Indistinguishable states I. Perfect model scenario

An accurate forecast of a nonlinear system will require an accurate estimat ion of the initial state. It is shown that even under the ideal conditions of a pet-feet model and infinite past observations of a deterministic nonli near system, uncertainty in the observations makes exact state estimation i s impossible. Consistent with the noisy observations there is a set of stat es indistinguishable from the true state. This implies that an accurate for ecast must be based on a probability density on the indistinguishable state s. This paper shows that this density can be calculated by first calculatin g a maximum likelihood estimate of the state, and then an ensemble estimate of the density of states that are indistinguishable from the maximum likel ihood state. A new method for calculating the maximum likelihood estimate o f the true state is presented which allows practical ensemble forecasting e ven when the recurrence time of the system is long. In a subsequent paper t he theory and practice described in this paper are extended to an imperfect model scenario. (C) 2001 Published by Elsevier Science B.V.