CONVERGENCE OF DATA ASSIMILATION BY PERIODIC UPDATING IN SIMPLE HAMILTONIAN AND DISSIPATIVE SYSTEMS

In this paper, we study the influence of the interval between data ins ertion events on the convergence of sequential data assimilation probl ems. An example of a conservative Hamiltonian system is presented (tha t of Henon and Heiles (1964)) where sequential assimilation with perio dic data insertion every Delta t achieves a more rapid convergence if data is not inserted at the smallest possible update interval, Delta t . It is shown analytically that this is true for all Hamiltonian syste ms when the updated variables produce convergence of the assimilation, because the resolvent matrix then varies as O(Delta t(2)) to highest order. The theory successfully predicts the turnover point for the Hi non and Heiles system when a larger Delta t lends to slower convergenc e and also the assimilation interval at which convergence may cease al together. The application to a simplified low order shallow water mode l describing coupled Rossby and gravity waves and with a forced-dissip ative perturbation extends the previous result to systems which are a more realistic model for the atmosphere and the ocean. Formally, the s ame behaviour still holds when a realistic dissipation scheme is appli ed with increasing amplitudes or when strongly dissipative systems, wh ich are not forced-dissipative perturbations of Hamiltonians, are used .