This thesis describes two novel Bayesian methods: the Iterative Ensemble Square Filter (IEnSRF) and the Warp Ensemble Square Root Filter (WEnSRF) for solving the barcode detection problem, the deconvolution problem in well testing and the history matching problem of facies patterns. For the barcode detection problem, at the expanse of overestimating the posterior uncertainty, the IEnSRF efficiently achieves successful detections with very challenging real barcode images which the other considered methods and commercial software fail to detect. It also performs reliable detection on low-resolution images under poor ambient light conditions. For the deconvolution problem in well testing, the IEnSRF is capable of quantifying estimation uncertainty, incorporating the cumulative production data and estimating the initial pressure, which were thought to be unachievable in the existing well testing literature. The estimation results for the considered real benchmark data using the IEnSRF significantly outperform the existing methods in the commercial software. The WEnSRF is utilised for solving the history matching problem of facies patterns. Through the warping transformation, the WEnSRF performs adjustment on the reservoir features directly and is thus superior in estimating the large-scale complicated facies patterns. It is able to provide accurate estimates of the reservoir properties robustly and efficiently with reasonably reliable prior reservoir structural information.

Accurate knowledge of water content in seasonal snow can be helpful for water resource management. In this study, a distributed temperature index snow model based on temperature and precipitation as forcing data, is used to estimate snow storage in the Jollie catchment approximately 20km east of the main divide of the central Southern Alps, New Zealand. The main objective is to apply a frequently used assimilation method, the ensemble Kalman square root filter, to assimilate remotely sensed snow covered area into the model and evaluate the impacts of this approach on simulations of snow water equivalent. A 250m resolution remotely sensed data from Moderate Resolution Imaging Spectroradiometer (MODIS), specifically tuned to the study location was used. Temperature and precipitation were given on a 0.055 latitude/longitude grid. Precipitation was perturbed as input into the model, generating 100 ensemble members, which represented model error. Only observations of snow covered area that had less that 25% cloud cover classification were used in the assimilation precess. The error in the snow covered area observations was assumed to be 0.1 and grow linearly with cloud cover fraction up to 1 for a totally cloud covered pixel. As the model was not calibrated, two withholding experiments were conducted, in which observations withheld from the assimilation process were compared to the results. Two model states were updated in the assimilation, the total snow accumulation state variable and the total snow melt state variable. The results of this study indicate that the model underestimates snow storage at the end of winter and/or does not detect snow fall events during the ablation period. The assimilation method only affected simulated snow covered area and snow storage during the ablation period. That corresponded to higher correlation between modelled snow cover area and the updated state variables. Withholding experiments show good agreement between observations and simulated snow covered area. This study successfully applied the ensemble Kalman square root filter and showed its applicability for New Zealand conditions.

This thesis introduces novel ideas in subsurface reservoir model calibration known as History Matching in the reservoir engineering community. The target of history matching is to mimic historical pressure and production data from the producing wells with the output from the reservoir simulator for the sole purpose of reducing uncertainty from such models and improving confidence in production forecast. Ensemble based methods such as the Ensemble Kalman Filter (EnKF) and Ensemble Smoother with Multiple Data Assimilation (ES-MDA) as been proposed for history matching in literature. EnKF/ES-MDA is a Monte Carlo ensemble nature filter where the representation of the covariance is located at the mean of the ensemble of the distribution instead of the uncertain true model. In EnKF/ES-MDA calculation of the gradients is not required, and the mean of the ensemble of the realisations provides the best estimates with the ensemble on its own estimating the probability density. However, because of the inherent assumptions of linearity and Gaussianity of petrophysical properties distribution, EnKF/ES-MDA does not provide an acceptable history-match and characterisation of uncertainty when tasked with calibrating reservoir models with channel like structures. One of the novel methods introduced in this thesis combines a successive parameter and shape reconstruction using level set functions (EnKF/ES-MDA-level set) where the spatial permeability fields' indicator functions are transformed into signed distances. These signed distances functions (better suited to the Gaussian requirement of EnKF/ES-MDA) are then updated during the EnKF/ES-MDA inversion. The method outperforms standard EnKF/ES-MDA in retaining geological realism of channels during and after history matching and also yielded lower Root-Mean-Square function (RMS) as compared to the standard EnKF/ES-MDA. To improve on the petrophysical reconstruction attained with the EnKF/ES-MDA-level set technique, a novel parametrisation incorporating an unsupervised machine learning method for the recovery of the permeability and porosity field is developed. The permeability and porosity fields are posed as a sparse field recovery problem and a novel SELE (Sparsity-Ensemble optimization-Level-set Ensemble optimisation) approach is proposed for the history matching. In SELE some realisations are learned using the K-means clustering Singular Value Decomposition (K-SVD) to generate an overcomplete codebook or dictionary. This dictionary is combined with Orthogonal Matching Pursuit (OMP) to ease the ill-posed nature of the production data inversion, converting our permeability/porosity field into a sparse domain. SELE enforces prior structural information on the model during the history matching and reduces the computational complexity of the Kalman gain matrix, leading to faster attainment of the minimum of the cost function value. From the results shown in the thesis; SELE outperforms conventional EnKF/ES-MDA in matching the historical production data, evident in the lower RMS value and a high geological realism/similarity to the true reservoir model.

To evaluate the traffic state over time and space, several models can be used. A typical model for estimating the state of the traffic for a stretch of road or a road network is the cell transmission model, which is a form of state space model. This kind of model typically needs to be calibrated since the different roads have different properties. This thesis will present a calibration framework for the velocity based cell transmission model, the CTM-v. The cell transmission model for velocity is a discrete time dynamical system that can model the evolution of the velocity field on highways. Such a model can be fused with an ensemble Kalman filter update algorithm for the purpose of velocity data assimilation. Indeed, enabling velocity data assimilation was the purpose for ever developing the model in the first place and it is an essential part of the Mobile Millennium research project. Therefore a systematic methodology for calibrating the cell transmission is needed. This thesis presents a framework for calibration of the velocity based cell transmission model that is combined with the ensemble Kalman filter. The framework consists of two separate methods, one is a statistical approach to calibration of the fundamental diagram. The other is a black box optimization method, a simplification of the complex method that can solve inequality constrained optimization problems with non-differentiable objective functions. Both of these methods are integrated with the existing system, yielding a calibration framework, in particular highways were stationary detectors are part of the infrastructure. The output produced by the above mentioned system is highly dependent on the values of its characterising parameters. Such parameters need to be calibrated so as to make the model a valid representation of reality. Model calibration and validation is a process of its own, most often tailored for the researchers models and purposes. The combination of the two methods are tested in a suit of experiments for two separate highway models of Interstates 880 and 15, CA which are evaluated against travel time and space mean speed estimates given by Bluetooth detectors with an error between 7.4 and 13.4 % for the validation time periods depending on the parameter set and model.

This thesis presents an investigation of ensemble Gaussianity, the effect of non- Gaussianity on covariance structures, storm-centered data assimilation techniques, and the relationship between commonly used data assimilation variables and the underlying dynamics for the case of Hurricane Humberto. Using an Ensemble Kalman Filter (EnKF), a comparison of data assimilation results in Storm-centered and Eulerian coordinate systems is made. In addition, the extent of the non-Gaussianity of the model ensemble is investigated and quantified. The effect of this non-Gaussianity on covariance structures, which play an integral role in the EnKF data assimilation scheme, is then explored. Finally, the correlation structures calculated from a Weather Research Forecast (WRF) ensemble forecast of several state variables are investigated in order to better understand the dynamics of this rapidly intensifying cyclone. Hurricane Humberto rapidly intensified in the northwestern Gulf of Mexico from a tropical disturbance to a strong category one hurricane with 90 mph winds in 24 hours. Numerical models did not capture the intensification of Humberto well. This could be due in large part to initial condition error, which can be addressed by data assimilation schemes. Because the EnKF scheme is a linear theory developed on the assumption of the normality of the ensemble distribution, non-Gaussianity in the ensemble distribution used could affect the EnKF update. It is shown that multiple state variables do indeed show significant non-Gaussianity through an inspection of statistical moments. In addition, storm-centered data assimilation schemes present an alternative to traditional Eulerian schemes by emphasizing the centrality of the cyclone to the assimilation window. This allows for an update that is most effective in the vicinity of the storm center, which is of most concern in mesoscale events such as Humberto. Finally, the effect of non-Gaussian distributions on covariance structures is examined through data transformations of normal distributions. Various standard transformations of two Gaussian distributions are made. Skewness, kurtosis, and correlation between the two distributions are taken before and after the transformations. It can be seen that there is a relationship between a change in skewness and kurtosis and the correlation between the distributions. These effects are then taken into consideration as the dynamics contributing to the rapid intensification of Humberto are explored through correlation structures.

Filtering problems in high dimensional geophysical applications often require spatially continuous models to interpolate spatially and temporally sparse data. Many applications in numerical weather and ocean state prediction are concerned with tracking and assessing the uncertainty in the position of large scale vorticity features, such as storm fronts, jets streams, and hurricanes. Quantifying the amplitude variance in these features is complicated by the fact that both height and lateral perturbations in the feature geometry are represented in the same covariance estimate. However, when there are sufficient observations to detect feature information like spatial gradients, the positions of these features can be used to further constrain the filter, as long as the statistical model (cost function) has provisions for both height perturbations and lateral displacements. Several authors since the 1990s have proposed various formalisms for the simultaneous modeling of position and amplitude errors, and the typical approaches to computing the generalized solutions in these applications are variational or direct optimization. The ensemble Kalman filter is often employed in large scale nonlinear filtering problems, but its predication on Gaussian statistics causes its estimators suffer from analysis deflation or collapse, as well as the usual curse of dimensionality in high dimensional Monte Carlo simulations. Moreover, there is no theoretical guarantee of the performance of the ensemble Kalman filter with nonlinear models. Particle filters which employ importance sampling to focus attention on the important regions of the likelihood have shown promise in recent studies on the control of particle size. Consider an ensemble forecast of a system with prominent feature information. The correction of displacements in these features, by pushing them into better agreement with observations, is an application of importance sampling, and Monte Carlo methods, including particle filters, and possibly the ensemble Kalman filter as well, are well suited to applications of feature displacement correction. In the present work, we show that the ensemble Kalman filter performs well in problems where large features are displaced both in amplitude and position, as long as it is used on a statistical model which includes both function height and local position displacement in the model state. In a toy model, we characterize the performance-degrading effect that untracked displacements have on filters when large features are present. We then employ tools from classical physics and fluid dynamics to statistically model displacements by area-preserving coordinate transformations. These maps preserve the area of contours in the displaced function, and using strain measures from continuum mechanics, we regularize the statistics on these maps to ensure they model smooth, feature-preserving displacements. The position correction techniques are incorporated into the statistical model, and this modified ensemble Kalman filter is tested on a system of vortices driven by a stochastically forced barotropic vorticity equation. We find that when the position correction term is included in the statistical model, the modified filter provides estimates which exhibit substantial reduction in analysis error variance, using a much smaller ensemble than what is required when the position correction term is removed from the model.

The standard ensemble filtering schemes such as Ensemble Kalman Filter (EnKF) and Sequential Monte Carlo (SMC) do not properly represent states of low priori probability when the number of samples is too small and the dynamical system is high dimensional system with highly non-Gaussian statistics. For example, when the standard ensemble methods are applied to two well-known simple, but highly nonlinear systems such as a one-dimensional stochastic diffusion process in a double-well potential and the well-known three-dimensional chaotic dynamical system of Lorenz, they produce erroneous results to track transitions of the systems from one state to the other.In this dissertation, a set of new parametric resampling methods are introduced to overcome this problem. The new filtering methods are motivated by a general H-theorem for the relative entropy of Markov stochastic processes. The entropy-based filters first approximate a prior distribution of a given system by a mixture of Gaussians and the Gaussian components represent different regions of the system. Then the parameters in each Gaussian, i.e., weight, mean and covariance are determined sequentially as new measurements are available. These alternative filters yield a natural generalization of the EnKF method to systems with highly non-Gaussian statistics when the mixture model consists of one single Gaussian and measurements are taken on full states.In addition, the new filtering methods give the quantities of the relative entropy and log-likelihood as by-products with no extra cost. We examine the potential usage and qualitative behaviors of the relative entropy and log-likelihood for the new filters. Those results of EnKF and SMC are also included. We present results of the new methods on the applications to the above two ordinary differential equations and one partial differential equation with comparisons to the standard filters, EnKF and SMC. These results show that the entropy-based filters correctly track the transitions between likely states in both highly nonlinear systems even with small sample size N=100.

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Accurate knowledge of water content in seasonal snow can be helpful for water resource management. In this study, a distributed temperature index snow model based on temperature and precipitation as forcing data, is used to estimate snow storage in the Jollie catchment approximately 20km east of the main divide of the central Southern Alps, New Zealand. The main objective is to apply a frequently used assimilation method, the ensemble Kalman square root filter, to assimilate remotely sensed snow covered area into the model and evaluate the impacts of this approach on simulations of snow water equivalent. A 250m resolution remotely sensed data from Moderate Resolution Imaging Spectroradiometer (MODIS), specifically tuned to the study location was used. Temperature and precipitation were given on a 0.055 latitude/longitude grid. Precipitation was perturbed as input into the model, generating 100 ensemble members, which represented model error. Only observations of snow covered area that had less that 25% cloud cover classification were used in the assimilation precess. The error in the snow covered area observations was assumed to be 0.1 and grow linearly with cloud cover fraction up to 1 for a totally cloud covered pixel. As the model was not calibrated, two withholding experiments were conducted, in which observations withheld from the assimilation process were compared to the results. Two model states were updated in the assimilation, the total snow accumulation state variable and the total snow melt state variable. The results of this study indicate that the model underestimates snow storage at the end of winter and/or does not detect snow fall events during the ablation period. The assimilation method only affected simulated snow covered area and snow storage during the ablation period. That corresponded to higher correlation between modelled snow cover area and the updated state variables. Withholding experiments show good agreement between observations and simulated snow covered area. This study successfully applied the ensemble Kalman square root filter and showed its applicability for New Zealand conditions.