The parameters of a two-dimensional numerical tidal model of the Arabian Gulf are estimated by optimal assimilation of data from tide gauges at several of the Saudi Aramco tide stations. The parameters estimated are the bottom drag coefficient and a correction to the bathymetry and are in general allowed to be position dependent. Significant improvements in the predictions of the model are obtained. The method used is based on the minimization of a cost function measuring the difference between the observed and computed water heights at the tide stations, the gradient of the cost function being computed by the adjoint method. The parameter functions are approximated by piecewise constant or piecewise linear approximations, and the cost function is minimized using the quasi-Newton algorithm CONMIN. It is found in general that in order to obtain stable estimates, it is necessary to include in the cost function a term that penalizes large variations in the parameter functions. Techniques are developed to reduce the storage demands of the algorithm sufficiently to permit the assimihation of 29 days of tidal data.
INTRODUCFION
Numerical models of flows in lakes, seas and oceans involve certain parameters, such as water depth, bottom friction coefficient, and eddy viscosity, and certain boundary values, such as the surface elevations on open boundaries, whose values may not be very well known. There has recently been considerable interest in the "inverse problem" of determining such values byincorporating measured data into the numerical model. In recent years, systematic techniques of such data assimilation based on optimal control methods have been developed, particularly in the field of meteorology. These methods were originated by $asaki [1955, 1970] and Marchuk [1974] and more recently have been reviewed by Lorenc [1986], Nayon [1986], and Le Dimet and Nayon [1989]. See also Zou, et al. [1992a, 1992b]. Similar methods have also been used by Chavent et al. [1975] and Carrera and Neumann [1986a, b, c] to estimate the parameters in models of flow in porous media. In the field of oceanography such optimal control methods have also recently come into use. Bennett and Mcintosh [1982] and Prevost and Salmon [1986] have applied the weak-constraint formalism of Sasaki [ 1970] to a tidal flow problem and a geostrophic flow problem, respectively. A general review of the strong-constraint formalism and an application to a model of wind-driven equatorial circulation have been given by Thacker and Long [1988]. Subsequently, the strong-constraint formalism and some earlier experimental results have been used by Panchang and O'Brien [1989] to determine the bottom friction coefficient in a problem of flow in a channel. Smedstad [ 1989] and Smedstad and O'Brien [1991] have extended this approach and used it to determine the effective phase speed in a model of the equatorial Pacific Ocean based on observations of sea level. Yu and O'Brien [1991] have used a similar method to estimate the eddy viscosity and surface drag coefficient ...
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