1. Estimating observing system error thresholds using an asymptotic approach Gary Brassington and Prasanth Divakaran Centre for Australian Weather and Climate Research Bureau of Meteorology

2. Acknowledgements We gratefully acknowledge the valuable contributions from the BLUElink> science and technical team, Bureau of Meteorology, CSIRO the Royal Australian Navy, NASA, CNES, ESA, AVISO, NOAA, US Navy, the Argo Science team, Geoscience Australia, OSTST, GODAE and GHRSST. Brassington, G. B. and P. Divakaran, 2009: The theoretical impact of remotely sensed sea surface salinity observations in a multi-variate assimilation system, Ocean Modelling, 27, 70-81 10.1016/j.ocemod.2008.12.005 Initial motivation “What error is sufficiently large that it will have a negligible impact to the analysis system?” Fact: SMOS and Aquarius has been estimated to have a large error Can an error ever be regarded as too large?

3. Acknowledgements We gratefully acknowledge the valuable contributions from the BLUElink> science and technical team, Bureau of Meteorology, CSIRO the Royal Australian Navy, NASA, CNES, ESA, AVISO, NOAA, US Navy, the Argo Science team, Geoscience Australia, OSTST, GODAE and GHRSST. Performance We can pose this formally asymptotically. Consider the simplest tri-variate problem

4. Acknowledgements We gratefully acknowledge the valuable contributions from the BLUElink> science and technical team, Bureau of Meteorology, CSIRO the Royal Australian Navy, NASA, CNES, ESA, AVISO, NOAA, US Navy, the Argo Science team, Geoscience Australia, OSTST, GODAE and GHRSST. The analysis for salinity can expressed as, where

5. Acknowledgements We gratefully acknowledge the valuable contributions from the BLUElink> science and technical team, Bureau of Meteorology, CSIRO the Royal Australian Navy, NASA, CNES, ESA, AVISO, NOAA, US Navy, the Argo Science team, Geoscience Australia, OSTST, GODAE and GHRSST. Large error is when the analysis equation asymptotically converges to the same analysis in the absence of the observations (i.e., )

6. Acknowledgements We gratefully acknowledge the valuable contributions from the BLUElink> science and technical team, Bureau of Meteorology, CSIRO the Royal Australian Navy, NASA, CNES, ESA, AVISO, NOAA, US Navy, the Argo Science team, Geoscience Australia, OSTST, GODAE and GHRSST. This occurs when, which is equivalent to the condition,

7. Acknowledgements We gratefully acknowledge the valuable contributions from the BLUElink> science and technical team, Bureau of Meteorology, CSIRO the Royal Australian Navy, NASA, CNES, ESA, AVISO, NOAA, US Navy, the Argo Science team, Geoscience Australia, OSTST, GODAE and GHRSST. The condition on for observation error for this simplified problem is then,

8. Acknowledgements We gratefully acknowledge the valuable contributions from the BLUElink> science and technical team, Bureau of Meteorology, CSIRO the Royal Australian Navy, NASA, CNES, ESA, AVISO, NOAA, US Navy, the Argo Science team, Geoscience Australia, OSTST, GODAE and GHRSST. R is uncorrelated Assume and

9. Acknowledgements We gratefully acknowledge the valuable contributions from the BLUElink> science and technical team, Bureau of Meteorology, CSIRO the Royal Australian Navy, NASA, CNES, ESA, AVISO, NOAA, US Navy, the Argo Science team, Geoscience Australia, OSTST, GODAE and GHRSST. Lets consider four conditions Univariate Bivariate Tri-variate, simplified Tri-variate R is uncorrelated

10. Acknowledgements We gratefully acknowledge the valuable contributions from the BLUElink> science and technical team, Bureau of Meteorology, CSIRO the Royal Australian Navy, NASA, CNES, ESA, AVISO, NOAA, US Navy, the Argo Science team, Geoscience Australia, OSTST, GODAE and GHRSST. Lets corrupt illustrate these conditions with an analysis scheme, BODAS (Up to this point the analysis is system independent)

11. Acknowledgements We gratefully acknowledge the valuable contributions from the BLUElink> science and technical team, Bureau of Meteorology, CSIRO the Royal Australian Navy, NASA, CNES, ESA, AVISO, NOAA, US Navy, the Argo Science team, Geoscience Australia, OSTST, GODAE and GHRSST. Performance Interpretation or an impact will occur when

12. Acknowledgements We gratefully acknowledge the valuable contributions from the BLUElink> science and technical team, Bureau of Meteorology, CSIRO the Royal Australian Navy, NASA, CNES, ESA, AVISO, NOAA, US Navy, the Argo Science team, Geoscience Australia, OSTST, GODAE and GHRSST. Performance Alternatively observations will have an impact when Univariate Tri-variate R is uncorrelated Aquarius SMOS

13. Acknowledgements We gratefully acknowledge the valuable contributions from the BLUElink> science and technical team, Bureau of Meteorology, CSIRO the Royal Australian Navy, NASA, CNES, ESA, AVISO, NOAA, US Navy, the Argo Science team, Geoscience Australia, OSTST, GODAE and GHRSST. Lets corrupt illustrate these conditions with an analysis scheme, BODAS (Up to this point the analysis is system independent)

14. Acknowledgements We gratefully acknowledge the valuable contributions from the BLUElink> science and technical team, Bureau of Meteorology, CSIRO the Royal Australian Navy, NASA, CNES, ESA, AVISO, NOAA, US Navy, the Argo Science team, Geoscience Australia, OSTST, GODAE and GHRSST. Tri-variate assimilationUni-variate assimilation

15. Acknowledgements We gratefully acknowledge the valuable contributions from the BLUElink> science and technical team, Bureau of Meteorology, CSIRO the Royal Australian Navy, NASA, CNES, ESA, AVISO, NOAA, US Navy, the Argo Science team, Geoscience Australia, OSTST, GODAE and GHRSST. Tri-variate assimilationBi-variate assimilation

16. Acknowledgements We gratefully acknowledge the valuable contributions from the BLUElink> science and technical team, Bureau of Meteorology, CSIRO the Royal Australian Navy, NASA, CNES, ESA, AVISO, NOAA, US Navy, the Argo Science team, Geoscience Australia, OSTST, GODAE and GHRSST. Performance

17. Acknowledgements We gratefully acknowledge the valuable contributions from the BLUElink> science and technical team, Bureau of Meteorology, CSIRO the Royal Australian Navy, NASA, CNES, ESA, AVISO, NOAA, US Navy, the Argo Science team, Geoscience Australia, OSTST, GODAE and GHRSST. Performance How does this compare to the increment variance?

18. Acknowledgements We gratefully acknowledge the valuable contributions from the BLUElink> science and technical team, Bureau of Meteorology, CSIRO the Royal Australian Navy, NASA, CNES, ESA, AVISO, NOAA, US Navy, the Argo Science team, Geoscience Australia, OSTST, GODAE and GHRSST. Performance Conclusions Multi-variate is always a tougher constraint for observation impact (i.e., GODAE type systems cannot tell you it is more valuable than a univariate objective analysis) Asymptotic conditions are system independent, applicable to hypothesis testing and diagnosing impact to any system Results will be assumption and system dependent and open to interpretation, although the “good” systems will always pose tougher constraints. For BLUElink, Aquarius will impact the tropics and equatorial band Less impact at mid-latitude with good coverage from SST SMOS will have considerably less impact for BLUElink

19. Acknowledgements We gratefully acknowledge the valuable contributions from the BLUElink> science and technical team, Bureau of Meteorology, CSIRO the Royal Australian Navy, NASA, CNES, ESA, AVISO, NOAA, US Navy, the Argo Science team, Geoscience Australia, OSTST, GODAE and GHRSST. Conclusions The asymptotic conditions can be reposed for other variables The conditions are extendible to larger matrices It can tell you where to expect impact It can tell the relative impact

20. Acknowledgements We gratefully acknowledge the valuable contributions from the BLUElink> science and technical team, Bureau of Meteorology, CSIRO the Royal Australian Navy, NASA, CNES, ESA, AVISO, NOAA, US Navy, the Argo Science team, Geoscience Australia, OSTST, GODAE and GHRSST. Performance Extendable C=

21. Acknowledgements We gratefully acknowledge the valuable contributions from the BLUElink> science and technical team, Bureau of Meteorology, CSIRO the Royal Australian Navy, NASA, CNES, ESA, AVISO, NOAA, US Navy, the Argo Science team, Geoscience Australia, OSTST, GODAE and GHRSST. Performance Extendable

22. Acknowledgements We gratefully acknowledge the valuable contributions from the BLUElink> science and technical team, Bureau of Meteorology, CSIRO the Royal Australian Navy, NASA, CNES, ESA, AVISO, NOAA, US Navy, the Argo Science team, Geoscience Australia, OSTST, GODAE and GHRSST. Computing – Cramer’s rule Cost of one inversion of the reduced observation space

23. Acknowledgements We gratefully acknowledge the valuable contributions from the BLUElink> science and technical team, Bureau of Meteorology, CSIRO the Royal Australian Navy, NASA, CNES, ESA, AVISO, NOAA, US Navy, the Argo Science team, Geoscience Australia, OSTST, GODAE and GHRSST. What about Argo?

24. Acknowledgements We gratefully acknowledge the valuable contributions from the BLUElink> science and technical team, Bureau of Meteorology, CSIRO the Royal Australian Navy, NASA, CNES, ESA, AVISO, NOAA, US Navy, the Argo Science team, Geoscience Australia, OSTST, GODAE and GHRSST. Performance

25. Acknowledgements We gratefully acknowledge the valuable contributions from the BLUElink> science and technical team, Bureau of Meteorology, CSIRO the Royal Australian Navy, NASA, CNES, ESA, AVISO, NOAA, US Navy, the Argo Science team, Geoscience Australia, OSTST, GODAE and GHRSST. Reposed for eta Tri-variate assimilation

26. Acknowledgements We gratefully acknowledge the valuable contributions from the BLUElink> science and technical team, Bureau of Meteorology, CSIRO the Royal Australian Navy, NASA, CNES, ESA, AVISO, NOAA, US Navy, the Argo Science team, Geoscience Australia, OSTST, GODAE and GHRSST.