Theoretical aspect of suitable spatial boundary condition specified for adjoint model on limited area

Theoretical argumentation for so-called suitable spatial condition is condu cted by the aid of homotopy framework to demonstrate that the proposed boun dary condition does guarantee that the over-specification boundary conditio n resulting from an adjoint model on a limited-area is no longer an issue, and yet preserve its well-poseness and optimal character in the boundary se tting. The ill-poseness of over-specified spatial boundary condition is in a sense, inevitable from an adjoint model since data assimilation processes have to adapt prescribed observations that used to be over-specified at th e spatial boundaries of the modeling domain. In the view of pragmatic implement, the theoretical framework of our propos ed condition for spatial boundaries indeed can be reduced to the hybrid for mulation of nudging filter, radiation condition taking account of ambient f orcing, together with Dirichlet kind of compatible boundary condition to th e observations prescribed in data assimilation procedure. All of these trea tments, no doubt, are very familiar to mesoscale modelers.