In-flight Assessment of the end-to-end spectral responses of the GERB radiometer detectors

23rd to 24th October 2006 GIST 25, UK Met Office, Exeter

In-flight Assessment of the end-to-end spectral responses of the GERB radiometer

detectors

Glyn Spencer and David Llewellyn-Jones

Earth Observation Science


In-flight Assessment of the GERB radiometer detectors end-to-end spectral responses

• Objectives of this work

• The end-to-end Spectral Response of a GERB radiometer detector

• A naïve recovery attempt

• The problem of inversion

• Inversion with regularisation

• First Results

• Further work

• Conclusions


Why make an in-flight assessment of the individual spectral responses?

• Pre-flight spectral measurements were made at component level only

• Need to investigate possibility changes due to launch and associated activities

• Need to assess self-consistency of all elements of measurement system

• How can this be done?


GERB radiometer detectors end-to-end spectral responses

• The signals from the ith GERB detector element correspond to the ToA radiances, expressed as:

• Re-expressed, using a quadrature rule:

• This is in order to generate a set of linear equations

b

a

dsLL NuimToA

fimToA

)()(,,

,,

CsLWLn

j

ijN

uimjToAj

fimToA

1

,,

,, )()(


• The ToA signals for the ith GERB detector element form the following set of linear equations:

• In matrix notation: L = As + C, where

• This leads to an initial solution of the end-to-end spectral response as when m n:

s A-1L

nin

i

i

i

imn

im

im

in

ii

in

ii

in

ii

fim

fi

fi

fi

C

C

C

C

s

s

s

s

AAA

AAA

AAA

AAA

L

L

L

L

3

2

1

3

2

1

,,2,1

3,3,23,1

2,2,22,1

1,1,21,1

,

,3

,2

,1

uijkk

ijk LWA ,

,,

GERB radiometer detectors end-to-end spectral responses


A naïve recovery attempt

• Generation of the coefficients of A:-

Atmospheric profiles obtained from ECMWF Surface types and properties from CERES/SARB working

group Calculate radiances at 20cm-1 intervals for 20cm-1-2500cm-1

Use STREAMER(DISORT v2) with RFM Use a quadrature rule A 105 105 matrix is formed Can apply standard numerical methods to solve for s


-400000

-300000

-200000

-100000

0

100000

200000

300000

0 500 1000 1500 2000 2500

Wavenumber (cm-1)

No

rma

lise

d s

pe

ctr

al r

es

po

ns

e




• What has gone wrong?

• Did not take account of the impact of the measurement errors on the inversion method.

• The matrix equation should take the form: L = As + e

• Are the errors correlated or not?

• Are the errors solely associated with the measurement vector L?

• What about A?


The problem of inversion

• The integral equation for filtered ToA radiances is a Fredholm equation of the first kind.

• The general form is:

• Where K(x,y) is known as the Kernel function, g(y) is a set of measurements and f(x) is the unknown function of interest.

b

a

dsLL NuimToA

fimToA

)()(,,

,,

b

a

dxxfyxKyg )(),()(


The problem of inversion

• The problem is ill-posed. Hadamard (1902) proposed that a problem is well-posed if:

1) A solution exists.

2) The solution is unique.

3) The solution is stable and continuous.

The coefficient matrix A is ill-conditioned:

1) One or more of the eigenvalues of A is zero or close to zero. Leading to A being singular or close to singular.

• Need a method by which the ill-conditioning of A and measurement errors e are controlled or constrained.


Inversion with regularisation

• Use Tikhonov-Twomey regularisation (TTR).

• Seek a solution by minimising an augmented least squares method:

min { = ||L – As|| + ||K(s-a)|| }

• K is the an operator matrix, a is an a priori solution estimate and is the regularisation parameter.

• TTR in general form: s=(ATSe-1A+KTK) -1(ATSe

-1L+KTKa)

• Se=CeCeT is an error covariance matrix. Instead of using KTK, it

can be replaced by Sa-1. Sa=CaCaT being an a priori covariance

matrix

• TTR in standard form: s=(ATA+I)-1(ATL+a), A=C-1AK-1 & L=C-

1A

22

22


• Choice of regularisation parameter : The L-curve method

Inversion with regularisation

Numerical implementation:

• Use P. C. Hansen’s, Regularization Tools: A Matlab package for analysis and

solution of discrete ill-posed problems, Numerical Algorithms 6 (1994), pp. 1-35 (revised in 2001)

• There are 53 documented Matlab functions for analysis and solution of discrete ill-posed, ill-conditioned, noisy problems

• Both direct and iterative regularization methods are available

• Can apply Singular Value Decomposition to TTR in standard form


• Basic selection criteria for candidate pixels:

1. GERB pixel scenes that are cloudy and cloud free

2. Have no distinct land/sea boundaries and surface type is ‘homogeneous’

3. Suitable number of pixel scenes available.

• Use GERB detector element 170.

• Location is ~17o North and from 13o West to 33o East.

First results


Results

Recall the generation of the coefficients of A - GERB ToA total ARG filtered radiances for detector element on 24/01/2005 at 0000hrs and corresponding

ToA unfiltered RTM calculations

First results

70

75

80

85

90

95

90 110 130 150 170 190

GERB scan column number

To

A r

ad

ian

ce

s (

W m

-2 s

r-1)

GERB data

RTM data


GERB ToA total ARG filtered radiances for detector element on 24/07/2005 at 0000hrs and corresponding ToA unfiltered RTM calculations

50

55

60

65

70

75

80

85

90

95

100

90 110 130 150 170 190

GERB scan column number

To

A r

adia

nce

s (W

m-2 s

r-1)

GERB data

RTM data

First results


First results

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0 250 500 750 1000 1250 1500 1750 2000 2250 2500

Wavenumber (cm-1)

Nor

mal

ised

spe

ctra

l res

pons

e

No a priori used


0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

0 250 500 750 1000 1250 1500 1750 2000 2250 2500

Wavenumber (cm-1)

No

rma

lise

d s

pe

ctr

al r

es

po

ns

e

First results

Half sized GERB NSR a priori used


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 250 500 750 1000 1250 1500 1750 2000 2250 2500

Wavenumber (cm-1)

Nor

mal

ised

spe

ctra

l res

pons

e

Calculated NSR

GERB a priori NSR

First results

Full sized GERB NSR a priori used


0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

2.00

0 250 500 750 1000 1250 1500 1750 2000 2250 2500

Wavenumber (cm-1)

No

rma

lise

d s

pe

ctr

al r

es

po

ns

e

Calculated NSR

GERB a priori NSR

First results

Inverted GERB NSR a priori used


y = 0.8721x + 11.285R2 = 0.9012

50

55

60

65

70

75

80

85

90

95

100

50 55 60 65 70 75 80 85 90 95 100

GERB ToA radiances (W m -2 sr -1)

RT

M T

oA

rad

ian

ces

(W m

-2 s

r-1)

First resultsPlot of GERB ToA total ARG filtered radiances for detector element 170 against

corresponding ToA unfiltered RTM calculations


• Build up a data base of GERB pixel scenes. Extend current analysis to a representative range of pixel scene types and atmospheric conditions. Select pixel scenes that are distinct and separate.

• Repeat for each scan-line (detector element) if possible.

• Extend to the shortwave and total channels.

• Which RTM to use for above?

• Try method on GERB-1 radiance data and other sources.

• Compare derived unfiltered ToA radiances with other methods

Further work


• An inversion technique has been developed which gives stable and physically realistic Results

• Indications are that the longwave end-to-end spectral responses can be recovered and estimated.

• Nevertheless, the problem is ill-posed. The problem is ill-conditioned.

• Standard numerical techniques to solve L = As + e for s fail.

• A preferred numerical technique is SVD with regularisation.

• Further Improvement in the kernel/forward problem are required in order to the analyze shortwave and total channels

Conclusions

Documents

In-flight Assessment of the end-to-end spectral responses of the GERB radiometer detectors