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

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 2

Limb scatteringOccultation

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

Many locationsAll spectral data

One locationAll spectral data Spectral

normalised data

Data selection for retrieval

Tomography

Spectral calibrated

data

Radiancecomparisons

λ -windows DOAS

inversionAll zOne step inversionAll λ

One z Spectrally global inversionAll λ

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 4

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)

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 5

Limb scattering

!

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

Iref (zref ,")

!

Rmod(z,") =

Imod(#, z,")

Imod

ref(#ref , zref ,")

A priori information + radiancedifficult to calculate

Additional bonuses: Ratio is insensitive to albedo

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 6

Retrieval choices

1. Forward model

2. Inverse modelling

3. Estimation

instrumenttarget data

Forward model

Inverse model

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 7

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

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 8

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.

Best

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)

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 9

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 10

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

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 11

!

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

!

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

P(y)=

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

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 12

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.

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 13

MCMCmethod

Maximumlikelihood

LSQ

Gaussian errors

max of

max of MAP

Linearmodel

LMmethod

Closedsolution

Whole distributionPoint estimation

!

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

Estimation choices

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 14

Prior information

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

• Smoothness: Tikhonov constraint

• A priori profile

• Positivity constraint or similar

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 15

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

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 16

Iref

occI

OCCULTATION

calibration free principle

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 17

GOMOS: Measured Sirius reference spectrum

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 18

GOMOS: Measured Sirius transmitted spectrum

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 19

GOMOS: Calculated Sirius transmissions

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 20

!

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

Occultation inversion is simple because...

But ...

Occultation inversion

s

!

" = cross section

z(s)

!

" = number density

= temperature

!

T

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 21

Weak scintillations:intensitymaxima and minima

Densityfluctuation

Strong scintillations: multiple stars

Stellar occultations: dilution & scintillations

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 22

Chromatic effects

Different colors different refraction angles

Same altitude

Same det. times

Different det.times

differentaltitudes

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 23

!

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).

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 24

GOMOS: Horizontal transmissions 5-100 km

O3 inmesosphere

O3 in stratosphere

NO2 instratosphere

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 25

!

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.

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 26

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))

TC

"1((T

obs"Tmod (N))

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 27

Aspects of spectral inversion in UV-VIS

• Linearization• Non-linear approach

• Spectrally global• Spectral windows

• Absolute cross sections• DOAS

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 28

10-23

10-22

10-21

10-20

10-19

10-18

10-17

Cro

ss s

ectio

n (c

m 2

)

6000500040003000Wavelength (Å)

O3

NO2

O3

NO3

OClO

BrO

Cross sections

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 29

Transmission components at 27 km

ozone

NO2

NO3

Rayleigh

aerosol

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 30

GOMOS vertical inversion

!

K =

d11

2d21

d22

2d31

2d32

d33

"

#

$ $ $ $ $ $

%

&

' ' ' ' ' '

Discretize

!

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

!

N = K"

where the kernel matrix is

Onion peel solution

d11d22 d21

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 31

Tikhonov regularization

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 32

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

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 33

GOMOS level 2

Crosssections

Spectralinversion

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

Verticalinversion

Line densities

Level 1transmissions

Dilution & scintillation corrections

Level 1photometer

data

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 34

LIMB SCATTERING RETRIEVAL

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 35

OSIRIS radiances

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 36

OSIRIS radiance ratios

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 37

Scattered limb radiances

Total radiance= single scattering + multiple scattering

!

I = Isun

Tsun" (s)(#

a(s)$

a(%)P

a+ #

R(s)$

R(%)P

R)Tdet (s)ds+ I

ms

s

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 38

Single and multiple scattering

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 39

Difficulties in limb radiative transfer

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

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 40

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,")

Imod

ref(#ref , zref ,")

Minimize

with onion peel type inversion or with one-step inversion

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 41

Multiple scattering

!

M =Itotal

Iss

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

!

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

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 42

GOMOS/OSIRISlimb processing scheme

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 43

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

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 44

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

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 45

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

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 46

Markov chain Monte Carlo

<xi>= Σ zti

Estimators from MCMC

1N t

N

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 47

Marginal posterior distributions at 30 km for different gases

Bright star

Weak star

MCMC examples (GOMOS)

by J. Tamminen, FMI

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 48

MCMC example: Model selection: aerosols

by M. Laine, FMI

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 49

ESA AATC Oxford 2008 Day 3 Limb retrieval - Erkki Kyrölä 50

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