Fault diagnosis by refining the parameter uncertainty space of nonlinear dynamic systems

This paper deals with fault detection and isolation problems for nonlinear dynamic systems. Both problems are stated as constraint satisfaction problems (CSP) and solved using consistency techniques. The main contribution is the isolation method based on consistency techniques and uncertainty space refining of interval parameters. The major advantage of this method is that the isolation speed is fast even taking into account uncertainty in parameters, measurements, and model errors. Interval calculations bring independence from the assumption of monotony considered by several approaches for fault isolation which are based on observers. An application to a well known alcoholic fermentation process model is presented

Fault diagnosis by refining the parameter uncertainty space of nonlinear dynamic systems

This paper deals with fault detection and isolation problems for nonlinear dynamic systems. Both problems are stated as constraint satisfaction problems (CSP) and solved using consistency techniques. The main contribution is the isolation method based on consistency techniques and uncertainty space refining of interval parameters. The major advantage of this method is that the isolation speed is fast even taking into account uncertainty in parameters, measurements, and model errors. Interval calculations bring independence from the assumption of monotony considered by several approaches for fault isolation which are based on observers. An application to a well known alcoholic fermentation process model is presented

Környezeti kockázatok felmérése. Paraméterek bizonytalanságának hatása a kockázatkezelési döntéshozatalra = Assessment of environmental risks – effects of parameter uncertainty to the risk management decisions

A környezeti kockázatok megfelelő felmérése és kezelése napjaink egyik legfontosabb kérdése, nemcsak a szakmai, hanem a széles értelemben vett közvélemény számára. A szerző cikkében azt vizsgálja, hogy a környezeti kockázatok felmérésének milyen megközelítései vannak. Kulcskérdésként pedig arra koncentrál, hogy a kockázatkezelési döntéseket hogyan befolyásolja a becslések bizonytalansága. Először a környezeti kockázat definícióját adja meg, majd azt mutatja be, hogy a környezeti kockázatok kezelésére vonatkozó megközelítések milyen párhuzamban állnak a pénzügyi rendszerrel, mint komplex rendszerre vonatkozó megközelítésekkel. Végül a jelenleg legnagyobb kockázatoknak tartott környezeti kockázatokat ismerteti röviden. A cikk második részében kockázatkezelési alternatívákat mutat be, és azt, hogy a kockázatkezelési lépések kiválasztását befolyásolja a bizonytalanság. Ezt illusztrálandó Brouwer-Blois (2008) modelljét használva a soklépéses szimulációt és alternatív döntési kritériumot – a kritikus (extrém) költség-hatás mutatót – alkalmazza. _____________ Adequate assessment and management of environmental risks is a key question nowadays also for professional experts and also for the overall public. In this article the author examines the different approaches concerning environmental risks. He concentrates as a key question the influence on risk management decisions of uncertainties raised by our estimations. First he analyses the definition of environmental risks, and he shows the similarities and differences between approaches concerning environmental risks and risks threatening financial system, and finally he gives short overview on the most current environmental risks. In the second part of the paper he presents risk management alternatives and analyses the influential power of uncertainty on risk management decisions. In order to illustrate this phenomenon the author applies the model of Brouwer-Blois (2008) with multistep simulation and an alternative decisive criterion, the ranking based on critical (extreme) cost to effect measure.

An experiment on the parameter uncertainty of hydrological models with different levels of complexity in a climate change context

La possibilité d’estimer l’impact du changement climatique en cours sur le comportement hydrologique des hydro-systèmes est une nécessité pour anticiper les adaptations inévitables et nécessaires que doivent envisager nos sociétés. Dans ce contexte, ce projet doctoral présente une étude sur l’évaluation de la sensibilité des projections hydrologiques futures à : (i) La non-robustesse de l’identification des paramètres des modèles hydrologiques, (ii) l’utilisation de plusieurs jeux de paramètres équifinaux et (iii) l’utilisation de différentes structures de modèles hydrologiques. Pour quantifier l’impact de la première source d’incertitude sur les sorties des modèles, quatre sous-périodes climatiquement contrastées sont tout d’abord identifiées au sein des chroniques observées. Les modèles sont calés sur chacune de ces quatre périodes et les sorties engendrées sont analysées en calage et en validation en suivant les quatre configurations du Different Splitsample Tests (Klemeš, 1986;Wilby, 2005; Seiller et al. (2012);Refsgaard et al. (2014)). Afin d’étudier la seconde source d’incertitude liée à la structure du modèle, l’équifinalité des jeux de paramètres est ensuite prise en compte en considérant pour chaque type de calage les sorties associées à des jeux de paramètres équifinaux. Enfin, pour évaluer la troisième source d’incertitude, cinq modèles hydrologiques de différents niveaux de complexité sont appliqués (GR4J, MORDOR, HSAMI, SWAT et HYDROTEL) sur le bassin versant québécois de la rivière Au Saumon. Les trois sources d’incertitude sont évaluées à la fois dans conditions climatiques observées passées et dans les conditions climatiques futures. Les résultats montrent que, en tenant compte de la méthode d’évaluation suivie dans ce doctorat, l’utilisation de différents niveaux de complexité des modèles hydrologiques est la principale source de variabilité dans les projections de débits dans des conditions climatiques futures. Ceci est suivi par le manque de robustesse de l’identification des paramètres. Les projections hydrologiques générées par un ensemble de jeux de paramètres équifinaux sont proches de celles associées au jeu de paramètres optimal. Par conséquent, plus d’efforts devraient être investis dans l’amélioration de la robustesse des modèles pour les études d’impact sur le changement climatique, notamment en développant les structures des modèles plus appropriés et en proposant des procédures de calage qui augmentent leur robustesse. Ces travaux permettent d’apporter une réponse détaillée sur notre capacité à réaliser un diagnostic des impacts des changements climatiques sur les ressources hydriques du bassin Au Saumon et de proposer une démarche méthodologique originale d’analyse pouvant être directement appliquée ou adaptée à d’autres contextes hydro-climatiques.

Term Structure Dynamics, Macro-Finance Factors and Model Uncertainty

This paper extends the Nelson-Siegel linear factor model by developing a flexible macro-finance framework for modeling and forecasting the term structure of US interest rates. Our approach is robust to parameter uncertainty and structural change, as we consider instabilities in parameters and volatilities, and our model averaging method allows for investors' model uncertainty over time. Our time-varying parameter Nelson-Siegel Dynamic Model Averaging (NS-DMA) predicts yields better than standard benchmarks and successfully captures plausible time-varying term premia in real time. The proposed model has significant in-sample and out-of-sample predictability for excess bond returns, and the predictability is of economic value.

Estimating uncertainty in terrestrial critical loads and their exceedances at four sites in the UK

Critical loads are the basis for policies controlling emissions of acidic substances in Europe and elsewhere. They are assessed by several elaborate and ingenious models, each of which requires many parameters, and have to be applied on a spatially-distributed basis. Often the values of the input parameters are poorly known, calling into question the validity of the calculated critical loads. This paper attempts to quantify the uncertainty in the critical loads due to this "parameter uncertainty", using examples from the UK. Models used for calculating critical loads for deposition of acidity and nitrogen in forest and heathland ecosystems were tested at four contrasting sites. Uncertainty was assessed by Monte Carlo methods. Each input parameter or variable was assigned a value, range and distribution in an objective a fashion as possible. Each model was run 5000 times at each site using parameters sampled from these input distributions. Output distributions of various critical load parameters were calculated. The results were surprising. Confidence limits of the calculated critical loads were typically considerably narrower than those of most of the input parameters. This may be due to a "compensation of errors" mechanism. The range of possible critical load values at a given site is however rather wide, and the tails of the distributions are typically long. The deposition reductions required for a high level of confidence that the critical load is not exceeded are thus likely to be large. The implication for pollutant regulation is that requiring a high probability of non-exceedance is likely to carry high costs. The relative contribution of the input variables to critical load uncertainty varied from site to site: any input variable could be important, and thus it was not possible to identify variables as likely targets for research into narrowing uncertainties. Sites where a number of good measurements of input parameters were available had lower uncertainties, so use of in situ measurement could be a valuable way of reducing critical load uncertainty at particularly valuable or disputed sites. From a restricted number of samples, uncertainties in heathland critical loads appear comparable to those of coniferous forest, and nutrient nitrogen critical loads to those of acidity. It was important to include correlations between input variables in the Monte Carlo analysis, but choice of statistical distribution type was of lesser importance. Overall, the analysis provided objective support for the continued use of critical loads in policy development. (c) 2007 Elsevier B.V. All rights reserved.

A note on non-linear phenomena in a non-ideal oscillator, with a snap-through truss absorber, including parameter uncertainties

This work considers the vibrating system that consists of a snap-through truss absorber coupled to an oscillator under excitation of an electric motor with an eccentricity and limited power, characterizing a non-ideal oscillator. It is aimed to use the non-linearity and quasi-zero stiffness of absorber (snap-through truss absorber) to obtain a significantly attenuation the jump phenomenon. There is also an interest to exhibit the reduction of Sommerfeld effect, to confirm the saturation phenomenon occurrence and show the power transfer in a non-linear structure, evidencing the pumping energy. As shown by simulations in this work, this absorber allows the energy pumping before and during the jump phenomenon, decreasing the higher amplitudes of considered system. Additionally, the occurrence of saturation phenomenon due use of snap-through truss absorber is verified. The analysis of parameter uncertainties was introduced. Sensitivity of system with parametric errors demonstrated a trustable system. © IMechE 2012.

A comparison of deterministic, reliability-based and risk-based structural optimization under uncertainty

In this paper, the effects of uncertainty and expected costs of failure on optimum structural design are investigated, by comparing three distinct formulations of structural optimization problems. Deterministic Design Optimization (DDO) allows one the find the shape or configuration of a structure that is optimum in terms of mechanics, but the formulation grossly neglects parameter uncertainty and its effects on structural safety. Reliability-based Design Optimization (RBDO) has emerged as an alternative to properly model the safety-under-uncertainty part of the problem. With RBDO, one can ensure that a minimum (and measurable) level of safety is achieved by the optimum structure. However, results are dependent on the failure probabilities used as constraints in the analysis. Risk optimization (RO) increases the scope of the problem by addressing the compromising goals of economy and safety. This is accomplished by quantifying the monetary consequences of failure, as well as the costs associated with construction, operation and maintenance. RO yields the optimum topology and the optimum point of balance between economy and safety. Results are compared for some example problems. The broader RO solution is found first, and optimum results are used as constraints in DDO and RBDO. Results show that even when optimum safety coefficients are used as constraints in DDO, the formulation leads to configurations which respect these design constraints, reduce manufacturing costs but increase total expected costs (including expected costs of failure). When (optimum) system failure probability is used as a constraint in RBDO, this solution also reduces manufacturing costs but by increasing total expected costs. This happens when the costs associated with different failure modes are distinct. Hence, a general equivalence between the formulations cannot be established. Optimum structural design considering expected costs of failure cannot be controlled solely by safety factors nor by failure probability constraints, but will depend on actual structural configuration. (c) 2011 Elsevier Ltd. All rights reserved.

Robust linear semi-infinite programming duality under uncertainty

In this paper, we propose a duality theory for semi-infinite linear programming problems under uncertainty in the constraint functions, the objective function, or both, within the framework of robust optimization. We present robust duality by establishing strong duality between the robust counterpart of an uncertain semi-infinite linear program and the optimistic counterpart of its uncertain Lagrangian dual. We show that robust duality holds whenever a robust moment cone is closed and convex. We then establish that the closed-convex robust moment cone condition in the case of constraint-wise uncertainty is in fact necessary and sufficient for robust duality. In other words, the robust moment cone is closed and convex if and only if robust duality holds for every linear objective function of the program. In the case of uncertain problems with affinely parameterized data uncertainty, we establish that robust duality is easily satisfied under a Slater type constraint qualification. Consequently, we derive robust forms of the Farkas lemma for systems of uncertain semi-infinite linear inequalities.

Integrated sensitivity analysis, calibration, and uncertainty propagation analysis approaches for supporting hydrological modeling

The successful performance of a hydrological model is usually challenged by the quality of the sensitivity analysis, calibration and uncertainty analysis carried out in the modeling exercise and subsequent simulation results. This is especially important under changing climatic conditions where there are more uncertainties associated with climate models and downscaling processes that increase the complexities of the hydrological modeling system. In response to these challenges and to improve the performance of the hydrological models under changing climatic conditions, this research proposed five new methods for supporting hydrological modeling. First, a design of experiment aided sensitivity analysis and parameterization (DOE-SAP) method was proposed to investigate the significant parameters and provide more reliable sensitivity analysis for improving parameterization during hydrological modeling. The better calibration results along with the advanced sensitivity analysis for significant parameters and their interactions were achieved in the case study. Second, a comprehensive uncertainty evaluation scheme was developed to evaluate three uncertainty analysis methods, the sequential uncertainty fitting version 2 (SUFI-2), generalized likelihood uncertainty estimation (GLUE) and Parameter solution (ParaSol) methods. The results showed that the SUFI-2 performed better than the other two methods based on calibration and uncertainty analysis results. The proposed evaluation scheme demonstrated that it is capable of selecting the most suitable uncertainty method for case studies. Third, a novel sequential multi-criteria based calibration and uncertainty analysis (SMC-CUA) method was proposed to improve the efficiency of calibration and uncertainty analysis and control the phenomenon of equifinality. The results showed that the SMC-CUA method was able to provide better uncertainty analysis results with high computational efficiency compared to the SUFI-2 and GLUE methods and control parameter uncertainty and the equifinality effect without sacrificing simulation performance. Fourth, an innovative response based statistical evaluation method (RESEM) was proposed for estimating the uncertainty propagated effects and providing long-term prediction for hydrological responses under changing climatic conditions. By using RESEM, the uncertainty propagated from statistical downscaling to hydrological modeling can be evaluated. Fifth, an integrated simulation-based evaluation system for uncertainty propagation analysis (ISES-UPA) was proposed for investigating the effects and contributions of different uncertainty components to the total propagated uncertainty from statistical downscaling. Using ISES-UPA, the uncertainty from statistical downscaling, uncertainty from hydrological modeling, and the total uncertainty from two uncertainty sources can be compared and quantified. The feasibility of all the methods has been tested using hypothetical and real-world case studies. The proposed methods can also be integrated as a hydrological modeling system to better support hydrological studies under changing climatic conditions. The results from the proposed integrated hydrological modeling system can be used as scientific references for decision makers to reduce the potential risk of damages caused by extreme events for long-term water resource management and planning.

A Parameter Estimation and Identifiability Analysis Methodology Applied to a Street Canyon Air Pollution Model

Mathematical models are increasingly used in environmental science thus increasing the importance of uncertainty and sensitivity analyses. In the present study, an iterative parameter estimation and identifiability analysis methodology is applied to an atmospheric model – the Operational Street Pollution Model (OSPMr). To assess the predictive validity of the model, the data is split into an estimation and a prediction data set using two data splitting approaches and data preparation techniques (clustering and outlier detection) are analysed. The sensitivity analysis, being part of the identifiability analysis, showed that some model parameters were significantly more sensitive than others. The application of the determined optimal parameter values was shown to succesfully equilibrate the model biases among the individual streets and species. It was as well shown that the frequentist approach applied for the uncertainty calculations underestimated the parameter uncertainties. The model parameter uncertainty was qualitatively assessed to be significant, and reduction strategies were identified.

Methodologies for calibration and predictive analysis of a watershed model

The use of a fitted parameter watershed model to address water quantity and quality management issues requires that it be calibrated under a wide range of hydrologic conditions. However, rarely does model calibration result in a unique parameter set. Parameter nonuniqueness can lead to predictive nonuniqueness. The extent of model predictive uncertainty should be investigated if management decisions are to be based on model projections. Using models built for four neighboring watersheds in the Neuse River Basin of North Carolina, the application of the automated parameter optimization software PEST in conjunction with the Hydrologic Simulation Program Fortran (HSPF) is demonstrated. Parameter nonuniqueness is illustrated, and a method is presented for calculating many different sets of parameters, all of which acceptably calibrate a watershed model. A regularization methodology is discussed in which models for similar watersheds can be calibrated simultaneously. Using this method, parameter differences between watershed models can be minimized while maintaining fit between model outputs and field observations. In recognition of the fact that parameter nonuniqueness and predictive uncertainty are inherent to the modeling process, PEST's nonlinear predictive analysis functionality is then used to explore the extent of model predictive uncertainty.

Time-lapse and probabilistic inversion strategies for plane-wave electromagnetic data.

L'utilisation efficace des systèmes géothermaux, la séquestration du CO2 pour limiter le changement climatique et la prévention de l'intrusion d'eau salée dans les aquifères costaux ne sont que quelques exemples qui démontrent notre besoin en technologies nouvelles pour suivre l'évolution des processus souterrains à partir de la surface. Un défi majeur est d'assurer la caractérisation et l'optimisation des performances de ces technologies à différentes échelles spatiales et temporelles. Les méthodes électromagnétiques (EM) d'ondes planes sont sensibles à la conductivité électrique du sous-sol et, par conséquent, à la conductivité électrique des fluides saturant la roche, à la présence de fractures connectées, à la température et aux matériaux géologiques. Ces méthodes sont régies par des équations valides sur de larges gammes de fréquences, permettant détudier de manières analogues des processus allant de quelques mètres sous la surface jusqu'à plusieurs kilomètres de profondeur. Néanmoins, ces méthodes sont soumises à une perte de résolution avec la profondeur à cause des propriétés diffusives du champ électromagnétique. Pour cette raison, l'estimation des modèles du sous-sol par ces méthodes doit prendre en compte des informations a priori afin de contraindre les modèles autant que possible et de permettre la quantification des incertitudes de ces modèles de façon appropriée. Dans la présente thèse, je développe des approches permettant la caractérisation statique et dynamique du sous-sol à l'aide d'ondes EM planes. Dans une première partie, je présente une approche déterministe permettant de réaliser des inversions répétées dans le temps (time-lapse) de données d'ondes EM planes en deux dimensions. Cette stratégie est basée sur l'incorporation dans l'algorithme d'informations a priori en fonction des changements du modèle de conductivité électrique attendus. Ceci est réalisé en intégrant une régularisation stochastique et des contraintes flexibles par rapport à la gamme des changements attendus en utilisant les multiplicateurs de Lagrange. J'utilise des normes différentes de la norme l2 pour contraindre la structure du modèle et obtenir des transitions abruptes entre les régions du model qui subissent des changements dans le temps et celles qui n'en subissent pas. Aussi, j'incorpore une stratégie afin d'éliminer les erreurs systématiques de données time-lapse. Ce travail a mis en évidence l'amélioration de la caractérisation des changements temporels par rapport aux approches classiques qui réalisent des inversions indépendantes à chaque pas de temps et comparent les modèles. Dans la seconde partie de cette thèse, j'adopte un formalisme bayésien et je teste la possibilité de quantifier les incertitudes sur les paramètres du modèle dans l'inversion d'ondes EM planes. Pour ce faire, je présente une stratégie d'inversion probabiliste basée sur des pixels à deux dimensions pour des inversions de données d'ondes EM planes et de tomographies de résistivité électrique (ERT) séparées et jointes. Je compare les incertitudes des paramètres du modèle en considérant différents types d'information a priori sur la structure du modèle et différentes fonctions de vraisemblance pour décrire les erreurs sur les données. Les résultats indiquent que la régularisation du modèle est nécessaire lorsqu'on a à faire à un large nombre de paramètres car cela permet d'accélérer la convergence des chaînes et d'obtenir des modèles plus réalistes. Cependent, ces contraintes mènent à des incertitudes d'estimations plus faibles, ce qui implique des distributions a posteriori qui ne contiennent pas le vrai modèledans les régions ou` la méthode présente une sensibilité limitée. Cette situation peut être améliorée en combinant des méthodes d'ondes EM planes avec d'autres méthodes complémentaires telles que l'ERT. De plus, je montre que le poids de régularisation des paramètres et l'écart-type des erreurs sur les données peuvent être retrouvés par une inversion probabiliste. Finalement, j'évalue la possibilité de caractériser une distribution tridimensionnelle d'un panache de traceur salin injecté dans le sous-sol en réalisant une inversion probabiliste time-lapse tridimensionnelle d'ondes EM planes. Etant donné que les inversions probabilistes sont très coûteuses en temps de calcul lorsque l'espace des paramètres présente une grande dimension, je propose une stratégie de réduction du modèle ou` les coefficients de décomposition des moments de Legendre du panache de traceur injecté ainsi que sa position sont estimés. Pour ce faire, un modèle de résistivité de base est nécessaire. Il peut être obtenu avant l'expérience time-lapse. Un test synthétique montre que la méthodologie marche bien quand le modèle de résistivité de base est caractérisé correctement. Cette méthodologie est aussi appliquée à un test de trac¸age par injection d'une solution saline et d'acides réalisé dans un système géothermal en Australie, puis comparée à une inversion time-lapse tridimensionnelle réalisée selon une approche déterministe. L'inversion probabiliste permet de mieux contraindre le panache du traceur salin gr^ace à la grande quantité d'informations a priori incluse dans l'algorithme. Néanmoins, les changements de conductivités nécessaires pour expliquer les changements observés dans les données sont plus grands que ce qu'expliquent notre connaissance actuelle des phénomenès physiques. Ce problème peut être lié à la qualité limitée du modèle de résistivité de base utilisé, indiquant ainsi que des efforts plus grands devront être fournis dans le futur pour obtenir des modèles de base de bonne qualité avant de réaliser des expériences dynamiques. Les études décrites dans cette thèse montrent que les méthodes d'ondes EM planes sont très utiles pour caractériser et suivre les variations temporelles du sous-sol sur de larges échelles. Les présentes approches améliorent l'évaluation des modèles obtenus, autant en termes d'incorporation d'informations a priori, qu'en termes de quantification d'incertitudes a posteriori. De plus, les stratégies développées peuvent être appliquées à d'autres méthodes géophysiques, et offrent une grande flexibilité pour l'incorporation d'informations additionnelles lorsqu'elles sont disponibles. -- The efficient use of geothermal systems, the sequestration of CO2 to mitigate climate change, and the prevention of seawater intrusion in coastal aquifers are only some examples that demonstrate the need for novel technologies to monitor subsurface processes from the surface. A main challenge is to assure optimal performance of such technologies at different temporal and spatial scales. Plane-wave electromagnetic (EM) methods are sensitive to subsurface electrical conductivity and consequently to fluid conductivity, fracture connectivity, temperature, and rock mineralogy. These methods have governing equations that are the same over a large range of frequencies, thus allowing to study in an analogous manner processes on scales ranging from few meters close to the surface down to several hundreds of kilometers depth. Unfortunately, they suffer from a significant resolution loss with depth due to the diffusive nature of the electromagnetic fields. Therefore, estimations of subsurface models that use these methods should incorporate a priori information to better constrain the models, and provide appropriate measures of model uncertainty. During my thesis, I have developed approaches to improve the static and dynamic characterization of the subsurface with plane-wave EM methods. In the first part of this thesis, I present a two-dimensional deterministic approach to perform time-lapse inversion of plane-wave EM data. The strategy is based on the incorporation of prior information into the inversion algorithm regarding the expected temporal changes in electrical conductivity. This is done by incorporating a flexible stochastic regularization and constraints regarding the expected ranges of the changes by using Lagrange multipliers. I use non-l2 norms to penalize the model update in order to obtain sharp transitions between regions that experience temporal changes and regions that do not. I also incorporate a time-lapse differencing strategy to remove systematic errors in the time-lapse inversion. This work presents improvements in the characterization of temporal changes with respect to the classical approach of performing separate inversions and computing differences between the models. In the second part of this thesis, I adopt a Bayesian framework and use Markov chain Monte Carlo (MCMC) simulations to quantify model parameter uncertainty in plane-wave EM inversion. For this purpose, I present a two-dimensional pixel-based probabilistic inversion strategy for separate and joint inversions of plane-wave EM and electrical resistivity tomography (ERT) data. I compare the uncertainties of the model parameters when considering different types of prior information on the model structure and different likelihood functions to describe the data errors. The results indicate that model regularization is necessary when dealing with a large number of model parameters because it helps to accelerate the convergence of the chains and leads to more realistic models. These constraints also lead to smaller uncertainty estimates, which imply posterior distributions that do not include the true underlying model in regions where the method has limited sensitivity. This situation can be improved by combining planewave EM methods with complimentary geophysical methods such as ERT. In addition, I show that an appropriate regularization weight and the standard deviation of the data errors can be retrieved by the MCMC inversion. Finally, I evaluate the possibility of characterizing the three-dimensional distribution of an injected water plume by performing three-dimensional time-lapse MCMC inversion of planewave EM data. Since MCMC inversion involves a significant computational burden in high parameter dimensions, I propose a model reduction strategy where the coefficients of a Legendre moment decomposition of the injected water plume and its location are estimated. For this purpose, a base resistivity model is needed which is obtained prior to the time-lapse experiment. A synthetic test shows that the methodology works well when the base resistivity model is correctly characterized. The methodology is also applied to an injection experiment performed in a geothermal system in Australia, and compared to a three-dimensional time-lapse inversion performed within a deterministic framework. The MCMC inversion better constrains the water plumes due to the larger amount of prior information that is included in the algorithm. The conductivity changes needed to explain the time-lapse data are much larger than what is physically possible based on present day understandings. This issue may be related to the base resistivity model used, therefore indicating that more efforts should be given to obtain high-quality base models prior to dynamic experiments. The studies described herein give clear evidence that plane-wave EM methods are useful to characterize and monitor the subsurface at a wide range of scales. The presented approaches contribute to an improved appraisal of the obtained models, both in terms of the incorporation of prior information in the algorithms and the posterior uncertainty quantification. In addition, the developed strategies can be applied to other geophysical methods, and offer great flexibility to incorporate additional information when available.

Impacts of upland open drains upon runoff generation: a numerical assessment of catchment-scale impacts

Shallow upland drains, grips, have been hypothesized as responsible for increased downstream flow magnitudes. Observations provide counterfactual evidence, often relating to the difficulty of inferring conclusions from statistical correlation and paired catchment comparisons, and the complexity of designing field experiments to test grip impacts at the catchment scale. Drainage should provide drier antecedent moisture conditions, providing more storage at the start of an event; however, grips have higher flow velocities than overland flow, thus potentially delivering flow more rapidly to the drainage network. We develop and apply a model for assessing the impacts of grips on flow hydrographs. The model was calibrated on the gripped case, and then the gripped case was compared with the intact case by removing all grips. This comparison showed that even given parameter uncertainty, the intact case had significantly higher flood peaks and lower baseflows, mirroring field observations of the hydrological response of intact peat. The simulations suggest that this is because delivery effects may not translate into catchment-scale impacts for three reasons. First, in our case, the proportions of flow path lengths that were hillslope were not changed significantly by gripping. Second, the structure of the grip network as compared with the structure of the drainage basin mitigated against grip-related increases in the concentration of runoff in the drainage network, although it did marginally reduce the mean timing of that concentration at the catchment outlet. Third, the effect of the latter upon downstream flow magnitudes can only be assessed by reference to the peak timing of other tributary basins, emphasizing that drain effects are both relative and scale dependent. However, given the importance of hillslope flow paths, we show that if upland drainage causes significant changes in surface roughness on hillslopes, then critical and important feedbacks may impact upon the speed of hydrological response. Copyright (c) 2012 John Wiley & Sons, Ltd.

Two-dimensional probabilistic inversion of plane-wave electromagnetic data: Methodology, model constraints and joint inversion with electrical resistivity data

Probabilistic inversion methods based on Markov chain Monte Carlo (MCMC) simulation are well suited to quantify parameter and model uncertainty of nonlinear inverse problems. Yet, application of such methods to CPU-intensive forward models can be a daunting task, particularly if the parameter space is high dimensional. Here, we present a 2-D pixel-based MCMC inversion of plane-wave electromagnetic (EM) data. Using synthetic data, we investigate how model parameter uncertainty depends on model structure constraints using different norms of the likelihood function and the model constraints, and study the added benefits of joint inversion of EM and electrical resistivity tomography (ERT) data. Our results demonstrate that model structure constraints are necessary to stabilize the MCMC inversion results of a highly discretized model. These constraints decrease model parameter uncertainty and facilitate model interpretation. A drawback is that these constraints may lead to posterior distributions that do not fully include the true underlying model, because some of its features exhibit a low sensitivity to the EM data, and hence are difficult to resolve. This problem can be partly mitigated if the plane-wave EM data is augmented with ERT observations. The hierarchical Bayesian inverse formulation introduced and used herein is able to successfully recover the probabilistic properties of the measurement data errors and a model regularization weight. Application of the proposed inversion methodology to field data from an aquifer demonstrates that the posterior mean model realization is very similar to that derived from a deterministic inversion with similar model constraints.