On predictability of nonlinear models with large uncertainty in the evolution law

We use the exit statistics approach (Ivanov et al., 1994) to qualify model predictability and understand non-trivial collective behavior of finite-amplitude perturbations inserted in the evolution law on different scales. Two kinds of perturbations are discussed: those inserted in wind forcing and/or perturbations added to open boundary conditions. In both the cases the predictability problem is reformulated as the stochastic stability problem of a generic system with many degrees of freedom randomized by external noise.

Such an approach allows us to (a) obtain the general equations (Pontryagin - Kolmogorov - Stratonovich type equations) for the predictability time (introduced as a first passage time) and its moments, and (b) develop analytical and numerical methods for estimating the predictability time both analytically and numerically, using asymptotic methods and Monte-Carlo method, respectively. These equations play a fundamental role in our analysis, and are the mathematical basis for small ensemble modeling and description of non-Gaussian prediction error (PE) statistics through the probability weighted moments and a variation principle for the relative entropy (the Kullback-Lieber distance).

Our methodology is to use Latin Hypercube design strategy for the generation of finite-amplitude perturbations, an original technique for the reconstruction of PE probability density functions from small statistical ensembles and analytical solutions for the predictability time obtained by asymptotical methods.

The following important question is discussed in the present study: what non-trivial effects of collective behavior of different scales can affect the model predictability and how they influent on high-order statistics (moments) of the prediction error? We demonstrate that prediction error growth strongly depends on the intensity and degree of spatial inhomogeneity of perturbations. For highly inhomogeneous perturbations the PE grows rather along a power law than the exponential one. Coherent growth of perturbations can be found for different scales at various stages of PE evolution.

We demonstrate that statistics of a prediction time scale quickly depart from Gaussian (the linear predictability regime) and becomes Weibullian (the non-linear predictability regime) as amplitude of initial perturbations grows. Bifurcations of the variance, skewness and kurtosis of the prediction time scale may accompany a transition from linear to non-linear predictability regime. A new analytical formula for the model predictability horizon is introduced and applied to estimate the limits of predictability for nonlinear models.

Another important question discussed here is how interference effects of perturbations from different sources of uncertainty contribute to model predictability. We suggest estimating such contributions through the so-called Sobol - Saltelli sensitivity indices. The main result of our calculation is that a model response to a group of finite-amplitude perturbations generated by different sources of uncertainty can be either stronger or weaker than when contributions of these perturbations are summarized in the additive manner.

Simplified models are involved to illustrate capability of the developed computer technologies and analytical methods, and to reveal the general trends in prediction error behavior and the model reconstructed error statistics.