An Optimal Estimation Spectral Retrieval Approach for Exoplanet
AtmospheresM.R. Line1, X. Zhang1, V. Natraj2, G.
Vasisht2, P. Chen2, Y.L. Yung1
1California Institute of Technology2Jet Propulsion Laboratory, California Institute of Technology
EPSC-DPS 2011, Nantes France
Line et al. in prep
Goals
• Find a robust technique for retrieving atmospheric compositions and temperatures from exoplanet spectra
• Determine the number of allowable atmospheric parameters that can be retrieved from a given spectral dataset
Method: Optimal Estimation (Rodgers 2000)
€
ds = tr(A)
€
H =1
2ln ˆ S −1Sa
Degrees of Freedom
Information Content
Bayes Theorem:
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P(x | y)∝ P(y | x)P(x)
y - measurement vectorx - state vector€
(x − ˆ x )T ˆ S −1(x − ˆ x ) = (y − Kx)TSe−1(y − Kx) + (x − xa )
TSa−1(x − xa )
Cost Function:
F(x) = Kx - forward modelK -Jacobian matrix—Se- data error matrix
€
K ij =∂Fi(x)
∂x j
xa- prior state vectorSa - prior uncertainty matrix
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ˆ x = xa + ˆ S KTSe−1(y − Kx)
€
ˆ S = (KTSe−1K + Sa
−1)−1Retrieval Uncertianty
Retrieved State
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A =∂ˆ x
∂x= ˆ S KTSe
−1K Averaging Kernel
Forward Model F(x)
• Parmentier & Guillot 2011 Analytical TPκv1,κv2, α, κIR ,Tirr , Tint
• Constant with Altitude Mixing RatiosH2O, CH4, CO, CO2, H2, He
• Reference Forward Model (http://www.atm.ox.ac.uk/RFM/)
-HITEMP Database for H2O, CO, CO2
-HITRAN Database for CH4
-H2-H2, H2-He Opacities (from A. Borysow)
HD189733b Jacobian
HD189733b Retrieval
DOF~ 5
Χ2=0.86
A priori StateRetrieved StateRetrieved State (Hi Res)
Degrees of Freedom and Information Content
€
ds ~(SN)2
(SN)2 +F 2
K 2σ a2
€
H ~ ln(1+σ a
2
F 2K 2(SN)2)
FINESSE
NICMOS
Conclusions
• Rodgers’ optimal estimation technique can provide a robust retrieval of exoplanetary atmospheric properties
• Quality of the retrieval of each parameter can be determined
• Knowledge of the Jacobian, Information content, and degrees of freedom can aid future instrument design
Synthetic Data Test
Model AtmosphereTirr=1220 K fH2=0.86Tint=100 K fHe=0.14κv1=4×10-3 cm2g-1 fH2O=5×10-4
κv2=4×10-3 cm2g-1 fCH4=1×10-6
α=0.5 fCO=3×10-4
κIR= 1×10-2 cm2g-1 fCO2=1×10-7
“Instrumental” SpecsR~40 at 2μm (Δλ=0.05 μm)S/N~10
Synthetic Data Jacobian
Synthetic Data Retrieval
Χ2=0.01
DOF= 6
Method: Optimal Estimation(Rodgers 2000)
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J(x) = (x − ˆ x )T ˆ S −1(x − ˆ x ) = (y − Kx)T Se−1(y − Kx) + (x − xa )
T Sa−1(x − xa )
Minimize Cost Function from Bayes:
Likelihood that data exists given some model
Prior Information
y - measurement vectorx - true state vector - retrieved state vectorxa- prior state vectorF(x)=Kx-forward modelK -Jacobian matrix—Se- data error matrixSa - prior uncertainty matrixŜ-retrieval uncertainty matrix
€
K ij =∂Fi(x)
∂x j
€
P(x | y)∝ P(y | x)P(x)
€
ˆ x = xa + ˆ S KTSe−1(y − Kx)
€
ˆ S = (KTSe−1K + Sa
−1)−1
€
ˆ x
€
A =∂ˆ x
∂x= ˆ S KTSe
−1K
€
ds = tr(A)
€
H =1
2ln ˆ S −1Sa
Degrees of Freedom
Information Content