The standard Kalman filter and even the Extended Kalman filter (for nonlinear problems) proved inadequate. I’ve now placed my hope in what’s known as the Ensemble Kalman Filter:
Another sequential data assimilation method which has received a lot of attention is named the Ensemble Kalman Filter (EnKF). The method was originally proposed as a stochastic or Monte Carlo alternative to the deterministic [Extended Kalman filter] by Evensen (1994a). The EnKF was designed to resolve the two major problems related to the use of the [Extended Kalman filter] with nonlinear dynamics in large state spaces, i.e. the use of an approximate closure scheme and the huge computational requirements associated with the storage and forward integration of the error covariance matrix.
The EnKF gained popularity because of its simple conceptual formulation and relative ease of implementation, e.g. it requires no derivation of a tangent linear operator or adjoint equations and no integrations backward in time. Furthermore, the computational requirements are affordable and comparable to other popular sophisticated assimilation methods […].*
* Excerpt from Geir Evensen’s Data Assimilation: The Ensemble Kalman Filter, 2007, p. 38.