In Chapters 9 and 11 we discussed finite ensemble filtering of nonlinear dynamical systems with Kalman-based (or linear) techniques and we demonstrated their advantages in the less turbulent regime and their limitation in the sparsely observed fully turbulent regime. We also showed their sensitivity toward variations of filtering parameters such as the variance inflation and ensemble size. Subsequently in Chapters 12–14, we discussed alternative strategies based on cheap reduced stochastic models to avoid all the disadvantages of ensemble Kalman filtering strategies and demonstrated their skill for filtering various turbulent nonlinear systems ranging from the idealized toy models like L-96 to quasi-geostrophic turbulence to turbulent diffusion models with non-Gaussian statistics.

Recently, there is an emerging need to develop nonlinear filtering methods beyond the ensemble Kalman filters since most physical problems are nonlinear and sometimes have highly non-Gaussian distributions. One of the most difficult problems is to devise a practically useful ensemble method based on the theoretically well-established particle filtering (or sequential Monte Carlo) methods (Del Moral, 1996; Del Moral and Jacod, 2001; Doucet et al., 2008). The major challenge is to have a particle filter that is practically skillful for high-dimensional complex turbulent systems with only small ensemble size (small number of particles); this is very important because of the large overload in generating individual ensemble members through the forward dynamical operator (Haven et al., 2005).