Ensemble Transport Smoothing - Part II: nonlinear updates
This study builds on the preceding study, and explores the performance of a nonlinear backward ensemble transport smoothers (EnTS) in greater detail. We demonstrate this performance in a number of different nonlinear and chaotic test cases:
a one-dimensional, periodically-bimodal sinusoidal system,
the three-dimensional Lorenz-63 system, and
the 40-dimensional Lorenz-96 system.
In our experiments, the nonlinear backward EnTS not only outperforms its linear counterpart - the ensemble Rauch-Tung-Striebel smoother - but even a state-of-the-art reference, the iterative ensemble Kalman Smoother.
The greatest strength of the EnTS is its adaptability, based on the freedom to adjust its parameterization to the demands of the problem at hand. We demonstrate the differing demand in nonlinear complexity between Lorenz-63 and Lorenz-96. The EnTS also allows us to exploit conditional independence properties, which not only makes the implementation of localization trivial, but increases the computational efficiency of the method, allowing it - in principle - to scale to even higher-dimensional systems.
Ensemble Transport Smoothing - Part I: unified framework
Smoothers are specialized Bayesian inference algorithms, seeking to estimate the posterior of a sequence of states x given a sequence of observations y. Where filters only concern themselves with the latest state, smoothers update past states as well. This makes them highly useful for systems with an interest in hindsight. The practical implementation of smoothers requires ensemble methods, which come with their own limitations:
Kalman-type smoothers such as the Ensemble Kalman Smoother (EnKS) or the Ensemble Rauch-Tung-Striebel smoother (EnRTSS) are sample-efficient, but linear. This means they are only exact if all distributions involved are Gaussian.
Methods based on importance sampling such as Particle Smoothers can implement highly nonlinear updates, but are very sample-inefficient.
This is a problem for models which are at once nonlinear (thus non-Gaussian) and computationally expensive. In the first part of our two-part smoothing study, we propose a general ensemble transport smoother (EnTS), which can accommodate a wide range of different smoothing strategies. We derive existing Kalman-type smoothers as linear special cases and investigate their operational differences.
Hydrogeological Uncertainty Estimation with the Analytic Element Method
In this study, we examine the potential of the Analytic Element Method (AEM) for Bayesian inference. Where most groundwater models solve the flow equations on a grid, AEM constructs a single equation which solves the flow equation everywhere at once. The method also does not require confinement by finite boundaries, making it well-suited for the exploration of forcing or boundary uncertainty.
To further capitalize on this property, we propose an analytic element based on a Möbius transformation (see the interactive element to the left), which permits the induction of flexible regional flow without the need for prescribed hydraulic head or flow boundaries.
For this study, we also created a Python toolbox which includes most of the commonly used analytic elements as well as the new Möbius element. We have also created an auxiliary toolbox with expands its capabilities to Bayesian uncertainty estimation using a Differential Evolution Markov Chain (DE-MC) algorithm. The toolbox is described in greater detail in the Software section.
Dissertation: Data Assimilation and Non-Gaussian Parameter Inference for Hydrogeological Models
I conducted my doctoral research at the Swiss Federal Institute of Aquatic Science and Technology (Eawag, part of the ETH domain) in Zürich under the supervision of Prof. Mario Schirmer, and obtained my doctorate from the University of Neuchâtel in December 2020. My dissertation explored different non-Gaussian Bayesian inference methods for hydrogeological uncertainty estimation, investigating their individual strengths and caveats.
This research was motivated by the importance of safeguarding water supply, an issue of growing worldwide concern. To manage groundwater sustainably, it is important to understand its dynamics. However, since most of the subsurface is unobservable, hydrogeology invariably operates in an environment of limited information. To ensure that this limitation does not compromise the rigour of our predictions, uncertainty estimation is an integral part of this discipline.
Unfortunately, most operational uncertainty quantification methods in hydrogeology are very simple. With such limited complexity, it is possible to underestimate the ambiguities inherent in the system, blinding us to potential risks with unpredictable consequences. The contribution of my research was to explore and adapt methods which permit greater complexity while remaining efficient enough to be viable for practical use, ultimately increasing the reliability of our decision support. This dissertation comprises of the three Water Resources Research publications listed below.
Non-Gaussian Parameter Inference for Hydrogeological Models Using Stein Variational Gradient Descent
In this study, we explore a novel algorithm called Stein Variational Gradient Descent (SVGD). This variational approach deterministically transforms an ensemble of almost arbitrarily distributed initial hypotheses into samples of the posterior, delineating the uncertainty in the system. To do so, the ensemble's particles exchange information with their neighbors to inform the next gradient descent directions. Since this algorithm is not based on the assumption of Gaussianity, it can recover more complex forms of uncertainty, such as Pareto frontiers (functionally equivalent parameter trade-offs) and multi-modality (distinct solutions).
We propose an ensemble-based approximation of the Jacobian matrix to make SVGD computationally tractable in hydrogeological practice, and demonstrate its use in a simple synthetic test case, successfully recovering the bi-modal distribution. We then apply it for a model of the complex, real groundwater catchment of Fehraltorf.
Quasi‐Online Groundwater Model Optimization Under Constraints of Geological Consistency Based on Iterative Importance Sampling
In this study, we explore a variation of Iterated Batch Importance Sampling (IBIS) for sequential parameter estimation. The motivation for this algorithm is the sequential optimization of parameter fields obtained by multiple-point statistics (MPS). MPS is an extremely flexible method to re-create parameter fields from patterns in a training image. The drawback is that the method creates highly non-Gaussian priors.
We adapt the IBIS algorithm for sequential, quasi-real-time parameter optimization, improving predictive performance as new data is assimilated. We test the algorithm with data from a field site near Valdobbiadene, Italy, inferring the possible locations of high-conductive, buried paleo-channels. We find that the algorithm can successfully identify channel configurations which closely reproduce the observed water table dynamics. However, the price for this is increasing computational effort as the time series grows. We discuss the strengths and weaknesses of the approach.
Data Assimilation and Online Parameter Optimization in Groundwater Modeling Using Nested Particle Filters
We investigate the use of Nested Particle Filters (NPF) for sequential state and parameter inference in hydrogeological models. Particle filters work similarly to biological evolution, randomly mutating an ensemble of alternative hypotheses, then selecting and multiplying them based on performance. This can even optimize systems following complex rules without human supervision. We explore its performance with hydrogeological conductivity fields created by parameterized field generators.
Specifically, we investigate how the algorithm performs if the geological patterns are deliberately or accidentally misspecified, for example when high-conductive lenses are used to reconstruct the flow response of a high-conductive paleo-channel. We discuss the strengths and limitations of the approach.