## Abstract

A data assimilation method for positive-definite variables is investigated and applied to a 1-Dimensional (1D) advection-diffusion model and a 3-variable nutrient-phytoplankton-zooplankton (NPZ) model. Conventional data assimilation methods that assume Gaussian distributed errors are problematic for most biogeochemical models because they do not constrain posterior estimates for concentration-based variables to be positive-definite. We apply the approach outlined by Fletcher (2010) that formulates the 4-dimensional variational (4DVAR) assimilation problem assuming lognormally distributed errors. This approach is sensible because many biogeochemical variables are better represented by lognormal than by Gaussian statistics, and it ensures positive-definite state variables. We introduce the incremental formulation of lognormal 4DVAR (L4DVAR) and consider two solutions - incremental mode (imode) and incremental median (imedian) which approximate the mode and the median of different posterior probability density functions. In a simple 0D test case, the two solutions performed similarly with small observation and background uncertainty, but the imedian solution resulted in smaller geometric bias and root-mean-squared error as uncertainty increased. Both solutions of incremental L4DVAR using a 1D linear advection-diffusion model and a nonlinear NPZ model reduce misfit between the model and observations significantly in various assimilation settings and yield a positive-definite adjusted state. We report also on the success of the incremental L4DVAR approach when model error is introduced.

Original language | English |
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Pages (from-to) | 1-17 |

Number of pages | 17 |

Journal | Ocean Modelling |

Volume | 54-55 |

DOIs | |

Publication status | Published - 2012 Sept |

### Bibliographical note

Funding Information:We are grateful for a grant from the Gordon and Betty Moore Foundation that supported this research. Two anonymous reviewers significantly improved the manuscript with several helpful comments.

## All Science Journal Classification (ASJC) codes

- Computer Science (miscellaneous)
- Oceanography
- Geotechnical Engineering and Engineering Geology
- Atmospheric Science