Journal ArticleDOI

# A systematic retrieval analysis of secondary eclipse spectra. i. a comparison of atmospheric retrieval techniques

, Xi Zhang1, D. Crisp1
01 Oct 2013-The Astrophysical Journal (American Astronomical Society)-Vol. 775, Iss: 2, pp 137

AbstractExoplanet atmosphere spectroscopy enables us to improve our understanding of exoplanets just as remote sensing in our own solar system has increased our understanding of the solar system bodies. The challenge is to quantitatively determine the range of temperatures and molecular abundances allowed by the data, which is often difficult given the low information content of most exoplanet spectra that commonly leads to degeneracies in the interpretation. A variety of spectral retrieval approaches have been applied to exoplanet spectra, but no previous investigations have sought to compare these approaches. We compare three different retrieval methods: optimal estimation, differential evolution Markov chain Monte Carlo, and bootstrap Monte Carlo on a synthetic water-dominated hot Jupiter. We discuss expectations of uncertainties in abundances and temperatures given current and potential future observations. In general, we find that the three approaches agree for high spectral resolution, high signal-to-noise data expected to come from potential future spaceborne missions, but disagree for low-resolution, low signal-to-noise spectra representative of current observations. We also compare the results from a parameterized temperature profile versus a full classical Level-by-Level approach and discriminate in which situations each of these approaches is applicable. Furthermore, we discuss the implications of our models for the inferred C-to-O ratios of exoplanetary atmospheres. Specifically, we show that in the observational limit of a few photometric points, the retrieved C/O is biased toward values near solar and near one simply due to the assumption of uninformative priors.

Topics: Exoplanet (51%), Markov chain Monte Carlo (51%), Monte Carlo method (51%)

### 1. INTRODUCTION

• Thermal emission spectra (∼1–30 μm) of extrasolar planets can tell us about their atmospheric temperatures and compositions (see, e.g., Charbonneau et al.
• The authors call their three-pronged retrieval approach CHIMERA (CaltecH Inverse ModEling and Retrieval Algorithms).

### 2.1. The Retrieval Techniques

• The authors use three different retrieval techniques to infer the compositions and temperatures from a spectrum.
• The techniques are inherently Bayesian as they attempt to solve the inverse problem by summarizing the full shape of the posterior in terms of the location in parameter space of the maximum likelihood and the uncertainties about that location.
• The first, and the fastest (least number of forward model calls), of these approaches is OE, the second is the model-dependent BMC, and the third is DEMC.

### 2.1.1. Optimal Estimation (OE)

• The OE retrieval approach is well established in the fields of Earth atmosphere remote sensing (Rodgers 1976, 2000; Twomey 1996; Kuai et al. 2013), solar system atmosphere remote sensing (Conrath et al.
• The basic approach is to minimize a cost function to obtain the maximum a posteriori (MAP) solution.
• The second term represents the prior uncertainties before making the measurements, which has less influence for higher quality data.
• This assumption is only valid when the region in phase space over which the forward model can be linearized is broader than the parameter uncertainties.

### 2.1.2. Model-dependent Bootstrap Monte Carlo (BMC)

• A common way to more robustly characterize errors is through a Monte Carlo resampling of the data (see, e.g., Press et al.
• These synthetic data are then refit using, say, OE, and the resulting best-fit parameter distributions represent the uncertainties.
• This new realization is then fit and the process is repeated many times.
• The approach the authors take is similar, but rather than generate a new spectrum using the residual, they simply take the best fit, from OE, and then resample each point by drawing it from a normal distribution with a mean given by the best-fit value and the width given by the data error bar for that point.

### 2.1.3. Differential Evolution Markov Chain Monte Carlo (DEMC)

• The MCMC approach has revolutionized parameter estimation and error analysis in many fields.
• If the proposed state (xp) has an improved likelihood over the current state, then that state is kept (xi+1 = xp) and a new proposal is made from that location.
• Rather than standard MCMC approaches, the authors use an adaptive algorithm known as DEMC (ter Braak 2006; ter Braak & Vrugt 2008).
• Set the last link in one of the chains to the best-fit state vector obtained in step 1.
• Repeat this process for the other Nchains − 1 chains.

### 2.2. The Forward Model

• The forward model, F(x), is the most important part of any retrieval algorithm.
• In the case of atmospheric retrieval, the forward model takes temperatures and compositions and generates a model spectrum.
• On that note, the authors use the HITEMP database (Rothman et al. 2010) to compute the tabulated cross sections for CO2, CO, and H2O and the STDS database for CH4 (Wenger & Champion 1998).
• The authors feel this is appropriate for two reasons.

### 2.2.1. Parameterized versus Level-by-Level Temperature Profile

• The authors employ two approaches to retrieve the temperature profiles.
• The Level-by-Level approach is only appropriate when the information content of the spectra is sufficiently high such that the addition of the Nlev additional parameters is justified.
• The second temperature profile retrieval approach makes use of a parameterization.
• This τ–P mapping assumes a linear relation between the optical depth and pressure, or a pressure-independent IR opacity.
• The authors can think of the prior as an artificial set of “data” on which the retrieval (all retrieval approaches) can rely when the measurements are insufficient to constrain a given parameter.

### 3.1. Synthetic Observations

• The authors create a generic hydrogen-dominated hot Jupiter planet and derive its emission spectrum in three different observing scenarios.
• Figure 2 shows the model atmosphere and spectrum of the synthetic planet.
• The thermal contribution functions indicate that the emission from shorter wavelengths comes from deeper layers in the atmosphere, and regions of high opacity tend to push the emission to higher altitudes.
• The size of the error bars are representative of typical errors of IRAC observations (e.g., Machalek et al. 2009).
• The second observational scenario is a multi-instrument case combining both Spitzer photometry, ground-based photometry, and Hubble WFC3 spectra .

### 3.2. The Prior

• As mentioned in Section 2.1.1, the prior is important when the spectral information content is limited.
• The authors choose extremely broad Gaussian priors in order to mitigate the influence they have on the retrievals.
• This value could go higher in the presence of very low emissivity and redistribution.
• Generally the opacity may have a pressure dependence, and hence solving self-consistently for this constant-with-altitude opacity would not actually recover the pressure dependence of this opacity.
• The prior temperature profile distributions are reconstructed by propagating the Gaussian prior probability distributions (including the above limits) of κIR, γv1 , γv2 , β, and α in Table 2 through Equations (13)–(16).

### 3.3. Results from the Parameterized Temperature Profile

• The authors apply the three retrieval techniques to the three synthetic observations in Figure 3 under the radiative equilibrium temperature profile parameterization.
• The temperature prior in Figure 4 is used in all three techniques for all three retrieval cases.
• The authors must be careful in interpreting the confidence interval values when the posteriors extend to the imposed upper and lower limits, especially when those limits are somewhat arbitrary.
• The second and third rows show the fits from the BMC and DEMC, respectively.
• In the following subsections, the authors summarize posteriors for the gas compositions, temperatures, and C to O ratios for each observational scenario.

### 3.3.1. Gas Abundance Retrievals

• The gas mixing ratio retrieval results are summarized in Figures 6 and 7.
• The first column of Figure 6 shows the marginalized gas posteriors for the broadband observational scenario and the top set of panels of Figure 7 show the correlations amongst the gases.
• CH4 has a better defined upper edge than CO2 because both the 3.6 μm and 8 μm channels overlap with the strongest methane absorption bands.
• BMC underestimates the uncertainties in all species with the exception of water.
• The differences in the 1σ uncertainties derived from OE are less than ∼10% than the uncertainties derived from DEMC.

### 3.3.2. Temperature Profile Retrievals

• The marginalized posteriors for the five parameters that govern the shape of the temperature profile for each observational scenario are shown in Figure 8.
• The dark and light red swaths show the 1σ and 2σ bounds on the reconstructed temperature profiles.
• Again, the BMC approach completely underestimates the error when compared with the other two approaches because of its inability to fully characterize the posterior outside of a small region of phase space localized around the OE original best fit.
• OE and BMC underestimate the temperature uncertainties relative to DEMC at 100 mbar, but the OE and DEMC have reasonable agreement over the entire profile.
• This correlation is prevalent in all three observational scenarios, as even the broadband points are strongly effected by water vapor absorption .

### 3.3.3. C to O Ratios

• Determination of the C-to-O ratios of explanatory atmospheres is critical to the understanding of their atmospheric chemistry (Lodders & Fegley 2002; Moses et al. 2011) and formation environments (Öberg et al. 2011; Madhusudhan et al. 2011).
• Before inspecting the posteriors derived from CHIMERA, the authors find it illustrative to investigate the prior.
• Upon propagating the Gaussian priors (with the limits) of the gases through Equation (19), the authors obtain the C/O prior in Figure 10.
• The authors would expect then that one gas will have a larger abundance than the other three 25% of the time.
• Since the OE best-fit gas abundances are very near truth , the BMC posteriors, which are highly localized about the OE best-fit parameters, will overemphasize the C/O derived by that best fit.

### 3.4. Results from the Level-by-Level Temperature Profile

• The Level-by-Level temperature profile approach attempts to determine the temperature for each model layer.
• The uncertainties in temperature at 100 mbar are reduced to ±177 K, though smaller uncertainties are achieved at deeper levels due to the addition of the WFC3 data which probe deeper atmospheric levels.
• This is why the averaging kernel profile peaks at a deeper level.
• As before, in both cases the BMC approach underestimates the temperature uncertainties relative to the OE derived uncertainties.
• For this, the authors investigate the effect of three different temperature priors (different prior profiles, xa, but same widths, Sa) on the retrieved profiles and check to see if they are consistent with the estimated errors .

### 4. DISCUSSION AND CONCLUSIONS

• The authors have developed a new statistically robust suite of exoplanet atmospheric retrieval algorithms known as CHIMERA.
• In the high signal-to-noise and high spectral resolution regime, both the BMC and OE methods provide reasonable parameter uncertainties.
• The temperature profiles and corresponding uncertainties can only be trusted for the region over which the thermal emission contribution functions peak, typically between a few bars and a few mbar .
• The authors also thank John Johnson and Jonathan Fortney for useful conversations.

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The Astrophysical Journal, 775:137 (22pp), 2013 October 1 doi:10.1088/0004-637X/775/2/137
C
A SYSTEMATIC RETRIEVAL ANALYSIS OF SECONDARY ECLIPSE SPECTRA. I.
A COMPARISON OF ATMOSPHERIC RETRIEVAL TECHNIQUES
Michael R. Line
1
, Aaron S. Wolf
1
, Xi Zhang
1
, Heather Knutson
1
, Joshua A. Kammer
1
, Elias Ellison
2
,
Pieter Deroo
3
, Dave Crisp
3
, and Yuk L. Yung
1
1
Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA; mrl@gps.caltech.edu
2
Flintridge Preparatory School, La Ca
˜
3
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109-8099, USA
Received 2013 April 19; accepted 2013 August 19; published 2013 September 16
ABSTRACT
Exoplanet atmosphere spectroscopy enables us to improve our understanding of exoplanets just as remote sensing in
our own solar system has increased our understanding of the solar system bodies. The challenge is to quantitatively
determine the range of temperatures and molecular abundances allowed by the data, which is often difﬁcult given
the low information content of most exoplanet spectra that commonly leads to degeneracies in the interpretation. A
variety of spectral retrieval approaches have been applied to exoplanet spectra, but no previous investigations have
sought to compare these approaches. We compare three different retrieval methods: optimal estimation, differential
evolution Markov chain Monte Carlo, and bootstrap Monte Carlo on a synthetic water-dominated hot Jupiter. We
discuss expectations of uncertainties in abundances and temperatures given current and potential future observations.
In general, we ﬁnd that the three approaches agree for high spectral resolution, high signal-to-noise data expected
to come from potential future spaceborne missions, but disagree for low-resolution, low signal-to-noise spectra
representative of current observations. We also compare the results from a parameterized temperature proﬁle versus
a full classical Level-by-Level approach and discriminate in which situations each of these approaches is applicable.
Furthermore, we discuss the implications of our models for the inferred C-to-O ratios of exoplanetary atmospheres.
Speciﬁcally, we show that in the observational limit of a few photometric points, the retrieved C/Oisbiasedtoward
values near solar and near one simply due to the assumption of uninformative priors.
Key words: methods: data analysis methods: statistical planets and satellites: atmospheres radiative transfer
Online-only material: color ﬁgures
1. INTRODUCTION
Thermal emission spectra (1–30 μm) of extrasolar plan-
ets can tell us about their atmospheric temperatures and com-
positions (see, e.g., Charbonneau et al. 2005; Tinetti et al.
2007, 2010a; Grillmair et al. 2007, 2008; Swain et al. 2009a,
2009b; Madhusudhan & Seager 2009; Stevenson et al. 2010;
Madhusudhan et al. 2011; Lee et al. 2012; Line et al. 2012). At
the moment, these observations come in two types, broadband
photometry mainly from the Spitzer Space Telescope (see, e.g.,
Knutson et al. 2010) and ground-based instruments (Croll et al.
2010; Anderson et al. 2010; Gibson et al. 2010; Deming et al.
2012; Gillon et al. 2012), as well as higher resolution spectra
such as Hubble Space Telescope (HST) Wide Field Camera 3
(WFC3; Berta et al. 2012; Swain et al. 2012; Deming et al.
2013) and Near-Infrared Camera and Multi-Object Spectrom-
eter (Swain et al. 2009a, 2009b; Tinetti et al. 2010a;Gibson
et al. 2011; Crouzet et al. 2012). From these observations,
signatures of a variety of molecules have been detected in-
cluding H
2
O, CH
4
, CO, and CO
2
(Tinetti et al. 2007;Swain
et al. 2009a, 2009b; Tinetti et al. 2010a), although the ro-
bustness of some of these detections have recently been called
into question (Gibson et al. 2011). These same data have been
used to infer the presence of atmospheric temperature inver-
sions for a subset of hot Jupiters (e.g., Burrows et al. 2007;
Knutson et al. 2008, 2010; Fortney et al. 2008; Madhusudhan &
Seager 2009).
While the above studies have given us insight into the nature
of these planetary atmospheres, very few have focused on the
uncertainties in temperatures and compositions. Until relatively
2011; Lee et al. 2012; Line et al. 2012), most compositions and
temperatures, and thus the subsequent conclusions, were de-
termined through self-consistent forward modeling approaches
that only explore a few potential solutions without a well-deﬁned
characterization of the uncertainty distributions of the physical
parameters (e.g., Burrows et al. 2005, 2007; Fortney et al. 2005).
Furthermore, some self-consistent solutions make physical as-
sumptions that may not necessarily be valid in exoplanetary
atmospheres such as the assumption of thermochemical equi-
librium gas concentrations or radiative–convective temperature
structures (that is, they may ignore other potentially important
processes such as vertical mixing, photochemistry, zonal winds,
etc.). Additionally, this forward modeling approach is often un-
guided by the data and primarily driven by preconceived notions
of how the atmosphere “should” look (as pointed out by Lee et al.
2012 and Benneke & Seager 2012) with the best solutions being
the few that provide the lowest values of chi-squared.
In order to more rigorously characterize the ranges of
allowable temperatures and compositions, Madhusudhan &
Seager (2009) developed a multidimensional grid search ap-
proach which can fully characterize the uncertainty distributions
for each parameter. Subsequent studies (Madhusudhan et al.
2011; Benneke & Seager 2012) used the more sophisticated
Markov chain Monte Carlo (MCMC) approach to accomplish
this goal. However, such approaches require the computation
of many millions of models in order to fully characterize the
parameter uncertainties which may not be feasible for more so-
phisticated forward models with many free parameters. In order
to remedy this problem, Lee et al. (2012) and Line et al. (2012)
1

The Astrophysical Journal, 775:137 (22pp), 2013 October 1 Line et al.
used the much faster optimal estimation (OE; e.g., Rodgers
2000) approach to estimate the error distributions of each pa-
rameter. This approach is much faster due to the assumption
that the parameter error distributions are Gaussian. However,
this Gaussian assumption may result in an incorrect estimate of
the error distributions (Benneke & Seager 2012).
The goals of this paper are to ﬁrst understand the composition
and temperature uncertainty distributions for different degrees
of observational quality, and second to understand how those
derived uncertainty distributions differ between the two fun-
damental parameter estimation approaches, OE and MCMC.
This investigation represents the ﬁrst direct comparison and
synthesis of these retrieval approaches as applied to exoplanet
atmospheres. A secondary goal is to understand how the de-
rived composition uncertainties propagate into the C/O uncer-
tainty. We accomplish these goals by comparing three different
retrieval algorithms: OE, a new MCMC algorithm known as
differential evolution Markov chain Monte Carlo (DEMC), and
the model-dependent bootstrap Monte Carlo (BMC) approach.
This investigation is analogous to the investigation carried out
by Ford (2005) on radial velocity data. First, we will describe
the three different retrieval techniques as well as our forward
model in Section 2. We call our three-pronged retrieval ap-
proach CHIMERA (CaltecH Inverse ModEling and Retrieval
Algorithms). Second, we compare the three spectral retrieval
methods on different synthetic spectral data sets of varying
observational quality in order to assess the robustness of the
error estimations from each approach in Section 3. We will also
compare the parameterized temperature proﬁle approach (e.g.,
Madhusudhan & Seager 2009; Line et al. 2012) with the Level-
by-Level proﬁle approach (Lee et al. 2012). Finally, we will
discuss the implications of these uncertainties for the estimated
C to O ratios.
2. METHODS
In this section, we describe the retrieval techniques, the
forward model, and the parameterizations we use to retrieve the
temperatures and compositions from thermal emission spectra.
2.1. The Retrieval Techniques
We use three different retrieval techniques to infer the
compositions and temperatures from a spectrum. The techniques
are inherently Bayesian as they attempt to solve the inverse
problem by summarizing the full shape of the posterior in terms
of the location in parameter space of the maximum likelihood
and the uncertainties about that location. The ﬁrst, and the fastest
(least number of forward model calls), of these approaches is
OE, the second is the model-dependent BMC, and the third is
DEMC.
2.1.1. Optimal Estimation (OE)
The OE retrieval approach is well established in the ﬁelds
of Earth atmosphere remote sensing (Rodgers 1976, 2000;
Twomey 1996; Kuai et al. 2013), solar system atmosphere
remote sensing (Conrath et al. 1998; Irwin et al. 2008; Nixon
et al. 2007; Fletcher et al. 2007; Greathouse et al. 2011), and
recently exoplanet atmosphere remote sensing (Lee et al. 2012;
Line et al. 2012). The basic approach is to minimize a cost
function to obtain the maximum a posteriori (MAP) solution.
Using Bayes theorem and the assumption that the data likelihood
and the prior are Gaussian, one can derive the following cost
function (or log likelihood):
J (x) = (y F(x))
T
S
1
e
(y F(x)) + (x x
a
)
T
S
1
a
(x x
a
), (1)
where y is the set of n observations, x is the set of m parameters
which we wish to retrieve or the state vector, F(x)istheforward
model that maps the state vector onto the observations (described
in Section 2.2), and S
e
is the n ×n data error covariance matrix
(typically off diagonal terms are zero and the diagonal elements
are the square of the 1σ errors of the observations). x
a
is the a
priori state vector and S
a
is the m×m a priori covariance matrix.
The ﬁrst term in Equation (1) is simply “chi-squared” and the
second term represents the prior knowledge of the parameter
distribution before we make the observations. For high-quality
observations the second term is generally not important as
most of the information in constraining the state vector comes
from the observations. For low-quality observations it is just
the opposite. Following Irwin et al. (2008), we minimize
Equation (1) with Newton’s iteration method given by
x
i+1
= x
a
+ S
1
a
K
T
i
K
i
S
1
a
K
T
i
+ S
1
e
(F(x) y K
i
(x
a
x
i
)), (2)
where i is the iteration index and K
i
is the Jacobian matrix at i
(K
nm
= ∂F
n
/∂x
m
). Rather than taking the full Newton step, we
damp the solution with
x
i+1
= x
i
+
x
i+1
x
i
1 + ζ
, (3)
where ζ is the damping parameter. At each iteration, we evaluate
J(x
i+1
) and J(x
i+1
). If the latter is smaller, we set the state vector
for the next iteration to x
i+1
and decrease ζ by 0.3. Otherwise, we
keep increasing ζ by a factor of 10 and re-evaluate Equations (1)
and (3) until J(x
i+1
) becomes less than J(x
i+1
). Convergence is
achieved when J changes by less than 1×10
6
from the previous
iteration, which typically occurs after 10 s of iterations. The
resulting state vector is the MAP solution, or the “best ﬁt.
Assuming that the posterior is normal, which is achieved by
linearizing the forward model about the best-ﬁt solution, the
uncertainties on the state vector parameters are given by the
posterior covariance matrix:
ˆ
S =
K
T
S
1
e
K + S
1
a
1
. (4)
Again, this matrix represents a multi-dimensional normal dis-
tribution (see Rodgers 2000 for the derivation). The diagonal
elements are the square of the marginalized errors, whereas
the off diagonal terms describe the correlations/degeneracies
amongst the parameters. The ﬁrst term, K
T
S
1
e
K, represents the
uncertainties due to the measurement errors. This term uses the
local gradient information to estimate the parameter uncertain-
ties. The second term represents the prior uncertainties before
making the measurements, which has less inﬂuence for higher
quality data. Again, the major assumption in Equations (1)
and (4) is that the posterior for each parameter is Gaussian.
This assumption is only valid when the region in phase space
over which the forward model can be linearized is broader than
the parameter uncertainties. However, it is this assumption that
allows this approach to be extremely fast requiring only tens
of forward model calls, which given the speed of our forward
model (5 s), results in a full retrieval in only a few minutes. As
we shall see in Section 3, this assumption is valid for data that
is of high resolution and signal to noise, but breaks down for
2

The Astrophysical Journal, 775:137 (22pp), 2013 October 1 Line et al.
low-resolution, low signal-to-noise data. In order to ensure that
the global minimum of Equation (1) is found, multiple start-
ing guesses are used. They generally all converge to the same
solution.
2.1.2. Model-dependent Bootstrap Monte Carlo (BMC)
A common way to more robustly characterize errors is
through a Monte Carlo resampling of the data (see, e.g., Press
et al. 1995, Chapter 15.6; Ford 2005, Section 4.2) in which
many thousands of realizations of the original data (in our
case, the spectra) are created using the uncertainties from the
original data set. These synthetic data are then reﬁt using, say,
OE, and the resulting best-ﬁt parameter distributions represent
the uncertainties. There are multiple ways of generating the
synthetic data realizations. The most common way is the residual
resampling approach in which data realizations are created
by adding the randomly shufﬂed residual between the best-ﬁt
model and the data back to the original best-ﬁt model. This new
realization is then ﬁt and the process is repeated many times.
The approach we take is similar, but rather than generate a new
spectrum using the residual, we simply take the best ﬁt, from
OE, and then resample each point by drawing it from a normal
distribution with a mean given by the best-ﬁt value and the width
given by the data error bar for that point. We chose this approach
over the residual resampling approach because sparse coverage
spectra, like those from broadband observations, have virtually
no residual as they can be ﬁt perfectly due to the greater number
of parameters than data points. We typically generate 1000
spectra realizations that are then reﬁt by OE to obtain the state
vector parameter distributions.
2.1.3. Differential Evolution Markov Chain Monte Carlo (DEMC)
The MCMC approach has revolutionized parameter estima-
tion and error analysis in many ﬁelds. It is routinely used
in radial velocity (Ford 2005) and transit light curve (e.g.,
Eastman et al. 2013) error analysis. Results from a well con-
verged MCMC analysis can generally be considered as the best
possible representation of the parameter uncertainties. Recently,
this approach has been applied to the exoplanet atmosphere re-
trieval problem (Madhusudhan et al. 2011; Benneke & Seager
2012). Unlike OE, MCMC approaches make no assumptions
about the shape of the posterior, but rather evaluate the posterior
with millions of samples.
The basic approach of MCMC is to sample the posterior
through a random walk process. The random walk is carried
out by drawing states from some proposal distribution and
evaluating whether or not the proposed state has an increased
likelihood over the previous. Typically, the proposal distribution
is a normal distribution with a mean given by the current state
in the chain (x
i
) and a user-deﬁned width to achieve a particular
acceptance rate (Gibbs sampling or Metropolis–Hastings). If the
proposed state (x
p
) has an improved likelihood over the current
state, then that state is kept (x
i+1
= x
p
) and a new proposal is
made from that location. If the proposal state has not improved
the likelihood, then that state is either rejected or accepted with
some probability. This previous state-dependent random walk
constitutes a Markov chain. Given enough samples, this Markov
chain will converge to the target posterior (see Ford 2005 for a
more detailed explanation).
Rather than standard MCMC approaches, we use an adaptive
algorithm known as DEMC (ter Braak 2006; ter Braak &
Vrugt 2008). The purpose of this approach is to obtain more
appropriate proposal states by identifying the proper scale and
orientation of the current estimate of the posterior. This scale
and orientation information comes from the chain history. This
approach gives a more efﬁcient probing method for highly
correlated parameter spaces and yields improved convergence
rates. Our DEMC procedure is as follows.
1. Apply the OE technique to the measurements to obtain the
best-ﬁt state vector and posterior covariance matrix,
ˆ
S.This
step provides an initial estimate of the posterior.
2. Initialize N
init
i=0N
init
) in each of the N
chains
(typ-
ically three chains, more chains will slow convergence)
independent chains (arrays) by randomly drawing state vec-
tors from the multivariate normal described by the posterior
covariance matrix from step 1. Set the last link in one of
the chains to the best-ﬁt state vector obtained in step 1.
This step provides a good starting history from which our
initial proposal states can be drawn. Combine each of the
independent chains into one long chain that composes the
history, X
history
.
3. Evaluate the cost function, J, in Equation (1) for the last link
in each of the chains. If using a ﬂat prior, ignore the second
term. Again, this is simply the equivalent of evaluating chi-
squared.
4. Draw two random numbers, R
1
and R
2
, between zero and
N
chains
×i, where i is the current state in the chain. Initially,
i = N
init
. Evaluate the proposed jump state given by
x
p
= x
i
+ γ (x
R1
x
R2
)+e, (5)
where x
R
1
and x
R
2
are the states from different points in
the chain history, X
history
. γ is a scale factor typically set to
2.38/
(2 m) (ter Braak 2006), where m is the number of
parameters. This factor is meant to give acceptance rates of
0.23 for large m. e is a vector drawn from a multivariate
normal distribution with a small variance relative to the
chain variance in order to introduce a small amount of
additional randomness. Repeat this process for the other
N
chains
1 chains.
5. Evaluate the Metropolis (Metropolis et al. 1953) ratio,
r = P (x
p
)/P (x
i
) = e
1
2
(J(x
p
)J(x
i
))
.Ifr is larger than 1,
set x
i+1
= x
p
and if it is smaller only accept if it is larger
than a random number between 1 and 0. Otherwise, do
not update the chain, set x
i+1
= x
i
. Repeat for the other
N
chains
chains
to
X
history
.
6. Repeat steps 4 and 5 until convergence is met. Convergence
can be determined by looking at the trace plots of X
history
for
each parameter or by using the Gelman–Rubin statistic on
the set of N
chains
chains. For this we use the algorithm from
Eastman et al. (2013) which requires the Gelman–Rubin
statistic to be less than 1.01 and the number of independent
draws to be greater than 1000 for each parameter. Conver-
gence typically occurs in less than 10
5
N
chains
for a total of N
chains
× 10
5
speed of our forward model, takes 5 days for a typical
run. This is about an order of magnitude less than parallel
tempering or pure Metropolis–Hastings.
2.2. The Forward Model
The forward model, F(x), is the most important part of any re-
trieval algorithm. It is what maps the state vector of retrievable
parameters onto the observations. In the case of atmospheric
retrieval, the forward model takes temperatures and composi-
tions and generates a model spectrum. Our particular forward
3

The Astrophysical Journal, 775:137 (22pp), 2013 October 1 Line et al.
model numerically solves the planet-parallel thermal infrared
(IR) radiation problem for an absorbing, emitting atmosphere
(we neglect scattering). We ﬁrst divide the atmosphere into N
lev
discretized atmospheric layers. The absorption optical depth for
the kth gas in the zth layer at wavelength λ is
Δτ
k,z,λ
= f
k,z
σ
k,z,λ
ΔP
z
μ
atm
g
, (6)
where f
k,z
is the volume mixing ratio of the kth gas in the zth
layer, σ
k,z,λ
is the absorption cross section per molecule of the kth
gas in zth layer at wavelength λ, ΔP
z
is pressure thickness of the
zth slab, μ
atm
is the mean molecular weight of the atmosphere,
and g is the gravity. The absorption cross sections are pre-
computed on a 1 cm
1
wavenumber grid at 20 evenly spaced
temperature and log-pressure points from 500–3000 K and
50–10
6
bars, respectively (similar to Sharp & Burrows 2007).
The cross sections for each wavelength on the pre-computed grid
are interpolated to the atmospheric temperatures and pressures
in the zth slab. To compute the total slab optical depth, we sum
the contribution from each gas to obtain
Δτ
z,λ
=
N
gas
k=1
Δτ
k,z,λ
. (7)
Upon computing the optical depths at each level, we can now
solve for the upwelling irradiance with
I
λ
=
N
lev
z=0
B
λ
(T
z
)e
N
lev
j=z
Δτ
j,λ
Δτ
z,λ
, (8)
where N
lev
is the number of atmospheric levels and B
λ
(T
z
)is
the Planck function at wavelength λ and temperature in the zth
slab. We use 90 atmospheric layers to compute the upwelling
ﬂux.
An important part of the forward model when using the OE
approach is the computation of the Jacobian, or the sensitivity to
the upwelling irradiance with respect to the desired retrievable
parameters. When possible, it is preferable that the Jacobian
be calculated analytically for both improvements in speed
and in accuracy. We are interested in the retrieval of both
abundances and temperatures so we must compute the Jacobian
with respect to both the abundances and temperatures. We
make the assumption of vertically uniform gas mixing ratios
throughout the atmosphere, and hence f
k,z
is independent of z.
We now differentiate Equation (8) with respect to the uniform
gas mixing ratios for each gas f
k
to obtain
∂I
λ
∂f
k
=
N
lev
z=0
B
λ
(T
z
)e
N
lev
j=z
Δτ
j,λ
Δτ
k,z,λ
f
k
N
lev
z=0
B
λ
(T
z
)e
N
lev
j=z
Δτ
j,λ
Δτ
z,λ
N
lev
j=z
Δτ
k,j,λ
f
k
. (9)
The ﬁrst term is due to the changing emissivity of the emitting
slab and the second term is how the change in transmittance
The sensitivity of the irradiance to a change in temperature
in the zth slab is given by
∂I
λ
∂T
z
=
e
N
lev
j=z+1
Δτ
j,λ
e
N
lev
j=z
Δτ
j,λ
∂B
λ
(T
z
)
∂T
z
. (10)
This equation is similar to Equation (14) in Irwin et al. (2008)
but we have neglected the ﬁrst and last terms in their formula as
they are small.
Since the observations are reported as the ratio of the planet
ﬂux to the stellar ﬂux and not the irradiance, we perform a
disk integration of Equations (8)–(10) using four point Gaussian
quadrature and then divide by an interpolated PHOENIX stellar
ﬂux grid model (Allard et al. 2000).
We include only CH
4
,CO
2
,CO,H
2
O, H
2
, and He in our
model. H
2
and He are ﬁxed in our models at thermochemically
justiﬁable abundances. The exact abundances of these species
are not critical as the sensitivity of the spectrum to H
2
and
He is minimal. We retrieve only CH
4
,CO
2
, CO, and H
2
O. We
choose these species because they are the most spectroscopically
active and abundant species. Admittedly, we could/should
include every possible atmospheric constituent but this would
be unwieldy and reliable high-temperature absorption line lists
only exist for a few. On that note, we use the HITEMP
database (Rothman et al. 2010) to compute the tabulated cross
sections for CO
2
, CO, and H
2
O and the STDS database for
CH
4
(Wenger & Champion 1998). Below 1.7 μmforCH
4
,
we simply use the HITRAN (Rothman et al. 2009) database
for lack of anything better (to the best of our knowledge).
We use the Borysow et al. (2001), Borysow (2002), and
Jørgensen et al. (2000) databases for the computation of the
H
2
–H
2
/He collision induced opacities. The Reference Forward
Model (http://www.atm.ox.ac.uk/RFM/) was used to compute
the tabulated cross sections from the line strength databases. We
have validated our forward model through a detailed comparison
with the Oxford NEMESIS group (e.g., Lee et al. 2012) and our
results agree to better than 5% (see Figure 1).
An additional component of the forward model is the instru-
mental function used to convolve with the high-resolution model
spectrum. For the broadband points we simply integrate the ﬂux
from the high-resolution model spectrum with the appropriate
ﬁlter function for that point. When ﬁtting higher resolution ob-
servations, the instrumental function is assumed to be a Gaussian
(valid for grating spectrometers) in wavelength with an FWHM
determined by observations.
Now that we have a well-deﬁned forward model we can deﬁne
our state vector. Again, we wish to retrieve the abundances of
CH
4
,CO
2
, CO, and H
2
O and the temperature proﬁle. More
speciﬁcally, we choose to retrieve the log of the abundances as
they can vary by orders of magnitude and to prevent negative
mixing ratios. Our state vector is given by
x = [log(f
H
2
O
), log(f
CH
4
), log(f
CO
), log(f
CO
2
),T]
T
, (11)
where the f
k
s are all assumed constant with altitude. We feel this
is appropriate for two reasons. First, vertical mixing will smooth
out the mixing ratio proﬁles over the thermal IR photosphere
(Line et al. 2010, 2011; Moses et al. 2011), and second, current
observations simply do not have the information content to
warrant the retrieval of vertical mixing ratio information (see
Lee et al. 2012). In the next section, we describe how to go
2.2.1. Parameterized versus Level-by-Level Temperature Proﬁle
We employ two approaches to retrieve the temperature
proﬁles. The ﬁrst, and the most commonly used in Earth and
solar system atmosphere retrieval problems, is the Level-by-
Level approach. This is the approach used in Lee et al. (2012).
The second is a parameterized temperature proﬁle approach
4

The Astrophysical Journal, 775:137 (22pp), 2013 October 1 Line et al.
NEMESIS Forward Model
CHIMERA Forward Model
CH
4
CO
2
CO
H
2
O
Figure 1. Comparison of the thermal emission spectrum from our forward model (black) with the NEMESIS forward model (red). The temperature–pressure proﬁle
is shown in the inset. For this comparison we assume uniform mixing ratios of 10
4
for CH
4
,CO
2
,CO,andH
2
O. H
2
is set to 0.85 and He is set to 0.15. This planet
is assumed to be hydrogen dominated (mean molecular weight of 2.3 amu) with a radius of 1 R
J
, a gravity of 22 ms
2
, orbiting a 5700 K pure blackbody star with a
sun
.
(A color version of this ﬁgure is available in the online journal.)
similar to the approach used in Madhusudhan & Seager (2009)
described below.
The Level-by-Level temperature retrieval approach seeks
an estimate of the temperature at each of the N
lev
model
layers. This approach is advantageous in that there are no
temperature should be parameterized. If the spectral signal to
noise and resolution are high enough, there is generally enough
sensitivity to obtain information about the temperature in
individual atmospheric layers. However, there is a ﬁnite vertical
resolution given the quality of the observations. Typically,
this resolution is set by the width of the thermal emission
weighting functions and how much they overlap. Generally,
when the spectra are noisy the Level-by-Level approach ﬁts
the noise which results in unphysical structure in the retrieved
temperature proﬁle. This is analogous to ﬁtting a high-degree
polynomial to only a few points. There are ways to smooth
unphysical structure, one of them being to assume a correlation
among the atmospheric layers (Rodgers 2000; Irwin et al. 2008)
implemented through the prior covariance matrix, S
a
, with
S
a,ij
= (S
a,ii
S
a,jj
)
1/2
e
−|ln(P
i
/P
j
)|
h
. (12)
Here P
i
and P
j
are the pressures at the ith and jth levels,
respectively, and h is the correlation length that controls the
level of smoothing. The correlation length can be thought of
as the number of scale heights over which the temperatures
are correlated. For our simulations we choose h = 7asthis
provides a sufﬁcient level of detail without producing unphysical
oscillations. When using this approach, our state vector is
exactly as it is in Equation (11) with T being an N
lev
vector
of temperatures at each level. The Level-by-Level approach is
only appropriate when the information content of the spectra
is sufﬁciently high such that the addition of the N
lev
parameters is justiﬁed. For most current exoplanet spectra, this
is an invalid approach.
The second temperature proﬁle retrieval approach makes use
of a parameterization. This approach is advantageous when the
information content of a spectrum is low as the number of free
variables is greatly reduced. However, the parameterization does
force the retrieved atmospheric temperature structure to conform
only to the proﬁle shapes and physical approximations allowed
by that parameterization. For our particular parameterization,
we assume the atmosphere to be in radiative equilibrium based
upon the analytic radiative equilibrium temperature proﬁle of
Guillot (2010, and others such as Hansen 2008; Heng et al. 2012;
Robinson & Catling 2012). This is the same parameterization
5

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##### References
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Book
17 Jul 2000
TL;DR: This book treats the inverse problem of remote sounding comprehensively, and discusses a wide range of retrieval methods for extracting atmospheric parameters of interest from the quantities such as thermal emission that can be measured remotely.
Abstract: Remote sounding of the atmosphere has proved to be a fruitful method of obtaining global information about the atmospheres of the earth and planets. This book treats the inverse problem of remote sounding comprehensively, and discusses a wide range of retrieval methods for extracting atmospheric parameters of interest from the quantities such as thermal emission that can be measured remotely. Inverse theory is treated in depth from an estimation-theory point of view, but practical questions are also emphasized, for example designing observing systems to obtain the maximum quantity of information, efficient numerical implementation of algorithms for processing of large quantities of data, error analysis and approaches to the validation of the resulting retrievals, The book is targeted at both graduate students and working scientists.

3,806 citations

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[...]

• ...More complicated mappings that account for the pressure dependence of κIR can also be used (see e.g., Robinson & Catling 2012)....

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, H. Dothe3
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