ofa~s and oceans ELSEVIER Dynamicsof Atmospheres and Oceans 27 (1997) 301-332 Extended Kalman filtering for vortex systems. Part I: Methodology and point vortices Kayo Ide *, Michael Ghil Department of Atmospheric Sciences and Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA 90095-1565, USA Received 13 February 1996;revised 18 July 1996; accepted 18 July 1996 Abstract Planetary flows--atmospheric and oceanic--are approximately two-dimensional and domi- nated by coherent concentrations of vorticity. Data assimilation attempts to determine optimally the current state of a fluid system from a limited number of current and past observations. In this two-part paper, an advanced method of data assimilation, the extended Kalman filter, is applied to the Lagrangian representation of a two-dimensional flow in terms of vortex systems. Smaller scales of motion are approximated here by stochastic forcing of the vortices. In Part I, the systems studied have either two point vortices, leading to regular motion or four point vortices and chaotic motion, in the absence of stochastic forcing. Numerical experiments are performed in the presence or absence of stochastic forcing. Point-vortex systems with both regular and chaotic motion can be tracked by a combination of Lagrangian observations of vortex positions and of Eulerian observations of fluid velocity at a few fixed points. Dynamically, the usual extended Kalman filter tends to yield insufficient gain if stochastic forcing is absent, whether the underlying system is regular or chaotic. Statistically, the type and accuracy of observations are the key factors in achieving a sufficiently accurate flow description. A simple analysis of the update mechanism supports the numerical results and also provides geometrical insight into them. In Part II, tracking of Rankine vortices with a finite core area is investigated and the results are used for observing-system design. © 1997 Elsevier Science B.V. 1. Introduction Data assimilation is a technique that describes the state of a dynamical system, subject to possible stochastic forcing, using limited knowledge about the system--such * Correspondingauthor. 0377-0265/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S0377-0265(97)00016-X