an optimal estimation spectral retrieval approach for exoplanet atmospheres

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:

€

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

€

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

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ds ~(SN)2

(SN)2 +F 2

K 2σ a2

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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)

€

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