15.1 Ensemble Prediction and Relative Entropy as a Measure of Predictability

Complex systems with many spatial degrees of freedom arise in diverse context, such as atmosphere/ocean general circulation models (GCMs) for climate or weather prediction, pollution models, and the models for the spread of hazardous biological, chemical, or nuclear plumes. These non-linear models are intrinsically chaotic over many time scales with sensitive dependence on initial conditions. In this chapter, as in Chapters 7 and 8 earlier, such models are represented discretely as a large system of ODEs for a vector ? ^N given by

Given both the uncertainty in a deterministic initial condition, ₀, as well as the intrinsic chaos in solutions of (15.1), it is natural instead to consider an ensemble of initial data characterized by a probability density, p ₀( ), satisfying p ₀ ? 0, ? p ₀ = 1 and with mean given by ₀, i.e.

The idea is to utilize the ensemble of solutions of (15.1) drawn from the initial data p ₀( ) to quantify the uncertainty and measure the confidence interval and predictive power of the deterministic solution beginning at ₀. For example, consider ₀, and an associated probability density consisting of small random perturbations of these initial data. This probability density measures the uncertainty in the measurement of the initial data. For instance, the random initial data might be sampled from a Gaussian probability distribution centered about ₀ (see Section 6.4)

where