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.