## Abstract

A direct inverse method is presented for inferring numerical model open boundary conditions from interior observational data. The dynamical context of the method is the frequency-domain 3D linear shallow water equations. A set of weight matrices is derived via finite-element discretization of the dynamical equations. The weight matrices explicitly express any interior solution of elevations or velocities as a weighted sum of boundary elevations. The interior data assimilation is then cast as a regression problem.

The weight matrix may be singular, which implies there may be an infinite set of boundary conditions that fit the data equally well. With the singular value decomposition technique, a general solution is provided for this infinite set of minimum-squared-misfit boundary conditions. Among them, a particular boundary condition, which minimizes potential energy on the boundary (hence the whole domain), is studied in detail: its confidence interval is defined and a way to smooth it is discussed.

Green’s function maps for the weight matrix provide insights into the dynamics inherent to the model domain. Such maps should be useful for many purposes. One of their usages demonstrated in this paper is to provide a physical explanation for the singularity of the system. Also discussed are how to assess the compatibility between data and the model and how to design a null-space smoothing device for smoothing the potential energy minimum boundary condition. While maintaining the goodness of fit between data and model, smoothing of the boundary condition may improve the interior solutions at the locations where the data have poor control.

The method is tested in a realistic domain but with synthetic data. The test yields very satisfactory results. Application of the method to Lardner’s open bay tidal problem demonstrates its advantages in computational accuracy and inexpensiveness.

*Corresponding author address:* Dr. Zhigang Xu, Ocean Sciences Division, Fisheries and Ocean Canada, Bedford Institute of Oceanography, P.O. Box 1006, Dartmouth, NS B2Y 4A2 Canada.