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ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 1

UV/VIS Limb Retrieval

Erkki Kyrölä Finnish Meteorological Institute

1. Data selection2. Forward and inverse possibilities3. Occultation: GOMOS inversion

4. Limb scattering: OSIRIS inversion5. Summary

6. Dessert: MCMC

Limb scatteringOccultation

Many locationsAll spectral data

One locationAll spectral data Spectral

normalised data

Data selection for retrieval

Tomography

Spectral calibrated

Radiancecomparisons

λ -windows DOAS

inversionAll zOne step inversionAll λ

One z Spectrally global inversionAll λ

Data normalisation

UV/VIS instruments are difficult to calibrate. Add ageing and stray light. Big trouble.

T(λ)=Iocc(λ)

Iref(λ)

Occultation

T(",z) = exp(# $ j (",T(z(s))% j&' (z(s))ds)

Observations

Modelling

Iocc(λ)Iref(λ)

Limb scattering

Robs(z,") =Iobs(z,")

Iref (zref ,")

Rmod(z,") =

Imod(#, z,")

ref(#ref , zref ,")

A priori information + radiancedifficult to calculate

Additional bonuses: Ratio is insensitive to albedo

Retrieval choices

1. Forward model

2. Inverse modelling

3. Estimation

instrumenttarget data

Forward model

Inverse model

Hierachy offorward models

“True” nature

z=all other pertinent variables

G(x,z) + "

Gknown

(x,zknown

= zfix) + "

The best forward model available.Uninteresting variables fixed.

Gapp(x,z

known= z

fix) + "

Model used in signalsimulation

Ginv(x,z

known= z

fix) + "

Model used for inversion

Single scattering. Multiplescattering only by LUT.

Turbulence seen asscintillations. Removed asnoise.

Inversion

Multiple scattering in 3-D.Clouds as an elevated surfacealbedo. Polarisation.

Light propagates throughphase screen, whererefraction takes place.

Simulation

Light propagates in 3-Datmosphere with absorptionand multiple scattering.Polarisation, simple broken,clouds and albedos, emissions.

Light propagates through2-D layered but fluctuatingatmosphere. Refraction,absorption, scattering,emissions.

Light propagates in 3-Datmosphere with absorptionsand multiple scatterings.Polarisation, clouds, groundsurface, emissions.

Light propagates throughturbulent 3-D atmosphere.Refraction, absorption,scattering, emissions.

NaturePhoton vsclassical?

OSIRISGOMOSForwardmodel

Forward modelling levels (draft only)

Contaminationmust becontrolled

Spectrallysmoothconstituentsneglected

Need to iterateto correct theapproximation

Noise alsotransformed

SmoothnessActive profile(optimal est.)Initial values

NoneA prioriinformation

DOASAbsoluteCross sections

Separate spatialand spectral

No factorisationOne-stepinversion

Modelfactorisation tospatial xspectral

LinearisedOriginalnonlinear

Modeltransformation

Inverse modelling choices

P(x | y)P(y) = P(y | x)P(x)

P(x | y) =P(y | x)P(x)

P(y | x)P(x)

P(y | x)P(x)dx"

P(x|y) = Conditional probability distribution for modelparameters x given data y

P(x) = A priori probability for model parametersP(y|x) = Conditional pdf for data y when x given. Also

called as likelihood.P(y) = The normalization. It can usually be ignored.

Bayesian method

A systematic basisfor inversion theory is given by

the Bayesian approach

• Model parameters are random variables• Probability distribution of model parameters is retrieved• Prior information is needed. This has led to many

controversies about the Bayesian approach.

Wiki: Thomas Bayes was born in London. In 1719 he enrolled at the University of Edinburgh to study logic and theology: Because he was a Nonconformist, Oxford and Cambridge were closed to him.

MCMCmethod

Maximumlikelihood

Gaussian errors

max of

max of MAP

Linearmodel

LMmethod

Closedsolution

Whole distributionPoint estimation

P(x | y) = P(y | x)P(x)

Estimation choices

Prior information

• Discrete grid: Assume that profile has only afinite number of free parameters

• Smoothness: Tikhonov constraint

• A priori profile

• Positivity constraint or similar

Literature and a referenceTarantola: Inverse problem theory, Methods for data fitting and model parameter estimation, Elsevier, 1987

Rodgers: Inverse Methods for Atmospheric Sounding:Theory and Practice, World Scientific, 2000

Menke: Geophysical data analysis: discrete inverse theory,Academic Press, 1984

OCCULTATION

calibration free principle

GOMOS: Measured Sirius reference spectrum

GOMOS: Measured Sirius transmitted spectrum

GOMOS: Calculated Sirius transmissions

T(",z) = exp(# $ j (",T(z(s))% j&' (z(s))ds) Beer-Lambert law

Occultation inversion is simple because...

But ...

Occultation inversion

" = cross section

" = number density

= temperature

Weak scintillations:intensitymaxima and minima

Densityfluctuation

Strong scintillations: multiple stars

Stellar occultations: dilution & scintillations

Chromatic effects

Different colors different refraction angles

Same altitude

Same det. times

Different det.times

differentaltitudes

T(z,") = Tref Text

We can, however, write

Transmission from refractive effects can be estimated fromray tracing calculations (dilution, chromatic effects). Inaddition, we need photometer measurements to estimate therandom part (scintillations).

GOMOS: Horizontal transmissions 5-100 km

O3 inmesosphere

O3 in stratosphere

NO2 instratosphere

T(",z) = exp(# $ j (")N%j(z))

N j (z) = " j# (z(s))ds

Occultation inversion using Beer-Lambert: Two step

Spectral inversion

Vertical inversion

This separation is not true if cross sections depend on temperature. In these cases we can use iteration over spectral and vertical inversion or one-step inversion.

C = covariance matrixT = transmission vector (all wavelengths)N = column density vector (different constituents)

We aim to minimize

Solution by Levenberg-Marquardt algorithm

Spectral inversion

S(N) = (Tobs"Tmod (N))

obs"Tmod (N))

Aspects of spectral inversion in UV-VIS

• Linearization• Non-linear approach

• Spectrally global• Spectral windows

• Absolute cross sections• DOAS

6000500040003000Wavelength (Å)

Cross sections

Transmission components at 27 km

Rayleigh

aerosol

GOMOS vertical inversion

$ $ $ $ $ $

' ' ' ' ' '

Discretize

N(z) = "# (z(s))ds

N = K"

where the kernel matrix is

Onion peel solution

d11d22 d21

Tikhonov regularization

GOMOS level 1Raw data

Geolocation & ray tracing

Instrumentalcorrections

Photometerdata

Transmissiondata

Limbdata

ECMWFprediction/analysisMSIS90

Calibrationdatabase

• Data extraction• Datation• Geolocation (ECMWF+MSIS90)• Wavelength assignment• Spectrometer samples correction• Photometer data processing• Central band background estimation• Star spectrum computation• Transmission computation• Products generation

GOMOS level 2

Crosssections

Spectralinversion

Local densitiesO3, NO2, NO3aerosols, AirT, H2O, O2

Verticalinversion

Line densities

Level 1transmissions

Dilution & scintillation corrections

Level 1photometer

LIMB SCATTERING RETRIEVAL

OSIRIS radiances

OSIRIS radiance ratios

Scattered limb radiances

Total radiance= single scattering + multiple scattering

I = Isun

Tsun" (s)(#

R)Tdet (s)ds+ I

Single and multiple scattering

Difficulties in limb radiative transfer

• MS time consuming• Albedo• Clouds• Aerosols• Polarization• Raman scattering

Modified onion peeling method

S(")= Rmod # Robs[ ]T

$C#1 $ Rmod # Robs[ ]

Measured ratiospectra:

Modelled ratio

spectra:

Robs(z,") =Iobs(z,")

Iref (zref ,")

Rmod(z,") =

Imod(#, z,")

ref(#ref , zref ,")

Minimize

with onion peel type inversion or with one-step inversion

Multiple scattering

M =Itotal

tabulated from a fullradiative transfer code likeFMI’s Monte Carlo modelSiro.

Imod (", z,#) = Imodss(", z,#) $M("apr, z,#)

GOMOS/OSIRISlimb processing scheme

Summary

DOAS with spectral windowsFlittner for limb scattering; 3 wavelengths

Occultation and limb scattering retrievals can be approached with similar methods. They are based on:-non-linear approach-using relative quantities, not directly measured quantities-original cross sections-all wavelengths

Other methods

Difficulties : Aerosol modelling, scintillations, multiple scattering

ReferencesThis presentation has followed:Kyrölä, E., E. Sihvola, M. Tikka, Y. Kotivuori T. Tuomi, and H. Haario, Inverse Theory forOccultation Measurements 1. Spectral Inversion, J. Geophys. Res., 98, 7367-7381, 1993.

Oikarinen, L., E. Sihvola, and E. Kyrölä, Multiple scattering radiance in limb-viewinggeometry, J. Geophys. Res., 104, 31261-31274, 2000.

Auvinen, H., L. Oikarinen and E. Kyrölä, Inversion algorithms for limb measurements, J.Geophys. Res., 107, D13, 2001JD000407, ACH 7-1: 7-7, 2002

Tukiainen, S., S. Hassinen, A. Seppälä, E. Kyrölä, J. Tamminen, P. Verronen,H. Auvinen, C. Haley, and N. Lloyd, Description and validation of a limb scatter inversionmethod for Odin/OSIRIS, J. Geophys. Res 113, D04308, 2008.

Haley, C., S. M. Brohede, C. E. Sioris, E. Griffioen,D. P. Murtagh, I. C. McDade,1 P.Eriksson, E. J. Llewellyn, A.Bazureau, and F. Goutail, Retrieval of stratospheric O3 andNO2 profiles from Odin Optical Spectrograph and Infrared ImagerSystem (OSIRIS) limb-scattered sunlight measurements ,J. Geophys. Res. 109, D16303,2004.

Numerical examples: FMI’s GomLab and LimbLab

Ultimate estimators: Markov chain Monte Carlo

Blind Mr. Levenberg: That’s it!

Mr. Markov: Hold your horses

Twin peaks drama

Top guy: Yes! Mean guy: <Sorry but...>

Flatness dullness

Markov chain Monte Carlo

<xi>= Σ zti

Estimators from MCMC

Marginal posterior distributions at 30 km for different gases

Bright star

Weak star

MCMC examples (GOMOS)

by J. Tamminen, FMI

MCMC example: Model selection: aerosols

by M. Laine, FMI

Tamminen and Kyrölä, JGR, 106, 14377, 2001

Tamminen: Ph.D. thesis, FMI contributions 47, 2004

Laine and Tamminen: Aerosol model selection, ACPD 2008

MCMC references

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