Improving Data Assimilation Algorithms for Enhanced Environmental Predictions

Data Assimilation (DA) methods provide a means of combining model output with observations based on their respective uncertainties. They are considered an invaluable tool in a wide variety of disciplines, particularly in hydrologic and meteorological forecasting. There is significant potential to improve existing DA methods, which have predominantly been developed in an ad-hoc manner to enhance their applicability to complex real world problems. In particular, relatively little attention has been devoted to one of the most fundamental aspects of DA: Model uncertainty quantification. This thesis aims to develop improved DA based methods for highly non-Gaussian/non-linear systems, with a particular focus on hydrologic and atmospheric systems. It also examines how DA methods can be enhanced to solve problems outside of their traditional application domain. Specifically, two overarching aims are investigated: 1) the development of DA based methods for estimating time varying model parameters, with the ultimate goal of improving hydrologic predictions in dynamic catchments; and 2) the development of objective model uncertainty quantification techniques for use in state-estimation DA. Firstly, a DA based method for sequentially estimating time varying model parameters is investigated. Two new methods for proposing prior parameter distributions are developed, which can be utilised depending on the amount of a priori information available regarding the form of temporal variations in model parameters. The methods are verified against synthetic data and applied to a number of real catchments with land use change, without relying on prior information of such changes. This approach represents a promising modelling paradigm for hydrologists faced with providing predictions in rapidly changing catchments. In addressing the second objective, two model uncertainty quantification methods are developed for DA in partially observed systems with highly non-Gaussian uncertainties. The methods proposed in this thesis address some of the major shortcomings in existing methods related to objectivity and ability to characterise non-Gaussian errors. Their efficacy is demonstrated through application to flood forecasting problems, and also for state estimation in a partially observed multi-scale atmospheric toy model. In all cases, the proposed methods are shown to provide improved forecasts and updates compared to standard approaches

Hydrologic Modeling in Dynamic Catchments: A Data Assimilation Approach

The transferability of conceptual hydrologic models in time is often limited by both their structural deficiencies and adopted parameterizations. Adopting a stationary set of model parameters ignores biases introduced by the data used to derive them, as well as any future changes to catchment conditions. Although time invariance of model parameters is one of the hallmarks of a high quality hydrologic model, very few (if any) models can achieve this due to their inherent limitations. It is therefore proposed to consider parameters as potentially time varying quantities, which can evolve according to signals in hydrologic observations. In this paper, we investigate the potential for Data Assimilation (DA) to detect known temporal patterns in model parameters from streamflow observations. It is shown that the success of the DA algorithm is strongly dependent on the method used to generate background (or prior) parameter ensembles (also referred to as the parameter evolution model). A range of traditional parameter evolution techniques are considered and found to be problematic when multiple parameters with complex time variations are estimated simultaneously. Two alternative methods are proposed, the first is a Multilayer approach that uses the EnKF to estimate hyperparameters of the temporal structure, based on apriori knowledge of the form of nonstationarity. The second is a Locally Linear approach that uses local linear estimation and requires no assumptions of the form of parameter nonstationarity. Both are shown to provide superior results in a range of synthetic case studies, when compared to traditional parameter evolution techniques

Perturbations and Projections of Kalman-Bucy Semigroups Motivated by Methods in Data Assimilation

The purpose of this work is to analyse the effect of various perturbations and projections of Kalman-Bucy semigroups and Riccati equations. The original motivation was to understand the behaviour of various regulation methods used in ensemble Kalman filtering (EnKF). For example, covariance inflation-type methods (perturbations) and covariance localisation methods (projections) are commonly used in the EnKF literature to ensure well-posedness of the sample covariance (e.g. sufficient rank) and to 'move' the sample covariance closer (in some sense) to the Riccati flow of the true Kalman filter. In the limit, as the number of samples tends to infinity, these methods drive the sample covariance toward a solution of a perturbed, or projected, version of the standard (Kalman-Bucy) differential Riccati equation. The behaviour of this modified Riccati equation is investigated here. Results concerning continuity (in terms of the perturbations), boundedness, and convergence of the Riccati flow to a limit are given. In terms of the limiting filters, results characterising the error between the perturbed/projected and nominal conditional distributions are given. New projection-type models and ideas are also discussed within the EnKF framework; e.g. projections onto so-called Bose-Mesner algebras. This work is generally important in understanding the limiting bias in both the EnKF empirical mean and covariance when applying regularisation. Finally, we note the perturbation and projection models considered herein are also of interest on their own, and in other applications such as differential games, control of stochastic and jump processes, and robust control theory, etc

Detecting Non-Stationary Hydrologic Model Parameters in a Paired Catchment

Non-stationarity represents one of the major challenges facing hydrologists. There exists a need to develop modelling systems that are capable of accounting for potential catchment changes, in order to provide useful predictions for the future. Such changes may be due to climatic temporal variations or human induced changes to land cover. Extensive research has been undertaken on the impacts of land-use change on hydrologic behaviour, however, few studies have examined this issue in a predictive modelling context. In this paper, we investigate whether a time varying model parameter estimation framework that uses the principles of Data Assimilation can improve prediction for two pairs of experimental catchments in Western Australia. All catchments were initially forested, but after three years one catchment was fully cleared whilst another had only 50% of its area cleared. Their adjacent catchments remained unchanged as a control. Temporal variations in parameters were detected for both treated catchments, with no comparable variations for the control catchments. Improved streamflow prediction and representation of soil moisture dynamics were also seen for the time varying parameter case, compared to when a time invariant parameter set from the calibration period was used. While we use the above mentioned catchments to illustrate the usefulness of the approach, the methods are generic and equally applicable in other settings. This study serves as an important validation step to demonstrate the potential for time varying model structures to improve both predictions and modelling of changing catchments

Hemodynamic Data Assimilation in a Subject-specific Circle of Willis Geometry

PURPOSE: The anatomy of the circle of Willis (CoW), the brain's main arterial blood supply system, strongly differs between individuals, resulting in highly variable flow fields and intracranial vascularization patterns. To predict subject-specific hemodynamics with high certainty, we propose a data assimilation (DA) approach that merges fully 4D phase-contrast magnetic resonance imaging (PC-MRI) data with a numerical model in the form of computational fluid dynamics (CFD) simulations. METHODS: To the best of our knowledge, this study is the first to provide a transient state estimate for the three-dimensional velocity field in a subject-specific CoW geometry using DA. High-resolution velocity state estimates are obtained using the local ensemble transform Kalman filter (LETKF). RESULTS: Quantitative evaluation shows a considerable reduction (up to 90%) in the uncertainty of the velocity field state estimate after the data assimilation step. Velocity values in vessel areas that are below the resolution of the PC-MRI data (e.g., in posterior communicating arteries) are provided. Furthermore, the uncertainty of the analysis-based wall shear stress distribution is reduced by a factor of 2 for the data assimilation approach when compared to the CFD model alone. CONCLUSION: This study demonstrates the potential of data assimilation to provide detailed information on vascular flow, and to reduce the uncertainty in such estimates by combining various sources of data in a statistically appropriate fashion