1. Mathematical/Numerical optimization

2. What are the effects of including correlated observation errors on the minimization? How does it affect the hessian conditionning – It affects the eigen spectrum. We are making the observation very accurate. – It would be interesting to identify the eigenvectors corresponding to these very accurate obs. – How does the eigenvalues and vectors of the hessian change when we account for correlated obs – Sensitivity of scales in B respect to those in R – Special case when the correlation does not decrease in space/or time (e.g. diurnal cycle), how does it affect the above? How does it affect the statistics in ensemble methods?

3. How should we regularize R to improve the numerical behaviour of the problem? Isn’t it dangerous to fiddle with the statistical just to improve the numerical aspects? – It is a matter of balance between accuracy and computing time – Should we be that confident to the diagnose R anyway? – It is probably not too harmful to bump-up the std dev, if they are quite small. There is already a literature on covariance regularization (e. g. in finance), maybe we should look into it. Should we use raw estimates or try to fit a correlation function?

4. What preconditioning techniques should we use? Do we need to completely rethink the whole preconditioning? – The second level preconditioning should still apply – Probably very different answers depending on the correlation structures in R