the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Applying prior correlations for ensemble-based spatial localization
- Department of Atmospheric and Oceanic Science, University of Maryland, College Park, United States
- Department of Atmospheric and Oceanic Science, University of Maryland, College Park, United States
Abstract. Localization is an essential technique for ensemble-based data assimilations (DA) to reduce the sampling errors due to limited ensembles. Unlike traditional distance-dependent localization, the correlation cutoff method (Yoshida and Kalnay, 2018; Yoshida 2019) tends to localize the observation impacts based on their background error correlations. This method was initially proposed as a variable localization strategy for coupled systems, but it also can be extensively utilized as a spatial localization. This study introduced and examined the feasibility of the correlation cutoff method as an alternative spatial localization preliminary on the Lorenz (1996) model. We compared the accuracy of the distance-dependent and Abstract. Localization is an essential technique for ensemble-based data assimilations (DA) to reduce the sampling errors correlation-dependent localizations and extensively explored the potential of integrative localization strategies. Our results suggest that the correlation cutoff method can deliver comparable analysis to the traditional localization more efficiently and with a faster spin-up. These benefits would become even more pronounced under a more complicated model, especially when the ensemble and observation sizes are reduced.
Chu-Chun Chang and Eugenia Kalnay
Status: final response (author comments only)
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RC1: 'Comment on npg-2022-5', Zheqi Shen, 26 Mar 2022
This manuscript applies a newly developed spatial localization method (YK18) for the ensemble-based data assimilation using Lorenz (1996) model. This correlation cutoff method is developed in their previous work as a variable localization strategy for coupled systems. They declaimed that it can be further utilized as a spatial localization method. They performed twin experiments with Lorenz 1996 model, and compared the YK18 method with the conventional spatial localization method. Overall, the work is useful and the manuscript is well written. However, some details of the method should be better explained. And I still have questions about the foundation of the method. The authors should answer my questions and make some revisions before it could be accepted for publication. Please see my questions and comments below.1. In section 2.3, the authors imply that “The prior square error correlations are collected from a preceding offline run.” According to section 3.2 (line 185), I realize the offline run is a DA experiment using a relatively large ensemble (50 in this case) without localization for a long period. Of cause it works with the toy models such as the Lorenz model in this work and that in YK18. However, for more complicated models (such as GCMs), it is meaningless to perform DA experiment without localization, and it is impossible to use an ensemble large enough to get rid of localization. That would weaken the argument about the usefulness of the method.2. Equation (6) implies that the temporal mean of the squared correlation over all analysis steps is computed to serve as "prior error correlation" to estimate the localization function. I have some questions about that:
- 2.1 Why do you compute the temporal mean over all analysis steps, instead of all steps including forecast and analysis? Considering the analyses are from the data assimilation without localization, does that indicate the ensemble size is large enough such that the true correlation can be recovered without localization? This is still impossible for large models.
- 2.2 You use the temporal mean of the squared correlation. So does the period of the offline run have an impact on the correlations?
- 2.3 Is the assimilation process necessary in the offline run? I wonder, is it possible to compute the correlation using an EnOI-like idea? i.e., running a single model and computing the correlation using members at different time steps. This seems much more practical for real applications. You have already use the temporal mean in the current method anyway.
3. About Figure 2a. I cannot see any connection between error correlation and observation size from equations (5) and (6). But the connection between error correlation and ensemble size is very clear from equation (5). Do you use this figure to explain the parameters in the offline run?4. The comparison results in figure 5 and figure 6 are not impressive. Though the authors declaim that YK18 can accelerate the spin-up. However, the parameters in GDL and YK18 may be not optimal, so the conclusions are not very persuasive.Detailed comments-
Line 175 "GDL: Distance-dependent localization introduced in Section 2.3.” I think it is section 2.2.
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For equations (1) - (3), there are linear observation operator H, for equation (5), it is a potentially nonlinear operator h(x)
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Line 99 "Equation (4) is a smooth and static Gaussian-like function that offers the same localization effect as the GC99 when applied to LETKF.” It is inaccurate, because GC99 uses a compact-support function, and it cutoff at some distance, but Eq. (4) does not cutoff.
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Line 91 and line 102, whether the R localization multiplies the elements of R inverse or R itself? Please clarify that.
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RC2: 'Comment on npg-2022-5', Anonymous Referee #2, 28 Mar 2022
The authors propose a localization scheme for prior correlations and compare it with the traditional localization scheme based on distance dependence. This localization scheme is of interest for the implementation of ensemble data assimilation methods. The manuscript is quite well written and meets the submission requirements of the journal NPG. Nevertheless, the manuscript has the following issues that need further clarification and improvement.
Specific comments:
- This new localization scheme for YK18 relies heavily on the statistical formulation of Equation (5), so is there any similarity between this formulation and Anderson's work, and what are their similarities and differences? Please elaborate explicitly.
- Does the statistical result of Equation (5) depend on the number of samples? If so, how much does this sample dependence affect the final results?
- Equation (5) counts the correlation coefficients between the model grid points and the observed points, but we know that the observed variables are hardly fixed in their positions at different moments. This situation is especially prominent when assimilating satellite data in NWP. Since the position of the observed data is difficult to be fixed, the observation operator H is actually difficult to be fixed as well. Then how should the correlation coefficients between the model grid points and the observed points, which are calculated by Eq. (5), be applied to other moments?
- Similarly, the model in the validation experiment given so far is very simple, with only one variable. For a true NWP model, there are perhaps multiple model state variables such as U, V, P, T, Q, on the same model grid point. And due to the use of different grid schemes, these variables may not appear at the same location of the grid. So how to use Eq. (5) for statistics in this case and apply it to the real situation?
- The authors elaborate that one of the advantages of YK18 is that it is more computationally efficient. However, it can be seen from their analysis that in fact YH18 should essentially provide some new calculations of localization correlation matrices as well, so why does it make the improvement of computational efficiency?
- As for the "a faster spin-up" proposed in the manuscript, I do not quite understand it. The purpose of our data assimilation is to give a more accurate initial field and then drive the model to forecast. The spin-up seems to be more appropriate in the simulation of climate models.
- It seems that Section 2.3 of GDL appearing in Page7 should be Section 2.2.
REFERENCES:
- Anderson, J. L. (2007). Exploring the need for localization in ensemble data
assimilation using a hierarchical ensemble filter. Phys. D 230, 99–111. doi:
10.1016/j.physd.2006.02.011
- Anderson, J. L., and Lei, L. (2013). Empirical localization of observation impact
in ensemble Kalman filters. Mon. Weather Rev. 141, 4140–4153. doi: 10.1175/MWR-D-12-00330.1
Chu-Chun Chang and Eugenia Kalnay
Chu-Chun Chang and Eugenia Kalnay
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