Smoothing is a specialized form of Bayesian inference for state-space models that characterizes the posterior distribution of a collection of states given an associated sequence of observations. Ramgraber et al. [38] proposes a general framework for transport-based ensemble smoothing, which includes linear Kalman-type smoothers as special cases. Here, we build on this foundation to realize and demonstrate nonlinear backward ensemble transport smoothers. We discuss parameterization and regularization of the associated transport maps, and then examine the performance of these smoothers for nonlinear and chaotic dynamical systems that exhibit non-Gaussian behavior. In these settings, our nonlinear transport smoothers yield lower estimation error than conventional linear smoothers and state-of-the-art iterative ensemble Kalman smoothers, for comparable numbers of model evaluations.

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Smoothers are algorithms for Bayesian time series re-analysis. Most operational smoothers rely either on affine Kalman-type transformations or on sequential importance sampling. These strategies occupy opposite ends of a spectrum that trades computational efficiency and scalability for statistical generality and consistency: non-Gaussianity renders affine Kalman updates inconsistent with the true Bayesian solution, while the ensemble size required for successful importance sampling can be prohibitive. This paper revisits the smoothing problem from the perspective of measure transport, which offers the prospect of consistent prior-to-posterior transformations for Bayesian inference. We leverage this capacity by proposing a general ensemble framework for transport-based smoothing. Within this framework, we derive a comprehensive set of smoothing recursions based on nonlinear transport maps and detail how they exploit the structure of state-space models in fully non-Gaussian settings. We also describe how many standard Kalman-type smoothing algorithms emerge as special cases of our framework. A companion paper [35] explores the implementation of nonlinear ensemble transport smoothers in greater depth.

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Uncertainty estimation plays an important part in practical hydrogeology. With most of the subsurface unobservable, attempts at system characterization will invariably be incomplete. Uncertainty estimation, then, must quantify the influence of unknown parameters, forcings, and structural deficiencies. In this endeavor, numerical modeling frameworks can resolve a high degree of subsurface complexity and its associated uncertainty. Where boundary uncertainty is concerned, however, numerical frameworks can be restrictive. The interdependence of grid discretization and its enclosing boundaries render exploration of uncertainties in their extent or nature challenging. The analytic element method (AEM) may be an interesting complement, as it is computationally efficient, economic with its parameter count, and does not require enclosure through finite boundaries. These properties make AEM well suited for uncertainty estimation, particularly in data-scarce settings or exploratory studies. In this study, we explore the use of AEM for flow field uncertainty estimation, with a particular focus on boundary uncertainty. To induce diverse, uncertain regional flow more easily, we propose a new element based on a Möbius transformation. We include this element in a simple Python-based AEM toolbox and benchmark it against MODFLOW. Coupling AEM with a Markov Chain Monte Carlo routine using adaptive proposals, we explore its use in a synthetic case study. We find that AEM permits efficient uncertainty estimation for groundwater flow fields, which may form a basis for stochastic Lagrangian transport modeling or can support advanced model design by informing the placement of numerical model boundaries.@en

The sustainable management of groundwater demands a faithful characterization of the subsurface. This, in turn, requires information which is generally not readily available. To bridge the gap between data need and availability, numerical models are often used to synthesize plausible scenarios not only from direct information but also from additional, indirect data. Unfortunately, the resulting system characterizations will rarely be unique. This poses a challenge for practical parameter inference: computational limitations often force modelers to resort to methods based on questionable assumptions of Gaussianity, which do not reproduce important facets of ambiguity such as Pareto fronts or multimodality. In search of a remedy, an alternative could be found in Stein Variational Gradient Descent (SVGD), a recent development in the field of statistics. This ensemble-based method iteratively transforms a set of arbitrary particles into samples of a potentially non-Gaussian posterior, provided the latter is sufficiently smooth. A prerequisite for this method is knowledge of the Jacobian, which is usually exceptionally expensive to evaluate. To address this issue, we propose an ensemble-based, localized approximation of the Jacobian. We demonstrate the performance of the resulting algorithm in two cases: a simple, bimodal synthetic scenario, and a complex numerical model based on a real world, prealpine catchment. Promising results in both cases—even when the ensemble size is smaller than the number of parameters—suggest that SVGD can be a valuable addition to hydrogeological parameter inference.

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The increasing use of wireless sensor networks and remote sensing permits real-time access to environmental observations. Data assimilation frameworks tap into such data streams to autonomously update and gradually improve numerical models. In hydrogeology, such methods are relevant in areas of long-term interest in water quality and quantity, for example, in drinking water production. Unfortunately, accurate hydrogeological predictions often demand a degree of geological realism, which is difficult to reconcile with the operational limitations of many data assimilation frameworks. Alluvial aquifers, for example, are sometimes characterized by paleo-channels of unknown extent and properties, which may act as preferential flow paths. Gradually optimizing such fields in real-time or quasi-real-time settings is a formidable task. Besides subsurface properties, ill-specified model forcings are a further source of predictive bias, which an optimizer could learn to compensate. In this study, we explore the use of a quasi-online optimizer based on the iterative batch importance sampling framework for a groundwater model of a field site near Valdobbiadene, Italy. This site is characterized by the presence of paleo-channels and heavily exploited for drinking water production and irrigation. We use Markov chain Monte Carlo steps to explore new parameterizations while maintaining consistency between states and parameters as well as conformance to a multipoint statistics training image. We also optimize a preprocessor designed to compensate for potential bias in the model forcing. We achieve promising and geologically consistent quasi-real-time optimization, albeit at the loss of parameter uncertainty.

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Over the past decades, advances in data collection and machine learning have paved the way for the development of autonomous simulation frameworks. Among these, many are capable not only of assimilating real-time data to correct their predictive shortcomings but also of improving their future performance through self-optimization. In hydrogeology, such techniques harbor great potential for informing sustainable management practices. Simulating the intricacies of groundwater flow requires an adequate representation of unknown, often highly heterogeneous geology. Unfortunately, it is difficult to reconcile the structural complexity demanded by realistic geology with the simplifying assumptions introduced in many calibration methods. The particle filter framework would provide the necessary versatility to retain such complex information but suffers from the curse of dimensionality, a fundamental limitation discouraging its use in systems with many unknowns. Due to the prevalence of such systems in hydrogeology, the particle filter has received little attention in groundwater modeling so far. In this study, we explore the combined use of dimension-reducing techniques and artificial parameter dynamics to enable a particle filter framework for a groundwater model. Exploiting freedom in the design of the dimension-reduction approach, we ensure consistency with a predefined geological pattern. The performance of the resulting optimizer is demonstrated in a synthetic test case for three such geological configurations and compared to two Ensemble Kalman Filter setups. Favorable results even for deliberately misspecified settings make us hopeful that nested particle filters may constitute a useful tool for geologically consistent real-time parameter optimization.

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