Nonlinear filtering with particle filters

Convective phenomena in the atmosphere, such as convective storms, are characterized by very fast, intermittent and seemingly stochastic processes. They are thus difficult to predict with Numerical Weather Prediction (NWP) models, and difficult to estimate with data assimilation methods that combine prediction and observations. In this thesis, nonlinear data assimilation methods are tested on two idealized convective scale cloud models, developed in [58] and [59]. The aim of this work was to apply the particle filter, a method specifically designed for nonlinear models, to the two toy models that resemble some properties of convection. Potential problems and characteristic features of particle filter methodology were analyzed, having in mind possible tests on far more elaborate NWP models. The first model, the stochastic cloud model, is a one-dimensional birth-death model initialized by a Poisson distribution, where clouds randomly appear or die independently from each other on a set of grid-points. This purely stochastic model is physically unreal- istic, but since it is highly nonlinear and non-Gaussian, it contains minimal requirements for representing main features of convection. The derivation of the transition probability density function (PDF) of the stochastic cloud model made it possible to understand better the weighting mechanism involved in the particle filter. This mechanism, which associates a weight to particles (state vectors) according to their likelihood with respect to observa- tions and to their evolution in time, is followed by resampling, where particles with high probability are replicated, and others eliminated. The ratio between magnitudes of the ob- servation probability distribution and the transition probability is shown to determine the selection process of particles at each time step, where data and prediction are combined. Further, a strong sensitivity of the filter to the observation density and to the speed of the cloud variability (given by the cloud life time) is demonstrated. Thus, the particle filter can outperform some simpler methods for certain observation densities, whereas it does not bring any improvement when using others. Similarly, it leads to good results for stationary cloud fields while having difficulties to follow fast varying cloud fields, because any change in the model state variable is potentially penalized. The main difficulty for the filter is the fact that this model is discrete, while the filter was designed for data assimilation of continuous fields. However, by representing an extreme testbed for the particle filter, the stochastic cloud model shows the importance of the observation and model error densities for the selection of particles, and it stresses the influence of the chosen model parameters on the filter’s performance. The second model considered was the modified shallow water model. It is based on the shallow water equations, to which is added a small stochastic noise in order to trigger convection, and an equation for the evolution of rain. It contains spatial correlations and is represented by three dynamical variables - wind speed, water height and rain concentration - which are linked together. A reduction of the observation coverage and of the number of updated variables leads to a relative error reduction of the particle filter compared to an ensemble of particles that are only continuously pulled (nudged) to observations, for a certain range of nudging parameters. But not surprisingly, reducing data coverage in- creases the absolute error of the filter. We found that the standard deviation of the error density exponents is a quantity that is responsible for the relative success of the filter with respect to nudging-only. In the case where only one variable is assimilated, we formulated a criterion that determines whether the particle filter outperforms the nudged ensemble. A theoretical estimate is derived for this criterion. The theoretical values of this estimate, which depends on the parameters involved in the assimilation set up (nudging intensity, model and observation error covariances, grid size, ensemble size,.), are roughly in accor- dance with the numerical results. In addition, comparing two different nudging matrices that regulate the magnitude of relaxation of the state vectors towards the observations, showed that a diagonally based nudging matrix leads to smaller errors, in the case of assimilating three variables, than a nudging matrix based on stochastic errors added in each integration time step. We conclude that the efficient particle filter could bring an improvement with respect to conventional data assimilation methods, when it comes to the convective scale. Its success, however, appears to depend strongly on the parameters of the test setting.