Wolfgang Nowak

Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion

We discuss the arbitrary polynomial chaos (aPC), which has been subject of research in a few recent theoretical papers. Like all polynomial chaos expansion techniques, aPC approximates the dependence of simulation model output on model parameters by expansion in an orthogonal polynomial basis. The aPC generalizes chaos expansion techniques towards arbitrary distributions with arbitrary probability measures, which can be either discrete, continuous, or discretized continuous and can be specified either analytically (as probability density/cumulative distribution functions), numerically as histogram or as raw data sets. We show that the aPC at finite expansion order only demands the existence of a finite number of moments and does not require the complete knowledge or even existence of a probability density function. This avoids the necessity to assign parametric probability distributions that are not sufficiently supported by limited available data. Alternatively, it allows modellers to choose freely of technical constraints the shapes of their statistical assumptions. Our key idea is to align the complexity level and order of analysis with the reliability and detail level of statistical information on the input parameters. We provide conditions for existence and clarify the relation of the aPC to statistical moments of model parameters. We test the performance of the aPC with diverse statistical distributions and with raw data. In these exemplary test cases, we illustrate the convergence with increasing expansion order and, for the first time, with increasing reliability level of statistical input information. Our results indicate that the aPC shows an exponential convergence rate and converges faster than classical polynomial chaos expansion techniques.

Efficient Computation of Linearized Cross-Covariance and Auto-Covariance Matrices of Interdependent Quantities

In many geostatistical applications, spatially discretized unknowns are conditioned on observations that depend on the unknowns in a form that can be linearized. Conditioning takes several matrix–matrix multiplications to compute the cross-covariance matrix of the unknowns and the observations and the auto-covariance matrix of the observations. For large numbers n of discrete values of the unknown, the storage and computational costs for evaluating these matrices, proportional to n2, become strictly inhibiting. In this paper, we summarize and extend a collection of highly efficient spectral methods to compute these matrices, based on circulant embedding and the fast Fourier transform (FFT). These methods are applicable whenever the unknowns are a stationary random variable discretized on a regular equispaced grid, imposing an exploitable structure onto the auto-covariance matrix of the unknowns. Computational costs are reduced from O(n2) to O(nlog2n) and storage requirements are reduced from O(n2) to O(n).

Uncertainty and data worth analysis for the hydraulic design of funnel-and-gate systems in heterogeneous aquifers

[1] Hydraulic failure of a funnel-and-gate system may occur when the contaminant plume bypasses the funnels rather than being captured by the gate. We analyze the uncertainty of capturing the plumes by funnel-and-gate systems in heterogeneous aquifers. Restricting the analysis to two-dimensional, steady state flow, we characterize plume capture by the values of the stream function at the boundaries of the plume and the funnels. On the basis of the covariance of the log conductivity distribution we compute the covariance matrix of the relevant stream function values by a matrix-based first-order second-moment method, making use of efficient matrix-multiplication techniques. From the covariance matrix of stream function values, we can approximate the probability that the plume is bypassing the funnels. We condition the log conductivity field to measurements ofthe logconductivity and the hydraulic head. Prior to performing additional measurements, we estimate their worth by the expected reduction in the variance of stream function differences. In an application to a hypothetical aquifer, we demonstrate that our method of uncertainty propagation and our sampling strategy enable us to discriminate between cases of success and failure of funnel-and-gate systems with a small number of additional samples. INDEX TERMS: 1829 Hydrology: Groundwater hydrology; 1869 Hydrology: Stochastic processes; 1832 Hydrology: Groundwater transport; KEYWORDS: conditioning, data worth, funnel-and-gate systems, heterogeneous aquifers, stream function, uncertainty propagation

On the link between contaminant source release conditions and plume prediction uncertainty.

The initial width of contaminant plumes is known to have a key influence on expected plume development, dispersion and travel time statistics. In past studies, initial plume width has been perceived identical to the geometric width of a contaminant source or injection volume. A recent study on optimal sampling layouts (Nowak et al., 2009) showed that a significant portion of uncertainty in predicting plume migration stems from the uncertain total hydraulic flux through the source area. This result points towards a missing link between source geometry and plume statistics, which we denote as the effective source width. We define the effective source width by the ratio between the actual and expected hydraulic fluxes times the geometric source width. The actual hydraulic flux through the source area is given by individual realizations while the expected one represents the mean over the ensemble. It is a stochastic quantity that may strongly differ from the actual geometric source width for geometrically small sources, and becomes identical only at the limit of wide sources (approaching ergodicity). We derive its stochastic ensemble moments in order to explore the dependency on source scale. We show that, if the effective source width is known rather than the geometric width, predictions of plume development can greatly increase in predictive power. This is illustrated on plume statistics such as the distribution of plume length, average width, transverse dispersion, total mass flux and overall concentration variance. The analysis is limited to 2D depth-averaged systems, but implications hold for 3D cases.

Flow Radar Glyphs—Static Visualization of Unsteady Flow with Uncertainty

A new type of glyph is introduced to visualize unsteady flow with static images, allowing easier analysis of time-dependent phenomena compared to animated visualization. Adopting the visual metaphor of radar displays, this glyph represents flow directions by angles and time by radius in spherical coordinates. Dense seeding of flow radar glyphs on the flow domain naturally lends itself to multi-scale visualization: zoomed-out views show aggregated overviews, zooming-in enables detailed analysis of spatial and temporal characteristics. Uncertainty visualization is supported by extending the glyph to display possible ranges of flow directions. The paper focuses on 2D flow, but includes a discussion of 3D flow as well. Examples from CFD and the field of stochastic hydrogeology show that it is easy to discriminate regions of different spatiotemporal flow behavior and regions of different uncertainty variations in space and time. The examples also demonstrate that parameter studies can be analyzed because the glyph design facilitates comparative visualization. Finally, different variants of interactive GPU-accelerated implementations are discussed.

Stochastic evaluation of mixing-controlled steady-state plume lengths in two-dimensional heterogeneous domains

We study plumes originating from continuous sources that require a dissolved reaction partner for their degradation. The length of such plumes is typically controlled by transverse mixing. While analytical expressions have been derived for homogeneous flow fields, incomplete characterization of the hydraulic conductivity field causes uncertainty in predicting plume lengths in heterogeneous domains. In this context, we analyze the effects of three sources of uncertainty: (i) The uncertainty of the effective mixing rate along the plume fringes due to spatially varying flow focusing, (ii) the uncertainty of the volumetric discharge through (and thus total mass flux leaving) the source area, and (iii) different parameterizations of the Darcy-scale transverse dispersion coefficient. The first two are directly related to heterogeneity of hydraulic conductivity. In this paper, we derive semi-analytical expressions for the probability distribution of plume lengths at different levels of complexity. The results are compared to numerical Monte Carlo simulations. Uncertainties in mixing and in the source strength result in a statistical distribution of possible plume lengths. For unconditional random hydraulic conductivity fields, plume lengths may vary by more than one order of magnitude even for moderate degrees of heterogeneity. Our results show that the uncertainty of volumetric flux through the source is the most relevant contribution to the variance of the plume length. The choice of different parameterizations for the local dispersion coefficient leads to differences in the mean estimated plume length.

A hypothesis‐driven approach to optimize field campaigns

Most field campaigns aim at helping in specified scientific or practical tasks, such as modeling, prediction, optimization, or management. Often these tasks involve binary decisions or seek answers to yes/no questions under uncertainty, e.g., Is a model adequate? Will contamination exceed a critical level? In this context, the information needs of hydro(geo)logical modeling should be satisfied with efficient and rational field campaigns, e.g., because budgets are limited. We propose a new framework to optimize field campaigns that defines the quest for defensible decisions as the ultimate goal. The key steps are to formulate yes/no questions under uncertainty as Bayesian hypothesis tests, and then use the expected failure probability of hypothesis testing as objective function. Our formalism is unique in that it optimizes field campaigns for maximum confidence in decisions on model choice, binary engineering or management decisions, or questions concerning compliance with environmental performance metrics. It is goal oriented, recognizing that different models, questions, or metrics deserve different treatment. We use a formal Bayesian scheme called PreDIA, which is free of linearization, and can handle arbitrary data types, scientific tasks, and sources of uncertainty (e.g., conceptual, physical, (geo)statistical, measurement errors). This reduces the bias due to possibly subjective assumptions prior to data collection and improves the chances of successful field campaigns even under conditions of model uncertainty. We illustrate our approach on two instructive examples from stochastic hydrogeology with increasing complexity.

Anthropogenic Trace Compounds (ATCs) in aquatic habitats - research needs on sources, fate, detection and toxicity to ensure timely elimination strategies and risk management.

Anthropogenic Trace Compounds (ATCs) that continuously grow in numbers and concentrations are an emerging issue for water quality in both natural and technical environments. The complex web of exposure pathways as well as the variety in the chemical structure and potency of ATCs represents immense challenges for future research and policy initiatives. This review summarizes current trends and identifies knowledge gaps in innovative, effective monitoring and management strategies while addressing the research questions concerning ATC occurrence, fate, detection and toxicity. We highlight the progressing sensitivity of chemical analytics and the challenges in harmonization of sampling protocols and methods, as well as the need for ATC indicator substances to enable cross-national valid monitoring routine. Secondly, the status quo in ecotoxicology is described to advocate for a better implementation of long-term tests, to address toxicity on community and environmental as well as on human-health levels, and to adapt various test levels and endpoints. Moreover, we discuss potential sources of ATCs and the current removal efficiency of wastewater treatment plants (WWTPs) to indicate the most effective places and elimination strategies. Knowledge gaps in transport and/or detainment of ATCs through their passage in surface waters and groundwaters are further emphasized in relation to their physico-chemical properties, abiotic conditions and biological interactions in order to highlight fundamental research needs. Finally, we demonstrate the importance and remaining challenges of an appropriate ATC risk assessment since this will greatly assist in identifying the most urgent calls for action, in selecting the most promising measures, and in evaluating the success of implemented management strategies.

Bayesian updating via bootstrap filtering combined with data-driven polynomial chaos expansions: methodology and application to history matching for carbon dioxide storage in geological formations

Model calibration and history matching are important techniques to adapt simulation tools to real-world systems. When prediction uncertainty needs to be quantified, one has to use the respective statistical counterparts, e.g., Bayesian updating of model parameters and data assimilation. For complex and large-scale systems, however, even single forward deterministic simulations may require parallel high-performance computing. This often makes accurate brute-force and nonlinear statistical approaches infeasible. We propose an advanced framework for parameter inference or history matching based on the arbitrary polynomial chaos expansion (aPC) and strict Bayesian principles. Our framework consists of two main steps. In step 1, the original model is projected onto a mathematically optimal response surface via the aPC technique. The resulting response surface can be viewed as a reduced (surrogate) model. It captures the model’s dependence on all parameters relevant for history matching at high-order accuracy. Step 2 consists of matching the reduced model from step 1 to observation data via bootstrap filtering. Bootstrap filtering is a fully nonlinear and Bayesian statistical approach to the inverse problem in history matching. It allows to quantify post-calibration parameter and prediction uncertainty and is more accurate than ensemble Kalman filtering or linearized methods. Through this combination, we obtain a statistical method for history matching that is accurate, yet has a computational speed that is more than sufficient to be developed towards real-time application. We motivate and demonstrate our method on the problem of CO2 storage in geological formations, using a low-parametric homogeneous 3D benchmark problem. In a synthetic case study, we update the parameters of a CO2/brine multiphase model on monitored pressure data during CO2 injection.

Probability density function of steady state concentration in two‐dimensional heterogeneous porous media

Spatial variability of hydraulic aquifer parameters causes meandering, squeezing, stretching, and enhanced mixing of steady state plumes in concentrated hot-spots of mixing. Because the exact spatial distribution of hydraulic parameters is uncertain, the spatial distribution of enhanced mixing rates is also uncertain. We discuss how relevant the resulting uncertainty of mixing rates is for predicting concentrations. We develop analytical solutions for the full statistical distribution of steady state concentration in two-dimensional, statistically uniform domains with log-hydraulic conductivity following an isotropic exponential model. In particular, we analyze concentration statistics at the fringes of wide plumes, conceptually represented by a solute introduced over half the width of the domain. Our framework explicitly accounts for uncertainty of streamline meandering and uncertainty of effective transverse mixing (defined at the Darcy scale). We make use of existing low-order closed-form expressions that lead to analytical expressions for the statistical distribution of local concentration values. Along the expected position of the plume fringe, the concentration distribution strongly clusters at its extreme values. This behavior extends over travel distances of up to tens of integral scales for the parameters tested in our study. In this regime, the uncertainty of effective transverse mixing is substantial enough to have noticeable effects on the concentration probability density function. At significantly larger travel distances, intermediate concentration values become most likely, and uncertainty of effective transverse mixing becomes negligible. A comparison to numerical Monte Carlo simulations of flow and solute transport show excellent agreement with the theoretically derived expressions.