Abstract
Abstract. Localization is widely used in data assimilation schemes to mitigate the impact of sampling errors on ensemble-derived background error covariance matrices. Strongly coupled data assimilation allows observations in one component of a coupled model to directly impact another component through the inclusion of cross-domain terms in the background error covariance matrix. When different components have disparate dominant spatial scales, localization between model domains must properly account for the multiple length scales at play. In this work, we develop two new multivariate localization functions, one of which is a multivariate extension of the fifth-order piecewise rational Gaspari–Cohn localization function; the within-component localization functions are standard Gaspari–Cohn with different localization radii, while the cross-localization function is newly constructed. The functions produce positive semidefinite localization matrices which are suitable for use in both Kalman filters and variational data assimilation schemes. We compare the performance of our two new multivariate localization functions to two other multivariate localization functions and to the univariate and weakly coupled analogs of all four functions in a simple experiment with the bivariate Lorenz 96 system. In our experiments, the multivariate Gaspari–Cohn function leads to better performance than any of the other multivariate localization functions.
Highlights
An essential part of any data assimilation (DA) method is the estimation of the background error covariance matrix Pb
In estimating the optimal cross-localization weight factor, we find that, since the only updates to X are through observations of Y, smaller cross-localization weight factors lead to degraded performance (Appendix B)
With weakly coupled localization functions, no information is shared in the update step between the observed Y process and the unobserved X process
Summary
An essential part of any data assimilation (DA) method is the estimation of the background error covariance matrix Pb. The background error covariance statistics stored in Pb provide a structure function that determines how observed quantities affect the model state variables, which is of particular importance when the state space is not fully observed (Bannister, 2008). Using an ensemble to estimate Pb allows the estimates of the background error statistics to change with the model state, which is desirable in many geophysical systems (Smith et al, 2017; Frolov et al, 2021). This estimate of Pb will always include noise due to sampling errors because the ensemble size is finite. Positive semidefiniteness of estimates of Pb is essential for the convergence of variational schemes and interpretability of schemes, like the Kalman fil-
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