error propagation from line parameters to spectra simulations illustration on high temperature...

ERROR PROPAGATION FROM LINE PARAMETERS TO SPECTRA SIMULATIONS

Illustration on High Temperature Methane.

Jean Paul Champion & Christian Wenger

Laboratoire Interdisciplinaire Carnot de BourgogneUMR 5209 CNRS - Université de Bourgogne

Dijon - France

11th HITRAN Conference Harvard-Smithsonian Center forAstrophysics Cambridge 16-18 June 2010 1 / 49

What to do when high-resolution spectroscopic information

is missing ?

Simply ignore missing data !

Are lower-resolution studies useless ?

What about error bars !


Methane at high-temperature

A typical illustration


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0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

Methane | Trot : 296 K | Tvib : 2000 K | Doppler regime

3 cold lines form "Pentad - GS" : 0.65 cm-2.atm-1

27 hot lines from "Octad - Dyad" : 0.30 cm-2.atm-1

126 hot lines from "Tetradecad - Pentad" : 0.43 cm-2.atm-1

Abs

orpt

ion

Coe

ffici

ent /

cm

-1

Wavenumber / cm-1

STDS prediction - extrapolation


Hot methane statistics (3057 - 3059 cm-1) STDS Tvib = 2000 K | Trot = 296 K


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126 hot lines from "Tetradecad - Pentad" : 0.43 cm-2.atm-1

Abs

orpt

ion

Coe

ffici

ent /

cm

-1

Wavenumber / cm-1

What to do with the 126 inacurrate hot lines ?


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0.005

0.010

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0.035




126 hot lines from "Tetradecad - Pentad" ignored

Abs

orpt

ion

Coe

ffici

ent /

cm

-1

Wavenumber / cm-1

Simply ignore inaccurate lines !


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0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0040

0.0045

0.0050

Methane | Trot : 296 K | Tvib : 2000 K | Res = 0.1 cm-1

"Tetradecad - Pentad" included : 1.38 cm-2.atm-1

"Tetradecad - Pentad" ignored : 0.95 cm-2.atm-1

Abs

orpt

ion

Coe

ffici

ent /

cm

-1

Wavenumber / cm-1

Low-resolution informationUseful ?


The scope of this presentation

How to provide completeness and confidence needed for astrophysical applications?

How to combine« high and low-resolution » information ?

How to provide rigorous estimates of the accuracy of spectrum predictions ?


Methane

present in the atmosphere of many astrophysical objects

specific energy structure due to symmetry

adapted for polyad simultaneous modeling

challenging ab initio approaches


Titan | Cassini - Huygens


Huygens DISR data


High temperature


Methane polyads and rovibrational energy levels

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

Statisticaldescription ...

Line by line in progress

Line by line close to experimental precision

Heptacontad

Pentacontakaipentad

Tetracontad

Triacontad

P9

P8

P7

P6P5 : Icosad

P4 : Tetradecad

P3 : Octad

P2 : Pentad

P1 : Dyad

Ene

rgy

/ cm

-1

J Quantum Number

P0 : Ground State


Starting point : Position and Intensity predictionswith individual standard deviations

Global / simultaneous analyses

Polyads

Effective hamiltonians

Variational / Ab initio calculations

(High resolution – highly sensitive lab. experiments)


Propagation of information within the polyads« Vibrational extrapolation schema »

Simultaneous fit of 5 sets of energies taken from higher order fits and / or « ab initio » results

58 adjustable parameters only for all polyads up to 2nd order …… with an accuracy of the order of 1 cm-1 !

Nb Jmax Wtd SD Bands Weight RMS Input Data 1 919 20 0.529 P1-P0 1.0 0.5 calc energies (order 6) 2 3492 20 0.693 P2-P0 1.0 O.6 calc energies (order 6) 3 2774 10 0.929 P3-P0 1.0 0.9 calc energies (order 4) 4 18 0 15.2 P4-P0 0.1 1.5 «observed» band centers 5 60 0 13.1 P4-P0 0.2 2.6 «ab initio» predictions cm-1 7263 Data fitted : Weighted Standard Deviation 1.617


Propagation of information within the polyads

Influence of various «ab initio» results

5800 5900 6000 6100 6200

- Simple extrapolation from P0 – P3 - Including P4 band centers predicted by Georges et al. J. Mol. Spectrosc. (1998) - Including P4 band centers predicted by Schwenke et al. Spec Chemica Acta (2001)- Including P4 band centers predicted by Wang et al. J. Chem. Phys. (2003)

Significant discrepancies

@ low resolution



from Dyad, Pentad, Octad, Tetradecad, [ ] data …… into «Dyad» effective parameters

Relative weights of Data from Parameter Value(St.Dev) Dyad Pentad Octad Tetradecad obs ab initio

11 0(0,0A1) 0100E 0100E 1534.002(72) 2.5 21.7 32.2 12.2 31.4 12 2(0,0A1) 0100E 0100E -6.79(84)x10-3 9.4 81.3 9.3 0.0 0.0 13 2(2,0E ) 0100E 0100E -2.909(76)x10-2 13.7 77.9 8.4 0.0 0.0 26 1(1,0F1) 0100E 0001F2 -9.144(42) 11.5 77.4 10.1 0.0 0.0 27 2(2,0F2) 0100E 0001F2 -5.735(87)x10-2 14.2 83.5 2.3 0.0 0.0 46 0(0,0A1) 0001F2 0001F2 1309.969(57) 2.4 23.4 35.8 8.2 30.2 47 1(1,0F1) 0001F2 0001F2 10.2468(83) 10.8 65.7 23.6 0.0 0.0 48 2(0,0A1) 0001F2 0001F2 -6.3(5.5)x10-4 8.4 82.0 9.6 0.0 0.0 49 2(2,0E ) 0001F2 0001F2 -7.50(88)x10-3 14.8 79.3 5.9 0.0 0.0 50 2(2,0F2) 0001F2 0001F2 -2.731(65)x10-2 13.8 80.1 6.1 0.0 0.0

from covariance matrix



Relative weights of Data from Parameter Value(St.Dev) Dyad Pentad Octad Tetradecad obs ab initio

207 0(0,0A1) 1100 E 1100 E -1.12(28) 0.0 0.0 8.8 38.5 52.7 217 0(0,0A1) 1100 E 0011 E -2.62(73) 0.0 0.0 5.6 41.4 53.0 235 0(0,0A1) 1001F2 1001F2 -10.94(26) 0.0 0.0 26.6 3.6 69.7 243 0(0,0A1) 1001F2 0110F2 7.04(88) 0.0 0.0 15.4 36.9 47.7 254 0(0,0A1) 1001F2 0011F2 -22.41(34) 0.0 0.0 17.5 41.9 39.6 277 0(0,0A1) 0110F1 0110F1 -18.93(15) 0.0 0.0 14.6 58.6 26.8 285 0(0,0A1) 0110F2 0110F2 -24.36(20) 0.0 0.0 15.0 55.1 29.8 293 0(0,0A1) 0110F1 0011F1 13.21(52) 0.0 0.0 20.0 48.5 31.5 307 0(0,0A1) 0110F2 0011F2 23.03(27) 0.0 0.0 11.7 29.6 57.7 347 0(0,0A1) 0011A1 0011A1 -37.20(48) 0.0 0.0 10.1 58.9 31.0 352 0(0,0A1) 0011 E 0011 E -12.03(30) 0.0 0.0 11.5 42.5 46.0 359 0(0,0A1) 0011F1 0011F1 -22.51(30) 0.0 0.0 23.2 40.2 36.6 367 0(0,0A1) 0011F2 0011F2 -11.63(27) 0.0 0.0 18.6 36.9 44.5 460 0(0,0A1) 2000A1 2000A1 -48.82(52) 0.0 0.0 0.0 50.0 50.0

from covariance matrix

from Octad, Tetradecad, [ ] data …… into «Octad» effective parameters


When high-resolution is not possible …

Band profile fitting

Degrading high-resolution experimental spectra !

Variational | ab initio approaches


« Dual Core »STDS

Frequencies IntensitiesEnergy levels

Global Effective Polyad Hamiltonian

Global Effective Polyad Dipole Moment

High resolution spectra PES / DMS

Line parameter reduction Variational calculationConvolutionHigh resolution (10-4 cm-1) Medium to low resolution / precision (1 to 10 cm-1)

Band profiles

Transition moments

Multi-resolution modeling road map


Several sets of model parameters

Orders of approximation depending on vibrational and rotational excitation

Exhaustive use of external information @ relevant resolutions



« Dual Core »STDS

Frequencies IntensitiesEnergy levels

Global Effective Polyad Hamiltonian

Global Effective Polyad Dipole Moment

High resolution spectra PES / DMS

Band profiles

Transition moments


Irreducible tensor formExplicite contact transformation

Promissing works about XY3 pyramidal molecules (Tyuterev et al Reims)


Multi-resolution investigations

How to provide rigorous estimates of the precision of spectrum predictions ?

Propagation of line parameter uncertaintiesinto absorption coefficient uncertainty

Statistical simulations + Analytical modeling


Absorption coefficient

@ resolution


Methane R(3) 3 region – High temperature

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0.1

0.2

0.3

: 0.200S : 50%

: 0.010S : 10%

: 0.001S : 3%

Trot : 296 K | Tvib : 2000 KIntensity threshold : 0.025

Pentad - GS linesOctad - Dyad lines

Tetradecad - Pentad lines


Absorption coefficient : Accuracy @ various resolutions ?

Resolution = 0.003 cm-1



?

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Typical Results

90 % confidence bands @

3 typical resolutions

0.003 ; 0.020 ; 0.100 cm-1


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Wavenumber / cm-1

Ab

sorp

tion

Co

effi

cie

nt /

cm

-1

Absorption coefficient90 % confidence band

@ res = 0.100 cm-1


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Wavenumber / cm-1

Abs

orp

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Coe

ffici

ent /

cm

-1


@ res = 0.020 cm-1


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bsor

ptio

n C

oeffi

cien

t / c

m-1

Wavenumber / cm-1


@ res = 0.003 cm-1


Procedure

Relationship between absorption coefficient and line parameters highly non-linear !

Numerical statistical simulationsPseudo random generator


Procedure

Normal random variables are associated to line parameters (and numerically simulated)

Statistical properties of the corresponding random function associated to the absorption coefficient are

deduced from the (numerical) analysis of histograms built at each spectral elements of the window


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0.002

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Wing

max prob

mean

min

max

Side Center

Wavenumber / cm-1

Abs

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ion

Coe

ffici

ent /

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Zoom @ 3058.204 cm-1

Statistical properties


Procedure

Numerical determination of the confidence intervals


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Wing

max prob

mean

min

max

Side Center

Wavenumber / cm-1

Abs

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Coe

ffici

ent /

cm

-1

Histogram


Confidence band@

different levels


Predicted signal in black 90% Confidence Band : Sup in red & Inf in blue


« Warnings » for astrophysicists and others …

Predicted signal inside confidence bandand baseline outside.

Idem but …baseline inside

Predicted signal outside confidence band


« Warnings » for astrophysicists and others …

Prediction exploitable without restriction

Exploitable with caution

Exploitable with circumspection !


Transmission Confidence band

Simulated Exp. Cond.Resolution 0.050 cm-1

Pressure 1 TorrPath length 1 km

Tvib 2000 K ; Trot 296 K


Summary

Multi-resolution investigations relevant for astrophysical applications

Completeness All sources of information @ relevant resolutions

line by line | band profile | ab initio

ReliabilityConfidence band @ relevant resolutions


Prospects

Accounting for correlations among predicted line parameters

Line shape uncertainties …

Computer implementation : « expert system »

User input : spectral range | pressure | temperature Output : confidence band @ various resolutions


Computer implementations

Open access

…

In project

http://hal.archives-ouvertes.fr/JP-CHAMPION


Connected work

about multi-resolution investigations

Open access…

Partition function

already available

http://hal.archives-ouvertes.fr/JP-CHAMPION


Acknowledgments

Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB) Dijon (France)

Christian Wenger, Vincent Boudon … Methane

Jet Propulsion Laboratory (JPL) Pasadena (USA)Linda Brown … FTIR spectra

Institut of Atmospheric Optics (IAO) Tomsk (Russia)Andrei Nikitin, Vladimir Tyuterev … MIRS … octad …

Institut de Physique de Rennes (France)Robert Georges … High temperature emission

Niels Bohr Institut Copenhagen (Denmark)Uffe Jørgensen … Brown dwarfs

LEFE-CHAT National Program CNRS (France) ANR : CH4@Titan (France)

GDRI : SAMIA Tomsk (Russia), Hefei (China), CNRS (France)11th HITRAN Conference Harvard-Smithsonian Center forAstrophysics Cambridge 16-18 June 2010 48 / 49

Thank You for attention


error propagation from line parameters to spectra simulations illustration on high temperature...

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