2. 23 rd to 24 th October 2006GIST 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

3. 23 rd to 24 th October 2006GIST 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

4. 23 rd to 24 th October 2006GIST 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?

5. 23 rd to 24 th October 2006GIST 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

6. 23 rd to 24 th October 2006GIST 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 -1 L GERB radiometer detectors end-to-end spectral responses

7. 23 rd to 24 th October 2006GIST 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

8. 23 rd to 24 th October 2006GIST 25, UK Met Office, Exeter A naïve recovery attempt

9. 23 rd to 24 th October 2006GIST 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?

10. 23 rd to 24 th October 2006GIST 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.

11. 23 rd to 24 th October 2006GIST 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.

12. 23 rd to 24 th October 2006GIST 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  =(A T S e -1 A+  K T K) -1 (A T S e -1 L+  K T Ka) S e =C e C e T is an error covariance matrix. Instead of using  K T K, it can be replaced by S a -1. Sa=C a C a T being an a priori covariance matrix TTR in standard form: s  =(A T A+  I) -1 (A T L+  a), A=C -1 AK -1 & L=C -1 A

13. 23 rd to 24 th October 2006GIST 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

14. 23 rd to 24 th October 2006GIST 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 ~17 o North and from 13 o West to 33 o East. First results

15. 23 rd to 24 th October 2006GIST 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

16. 23 rd to 24 th October 2006GIST 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 First results

17. 23 rd to 24 th October 2006GIST 25, UK Met Office, Exeter First results No a priori used

18. 23 rd to 24 th October 2006GIST 25, UK Met Office, Exeter First results Half sized GERB NSR a priori used

19. 23 rd to 24 th October 2006GIST 25, UK Met Office, Exeter First results Full sized GERB NSR a priori used

20. 23 rd to 24 th October 2006GIST 25, UK Met Office, Exeter First results Inverted GERB NSR a priori used

21. 23 rd to 24 th October 2006GIST 25, UK Met Office, Exeter First results Plot of GERB ToA total ARG filtered radiances for detector element 170 against corresponding ToA unfiltered RTM calculations

22. 23 rd to 24 th October 2006GIST 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

23. 23 rd to 24 th October 2006GIST 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