Mathematical issues arising when applying four-dimensional variational (4DVAR) data assimilation to limited-area problems are studied. The derivation of the adjoint system for the initial-boundary value problem for a general hyperbolic system using the standard variational approach requires that the inflow adjoint variables at an open boundary be zero. However, in general, these ``natural'' boundary conditions will lead to a different solution than that provided by the global assimilation problem. The impact of using natural boundary conditions when there are errors (on the boundary) in the initial guess on the assimilated initial conditions is discussed. A proof of the uniqueness of the solution for both forward and adjoint equations in the presence of open boundaries at each iteration of the minimization procedure is provided, along with an assessment of the convergence of numerical solutions. Numerical experiments with a simple advection equation support the theoretical analyses. Numerical results show that if observational data are perfect, 4DVAR data assimilation using a limited-area model can produce a reasonable initial condition. However, if there are errors in the observational data at the open boundaries and if natural boundary conditions are assumed, boundary errors can contaminate the assimilated solutions.}
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