It has long been recognized that one needs to combine observations with dynamic models to achieve an optimal description of a turbulent global fluid system such as the ocean. This model/data combination is referred to as ``data assimilation'' following the terminology in meteorology. The development of data assimilation into numerical ocean models has been hampered by the lack of well-sampled data. For a review of the status of the subject the reader is referred to Ghil and Malanotte-Rizzoli [1991] and Bennett [1992]. The availability of satellite altimetry data has spurred significant progress in ocean data assimilation over the past several years.
One approach to data assimilation is direct data insertion, or reinitialization. Berry and Marshall [1989] inserted data only to the surface layer and let the information be dynamically transferred to the lower layers. De Mey and Robinson [1987], De Mey and Menard [1989], and Hurlburt et al. [1990] projected the surface information to the deep layers using statistical methods such as empirical orthogonal functions. Altimetry data along with the subsurface projection were then directly inserted into a model, resulting in a more efficient vertical information transfer. Robinson and Walstad [1987] used a feature model for the Gulf Stream and its eddies to initialize the 3-dimensional field for data insertion. Haines [1991] demonstrated a novel approach in which the data insertion was applied to the surface stream function while the potential vorticity of the lower layers were kept unchanged so as to create an effective adjustment of the lower layers to the data insertion at the top layer.
A relatively simple and effective scheme for assimilating data continuously into a model is the so-called nudging technique [ Holland and Malanotte-Rizzoli, 1989; Verron, 1990, 1992]. The information of the altimetry data is ingested into the surface layer by placing in the model equations a term that is proportional to the difference between the model sea surface height and the observation. The information of the data is transferred to the deep layers by the model dynamics. This technique works quite well for quasi-geostrophic models and has been applied to the Geosat data in the Agulhas Current [ Holland et al., 1991] and the western North Atlantic [ Verron et al., 1992]. Blayo et al. [1994] applied the same approach to the TOPEX/POSEIDON data in the North Atlantic. Smedstad and Fox [1994] demonstrated the use of the technique with a primitive equation model . However, the effects of observation error and model error are generally not taken into account in the nudging approach.
White et al. [1990 c] employed an optimal interpolation (OI) technique to assimilate the Geosat data into a quasi-geostrophic model of the California Current, specifically taking into account the model and observation errors in the assimilation step. However, the information of the data was assimilated only to the surface layer, while the vertical transfer of information was via the model dynamics. Dombrowski and De Mey [1992] used the empirical orthogonal function technique to project the altimetry information to the lower layers of a quasi-geostrophic model and applied an OI scheme to assimilating the Geosat data into all the vertical layers simultaneously. Using statistical correlation in the transfer of information from the sea surface height to subsurface fields, Mellor and Ezer [1991] and Ezer and Mellor [1994] demonstrated the use of an OI scheme in assimilating altimetry data into a primitive equation model of the Gulf Stream region. The statistical correlation between the sea surface height and subsurface temperature was demonstrated by Ezer et al. [1993]. The OI approach is particularly effective in mapping the horizontal fields, but the vertical transfer of information to subsurface fields is somewhat problematic.
Variational methods have been applied to seeking model solutions that minimize a ``cost function'' related to the model-data misfit [ Moore, 1991; Bourles et al., 1992; Schroter et al., 1993; Greiner and Perigaud, 1994]. The solutions obtained are exact solutions to the model equations in the so-called strong constraint approach. Using a ``weak-constraint'' approach, Egbert et al. [1994] demonstrated the use of the variational method to obtain an optimal estimation of the global ocean tides by combining the TOPEX/POSEIDON data with the Laplace tidal equation. In addition to the optimal model solution, other parameters such as friction coefficient, wind forcing amplitude, and model initial conditions can also be estimated as part of the solutions. Although the approach is quite effective in testing the consistency between models and data, error estimates for the solutions were seldom discussed because of the required large computational burden.
The popular method of the ``Kalman filter/smoother'' in the engineering community is a sequential approach to optimal estimation by combining model prediction with observations in a way to minimize the estimation error. Miller [1989] demonstrated the use of the Kalman filter to assimilate the altimeter differences into linear ocean models. Gaspar and Wunsch [1989] applied the technique to extracting Rossby wave signals from the Geosat data in the western North Atlantic. They found that only about 6% of the data variance was consistent with the Rossby wave dynamics. Fu et al. [1991 b, 1993] showed that a significant portion (68%) of the Geosat data in the tropical Pacific was consistent with the dynamics of linear Kelvin waves and Rossby waves. Gourdeau et al. [1992] obtained similar results in the Tropical Atlantic. The computational burden of the Kalman filter/smoother is notoriously heavy, because the error covariance also needs to be carried forward in time by the model dynamics. Noting the rapid approach to a steady state of the error covariance in many Kalman filter applications, Fukumori et al. [1993] developed a steady-state Kalman filter/smoother and demonstrated its use in assimilating the Geosat data into a simple, coarse-resolution general circulation model of the North Atlantic. The technique was also evaluated by Fu et al. [1993] by comparing it to an exact, time-varying Kalman filter calculation. To make the technique feasible to a full- scale ocean general circulation model, Fukumori and Malanotte-Rizzoli [1995] demonstrated a further approximation of the Kalman filter by assimilating only the large-scale content of the data. Such approach holds promise for assimilating data into truly global models.