ABSTRACT
Probably, the most well known and widely used sequential data assimilation technique is the Kalman filter. It was introduced by Kalman [37] in 1960 within the field of control theory. The Kalman filter is the statistically optimal method for the sequential assimilation of data into linear numerical models and provides an estimate of the state of the system at the current time step based on all measurements of the system available up to and including the current time. Typical applications of the Kalman filter in linear models for storm surge forecast can be found in Heemink [31], Vested et al. [65] and Pietrazk and Bolding [49]. In these cases, a linearised hydrodynamic model was coupled to a stationary Kalman filter which could be calculated off-line. The Kalman filter can also be applied to weakly non-linear systems using the so-called extended Kalman filter (Kalman and Bucy [38]). The computational burden associated with the propagation of the error statistics forward in time, however makes the application of the extended Kalman filter to large systems untenable. As a result, new Kalman-filter-based algorithms have been developed. Within these new algorithms, a first approach posed the problem of propagating the error statistics in a more efficient manner. This leads to the so called Suboptimal Schemes (SOS) which can in turn be grouped into two approaches: simplification of model dynamics (Todling and Cohn [59]) which makes use of a state-transition matrix with lower rank; and simplification of covariance modelling as the Reduced Rank Square Root Filter (Verlaan and Heemink [63]) which approximates the error covariance matrix by a matrix of lower rank. A different approach for solving this problem is the ensemble Kalman filter of Evensen [23] where the error statistics are calculated using Monte Carlo methods.
