Guidance on how to improve vertical covariance localization based on a 1000-member ensemble

. 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 convection-permitting 1000-member ensemble simulation. 1000-member ensemble correlations serve as truth for examining vertical correlations and their sampling error. We discuss 5 requirements for vertical localization by deriving an empirical optimal localization (EOL) that minimizes the sampling error in 40-member sub-sample correlations with respect to the 1000-member 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 cross-correlations). Comparing the empirical optimal localization with common distance-dependent localization approaches highlights that finding suitable localization 10 functions bears substantial room for improvement. Furthermore, we discuss the gain of combining different localization approaches with an adaptive statistical sampling error correction.

Weak pressure gradients and slowly moving convective cells resulted in high local precipitation rates and flash flooding. Due to these severe weather events, several studies focused on this exceptional period (e.g., Piper et al., 2016). Necker et al. (2020a, b), 125 Nomokonova et al. (2022), andCraig et al. (2022) provide further details on the weather situation in this period as these studies also explore the 1000-member ensemble simulation with a different purpose.

Vertical localization
Error covariances are a key component in data assimilation and determine how assimilated information is weighted and distributed in state space. Given a sample of state vectors x provided by a background forecast ensemble the flow-dependent 130 sample error covariance matrix P can be computed as follows where N is the ensemble size and x is the ensemble mean state. The covariance matrix P per definition is a symmetric, positive semi-definite matrix with variances on its diagonal and covariances on its off-diagonal entries. Each off-diagonal element contains a sample covariance cov of two state variables x i 135 cov(x 1 , x 2 ) = r(x 1 , x 2 )σ(x 1 )σ(x 2 ) where r ∈ [−1, 1] is the sample correlation and σ the sample standard deviation.
Usually, the number of affordable ensemble members is limited in NWP due to a huge state space and computational restrictions. This deficit causes severe sampling errors. Consequently, all ensemble filters require a correction of sampling errors, often referred to as localization. For example Anderson (2012) highlighted that the sampling error in covariances is 140 dominated by sampling error in the sample correlation r, not by sampling error in the variances. Therefore, our analysis will solely tackle sampling errors in sample correlations. Sample correlations are normalized with standard deviations and possess no unit. The normalization allows comparing or combining correlations of different variables facilitating the interpretation.
The implementation of localization depends on various factors determined by the type of ensemble filter. Usually, localization is applied directly to the background error covariance matrix using a Schur-product where C is the localization matrix. The matrix C consists of tapering factors α that are determined using the localization approach of choice.

Distance-dependent localization
The most common localization approach is a distance-dependent localization that determines tapering factors α based on 150 distance Mitchell, 1998, 2001). The vertical separation distance in our study is defined in ln(p). We consider the widely used Gaussian-shaped Gaspari-Cohn function (GC; Eq. 4.10, Gaspari and Cohn, 1999) for comparison with other methods. Applying a GC function always requires the selection of the separation distance. The separation distance is often referred to as the localization scale, while the cut-off radius is usually twice the localization scale. In our study, we apply vertical localization according to the definition of Deutscher Wetterdienst (DWD) (Schraff et al., 2016). For DWD, the localization scale 155 is determined by a pre-selected localization length that is multiplied by a factor of ( 10/3). Operationally, the localization length of DWD is height-dependent and increases linearly in ln(p) from the surface (0.075) to 300 hPa (0.5).
In Sect. 3.3, we apply three different domain-uniform GC localization setups: a) "GC": An optimally tuned GC localization scale that applies a uniform localization scale for all variables and heights. b) "GCLEV": An height-dependent optimally tuned GC localization scale that is uniform for all variables. c) "DWD": A localization setting similar to DWD as described above 160 that is also domain-and variable-uniform.

Sampling error correction (SEC)
Necker et al. (2020b) showed that an adaptive statistical sampling error correction (SEC, Anderson, 2012Anderson, , 2016 substantially reduces the sampling error in sample correlations and ensemble sensitivities. The SEC is a look-up

Sub-sampling and vertical correlations
The sampling noise expected for zero correlation estimates and sample size N is ( (Houtekamer and Mitchell, 1998).
For the 1000-member ensemble (N=1000), this estimation yields a very small sampling noise of approximately 3%. In comparison, a 40-member ensemble reveals an expected sampling noise of approximately 16%. Throughout this study, correlations computed using the full 1000-member ensemble serve as truth (r 1000 ) for the interpretation of vertical correlations and the 175 evaluation of sampling errors and localization in smaller subsamples of the full ensemble. We focus on vertical correlations and sampling errors in 40-member subsamples as this is a typical ensemble size applied by operational weather services such as, e.g., Deutscher Wetterdienst. Preceding studies applied a similar approach for studying sampling errors (Hamill et al., 2001;Poterjoy et al., 2014;Bannister et al., 2017;Necker et al., 2020a, b).
The present study will adopt the sub-sampling approach from Necker et al. (2020a) and Craig et al. (2022). The 1000-180 member ensemble provides 25 random 40-member subsamples with unique members (illustrated in Fig. 1 (a)). We assume that the 40-member sub-ensembles of the 1000-member ensemble statically have sampling errors similar to those independent 40-member ensemble EnKF systems would have. As mentioned above, we will analyze ten 3 h forecasts. This setup results in a sample of 250 ensemble forecasts with 40 members that we can compare to the ten ensemble forecasts with 1000 members.
The model domain has 250 × 230 grid points yielding 57.500 vertical columns in our domain. We will, therefore, analyze 185 approximately 11.5 × 10 6 true and 287.5 × 10 6 40-member vertical correlation profiles per variable pair, accounting for all 20 reference levels. This data set allows robust statistical analysis of error correlations, but it should be noted that error correlations may differ for other periods and regions.
In the present study, we will analyze four prognostic variables: temperature (T), specific humidity (Q), zonal wind (U), and meridional wind (V). This setup yields 16 correlation pairs (Tab. 1) that we will inspect for different reference levels.  thermore, we will group correlations in "self" (e.g., temperature-temperature as shown in Fig. 1) or "cross" (e.g., temperaturehumidity) correlations for highlighting common behavior. Subsequently, we will use the correlation coding shown in Tab. 1.
For example, "TQ" combines all temperature correlations from the reference level to specific humidity at all other vertical levels in a column. Throughout the manuscript, we will mainly present results for the U-wind component as conclusions for the V-wind component are similar.

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Example of vertical correlations Fig. 1 (a) shows an example of vertical self-correlations of temperature (TT) from reference level 500 hPa to all other levels in a single random vertical column. The 1000-member correlation (also referred to as true correlation) is one at the reference level and drops to half after approximately 100 hPa vertical distance. Given the true correlation, the temperature at 500 hPa weakly correlates with the temperature in the boundary layer. This weak correlation is linked to cloud shadowing by mid-tropospheric clouds and resulting colder near-surface temperatures. Almost no correlation is 200 visible to levels above the tropopause, which lies around 200 hPa. Most 40-member sample correlations strongly deviate from the true correlation, highlighting the severe under-sampling issue. Sampling errors appear to be larger with increasing distance and smaller correlation values. This behavior motivates most distance-based localization approaches with a predefined tapering function that damp distant correlations. However, such an approach might cut off significant non-zero correlations, as seen for the boundary layer close to the surface in this example.

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Throughout this manuscript we will analyze the 1000-member horizontally averaged absolute vertical correlation to support the discussion of the empirical optimal localization. Averaged absolute correlations are computed as follows: where K is the number of vertical columns in the domain. This analysis will be done separately for different forecasts t, reference levels z, pressure levels p, and variable pairs A.

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This behavior explains why distance-dependent vertical localization scales are defined in logarithmic pressure coordinates.

Empirical optimal localization (EOL)
Our goal is to empirically find the optimal localization factor α that minimizes the sampling error or cost function J where the minimization is done separately for each forecast time t, reference level z, pressure level p, and variable pair A. This 220 is equivalent to finding the α that minimizes Taking a derivative with respect to α and finding the minimum gives us In other words, the empirical optimal localization (EOL) minimizes the Root Mean Square Difference (RMSD) between 225 the 1000-member correlation and all 25 40-member sub-sample correlations for a chosen setting. For technical reasons, we minimized the cost function using the Brents method as implemented in scipy.optimize (Virtanen et al., 2020). Note that the range of localization is not confined to [0, 1], which means that the EOL could inflate correlations if required. 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. Negative EOL values can be observed when the EOL is computed for a 230 small correlation sample (e.g., a single vertical column), which is not the case in the present study. However, we suggest setting negative EOL values to zero in case they might occur.
Applying the EOL by construction yields a symmetric but not necessarily a positive semi-definite localization matrix. In our case, the computed localization matrices are not symmetric positive semi-definite (SPSD), which can result in non-SPSD localized covariance matrices. As some DA algorithms require SPSD covariance matrices (Gaspari and Cohn, 1999;Bannister, 235 2008), additional steps would be required to apply the EOL results to such algorithms.
Our approach for empirically estimating localization is inspired by Lei and Anderson (2014) who compare two methods: The Global Group Filter (GGF) and Empirical Localization Functions (ELF). The GGF minimizes the RMS difference between the estimated regression coefficients in subsets of the ensemble using a hierarchical ensemble filter (Anderson, 2007;Lei et al., 2016). ELFs are derived from an Observing System Simulation Experiment (OSSE) by minimizing the RMS difference 240 between the true values of the state variables and the posterior ensemble mean (Anderson and Lei, 2013). In contrast to ELFs, the GGF and EOL purely judge localization based on ensemble sampling error without an OSSE. Furthermore, in contrast to the GGF, the EOL assumes the large ensemble correlation as truth for minimizing the sampling error. The EOL presented in our study corresponds to a non-adaptive distant-dependent domain-uniform vertical localization that is common for operational convective-scale regional data assimilation systems.
245 Figure 1 (c) displays the EOL (α(p)) as estimated for the example of TT correlations introduced above and reference level 500 hPa. The domain-uniform EOL equals one at the reference level 500 hPa as no correction is needed. The EOL appears broader and follows the shape of the mean absolute correlation. For example, this localization behavior was also described by Flowerdew (2015). The error reduction is largest for weak and distant correlations.

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This section presents mean absolute 1000-member vertical correlations and EOLs for various settings. First, we will evaluate how vertical localization for various single variable pairs should be constructed. Afterward, we will group variable pairs based on similar behavior. Finally, at the end of the results section, we will evaluate the error reduction of all discussed localization approaches, including combinations with the SEC.

Vertical localization for single variable pairs 255
As discussed in Sect. 2.4, the domain-averaged absolute vertical correlation can aid the interpretation of the EOL. For this reason, we will first evaluate the mean absolute vertical correlation and then the EOL. Figure 2 shows the mean absolute vertical correlation for all possible variable combinations and reference level 500 hPa. Self-correlations of the same variable all peak at the reference level. In contrast, cross-correlations are weaker and do not always exhibit a maximum correlation at 500 hPa. The TU correlation, for example, peaks around the tropopause, while the UT correlation reveals a minimum at that 260 height. The mean vertical correlation length is variable dependent, shortest for specific humidity and longest for wind. The domain-averaged absolute vertical correlation only exhibits a fairly small variability within the five-day experimental period.
The variability between day to night also appears to be small. Results could, however, differ for other conditions or seasons, e.g., situations with strong atmospheric stability.
Next, we focus on the EOL derived for 40-member subsamples from all forecasts. Figure 3 displays the EOL for all variable 265 combinations and reference level 500 hPa. The EOL depends on the prevailing correlation but has a different shape and vertical extent. As seen for the single-forecast example in Sect. 2.5, weaker correlations are more affected by sampling errors and require stronger correction. Consequently, all cross-correlations require a stronger localization. The localization for crosscorrelations reveals an amplitude smaller than one at the reference level. Given this behavior, tapering functions for crosscorrelations should not be one at zero distance when applying a distance-dependent localization. Self-correlations are less 270 affected by sampling error and require only a weaker correction, especially close to the reference level. EOLs for humidity correlations all peak at the reference level 500 hPa ( Fig. 3(a)). However, temperature and wind EOLs behave differently (Fig. 3(b,c)) and do not peak at the reference level following the correlation pattern. (Fig. 2(b,c)). For example, the TU EOL peaks around the tropopause, where winds are typically strongest. Wind correlations (e.g., UU; Fig. 3(c))) require only a small correction. The EOL for UV correlations is almost constant with height and does not show a distinct maximum.

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All self-and cross-correlations involving humidity peak at the reference level 500 hPa (for example, see UQ localization).
Overall, the variability of domain-averaged correlations from forecast to forecast is small (Fig. 2). EOLs exhibit a larger variability than domain-averaged correlations. For most variables, the variability is larger close to the surface, especially for temperature correlations (Fig. 3(b)). Results should be treated with caution where changes of the EOL with height are smaller than the variability from forecast to forecast.

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Subsequently, we will discuss the EOL for two additional reference levels to highlight changes in height within the troposphere. Figure 4 shows the EOL for a reference level 300 hPa. For reference level 300 hPa, EOLs appear to be broader compared to 500 hPa. This height-dependence is in line with larger vertical correlation length scales found for the upper troposphere in contrast to the lower troposphere, boundary layer, or close to the surface. Similar to other reference levels in the middle and upper troposphere, EOLs for correlations between wind and temperature reveal a maximum (TU; Fig. 4(b)) and 285 minimum (UT; Fig. 4(c)) above the tropopause level.
All reference levels within the boundary layer show similar behavior of the EOL (see, for example, Fig. 5 using reference level 900 hPa). The EOL shows a narrow optimal vertical localization for reference levels close to the surface. In contrast to higher reference levels, also the EOL of cross-correlations peaks at the reference level (Fig. 5). The EOL drops to different   distances where the EOL converges to a small constant value could indicate suitable cut-off distances. A common aspect of the choice of cut-off distance is the signal-to-noise ratio that depends on the ensemble size and correlation strength.

Error reduction for different variables 295
Assessing the EOL for single variable pairs revealed several requirements for vertical localization. Now, we evaluate the error reduction by the EOL, considering each possible correlation pair separately. The 1000-member ensemble correlation serves as truth to compute the RMSD of each 40-member sub-sample correlation. Figure 6 displays the RMSD before and after applying the EOL. The applied EOL varies for each forecast and height level for the error evaluation. The final result shows the average RMSD of all 40-member subsamples, forecasts, and height levels. The results can be interpreted as a benchmark of 300 the maximum possible correlation error reduction achieved by a domain-uniform height and variable-dependent localization.
Note that results for optimizing the analysis may lead to different optimal localization values under some circumstances, but this analysis is beyond the scope of this paper.
The sampling error of the 40-member correlation of most correlations lies within the expected range and close to ( √ 40) −1 (Fig. 6). Self-correlations exhibit a smaller sampling error as, on average, they are stronger and less affected by spurious cor-305 relations. The error reduction achieved by the EOL ranges approximately from 10 to 40%, depending on the variable pair. The QQ self-correlation benefits most from localization, whereas the UU self-correlation benefits the least. Correlations involving humidity are weaker and, therefore, benefit most from localization. On the other hand, correcting temperature correlations seems most challenging. Temperature correlations exhibit the largest RMSD, even after applying the EOL. The error is larger than for wind correlations, which is surprising considering a larger correlation strength and length for wind. This result could different localization scales within the domain depending on, e.g., vertical velocity. First tests showed promising results for such a situation-dependent approach, but a thorough evaluation will be left for subsequent study.

Vertical localization for grouped variable pairs
315 Some operational DA systems apply a uniform distance-based vertical localization that does not change with time, height, variable, or observation type. In this case, appropriate localization needs to meet several requirements using a suitable uniform localization approach. Results in Sect. 3.1 showed that cross-correlations systematically behave differently than self-correlations.
For this reason, we will now evaluate the mean absolute correlation and EOL of three groups of variables: self, cross, or all correlations combined. Derived EOLs now minimize the sampling error for all gathered correlations of each group. 320 Fig. 7 displays the mean absolute correlation for the three groups of correlations. The results show the average correlation and its variability over the ten forecasts. Self-correlations again highlight the height dependence of the vertical correlation length and always exhibit a peak that is one. Cross-correlations are weaker and only exhibit a narrow peak at the reference level. For all correlations combined, the peak amplitude is closer to the peak of cross-correlations as there are more crossthan self-correlations. Combining all correlations or only cross-correlations results in a peak amplitude smaller than one at the 325 reference level.
In contrast, the peak amplitude of the EOL for all correlations is closer to the peak of self-correlation (Fig. 8). The shape of EOLs substantially differs from the single variable pair cases. The EOL is weaker due to wind correlations that account for half of all correlations. The change in the shape of the EOL indicates that different tapering functions could be needed for different variables. Minimizing the error for grouped correlations, the strength of the EOL is always weaker than 0.4. Finally, domain 330 averaged absolute correlations reveal a small variability from forecast to forecast (Fig. 7). The same applies to EOLs. Only the EOL of self-correlations exhibits a slightly larger variability, especially far from the reference level (Fig. 8).

Setting
As discussed in Sect. 3.1, the maximum reduction of sampling errors achieved by an EOL ranges from 11 to 44 % depending 335 on the variable pair. Now, we will compare the performance of the EOL with different localization setups that use two common localization approaches, a distance-dependent localization using a Gaspari-Cohn tapering function (GC; Houtekammer1998, Gaspari1999) and a statistical sampling error correction (SEC; Anderson 2012). Furthermore, we investigate the benefit of combining non-adaptive localization approaches with the adaptive SEC. Compared to Sect. 3.1, the improvement will be evaluated using 1000-member correlations from independent background forecasts. Again we will analyze the improvement 340 relative to uncorrected 40-member ensemble sub-sample correlations (REF40, Fig. 9). The first eight forecasts (29th May to 1st June 2016) serve as training data for estimating EOLs. Similarly, localization scales for distance-dependent localization are tuned using the same training period. We then verified the performance using the last two independent forecasts on 2nd June 2016.

Empirical optimal localization (EOL)
345 Figure 9 displays the error reduction achieved by all considered vertical localization setups. REF40 shows the RMSD found when modeling error correlations using small 40-member ensembles without localization. First, we will evaluate the performance of different EOL settings. Applying a different EOL for each variable pair and height (as presented in Sect. 3.1) gives the largest error reduction of all setups (SINGLE, 26.7 %). Only small differences are visible between day and night time. Using different EOLs only for self-and cross-correlations leads to a slightly reduced performance but still gives about 23 % error 350 reduction (SELF). Finally, applying an EOL that was estimated for all correlations at once reduces the error by 17 % (ALL).
Given these results, treating variable pairs, self-or cross-correlations differently enables substantial improvements. Besides, we tested the error reduction for applying the EOL estimated for self-correlations to both self-and cross-correlations 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 benefit of treating self-and cross-correlations differently.

Distance-dependent localization (GC)
Now, we will compare the performance of EOLs to three different domain-uniform distance-dependent localization approaches using Gaspari-Cohn functions (GC). Sec 2.3.1 lists details on all considered GC setups. We will first evaluate two optimal GC localization setups with tuned localization scales and then apply a localization similar to DWD. The first GC setup uses a uniform localization scale for all levels and variable pairs (GC), and the second setup uses a height-dependent optimal 360 localization scale that changes with the reference level (GCLEV). The optimally tuned GC reduces the sampling error by about gain of the height-dependent localization is partly associated with a sub-optimal shape of a Gaussian-shaped tapering function, given the error reduction achieved by the uniform EOL (ALL). This comparison highlights that finding suitable tapering scales and functions bears great potential for improving vertical localization.

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In contrast, a vertical localization constructed similar to the regional DA system of Deutscher Wetterdienst (DWD) increases the difference of the 40-member ensemble correlation with respect to the 1000-member ensemble. The increased difference originates from the damping of meaningful error correlations. The DWD system employs a LETKF that uses observation-space localization, tuned to function in all seasons and weather situations that may differ from our investigation period. Furthermore, it needs to be considered that localization in the LETKF also affects the degrees of freedom of the analysis (Hotta and Ota, Now, we will evaluate the benefit of using a look-up table-based sampling error correction (SEC) that adjusts correlations based on predefined statistical assumptions. The SEC is an adaptive localization approach that corrects sampling errors as a function of the correlation value. Therefore, the SEC applies an individual correction for each correlation within the domain.
An adaptive localization (SEC) achieves 17.5 % error reduction and outperforms a optimal domain-uniform GC localization.
The SEC exhibits a similar error reduction as seen for ALL but can not outperform the SELF or SINGLE setup. An optimal 380 domain-uniform localization can compete with an adaptive statistical sampling error correction for the evaluated period.

Combined approaches
Finally, we investigate the benefit of combining the statistical SEC with an EOL or a distance-dependent localization. For this analysis, EOLs have been estimated after applying the SEC to highlight the maximum error reduction achieved by combining SEC with an optimal localization. The localization scale of the distance-dependent localization is kept the same as for the GC 385 setup to emphasize required changes for the localization scale. SEC+GC reveals a similar performance as the SEC alone but outperforms the GC setup. Combining SEC with GC requires a re-tuning of localization scales to larger values (not shown).
Combining the SEC with a uniform EOL (SEC+ALL) reduces the sampling error by about 20 %. However, combining the SEC with the SELF or SINGLE EOL derogates the error reduction. The poor performance could originate from sub-optimal assumptions made in the derivation of the SEC (Anderson, 2016;Necker et al., 2020b). For example, the EOL exhibited values 390 larger than one when estimated after applying the SEC. This inflation compensated for an over-correction of sampling errors by the SEC, especially close to the reference level (not shown). In this study, we apply the most general SEC look-up table as provided in the Data Assimilation Research Test (DART; Anderson et al., 2009), which assumes that each correlation value is equally likely. Studying more informed prior assumptions in the SEC may lead to better results but is beyond the scope of the present study.

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-Domain-uniform localization: An tuned uniform distance-dependent localization using Gaspari-Cohn functions reduces the sampling error by about 10 %. Using tapering functions with an optimal shape could improve the localization 430 substantially. The maximum error reduction was found for domain-uniform, variable, and height-dependent EOLs with about 27 % improvement. Distinguishing between self-and cross-correlations leads to a similar but slightly smaller error reduction.
-Adaptive localization: A statistical sampling error correction (SEC) achieves similar error reduction as a variableand domain-uniform localization. Combining the SEC with a Gaspari-Cohn localization improves the error reduction.

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However, combining distance-dependent and statistical approaches requires re-tuning of localization scales. Combining SEC and EOLs led to an over-correction of correlations, which slightly degraded the error reduction. This change could be related to sub-optimal prior assumptions when deriving SEC, as discussed by Anderson (2016) and Necker et al. (2020b).
Our results allow a better understanding of the requirements for vertical localization. When employing these conclusions,

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it is important to consider the specific demands of different ensemble filter algorithms. In ensemble transform Kalman filters, localization increases the degrees of freedom of the analysis and thereby enables the assimilation of more observations (Hotta and Ota, 2021). Furthermore, our evaluation excluded considerations about the rank of the error covariance matrix and computational efficiency. Hence, our findings might need to be adapted to improve the analysis performance depending on the data assimilation system. Localization in operational NWP has many system-dependent requirements and is tuned to avoid 445 bad signal-to-noise ratios during assimilation. For example, while we find no strong support for a vertical cut-off within the troposphere for some variables, this could be beneficial due to the reasons discussed above.
Our study solely judges localization based on ensemble sampling error, assuming the 1000-member ensemble correlation as truth. It is difficult to predict the number of ensembles needed to apply our method, as it will vary for differing scenarios.
However, we do not expect our results to change drastically if we had a larger ensemble. Besides, it would be interesting 450 to compare the EOL with the ELF or GGF approach. For example, comparing ELF and EOL could allow to investigate other error sources in the assimilation that can influence localization (Anderson and Lei, 2013). However, a proper comparison would require an OSSE with a sufficiently large ensemble.
We have found robust results for a mid-latitude convective summer period. The ever-increasing computational capabilities will enable extended data sets and a higher vertical resolution that is comparatively coarse in the current setup. Furthermore, our 455 approach can be easily applied to other large ensemble simulations to study additional aspects, including horizontal localization.
Extending this analysis is desirable given that localization can depend on the underlying weather condition (Lei et al., 2015;Destouches et al., 2021). For example, using a global simulation with a higher model top would allow studying different geographical regions, seasons, and stratospheric correlations that are particularly important for satellite data assimilation (Lei et al., 2018;Scheck et al., 2020).
localization. For example, covariances or localization matrices often need to be symmetric positive semi-definite, 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 semi-definite 465 matrix.
Code and data availability. Code and processed data such as derived empirical optimal localizations are be shared via zenodo: (Necker, 2022). The 1000-member ensemble data-set and derived covariances and correlations (approximately 60 TB of data) are too large for an upload but available upon request.
Author contributions. Competing interests. Co-author Takemasa Miyoshi is editor at the copernicus journal Nonlinear Processes in Geophysics (NPG). However, the peer-review process was guided by an independent editor. The authors have no other competing interests to declare.