Some theoretical considerations on predictability of linear stochastic dynamics

Predictability is a measure of prediction error relative to observed variability and so depends on both the physical and prediction systems. Here predictability is investigated for climate phenomena described by linear stochastic dynamics and prediction systems with perfect initial conditions and perfect linear prediction dynamics. Predictability is quantified using the predictive information matrix constructed from the prediction error and climatological covariances. Predictability measures defined using the eigenvalues of the predictive information matrix are invariant under linear state-variable transformations and for univariate systems reduce to functions of the ratio of prediction error and climatological variances. The predictability of linear stochastic dynamics is shown to be minimized for stochastic forcing that is uncorrelated in normal-mode space. This minimum predictability depends only on the eigenvalues of the dynamics, and is a lower bound for the predictability of the system with arbitrary stochastic forcing. Issues related to upper bounds for predictability are explored in a simple theoretical example.