Empirical normal modes versus empirical orthogonal functions for statistical prediction

Abstract

The theory of empirical normal modes (ENMs) for a shallow water fluid is developed. ENMs are basis functions that both have the statistical properties of empirical orthogonal functions (EOFs) and the dynamical properties of normal modes. In fact, ENMs are obtained in a similar manner as EOFs but with the use of a quadratic form instead of the Euclidean norm. This quadratic form is a global invariant, the wave activity, of the linearized equations about a basic state. A general formulation is proposed for calculating normal modes from a generalized hermitian problem, even when the basic state is not zonal. The projection coefficients of the flow onto a few leading ENMs generally have a more monochromatic behavior than that obtained for EOFs, which give them an intrinsically more predictable character. This property is illustrated by numerical experiments using the shallow water model on the sphere. It is shown, in particular, that the ENM coefficients, when used as predictors in a statistical linear model, provide better predictions of the behavior of the shallow water atmosphere than EOF coefficients. It is also shown that the choice of the basic state itself is crucial.