Three- and Four-Dimensional Variational Assimilation with a General Circulation Model of the Tropical Pacific Ocean. Part I: Formulation, Internal Diagnostics, and Consistency Checks

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Abstract
Three- and four-dimensional variational assimilation (3DVAR and 4DVAR) systems have been developed for the Océan Parallélisé (OPA) ocean general circulation model (OGCM) of the Laboratoire d'Océanographie Dynamique et de Climatologie. An iterative incremental approach is used to minimize a cost function that measures the statistically weighted squared differences between the observational information and their model equivalent. The control variable of the minimization problem is an increment to the background estimate of the model initial conditions at the beginning of each assimilation window. In 3DVAR, the increment is transported between observation times within the window using a persistence model, while in 4DVAR a dynamical model derived from the tangent linear (TL) of the OGCM is used. Both the persistence and TL models are shown to provide reasonably good descriptions of the evolution of typical errors over the 10- and 30-day widths of the assimilation windows used in the authors' 3DVAR and 4DVAR experiments, respectively. The present system relies on a univariate formulation of the background-error covariance matrix. In practice, the background-error covariances are specified implicitly within a change of control variable designed to improve the conditioning of the minimization problem. Horizontal and vertical correlation functions are modeled using a filter based on a numerical integration of a diffusion equation. The background-error variances are geographically dependent and specified from the model climatology. Single observation experiments are presented to illustrate how the TL dynamics act to modify these variances in a flow-dependent way by diminishing their values in the mixed layer and by displacing the maximum value of the variance to the level of the background thermocline. The 3DVAR and 4DVAR systems have been applied to a tropical Pacific version of OPA and cycled over the period 1993–98 using in situ temperature observations from the Global Temperature and Salinity Pilot Programme. The overall effect of the data assimilation is to reduce a large bias in the thermal field, which was present in the control. The fit to the data in 4DVAR is better than in 3DVAR, and within the specified observation-error standard deviation. Intermittent updating of the linearization state of the TL model is shown to be an important feature of the incremental 4DVAR algorithm and contributes significantly to improving the fit to the data.

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