The “toy” forward model introduced in Chapter 1 defines a well-posed mixed initial value–boundary value problem. The associated operator (wave operator plus initial operator plus boundary operator) is invertible, or nonsingular. Specifying additional data in the interior of the model domain renders the problem overdetermined. The operator becomes uninvertible, or singular. The difficulty may be resolved by constructing the generalized inverse of the operator, in a weighted least-squares sense.

An important class of regional models of the ocean or atmosphere defines an ill-posed initial-boundary value problem, regardless of the choice of open boundary conditions. All flow variables may as well be specified on the open boundaries. The excess of information may be regarded as data on a bounding curve, rather than at an interior point. The difficulty may again be resolved by constructing the generalized inverse in the weighted least-squares sense. The Euler–Lagrange equations form a well-posed boundary value problem in space–time. Solving them by forward and backward integrations is precluded, since no partitioning of the variational boundary conditions yields well-posed integrations. The penalty functional must be minimized directly.

Things are different if the open region is moving with the flow.