Data Assimilation in High Dimensional Systems Using Local Particle Filters: Overcoming the curse of dimensionality in hydrology

This dissertation's ultimate goal is to provide solutions to two problems that the promising data assimilation method, called the Particle Filter, has when applied to high dimensional non-linear models, such as those often used in hydrological research and forecasting. Two local particle filters have been proposed to overcome three major issues. Firstly, the curse of dimensionality caused by high dimensional models. Secondly, the uncertainty brought by the data assimilation method itself and finally the problem of nonlinearity in observation operators that link model states to observations. Both newly introduced data assimilation algorithms have been assessed using the Lorenz model (1996), a toy model that provides a perfect evaluation environment for such methods because it is a one-dimensional discrete chaotic model, which can simulate the behavior of changes of atmosphere. One local particle filter has been used in a practical application in hydrology to improve discharge accuracy in the Rhine river basin by assimilating satellite soil moisture into the PCR-GLOWB hydrological model.

The curse of dimensionality is well-known in particle filters. It happens in high dimensional models because, to remain accurate, the number of particles needs to increase exponentially with the increase of the model scale (ie. model dimension). One possible solution to avoid this curse is to apply localization in particle filters. Both proposed particle filters are based on a localization method. Uncertainty sources in data assimilation are many, and it is not easy to separate all of them clearly and directly. The two variants of the particle filter proposed in this thesis focus on different issues.

The localization used in the first particle filters divided the whole analysis of data assimilation into small batches for each model state. Each local analysis is independent, and it only assimilates observations within the localization scale. In the process it quantifies the uncertainty that is introduced by the data assimilation process itself. The localization method for the second local particle filter variant used another strategy. In its procedure, all observations are assimilated one by one, and each observation only affects near model states within the localization radius. When all observations are assimilated sequentially, all model states are updated. In addition, the second particle filter variant tried to solve the problem caused by nonlinear observation operators. To overcome the latter problems, the nonlinear observation operator was replaced by a surrogate model, named the Gaussian process regression model. For the calculation of the weights for each particle, model states needed to be transferred into the observation space. A Gaussian process regression surrogate model makes the transition process more straightforward in the nonlinear case because it provides the mean and standard deviation of estimates. Both local particle filter variants introduced in this thesis were evaluated thoroughly, and all results demonstrated that they performed satisfactorily in the specific nonlinear case and can be applied in high dimensional systems.

In addition to testing both local particle filters in the controlled Lorenz model, LPF-GT has also been verified as beneficial in a case study with the hydrological model PCR-GLOBWB. The specific study area focused on the Rhine river basin. The local particle filters have been applied to assimilate satellite soil moisture from the SMAP mission into the PCR-GLOBWB model to improve discharge estimates. Results show that the local particle filter performed well and significantly improved discharge accuracy by assimilating SMAP soil moisture. The new LPF-GT only requires a handful of particles to reach better performance in the Rhine river basin. This is particularly useful and practical for large-scale models that are often used in hydrology. Only requiring a small number of particles is the primary advantage of this data assimilation method because it saves lots of computational costs. In addition, the use of the localization in this particle filter makes the update for each model state independent from each other and can be conducted in parallel. Thus, the efficiency of this data assimilation method can be improved further.


In conclusion, the new additions to the particle filter proposed in this thesis are stable and can provide satisfying accuracy in nonlinear cases and for high dimensional models. Both of them have been proven to perform well in a toy model with many dimensions where they have direct value in solving the curse of dimensionality and nonlinearity. More importantly, they are valuable data assimilation methods to give direct insights into how to cope with uncertainty in nonlinear cases and to offer data assimilation frameworks for developing new particle filters in the future. The successful hydrological application of data assimilation using local particle filters in this research shows its considerable potential in hydrology.