Stochastic inversion for soil hydraulic parameters in the presence of model error: An example involving ground-penetrating radar monitoring of infiltration

International audienceProxy forward solvers are commonly used in Bayesian solutions to inverse problems in hydrology and geophysics in order to make sampling of the posterior distribution, for example using Markov-chain-Monte-Carlo (MCMC) methods, computationally tractable. However, use of these solvers introduces model error into the problem, which can lead to strongly biased and overconfident parameter estimates if left uncorrected. Focusing on the specific example of estimating unsaturated hydraulic parameters in a layered soil from time-lapse ground-penetrating radar data acquired during a synthetic infiltration experiment, we show how principal component analysis, conducted on a suite of stochastic model-error realizations, can for some problems be used to build a sparse orthogonal basis for the model error arising from known forward solver approximations and/or simplifications. Projection of the residual onto this basis during MCMC permits identification and removal of the model error before calculation of the likelihood. Our results indicate that, when combined with an informal likelihood metric based on the expected behaviour of the -norm of the residual, this methodology can yield posterior parameter estimates exhibiting a marked reduction in bias and overconfidence when compared to those obtained with no model-error correction, at reasonable computational cost

Hydrogeophysical parameter estimation using iterative ensemble smoothing and approximate forward solvers

In iterative ensemble smoother approaches and ensemble methods in general, the ensemble size governs the accuracy of the parameter estimates obtained. However, employing large ensembles may be computationally infeasible in applications with expensive forward solvers. Here, we reduce the computational cost of using large ensembles in iterative ensemble smoothing through the use of a proxy solver. To correct the proxy response for the corresponding model error, the latter of which can bias posterior parameter estimates if left untreated, we propose a local basis approach. With this approach, the discrepancy between the detailed and proxy solvers is learned for a subset of the ensemble and collected in a dictionary that grows with each iteration. For each ensemble member, the K-nearest neighbors in the dictionary are employed to build an orthonormal basis which is used to identify the model-error component of the residual by projection. The proposed methodology reduces the effects of overfitting the data with the proxy solver, but may lead to underfitting of the data in the absence of a sufficient number of dictionary entries, meaning that the number of ensemble members relative to the number of detailed-solver runs cannot be inflated arbitrarily. We present our approach in the context of the ensemble smoother with multiple data assimilations (ES-MDA) algorithm, and show its successful application to a high-dimensional synthetic example that involves crosshole ground-penetrating radar (GPR) travel-time tomography

Impact of poroelastic effects on the inversion of fracture properties from amplitude variation with offset and azimuth data in horizontal transversely isotropic media

<jats:p> The identification and characterization of fractures is an important objective in many areas of earth and environmental sciences. Amplitude variation with offset and azimuth (AVOAz) analysis of seismic reflection data is a key method for achieving these tasks. Theoretical and experimental studies have shown that the presence of pore fluids together with the strong mechanical contrast between the fractures and their embedding background give rise to wave-induced fluid flow (WIFF) effects. This implies that the effective stiffness tensor of a fluid-saturated fractured rock defining its seismic response becomes viscoelastic and frequency-dependent. In spite of this, AVOAz analysis typically relies on end-member-type elastic stiffness models that either assume a relaxed (i.e., equilibrated) or unrelaxed (i.e., unequilibrated) state of the wave-induced fluid pressure in the rock. In general, however, neither the appropriateness of the chosen model nor the associated errors in the inversion process are known. To investigate this topic, we have considered a poroelastic medium containing parallel vertical fractures and generate synthetic seismic AVOAz data using the classic Rüger approximations for PP-wave reflection coefficients in horizontally transversely isotropic media. A Markov chain Monte Carlo method is used to perform a Bayesian inversion of the synthetic seismic AVOAz data. We quantify the influence of WIFF effects on the AVOAz inversion results when elastic relaxed and unrelaxed models are used as forward solvers of inversion schemes to estimate the fracture volume fraction, the elastic moduli, and the porosity of the background rock, as well as the overall weakness of the medium due to the presence of fractures. Our results indicate that, when dealing with single-frequency data, relaxed elastic models provide biased but overall better inversion results than unrelaxed ones, for which some fracture parameters cannot be resolved. Improved inversion performance is achieved when using frequency-dependent data, which illustrates the importance of accounting for poroelastic effects. </jats:p&gt

Bayesian Network Model for Accessing Safety and Security of Offshore Wind Farms

The offshore wind industry experience a rising importance for the worldwide energy production, which is accompanied by increasing amount of wind turbines and Offshore Wind Farms (OWFs). However, due to its harsh environment and its role for energy provision, OWFs are confronted with several threats that are impacting its safety and security. Consequently, decision making at design as well as run time plays an important role for providing safe and secure operation in OWFs. We propose in this work the application of a Bayesian Network (BN) for a high-level representation of the safety and security state of an OWF. The developed BN-model is based on the safety and security goals and related functions defined in Kopke et al. (2019). The derived model enables a user to analyze the overall importance of high impact functions, like compliance, environmental protection, supply reliability or accident prevention. Obtained results indicate that the proposed BN-model enables decision makers to explore cross-system interrelations, and thus, to define requirements for implementations on lower design levels

Resilience management processes in the offshore wind industry: Schematization and application to an export cable attack

Offshore wind energy (OWE) production is a crucial element for increasing the amount of renewable energy. Consequently, one can observe a strong and constant rise of the OWE industry, turning it to an important contributor of national energy provision. This trend, however, is accompanied by increasing pressure on the reliability, safety, and security of the OWE infrastructure. Related security threats are characterized by high uncertainty regarding impact and probability leading to considerable complication of the risk assessment. On the other hand, the resilience concept emphasizes the consideration of the system’s response to such threats, and thus, offers a solution for dealing with the high uncertainty. In this work, we present an approach for combining the strengths of risk and resilience management to provide a solution for handling security threats in OWE infrastructures. Within this context, we introduce a quality assessment enabling the quantification of the trustworthiness of obtained results

Bewertung von Lebensmitteln verschiedener Produktionsverfahren - Statusbericht 2003

INHALTSVERZEICHNIS: 1 Einleitung 2 Zur Struktur der Studie 3 Qualität von Lebensmitteln nach Produktionsverfahren 3.1 Prozessqualität 3.1.1 Verfahrensweisen und Elemente der Prozessqualität bei der landwirtschaftlichen Erzeugung 3.1.1.1 Ökobilanzen 3.1.1.2 Vergleich der Produktionsverfahren für einzelne Umweltwirkungsbereiche 3.1.2 Prozessqualität unter besonderer Berücksichtigung der Verarbeitung 3.1.3 Prozessqualität der Erzeugnisse - Bewertung durch Verbraucherinnen und Verbraucher 3.1.4 Schlussfolgerungen, Empfehlungen und Forschungsbedarf im Bereich der Prozessqualität 3.2 Produktqualität 3.2.1 Vom Produktionsverfahren unabhängige Einflüsse auf die Produktqualität 3.2.2 Gesetzlich vorgeschriebene Qualität (Lebensmittelsicherheit) 3.2.3 Ernährungsphysiologische Qualität 3.2.4 Genusswert 3.2.5 Eignungswert 3.2.6 Schlussfolgerungen, Empfehlungen und Forschungsbedarf im Bereich der Produktqualität 4 Komplementäre Ansätze zur Erfassung der Lebensmittelqualitäten 4.1 Bildschaffende Methoden 4.2 Nachernteverhalten 4.3 Fluoreszenz-Anregungs-Spektroskopie 4.4 Physiologischer Aminosäurenstatus 4.5 Elektrochemische Methoden 4.6 Futterwahl und Fütterungsversuche 4.7 Konsequenzen für die Forschung zur Erfassung der Lebensmittel- qualität - Modellvorstellungen 5. Sozioökonomische Aspekte ökologisch erzeugter Lebensmittel in Deutschland 5.1 Ökologisch erzeugte Lebensmittel aus Verbrauchersicht 5.2 Auswirkungen von ökologischen Ernährungsstilen auf die Kosten im Gesundheitswesen und auf den Ressourcenverbrauch 5.2.1 Auswirkungen auf die Kosten im Gesundheitswesen 5.2.2 Auswirkungen auf die Kosten in den Bereichen Umwelt und Ressourcen 5.3 Nachhaltige Entwicklung im Bedürfnisfeld Ernährung 5.4 Aspekte des Marktes für ökologisch erzeugte Lebensmittel 5.5 Ökologisch erzeugte Lebensmittel in der Gemeinschaftsverpflegung (GV) 5.6 Schlussfolgerungen, Empfehlungen und Forschungsbedarf zu sozioökonomischen Aspekten bei Bio-Lebensmitteln 6 Schlussfolgerungen, Empfehlungen und Forschungsbedarf 6.1 Prozessqualität 6.1.1 Ökobilanzen über Umweltwirkungsbereiche 6.1.2 Erzeugung von Lebensmitteln (Tierhaltung) 6.1.3 Lebensmittelverarbeitung 6.1.4 Bewertung durch den Verbraucher 6.2 Produktqualität 6.2.1 Produktspezifischer Forschungsbedarf bei pflanzlichen Erzeugnissen 6.2.2 Produktspezifischer Forschungsbedarf für Bio-Lebensmittel 6.2.3 Produktspezifische Qualitätssicherung bei vom Tier stammenden Erzeugnissen 6.3 Komplementäre Methoden der Qualitätserfassung 6.4 Sozioökonomische Aspekte 6.4.1 Ökologisch erzeugte Lebensmittel aus Verbrauchersicht 6.4.2 Auswirkungen von ökologischen Ernährungsstilen auf die Kosten im Gesundheitswesen und auf den Ressourcenverbrauch 6.4.3 Aspekte des Marktes 6.4.4 Ökologische Erzeugnisse in der Gemeinschaftsverpflegung (GV) 6.5 Schlussbetrachtung Anhang 1: Literatur Anhang 2: Begriffserläuterungen/Rechtliche Rahmenbedingungen Anhang 3: Grundlagen des Lebensmittelrechts Anhang 4: Ganzheitlichkeit in der Lebensmittelsmittelforschun

An Expert-Driven Probabilistic Assessment of the Safety and Security of Offshore Wind Farms

Offshore wind farms (OWFs) are important infrastructure which provide an alternative and clean means of energy production worldwide. The offshore wind industry has been continuously growing. Over the years, however, it has become evident that OWFs are facing a variety of safety and security challenges. If not addressed, these issues may hinder their progress. Based on these safety and security goals and on a Bayesian network model, this work presents a methodological approach for structuring and organizing expert knowledge and turning it into a probabilistic model to assess the safety and security of OWFs. This graphical probabilistic model allowed us to create a high-level representation of the safety and security state of a generic OWF. By studying the interrelations between the different functions of the model, and by proposing different scenarios, we determined the impacts that a failing function may have on other functions in this complex system. Finally, this model helped us define the performance requirements of such infrastructure, which should be beneficial for optimizing operation and maintenance

Identification and characterization of model error arising from simplified forward solvers in hydrogeophysical inverse problems

Le domaine de l'hydrogéophysique se définit comme l'application de méthodes géophy¬siques à des problèmes hydrologiques. Dans le but de caractériser les propriétés importantes du sous-sol ainsi que leur incertitude relative à partir de mesures géophysiques, les proprié¬tés du sous-sol sont décrites par un ensemble de paramètres et, basé sur les phénomènes physiques sous-jacents, un solveur numérique capable de reproduire le type de données observées en fonction de ces paramètres est retenu. Bien souvent, plusieurs solveurs numé¬riques existent et diffèrent par leur précision et leur justesse. L'emploi de solveur numérique simplifiant grandement les phénomènes physiques sous-jacents permet certes de réduire drastiquement les coûts de calcul liés à la formulation du problème inverse, mais conduit à des erreurs de modèle. Si ces erreurs de modèle ne sont pas prises en compte, cela peut augmenter considérablement le biais des paramètres estimés et mener à une sous-estimation de l'incertitude. Les approches existantes pour traiter l'erreur de modèle se restreignent essentiellement à des problèmes de petite dimension ou bien reposent sur des hypothèses très fortes concernant les propriétés statistiques des erreur de modèle. Dans cette thèse, je développe une nouvelle méthode qui prend en compte l'erreur de modèle dans une inversion Bayésienne, et je l'applique à divers exemples qui traduisent des pro¬blèmes pertinents en hydrologie. Contrairement aux approches traditionnelles qui caracté¬risent l'erreur de modèle par des distributions statistiques paramétriques ou par construction d'un modèle d'erreur, ma méthode se focalise sur l'identification de composantes d'erreur de modèle de résidus par projection orthogonale. Plus spécifiquement, le résidu est projeté sur une base orthonormale d'erreur de modèle qui est construite par analyse de différence de résultats obtenus par solveur de physique simplifiée ou détaillée à partir d'un ensemble de réalisations stochastiques. Cette approche repose sur l'hypothèse que l'erreur de modèle est orthogonale aux autres composantes des résidus et de fait séparable. Dans cette thèse, la méthode que je propose est tout d'abord appliquée à l'inversion de données synthétiques de radar à pénétration de sol (GPR), relatives à la teneur en eau, acquises lors d'une expérience d'infiltration forcée en milieu non-saturé. Dans ce cas, l'erreur de modèle est construite comme une base globale unique.Ensuite, la méthode est utilisée pour inverser des données synthétiques de temps d'arrivée GPR entre forages, afin d'inférer à la vitesse de propagation des ondes entre les forage. Dans ce cas, il a été nécessaire d'apporter une modification, pour permettre le développement d'une base d'erreur de modèle locale. Cette base est développée lors de la procédure d'inversion et repose sur l'utilisation des K réalisations d'erreur de modèle les plus proches de l'état actuel dans l'espace des paramètres. Finalement, l'approche proposée est appliquée à l'inversion de données synthétiques de temps d'arrivée de GPR entre forages à partir d'une technique d'atténuation d'ensemble itérative. L'approche d'erreur de modèle proposée peut être appliquée à des problèmes d'estimation de paramètres à grande échelle, sans avoir besoin de formuler d'hypothèses sur les statistiques d'erreur de modèle. Les résultats obtenus dans cette thèse sont prometteurs quant à l'application de la méthode à d'autres exemples et à des problèmes inverses issus de données réelles. -- The field of hydrogeophysics concerns the application of geophysical methods to hydrological problems. To estimate hydrologically relevant subsurface properties and their corresponding uncertainties from geophysical data, the subsurface must be parameterized and a forward solver must be chosen to represent the physics of the underlying measurement process. In this regard, several forward solvers often exist having différent levels of accuracy. Employing simplified forward solvers can reduce the computational costs of the inverse problem but leads to model error. If this model error is not accounted for, parameter estimâtes can become strongly biased and overconfident. Existing approaches to deal with model error are mainly limited to low-dimensional problems or based on strong assumptions about the statistical distribution of the model error. In this thesis I develop an alternative method to account for model error in Bayesian inver¬sion, and I apply it to a number of hydrologically relevant example problems. In contrast to traditional approaches aimed towards characterizing the model error through parametric sta-tistical distributions or construction of an error model, my method focuses on identification ofthe model-error component of the residual through orthogonal projection. Specifically, the residual is projected onto an orthonormal model-error basis that is constructed through the analysis of stochastic realizations of the différence between the simplified and detailed for¬ward solvers. The approach is based on the assumption that the model error lies orthogonal to the other components of the residual and is therefore separable. In the thesis, my proposed method is first applied to the inversion of synthetic ground- penetrating-radar- (GPR-) derived water content data, acquired during a forced infiltration experiment, for unsaturated soil hydraulic properties. In this case, a single global basis is constructed for the model error. Next, the method is used to invert synthetic crosshole GPR travel-time data for the distribution of the radar wave speed between the boreholes, where a modification was necessary to allow for the development of a local model error basis. This basis is developed during the inversion procédure and is constructed using the K-nearest- neighboring model-error realizations to current location in the parameter space. Finally, the proposed approach is applied to synthetic crosshole GPR travel-time inversion performed with an iterative ensemble smoothing technique. The proposed model-error approach can be applied to large scale parameter-estimation problems without assumptions regarding the model-error statistics. The results in the thesis motivate the application to other examples and real-world inverse problems