## 1. Introduction

[2] One of the primary goals of numerical circulation model development is to simulate and predict hydrographic and flow fields in the ocean. The basic idea of four-dimensional (4-D) data assimilation is to make a synthesis of existing data and model dynamics to provide a systematically better model simulation consistent with the observed fields that are normally noisy and incomplete in space and time. Data assimilation systems consist of three components: a set of observed data, a dynamical model, and an assimilation scheme. Since the data have errors and models are imperfect, a well-constructed assimilation scheme should provide a better match between data and model within the bounds of observational and modeling errors [*Ghil*, 1989]. Therefore, once a dynamical model is fully developed, the appropriate assimilation scheme becomes more critical in ensuring successful model applications.

[3] The most widely used data assimilation techniques are nudging, optimal interpolation (OI), adjoint-based methods, and Kalman filters (KFs). Nudging and OI are the simplest and least computationally intensive data assimilation methods. They work efficiently in applications with full data coverage of the model domain, but can lead to unrealistic density gradients and current shears at the data boundary when the data coverage is limited to a discrete portion of the model domain [*Lorenc*, 1981; *Chen et al.*, 2007, 2008]. The adjoint-based methods are based on control theory in which a cost function defined by the difference between model-derived and measured quantities is minimized in a least squares sense under the constraints of the model equations [*Le Dimet and Talagrand*, 1986; *Thacker and Long*, 1988; *Tziperman and Thacker*, 1989; *Bergamasco et al.*, 1993; *Morrow and De Mey*, 1995]. This method has been widely used in both linear and nonlinear systems for the estimation of model parameters, initial and boundary conditions, and surface forcing such as wind stress and heat/moisture fluxes. KFs are sequential data assimilation methods based on estimation theory [*Kalman*, 1960]. Since they provide prediction of both analysis and forecast error covariances, they can be used for nowcasting and forecasting with global- to basin-scale ocean and atmospheric models [*Evensen*, 1992, 1993, 2003; *Blanchet et al.*, 1997; *Ghil and Malanotte-Rizzoli*, 1991].

[4] Malanotte-Rizzoli and collaborators have constructed a package of KFs, including a reduced rank Kalman filter (RRKF) [*Buehner and Malanotte-Rizzoli*, 2003; *Buehner et al.*, 2003], an ensemble Kalman filter (EnKF) [*Evensen*, 1994; *Zang and Malanotte-Rizzoli*, 2003], a deterministic and stochastic ensemble square-root Kalman filter (EnSKF), and an ensemble transform Kalman filter (EnTKF) [*Bishop et al.*, 2001]. This package was first applied to idealized cases as proof-of-concept tests and subsequently has been successively incorporated into fully realistic, primitive equation basin-scale ocean models. In a linear dynamical system, when the temporal variability of model forecast errors are characterized by the dominant modes, the error evolution can be resolved in a subspace defined by a few leading empirical orthogonal functions (EOFs). This RRKF based on EOF theory is not computationally intensive, and can effectively reduce model errors with better performance than OI [*Buehner and Malanotte-Rizzoli*, 2003]. *Evensen* [1994] first suggested that the error covariance relative to the mean of ensemble members could provide a better estimation of the error covariance defined in the classical Kalman filter. The EnKF is traditionally constructed by running a forecast model driven by an ensemble of initial conditions with random perturbations generated using a Monte Carlo approach and then the error covariance relative to the ensemble mean is estimated to determine the ensemble analysis values for the next forecast [*Evensen*, 1994]. Recently, the singular evolutive extended Kalman (SEEK) [*Pham et al.*, 1998a] and singular evolutive interpolated Kalman (SEIK) [*Pham et al.*, 1998b] filters were developed, in which the ensemble members are specified initially using either dominant eigenvectors (or EOFs) or reinitialized periodically in suberror space to improve the computational efficiency by optimally constructing the error covariance using lower ensemble numbers [*Nerger et al.*, 2005]. The EnSKF and EnTKF are derivatives of EnKF that are applied using deterministic observations [*Bishop et al.*, 2001; *Whitaker and Hamill*, 2002]. The EnTKF was introduced by *Bishop et al.* [2001] for optimally deploying adaptive observations. This filter was used by S. J. Lyu et al. (A comparison of data assimilation results from the deterministic and stochastic ensemble Kalman filters, unpublished manuscript, 2009; Optimal fixed and adaptive observation arrays in an idealized wind-driven ocean model, unpublished manuscript, 2009) to design optimal fixed and adaptive observational arrays in an idealized model of the wind-driven circulation in a double gyre ocean.

[5] Recently, a multi-institutional (UMASSD-MIT-WHOI) effort was made to implement KF methods into the unstructured grid Finite-Volume Coastal Ocean Model (FVCOM) for coastal and estuarine applications [*Chen et al.*, 2006a]. A series of proof-of-concept tests were conducted to compare the abilities of various KFs to restore a dynamical coastal or estuarine system after they had been subjected to random perturbations. Experiments were made to examine the sensitivity of the convergence rate to the sampling location and assimilation configuration (univariate or a multivariate covariance approach related to the number of state variables in assimilation). Three idealized coastal and estuarine problems were selected: (1) tidal oscillations in a flat bottom circular basin, (2) a low-salinity plume over an idealized continental shelf, and (3) tidal flushing in addition to freshwater discharge in an idealized rectangular estuary with intertidal zones. Unlike global- and basin-scale ocean model systems, KFs have not been widely used in coastal ocean and estuarine models. The three idealized cases presented in this study represent fundamental processes that occur widely in the coastal ocean, and results obtained from these experiments can provide useful guidance for future application of KFs to realistic coastal and estuarine forecast or hindcast systems.

[6] This paper summarizes the validation experiment results with a focus on comparisons of RRKF and EnKF for the three idealized cases. Although the experiments were made using FVCOM, the results are applicable in general to any unstructured or structured grid ocean model. The paper is organized as follows: FVCOM and schematics of KF-FVCOM coupling are briefly described in section 2, the results for the three idealized cases are presented in sections 3, 4 and 5, and the discussion and conclusions are presented in section 6.