Commentary: On the Efficiency of Covariance Localisation of the Ensemble Kalman Filter Using Augmented Ensembles

(i) The predecessor of the GETKF eigenvalue form of themodified gainmatrix equation appeared in Posselt and Bishop [4, 5]—before Bocquet [1]. (ii) The spectral shift theorem reduces the differences in the numerical cancellation errors referred to by Farchi and Bouquet. (iii) The eigenvalue form enables Wang et al.’s [6] corrections for ensemble rank deficiency. (iv) A proof of the equivalence of the eigenvalue form and Bouquet’s form.


INTRODUCTION
In discussing Equation (39) (Equation (25) of Bocquet [1]), Farchi and Bouquet [2] state that "This perturbation update has been rediscovered by Bishop et al. [3] and included in their gain ETKF (GETKF) algorithm. However, the update formula used in the GETKF is prone to numerical cancellation errors as opposed to Equation (39)". Here, we note: (i) The predecessor of the GETKF eigenvalue form of the modified gain matrix equation appeared in Posselt and Bishop [4,5]-before Bocquet [1]. (ii) The spectral shift theorem reduces the differences in the numerical cancellation errors referred to by Farchi and Bouquet. (iii) The eigenvalue form enables Wang et al.'s [6] corrections for ensemble rank deficiency. (iv) A proof of the equivalence of the eigenvalue form and Bouquet's form.
On page 12, Farchi and Bouquet [2] also state that "Such an extension had been discussed by Bishop et al. [3] but without numerical illustration." This is incorrect. Lei et al. [7] used the GETKF to show that model space ensemble covariance localization provided satellite data assimilation (DA) performance comparable to 3DEnsVar.
To be specific about the forms of the modified gain matrix, let , and where K is the total number of ensemble members in the ensemble forecast and where the n-vector x f i is the ith member of the prior ensemble forecast and where the p-vector H x f i is the ith member of the prior ensemble forecast of the p-vector y of p observations. When p<K, the numerical cost of the pxp eigen decompositioñ is less than the K × K eigen decomposition In (2), E is a pxp eigenvector matrix for which EE T = E T E = I pxp , Ŵ pxp is a pxp diagonal matrix of eigenvalues. In (3), C is a K × K orthonormal matrix of eigenvectors CC T = C T C = I K×K and Ŵ KxK is a K × K diagonal matrix of eigenvalues. At least Kp of the eigenvalues in Ŵ K×K will be equal to zero in the case of K>p. Equation's (2) and (3) are directly connected to the verbose singular value decompositionHZ However, since the columns of C associated with zero eigenvalues cannot contribute to products of the matrixHZ f with other vectors, it is more efficient to work with the concise svd where L Kxp lists the p columns of C K×K having non-zero eigenvalues. Posselt and Bishop [4,5] note that L Kxp is given by and hence can be computed without performing an eigen decomposition of the larger K × K matrix in (3). Posselt and Bishop [4,5] prove that for a linear observation operatorH, if (see [5], Equation A10) then The analysis perturbations are given by X a = Z a √ K − 1, hence, is the perturbation update equation implied by Posselt and Bishop [4,5].
In the above notation, and when propagation of small amplitude ensemble perturbations by the non-linear model is replaced by the propagation of raw ensemble perturbations by the non-linear model (i.e., no tangent linear model approximation is made), Bocquet's Equation (25) [1] for the ensemble perturbation update takes the form, A fundamental difference between (7) and (8) (7) becomes Dividing Equation (9) by √ K − 1 recovers Equation (24) of Bishop et al. [3].
The above shows that Bishop et al.'s Equation (24) [3] was not "rediscovered" from Bocquet's [1] form as implied by Farchi and Bocquet [2]. It is an extension of Posselt and Bishop's [4,5] eigenvalue form to the case of K>p. Equation (9)

NUMERICAL ISSUES, CONDITION NUMBERS, AND UNDERSTANDING
Numerical cancellation errors increase when the condition number of the matrix increases. Let us define the scalars γ max i and γ min i to, respectively, denote the maximum and minimum of the eigenvalues listed in the eigenvalue matrix Ŵ pxp . The Because γ min i can be zero, κ H Z TH Z can be infinite.
and hence has κ H Z TH Z + αI = . Once the eigen decomposition C C T of (14) has been obtained, one obtains the eigenvalues required by the GETKF or ETKF using Ŵ = − αI. Thus, condition number differences between the Bouquet and eigenvalue form are easily eliminated. The eigenvalue form lends understanding to the performance of DA schemes in much the same way that Empirical Orthogonal Functions lend understanding to climate variability. Wang et al. [6] used this understanding to correct gross aspects of the eigenvalue overestimation that occurs when the size of the ensemble is much smaller than the rank of the true observation space forecast error covariance matrix.