EGU General Assembly 2021
© Author(s) 2021. This work is distributed under
the Creative Commons Attribution 4.0 License.

Insights about the hybrid error covariance models

Francesco Sardelli1,2,3 and Craig Bishop1,2,3
Francesco Sardelli and Craig Bishop
  • 1University of Melbourne, Faculty of Science, School of Earth Sciences, Australia
  • 2ARC Centre of Excellence for Climate Extremes, Australia
  • 3Bureau of Meteorology, Australia

Hybrid error covariance models construct the covariance matrix to be used in variational data assimilation methods through a linear combination Ph= αcPc + αePl of the climatological error covariance matrix Pc and the localized ensemble covariance matrix Pl = CP with scalar weights αc and αe.

This work aims to provide a theoretical justication for current hybrid error covariance models and identify a critical issue in them in order to improve them in future research. In the framework of Bayes' theorem, a theory is developed by modelling the climatological distribution of true forecast error covariance matrix Pf as an inverse matrix gamma distribution (prior distribution) and the distribution of the localized ensemble covariance matrix Pl given a true forecast error covariance matrix Pf as a Wishart or matrix gamma distribution (likelihood distribution). The following formulas for the expected values of the prior and likelihood distributions are assumed: E [Pf ] = Pc and E [Pl Pf ]= Pf , respectively. The posterior distribution for the true forecast error covariance matrix Pf given the localised covariance matrix Pl is derived: it turns out to be an inverse matrix gamma distribution. Within this theory, a formula for the expected value E [PfP ] of the true forecast error covariance matrix Pf given the ensemble covariance matrix P is derived: E [PfP]= βcPc+ βePl (where βc  and βe are scalar weights). This provides a theoretical justication for hybrid error covariance models. Moreover, expressions (and thus an interpretation) for the scalar weights βc and βe in terms of the relative variances of the diagonal elements of the prior and likelihood distributions are obtained.

Hence, the consistency of current hybrid covariance models with the assumption E [Pl Pf ]= Pf is showed. This assumption is, in turn, inconsistent with E [PPf]= Pf , which ensemble DA schemes are meant to satisfy, and it is falsiable.

To illustrate the above theory, an experiment is run to simulate 3200 replicate Earth's all having the same true state trajectory, weather prediction system and observational network, but different realizations of the observations. Each replicate Earth is simulated through a 10-variable Lorenz '96 model with an ETKF data assimilation system. From the set of the true forecast errors of all replicate Earth's, the (otherwise hidden) true forecast error covariance matrix Pf is computed at each time step and the (dis)similarity of its climatological distribution from the best-fit inverse matrix gamma distribution is considered. It is found that (i) the inverse-matrix gamma distribution overestimates the probability of signicant error correlations between widely separated model variables; (ii) it is the un-localized ETKF ensemble covariance matrix that equals the mean climatological covariance matrix, not the localized ensemble covariance matrix. These findings motivate research to discover more accurate approximations to the climatological distribution of the true forecast error covariance matrix and more accurate hybrid covariance models.

How to cite: Sardelli, F. and Bishop, C.: Insights about the hybrid error covariance models, EGU General Assembly 2021, online, 19–30 Apr 2021, EGU21-13764,, 2021.

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