Guidance on how to improve vertical covariance localization based on a 1000member ensemble
 ^{1}Institut für Meteorologie und Geophysik, Universität Wien, Vienna, Austria
 ^{2}RIKEN Center for Computational Science, Kobe, Japan
 ^{1}Institut für Meteorologie und Geophysik, Universität Wien, Vienna, Austria
 ^{2}RIKEN Center for Computational Science, Kobe, Japan
Abstract. The success of ensemble data assimilation systems substantially depends on localization, which is required to mitigate sampling errors caused by modeling background error covariances with undersized ensembles. However, finding an optimal localization is highly challenging as covariances, sampling errors, and appropriate localization depend on various factors. Our study investigates vertical localization based on a unique convectionpermitting 1000member ensemble simulation. 1000member ensemble correlations serve as truth for examining vertical correlations and their sampling error. We discuss requirements for vertical localization by deriving an empirical optimal localization (EOL) that minimizes the sampling error in 40member subsample correlations with respect to the 1000member reference. Our analysis covers temperature, specific humidity, and wind correlations on various pressure levels. Results suggest that vertical localization should depend on several aspects, such as the respective variable, vertical level, or correlation type (self or crosscorrelations). Comparing the empirical optimal localization with common distancedependent localization approaches highlights that finding suitable localization functions bears substantial room for improvement. Furthermore, we discuss the gain of combining different localization approaches with an adaptive statistical sampling error correction.
Tobias Necker et al.
Status: final response (author comments only)

RC1: 'Comment on egusphere2022434', Lili Lei, 04 Jul 2022
Summary
This manuscript uses a convectionpermitting 1000member ensemble simulation to examine the vertical localization. An empirical optimal localization is proposed, which minimizes the sampling error of correlations estimated from a 40member ensemble comparing to those from the 1000member ensemble. Vertical correlations and localization functions for different state variable and cross variables are systematically examined. Results show that different vertical localization functions are required for different variables and vertical height. Combination of the empirical optimal localization with adaptive sampling error correction is also investigated. The manuscript is wellwritten, and could be a valuable contribution for the data assimilation community. I have several comments as below.
 l22, It is more appropriate to say members of O(100), since Canadian Center has 2 groups of 128member ensembles.
 l100105, the description of BCs is confusing. Is the GEFS 20member analysis used 50 times to get 1000 BCs? Or climatological GEFS is sampled for 1000 BCs?
 l197, there are increased correlations below 800hPa. Are there any physical explanations for this?
 Eq. (5), is s the index for 40member groups with S=25? Eq. (5) is similar to Eq. (1) in Lei and Anderson (2014).
 If the sample correlation r_40 tends to overestimate the true correlation r_1000, the EOL computed from Eq. (7) should be no larger than 1.0? As discussed by Lei and Anderson (2014), ELF can count inflation compared to GGF. But I am not sure about the EOL in Eq. (7), which could be true since the true correlation is known. Can the authors provide some derivations on this statement?
 Figs 2 and 7, how about the sample correlations estimated for cross variables?
 Figs 35, the UU EOL seems have values larger than 1.0. What is the exact value at the reference level? Why EOL estimates localization larger than 1.0 when sample correlations are close to 1.0? Intuitively, when sample correlations are close to the true correlations as 1.0, localization value goes to 1.
 l258260, this discussion is based on sampling errors in correlations. But for cycling data assimilation experiments, too strong taper for cross variables may result in too weak corrections.
 l290, the “error reduction” is for estimated correlation, not for prior/posterior errors by using the EOL. Also it would be helpful to have some discussions about the estimated localization and localization applied for cycling data assimilation in the section of conclusions and discussions.
 Section 3.1, a curious question, if the directvariable EOL is applied rather than crossvariable EOLs, i.e., UU is applied for UU, UV, UT, and UQ, how about the correlation error reduction?
 l396399, it would be helpful to add some dynamical explanations for these results.
 l407, the computational efficiency issue can be treated by modelspace localization (Lei et al. 2018).

AC1: 'Reply on RC1', Tobias Necker, 23 Jul 2022
We would like to thank the reviewer for valuable comments and suggestions. Below we respond to the reviewer's comments and summarise how we plan to address them in the revised manuscript. The original queries are bold, and our responses are normal text. Planned changes in the manuscript are italic.
RC1: 'Comment on egusphere2022434', Lili Lei, 04 Jul 2022
Summary
This manuscript uses a convectionpermitting 1000member ensemble simulation to examine the vertical localization. An empirical optimal localization is proposed, which minimizes the sampling error of correlations estimated from a 40member ensemble comparing to those from the 1000member ensemble. Vertical correlations and localization functions for different state variable and cross variables are systematically examined. Results show that different vertical localization functions are required for different variables and vertical height. Combination of the empirical optimal localization with adaptive sampling error correction is also investigated. The manuscript is wellwritten and could be a valuable contribution for the data assimilation community. I have several comments as below.
1) l22, It is more appropriate to say members of O(100), since Canadian Center has 2 groups of 128member ensembles.
Reply: We are aware that Environment Canada has a larger ensemble than other forecasting centres and tried to address this by using the word “usually”. To our knowledge Environment Canada is the only centre that exceeds 100 members, while most other centres use about 4050 members. For this reason, we think that O(100) might be misleading.
Some ensemble sizes:
En. Ca.  EnKF  256 members
NCEP  deterministic EnSRF  80 members
ECMWF  hybrid 4DVAR  50 members
JAM  hybrid LETKF/4DVar 50 members
MeteoFrance  hybrid 4DVAR  50 members
Korea  KIAPSLETKF  50 members
DWD  LETKF  40 members2) l100105, the description of BCs is confusing. Is the GEFS 20member analysis used 50 times to get 1000 BCs? Or climatological GEFS is sampled for 1000 BCs?
Reply: The 20member analysis ensemble is used 50 times to reach 1000 BCs and afterwards combined with 1000 random climatologically scaled perturbations.
To avoid confusion, we will change the following sentences: “These BCs combine 1000 climatologically scaled random perturbations with an analysis ensemble of the NCEP Global Ensemble Forecast System (GEFS). The GEFS 20member analysis ensemble is used 50 times to reach 1000 BCs and afterwards combined with 1000 random climatologically scaled perturbations.”3) l197, there are increased correlations below 800hPa. Are there any physical explanations for this?
Reply: Following your question, we analysed temperature and hydrometeor profiles in the ensemble. Members with colder midtropospheric temperatures exhibited more upperlevel clouds resulting in colder nearsurface temperatures that likely are caused by stronger cloud shadowing.
We will add the following comment: “This weak correlation is linked to cloud shadowing by midtropospheric clouds and resulting colder nearsurface temperatures.”4) Eq. (5), is s the index for 40member groups with S=25? Eq. (5) is similar to Eq. (1) in Lei and Anderson (2014).
Reply: Yes, the index for S is 25 as we use 25 40member subsamples (groups) to compute the EOL.
We will add the following sentence: K is the number of vertical columns in the domain, and S is the number of subsamples.
Eq. (5) is similar to Eq. (2) in Anderson (2007) and to Eq. (1) in Lei and Anderson (2014) but exhibits two essential differences: On the one hand, we consider the correlation coefficient instead of the regression coefficient. On the other hand, our cost function minimises the sampling error with respect to the 1000member truth, which also results in different sums.5) If the sample correlation r_40 tends to overestimate the true correlation r_1000, the EOL computed from Eq. (7) should be no larger than 1.0? As discussed by Lei and Anderson (2014), ELF can account for inflation compared to GGF. But I am not sure about the EOL in Eq. (7), which could be true since the true correlation is known. Can the authors provide some derivations on this statement?
Reply: The EOL can inflate sample correlations similar to the ELF but only by optimising sample correlations. The EOL can reach values larger than 1.0 if the true correlation/r_1000 is larger than the sample correlation/r_40. In our setting, this is unlikely as we apply multiple sample correlations that are usually larger than the true correlation given a sufficient number of subsamples. However, we got EOL values larger than 1 when combining the SEC with the EOL (see, for example, Fig 2 in the Supplement/Appendix). The EOL inflated sample correlations when the SEC was applied first and damped sample correlations too strong.
We will add the following sentences: “Values larger than one can occur when the true correlation is larger than the sample correlation. For example, this can happen when estimating the EOL after applying other localization approaches.“6) Figs 2 and 7, how about the sample correlations estimated for cross variables?
Reply: It would also be possible to show and discuss the sample correlations. However, we believe that providing the sample correlations adds little additional information that is not already supplied by the curves in Fig 2 and the corresponding EOL in Fig 3 (Fig 7 and 8, respectively). We. therefore, believe that including extra lines or figures would rather distract the reader. Fig 3 in the Supplement/Appendix shows the sample correlations for single variable pairs and 500hPa.7) Figs 35, the UU EOL seems have values larger than 1.0. What is the exact value at the reference level? Why EOL estimates localization larger than 1.0 when sample correlations are close to 1.0? Intuitively, when sample correlations are close to the true correlations as 1.0, localization value goes to 1.
Reply: We will modify the xaxis (extended x range) in Figs 35, so the reader can see that the EOL values do not surpass 1.0 (see, for example, Fig 1 in the Supplement/Appendix). At the reference level, the EOL reaches a value of 1.0. Please also refer to our reply to comment 5, which addresses a similar point.8) l258260, this discussion is based on sampling errors in correlations. But for cycling data assimilation experiments, too strong taper for cross variables may result in too weak corrections.
Reply: Unfortunately, based on our experimental setting, we can only judge sampling errors in background sample correlations, which excludes a cycled assimilation environment. This means that the EOL is optimal in terms of sampling error in correlations but not necessarily optimal in terms of analysis or cycling performance.9) l290, the “error reduction” is for estimated correlation, not for prior/posterior errors by using the EOL. Also it would be helpful to have some discussions about the estimated localization and localization applied for cycling data assimilation in the section of conclusions and discussions.
Reply: As mentioned in the previous point, we only can estimate localization and error reduction based on the background correlation/covariance. We want to avoid speculation and therefore prefer not to discuss potential localisation changes that might impact the error in a single or cycled analysis.
For clarity, we will add the following sentence: “The result can be interpreted as a benchmark of the maximum possible correlation error reduction achieved by a domainuniform height and variabledependent localization. Note that results for optimizing the analysis may lead to different optimal localization values under some circumstances, but this is beyond the scope of this paper.”10) Section 3.1, a curious question, if the directvariable EOL is applied rather than crossvariable EOLs, i.e., UU is applied for UU, UV, UT, and UQ, how about the correlation error reduction?
Reply: This is an interesting question. We tested this approach in one experiment and found an error reduction similar to ALL or SEC.
We plan to add the following sentences: “Additionally, we tested the error reduction for applying the EOL estimated for selfcorrelations to both self and crosscorrelations of each variable (e.g., EOL derived from TT applied to TT, TQ, TU, and TV). For this setting, the error reduction was similar to ALL or SEC (not shown), which underlines the need to treat self and crosscorrelations differently.”11) l396399, it would be helpful to add some dynamical explanations for these results.
Reply: Finding dynamical explanations would be interesting. We attempted to incorporate such a dynamical interpretation, but it is hard to distinguish between a coincidental correlation or a dynamically induced correlation (correlation does not imply causation). Additionally, we show results averaged over a very large sample of profiles during different atmospheric conditions. Thus, it seems impossible to provide a short explanation of dynamical reasons without speculation.12) l407, the computational efficiency issue can be treated by modelspace localization (Lei et al. 2018).
Reply: Keeping in mind the full range of ensemble DA methods implemented in various places, we are convinced that it is generally correct to say that other factors, e.g. computational efficiency, may also need to be considered. However, we will add “may” in this sentence to weaken the statement.

RC2: 'Comment on egusphere2022434', Pavel Sakov, 08 Aug 2022

AC2: 'Reply on RC2', Tobias Necker, 14 Sep 2022
We would like to thank the referee for taking the time to review our manuscript and for providing valuable comments. Below we respond to the comments and mention how we plan to address them. The original queries are bold, and our responses are normal text. Planned changes in the manuscript are italic.
RC2: 'Comment on egusphere2022434', Pavel Sakov, 08 Aug 2022
Review of the manuscript egusphere2022434
“Guidance on how to improve vertical covariance localization based on a 1000member ensemble” by Tobias Necker, David Hinger, Philipp Johannes Griewank, Takemasa Miyoshi, and Martin Weissmann.
Pavel Sakov August 8, 2022
1 General comments
The manuscript presents a study conducted in a straightforward way. It considers a set of 1000member ensembles as representing the true state error covariance, and then investigates how various approaches to the vertical localisation can minimise the vertical correlation errors in 40member subensembles. This line of research is coherent with the previous efforts of the authors in the atmospheric ensemble DA.
In my opinion, for what it is, the study is done in a methodical and comprehensive way, and provides helpful material for further studies in that direction. However, the manuscript provokes a few questions in a more general context.
2 Questions
There are two main questions to the study for me: (1) how rigorous is the adopted methodology, and (2) how relevant are the results for other geophysical EnKF systems.
 On the concept of statistical ensemble.
The underlying assumption employed in the study is that the EnKF ensemble is a statistical ensemble, i.e. that it is composed of members drawn from the same pool. While this can be true to some degree for some EnKF systems, it also can be demonstrated to be wrong for other systems. The alternative view is that the EnKF ensemble is a unit carrying the state of the DA system, and that ensembles of different size can have rather different statistical properties. For example, it is possible that a 40member subensemble of a 1000member EnKF will have qualitatively different correlation errors to an ensemble of a properly set 40member EnKF.
This real or potential concern could be partly overcome by experimental testing of results with 40member systems. I say “partly” here because these experiments would still be conducted in a very specific environment.
Reply: Thanks for raising this point. Indeed, we assume that our EnKF ensemble is a statistical ensemble without mentioning it explicitly. Please also keep in mind that our convective ensemble is initialised from a downscaled analysis, which might impact the statistics, too.
Unfortunately, testing this assumption is beyond the scope of the present study, but we will consider investigating this assumption in future research. For clarity, we will add the following sentence in Sec. 2.4: “We assume that the 40member subensembles of the 1000member ensemble statically have sampling errors similar to those independent 40member ensemble EnKF systems would have.”
 On the importance of the “right” localisation.
While localisation is a necessary attribute of largescale EnKF systems, the sensitivity of the performance to the details of its implementation can be rather flat. From our experience with global ocean EnKF forecasting systems increasing or decreasing the horizontal localisation radius by say factor of 1.5 results to marginal changes in forecast innovation statistics. (Provided that the observation error variance is scaled proportionally to the localisation radius squared to keep the observation impact at the same level.) Therefore, I would suggest, firstly, to moderate claims of the importance of the choice of localisation technique for the forecasting skill of EnKF systems; and secondly, experimentally demonstrate the impact of the proposed taper functions.
Reply: Thank you for providing insights on your practical experience regarding localization in the ocean EnKF system. We are currently working on demonstrating the impact of EOLbased localization functions in an OSSE. These ongoing experiments will provide further insights and likely be part of a future manuscript. Especially for satellite DA, we already see a large influence of vertical localization.
Our present analysis only allows evaluating localization and error reduction based on the background error correlation judged by the 1000member ensemble truth. We want to avoid speculation and therefore do not discuss the potential impact the choice of localisation might have on a single or cycled analysis (or forecast).
For clarity, we will add the following sentence: “The result can be interpreted as a benchmark of the maximum possible correlation error reduction achieved by a domainuniform height and variabledependent localization. Note that results for optimizing the analysis may lead to different optimal localization values under some circumstances, but this is beyond the scope of this paper.”Furthermore, we will weaken the statement in line 71 (“is crucial” > “has the potential”): “Consequently, a better understanding of optimal vertical localization for convectionpermitting simulations has the potential to improve forecasts of convective precipitation and related hazards.”
3 Conclusion
I reiterate that in my view the study is conducted in a methodical and comprehensive way and would be interesting to specialists working on further advancements in that direction.
In a wider context, there remain grounds for scepticism in regard to the rigoursness of the underlying assumptions and applicability of the results to other systems. It seems to me that the study could benefit from experimental testing of the results. Also, it would be interesting to get some insight on implementation of the vertical localisation in the LETKF systems used.
I recommend to accept the paper for publication in NPG.

AC2: 'Reply on RC2', Tobias Necker, 14 Sep 2022

EC1: 'Comment on egusphere2022434', Olivier Talagrand, 17 Aug 2022
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/egusphere2022434/egusphere2022434EC1supplement.pdf

AC3: 'Reply on EC1', Tobias Necker, 14 Sep 2022
We would like to thank the editor for taking the time to review our manuscript and for providing additional comments. Below we respond to the editor's comments and explain how we plan to address them in the revised manuscript. The original queries are bold, and our responses are normal text. Planned changes in the manuscript are italic.
EC1: 'Comment on egusphere2022434', Olivier Talagrand, 17 Aug 2022
Following the comments of the two referees, I would like as Editor to raise a point that I think is important, and possibly critical for acceptance of the paper. It is the symmetric positive semidefinite character of the matrices that are defined in the paper for representing localized covariances and correlations.
As reminded by the authors, a covariance matrix (and in particular a correlation matrix) must be symmetric positive semidefinite (SPSD, meaning without negative eigenvalues). If that condition is not verified in an EnKF, the minimization of the variance of the estimation error that is implicit in the analysis step of the EnKF (as also in variational assimilation) may lead to negative ‘minima’ (actually saddlepoints) and to absurd results.
I understand that all localized correlations determined in the paper are obtained through formulas of the type of Eq. (3), i. e., through Schurmultiplication of an original covariance matrix P by a localization matrix C. For a given localization matrix C, the localized matrix Ploc defined by the Schurproduct will be SPSD for any covariance matrix P, if, and only if, the localization matrix is itself SPSD.
Is the condition that the obtained matrices are SPSD verified in the paper? As a precise example, do the quantities obtained through the EOL minimization Eq. (57) define (as it seems to be the authors’ purpose) a correlation matrix, i.e. an SPSD matrix with 1’s on its diagonal (the remark ((ll. 374375) … the EOL exhibited values larger than one when estimated after applying the SEC suggests that all ‘correlations’ defined in the paper are not proper correlations)?
These questions are not discussed, nor even mentioned, in the paper. I think they should be. It is possible that they have been discussed in previous papers (either by the authors of the present paper or by other authors), where responses that are relevant for the present paper can be found. If so, appropriate references and explanations must be given.
If not, I think it is necessary to check the SPSD character of either the localization matrix C or the localized matrix Ploc (or both). That can be done on the basis of theoretical considerations (it is not clear to me if the correlation matrices defined by the EOL minimization Eq. (57) are even symmetric). Or it can be done alternatively through numerical computations. There are in the present case 4 physical variables and 20 vertical levels, so that the relevant matrices have dimension 80 x 80, of which it must be possible to determine explicitly their full spectrum of eigenvalues. And if that is too costly, it is possible to consider submatrices, for instance by reducing the number of vertical levels.
If matrices that are meant to be SPSD, while being symmetric, turn out not to be SPSD, but with only a small number of small negative eigenvalues, one solution may be to set those eigenvalues to 0 (or to small positive values). If the negative character of the matrices is significant, there will be a real problem, which will have to be solved or at least discussed in depth. It may be that the conclusion of the paper will be that difficulties remain, to be solved in future works.
In any case, I consider as Editor that a proper discussion of those SPSD aspects is critical for acceptance of the paper.
Reply: Thank you for raising this important point. Overall, one central aim of our study was to analyse how an optimal vertical localization should look without given algorithmic constraints. The SPSD requirement is one of those constraints. We agree that it would be helpful to discuss this aspect to make the reader and future studies aware of this constraint. Following your comment, we performed some additional analysis in this regard, which we discuss below. Furthermore, we will discuss the SPSD aspect in the revised manuscript, given its importance for EnKF and variational systems.
The localization matrix C based on the SINGLE EOL is symmetric by definition given Eq. 7. The diagonal contains all 1’s. As suggested, we analysed the C matrix with dimensions 80x80 (see the Appendix/Supplement, Figure 1). In this example, the resulting localization matrix was symmetric but not PSD. The eigenvalues of this C matrix range from 0.5 to 50, while more than half of the eigenvalues are positive. Setting all negative eigenvalues to zero and constructing a C’ matrix that is SPSD (see Figure 2) changes the localization values by up to 15% (see Figure 3).
We will add discussion in Sec. 2.5: “Applying the EOL by construction yields a symmetric but not necessarily a positive semidefinite localization matrix. In our case, the computed localization matrices are not symmetric positive semidefinite (SPSD), which can result in nonSPSD localized covariance matrices. As some DA algorithms require an SPSD covariance matrix (Gaspari and Cohn 1999, Bannister 2008), additional steps would be required to apply the EOL results to such algorithms.”
We will add the following discussion in the conclusion section: “For a serial filter (e.g., the Ensemble Adjustment Kalman Filter (EAKF) by Anderson2001), an EOLbased localization can be applied directly and easily tested in future studies. Yet, each filter can exhibit algorithmspecific requirements for localization. For example, covariances or localization matrices often need to be symmetric positive semidefinite, which the EOL methodology might not fulfil. However, in all cases, EOL results can serve as guidance for finding better localization functions or methods that resemble the results of the EOL but also fulfil the criteria of a symmetric positive semidefinite matrix.”
“(the remark ((ll. 374375) … the EOL exhibited values larger than one when estimated after applying the SEC suggests that all ‘correlations’ defined in the paper are not proper correlations)?”
Reply: In our study, correlations are proper correlations. EOL values larger than one occur when the SEC was previously applied to statistically correct sampling error in correlations. Yet, correlations sometimes are damped too much due to the suboptimal behaviour of the SEC. In this case, the EOL allows the diagnosis of deficiencies in the applied localization approach. EOL values larger than one indicate that the SEC damps correlations too much, while EOL values smaller than one reveal that the SEC did not successfully correct sampling errors.
I add one remark. The authors mention on several occasions (e.g. ll. 4445) distancedependent tapering functions with a cutoff at finite distance. A distancedependent SPSD function that is continuous at the origin (i.e. at distance 0) cannot have a discontinuity elsewhere (that would be inconsistent with the requirement that the correlation between two close points must tend to 1 when the distance between those points tends to 0). It may be that people who have used such ‘cutoff’ functions have not run into difficulties because of the ‘small’ negativity of the corresponding covariancescorrelations, but those functions cannot mathematically be SPSD.
Reply: Thanks for this interesting remark. Indeed, a cutoff could lead to discontinuities. However, our paper only applies and refers to the GaspariCohn (GC) function when discussing cutoffs. The GC function is a piecewise rational function which is continuous (Gaspari and Cohn, 1999). This property explains why people do not run into difficulties caused by negativity when using the GC function that exhibits a cutoff (damps correlations to zero after a defined distance).
We changed the following sentences to be more precise and to make people aware of the importance of continuity:
“Distancedependent localization always requires tuning of localization scales and cutoff distances (deleted).”
“This behaviour motivates most distancebased localization approaches with a predefined tapering function that damp or cutoff (deleted) distant correlations.”
“However, other considerations, e.g., continuity, computational efficiency or matrix rank, also may need to be considered when deciding on a cutoff.”

EC2: 'Reply on AC3', Olivier Talagrand, 19 Sep 2022
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/egusphere2022434/egusphere2022434EC2supplement.pdf

EC2: 'Reply on AC3', Olivier Talagrand, 19 Sep 2022

AC3: 'Reply on EC1', Tobias Necker, 14 Sep 2022
Tobias Necker et al.
Tobias Necker et al.
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