|The revised paper has added material that attempts to support the (new) sentence (Lines 414-416) “The central argument of this paper is that applying a single localization function for the localization of covariances between multiple state variables in an EnKF scheme may lead to a rank deficient estimate of the background covariance matrix.” As I describe in the “Matters which require attention” portion of my review, much of the supporting material for this new sentence is either unsubstantiated or incorrect. As such, despite improvements that have been made to other parts of the paper and despite the fact that I still think that their suggested methods for inter-variable localization and their description of the generalized multi-variate Askey localization function would be of interest to readers of non-linear processes in geophysics, I do not think the paper should be published in its current form. My overall recommendation is to accept the paper after revisions that address the concerns outlined below.|
Matters which require attention
1. Lines 85-87 suggest that the Schur product between a semi-positive definite symmetric matrix (like that of the Gaspari-Cohn localization matrix) and a pre-existing positive definite matrix may result in a matrix that has zero eigenvalues. If understand localization correctly, it is incorrect to suggest that univariate localization will reduce the rank of the original covariance matrix. In most practical examples, it massively increases the rank of the covariance matrix (without creating negative eigenvalues) and this is one of the main reasons it has proven to be so useful. If the authors have a relevant example where univariate localization reduces the rank of the matrix to which the localization is applied, please include it in an Appendix. Otherwise drop this sentence.
2. Lines 87-88 suggest that the symmetry of a matrix implies that its eigenvalues are non-negative. This is also incorrect. For example, the negative of the identity matrix is symmetric but both of its eigenvalues are negative. Symmetry does not imply positive eigenvalues.
3. Lines 99 – 101. Do the author’s know that Kang et al. (2011) was not also motivated by the need to achieve a semi-positive definite background error covariance matrix? Zeroing out the inter-variable covariances does not produce negative eigenvalues. Many researchers have been trying to find localization functions that filter spurious correlations AND do not produce negative eigenvalues. I think you need to justify your claim that Kang was not interested in preserving semi-positive definiteness or drop the statement.
4. Lines 182-184. The statement “although C is rank-deficient and thus so is the localized covariance matrix” is incorrect. To see this, take the element wise-product of the localization matrix defined by (5) with the identity matrix – one then will recover the identity matrix – which has full rank and is not rank deficient.
5. Lines 414-416. Here we here that the central argument of the revised paper is “that applying a single localization function for the localization of covariances between multiple state variables in an EnKF scheme may lead to a rank deficient estimate of the background covariance matrix”. I do not believe that rank-deficiency is the primary problem with univariate localization so making this the central argument of the revised paper has decreased the papers appeal for me. Suppose I have many more ensemble members than variable types; e.g. I might have 5 model variables at each grid point and 80 ensemble members. In this case, I could use a univariate localization matrix of the form given by equation (5), where each of the C_0 matrices was the identity matrix and then apply this to my 80 member ensemble covariance matrix. The resulting covariance matrix would almost certainly have full rank even if the number of model grid points was in the tens of millions because the localization has zeroed out all inter-grid-point covariances leaving 80 ensemble members to describe the covariance between the 5 model variables on each grid point. Without doubt the rank of the matrix obtained using (5) would be very much higher than the rank 79 of the unlocalized ensemble covariance matrix. Note that it is easy to construct dynamical data assimilation systems where the rank of the true forecast error covariance matrix is rank deficient (e.g. Bishop et al., 2003, J.Atmos. Sci.). Hence, the best localization strategies may be those that stop short of making the localized ensemble covariance matrix full rank. I think there is broad agreement that the localization matrix should attenuate spurious correlations and also ensure that the localized covariance matrix has no negative eigenvalues. (To ensure this, one only needs to ensure that the localization matrix is semi-positive definite). The value of your study is that you have presented new and effective methods for generating multi-variate semi-positive definite localization matrices. In my view, that’s the only argument you have to make.
1. Line 16. Abstract: In order to be just a bit clearer, change “multiple state variables” to “multiple state variables that exist at the same location”
2. Line 165: Replace “of d” by “of the separation distance d”
3. Eq’s (9)-(11) please add more discussion of the meaning of the terms mu_12, mu_11 and mu_22 in equations (9) through (11). Perhaps mention the values you ended up choosing to use for these parameters in your experiments? In equation (9) and (11) can one think of the factors beta and B as simply being normalization factors that ensure that the correlation between the variables becomes equal to unity when the distance between the variables is equal to zero?
4. Figure 4. Please give more explanation of Figure 4 within the figure caption. What is the difference between the circles and the squares? Perhaps reiterate the percentile significance of the box and whiskers plots. Clarify that the reason that there are no grey lines for the S4, beta=1 case is because the Askey function is not defined in this case, etc. Consider adding more explanation of what the figure is showing to the other figures as well.
5. Figure 3. Consider rephrasing the caption to “For the partially observed case, locations of observations of X and Y are indicated by the black dots and grey circles, respectively.”