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.@en

Particle filters are non-Gaussian filters, which means that the assumption that the error distribution of the ensemble should be Gaussian is unnecessary. Like the ensemble Kalman filter, particle filters are based on the Monte Carlo approximation to represent the distribution of model states. It requires a substantial number of particles to approximate the probability density function of states in high-dimensional models, which is prohibitive for real applications. In order to overcome problems with high dimensionality, localization was applied in an Ensemble-type data assimilation system. This study combines the localization in LETKF (Local Ensemble Transformation Kalman Filter) with particle filters and proposes a new local particle filter with the model state space correction using Gamma test theory for high-dimensional models. A series of tests with various parameter settings, including different the numbers of particles, observation intervals, localization scale, inflation factors, and observation operators, were used to evaluate the performance of this new method using a Lorenz model with 40 variables. Besides, the proposed filter was applied in the Lorenz model with 1,000 variables to evaluate its performance in the model with higher dimensions. The results show that this approach can deal with the issue of dimensionality, which otherwise leads to the collapse of the particle filters in high-dimensional systems. The local particle filter is stable and has considerable potential for complex higher-dimensional models.

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In data assimilation(DA), various types of observations can be assimilated. Highly nonlinear observation operators are very common in geoscience, which goes against the linear and Gaussian assumptions of the Kalman filter. Particle filters are promising and possible solutions to non-linear issues in DA because they do not rely on Gaussian assumptions. However, traditional particle filters cannot be applied in high dimensional systems. As a method based on Bayes theory and Monte Carlo approximation, particle filters require numerous particles to represent the distribution of high-dimensional model states, which is called the curse of dimensionality. Because of this, it is prohibitive for realistic DA applications. In this study, a new local particle filter was proposed to deal with both the nonlinearity of observation operators and the dimensionality of particle filters. Localization methods are frequently used in Ensemble-type methods to solve the issues caused by limited ensemble size in high dimensional models. In this research, localization is applied to overcome the curse of dimensionality in particle filters. For the non-linear issue in observation operators, Gaussian process regression (GPR) is used to estimate the uncertainty of non-linear observation operators. At each update, the surrogate is adaptively trained and refined by current observations and model states. When the model states are transferred to the observation space, the surrogate can give the information about estimations and the corresponding uncertainty. A Lorenz model (1996) with 40 variables is used to evaluate the performance of this proposed local particle by conducting a set of experiments with different settings including the number of particles, the impact of localization scales, etc. To test its ability to deal with nonlinear issues, a highly nonlinear observation operator is designed and used in experiments. LETKF and local EAKF are two benchmarks in this research. The results show that the new method has a stable performance with high accuracy and it outperforms the two benchmarks. More importantly, for the non-linear case in this study, the new method only uses 25 particles to achieve a good performance. Although only the Lorenz model is considered in this study, it is highly likely to apply the proposed method to other models.@en

Particle filtering is a nonlinear and non-Gaussian dynamical filtering system. It has found widespread applications in hydrological data assimilation. In order to solve the loss of particle diversity exiting in resampling process of particle filter, this research proposes an improved particle filter algorithm using genetic algorithm optimization and Gamma test. This method combines the genetic algorithm and Gamma test into the resampling procedure of particle filter to improve the adaptability and performance of particle filter in data assimilation. First, the particles are classified to three different groups based on resampling method. The particles with high weight values remain unchanged. Then genetic algorithm is used to cross and variate the rest of the particles. In the process of the optimization, the Gamma test method is applied for monitoring the quality of the new generated particles. When the gamma statistic stays stable, the algorithm will end the optimization and continue to perturb next observations in particle algorithm. The algorithm is illustrated for the three-dimensional Lorenz model and the much more complex 40-dimensional Lorenz model. The results demonstrate this method can keep the diversity of the particles and enhance the performance of the particle filter, leading to the promising conjecture that the method is applicable to realistic hydrological problems.@en