Probabilistic Topological Analysis of Predictability, Sensitivity, and Assimilability: Theory and Simple Illustrative Examples

Abstract

A probabilistic topological theory is developed to clarify some fundamental aspects of the predictability, sensitivity, and assimilability of initial value problems. In order to measure predictability, sensitivity, and as-similability quantitatively, three indices are introduced. The predictability index is defined in terms of discretized observational error statistics and mapping classifications of the models. Similarly, in addition to the predictability index, the sensitivity and assimilability indices are also defined. The prediction model is simplified to make it easier to handle the nonlinearity of their models. The simplification is a piecewise linear approximation. The nonlinearity condenses into a singularity, fixed point, stationary state, or equilibrium state. The significance and possible advantages of the probabilistic topological analysis are demonstrated by the simple illustrative examples, such as the evolution of globally averaged temperature of a hypothetical planet and homoclinic phase-space trajectories of westerly jet-stream blocking. The result shows that the predictability is high and the sensitivity of the initial condition is low by the simplified nonlinear deterministic model in the regular domain of phase space where a stable singularity or no singularity exists. It also shows that in the singular domain where an unstable singularity exists, initially, infinitesimal separation of two phase-space trajectories can grow into a recognizable separation. In the latter case, the predictability decreases and the sensitivity increases. Uncertainty of the initial values that may be caused by an error in observation creates uncertainty in the prediction. The predictability, sensitivity, and assimilability indices become probabilistic. Furthermore, the importance of the Sobolev norm is demonstrated, suggesting that the inclusion of an observational tendency term in the norm may improve predictability. An outline of the continuing application of the probabilistic topological analysis to a shallow water system is also presented.