The calculus of variations uses Green's functions and representers to express the best fit to a linear model and data. Mathematical construction of the representers is devious, and the meaning of the representer solution to the “control problem” of Chapter 1 is not obvious. There is a geometrical interpretation, in terms of observable and unobservable degrees of freedom. Unobservability defines an orthogonality, and the representers span a finite-dimensional subspace of the space of all model solutions or “circulations”. The representers are in fact the observable degrees of freedom.

A statistical interpretation is also available: if the unknown errors in the model are regarded as random fields having prescribed means and covariances, then the representers are related, via the measurement processes, to the covariances of the circulations. Thus the representer solution to the variational problem is also the optimal linear interpolation, in time and space, of data from multivariate, inhomogeneous and nonstationary random fields. The minimal value of the penalty functional that defines the generalized inverse or control problem is a random number. It is the χ2 variable, if the prescribed error means or covariances are correct, and has one degree of freedom per datum. Measurements need not be pointwise values of the circulation; representers along with their geometrical and statistical interpretations may be constructed for all bounded linear measurement functionals.

Analysis of the conditioning of the determination of the representer amplitudes reveals those degrees of freedom which are the most stable with respect to the observations. […]