J5.4 4D ENSEMBLE KALMAN FILTERING FOR ASSIMILATION OF ASYNCHRONOUS OBSERVATIONS T. Sauer * George Mason University, Fairfax, VA 22030 B.R. Hunt, J.A. Yorke, A.V. Zimin, E. Ott, E.J. Kostelich † , I. Szunyogh, G. Gyarmati, E. Kalnay, D.J. Patil, University of Maryland, College Park, MD 20742 Abstract A 4-dimensional ensemble Kalman filter method (4DEnKF), which adapts ensemble Kalman filtering to the assimilation of observations that are asyn- chronous with the analysis cycle, is discussed. In the ideal case of linear dynamics between consecu- tive analyses, the algorithm is equivalent to Kalman filtering assimilation at each observation time. Tests of 4DEnKF on the Lorenz 40 variable model are conducted. 1. INTRODUCTION In ensemble Kalman filtering, a set of background trajectories is integrated by the dynamical model, and used to estimate the background covariance matrix. Numerical experiments have shown that en- semble Kalman filters (EnKF, e.g., Evensen 1994; Evensen and van Leewen 1996, Houtekamer and Mitchell 1998, 2001; Hamill and Snyder 2000) are efficient ways to carry out data assimilation from simple models to state-of-the-art operational nu- merical prediction models. The ensemble square- root Kalman filter approach (Tippett et al. 2002; Bishop et al. 2001; Anderson 2001; Whitaker and Hamill 2002; Ott et al. 2002) has attracted much re- cent attention. A further advantage of the ensemble square root Kalman filter is that it allows asynchrononous observations to be naturally assimilated. The Four- Dimensional Ensemble Kalman Filter (4DEnKF), first proposed in Hunt et al. (2003), is a practical way of unifying the ensemble Kalman filter and the four-dimensional variational approach. Instead of treating observations as if they occur only at as- similation times, we can take observations times into account in a natural way, even if they are dif- ferent from the assimilation times. The observa- tional increments are propagated at intermediate time steps using the ensemble of background fore- casts. This extension of the EnKF to a 4DEnKF can * Corresponding author address: George Mason University, Fairfax, VA 22030, USA † Current address: Arizona State University, Tempe, AZ 85287, USA be considered analogous to the extension of the three-dimensional variational technique (3D-Var) to the four dimensional variational technique (4D-Var). The idea is to infer the linearized model dynam- ics from the ensemble instead of the tangent-linear map, as done in conventional 4D-Var schemes. Furthermore, in the case of linear dynamics, our technique is equivalent to instantaneous assimila- tion of measured data. 2. ENSEMBLE KALMAN FILTERS To set notation, we recall the EnKF method when the observations are synchronous with the analysis. Let ˙ x m = G m (x 1 , ... ,x M ) (1) for m =1, ... ,M be a continuous dynamical system representing the background vector field, where x = (x 1 , ... ,x M ). The Ensemble Kalman Filter is de- signed to track the evolution, under this dynamical system, of an M-dimensional Gaussian distribution centered at x (t ) with covariance matrix P (t ). In the implementation of (Ott et al. 2003; Tippett et al. 2003), k + 1 trajectories of (1) are followed starting from initial conditions x a(1) , ... ,x a(k +1) over a time interval [t a ,t b ]. Since the system is typically high-dimensional, assume that k +1 ≤ M. The k + 1 initial conditions are chosen so that their sample mean and sample covariance are x (t a ) and P (t a ), respectively. After running the system over the time interval, we denote the trajectory points at the end of the interval by x b(1) , ... ,x b(k +1) , and compute a new sample mean x b and sample covariance P b from these k + 1 vectors. Define the mean vector x b = 1 k +1 k +1 ∑ i =1 x b(i ) and δx b(i ) = x b(i ) − x b . Set the matrix X b = 1 √ k [δx b(1) |···|δx b(k +1) ]