Measuring dynamical prediction utility using relative entropy

A new parameter of dynamical system predictability is introduced that measures the potential utility of predictions. It is shown that this parameter satisfies a generalized second law of thermodynamics in that for Markov processes utility declines monotonically to zero at very long forecast times. Expressions for the new parameter in the case of Gaussian prediction ensembles are derived and a useful decomposition of utility into dispersion (roughly equivalent to ensemble spread) and signal components is introduced. Earlier measures of predictability have usually considered only the dispersion component of utility. A variety of simple dynamical systems with relevance to climate and weather prediction is introduced, and the behavior of their potential utility is analyzed in detail. For the climate systems examined here, the signal component is at least as important as the dispersion in determining the utility of a particular set of initial conditions. The simple "weather" system examined (the Lorenz system) exhibited different behavior with the dispersion being more important than the signal at short prediction lags. For longer lags there appeared no relation between utility and either signal or dispersion. On the other hand, there was a very strong relation at all lags between utility and the location of the initial conditions on the attractor.