Reply on RC1

queries and


Korea -KIAPS-LETKF -50 members DWD -LETKF -40 members
2) l100-105, the description of BCs is confusing. Is the GEFS 20-member analysis used 50 times to get 1000 BCs? Or climatological GEFS is sampled for 1000 BCs? Reply: The 20-member 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 20-member 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 mid-tropospheric temperatures exhibited more upperlevel clouds resulting in colder near-surface temperatures that likely are caused by stronger cloud shadowing. We will add the following comment: "This weak correlation is linked to cloud shadowing by mid-tropospheric clouds and resulting colder near-surface temperatures."

4) Eq. (5), is s the index for 40-member 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 40-member sub-samples (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 1000-member 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 3-5, 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 x-axis (extended x range) in Figs 3-5, 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) l258-260, 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 variable-dependent 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."