# Improving strategies with constraints regarding non-Gaussian statistics in a three-dimensional variational assimilation method

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## Abstract

We assess validity of a Gaussian error assumption, the basic assumption in data assimilation theory, and propose two kinds of constraints regarding non-Gaussian statistics. In the mixed water region (MWR) off the east coast of Japan exhibiting complicated frontal structures, a probability density function (PDF) of subsurface temperature shows double peaks corresponding to the Kuroshio and Oyashio waters. The complicated frontal structures characterized by the temperature PDF sometimes cause large innovations, bringing about a non-Gaussianity of errors. It is also revealed that assimilated results with a standard three-dimensional variational (3DVAR) scheme have some issues in MWR, arising from the non-Gaussianity of errors. The Oyashio water sometimes becomes unrealistically cold. The double peaks seen in the observed temperature PDF are too smoothed. To improve the assimilated field in MWR, we introduce two kinds of constraints, *J* _{ c1} and *J* _{ c2}, which model the observed temperature PDF. The constraint *J* _{ c1} prevents the unrealistically cold Oyashio water, and *J* _{ c2} intends to reproduce the double peaks. The assimilated fields are significantly improved by using these constraints. The constraint *J* _{ c1} effectively reduces the unrealistically cold Oyashio water. The double peaks in the observed temperature PDF are successfully reproduced by *J* _{ c2}. In addition, not only subsurface temperature but also whole level temperature and salinity (T–S) fields are improved by adopting *J* _{ c1} and *J* _{ c2} to a multivariate 3DVAR scheme with vertical coupled T–S empirical orthogonal function modes.

## Keywords

Data assimilation Variational method Non-Gaussian Mixed water region Temperature front## Notes

### Acknowledgments

The authors would like to thank the members of the Oceanographic Division of the Meteorological Research Institute for fruitful discussions. Thanks are extended to two anonymous reviewers for helpful comments on a previous version of the manuscript. This work is funded by the Meteorological Research Institute. Part of this study is supported by the Research Program on Climate Change Adaptation (RECCA) and by MEXT Grant-in-Aid for Young Scientists (B).

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