Description

In the extended Kalman filter (EKF) for nonlinear systems (Jazwinski, 1970), approximate expressions are found for the propagation of the conditional mean and its associated covariance matrix. The structure of the propagation equations is similar to those of the classic Kalman filter for a linear system, as they are linearized about the conditional mean. In high-dimensional problems (such as storm surge forecasting or coastal models) the EKF often has to be simplified by a suboptimal scheme for reduction of the computational burden.

Software

• Matlab

Advantages

EKF may be applied to estimation of nonlinear multidimensional systems with small non-linearities. This method handles well small non-linearities. It has been successfully applied to the land data assimilation problem (Entekhabi et al., 1994; Walker et al., 2001); as well as in 2D hydrodynamics and oceanography (Canizares et al., 2001; Madsen and Canizares, 1999).

Disadvantages

EKF is rather inefficient in case of very nonlinear systems as explained in Julier and Uhlman (1996). Moreover the method is not suited for large dimension systems, as the calculation of the derivatives, using a finite difference method, demands n + 1 model evaluations for each time step (n is the dimension of the state vector), and q + 1 evaluations of the observation operator (q is dimension of the observation space). The other possibility is to write a tangent linear model, but it is generally difficult for complex models or impossible for highly non-linear models. The derivation of a tangent linear model to approximate a complex system may be very tedious, as well as techniques to treat the instabilities which might arise from such an approximation. (Evensen, 1992)

References and Further Reading

The basic discussion and derivation of the EKF is given by Jazwinski (1970) and Gelb (1974). A very good discussion on data assimilation techniques and their applications in environmental problems is given by Bertino et al. (2003). General introduction to EKF is given by Decourt (2003).

Bertino, L., Evensen, G. and Vackernagel, H., 2003. Sequential data assimilation techniques in oceanography. International Statistical Review, 71: 223-242.

Canizares, R., Madsen, H., Jensen, H.R. and Vested, H.J., 2001. Developments in operational shelf sea modelling in danish waters. Estuarine and Coastal Shelf Science, 53: 595-605.

Drecourt, J.-P., 2003. Kalman filtering in hydrological modeling - DAIHM Technical Report 2003-1.

Entekhabi, D., Nakamura, H. and Njoku, E., 1994. Solving the inverse problem for soil moisture and temperature profiles by sequential assimilation of multifrequency remotely sensed observations. IEEE Trans. Geosci. Remote Sens., 32: 4328-448.

Evensen, G., 1992. Using the extended Kalman filter with a multilayer quasi-geostrophic ocean model. Journal of Geophysical Research, 97(C11): 17905–17924.

Gelb, A., 1974. Applied Optimal Estimation. MIT Press, Cambridge, MA.

Jazwinski, A.H., 1970. Stochastic Processes and Filtering Theory. Elsevier, New York.

Julier, S.J. and Uhlmann, J.K., 1996. A general method for approximating nonlinear transformations of probability distributions.

Madsen, H. and Canizares, R., 1999. Comparison of extended and ensemble Kalman fitlers for data assimilation in coastal area modelling. International Journal For Numerical Methods in Fluids, 31: 961-981.

Walker, J.P., Willgoose, G.R. and Kalma, J.D., 2001. One-dimensional soil moisture profile retrieval by assimilation of near-surface observations: A comparison of retrieval algorithms. Advances in Water Resources, 24: 631– 650.