Nonlinear model evaluation: $\iota$ shadowing, probabilistic prediction and weather forecasting

Abstract

Physical processes are often modelled using nonlinear dynamical systems. If such models are relevant then they should be capable of demonstrating behaviour observed in the physical process. In this thesis a new measure of model optimality is introduced: the distribution of $\iota$ shadowing times defines the durations over which there exists a model trajectory consistent with the observations. By recognising the uncertainty present in every observation, including the initial condition, $\iota$ shadowing distinguishes model sensitivity from model error; a perfect model will always be accepted as optimal. The traditional root mean square measure may confuse sensitivity and error, and rank an imperfect model over a perfect one. In a perfect model scenario a good variational assimilation technique will yield an $\iota$ shadowing trajectory but this is not the case given an imperfect model; the inability of the model to $\iota$ shadow provides information on model error, facilitating the definition of an alternative assimilation technique and enabling model improvement. While the $\iota$ shadowing time of a model defines a limit of predictability, it does not validate the model as a predictor. Ensemble forecasting provides the preferred approach for evaluating the uncertainty in predictions, yet questions remain as to how best to construct ensembles. The formation of ensembles is contrasted in perfect and imperfect model scenarios in systems ranging from the analytically tractable to the Earth's atmosphere, thereby addressing the question of whether the apparent simplicity often observed in very high-dimensional weather models fails `even in or only in' low-dimensional chaotic systems. Simple tests of the consistency between constrained ensembles and their methods of formulation are proposed and illustrated. Specifically, the commonly held belief that initial uncertainties in the state of the atmosphere of realistic amplitude behave linearly for two days is tested in operational numerical weather prediction models and found wanting: nonlinear effects are often important on time scales of 24 hours. Through the kind consideration of the European Centre for Medium-range Weather Forecasting, the modifications suggested by this are tested in an operational model