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

€

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

€

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)

€

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