A systematic retrieval analysis of secondary eclipse spectra. i. a comparison of atmospheric retrieval techniques
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/O is biased toward values near solar and near one simply due to the assumption of uninformative priors.
Summary (4 min read)
- 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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...ExoMol data have been incorporated into many radiative transfer and retrieval codes including Tau-REx [13–15], Phoenix , NEMESIS , CHIMERA , HELIOS  and HELIOS-r2 , ATMO [21–23], ARCiS , ARTES , HyDRA , GENESIS , petitRADTRANS , PLATON , VSTAR [30, 31], BART  and Pyrat Bay ....
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...55 61 peratures and compositions (see e.g., Charbonneau et al. 2005; Tinetti et al. 2007; 2010; Grillmair et al. 2007; 2008; Swain et al. 2009a; 2009b; Madahusudhan & Seager 2009; Stevenson et al. 2011; Madhusudhan et al. 2011; Lee et al. 2012; Line et al. 2012)....