# The effect of spectroscopic binning on atmospheric retrievals

Jack J. Davey,1 Kai Hou Yip,1 Ahmed F. Al-Refaie1 and Ingo P. Waldmann1
1Department of Physics and Astronomy, University College London, Gower Street,
WC1E 6BT London, UK E-mail<EMAIL_ADDRESS>(JJD)

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

With the James Webb Space Telescope (JWST) offering higher resolution data in
space-based transmission spectroscopy, understanding the capabilities of our
current atmospheric retrieval pipelines is essential. These new data cover
wider wavelength ranges and at much higher spectral resolution than previous
instruments have been able to offer. Therefore, it is often appealing to bin
spectra to fewer points, better constrained in their transit depth, before
using them as inputs for atmospheric retrievals. However, little quantitative
analysis of the trade-off between spectral resolution and signal-to-noise
ratio has been conducted thus far. As such, we produce a simulation
replicating the observations of WASP-39b by the NIRSpec PRISM instrument on
board JWST and assess the accuracy and consistency of retrievals while varying
resolution and the average photometric error. While this probes a specific
case we also plot ‘binning paths’ in the resulting sensitivity maps to
demonstrate the best attainable atmospheric parameter estimations starting
from the position of the real JWST Early Release Science observation. We
repeat this analysis on three different simulation setups where each includes
an opaque cloud layer at a different height in the atmosphere. We find that a
much greater resolution is needed in the case of a high cloud deck since
features are already heavily muted by the presence of the clouds. In the other
two cases, there are large ‘safe zones’ in the parameter space. If these maps
can be generalised, binning paths could inform future observations on how to
achieve the most accurate retrieval results.

###### keywords:

exoplanets – exoplanet atmospheres – spectroscopic telescopes

††pubyear: 2023††pagerange: The effect of spectroscopic binning on atmospheric
retrievals–A.8

##  1 Introduction

There is now substantial diversity in the population of known exoplanets and
this is driving an ever increasing emphasis on characterising rather than just
identifying exoplanets. In this pursuit, analysis of atmospheric transmission
spectra has long been the most fruitful means of investigation. Through this
technique, constraints have been placed not only on atmospheric compositions
(Barman, 2007; Tinetti et al., 2007; Swain et al., 2008; Pinhas et al., 2018),
but also on the temperature-pressure profiles and cloud properties (Kreidberg
et al., 2014a; MacDonald & Madhusudhan, 2017) of many planets. However, from
its inception to the the most recent observations, transmission spectroscopy
has always been a field limited by the quality and quantity of data available.

Past spectra produced by the Hubble (HST) and Spitzer Space Telescopes (SST)
were inherently limited by the signal-to-noise ratio (SNR) and spectral
resolution that these instruments could achieve. As such, the chemical species
that we could identify and the confidence with which we could constrain
atmospheres was lower than necessary for detailed characterisation (Swain et
al., 2009; Crouzet et al., 2012; Berta et al., 2012; Deming et al., 2013;
Swain et al., 2014; Kreidberg et al., 2014b, 2015; Tsiaras et al., 2019; Saba
et al., 2022).

With the James Webb Space Telescope (JWST) now offering a vast improvement in
the quality of data available for transmission spectroscopy, we are making
great strides towards the goal of improved characterisation. Across several
instruments and observing modes, JWST is able to offer transmission spectra
covering a broader wavelength range and at a much higher spectral resolution
than has previously been possible (Batalha et al., 2015; Greene et al., 2016).
Since with these new data we can study more atmospheres and in greater detail,
it is possible to determine the atmospheric abundances of an ever increasing
number of chemical species (e.g. SO2, CO2 and CH4 Alderson et al., 2023; Ahrer
et al., 2023a; Bell et al., 2023) on a more diverse sample of planets (such as
Earth-sized planets studied theoretically by Bean et al. (2018), Morley et al.
(2017) and Barstow et al. (2015) and shown in early releases of JWST data by
Lustig-Yaeger et al. (2023) among others). Additionally, with a larger
population to analyse we can begin to employ statistics to make statements
concerning general trends observed (as has already been attempted by Sing et
al. (2016), Barstow et al. (2016a), Tsiaras et al. (2018), Fisher & Heng
(2018), Pinhas et al. (2018), Changeat et al. (2022) and Edwards et al. (2023)
using the available HST and SST data and discussed generally by Bean et al.
(2017)). Planned missions such as Ariel intend to exploit this large
population of exoplanets and will focus specifically on common formation and
evolution processes exhibited across the population (Tinetti et al., 2018).
However, even in this pursuit, we are reliant on our ability to characterise
individual planets first. Thus, with these new, more detailed spectra we face
new challenges. We must now question how best to use these data in order to
maximise the efficiency of our analysis.

Specifically, this investigation focuses on atmospheric retrieval frameworks.
Retrievals offer the current, state of the art method in the analysis of
transmission spectra. With the concept first being discussed by Madhusudhan &
Seager (2009), many retrieval codes have now been developed (e.g. NEMESIS
(Irwin et al., 2008), CHIMERA (Line et al., 2013), TauREx (Waldmann et al.,
2015a, b; Al-Refaie et al., 2021) and petitRADTRANS (Mollière et al., 2019),
with a more comprehensive list presented in Macdonald & Batalha (2023)).
Whilst HST and SST were marred by comparatively poor spectral resolution and
high uncertainties, JWST offers unprecedented observational data quality.
However, dealing with these large, complex data sets can present issues of
their own.

It is imperative that we understand the limits of results attainable from
studies of atmospheres on individual exoplanets and the remaining obstacles
with our current methods of analysis. Studies of temporal binning during the
lightcurve fitting procedure have been carried out for both transit (Kipping,
2010) and phase curve observations (Morello et al., 2022). Additionally,
studies of the sensitivity of high resolution spectroscopy using cross
correlation techniques to probe exoplanet atmospheres have recently been
conducted by Boldt-Christmas et al. (2023). However, to date, there has been
little attention given to binning data at the spectrum level (i.e. binning in
wavelength space) for use with medium and low resolution spectra observed with
space-based telescopes.

In this work, we evaluate the trade-off between spectral resolution and the
scale of photometric error when binning spectra. This is quantified by
considering the accuracy of parameter estimations made in atmospheric
retrievals on data binned to varying resolutions. When binning spectra, we
must consider three separate but related effects:

  1. 1.

The achievable uncertainty on the retrieved parameters

  2. 2.

The induced bias on the retrieved values of each parameter

  3. 3.

The change in correlations observed in the conditional posterior distributions

Given that a much lower SNR will be recovered by observations of smaller
planets, understanding the effectiveness of binning as a way of improving the
SNR of data by sacrificing spectral information will be a crucial step towards
probing this part of the exoplanet population. In an ideal scenario, given
that we know the observing resolution that facilities such as JWST and Ariel
offer, this type of research could advise on how long to observe a given
target in order to reach the photometric error which would maximise the
accuracy of its parameter predictions, possibly with the aid of subsequent
spectral binning. This would require a generalised framework but in this
initial investigation we use the specific example of planets similar to
WASP-39b (Faedi et al., 2011), a hot, Saturn-sized planet observed with JWST
(Alderson et al., 2023; Ahrer et al., 2023a; Rustamkulov et al., 2023; Ahrer
et al., 2023b; Feinstein et al., 2023).

We wish to determine the optimal binning regime which preserves sufficient
information for a retrieval while also reducing the computational demands of
the forward modelling procedure. Similarly, by investigating the trade-off of
higher-SNR spectral points at reduced spectral resolution, we can hope to
identify optimal observing strategies for JWST. Through simulations, we
analyse WASP-39b across the wavelength range of the NIRSpec PRISM instrument
(Jakobsen et al., 2022) in three distinct cases; one with a cloud-free
atmosphere, one with low altitude clouds and one with high altitude clouds. By
doing so, we are able to explore the importance of the height of spectral
features in a spectrum and comment on how it affects the previous three
considerations. (Additional details on WASP-39b can be found in section 1.2
and details concerning the simulations in section 2.)

We investigate the dependency of these effects on resolution and error using
two independent methods. The first is by generating simulated spectra across a
grid of resolutions and errors. These spectra have constant resolution across
the wavelength range of the NIRSpec PRISM instrument and the error bars on
each point are equal. While this is a simplistic assumption, it allows us to
accurately position each within a resolution-error grid and study the change
across this parameter space.

The second method uses a more realistic binning scheme which retains some of
the variation across the instrument. We start from the wavelength grid of the
original WASP-39b observation as part of the JWST Early Release Science (ERS)
program (Rustamkulov et al., 2023) and, propagating the errors reported, we
bin down to lower resolutions. (The method by which this is achieved is
described in section 2.3.)

In this investigation, we expect to be able to map the resolution-error space
and mark boundaries between poorly retrieved regimes and ‘safe zones’. In
these safe zones, binning will not significantly alter the retrieval results
but other regions (with larger errors or lower resolution) may see rapid
changes in the accuracy of retrieval predictions. Given the non-linear nature
of the effects of binning within the retrieval process, these safe zones must
be mapped out numerically as described above.

###  1.1 Motivation for binning spectra

Atmospheric retrievals employ forward modelling and Bayesian sampling
processes to find the set of input parameters which best replicate an observed
spectrum (Madhusudhan, 2018). This is accomplished through the process of
drawing a set of parameters from some pre-defined prior space, computing a
forward model using equations of radiative transfer and, then, evaluating a
likelihood function to quantify the level of agreement between the observed
and simulated data. The process then repeats until some limit or level of
accuracy is achieved.

Many factors affect the efficiency of this process. The complexity of the
forward model calculations, the number of parameters for which the retrieval
is fitting and the size of the data set are just a few but already provide
plenty of scope for investigations. When looking to increase efficiency, it is
always important to bear in mind any consequent sacrifice in the accuracy of
the output estimations.

Binning spectra in wavelength space can be a useful method when trying to
extract the maximum possible information from a spectrum while also giving
consideration to the computational cost of retrievals or other fitting
procedures. However, the consequences of binning can be severe. It is a
process which results in an inevitable loss of information. As such, a balance
should be struck between the reduction in the photometric error by binning an
observation and the loss of morphological information as spectral resolution
decreases.

Spectral resolution ($R$) can be calculated via the equation

$R=\frac{\lambda}{\Delta\lambda}$ (1)

which is the ratio of the observed wavelength ($\lambda$) and the minimum
separation to the next observable feature ($\Delta\lambda$). As such, a higher
resolution spectrum will have more points than a lower resolution spectrum
across the same wavelength range. All instruments will have some intrinsic
maximum resolution which is set by the dispersion element and detector
characteristics. However, binning down spectra remains an option later in the
data processing pipeline.

When a set of spectral points are binned together, their combined photon
counts contribute to the strength of the signal. Given that this is a counting
process, Poisson statistics apply and the result is a reduction in photometric
error proportional to $1/\sqrt{\text{N\textsubscript{photons}}}$ (where
Nphotons is the number of photons contributing to the signal). Thus, the
resulting set of spectral points can be better localised in their transit
depth but at the expense of being less well constrained in their wavelength.

In our analysis we compute approximate ‘binning paths’ to give an idea of how
a specific dataset can move in the resolution-error space under investigation
with additional binning. These paths are calculated based on the number of
bins remaining in our simulated spectra and the corresponding reduction in
photometric error assuming, naively, that there are equal photon counts in
every bin in a spectrum. The errors are then reduced by a factor

$\sigma_{2}=\sigma_{1}\left(\frac{\sqrt{N_{Bins}(R_{2})}}{\sqrt{N_{Bins}(R_{1})}}\right)$
(2)

where the $\sigma_{i}$ refer to the errors on the spectral bins at their
corresponding resolutions of $R_{i}$, and $N_{Bins}(R_{i})$ refers to the
number of bins (or spectral points) at a resolution of $R_{i}$. These paths
should not be seen as definitive. Instead, they should be used as guides to
indicate the resolution-error combination possible given a specific start
point in the parameter space.

| Parameter | Value
---|---|---
Planetary | Mass (MJup) | 0.28a
Values from | Radius (RJup) | 1.27a
Bonomo et al. (2017)a | Effective Temperature (K) | 1000
and Maciejewski et al. (2016)b | Impact Parameter | 0.45b
| Orbital Period (days) | 4.05a
Stellar | Mass (M$\odot$) | 0.93
All values from | Radius (R$\odot$) | 0.895
Bonomo et al. (2017) | Temperature (K) | 5400
Chemical | H2O | -3.5
abundances | CO2 | -5.5
(All in log scale) | CO | -3.5
| SO2 | -6
| Na | -5
Fill gases | H2 | 0.17
(Ratio of remaining | He | 1
atmosphere) |  |
Atmospheric layers | High clouds (Pa) | 10
(3 cloud cases) | Low clouds (Pa) | 1000
| No clouds (Pa) | N/A
| Min. pressure level (Pa) | 10-1
| Max. pressure level (Pa) | 106
Table 1: The inputs parameters used to generate the simulation spectrum. This simulation is designed to replicate the shape of the spectrum observed by the NIRSpec PRISM on JWST for the planet WASP-39b. A base spectrum was generated using parameters from the relevant literature but the chemical abundances were altered to best match the reference observations. | Name | Opacity reference
---|---|---
Atomic and molecular | H2O | Polyansky et al. (2018)
cross-sections | CO2 | Yurchenko et al. (2020)
| CO | Li et al. (2015)
| SO2 | Underwood et al. (2016)
| Na | Allard et al. (2019)
Collisionaly induced | H2-H2 | Borysow et al. (2001)
absorption | H2-He | Borysow, A. (2002)
|  | Richard et al. (2012)
|  | Rothman et al. (2013)
Table 2: A list of opacity sources used for the generation of forward models
in this investigation. All sources come from the ExoMol project (Tennyson &
Yurchenko, 2012; Tennyson et al., 2016; Chubb et al., 2021) or the HITRAN
database (Richard et al., 2012; Rothman et al., 2013). Figure 1: Simulated
spectra for the atmosphere of WASP-39b based on the JWST observations
presented in Rustamkulov et al. (2023) using the FIREFLy data reduction
pipeline (Rustamkulov et al. (2022)). In panel a) the three sets of points
represent different instances of the assumed cloud deck pressure and spectral
resolution. In each case the error bars are constant across the wavelength
range and have been set to 100 ppm in this figure to reflect the average error
quoted on the WASP-39b NIRSpec PRISM spectrum. In blue is the cloud-free case
at a resolution of R = 900; shown in orange is the case with low level clouds
(at a pressure of 1000 Pa) and R = 500; in green is the case with high clouds
(at a pressure of 10 Pa) and R = 100. The three arrows (in blue, green and
red) show the bins used when calculating the average error in particular
wavelength ranges. The limits of these regions are chosen arbitrarily but,
broadly speaking, align with prominent features in the spectrum. In b),
spectra demonstrating the extreme cases of both binning and photometric error
considered in this investigation are plotted. In pink is a spectrum using the
highest resolution, R = 1100 and with the largest error, $\sigma$ = 200 ppm,
while in red, the resolution is R = 10 and $\sigma$ = 50 ppm (the minimum
values). Note that both spectra in panel b) use the low cloud case and the
spectrum shown in black has been shifted by $(R_{p}/R_{\star})^{2}$ = 0.003
for visibility.

###  1.2 WASP-39b

Due to its large scale height and relatively inactive host star, the hot
Saturn exoplanet, WASP-39b (Faedi et al., 2011), is an ideal candidate for
transmission spectroscopy studies. It has been observed with multiple
instruments using HST and SST (Fischer et al., 2016; Wakeford et al., 2018),
the Very Large Telescope (VLT Nikolov et al., 2016), the William Herschel
Telescope (Kirk et al., 2019) and, more recently, with JWST (Alderson et al.,
2023; Rustamkulov et al., 2023; Ahrer et al., 2023b; Feinstein et al., 2023).
In combination, these observations provide wavelength coverage of
approximately 0.3 µm to 5.5 µm.

These observations have given a good deal of insight into the composition and
structure of WASP-39b’s atmosphere but degeneracies and discrepancies within
and between these analyses remain. In particular, there is disagreement in the
quoted metallicity and corresponding dependence on the presence of clouds
and/or hazes in the atmosphere (Fischer et al., 2016; Kirk et al., 2019;
Arfaux & Lavvas, 2024). Since both of these physical features can serve to
reduce the observed transit depth by suppressing the observable scale height
in transmission, they often lead to conflicting results in the literature.
WASP-39b was also found to be the first example of an exoplanet where
photochemistry is observed in the atmosphere (Tsai et al., 2023) and, as such,
it is a valuable target and a good testing ground for the capabilities of the
JWST NIRSpec instrument (Jakobsen et al., 2022) in observing atmospheres of
exoplanets in transit.

Given that there are multiple studies of this system, full details of the
system parameters used for these investigations are shown in the ‘Planetary’
and ‘Stellar’ sections of table 1 along with their appropriate references.
While this investigation will focus on the specific case of planets similar to
WASP-39b, future studies will consider how easily these results could be
generalised and applied to other planets (of the same or differing planetary
type) and other instruments.

##  2 Simulation and Retrieval Methodologies

For the purposes of this investigation, simulations were created to mimic the
form of the JWST Early Release Science (ERS) observations of WASP-39b over the
wavelength range of the NIRSpec PRISM instrument (Rustamkulov et al., 2023).
Multiple spectra are generated since we vary the spectral resolution (between
10 and 1000) and the photometric error on each data point (between 50 ppm and
200 ppm).

These spectra are then input as observations to the TauREx 3 retrieval code
(Waldmann et al., 2015a, b; Al-Refaie et al., 2021) and the resultant
parameter estimations are compared to the known values. We quantify accuracy
through the percentage deviation of the predictions compared to the know input
values using the equation,

$Deviation(\%)=\frac{|\theta_{fit}-\theta_{true}|}{|\theta_{true}|}\times
100,$ (3)

where $\theta_{fit}$ is the retrieved value and $\theta_{true}$ the value
input to the simulation for the parameter, $\theta$.

We then plot ‘bias maps’ using the deviation value. Doing so, we can study the
region (in resolution-error space) where the actual observations lie and
determine how well they can be retrieved at their native resolution and at
binned resolutions. To investigate the importance of spectral feature height
in the retrievals, spectra with different cloud decks have been considered and
are investigated using identical retrieval and analysis methods on each
(described in detail in the sections 2.1 and 2.2).

In addition to this basic investigation, we also give consideration to other
possible effects which could bias our resulting sensitivity maps.
Specifically, we look into model parameterisation, the influence of our choice
of wavelength grid and the binning strategy used to reduce the resolution.

###  2.1 Simulation set up

Figure 2: The different spectra obtained by binning the simulation of a planet
like WASP-39b to the wavelength grid of the JWST NIRSpec PRISM data from the
ERS program (Rustamkulov et al., 2023) and then reducing the number of points
in the spectrum before binning further. All spectra are produced by TauREx and
propagate the error bars from the ERS data to the new wavelength grid. This
method preserves more of the variation in resolution and error scale across
the wavelength coverage of the instrument than binning to a constant
resolution grid with constant error bars. Each spectrum receives an artificial
offset in transit depth in this plot so that they can be distinguished from
one another.

In order to study the deviation of retrieved values from the ‘true’ value,
simulated data sets were generated so that the specific input parameters were
known. Simulations were generated for three different cloud scenarios. These
included the case of no clouds being present in the atmosphere, a high
altitude (low pressure, 10 Pa) grey cloud deck and a low altitude (high
pressure, 1000 Pa) grey cloud deck. The cloud level is the only parameter
which differs between the three forward models, every other parameter was
fixed to a constant value.

The simulations used a typical set up for a hot gaseous planet. Excess
atmospheric content was filled with H2 and He in the ratio [0.17:1] once
dominant gas abundances had been specified. Molecular contributions were
included due to H2O, CO, CO2 and SO2. Na was also included as an atomic
feature at the short wavelength end of the spectrum. Contributions due to
collisionally induced absorption (C.I.A.) of H2-H2 and H2-He pairs are also
included. All opacities were taken from the ExoMol project (Tennyson &
Yurchenko, 2012; Tennyson et al., 2016; Chubb et al., 2021). Table 2 provides
the appropriate references for each opacity calculation.

We use an isothermal temperature profile through an atmosphere of 100 layers
with the pressure varying between 10-1 Pa and 106 Pa. While this is not the
most flexible model available (due to the development of two stream
approximations (Heng et al., 2014; Malik et al., 2017) or parametric models
such as Guillot (2010)), it suits the purpose of this investigation since we
are interested in the accuracy of a retrieval using common assumptions.
Additionally, we use a temperature slightly cooler than the calculated
equilibrium temperature in the literature (e.g. in Barstow et al. (2016b)) as
isothermal profiles have been shown to retrieve lower temperatures than
equilibrium temperatures by several studies including those by MacDonald et
al. (2020) and Skaf et al. (2020).

As with the isothermal temperature model, chemical abundances are assumed to
be constant throughout the atmosphere. More accurate profiles such as those
using equilibrium chemistry (discussed in Al-Refaie et al. (2022)) or the two-
layer model presented in Changeat et al. (2019) are now available and the
issue of day-night heterogeneity has been discussed in Pluriel et al. (2020).
Consideration of the effect of more complex chemical models is left to future
investigations.

The full set of inputs used in the simulations can be found in table 1. All
models were generated using the forward modelling capabilities of TauREx
(Waldmann et al., 2015a, b; Al-Refaie et al., 2021). Examples of the three
simulations (at varying resolutions) can be found in figure 1.

For each model, simulations at varying resolution and with varying photometric
errors were produced. The binning procedure used was to generate a spectrum of
constant resolution across the wavelength coverage of JWST’s NIRSpec PRISM
using the binning functionality already present in the TauREx code. Errors
were then simply set to be constant for each point across the grid. We give
consideration to the limitations in these simulations and their simple error
and resolution profiles in section 4 where we employ an alternative binning
method.

The resolution grid used steps of 10 from R = 10 to R = 100 and steps of 100
between R = 100 and R = 1000. The errors on the spectra followed steps of 25
ppm between a minimum of 50 ppm and a maximum of 200 ppm. These values were
chosen since the JWST ERS spectrum presented in Rustamkulov et al. (2023) has
an average resolution of approximately 137 and an average error of
approximately 111 ppm across its data. Thus, the chosen grid includes and
extrapolates beyond this point.

Given the large number of spectra this grid yields (and the additional factor
of three enforced by the study of three different instances of cloud decks) we
elect to perform retrievals on models without additional scatter applied to
simulate observational noise. Following the central limit theorem, Feng et al.
(2018) show that, when noise instances are drawn from a Gaussian distribution
the two available methods, simulating thousands of randomised noise instances
or taking a noise-free instance of the simulated spectrum, are equivalent in
the results they yield except that the noise-free method may produce
predictions which are better centred on the true value than they might
otherwise be; the shapes of the posterior distributions remain unchanged.

###  2.2 Retrieval set up

For each of the retrievals conducted in the initial investigation, we used the
Nested Sampling (Skilling, 2004) code, Multinest, through its PyMultinest
implementation (Feroz & Hobson, 2008; Feroz et al., 2009, 2019; Buchner, J. et
al., 2014) and 500 live points were set. In each case there were seven
parameters to fit: planetary radius, effective temperature, molecular
abundances of H2O, CO, CO2 and SO2 and the cloud deck pressure. Note that Na
is not fit for in these analyses. Table 3 provides all of the corresponding
prior ranges used in the investigation.

Parameter | Prior range | Prior type
---|---|---
Planetary Radius (RJup) | [1,1.5] | Uniform
Effective Temperature (K) | [500,1500] | Uniform
Log10(Chemical abundances) | [-1,-8] | Log uniform
Cloud deck (Pa) | [10-3,105] | Uniform
Table 3: The input priors are all uniform or log uniform (for chemical
abundances). Each contains the ’true’ value of the parameter used in the
simulation except in the ’No cloud’ case.

###  2.3 Additional investigations

Figure 3: The different spectra obtained by binning the simulation of a planet
like WASP-39b to the wavelength grid of the JWST NIRSpec PRISM data from the
ERS program (Rustamkulov et al., 2023) reduced by retaining every fifth point
in the original wavelength grid with and without an additional shift.

Having completed the basic retrievals (as described in the previous
subsection), several additional tests were conducted. These tests aimed to
study possible sources of bias based on the set up of our simulation and our
retrieval strategies.

Figure 4: Posterior width maps for the case of a high cloud deck. In each
subplot (labelled by which parameter is represented) the pink crosses mark the
position of data from retrievals. Interpolation has been used to fill in the
rest of the grid and generate a map. Retrievals are spaced by R = 10 below R =
100 and by 100 between R = 100 and R = 1000. The colour bar associated with
each map describes the width of the marginalised posterior distribution
obtained in retrievals for the given parameter. The white arrow marks the
approximate position of the JWST ERS data in the map. The error bar on this
point is set by the 25th and 75th percentiles of the variation in resolution
and error in the input spectrum. The blue, green, red and pink dashed lines
plot potential binning paths under the assumption that each bin (before
binning) contains an equal number of photon counts. The start position is set
by the average resolution of the ERS data and the average error in the
shortest (blue), middle (green) and longest (red) wavelength regions in the
data. The pink dashed line plots the potential binning path from the
approximate position of the ERS data point, taking the average error over the
whole spectrum. The lower right plot in the grid shows the same data as the
lower left but as a series of lines.

Bias due to binning method: To investigate the impact of our choice of binning
strategy, two additional types of binning grid were introduced. The first
additional type of grid was based on the FIREFLy (Rustamkulov et al., 2022)
reduction of the JWST data (Rustamkulov et al., 2023) and in these cases the
resolution is not constant across the spectrum. However, we are limited in the
maximum resolution we can achieve by the resolution of the input spectrum as
we can only bin down from the starting point of the input data. We scale down
the resolution of these grids by retaining only every Nth point in the
spectrum (where N is an integer between 2 and 8). In these cases, we use the
mean resolution and photometric error across the spectrum to plot them in
subsequent analyses. These tests were designed to validate our results from
constant resolution grids by comparing them to more instrument specific
examples. Example plots of these spectra can be seen in figure 2.

For this first type of grid, we extend the analysis further by comparing
predictions when binning directly from the resolution of the forward model to
the desired wavelength grid and when first binning to the observation’s (ERS
data’s) resolution as an intermediate step.

The second type of grid was the same as the first but for each point in the
wavelengths grids, a minor shift was added. This shift was positive in
wavelength space and applied to the grid prior to binning the spectrum. The
value of the shift was set such that the new points in the ‘shifted’ grids
fell between the points in the original, ‘unshifted’ grids. This was achieved
by adding half the difference in wavelength between subsequent points to the
shorter wavelength point. An example of these spectra is shown in figure 3.

Bias due to model complexity: Taking a few cases to sparsely sample the
resolution-error grid, retrievals were re-run while fixing all of the chemical
abundances other than that of H2O to their true values. We then fit only for
planet radius, effective temperature, the abundance of H2O and the cloud deck
pressure in the retrieval to perform a preliminary investigation into the
dependence of the results on model complexity. In doing so, we wish to
investigate whether or not the bias in predictions is reduced by knowing the
log-abundances of other molecules a-priori and to what extent this depends on
the resolved lines in the water absorption bands.

Bias due to stochastic sampling: Finally, we check for consistency in the
results by running repeat instances of a subset of the retrievals. The aim of
this study was to verify the stability of the sensitivity maps that we produce
considering the fact that nested sampling is a stochastic process.

Figure 5: Same as figure 4 but for the deviation of the mean value from the
ground truth of the retrieved parameters. The colour bar associated with each
map describes the percentage deviation of the value obtained in retrievals
from the true value of that parameter (input to the simulation). The bottom
panel shows a collapsed version of the grid for $\log$(H2O) across the entire
resolution grid. Note that the y-axis uses the absolute value of the retrieved
abundance rather than the deviation. The input value is marked by the black
dashed line.

##  3 Results: Constant resolution-error grid

Without any quantitative study, we would expect that as the resolution of the
spectrum increased and the scale of the photometric error decreased, the
retrieval would more confidently and more accurately converge on the ‘true’
value for each parameter. Increasing resolution provides better coverage
across the instrument’s wavelength range and decreasing the size of errors
offers us greater confidence in the levels of transmission in the spectrum.
However, certain cases in this analysis showed evidence of additional
structure in their sensitivity maps.

Below, we present maps focusing on three separate but related aspects: 1)
Parameter estimation uncertainty; 2) Induced biases in the mean retrieved
values; and 3) Change in correlations in the conditional posterior
distribution.

###  3.1 Retrieval confidence

Before assessing the bias of parameter estimations in the retrievals, we
consider the confidence with which the sampling returns predictions, i.e. the
retrieved uncertainty in the free-parameters fitted. In general, we observe a
logarithmic decrease in the width of the posterior distribution (measured by
the extent of the 1 $\sigma$ Bayesian credible interval on the marginalised
posterior) as the resolution of the spectrum increases. A map for this metric
for the high cloud case can be seen in figure 4 (with equivalent plots for the
low and no cloud cases demonstrating similar trends included in Appendix A,
figures 13 and 14 respectively). At the high resolution end, this means that
there is very little reduction in the size of the confidence interval for
resolutions above R = 100. In all cases, we observe parameter uncertainties to
be a smooth and continuous function of SNR and resolution, as expected.

Figure 6: Sensitivity maps for the case of a low cloud deck. In each subplot
(labelled by which parameter is represented) the blue crosses mark the
position of data from retrievals. Interpolation has been used to fill in the
rest of the grid and generate a map. A log scale is used for the spectral
resolution (x axis) since these data are spaced by 10 below R = 100 and by 100
between R = 100 and R = 500. The colour bar associated with each map describes
the percentage deviation of the value obtained in retrievals from the true
value of that parameter (input to the simulation). The white arrow marks the
approximate position of the JWST ERS data in the map (including error bars set
by the 25th and 75th percentiles of the variation across the observed spectrum
in resolution and error bar size). The green dashed lines plot potential
binning paths under the assumption that each bin (before binning) contains an
equal number of counts. The pink dashed line plots the potential binning path
from the approximate position of the ERS data point under the same
assumptions. The bottom panel shows a collapsed version of the grid for
$\log$(H2O) across the entire resolution grid. Note also that the y-axis uses
the absolute value of the retrieved abundance rather than the deviation. The
input value is marked by the black dashed line.

Considering only this information, it would suggest there is very little to be
gained from retrieving at higher resolutions for the broad band absorbers
considered here. One would not expect to achieve any significant improvement
in the confidence of one’s predictions at the expense of additional
computational cost. Thus, analysis of the posterior width maps alone would
suggest that it is beneficial to bin down higher resolution spectra to R = 100
as a limiting case.

Furthermore, the coloured lines in the lower right panel of figure 4 show the
trends of the width of the marginal posterior distributions for different
photometric errors on the spectra as resolution increases. We observe
convergence of these lines suggesting that the confidence becomes insensitive
(within reasonably expected SNRs) to the size of error bars on the input
spectra at high resolutions.

Here, the data enter into a resolution limited region and this suggests that,
at higher resolutions, if we assume that the scale of the photometric error on
a spectrum can be decreased with longer observing time (i.e. multiple
observations of the target), less time would be required for observations
since there is little improvement to be gained by substantially reducing the
size of error bars. In this regime, the resolution rather than the SNR has the
greater influence on the precision of our predictions. However, we should
still give consideration to both resolution and SNR when planning observations
as significant uncertainty on our data could still lead to uninformative
predictions. Furthermore, we treat the posterior widths as only the
theoretical lower bound on the widths that would be retrieved for real data
since we use the same forward model in producing our simulations and
performing the retrieval.

###  3.2 Bias maps (10 $\leq$ R $\leq$ 500)

Figure 7: Sensitivity maps for the case of a no clouds included in the
simulation. In each subplot (labelled by which parameter is represented) the
blue crosses mark the position of data from retrievals. Interpolation has been
used to fill in the rest of the grid and generate a map. A log scale is used
for the spectral resolution (x axis) since these data are spaced by 10 below R
= 100 and by 100 between R = 100 and R = 500. The colour bar associated with
each map describes the percentage deviation of the value obtained in
retrievals from the true value of that parameter (input to the simulation).
The white arrow marks the approximate position of the JWST ERS data in the map
(including error bars set by the 25th and 75th percentiles of the variation
across the observed spectrum in resolution and error bar size). The green
dashed lines plot potential binning paths under the assumption that each bin
(before binning) contains an equal number of counts. The pink dashed line
plots the potential binning path from the approximate position of the ERS data
point under the same assumptions. The bottom panel shows a collapsed version
of the grid for $\log$(H2O) across the entire resolution grid. Note also that
the y-axis uses the absolute value of the retrieved abundance rather than the
deviation. The input value is marked by the black dashed line. Figure 8: Plots
of the retrieved spectra in the case of a high cloud deck with errors of 100
ppm. Below a resolution of R = 500, only SO2 shows a reduction in the accuracy
of it’s predictions. Shown here are the contributions to the retrieved spectra
at resolutions of R = 10, R = 100 and R = 400. Coloured lines indicate
retrieved contributions from individual molecules while the dashed black line
presents the combined spectrum binned to the resolution of the input data.

Next, we turn our attention to the accuracy of our grid of retrievals. Figures
5, 6 and 7 show the sensitivity maps for each parameter in retrievals for the
high, low and no cloud scenarios respectively. In these plots, the colour bar
corresponds to the percentage deviation of the retrieved value from the true
value (as defined in section 2).

On each of the maps, blue crosses mark the positions of the retrievals in
resolution-error space. The horizontal and vertical dashed lines mark the
position of the average error and the average resolution of the ERS
observation. The solid lines at their intersection demonstrate the extent of
the variation of data error and resolution element across the ERS spectrum by
marking the 25th and 75th percentiles for each value. Additionally, ‘binning
paths’ (as defined in section 1.1) have been plotted in pink starting from the
ERS data point and in blue, green and red starting from the average error on
the spectrum at the shortest, middle and longest wavelength bins marked by
coloured arrows on the left panel of figure 1.

Note that these maps only plot the data up to a resolution of 500 since there
is less variation beyond this point and the inclusion of high resolution
retrievals can mask the structure seen at lower resolutions if included in the
maps. The high resolution end of the grids can be seen in appendix A.2,
figures 15, 16 and 17 for the high, low and no cloud cases respectively.

In the case of high clouds (shown in figure 5) there is a significant amount
of structure present in the bias maps across all resolutions and error scales.
Notably, the maps of log-abundances of CO or CO2 and that of SO2 appear to
present inverse patterns to one another.

Shown in figure 8, the most significant cause of this inverted structure
appears to be the underestimation of the CO2 abundance. The retrieved values
of CO2 vary by 0.42 in log-abundance between R = 10 and R = 400. In part, this
is due to the overlap in the spectra of CO2 and SO2 at low resolutions. At R =
10, both exhibit peaks at 4.35 µm and 2.91 µm. However, as resolution
increases, more of the structure in and around these features is sampled. Of
particular importance is the increase in sampling between the bins of 3.93 µm
and 4.35 µm for the SO2 spectrum. This reveals that the peak actually occurs
at a slightly shorter wavelength for SO2 than it does for CO2 (4.05 µm rather
than 4.35 µm).

SO2 is the only parameter of those retrieved where we observe a different
trend and the bias seems to worsen with increasing resolution in figure 5.
This is an artifact of the low volume mixing ratio of SO2 in this simulation
and the fact that SO2 and CO2 emission features are not separable at low
resolution and that SO2 can only be retrieved as an upper limit in our
simulation. If we consider the width of the SO2 marginal posterior
distribution (see figure 4), we can see that the values of the log-abundance
of SO2 are less well constrained and we are only ever able to place an upper
limit on its prediction whereas the log-abundance of other molecules become
well resolved at higher resolutions. (This is also shown in the corner plot
included in appendix A.3.)

Maps for all of the other parameters show their most significant differences
when the resolution is reduced below R = 100 and this holds true for the other
cloud scenarios as well (see figures 6 and 7). Though their patterns are
generally matching, it is worth noting the difference in the scales of these
bias maps. In general, planetary radius is well constrained and only deviates
by around 2 % in the worst case scenario. However, the molecular abundances
can experience deviations up to about 30 % across the different molecules and
cloud scenarios (with $\log$(CO) sitting as an outlier with a deviation as
high as 55 % in the high cloud case).

The common structure observed does follow the expectation that spectra with
low resolution and at high photometric error (i.e. the top left corner of each
panel) would be the worst retrieved.

The bottom right panel of each map presents a collapsed version of the maps
for the log-abundance of H2O where each line shows the retrieved value at
various spectral noise levels. This allows us to see if the retrieval is
inaccurate due to over- or under-estimations in the parameters.

We find that, in the case of H2O, the log-abundance is more often over-
estimated in cases including clouds but under-estimated when clouds are
absent. However, it is worth noting the change in scale between the cases
including clouds and the no cloud case. With clouds, the prediction for the
log-abundance of H2O varies by approximately $\pm$ 0.4 but in the case without
clouds the magnitude of the deviation is about ten times less ($\pm$ 0.04).

###  3.3 Correlation analysis

Figure 9: The correlation between specific retrieved parameters in the case of
a spectrum with a high cloud deck. The parameters under investigation are
marked by the text in each panel and the different coloured lines show the
dependence of these trends on the scale of photometric error under
investigation. Note that only the pairings of parameters where there is a
significant change in the correlation across the resolution grid are displayed
here. Figure 10: Same as in figure 9 but now focusing on pairings of retrieved
parameters in the case of spectra with a low cloud deck. In this case, the
pressure of the cloud deck is the parameter that causes the most significant
changes in the correlations.

For each possible pairing of parameters, we calculate the Pearson correlation
coefficient on their sampling traces. In doing so, we are studying the fitted
parameter degeneracies found in the conditional posterior distributions. As
individual parameters become more or less retrievable as a function of SNR and
resolution change, we expect the underlying degeneracies in the radiative
transfer solutions to reflect this. Different trends are seen for each cloud
case. Of particular interest are cases where the correlation changes as a
function of resolution. Several examples are discussed in this section but we
give consideration to the high cloud case first in order to further
investigate the structure seen in the bias maps (figure 5).

For the case of high clouds the most significant changes are found with the
fitting of the log-abundance of CO and its dependence on the log-abundances of
CO2 or H2O. Plots of these trends are given in figure 9’s top and bottom
panels respectively. In both cases, the correlation between the traces
increases with increasing resolution. At our lowest resolution in the
retrieval grid, we find that there is little to no resolvable correlation
between these parameters. However, at a resolution of 1000, the value
approaches +0.50 as we are better able to resolve the features in the spectrum
and when the value of correlation stabilises, we find the natural degeneracies
in the retrieval (rather than those caused by an undersampling of the
spectrum).

It is also worth noting that the spectra with smaller uncertainties appear to
see a more rapid increase in the correlation between parameters as spectral
resolution increases. Thus, with smaller uncertainties, we are able to resolve
the natural degeneracies between parameters at a lower spectral resolution
since we retain more information in the spectrum at any given resolution step.

In the case of our low cloud deck, we observe a reduction in the correlation
between several parameters with increasing spectral resolution. The three most
prominent cases of this effect are shown in figure 10. This shows clearly the
impact spectral resolution has in mitigating model degeneracies.

This is the opposite effect to that seen in the high cloud case where
correlation increased. One similarity comparing to the trends in the high
cloud case is the dependence on photometric error. We note in the low cloud
cases that the smaller the photometric error, the more rapid the change in the
correlation with increasing resolution for the reasons mentioned previously.

With the low cloud spectra, the parameter which experiences the greatest
change in correlation is the cloud top pressure. This is degenerate with the
temperature, planetary radius and H2O log-abundance at low resolution but
parameters become disentangled at higher resolutions as is demonstrated by the
reduction in the magnitude of the Pearson correlation coefficient.

An additional interesting feature of these correlation plots is the scale at
which the values stabilise and we uncover the natural degeneracies of the
model fit. The minimum resolution at which all of the lines in plots 9 and 10
reach a stable value is approximately 500. However, if we compare this to the
resolution at which the values of our input parameters are well predicted in
the bias maps, we can see that, even in the least favourable conditions (the
high cloud deck), parameters are well predicted above a resolution of 200.
This means that we require a much higher resolution to reveal degeneracies
between parameters than we do to fit the correct values in a retrieval.

##  4 Results: Binning from ERS data

In order to validate our methods, we also consider the case of retrieving from
a binned version of the JWST ERS data as described in section 2.3.

Binning factor | Average spectral resolution | Average error (ppm)
---|---|---
1 | 137 | 110
2 | 68 | 69
3 | 45 | 64
4 | 34 | 53
5 | 27 | 51
6 | 22 | 46
7 | 19 | 43
8 | 17 | 40
Table 4: The corresponding mean spectral resolution and photometric error for
the spectra when binned down from the JWST ERS observations by a given binning
factor. Note that the resolution and error values have been rounded to the
closest integer value.

The percentage deviations for each parameter in retrievals on these spectra
are shown in figure 11. Note that there is no separation by error scale in
this case since the errors are set by the propagation of error bars from the
ERS observations. Thus, they have pre-defined values for each wavelength grid.
A table of the average error for each grid can be found in table 4.

This method of binning perhaps better represents JWST NIRSpec data as it
retains some of the variation across the spectra considering both error and
resolution whereas our other simulations use constant resolution and
photometric error on every point. However, this alternative technique also
bears limitations. Most notably, you cannot bin up from the input data so
there is a maximum set on the resolution that can be used in such analysis.

###  4.1 Comparison to constant resolution-error grid results

In figure 11 we observe a general trend of increasing accuracy with increasing
spectral resolution as is expected. In most cases, there are some higher
binning factors (corresponding to a lower spectral resolution) where
predictions appear to show reduced biases for some parameters whilst increased
biases for other. We attribute this to lucky sampling that is better suited to
one parameter compared to others.

To further investigate the influence of binning strategy on the predictions of
retrievals, we investigate the consequences of multiple binning instances on a
single dataset and compare to a single instance of binning. In figure 11, the
dashed lines provide the predictions when data are binned to the desired
wavelength grid directly from the native resolution of the TauREx forward
model. Alongside these, the solid lines provide the results when binning from
the native resolution to the ERS wavelength grid and then binning further to
the desired grid. (Note that this means that the spectra at a binning factor
of 1 are identical with both methods and, as expected, agree in their
predictions.)

This additional test is designed to simulate the process of observation
through the initial binning step (to the ERS wavelength grid) by simulating
the noise induced by binning the spectra. Results between the two methods are
largely consistent with some variation between the predictions stemming from
their prediction uncertainties. In general we find that the scale of
deviations agree between our retrievals in this test and our initial
resolution-error grids and they follow the same broad trends. Additionally,
our results indicate that the decrease in resolution is more significant in
the accuracy of predictions than the corresponding decrease in photometric
error since predictions worsen at lower resolutions (higher binning factors).

Figure 11: The percentage deviation of retrieved parameters from spectra
obtained by binning down simulations of the JWST ERS data for WASP-39b using
the NIRSpec PRISM (Rustamkulov et al., 2023) in the low cloud case. The
wavelength grid was constructed by retaining only every Nth point in the
spectrum where N is the ‘Binning factor’. Each line represents a different
parameter and the top and bottom x-axes quantify the level of binning through
the binning factor and the average resolution respectively. All values for
average resolution are obtained by considering the mean value across the
spectrum. Solid lines show data from retrievals where the simulation was first
binned to the wavelength grid of the ERS data and then binned further whereas
the dashed lines bin directly from the native resolution of the TauREx
simulation to the appropriate reduced grid.

###  4.2 Shifted wavelength grids

Figure 12: The solid lines represent the same data as in figure 11. However,
the dot-dashed lines present results from retrievals where there initial
wavelength grids of spectra used to retrieve the solid lines have had an
additional wavelength shift applied prior to binning the spectrum. This shift
is set point-by-point across the spectrum such that the resulting points in
the new grid are halfway between the original point and the next in the un-
shifted grid. See figure 3 for examples of these spectra.

As described in section 2.3, we have performed additional retrievals testing
for the impact of a slight wavelength grid shift (see figure 3 for an example
spectrum). We continue to see broad consistency with the results from
‘shifted’ and ‘unshifted’ grids. These results are shown in figure 12 for
spectra of the low cloud case (with similar plots for the high and no cloud
spectra shown in appendix A.5, figures 19 and 20).

There appear to be offsets between some parameters when retrieving from
shifted and unshifted spectra but the same trend of worsening predictions with
decreasing spectral resolution is observed at similar magnitudes. Hence trends
described earlier are irrespective of the type of grid chosen.

##  5 Results: Additional tests

The final tests in this investigation were used to validate our retrieval
methodology and to investigate any bias emerging from model complexity or from
the stochasticity of our sampling procedure.

###  5.1 Simplified retrievals and models

In order to investigate the dependence of our results on model complexity we
perform two additional types of retrieval. In the first case, we leave the
base simulation as defined in section 2.1. However, we change the fitting
parameters in the retrieval. The log-abundances of all molecules other than
H2O are fixed to their correct input values and retrievals are run while
fitting only for temperature, planetary radius, cloud deck pressure and log-
abundance of H2O.

Type of | Molecule | Included in the | Fit in
---|---|---|---
retrieval |  | input spectrum | retrieval
Simplified Retrieval | H2O | True | Free
| CO2 | True | Fixed
| CO | True | Fixed
| SO2 | True | Fixed
| Na | True | Fixed
Simplified Model | H2O | True | Free
| CO2 | False | -
| CO | False | -
| SO2 | False | -
| Na | False | -
Table 5: The chemical input and fittable parameters for the simplified
retrieval and simplified model tests. In the final column, ‘Free’ indicates
that the value is fit during the retrieval, ‘Fixed’ Indicates that it is set
to its correct value a-priori and ‘-’ indicates that it is left out of the
retrieval all together.

We call this a simplified retrieval. Here, we allow the retrieval to vary only
a subset of the parameters that are present in the simulation. Thus, it is
restricted in the possible combinations of parameters it can sample due to the
reduction in the dimensionality of the prior space.

The second type we refer to as a simplified model. The simplified model refers
to a case in which the base simulation only contains H2O as a molecular
contribution to the atmospheric absorption. All the other chemical
contributions (CO, CO2, SO2 and Na) are neglected in these spectra. The
retrieval is then run on this spectrum fitting for temperature, planetary
radius, cloud deck pressure and log-abundance of H2O as in the previous test.

A breakdown of the chemical inputs and fitting parameters for these retrievals
can be seen in table 5.

In our results we find that there is little dependence on the model complexity
under these tests. Figures of these data are shown in the appendices (appendix
A.6, figure 23). At low resolution, the original grid performs slightly worse
than the simplified retrievals but towards higher resolutions, all retrievals
converge on the true input parameters. Intuitively, this makes sense since the
simplified retrieval has fewer parameters to deal with and, as such, converges
on the true solution faster and with greater confidence. By fixing other
molecular abundances to their correct values, we are removing the uncertainty
that arises from degeneracies between the log-abundance of H2O and other
molecular contributions.

###  5.2 Repeat retrieval instances

We run two tests on repeat retrieval instances. In the first case, we are
testing the stability of our parameter predictions given the stochastic nature
of the nested sampling procedure used. For this investigation we simply re-run
retrievals from the original retrieval grid and compare the percentage
deviations from each instance of the retrieval procedure.

Overall we find minimal difference when the retrieval is re-run. As expected,
the difference between the two instances of retrievals reduces with increasing
resolution since the confidence in predictions increases in this regime. See
appendix A.7 for figures displaying the results of this additional
investigation.

Then, we also test the variation in our results when we introduce some
artificial scatter into the spectrum. We do this for spectra in the low cloud
case across all resolutions. In the scattered spectra each data point is
resampled from a Gaussian distribution centred on the original point and with
a width set by the size of the 1 $\sigma$ photometric error. We run three
tests in this investigation using the best (smallest, 50 ppm), worst (largest,
200 ppm) and a mid-level (100 ppm) photometric error.

Overall we find that, as with the previous test, the difference in the
percentage deviation for each retrieval decreases with increasing resolution
since the data are then more heavily influenced by their wavelength coverage
than by their errors. There are larger discrepancies in these cases than in
the repeat cases with no scatter, particularly at the low resolution end where
we see differences of up to approximately 25 % in the worst case scenario
(where the photometric error is 200 ppm on every spectral point). This is due
to the reliance of well constrained transit depths in the lower resolution
spectra which is accounted for with a more finely sampled wavelength grid at
higher resolutions.

These data are available in the appendices (see appendix A.8).

##  6 Discussion and conclusions

The motivation for binning is indisputable. It is a useful tool when it comes
to analysis of exoplanet atmospheres using Bayesian retrieval methods,
reducing the necessary computing resources and improving the SNR of individual
points in the spectra. However, binning below R = 100, we observe a
logarithmic increase in the width of the posterior distribution. Thus, even in
cases where the correct abundance is retrieved, it’s credible interval remains
large and poorly constrained. It is necessary in such cases to consider how
informative these results are as, for molecular abundances, these
uncertainties can cover orders of magnitude.

This is particularly true for high cloud cases where spectral features are
muted. In these cases, reducing resolution of spectra comes at a significant
cost and the region where atmospheric parameters can be well retrieved is
shifted towards a higher resolution in our grid of resolution-error space.

It is pleasing to see that, in the low cloud case which is the one which most
accurately mimics the shape of the NIRSpec PRISM data observed in Rustamkulov
et al. (2023) for WASP-39b, at the native resolution of the observation, the
data fall into a relatively safe zone in the retrieval grid, suggesting the
right conditions for near optimal retrievals. Our analysis suggests similar
accuracy could be achieved with additional binning of this data set but the
results will only be as good, not better, and, again, the uncertainty on the
retrieved value would grow significantly.

While this initial investigation targets a specific case, the method should be
applicable to other planets. We would expect planets of similar type (e.g.
size and mass), orbital configuration (e.g. orbital period and separation) and
with similar host stars to follow similar trends as those observed in this
study. However, if such a planet were to have a high cloud deck, caution would
be strongly advised with JWST’s NIRSpec PRISM as the native resolution of the
instrument sits right on the boundary of the well and poorly retrieved cases
in resolution-error space. In this case it would be advisable to obtain
additional transit observations to reduce the average photometric error in the
initial data. This comment may also apply to studies of smaller planets where
atmospheric features are muted by a reduced atmospheric scale height rather
than a cloud deck.

###  6.1 Limitations and Future Work

The main limitation of this study is the fact that it targets a specific case,
namely, that of a planet similar to WASP-39b. All of the results presented in
this study apply to the simulated planet described in the methods sections.
How easily these results can be generalised across a wider exoplanet
population remains to be seen.

Another limitation emerges from the wavelength range and resolution-error grid
considered in this study. For most of the investigations presented in this
initial work, we use spectra of constant resolution and equal error bars
across their entire wavelength range. This is simplistic but informative as a
first analysis. However, we are pleased to see that our study of alternative
binning strategies seem to present results which are largely in agreement with
the data presented in our bias maps. Results shown are specific to the JWST
NIRSpec PRISM mode. Other instruments feature different spectral resolutions
and wavelength coverages and results presented here may not be directly
applicable to those modes.

We here presented a uniform binning grid across the full spectral range.
Wavelength-dependent binning that is sensitive to the presence of spectral
features may be a more optimal approach. Such considerations go beyond the
scope of this investigation and will be investigated in future publications.

###  6.2 Potential Applications

While the main purpose of this research was to see if and to what extent data
can be binned across a spectrum without significant losses in the accuracy of
a retrieval’s predictions, there are other potential applications of this type
of analysis. For instance, it would be interesting to try and asses the
benefit of this work to future missions and alternative instrument modes
available for the JWST.

In particular, this type of analysis could benefit the Ariel space mission
since the mission will require efficient observation and analysis of
atmospheres to probe the exoplanet population as broadly as possible within
its lifetime. Ariel will use a tiered observing strategy with the survey level
tier featuring binned spectra where appropriate. Understanding the maximum
attainable information given a specific binning strategy (such as has already
been considered in Changeat et al. (2020) and Mugnai et al. (2021)) is
important to the success of the mission.

With additional work, we hope that indicative binning paths may help to inform
the necessary number of transits needed for accurate parameter estimations by
identifying the photometric error that must be achieved. Providing a more
holistic understanding of optimal observing strategies that takes into account
optimal retrievability of planetary parameters given the minimally viable
resolution and signal-to-noise regime of the observation, promises to increase
the overall competitiveness of exoplanet transit and secondary eclipse
observations on competitive instruments such as JWST and the ELTs.

## Acknowledgements

This research received funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (grant
agreement n∘ 758892/ExoAI), and from the Science and Technology Facilities
Council (STFC; grant n∘ ST/W00254X/1 and grant n∘ ST/W50788X/1).

## Data Availability

Simulations generated for this study were based on the observational data
presented in Rustamkulov et al. (2023) which can be accessed through the
Mikulski Archive for Space Telescopes (MAST). Simulated data used in this
investigation can be made available upon request.

## References

  * Ahrer et al. (2023a) Ahrer E.-M., et al., 2023a, Nature, 614, 649–652
  * Ahrer et al. (2023b) Ahrer E.-M., et al., 2023b, Nature, 614, 653–658
  * Al-Refaie et al. (2021) Al-Refaie A. F., Changeat Q., Waldmann I. P., Tinetti G., 2021, ApJ, 917, 37
  * Al-Refaie et al. (2022) Al-Refaie A. F., Changeat Q., Venot O., Waldmann I. P., Tinetti G., 2022, ApJ, 932, 123
  * Alderson et al. (2023) Alderson L., et al., 2023, Nature, 614, 664–669
  * Allard et al. (2019) Allard N. F., Spiegelman F., Leininger T., Molliere P., 2019, A&A, 628, A120
  * Arfaux & Lavvas (2024) Arfaux A., Lavvas P., 2024, MNRAS, 530, 482
  * Barman (2007) Barman T., 2007, ApJ, 661, L191
  * Barstow et al. (2015) Barstow J. K., Aigrain S., Irwin P. G. J., Kendrew S., Fletcher L. N., 2015, MNRAS, 448, 2546–2561
  * Barstow et al. (2016a) Barstow J. K., Aigrain S., Irwin P. G. J., Sing D. K., 2016a, ApJ, 834, 50
  * Barstow et al. (2016b) Barstow J. K., Aigrain S., Irwin P. G. J., Sing D. K., 2016b, ApJ, 834, 50
  * Batalha et al. (2015) Batalha N., Kalirai J., Lunine J., Clampin M., Lindler D., 2015, arXiv e-prints, p. arXiv:1507.02655
  * Bean et al. (2017) Bean J. L., Abbot D. S., Kempton E. M.-R., 2017, ApJ, 841, L24
  * Bean et al. (2018) Bean J. L., et al., 2018, PASP, 130, 114402
  * Bell et al. (2023) Bell T. J., et al., 2023, Nature, 623, 709–712
  * Berta et al. (2012) Berta Z. K., et al., 2012, ApJ, 747, 35
  * Boldt-Christmas et al. (2023) Boldt-Christmas L., Lesjak F., Wehrhahn A., Piskunov N., Rains A. D., Nortmann L., Kochukhov O., 2023, A&A
  * Bonomo et al. (2017) Bonomo A. S., et al., 2017, A&A, 602, A107
  * Borysow, A. (2002) Borysow, A. 2002, A&A, 390, 779
  * Borysow et al. (2001) Borysow A., Jørgensen U. G., Fu Y., 2001, Journal of Quantitative Spectroscopy and Radiative Transfer, 68, 235
  * Buchner, J. et al. (2014) Buchner, J. et al., 2014, A&A, 564, A125
  * Changeat et al. (2019) Changeat Q., Edwards B., Waldmann I. P., Tinetti G., 2019, ApJ, 886, 39
  * Changeat et al. (2020) Changeat Q., Al-Refaie A., Mugnai L. V., Edwards B., Waldmann I. P., Pascale E., Tinetti G., 2020, AJ, 160, 80
  * Changeat et al. (2022) Changeat Q., et al., 2022, ApJS, 260, 3
  * Chubb et al. (2021) Chubb K. L., et al., 2021, A&A, 646, A21
  * Crouzet et al. (2012) Crouzet N., Mccullough P. R., Burke C., Long D., 2012, ApJ, 761, 7
  * Deming et al. (2013) Deming D., et al., 2013, ApJ, 774, 95
  * Edwards et al. (2023) Edwards B., et al., 2023, ApJS, 269, 31
  * Faedi et al. (2011) Faedi F., et al., 2011, A&A, 531, A40
  * Feinstein et al. (2023) Feinstein A. D., et al., 2023, Nature, 614, 670–675
  * Feng et al. (2018) Feng Y. K., Robinson T. D., Fortney J. J., Lupu R. E., Marley M. S., Lewis N. K., Macintosh B., Line M. R., 2018, AJ, 155, 200
  * Feroz & Hobson (2008) Feroz F., Hobson M. P., 2008, MNRAS, 384, 449
  * Feroz et al. (2009) Feroz F., Hobson M. P., Bridges M., 2009, MNRAS, 398, 1601
  * Feroz et al. (2019) Feroz F., Hobson M. P., Cameron E., Pettitt A. N., 2019, The Open Journal of Astrophysics, 2, 10
  * Fischer et al. (2016) Fischer P. D., et al., 2016, ApJ, 827, 19
  * Fisher & Heng (2018) Fisher C., Heng K., 2018, MNRAS, 481, 4698
  * Greene et al. (2016) Greene T. P., Line M. R., Montero C., Fortney J. J., Lustig-Yaeger J., Luther K., 2016, ApJ, 817, 17
  * Guillot (2010) Guillot T., 2010, A&A, 520, A27
  * Heng et al. (2014) Heng K., Mendonça J. M., Lee J.-M., 2014, ApJS, 215, 4
  * Irwin et al. (2008) Irwin P., et al., 2008, Journal of Quantitative Spectroscopy and Radiative Transfer, 109, 1136
  * Jakobsen et al. (2022) Jakobsen P., et al., 2022, A&A, 661, A80
  * Kipping (2010) Kipping D. M., 2010, MNRAS, 408, 1758
  * Kirk et al. (2019) Kirk J., López-Morales M., Wheatley P. J., Weaver I. C., Skillen I., Louden T., McCormac J., Espinoza N., 2019, AJ, 158, 144
  * Kreidberg et al. (2014a) Kreidberg L., et al., 2014a, Nature, 505, 69–72
  * Kreidberg et al. (2014b) Kreidberg L., et al., 2014b, ApJ, 793, L27
  * Kreidberg et al. (2015) Kreidberg L., et al., 2015, ApJ, 814, 66
  * Li et al. (2015) Li G., Gordon I. E., Rothman L. S., Tan Y., Hu S.-M., Kassi S., Campargue A., Medvedev E. S., 2015, ApJS, 216, 15
  * Line et al. (2013) Line M. R., et al., 2013, ApJ, 775, 137
  * Lustig-Yaeger et al. (2023) Lustig-Yaeger J., et al., 2023, Nature Astronomy
  * MacDonald & Madhusudhan (2017) MacDonald R. J., Madhusudhan N., 2017, MNRAS, 469, 1979
  * MacDonald et al. (2020) MacDonald R. J., Goyal J. M., Lewis N. K., 2020, ApJ, 893, L43
  * Macdonald & Batalha (2023) Macdonald R. J., Batalha N. E., 2023, Research Notes of the AAS, 7, 54
  * Maciejewski et al. (2016) Maciejewski G., et al., 2016, Acta Astron., 66, 55
  * Madhusudhan (2018) Madhusudhan N., 2018, Atmospheric Retrieval of Exoplanets. p. 2153–2182, doi:10.1007/978-3-319-55333-7_104
  * Madhusudhan & Seager (2009) Madhusudhan N., Seager S., 2009, ApJ, 707, 24
  * Malik et al. (2017) Malik M., et al., 2017, AJ, 153, 56
  * Mollière et al. (2019) Mollière P., Wardenier J. P., van Boekel R., Henning T., Molaverdikhani K., Snellen I. A. G., 2019, A&A, 627, A67
  * Morello et al. (2022) Morello G., Dyrek A., Changeat Q., 2022, MNRAS, 517, 2151–2164
  * Morley et al. (2017) Morley C. V., Kreidberg L., Rustamkulov Z., Robinson T., Fortney J. J., 2017, ApJ, 850, 121
  * Mugnai et al. (2021) Mugnai L. V., Al-Refaie A., Bocchieri A., Changeat Q., Pascale E., Tinetti G., 2021, AJ, 162, 288
  * Nikolov et al. (2016) Nikolov N., Sing D. K., Gibson N. P., Fortney J. J., Evans T. M., Barstow J. K., Kataria T., Wilson P. A., 2016, ApJ, 832, 191
  * Pinhas et al. (2018) Pinhas A., Madhusudhan N., Gandhi S., MacDonald R., 2018, MNRAS, 482, 1485
  * Pluriel et al. (2020) Pluriel W., Zingales T., Leconte J., Parmentier V., 2020, A&A, 636, A66
  * Polyansky et al. (2018) Polyansky O. L., Kyuberis A. A., Zobov N. F., Tennyson J., Yurchenko S. N., Lodi L., 2018, MNRAS, 480, 2597
  * Richard et al. (2012) Richard C., et al., 2012, Journal of Quantitative Spectroscopy and Radiative Transfer, 113, 1276
  * Rothman et al. (2013) Rothman L., et al., 2013, Journal of Quantitative Spectroscopy and Radiative Transfer, 130, 4
  * Rustamkulov et al. (2022) Rustamkulov Z., Sing D. K., Liu R., Wang A., 2022, ApJ, 928, L7
  * Rustamkulov et al. (2023) Rustamkulov Z., et al., 2023, Nature, 614, 659–663
  * Saba et al. (2022) Saba A., et al., 2022, AJ, 164, 2
  * Sing et al. (2016) Sing D. K., et al., 2016, Nature, 529, 59–62
  * Skaf et al. (2020) Skaf N., et al., 2020, AJ, 160, 109
  * Skilling (2004) Skilling J., 2004, in 8TH BRUNEI INTERNATIONAL CONFERENCE ON ENGINEERING AND TECHNOLOGY 2021. 8TH BRUNEI INTERNATIONAL CONFERENCE ON ENGINEERING AND TECHNOLOGY 2021, doi:10.1063/1.1835238
  * Swain et al. (2008) Swain M. R., Vasisht G., Tinetti G., 2008, Nature, 452, 329–331
  * Swain et al. (2009) Swain M. R., Vasisht G., Tinetti G., Bouwman J., Chen P., Yung Y., Deming D., Deroo P., 2009, ApJ, 690, L114–L117
  * Swain et al. (2014) Swain M. R., Line M. R., Deroo P., 2014, ApJ, 784, 133
  * Tennyson & Yurchenko (2012) Tennyson J., Yurchenko S. N., 2012, MNRAS, 425, 21
  * Tennyson et al. (2016) Tennyson J., et al., 2016, Journal of Molecular Spectroscopy, 327, 73
  * Tinetti et al. (2007) Tinetti G., et al., 2007, Nature, 448, 169–171
  * Tinetti et al. (2018) Tinetti G., et al., 2018, Experimental Astronomy, 46, 135–209
  * Tsai et al. (2023) Tsai S.-M., et al., 2023, Nature, 617, 483–487
  * Tsiaras et al. (2018) Tsiaras A., et al., 2018, AJ, 155, 156
  * Tsiaras et al. (2019) Tsiaras A., Waldmann I. P., Tinetti G., Tennyson J., Yurchenko S. N., 2019, Nature Astronomy, 3, 1086
  * Underwood et al. (2016) Underwood D. S., Tennyson J., Yurchenko S. N., Huang X., Schwenke D. W., Lee T. J., Clausen S., Fateev A., 2016, MNRAS, 459, 3890
  * Wakeford et al. (2018) Wakeford H. R., et al., 2018, AJ, 155, 29
  * Waldmann et al. (2015a) Waldmann I. P., Tinetti G., Rocchetto M., Barton E. J., Yurchenko S. N., Tennyson J., 2015a, ApJ, 802, 107
  * Waldmann et al. (2015b) Waldmann I. P., Rocchetto M., Tinetti G., Barton E. J., Yurchenko S. N., Tennyson J., 2015b, ApJ, 813, 13
  * Yurchenko et al. (2020) Yurchenko S. N., Mellor T. M., Freedman R. S., Tennyson J., 2020, MNRAS, 496, 5282

##  Appendix A Description of Appendices

Additional figures are included in these appendices. In most cases these
represent either additional tests of the reliability or stability of our
results under the scrutiny of further investigation or additional grids which
follow similar trends to those mentioned in the main text possibly for a
different set up (e.g. varying cloud levels).

Section A.1 provides the maps of the width of the posterior distribution for
each parameter in the low and no cloud cases. Similarly, section A.2 provides
supplementary grids for the bias of retrievals on the high resolution end of
the retrieval grid in each cloud case. A.3 provides three overlaid corner
plots to provide further evidence for the discussion of figure 8 in the main
text. In A.4 further plots considering retrievals on spectra binned from the
ERS wavelength grid are presented and A.5 shows similar plots for the
retrievals on a shifted wavelength grid. A.6 shows our investigation of the
effects of model complexity on the retrieval bias while A.7 and A.8 show plots
considering the difference in the retrieval bias under repeat retrievals with
or without additional scatter on the spectra.

###  A.1 Posterior width grids

Figures 13 and 14 show the maps of the posterior distribution widths plotted
for the cases of low and no clouds respectively. These plots are formatted in
the same way as figure 4 from the main text. They are included here for
completeness but only show the expected trend that the width of the posterior
distribution decreased as resolution increased and photometric error
decreased.

Figure 13: Posterior width maps for the case of a low cloud deck. In each
subplot (labelled by which parameter is represented) the pink crosses mark the
position of data from retrievals. Interpolation has been used to fill in the
rest of the grid and generate a map. A log scale is used for the spectral
resolution (x axis) since these data are spaced by 10 below R = 100 and by 100
between R = 100 and R = 500. The colour bar associated with each map describes
the width of the posterior distribution obtained in retrievals for the given
parameter. The white arrow marks the approximate position of the JWST ERS data
in the map (including error bars set by the 25th and 75th percentiles of the
variation across the observed spectrum in resolution and error bar size). The
green dashed lines plot potential binning paths under the assumption that each
bin (before binning) contains an equal number of counts. The pink dashed line
plots the potential binning path from the approximate position of the ERS data
point under the same assumptions. The lower right plot in the grid shows the
same data as the lower left but as a series of lines. Figure 14: The same as
figure 13 except in the case of no clouds rather than a low cloud deck.

###  A.2 High resolution bias maps

In figures 15, 16 and 17 we plot the bias maps for the high resolution end of
the retrieval grid in the cases of a high cloud deck, a low cloud deck and no
clouds respectively. These should be seen to compliment figures 5, 6 and 7 in
the main text which show the low resolution end of the grid (up to resolutions
of R = 500). Less deviation is seen at resolutions between 500 and 1000 so
they are left to the appendices rather than the main text.

Figure 15: Sensitivity maps for the case of a high cloud deck included in the
simulation. In each subplot (labelled by which parameter is represented) the
blue crosses mark the position of data from retrievals. Interpolation has been
used to fill in the rest of the grid and generate a map. A log scale is used
for the spectral resolution (x axis). The colour bar associated with each map
describes the percentage deviation of the value obtained in retrievals from
the true value of that parameter (input to the simulation). Note that the
faint dashed line indicates the position of the ERS data point in the
parameter space. The point itself is not visible since it lies outside of the
resolution range of these plots. Figure 16: Same as figure 15 except that here
we show data from retrievals on the low cloud spectra. Figure 17: Same as
figure 15 except that here we show data from retrievals on the cloud free
spectra.

###  A.3 Retrieval corner plots

Here we include corner plots for the same retrievals as those used to produce
the contribution plots shown in the main text, figure 8. Presented in figure
18, these retrievals use high cloud simulations at resolutions of 10, 100 and
400. They are plotted on the same set of corner plot axes in order to
demonstrate how we are better able to constrain most of the parameters with
increasing resolution but the log-abundance of SO2 remains poorly retrieved
since its features are masked by the cloud deck.

Figure 18: A corner plot of the retrievals used to produce the spectrum
contribution plots in figure 8. The three different colours represent
retrievals on spectra at three different resolutions.

###  A.4 Binning from ERS grid

Figure 19: The percentage deviation of retrieved parameters from spectra
obtained by binning down simulations of the JWST ERS data for WASP-39b using
the NIRSpec PRISM (Rustamkulov et al., 2023) with a high cloud deck. The
wavelength grid was constructed by retaining only every Nth point in the
spectrum where N is the ’Binning factor’. Each line represents a different
parameter and the top and bottom x-axes quantify the level of binning through
the binning factor and the average resolution respectively. All values for
average resolution are obtained by considering the mean value across the
spectrum. Solid lines show data from retrievals where the simulation was first
binned to the wavelength grid of the ERS data and then binned further whereas
the dashed lines bin directly from the native resolution of the TauREx
simulation to the appropriate reduced grid. Figure 20: The same as figure 19
except that these retrievals use cloud free spectra. Note that the values for
cloud pressure have been omitted since they cannot be accurately retrieved in
this set up and therefore their large deviations would make the variation in
other parameters difficult to view.

Using the same grids as discussed in section 4, we run retrievals on the
simulations for the high cloud and no cloud cases (with results shown in
figures 19 and 20 respectively). In both cases we observe similar trends when
retrieving parameters from simulations binned directly to the reduced grid
(dashed lines) and when binning first to the ERS resolution (solid lines).

###  A.5 Binning from ERS Grid with a shifted grid

As in the previous section, plots in this section use retrievals on spectra
which have been reduced using the methods discussed in section 4. However, in
this case some of the wavelength grids receive an artificial shift before the
spectra are binned. These shifts are set such that the wavelength points in
the shifted grid will fall halfway between the corresponding point and the
next point in the original wavelength grid.

Plots compare the percentage deviation (the accuracy) of the retrieved
parameters when retrieving from spectra that have been binned to grids with
and without the additional shift applied. The plot for data using a low cloud
simulation is included in the main text (figure 12) and the results for the
high and no cloud case are included here (figures 21 and 22 respectively).

Figure 21: The percentage deviation of retrieved parameters from spectra
obtained by binning down simulations of the JWST ERS data for WASP-39b using
the NIRSpec PRISM (Rustamkulov et al., 2023) with a high cloud deck. The
wavelength grid was constructed by retaining only every Nth point in the
spectrum where N is the ’Binning factor’. Each line represents a different
parameter and the top and bottom x-axes quantify the level of binning through
the binning factor and the average resolution respectively. All values for
average resolution are obtained by considering the mean value across the
spectrum. Solid lines show data from retrievals where the simulation was first
binned to the wavelength grid of the ERS data and then binned further. These
lines have a reduced opacity in order to highlight the new data shown by the
dot-dashed lines. These results also have a shift applied to the wavelength
grid prior to binning the simulation. Figure 22: The same as figure 21 except
that these retrievals use cloud free spectra. Note that the values for cloud
pressure have been omitted since they cannot be accurately retrieved in this
set up and therefore their large deviations would make the variation in other
parameters difficult to view.

###  A.6 Reduced model and retrieval complexity

In our investigation of changing model complexity, we compare the results in
the case of the original retrievals and only fitting for certain molecules in
the retrievals as well as cases where a model of reduced complexity is used in
the first place. Figure 23 shows results in the cases where we use a spectrum
only containing water and a spectrum containing all of the same molecules as
our initial investigation but only fitting for H2O in the retrieval. In this
second case, other molecules have their log-abundances fixed to their true
values. These two tests use the low cloud case and constant errors bars of 100
ppm (since these points in our grid probe near to the average error of the
JWST ERS data).

Figure 23: Comparing the retrieved values for parameters in reduced complexity
models and retrievals. On both the left and the right columns, data in black
show the results from the original resolution-error grid. Then, on the left,
coloured data represent the retrieved values when a spectrum containing H2O as
the only atmospheric molecule is input. In the right column, coloured data
represent retrieved values when a spectrum featuring the same complexity as
the original model is used but the retrieval only fits for the molecular log-
abundance of H2O. All of the other molecular abundances are fixed to their
correct values. All of these simulations used the low cloud scenario and
errors of 100 ppm.

###  A.7 Repeat retrieval instances

Plots in this section consider the results from repeated retrievals. By this
we mean cases where retrievals were re-run on spectra with constant resolution
grids and constant error bars (set at 100 ppm). We include plots for the high,
low and no cloud simulations (figures 24, 25 and 26 respectively). Overall we
do not see any significant differences between the original sets of retrievals
and these repeat instances. Our retrievals return the same predictions in both
cases (within error) indicating the stability of these results despite the
stochatsic sampling processes used.

Figure 24: Two set of retrieval results for identical input spectra (high
cloud simulations with errors of 100 ppm). The top panel plots the two sets of
results against one another with the original data using a solid line and
crosses and the repeat instance of the retrievals using a dashed line and
circles. Parameter are separated by colours in the plot. The lower panel shows
the difference in the percentage deviation for each parameter for each
retrieval. Figure 25: The same as figure 24 except that these retrievals use
low cloud spectra. Figure 26: The same as figure 24 except that these
retrievals use cloud free spectra.

###  A.8 Retrievals on scattered spectra

To investigate the effect of adding additional noise to the data, we scatter a
subset of the spectra from our original retrieval grid. The spectra used are
low cloud simulations with errors of 50, 100 and 200 ppm.

With dashed lines and circular markers, we plot the new data in the a) panels
of figure 27 alongside the corresponding retrievals from the original grid
plotted with solid lines and cross markers. Then, in the b) panels, we plot
the differences in the percentage deviation of each parameters prediction when
retrieving from a scattered or noise-free simulation.

Since we are only using one instance of the random resampling of spectra, we
expect there to be some outliers in the predictions due to particular noise
instances. However, these plots help to show the extent of the possible error
emerging from one specific noise instance. With a photometric error of 200 ppm
on every spectral point, the percentage deviations can differ by more than 25
% between the noise-free and noisy spectra. However, it is only about 5 % for
the worst case in the the spectra with errors of 50 ppm. Additionally, it is
worth noting that these significant differences are seen at the low resolution
ends of our tested grids since here the data is more dependent on its accuracy
in transit depth than at the high resolution end where this is accounted for
by finer wavelength coverage.

Figure 27: Comparison of two set of retrieval results for simulations of a low
cloud spectrum with errors of 50, 100 or 200 ppm. In each pair of plots (an a)
and a b) plot), one retrieval is run on a spectrum without any noise added and
one is run on a spectrum where the data have been scattered according to a
Gaussian distribution centred on the spectrum point and with a width set by
the size of the error bar. The a) panels plot the two sets of results against
one another with the non-scattered data using a solid line and crosses and the
perturbed instance of the retrievals using a dashed line and circles.
Parameter are separated by colours in their plots. The b) panels show the
difference in the percentage deviation for each parameter for each retrieval
between the scattered and non-scattered datasets.