A data space search is the most efficient way to solve a linear, least-squares smoothing problem defined over a fixed time interval. The method exploits linearity, and so is unavailable for nonlinear dynamics, or for penalties other than least-squares. As discussed in Chapter 3, a data space search may be conducted on linear iterates of the nonlinear Euler–Lagrange equations. The existence of the nonlinear equations implies that the penalty is a smooth functional of the state, in which case a state space search may always be initiated. The nature of state space searches is intuitively clear, and their use is widespread. Conditioning degrades as the size of the state space gets very large. Collapsing the size of the state space by assuming “perfect” dynamics is the basis of “the” variational adjoint method: only initial values, boundary values and parameter values are varied. Preconditioning may in principle be effected by use of second-order variational equations, but even iterative construction of the state space preconditioner is unfeasible for highly realistic problems. Technique for numerical integration of variational equations is not a paramount consideration, but deserving of attention since it can be consuming of human time.

Operational forecasting is inherently sequential; data are constantly arriving and forecasts must be issued regularly. In such an environment, it is more natural to filter a model and data sequentially than to smooth them over a fixed interval. […]