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Shunji Kotsuki
Craig H. Bishop


Recent numerical weather prediction systems have significantly improved medium-range forecasts by implementing hybrid background error covariance, for which climatological (static) and ensemble-based (flow-dependent) error covariance are combined. While the hybrid approach has been investigated mainly in variational systems, this study aims at exploring methods for implementing the hybrid approach for the local ensemble transform Kalman filter (LETKF). Following Kretschmer et al., the present study constructed hybrid background error covariance by adding collections of climatological perturbations to the forecast ensemble. In addition, this study proposes a new localization method that attenuates the ensemble perturbation (Z-localization) instead of inflating observation error variance (R-localization). A series of experiments with a simplified global atmospheric model revealed that the hybrid LETKF resulted in smaller forecast errors than the LETKF, especially in sparsely observed regions. Due to the larger ensemble enabled by the hybrid approach, optimal localization length scales for the hybrid LETKF were larger than those for the LETKF. With the LETKF, the Z-localization resulted in similar forecast errors as the R-localization. However, Z-localization has an advantage in enabling us to apply different localization scales for flow-dependent perturbation and climatological static perturbations with the hybrid LETKF. The optimal localization for climatological perturbations was slightly larger than that for flow-dependent perturbations. This study also proposes optimal eigendecomposition (OED) ETKF formulation to reduce computational costs. The computational expense of the OED ETKF formulation became significantly smaller than that of standard ETKF formulations as the number of climatological perturbations was increased beyond a few hundred.

Open access
Shunji Kotsuki
Steven J. Greybush
, and
Takemasa Miyoshi


With the serial treatment of observations in the ensemble Kalman filter (EnKF), the assimilation order of observations is usually assumed to have no significant impact on analysis accuracy. However, Nerger derived that analyses with different assimilation orders are different if covariance localization is applied in the observation space. This study explores whether the assimilation order can be optimized to systematically improve the filter estimates. A mathematical demonstration of a simple two-dimensional case indicates that different assimilation orders can cause different analyses, although the differences are two orders of magnitude smaller than the analysis increments if two identical observation error variances are the same size as the two identical state error variances. Numerical experiments using the Lorenz-96 40-variable model show that the small difference due to different assimilation orders could eventually result in a significant difference in analysis accuracy. Several ordering rules are tested, and the results show that an ordering rule that gives a better forecast relative to future observations improves the analysis accuracy. In addition, the analysis is improved significantly by ordering observations from worse to better impacts using the ensemble forecast sensitivity to observations (EFSO), which estimates how much each observation reduces or increases the forecast error. With the EFSO ordering rule, the change in error during the serial assimilation process is similar to that obtained by the experimentally found best sampled assimilation order. The ordering has more impact when the ensemble size is smaller relative to the degrees of freedom of the dynamical system.

Open access