Boundary Conditions for the Extended Kalman Filter With aQuasi - Geostrophic Ocean

The formulation of consistent boundary conditions for the quasi-geostrophic (QG) model with an extended Kalman lter in a data assimilation scheme is discussed. To form a well-posed boundary value problem for the QG model, the stream function must be speciied at all boundaries and the vorticity must be speciied at the innow boundaries. The situation becomes signiicantly more complicated when proper boundary conditions are to be speciied for the error covariance evolution equation. For closed or periodic boundaries no severe problems occur, but in general cases with open boundaries, only approximative methods can be used. Here a scheme is presented which allows for the stream function to be updated on the boundaries, e.g., from the use of measurements located close to the boundaries, or from meanders and eddies approaching the boundaries from the interior of the domain. Further, the boundary value problem for the error covariance evolution equation is treated extensively. It is demonstrated that numerical discretization of the error covariance evolution equation leads to severe numerical diiculties when open boundaries are used. An approximate numerical scheme that can be used to handle open boundaries with innow and outtow is proposed, and examples are given to illustrate the method. Is is shown that the boundary scheme is consistent and can be used even with data located at innow boundaries. However, the approximations used in the scheme may lead to loss of positive deeniteness for the error covariance matrix and an algorithm must be used to ensure positive deeniteness for long time integrations.