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Assessing the performance of the CFSR by an ensemble of analyses

Abstract

The Climate Forecast System Reanalysis (CFSR, Saha et al. in Bull Am Meteor Soc 91:1015–1057, 2010) is the latest global reanalysis from the National Centers of Environmental Prediction (NCEP). In this study, we compare the CFSR tropospheric analyses to two ensembles of analyses. The first ensemble consists of 12 h analyses from various operational analyses for the year 2007. This ensemble shows how well the CFSR analyses can capture the daily variability. The second ensemble consists of monthly means from the available reanalyses from the years 1979 to 2009 which is used to examine the trends. With the 2007 ensemble, we find that the CFSR captures the daily variability in 2007 better than the older reanalyses and is comparable to the operational analyses. With the ensemble of monthly means, the CFSR is often the outlier. The CFSR shows a strong warming trend in the tropics which is not seen in the observations or other reanalyses.

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Acknowledgments

Craig Long supplied the sonde data for Singapore and Accension Island. Suru Saha and the rest of the CFSR team for creating and supplying the data for the CFSR. Emily Becker downloaded much of the JRA-25 and MERRA data. Bhaskar Jha for supplying the CFSR-AMIP simulations and the reviewers for their comments which improved this paper.

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Correspondence to Wesley Ebisuzaki.

Appendix

Appendix

Part A: expected values of the correlations in Table 3

Suppose the ith ensemble member can be written as Ai + T where T is the truth. Then the ensemble mean can be written as M = (1/n) ∑Ai + T where n is the number of ensemble members. Assuming that the Ai are statistically independent, unbiased and have the same variance (σ2), then

$$ \begin{aligned} & {\text{E}}\left( {{\text{A}}_{\text{i}} } \right) = 0 \\ & {\text{E}}\left( {{\text{A}}_{\text{i}} {\text{A}}_{\text{i}} } \right) = \sigma^{2} \\ & {\text{E}}\left( {{\text{A}}_{\text{i}} {\text{A}}_{\text{j}} } \right) = 0\,{\text{for}}\,{\text{i}} \ne j \\ \end{aligned} $$

The temporal correlation of the perturbations from the ensemble mean can be written as

$$ {{{{\uprho}}_{\text{ij}} = \sum \left[ {\left( {{\text{A}}_{\text{i}} - {\text{M}}} \right)\left( {{\text{A}}_{\text{j}} - {\text{M}}} \right)} \right]} \mathord{\left/ {\vphantom {{{{\uprho}}_{\text{ij}} = \sum \left[ {\left( {{\text{A}}_{\text{i}} - {\text{M}}} \right)\left( {{\text{A}}_{\text{j}} - {\text{M}}} \right)} \right]} {\left( {\sum \left( {{\text{A}}_{\text{i}} - {\text{M}}} \right)^{2} \sum \left( {{\text{A}}_{\text{j}} - {\text{M}}} \right)^{2} } \right)^{1/2} }}} \right. \kern-\nulldelimiterspace} {\left( {\sum \left( {{\text{A}}_{\text{i}} - {\text{M}}} \right)^{2} \sum \left( {{\text{A}}_{\text{j}} - {\text{M}}} \right)^{2} } \right)^{1/2} }} $$

where the values are summed over the samples. Now the expected value of ∑(Ai−M)2 is

$$ {\text{E}}\left( {\sum \left( {{\text{A}}_{\text{i}} - {\text{M}}} \right)^{2} } \right) = {{\text{E}}\left( {\sum \left( {{\text{A}}_{\text{i}}^{2} } \right)} \right)} - {\text{E}}\left( {2\sum {\text{A}}_{\text{i}} {\text{M}}} \right) + {\text{E}}\left( {\sum {\text{M}}^{2} } \right) = {\text{m}}\left( {1 - 1/{\text{n}}} \right)\sigma^{2} $$

where m is the number of samples

The expected value of ∑ [(Ai−M)(Aj−M)] when i ≠ j is

$$ {\text{E}}\left( {\sum \left[ {\left( {{\text{A}}_{\text{i}} - {\text{M}}} \right)\left( {{\text{A}}_{\text{j}} - {\text{M}}} \right)} \right]} \right) = {\text{E}}\left( {\sum \left( {{\text{A}}_{\text{i}} {\text{A}}_{\text{j}} } \right)} \right) - {\text{E}}\left( {{\text{A}}_{\text{i}} {\text{M}}} \right) - {\text{E}}\left( {{\text{A}}_{\text{j}} {\text{M}}} \right) + {\text{E}}\left( {{\text{M}}^{2} } \right) = - {\text{m}}/{\text{n }}\sigma^{2} $$

So the expected value of the temporal correlation (when the time series is long) is −1/(n−1). For the high-frequency ensemble, n = 5 and the expected correlation of −0.25 which is in the middle of the range of calculated values in Table 3.

Part B: the expected values of the correlations in Table 5

Let AEC represent the EC analyses.

Let Ai i = 1,…,4 represent the CFSR, CMC, FNO, UK analyses

Assume,

$$ \begin{aligned} & {\text{E}}\left( {{\text{A}}_{\text{i}} {\text{A}}_{\text{i}} } \right) = \sigma_{\text{i}}^{2} \\ & {\text{E}}\left( {{\text{A}}_{\text{i}} {\text{A}}_{\text{j}} } \right) = 0\,{\text{for}}\,{\text{i}} \ne {\text{j}} \\ & {\text{E}}\left( {{\text{A}}_{\text{i}} {\text{A}}_{\text{EC}} } \right) = 0 \\ & {\text{E}}\left( {{\text{A}}_{\text{EC}} {\text{A}}_{\text{EC}} } \right) = \sigma_{\text{EC}}^{2} \\ \end{aligned} $$

The correlation between the terms in Table E is given by

$$ {{{{\uprho}}_{\text{ij}} = \sum \left[ {\left( {{\text{A}}_{\text{i}} - {\text{A}}_{\text{EC}} } \right)\left( {{\text{A}}_{\text{j}} - {\text{A}}_{\text{EC}} } \right)} \right]} \mathord{\left/ {\vphantom {{{{\uprho}}_{\text{ij}} = \sum \left[ {\left( {{\text{A}}_{\text{i}} - {\text{A}}_{\text{EC}} } \right)\left( {{\text{A}}_{\text{i}} - {\text{A}}_{\text{EC}} } \right)} \right]} {\left( {\sum \left( {{\text{A}}_{\text{i}} - {\text{A}}_{\text{EC}} } \right)^{2} \sum \left( {{\text{A}}_{\text{j}} - {\text{A}}_{\text{EC}} } \right)^{2} } \right)^{1/2} }}} \right. \kern-\nulldelimiterspace} {\left( {\sum \left( {{\text{A}}_{\text{i}} - {\text{A}}_{\text{EC}} } \right)^{2} \sum \left( {{\text{A}}_{\text{j}} - {\text{A}}_{\text{EC}} } \right)^{2} } \right)^{1/2} }} $$

Now for i ≠ j, the expected value of ∑ [(Ai−AEC)(Aj−AEC)] is m·σ 2EC and the expected valued of ∑(Ai−AEC)2 is m(σ 2i  + σ 2EC ). So the expected value of ρij when the m is large is

$$ {\text{E}}({{\uprho}}_{\text{ij}} ) = \sigma_{\text{EC}}^{2} \left[ {\left( {\sigma_{\text{i}}^{2} + \sigma_{\text{EC}}^{2} } \right)\left( {\sigma_{\text{j}}^{2} + \sigma_{\text{EC}}^{2} } \right)} \right]^{ - 1/2} $$

Using σ 2i estimated by Column A of Table 4, the expected values of the correlation when deviations from “truth” are random can be calculated are shown in parenthesis in Table 5.

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Ebisuzaki, W., Zhang, L. Assessing the performance of the CFSR by an ensemble of analyses. Clim Dyn 37, 2541–2550 (2011). https://doi.org/10.1007/s00382-011-1074-5

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Keywords

  • CFSR
  • CFSRR
  • Reanalysis
  • Ensemble