Data assimilation in a coupled physical–biogeochemical model of the California Current System using an incremental lognormal 4-dimensional variational approach: Part 1—Model formulation and biological data assimilation twin experiments

•A quadratic formulation for incremental lognormal 4DVar method is presented.•A quadratic lognormal 4DVar method is implemented to Regional Ocean Modeling System.•The new method is evaluated in ideal twin experiments along the U.S. west coast.•The new method outperforms conventional 4DVar based on Gaussian error distribution.

A quadratic formulation for an incremental lognormal 4-dimensional variational assimilation method (incremental L4DVar) is introduced for assimilation of biogeochemical observations into a 3-dimensional ocean circulation model. L4DVar assumes that errors in the model state are lognormally rather than Gaussian distributed, and implicitly ensures that state estimates are positive definite, making this approach attractive for biogeochemical variables. The method is made practical for a realistic implementation having a large state vector through linear assumptions that render the cost function quadratic and allow application of existing minimization techniques. A simple nutrient-phytoplankton-zooplankton-detritus (NPZD) model is coupled to the Regional Ocean Modeling System (ROMS) and configured for the California Current System. Quadratic incremental L4DVar is evaluated in a twin model framework in which biological fields only are in error and compared to G4DVar which assumes Gaussian distributed errors. Five-day assimilation cycles are used and statistics from four years of model integration analyzed. The quadratic incremental L4DVar results in smaller root-mean-squared errors and better statistical agreement with reference states than G4DVar while maintaining a positive state vector. The additional computational cost and implementation effort are trivial compared to the G4DVar system, making quadratic incremental L4DVar a practical and beneficial option for realistic biogeochemical state estimation in the ocean.