$ | $23.07^{+0.11}_{-0.1}$ | $23.71^{+0.12}_{-0.12}$ | $23.37^{+0.1}_{-0.1}$ | $22.69^{+0.06}_{-0.06}$ | $22.24^{+0.06}_{-0.04}$ | $22.12^{+0.03}_{-0.03}$
| 1.79$\sigma$ |  3.18$\mathbf{\sigma}$ $|$ 0.79 | 1.52$\sigma$ $|$ 0.63 |  3.31$\mathbf{\sigma}$ $|$ 1.11 |  8.09$\mathbf{\sigma}$ $|$ 1.26 |  6.31$\mathbf{\sigma}$ $|$ 1.05 | -0.82$\sigma$ $|$ 0.63 |  -8.32$\mathbf{\sigma}$ $|$ 0.53 | -20.64$\mathbf{\sigma}$ $|$ 0.35
Log10N${}_{\rm CO_{2}}$ [m-3] | $20.88^{+0.11}_{-0.1}$ | $20.93^{+0.11}_{-0.09}$ | $20.82^{+0.09}_{-0.08}$ | $21.43^{+0.11}_{-0.1}$ | $21.79^{+0.13}_{-0.13}$ | $21.75^{+0.09}_{-0.11}$ | $20.68^{+0.1}_{-0.1}$ | $19.84^{+0.07}_{-0.07}$ | $19.59^{+0.07}_{-0.07}$
| 1.41$\sigma$ | 2.12 $|$ 0.95 | 1.01$\sigma$ $|$ 0.81 |  6.91$\mathbf{\sigma}$ $|$ 1.0 |  8.08$\mathbf{\sigma}$ $|$ 1.24 |  9.19$\mathbf{\sigma}$ $|$ 0.95 | -0.59$\sigma$ $|$ 0.95 |  -12.84$\mathbf{\sigma}$ $|$ 0.67 | -16.42$\mathbf{\sigma}$ $|$ 0.67
Log10N${}_{\rm CH_{4}}$ [m-3] | $22.88^{+0.07}_{-0.08}$ | $22.82^{+0.09}_{-0.06}$ | $22.94^{+0.06}_{-0.05}$ | $23.31^{+0.1}_{-0.08}$ | $23.75^{+0.11}_{-0.1}$ | $23.64^{+0.09}_{-0.08}$ | $22.65^{+0.06}_{-0.05}$ | $21.48^{+0.07}_{-0.02}$ | $21.42^{+0.02}_{-0.02}$
| 1.76$\sigma$ | 1.35$\sigma$ $|$ 0.94 |  4.02$\mathbf{\sigma}$ $|$ 0.69 |  7.14$\mathbf{\sigma}$ $|$ 1.12 |  10.11$\mathbf{\sigma}$ $|$ 1.31 |  11.26$\mathbf{\sigma}$ $|$ 1.06 | -1.48$\sigma$ $|$ 0.69 |  -17.99$\mathbf{\sigma}$ $|$ 0.56 | -65.95$\mathbf{\sigma}$ $|$ 0.27
Log10N${}_{\rm O_{3}}$ [m-3] | $19.92^{+0.09}_{-0.09}$ | $19.84^{+0.11}_{-0.07}$ | $19.94^{+0.08}_{-0.07}$ | $20.35^{+0.1}_{-0.1}$ | $20.82^{+0.12}_{-0.11}$ | $20.66^{+0.11}_{-0.09}$ | $19.66^{+0.09}_{-0.07}$ | $19.05^{+0.08}_{-0.05}$ | $18.86^{+0.05}_{-0.05}$
| 2.01$\sigma$ | 1.44$\sigma$ $|$ 0.95 |  2.87$\mathbf{\sigma}$ $|$ 0.79 |  6.11$\mathbf{\sigma}$ $|$ 1.05 |  9.83$\mathbf{\sigma}$ $|$ 1.21 |  10.23$\mathbf{\sigma}$ $|$ 1.05 | -0.88$\sigma$ $|$ 0.84 |  -8.61$\mathbf{\sigma}$ $|$ 0.68 | -17.58$\mathbf{\sigma}$ $|$ 0.56
Log10N${}_{\rm H_{2}}$ [m-3] | $23.25^{+0.12}_{-0.12}$ | $23.4^{+0.13}_{-0.13}$ | $23.03^{+0.08}_{-0.09}$ | $23.36^{+0.18}_{-0.18}$ | $11.75^{+6.75}_{-7.96}$ | $23.39^{+0.21}_{-0.37}$ | $23.17^{+0.05}_{-0.04}$ | $23.38^{+0.06}_{-0.02}$ | $23.56^{+0.04}_{-0.04}$
| -1.17$\sigma$ | 0.07$\sigma$ $|$ 1.3 |  -4.51$\mathbf{\sigma}$ $|$ 0.85 | -0.17$\sigma$ $|$ 1.8 | -1.72$\sigma$ $|$ 73.55 | -0.0$\sigma$ $|$ 2.9 |  -4.41$\mathbf{\sigma}$ $|$ 0.45 | -0.17$\sigma$ $|$ 0.4 | 4.24$\mathbf{\sigma}$ $|$ 0.33
Red. $\chi^{2}_{\nu}$ | 0.942 | 1.002 | 1.079 | 0.974 | 1.179 | 0.985 | 0.940 | 6.504 | 4.89
SSD | - | 276.76 | 512.77 | 983.69 | 1243.21 | 2501.09 | 247.28 | 12624.09 | 12035.61
PSD | - | 4.74 | 8.91 | 39.47 | 33.68 | 50.2 | 5.79 | 245.63 | 383.04

$\dagger$ Retrieved parameters are introduced on the first line of a cell with
its upper and lower 1$\sigma$ confidence interval.

$\ddagger$ Observed biases are introduced on the second line of a cell as the
difference between the retrieved parameter and the nominal value divided by
the retrieved uncertainty interval. The second number reported on this line is
the ratio between retrieved uncertainty interval and nominal uncertainty
interval to highlight an artificial amplification or reduction of the
interval. Biases over 2.5$\sigma$ and interval amplification/reduction over
2/below 0.5 are shown in bold.

Supplementary figure 4: Corner plot of the posterior probability distribution
of the atmospheric parameters for a super-Earth. Corner plot showing the PPDs
of the retrieved parameters for the case of the super-Earth for self-retrieval
(CS-DFLT, red) and cross-retrieval (CS-MAXB, green). The only difference
between the two cross-sections relate to broadening; CS-DFLT assumes air-
broadening (geocentric) while CS-MAXB assumes twice the maximum between air-
and self-broadening. Strong biases are seen in the retrieved value of water
(-6.97$\sigma$), carbon dioxide (-6.23$\sigma$), methane (-6.23$\sigma$), and
ozone (-6.23$\sigma$). (Top Right) 500 random PT profiles constructed from the
posteriors are shown for comparison against the true profile shown in black.
The dotted line shows the contribution factor can change significantly due to
the changes in the broadening values. Table 4: Retrieved parameters$\dagger$
and their observed biases$\ddagger$ for the case of warm-Jupiter

| CS-DFLT | CS-1SUP | CS-1SDN | CS-SELF | CS-500W | CS-MAXB | CS-MINB | CS-EXML | CS-HTMP
---|---|---|---|---|---|---|---|---|---
T0 [K] | $418.94^{+66.68}_{-58.25}$ | $437.22^{+73.41}_{-61.21}$ | $418.88^{+69.2}_{-59.97}$ | $219.09^{+37.27}_{-32.61}$ | $200.77^{+37.91}_{-34.07}$ | $116.32^{+14.94}_{-10.63}$ | $674.88^{+100.0}_{-76.54}$ | $568.35^{+14.47}_{-50.67}$ | $599.42^{+120.0}_{-88.46}$
| -1.22$\sigma$ | -0.86$\sigma$ $|$ 1.08 | -1.17$\sigma$ $|$ 1.03 |  -7.54$\mathbf{\sigma}$ $|$ 0.56 |  -7.89$\mathbf{\sigma}$ $|$ 0.58 |  -25.68$\mathbf{\sigma}$ $|$ 0.2 | 2.28$\sigma$ $|$ 1.41 | 1.35$\sigma$ $|$ 0.52 | 1.12$\sigma$ $|$ 1.67
Log10HT [m-1] | $-1.75^{+0.38}_{-0.22}$ | $-1.66^{+1.15}_{-0.29}$ | $-1.7^{+1.11}_{-0.28}$ | $-1.57^{+0.26}_{-0.15}$ | $-1.13^{+0.92}_{-0.35}$ | $-1.5^{+0.2}_{-0.13}$ | $-0.9^{+1.3}_{-5.81}$ | $-3.21^{+1.89}_{-4.62}$ | $-0.96^{+1.26}_{-0.97}$
| 1.14$\sigma$ | 1.17$\sigma$ $|$ 2.4 | 1.07$\sigma$ $|$ 2.32 |  2.87$\mathbf{\sigma}$ $|$ 0.68 | 2.49$\sigma$ $|$ 2.12 |  3.85$\mathbf{\sigma}$ $|$ 0.55 | 0.19$\sigma$ $|$ 11.85 | -0.64$\sigma$ $|$ 10.85 | 1.07$\sigma$ $|$ 3.72
T∞ [K] | $602.77^{+6.41}_{-5.44}$ | $604.7^{+6.95}_{-5.72}$ | $599.2^{+8.11}_{-5.7}$ | $592.6^{+4.56}_{-4.52}$ | $599.15^{+4.72}_{-4.74}$ | $591.11^{+4.44}_{-4.21}$ | $600.22^{+21.54}_{-8.61}$ | $600.47^{+112.48}_{-22.18}$ | $579.1^{+2.86}_{-3.46}$
| 0.51$\sigma$ | 0.82$\sigma$ $|$ 1.07 | -0.1$\sigma$ $|$ 1.17 | -1.62$\sigma$ $|$ 0.77 | -0.18$\sigma$ $|$ 0.8 | -2.0$\sigma$ $|$ 0.73 | 0.03$\sigma$ $|$ 2.54 | 0.02$\sigma$ $|$ 11.36 | -7.31$\mathbf{\sigma}$ $|$ 0.53
Log10N${}_{\rm N_{2}}$ [m-3] | $9.98^{+6.85}_{-6.67}$ | $10.22^{+6.85}_{-7.04}$ | $11.18^{+6.96}_{-7.56}$ | $10.5^{+6.59}_{-6.98}$ | $9.96^{+7.75}_{-6.54}$ | $10.92^{+6.78}_{-7.36}$ | $10.92^{+6.73}_{-7.44}$ | $10.19^{+5.97}_{-6.52}$ | $11.38^{+5.42}_{-6.69}$
| -1.25$\sigma$ | -1.21$\sigma$ $|$ 1.03 | -1.05$\sigma$ $|$ 1.07 | -1.22$\sigma$ $|$ 1.0 | -1.1$\sigma$ $|$ 1.06 | -1.12$\sigma$ $|$ 1.05 | -1.13$\sigma$ $|$ 1.05 | -1.39$\sigma$ $|$ 0.92 | -1.66$\sigma$ $|$ 1.09
Log10NCO [m-3] | $19.32^{+0.26}_{-0.29}$ | $19.38^{+0.25}_{-0.28}$ | $19.17^{+0.25}_{-0.33}$ | $19.64^{+0.28}_{-0.3}$ | $18.42^{+0.19}_{-0.2}$ | $19.73^{+0.27}_{-0.28}$ | $19.19^{+0.29}_{-0.3}$ | $19.13^{+0.23}_{-0.3}$ | $19.07^{+0.27}_{-0.33}$
| -0.76$\sigma$ | -0.55$\sigma$ $|$ 0.96 | -1.39$\sigma$ $|$ 1.05 | 0.41$\sigma$ $|$ 1.05 |  -5.77$\mathbf{\sigma}$ $|$ 0.71 | 0.76$\sigma$ $|$ 1.0 | -1.13$\sigma$ $|$ 1.07 | -1.68$\sigma$ $|$ 0.96 | -1.66$\sigma$ $|$ 1.09
Log10N${}_{\rm H_{2}O}$ [m-3] | $18.84^{+0.07}_{-0.06}$ | $18.81^{+0.08}_{-0.06}$ | $18.83^{+0.08}_{-0.07}$ | $18.72^{+0.08}_{-0.08}$ | $18.9^{+0.09}_{-0.07}$ | $18.78^{+0.09}_{-0.1}$ | $18.78^{+0.04}_{-0.04}$ | $18.85^{+0.06}_{-0.04}$ | $18.84^{+0.07}_{-0.06}$
| 0.36$\sigma$ | -0.1$\sigma$ $|$ 1.08 | 0.17$\sigma$ $|$ 1.15 | -1.23$\sigma$ $|$ 1.23 | 1.17$\sigma$ $|$ 1.23 | -0.43$\sigma$ $|$ 1.46 | -0.96$\sigma$ $|$ 0.62 | 0.79$\sigma$ $|$ 0.77 | 0.36$\sigma$ $|$ 1.0
Log10N${}_{\rm CO_{2}}$ [m-3] | $18.54^{+0.07}_{-0.07}$ | $18.48^{+0.07}_{-0.07}$ | $18.58^{+0.08}_{-0.07}$ | $18.82^{+0.08}_{-0.09}$ | $18.62^{+0.09}_{-0.08}$ | $18.89^{+0.09}_{-0.1}$ | $18.49^{+0.06}_{-0.05}$ | $18.57^{+0.06}_{-0.05}$ | $18.62^{+0.06}_{-0.06}$
| 0.33$\sigma$ | -0.53$\sigma$ $|$ 1.0 | 0.9$\sigma$ $|$ 1.07 |  3.37$\mathbf{\sigma}$ $|$ 1.21 | 1.29$\sigma$ $|$ 1.21 |  3.73$\mathbf{\sigma}$ $|$ 1.36 | -0.45$\sigma$ $|$ 0.79 | 1.06$\sigma$ $|$ 0.79 | 1.72$\sigma$ $|$ 0.86
Log10N${}_{\rm CH_{4}}$ (m-3) | $18.82^{+0.06}_{-0.05}$ | $18.69^{+0.06}_{-0.05}$ | $18.94^{+0.07}_{-0.05}$ | $18.99^{+0.06}_{-0.08}$ | $18.86^{+0.08}_{-0.06}$ | $19.06^{+0.09}_{-0.1}$ | $18.74^{+0.03}_{-0.03}$ | $18.37^{+0.04}_{-0.03}$ | $18.4^{+0.04}_{-0.03}$
| 0.04$\sigma$ | -2.13$\sigma$ $|$ 1.0 | 2.44$\sigma$ $|$ 1.09 | 2.15$\sigma$ $|$ 1.27 | 0.7$\sigma$ $|$ 1.27 | 2.42$\mathbf{\sigma}$ $|$ 1.73 |  -2.6$\mathbf{\sigma}$ $|$ 0.55 |  -11.2$\mathbf{\sigma}$ $|$ 0.64 | -10.45$\mathbf{\sigma}$ $|$ 0.64
Log10N${}_{\rm O_{3}}$ (m-3) | $17.55^{+0.07}_{-0.07}$ | $17.45^{+0.07}_{-0.06}$ | $17.66^{+0.08}_{-0.07}$ | $17.72^{+0.07}_{-0.08}$ | $17.63^{+0.08}_{-0.07}$ | $17.79^{+0.09}_{-0.1}$ | $17.49^{+0.05}_{-0.05}$ | $17.56^{+0.06}_{-0.06}$ | $17.56^{+0.07}_{-0.06}$
| 0.47$\sigma$ | -0.96$\sigma$ $|$ 0.93 | 2.04$\sigma$ $|$ 1.07 |  2.54$\mathbf{\sigma}$ $|$ 1.07 | 1.61$\sigma$ $|$ 1.07 |  2.73$\mathbf{\sigma}$ $|$ 1.36 | -0.54$\sigma$ $|$ 0.71 | 0.72$\sigma$ $|$ 0.86 | 0.72$\sigma$ $|$ 0.93
Log10N${}_{\rm H_{2}}$ (m-3) | $23.45^{+0.04}_{-0.05}$ | $23.42^{+0.05}_{-0.04}$ | $23.45^{+0.06}_{-0.05}$ | $23.56^{+0.05}_{-0.07}$ | $23.45^{+0.08}_{-0.05}$ | $23.64^{+0.08}_{-0.09}$ | $23.36^{+0.01}_{-0.01}$ | $23.42^{+0.03}_{-0.02}$ | $23.42^{+0.03}_{-0.02}$
| -0.53$\sigma$ | -1.02$\sigma$ $|$ 1.0 | -0.35$\sigma$ $|$ 1.22 | 1.27$\sigma$ $|$ 1.33 | -0.26$\sigma$ $|$ 1.44 | 1.88$\sigma$ $|$ 1.89 |  -11.1$\mathbf{\sigma}$ $|$ 0.22 | -1.7$\sigma$ $|$ 0.56 | -1.7$\sigma$ $|$ 0.56
Red. $\chi^{2}_{\nu}$ | 1.038 | 1.065 | 1.031 | 1.068 | 1.076 | 1.069 | 1.046 | 1.996 | 1.788
SSD | $-$ | 89.79 | 139.84 | 613.77 | 93.83 | 1709.87 | 119.44 | 3655.73 | 3536.44
PSD | - | 4.90 | 7.88 | 19.37 | 6.44 | 27.67 | 6.56 | 108.42 | 116.45

$\dagger$ Retrieved parameters are introduced on the first line of a cell with
its upper and lower 1 sigma confidence interval.

$\ddagger$ Observed biases are introduced on the second line of a cell as the
difference between the retrieved parameter and the nominal value divided by
the retrieved uncertainty interval. The second number reported on this line is
the ratio between retrieved uncertainty interval and nominal uncertainty
interval to highlight an artificial amplification or reduction of the
interval. Biases over 2.5$\sigma$ and interval amplification/reduction over
2/below 0.5 are shown in bold.

Supplementary figure 5:  Propagation of the ensemble of opacity-model
perturbations to the level of retrieved atmospheric properties for the warm-
Jupiter scenario. Posterior probability distributions (PPDs) of the retrieved
atmospheric parameters for the warm-Jupiter scenario highlighting the biases
induced by perturbations to the opacity model (see Methods). Each cross-
section is identified by its color and label on the right. The dotted black
vertical lines represent the true values used in generating the synthetic
spectrum. Deviations with a statistical significance of up to 20$\sigma$ and
physical significance of over 1 dex are reported. Supplementary figure 6:
Corner plot of the posterior probability distribution of the atmospheric
parameters for a warm-Jupiter. Corner plot showing the PPDs of the retrieved
parameters for the case of the warm-Jupiter for self-retrieval (CS-DFLT, red)
and cross-retrieval (CS-SELF, blue). The only difference between the two
cross-sections relate to self-broadening; CS-DFLT assumes air-broadening
(geocentric) while CS-SELF assumes only self-broadening. Strong biases are
seen in the retrieved value of T0 (7.32$\sigma$), T∞ (2.64$\sigma$), carbon
dioxide (7.32$\sigma$), methane (7.32$\sigma$), water (5.55$\sigma$), ozone
(-5.57$\sigma$), and hydrogen (-3.97$\sigma$). (Top Right) 500 random PT
profiles constructed from the posteriors are shown for comparison against the
true profile shown in black. The dotted line shows the contribution factor can
change significantly due to the changes in the pressure broadening values.
Supplementary figure 7:  Retrieval sensitivity to CO affected by opacity
model. Plot highlighting the difference in sensitivity to CO’s number density
between CS-DFLT (red) and CS-HTMP (gray) for the super-Earth case. (Top) Best-
fit models are shown as solid lines while the models perturbed by -0.25 on
Log10NCO are shown as dashed lines. (Bottom) Difference between best-fit and
perturbed models, highlighting that the primary effect of a change in CO
abundance is a change in scale height (all molecular features are affected).
While a 0.25 change on Log10NCO from its best-fit value for CS-DFLT increases
the $\chi^{2}$ by $\sim$1,000, it does increases the $\chi^{2}$ for CS-HTMP by
$\sim$3,500 which explained the tighter constraint on CO reported in Figure 3
and Supplementary Table 1.