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
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Limb scatteringOccultation
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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
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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)
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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
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Retrieval choices
1. Forward model
2. Inverse modelling
3. Estimation
instrumenttarget data
Forward model
Inverse model
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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
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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)
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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
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!
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
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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.
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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
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Prior information
• Discrete grid: Assume that profile has only afinite number of free parameters
• Smoothness: Tikhonov constraint
• A priori profile
• Positivity constraint or similar
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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
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Iref
occI
OCCULTATION
calibration free principle
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GOMOS: Measured Sirius reference spectrum
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GOMOS: Measured Sirius transmitted spectrum
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GOMOS: Calculated Sirius transmissions
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!
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
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Weak scintillations:intensitymaxima and minima
Densityfluctuation
Strong scintillations: multiple stars
Stellar occultations: dilution & scintillations
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Chromatic effects
Different colors different refraction angles
Same altitude
Same det. times
Different det.times
differentaltitudes
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!
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).
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GOMOS: Horizontal transmissions 5-100 km
O3 inmesosphere
O3 in stratosphere
NO2 instratosphere
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!
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.
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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))
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Aspects of spectral inversion in UV-VIS
• Linearization• Non-linear approach
• Spectrally global• Spectral windows
• Absolute cross sections• DOAS
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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
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Transmission components at 27 km
ozone
NO2
NO3
Rayleigh
aerosol
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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
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Tikhonov regularization
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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
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GOMOS level 2
Crosssections
Spectralinversion
Local densitiesO3, NO2, NO3aerosols, AirT, H2O, O2
Verticalinversion
Line densities
Level 1transmissions
Dilution & scintillation corrections
Level 1photometer
data
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LIMB SCATTERING RETRIEVAL
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OSIRIS radiances
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OSIRIS radiance ratios
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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
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Single and multiple scattering
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Difficulties in limb radiative transfer
• MS time consuming• Albedo• Clouds• Aerosols• Polarization• Raman scattering
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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
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Multiple scattering
!
M =Itotal
Iss
tabulated from a fullradiative transfer code likeFMI’s Monte Carlo modelSiro.
!
Imod (", z,#) = Imodss(", z,#) $M("apr, z,#)
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GOMOS/OSIRISlimb processing scheme
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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
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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
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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
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Markov chain Monte Carlo
<xi>= Σ zti
Estimators from MCMC
1N t
N
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Marginal posterior distributions at 30 km for different gases
Bright star
Weak star
MCMC examples (GOMOS)
by J. Tamminen, FMI
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MCMC example: Model selection: aerosols
by M. Laine, FMI
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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