Four-dimensional variational data assimilation for limited-area models: Lateral boundary conditions, solution uniqueness, and numerical convergence

Mathematical issues arising when applying four-dimensional variational (4DV AR) 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 natu ral boundary conditions when there are errors (on the boundary) in the init ial guess on the assimilated initial conditions is discussed. A proof of the uniqueness of the solution for both forward and adjoint equa tions in the presence of open boundaries at each iteration of the minimizat ion procedure is provided, along with an assessment of the convergence of n umerical solutions. Numerical experiments with a simple advection equation support the theoreti cal analyses. Numerical results show that if observational data are perfect , 4DVAR data assimilation using a limited-area model can produce a reasonab le initial condition. However, if there are errors in the observational dat a at the open boundaries and if natural boundary conditions are assumed, bo undary errors can contaminate the assimilated solutions.