Michael Höpfner

MID-IR Nadir/Limb Retrieval

Michael Höpfner Karlsruhe Institute of Technology

IMK-ASF

Michael Höpfner

Thank’s to: • André Butz • Frank Hase • Thomas v. Clarmann

Michael Höpfner

Inversion of remote sensing observations

)ˆ(ˆˆ xFy =Signal (e.g. nadir spectrum)

Radiative transfer

)ˆ(ˆˆ 1 yFx −= ??? Inverse:

∫ ′′+=−′

−−

−↑↑

BodenBoden

deJeLL Boden

,, 1),,(),,(),,( ,

λ

λ

λλλλ τ

τλ

µττ

λλµ

ττ

λλλλ τµ

ϕµτϕµτϕµτ

0

20

40

60

80

100

150 200 250 300

polar summer polar winter

Temperature [K]

Altitu

de [k

m]

Forward:

Michael Höpfner

)ˆ(ˆˆ xFy = )ˆ(ˆˆ 1 yFx −=

dLλ / ds =

+ βs,λ(4π)-1 P(Ω,Ω sol) Fsol

- βa,λ Lλ - βs,λ Lλ

+ βa,λ Bλ + βs,λ(4π)-1 ∫ P(Ω‘,Ω) Ldiff(Ω‘) dΩ‘

Michael Höpfner

Emission and Absorption - Only emission and absorption of radiation, no scattering → most important for radiative transfer in MW and IR - Local thermodynamic equilibrium → Planckfunction as source function → Kirchhoff’s law: emission = absorption

dssBsdssLsdLdL

sLdssLdL

aa

emiabs

)()()()(

)()(

,,

,,

λλλλ

λλ

λλλ

ββ +−=

+=−+=

( ))()()(, sLsBsds

dLa λλλ

λ β −=Schwarzschild-equation:

Naa λλ σβ ,, =Absorption coefficient:

Absorption cross-section: λσ ,a

Number density: N)(sBλPlanck function:

Michael Höpfner

.

s1 s2 s

),( 2ssλτ

),( 21 ssλτ

Lλ(s1) Lλ(s2)

∫ −− +=2

1

221 ),(,

),(12 )()()()(

s

s

ssa

ss dsesBsesLsL λλ τλλ

τλλ β

Solution of Schwarzschild’s equation

∫=2

1

)(),( ,21

s

sa dssss λλ βτOptical depth:

Michael Höpfner

Emission of a layer with constant T ∫ −=),(

02

),(2

21

2 ),()()(ss

ss ssdesBsLλ

λ

τ

λτ

λλ τ

λλ BconstsB ==)(

layer theof Emissivity

layer theofon Transmissi

21),(

),(

02

),(2 )),(1()1(),()( 21

21

2 sstBeBssdeBsL ssss

ssλλ

τλ

τ

λτ

λλλ

λ

λ τ −=−== −−∫

For three layers:

)1()()()1()1()1(

3,3,3,2,3,2,3,2,1,3,2,1,

3,3,3,2,2,3,2,1,1,

λλλλλλλλλλλλ

λλλλλλλλλλ

tBtttBtttttBtBttBtttBL

−+−+−=

−+−+−=

1 2 3

Michael Höpfner

Thermal contrast in nadir sounding: the lowest layer

)1( 1,1,1,, λλλλλ tBtBL surface −+=

1,, λλ BB surface = 1,λλ BL =→ → No information about transmission of lowest layer

1,, λλ BB surface ≠ →

1,,

1,1,

λλ

λλλ BB

BLt

surface −−

=

1,λt1,λB

surfaceB ,λ

Michael Höpfner

Schwarzschild’s equation as function of transmission and weighting function

∫+=2

1

)()(),()()( 2112

s

s

dssWsBsstsLsL λλλλλ

dsssdtsW ),()( 2λ

λ =Weighting function:

∫ −− +=2

1

221 ),(,

),(12 )()()()(

s

s

ssa

ss dsesBsesLsL λλ τλλ

τλλ β

[ ]),(exp),( 2121 sssst λλ τ−=Transmission:

Michael Höpfner

dzzdt )(λ

)(ztλ

Michael Höpfner

Smith et al., 2009

Nadir sounding weighting functions

Michael Höpfner Petty216

Looking up or down?

Michael Höpfner Petty216

Looking up: where?

Michael Höpfner 15

‘IASI’ and ‘monochromatic’ spectrum

Michael Höpfner 16

Michael Höpfner 17

Michael Höpfner 18

Michael Höpfner

“Geometric” cross-section

Naa λλ σβ ,, =Absorption coefficient:

Absorption cross-section: λσ ,a

Number density: N

Michael Höpfner

“Radiative” effective cross-section at stronly absorbing wavelength

Michael Höpfner

“Radiative” effective cross-section at weakly absorbing wavelength

Michael Höpfner

P. W. Atkins, Physikalische Chemie, 1996 http://www.polymere.uni-koeln.de/11572.html

• Quantum mechanics: a bound microscopic system can only be in distinct rotational/vibrational/electronic states.

• Transfer from one to the other state can occur through emission/absorption of electromagnetic radiation (photon)

• Microwave: rotation of a molecule with static dipole moment

• IR: vibration of a molecule; changing dipole moment • UV-VIS: electronic transitions

∆Ε = h f

Michael Höpfner

Beispiel: Schwingung und Rotation des H2O-Moleküls 2.74 µm 6.27 µm 2.66 µm

Michael Höpfner

3,7 3,8 3,9 4,0 4,1

0,2

0,4

0,6

0,8

1,0

Tran

smis

sion

Wellenlänge [µm]

HBr 2 mbar2 cm Zelle21 °C

Ro-vibrational transition in the mid-IR

Michael Höpfner 25

Höpfner / Friedl-Vallon - Messverfahren: FTIR-Spektroskopie

780.2 780.4 780.6 780.8

6.0

8.0

10.0

12.0

solar

CO2

O3

ClNO3Ra

dianz

[W /

m2 sr

cm-1]

Wellenzahl [cm-1]

The finite width of spectral lines Doppler-broadening: • Thermal movement of molecules

along the line-of-sight • Gaussian shape • ~0.001 cm-1 @ 1000 cm-1 • Proportional to the frequency

Pressure-broadening: • Collisions with other molecules

disturb their oscillation • Transition frequency becomes

‘blurred’ • Lorentz shape • ~0.05 cm-1 @ 1000 hPa • Proportional to the collision rate

(pressure)

-5 -4 -3 -2 -1 0 1 2 3 4 5

0,0

0,2

0,4

0,6

0,8

1,0

distance from line center

Gauss Lorentz Voigt

Michael Höpfner

http://www.cfa.harvard.edu/hitran/

Michael Höpfner

ε

+= )(xFy

Discretisation and measurement error

Measurement vector (spectral channels)

Radiative transfer model

Vector with altitude profiles (n points)

Error (spectral noise)

)ˆ(ˆˆ xFy =

Michael Höpfner

Linearisation ε

+= )(xFy

εε

+−+=+−∂

∂+= )()()()()( 0000

0

xxxFxxxxFxFy

x

K

First guess Jacobi-Matrix (~weighting functions)

∂∂

∂∂

∂∂

∂∂

=

n

mm

n

xF

xF

xF

xF

...

...

...

...

...

1

1

1

1

K

Michael Höpfner

Case 1: m = n = 1

))(()(

11 i

x

ii xfy

dxxdf

xx

i

−+=+

ε

+−+= )()( 00 xxxFy K

ε+−+= )()()( 000

xxdx

xdfxfyx

Newton-Iteration:

Error estimation: 22

2 1yx

dxdf

σσ

=

Michael Höpfner

+−

=

2

10

2

1

2

1 )()(

)(

0

0

εε

xx

dxxdF

dxxdF

yy

x

x

K

K is vector and cannot be inverted

Way 1:

2. Calculate error for each xi as in example 1 21, xx σσ

3. Calculate the result x as weighted mean 22

22

21

21

21

11

xx

xx

xx

x

σσ

σσ

+

+

=

4. And the error of x as: 1

222

21

11−

+=

xxx σσ

σ

1. Calculate x separately for each y as in example 1

Case 2: m =2 n = 1

Michael Höpfner

Way2:

Least-squares inversion

Minimise the weighted quadratic differences:

[ ] [ ] ( )[ ] ( )[ ])()()()()()( 001

001 xxxFySxxxFyxFySxFy y

Ty

T−+−−+−=−− −− KK

Covariance matrix of observations:

= 2

2

2

1

00

y

yy σ

σS

Result of minimisation ( ) ( )

−

′+−

′′

+′

+= )()(

)()(

)()(1

022202

011201

2

202

2

201

0

21

21

xFyxF

xFyxF

xFxFxx

yy

yy

σσσσ

Same result as in way 1

Case 2: m =2 n = 1

Michael Höpfner

For non-linear problems this can be written as iteration:

( ) ( ))(1111 iy

Ty

Tii xFyxx

−+= −−−

+ SKKSK

( ) 112 −−= KSK yT

xσ

Variance

( ) ( )

−

′+−

′′

+′

+= )()(

)()(

)()(1

022202

011201

2

202

2

201

0

21

21

xFyxF

xFyxF

xFxFxx

yy

yy

σσσσ

Case 2: m =2 n = 1

Michael Höpfner

( ) ( ))(1111 iy

Ty

Tii xFyxx

−+= −−−+ SKKSK

( ) 11 −−= KSKS yT

x

General case for m,n

Michael Höpfner

Regularization

• More unknowns than observations n > m • m >= n but linear dependent

Introduction of constraints (Regularization)

Michael Höpfner

Tikhonov-Phillips regularization The altitude profile should be “smooth”

[ ] [ ])()( 1 xFySxFy yT

−− −Instead of:

[ ] [ ] xxxFySxFy TTy

T LLγ+−− − )()( 1

−

−−

=

110.......................0110...0011

L

Solution:

( ) ( ) ( )[ ]iaT

iyTT

yT

ii xxxFyxx

−+−++= −−−+ LLSKLLKSK γγ )(1111

Minimize:

Michael Höpfner

Linear statistical regularization “optimal estimation”

Use statistical knowledge about atmospheric parameter x

[ ] [ ])()( 1 xFySxFy yT

−− −

Solution:

[ ] [ ] )()()()( 11aa

Tay

TxxxxxFySxFy

−−+−− −− S

a-priori covariance matrix

( ) ( ) ( )[ ]iaaiyT

ayT

ii xxxFyxx

−+−++= −−−−−+

111111 )( SSKSKSK

Michael Höpfner

Tikhonov-Phillips Optimal estimation Column k: answer of the retrieval to a delta-function at altitude k . Row j: contribution of diffenent altitudes to the results in altitude j Smoothing described by

( ) KSKLLKSKA 111 −−− += yTT

yT γ

( ) KSKSKSKA 1111 −−−− += yT

ayT

)( 00 xxxx trueinv

−+= A

Averaging kernel matrix

MIPAS ClONO2 Tikhonov Regul.

Michael Höpfner

782.6 782.7 782.8

8.0

10.0

12.0 O3O3

Radi

anz

[W /

m2 s

r cm

-1]

Wellenzahl [cm-1]

0

10

20

30

40

50

60

0 2 4 6

O3-Startprofil

Invertiertes O3-Profil

Mischungsverhältnis [ppmv]Hö

he [k

m]

Ozone inversion from ground-based FTIR

jiij xyK ∂∂= /

Iterative inversion of linearized problem: Minimize:

xy ∆=∆ K

[ ] [ ] xxxFySxFy TTy

T LLγ+−− − )()( 1

)(xFy =

( ) ( ) ( )[ ]iaT

iyTT

yT

ii xxxFyxx

−+−++= −−−+ LLSKLLKSK γγ )(1111

Michael Höpfner

Averaging kernel and vertical resolution

10 20 30 40 50

10

20

30

40

50

Höhe [km]

Höhe

[km

]

0

10

20

30

40

50

-0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30

2 km 14 km 20 km 30 km 40 km

Antwort

Höh

e [k

m]

Ozone from ground-based FTIR ClONO2 from MIPAS