### 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 |
---|---|

Pages (from-to) | 1-17 |

Number of pages | 17 |

Journal | Ocean Modelling |

Volume | 54-55 |

DOIs | |

Publication status | Published - 2012 Sep 1 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*Ocean Modelling*,

*54-55*, 1-17. https://doi.org/10.1016/j.ocemod.2012.06.001

}

*Ocean Modelling*, vol. 54-55, pp. 1-17. https://doi.org/10.1016/j.ocemod.2012.06.001

**Incremental four-dimensional variational data assimilation of positive-definite oceanic variables using a logarithm transformation.** / Song, Hajoon; Edwards, Christopher A.; Moore, Andrew M.; Fiechter, Jerome.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Incremental four-dimensional variational data assimilation of positive-definite oceanic variables using a logarithm transformation

AU - Song, Hajoon

AU - Edwards, Christopher A.

AU - Moore, Andrew M.

AU - Fiechter, Jerome

PY - 2012/9/1

Y1 - 2012/9/1

N2 - 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.

AB - 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.

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U2 - 10.1016/j.ocemod.2012.06.001

DO - 10.1016/j.ocemod.2012.06.001

M3 - Article

AN - SCOPUS:84865364995

VL - 54-55

SP - 1

EP - 17

JO - Ocean Modelling

JF - Ocean Modelling

SN - 1463-5003

ER -