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In-flight Assessment of the end-to-end spectral responses of the GERB radiometer detectors. Earth Observation Science. Glyn Spencer and David Llewellyn-Jones. In-flight Assessment of the GERB radiometer detectors end-to-end spectral responses. Objectives of this work - PowerPoint PPT Presentation
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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
23rd to 24th October 2006 GIST 25, UK Met Office, Exeter
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
23rd to 24th October 2006 GIST 25, UK Met Office, Exeter
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?
23rd to 24th October 2006 GIST 25, UK Met Office, Exeter
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
,,
,, )()(
23rd to 24th October 2006 GIST 25, UK Met Office, Exeter
• 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
23rd to 24th October 2006 GIST 25, UK Met Office, Exeter
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
23rd to 24th October 2006 GIST 25, UK Met Office, Exeter
-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
A naïve recovery attempt
23rd to 24th October 2006 GIST 25, UK Met Office, Exeter
A naïve recovery attempt
• 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?
23rd to 24th October 2006 GIST 25, UK Met Office, Exeter
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 )(),()(
23rd to 24th October 2006 GIST 25, UK Met Office, Exeter
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.
23rd to 24th October 2006 GIST 25, UK Met Office, Exeter
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
23rd to 24th October 2006 GIST 25, UK Met Office, Exeter
• 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
23rd to 24th October 2006 GIST 25, UK Met Office, Exeter
• 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
23rd to 24th October 2006 GIST 25, UK Met Office, Exeter
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
23rd to 24th October 2006 GIST 25, UK Met Office, Exeter
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
23rd to 24th October 2006 GIST 25, UK Met Office, Exeter
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
23rd to 24th October 2006 GIST 25, UK Met Office, Exeter
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
23rd to 24th October 2006 GIST 25, UK Met Office, Exeter
-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
23rd to 24th October 2006 GIST 25, UK Met Office, Exeter
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
23rd to 24th October 2006 GIST 25, UK Met Office, Exeter
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
23rd to 24th October 2006 GIST 25, UK Met Office, Exeter
• 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
23rd to 24th October 2006 GIST 25, UK Met Office, Exeter
• 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