Bayesian maximum entropy approach and its applications: a review

The present paper reviews the conceptual framework and development of the Bayesian Maximum Entropy (BME) approach. BME has been considered as a significant breakthrough and contribution to applied stochastics by introducing an improved, knowledge-based modeling framework for spatial and spatiotemporal information. In this work, one objective is the overview of distinct BME features. By offering a foundation free of restrictive assumptions that limit comparable techniques, an ability to integrate a variety of prior knowledge bases, and rigorous accounting for both exact and uncertain data, the BME approach was coined as introducing modern spatiotemporal geostatistics. A second objective is to illustrate BME applications and adoption within numerous different scientific disciplines. We summarize examples and real-world studies that encompass the perspective of science of the total environment, including atmosphere, lithosphere, hydrosphere, and ecosphere, while also noting applications that extend beyond these fields. The broad-ranging application track suggests BME as an established, valuable tool for predictive spatial and space–time analysis and mapping. This review concludes with the present status of BME, and tentative paths for future methodological research, enhancements, and extensions.     

Modelling storage‐driven connectivity between landscapes and riverscapes: towards a simple framework for long‐term ecohydrological assessment

The importance of conceptualizing the dynamics of storage‐driven saturation area connectivity in runoff generation has been central to the development of TOPMODEL and similar low parameterized rainfall–runoff models. In this contribution, we show how we developed a 40‐year hydrometric data base to simulate storage–discharge relationships in the Girnock catchment in the Scottish Highlands using a simple conceptual model. The catchment is a unique fisheries reference site where Atlantic salmon populations have been monitored since 1966. The modelling allowed us to track storage dynamics in hillslopes, the riparian zone and groundwater, and explicitly link non‐linear changes of streamflows to landscape storage and connectivity dynamics. This provides a fundamental basis for understanding how the landscape and riverscape are hydrologically connected and how this regulates in‐stream hydraulic conditions that directly influence salmonids. We use the model to simulate storage and discharge dynamics over the 40‐year period of fisheries records. The modelled storage‐driven connectivity provides an ecohydological context for understanding the dynamics in stream flow generation which determine habitat hydraulics for different life stages of salmon population. This new, long‐term modelling now sets this variability in the riverscape in a more fundamental context of the inter‐relationships between storage in the landscape and stream flow generation. This provides a simple, robust framework for future ecohydrological modelling at this site, which is an alternative to more increasingly popular but highly parameterized and uncertain commercial ecohydrological models. It also provides a wider, novel context that is a prerequisite for any model‐based scenario assessment of likely impacts resulting from climate or land use change. Copyright © 2016 The Authors Hydrological Processes Published by John Wiley & Sons Ltd. Copyright © 2016 The Authors Hydrological Processes Published by John Wiley & Sons Ltd.     

Sensitivity analysis of prior model probabilities and the value of prior knowledge in the assessment of conceptual model uncertainty in groundwater modelling

A key point in the application of multi‐model Bayesian averaging techniques to assess the predictive uncertainty in groundwater modelling applications is the definition of prior model probabilities, which reflect the prior perception about the plausibility of alternative models. In this work the influence of prior knowledge and prior model probabilities on posterior model probabilities, multi‐model predictions, and conceptual model uncertainty estimations is analysed. The sensitivity to prior model probabilities is assessed using an extensive numerical analysis in which the prior probability space of a set of plausible conceptualizations is discretized to obtain a large ensemble of possible combinations of prior model probabilities. Additionally, the value of prior knowledge about alternative models in reducing conceptual model uncertainty is assessed by considering three example knowledge states, expressed as quantitative relations among the alternative models. A constrained maximum entropy approach is used to find the set of prior model probabilities that correspond to the different prior knowledge states. For illustrative purposes, a three‐dimensional hypothetical setup approximated by seven alternative conceptual models is employed. Results show that posterior model probabilities, leading moments of the predictive distributions and estimations of conceptual model uncertainty are very sensitive to prior model probabilities, indicating the relevance of selecting proper prior probabilities. Additionally, including proper prior knowledge improves the predictive performance of the multi‐model approach, expressed by reductions of the multi‐model prediction variances by up to 60% compared with a non‐informative case. However, the ratio between‐model to total variance does not substantially decrease. This suggests that the contribution of conceptual model uncertainty to the total variance cannot be further reduced based only on prior knowledge about the plausibility of alternative models. These results advocate including proper prior knowledge about alternative conceptualizations in combination with extra conditioning data to further reduce conceptual model uncertainty in groundwater modelling predictions. Copyright © 2009 John Wiley & Sons, Ltd.     

Subspace system identification of support excited structures—part II: gray‐box interpretations and damage detection

A theoretical framework is presented for the estimation of the physical parameters of a structure (i.e., mass, stiffness, and damping) from measured experimental data (i.e., input–output or output‐only data). The framework considers two state‐space models: a physics‐based model derived from first principles (i.e., white‐box model) and a data‐driven mathematical model derived by subspace system identification (i.e., black‐box model). Observability canonical form conversion is introduced as a powerful means to convert the data‐driven mathematical model into a physically interpretable model that is termed a gray‐box model. Through an explicit linking of the white‐box and gray‐box model forms, the physical parameters of the structural system can be extracted from the gray‐box model in the form of a finite element discretization. Prior to experimental verification, the framework is numerically verified for a multi‐DOF shear building structure. Without a priori knowledge of the structure, mass, stiffness, and damping properties are accurately estimated. Then, experimental verification of the framework is conducted using a six‐story steel frame structure under support excitation. With a priori knowledge of the lumped mass matrix, the spatial distribution of structural stiffness and damping is estimated. With an accurate estimation of the physical parameters of the structure, the gray‐box model is shown to be capable of providing the basis for damage detection. With the use of the experimental structure, the gray‐box model is used to reliably estimate changes in structural stiffness attributed to intentional damage introduced. Copyright © 2012 John Wiley & Sons, Ltd.     

Catchment properties,function, and conceptual model representation: is there a correspondence?

This study investigates the possible correspondence between catchment structure, as represented by perceptual hydrological models developed from fieldwork investigations, and mathematical model structures, selected on the basis of reproducing observed catchment hydrographs. Three Luxembourgish headwater catchments are considered, where previous fieldwork suggested distinct flow‐generating mechanisms and hydrological dynamics. A set of lumped conceptual model structures are hypothesized and implemented using the SUPERFLEX framework. Following parameter calibration, the model performance is examined in terms of predictive accuracy, quantification of uncertainty, and the ability to reproduce the flow–duration curve signature. Our key research question is whether differences in the performance of the conceptual model structures can be interpreted based on the dominant catchment processes suggested from fieldwork investigations. For example, we propose that the permeable bedrock and the presence of multiple aquifers in the Huewelerbach catchment may explain the superior performance of model structures with storage elements connected in parallel. Conversely, model structures with serial connections perform better in the Weierbach and Wollefsbach catchments, which are characterized by impermeable bedrock and dominated by lateral flow. The presence of threshold dynamics in the Weierbach and Wollefsbach catchments may favour nonlinear models, while the smoother dynamics of the larger Huewelerbach catchment were suitably reproduced by linear models. It is also shown how hydrologically distinct processes can be effectively described by the same mathematical model components. Major research questions are reviewed, including the correspondence between hydrological processes at different levels of scale and how best to synthesize the experimentalist's and modeller's perspectives. Copyright © 2013 John Wiley & Sons, Ltd.     

The application of Karhunen–Loéve,or principal component analysis method,to study the non‐linear seismic response of structures

The Karhunen–Loéve (K–L) method is used to interpret dynamic response data obtained from shaking table and pseudodynamic tests conducted on civil engineering structures subjected to earthquake loading. It is shown how the K–L method can be used to monitor on‐line, or a posteriori, the structural response of non‐linear dynamical systems. Results from these analyses make it possible to quantitatively verify the number and participation factors of non‐linear modes and how they correspond to physical behaviour of the structure. Comments are made regarding the use of this technique in various fields including numerical calculations, experiments and control. Copyright © 2000 John Wiley & Sons, Ltd.     

Transmissivity and storage coefficient estimation by coupling the Cooper–Jacob method and modified fuzzy least-squares regression

Traditionally the Cooper–Jacob equation is used to determine the transmissivity and the storage coefficient for an aquifer using pump test results. This model, however, is a simplified version of the actual subsurface and does not allow for analysis of the uncertainty that comes from a lack of knowledge about the heterogeneity of the environment under investigation. In this paper, a modified fuzzy least-squares regression (MFLSR) method is developed that uses imprecise pump test data to obtain fuzzy intercept and slope values which are then used in the Cooper–Jacob method. Fuzzy membership functions for the transmissivity and the storage coefficient are then calculated using the extension principle. The supports of the fuzzy membership functions incorporate the transmissivity and storage coefficient values that would be obtained using ordinary least-squares regression and the Cooper–Jacob method. The MFLSR coupled with the Cooper–Jacob method allows the analyst to ascertain the uncertainty that is inherent in the estimated parameters obtained using the simplified Cooper–Jacob method and data that are uncertain due to lack of knowledge regarding the heterogeneity of the aquifer.     

Conceptual fuzzy neural network model for water quality simulation

Artificial neural networks (ANNs) have been applied successfully in various fields. However, ANN models depend on large sets of historical data, and are of limited use when only vague and uncertain information is available, which leads to difficulties in defining the model architecture and a low reliability of results. A conceptual fuzzy neural network (CFNN) is proposed and applied in a water quality model to simulate the Barra Bonita reservoir system, located in the southeast region of Brazil. The CFNN model consists of a rationally‐defined architecture based on accumulated expert knowledge about variables and processes included in the model. A genetic algorithm is used as the training method for finding the parameters of fuzzy inference and the connection weights. The proposed model may handle the uncertainties related to the system itself, model parameterization, complexity of concepts involved and scarcity and inaccuracy of data. The CFNN showed greater robustness and reliability when dealing with systems for which data are considered to be vague, uncertain or incomplete. The CFNN model structure is easier to understand and to define than other ANN‐based models. Moreover, it can help to understand the basic behaviour of the system as a whole, being a successful example of cooperation between human and machine. Copyright © 2006 John Wiley & Sons, Ltd.     

Simulation/optimization modeling for robust pumping strategy design

A new simulation/optimization modeling approach is presented for addressing uncertain knowledge of aquifer parameters. The Robustness Enhancing Optimizer (REO) couples genetic algorithm and tabu search as optimizers and incorporates aquifer parameter sensitivity analysis to guide multiple-realization optimization. The REO maximizes strategy robustness for a pumping strategy that is optimal for a primary objective function (OF), such as cost. The more robust a strategy, the more likely it is to achieve management goals in the field, even if the physical system differs from the model. The REO is applied to trinitrotoluene and Royal Demolition Explosive plumes at Umatilla Chemical Depot in Oregon to develop robust least cost strategies. The REO efficiently develops robust pumping strategies while maintaining the optimal value of the primary OF-differing from the common situation in which a primary OF value degrades as strategy reliability increases. The REO is especially valuable where data to develop realistic probability density functions (PDFs) or statistically derived realizations are unavailable. Because they require much less field data, REO-developed strategies might not achieve as high a mathematical reliability as strategies developed using many realizations based upon real aquifer parameter PDFs. REO-developed strategies might or might not yield a better OF value in the field.     

Modelling stream flow for use in ecological studies in a large,arid zone river,central Australia

Australian arid zone ephemeral rivers are typically unregulated and maintain a high level of biodiversity and ecological health. Understanding the ecosystem functions of these rivers requires an understanding of their hydrology. These rivers are typified by highly variable hydrological regimes and a paucity, often a complete absence, of hydrological data to describe these flow regimes. A daily time‐step, grid‐based, conceptual rainfall–runoff model was developed for the previously uninstrumented Neales River in the arid zone of northern South Australia. Hourly, logged stage data provided a record of stream‐flow events in the river system. In conjunction with opportunistic gaugings of stream‐flow events, these data were used in the calibration of the model. The poorly constrained spatial variability of rainfall distribution and catchment characteristics (e.g. storage depths) limited the accuracy of the model in replicating the absolute magnitudes and volumes of stream‐flow events. In particular, small but ecologically important flow events were poorly modelled. Model performance was improved by the application of catchment‐wide processes replicating quick runoff from high intensity rainfall and improving the area inundated versus discharge relationship in the channel sections of the model. Representing areas of high and low soil moisture storage depths in the hillslope areas of the catchment also improved the model performance. The need for some explicit representation of the spatial variability of catchment characteristics (e.g. channel/floodplain, low storage hillslope and high storage hillslope) to effectively model the range of stream‐flow events makes the development of relatively complex rainfall–runoff models necessary for multisite ecological studies in large, ungauged arid zone catchments. Grid‐based conceptual models provide a good balance between providing the capacity to easily define land types with differing rainfall–runoff responses, flexibility in defining data output points and a parsimonious water‐balance–routing model. Copyright © 2004 John Wiley & Sons, Ltd.     

ANALYSIS OF AN ISOLATED STORM EVENT HYDROGRAPH BY IDENTIFYING THE INFLOW SEQUENCE TO A NON-LINEAR CONCEPTUAL RESERVOIR

In an isolated storm event model relating gross rainfall to the flow hydrograph, the shape transformation component can be composed of a number of subsystems. It is assumed here that this component contains, along with other subsystems, a non-linear conceptual reservoir expressed in general terms. Given an isolated storm event, a procedure is described that directly identifies the particular inflow sequence that needs to be applied to this reservoir to reproduce the observed discharge hydrograph as outflow. This procedure makes use of information contained only in the observed hydrograph and does not rely on prior knowledge of the gross or net rainfall hyetographs. Nor is it necessary to identify in full the storage–outflow relation of the non-linear reservoir used. Future comparisons between the gross rainfall hyetograph and the identified inflow sequence should be facilitated by the procedure's ability to remove the long trailing recession belonging to the outflow.     

从克拉通破坏到板块动力模型的研究

华北克拉通破坏动力机制研究导致了全球动力系统及“板块动力模型”研究.板块运动最有可能的动力是地幔物质流动,但由于地幔物质流动的成因至今尚未查明,所以板块构造学说研究仍处于“运动模型”阶段,而没有进入“动力模型”阶段.如果地幔密度异常是驱动地幔物质流动的成因,那么就有可能基于重力学方法以“板块动力模型”的形式建立地幔密度异常驱动模式;软流圈中可能存在着动力特性不同的区块,地幔密度正异常代表物质盈余、区块内的物质要向区块外移动,地幔密度负异常代表物质亏损、区块外的物质要向区块内移动.本项目采用重力和地震资料相结合研究地球的整体分层,根据重力大地水准面联合地震波速度结构反演求解地幔密度异常,再根据地球正常密度假说和板块运动重力学机制的观点并与现有“板块运动模型”相结合,分析地幔密度异常动力区块,初步建立“基于重力学机制的板块动力模型”;为最终建立多学科机制的“全球板块动力模型”,迈出重要一步.     

Look-ahead vertical seismic profiling inversion approach for density and compressional wave velocity in Bayesian framework

To reduce drilling uncertainties, zero-offset vertical seismic profiles can be inverted to quantify acoustic properties ahead of the bit. In this work, we propose an approach to invert vertical seismic profile corridor stacks in Bayesian framework for look-ahead prediction. The implemented approach helps to successfully predict density and compressional wave velocity using prior knowledge from drilled interval. Hence, this information can be used to monitor reservoir depth as well as quantifying high-pressure zones, which enables taking the correct decision during drilling. The inversion algorithm uses Gauss–Newton as an optimization tool, which requires the calculation of the sensitivity matrix of trace samples with respect to model parameters. Gauss–Newton has quadratic rate of convergence, which can speed up the inversion process. Moreover, geo-statistical analysis has been used to efficiently utilize prior information supplied to the inversion process. The algorithm has been tested on synthetic and field cases. For the field case, a zero-offset vertical seismic profile data taken from an offshore well were used as input to the inversion algorithm. Well logs acquired after drilling the prediction section was used to validate the inversion results. The results from the synthetic case applications were encouraging to accurately predict compressional wave velocity and density from just a constant prior model. The field case application shows the strength of our proposed approach in inverting vertical seismic profile data to obtain density and compressional wave velocity ahead of a bit with reasonable accuracy. Unlike the commonly used vertical seismic profile inversion approach for acoustic impedance using simple error to represent the prior covariance matrix, this work shows the importance of inverting for both density and compressional wave velocity using geo-statistical knowledge of density and compressional wave velocity from the drilled section to quantify the prior covariance matrix required during Bayesian inversion.     

Assessing uncertainties in urban drainage models

The current state of knowledge regarding uncertainties in urban drainage models is poor. This is in part due to the lack of clarity in the way model uncertainty analyses are conducted and how the results are presented and used. There is a need for a common terminology and a conceptual framework for describing and estimating uncertainties in urban drainage models. Practical tools for the assessment of model uncertainties for a range of urban drainage models are also required to be developed. This paper, produced by the International Working Group on Data and Models, which works under the IWA/IAHR Joint Committee on Urban Drainage, is a contribution to the development of a harmonised framework for defining and assessing uncertainties in the field of urban drainage modelling. The sources of uncertainties in urban drainage models and their links are initially mapped out. This is followed by an evaluation of each source, including a discussion of its definition and an evaluation of methods that could be used to assess its overall importance. Finally, an approach for a Global Assessment of Modelling Uncertainties (GAMU) is proposed, which presents a new framework for mapping and quantifying sources of uncertainty in urban drainage models.     

Solifluction meets vegetation: the role of biogeomorphic feedbacks for turf‐banked solifluction lobe development

Vegetation is an important factor influencing solifluction processes, while at the same time, solifluction processes and landforms influence species composition, fine‐scale distribution and corresponding ecosystem functioning. However, how feedbacks between plants and solifluction processes influence the development of turf‐banked solifluction lobes (TBLs) and their geomorphic and vegetation patterns is still poorly understood. We addressed this knowledge gap in a detailed biogeomorphic investigation in the Turtmann glacier foreland (Switzerland). Methods employed include geomorphic and vegetation mapping, terrain assessment with unmanned aerial vehicle (UAV) and temperature logging. Results were subsequently integrated with knowledge from previous geomorphic and ecologic studies into a conceptual model. Our results show that geomorphic and vegetation patterns at TBLs are closely linked through the lobe elements tread, risers and ridge. A conceptual four‐stage biogeomorphic model of TBL development with ecosystem engineering by the dwarf shrub Dryas octopetala as the dominant process can explain these interlinked patterns. Based on this model, we demonstrate that TBLs are biogeomorphic structures and follow a cyclic development, during which the role of their components for engineer and non‐engineer species changes. Our study presents the first biogeomorphic model of TBL development and highlights the applicability and necessity of biogeomorphic approaches and research in periglacial environments. Copyright © 2016 John Wiley & Sons, Ltd.     

全球地震层析地幔密度横向不均匀及大陆动力学

地震层析为我们提供了研究全球及区域地幔横向不均匀的有力工具。本文使用Ｓ１２－ＷＭ１３［１］层析模型,假设地震波速度异常与密度异常之间呈线性关系［２］,以ＰＲＥＭ模型［３］为基础,计算了全球和部分区域的球谐展开１２×１２阶地幔密度异常分布;分析了一系列的结果,特别分析了大陆下部地幔密度异常的基本格局;讨论了其对大陆动力学研究的意义。应用于中国大陆发现,整个中国大陆下部７０ｋｍ－１５０ｋｍ深部范围内是地震剪切波的低速带或密度低的地区,而在１５０－４００ｋｍ范围中又处于密度高的异常地带。本文从地幔动力学的观念出发讨论了中国大陆岩石层构造运动的地幔动力学问题。     

Is there a superior conceptual groundwater model structure for baseflow simulation?

Previous work has shown that streamflow response during baseflow conditions is a function of storage, but also that this functional relationship varies among seasons and catchments. Traditionally, hydrological models incorporate conceptual groundwater models consisting of linear or non‐linear storage–outflow functions. Identification of the right model structure and model parameterization however is challenging. The aim of this paper is to systematically test different model structures in a set of catchments where different aquifer types govern baseflow generation processes. Nine different two‐parameter conceptual groundwater models are applied with multi‐objective calibration to transform two different groundwater recharge series derived from a soil‐atmosphere‐vegetation transfer model into baseflow separated from streamflow data. The relative performance differences of the model structures allow to systematically improve the understanding of baseflow generation processes and to identify most appropriate model structures for different aquifer types. We found more versatile and more aquifer‐specific optimal model structures and elucidate the role of interflow, flow paths, recharge regimes and partially contributing storages. Aquifer‐specific recommendations of storage models were found for fractured and karstic aquifers, whereas large storage capacities blur the identification of superior model structures for complex and porous aquifers. A model performance matrix is presented, which highlights the joint effects of different recharge inputs, calibration criteria, model structures and aquifer types. The matrix is a guidance to improve groundwater model structures towards their representation of the dominant baseflow generation processes of specific aquifer types. Copyright © 2014 John Wiley & Sons, Ltd.     

Dealing with uncertainty in risk assessments in early stages of a CO<Subscript>2</Subscript> geological storage project: comparison of pure-probabilistic and fuzzy-probabilistic frameworks

CO₂ capture and storage is recognized as a promising solution among others to tackle greenhouse gas emissions. This technology requires robust risk assessment and management from the early stages of the project (i.e. during the site selection phase, prior to injection), which is a challenging task due to the high level of aleatory and epistemic uncertainties. This paper aims at implementing and comparing two frameworks for dealing with uncertainties: a classical probabilistic framework and a probabilistic-fuzzy-based (i.e. jointly combining fuzzy sets and probabilities) one. The comparison of both frameworks is illustrated for assessing the risk related to leakage of brine through an abandoned well on a realistic site in the Paris basin (France). For brine leakage flow computation, a semi-analytical model, requiring 25 input parameters, is used. Depending on the framework, available data is represented in a different manner (either using classical probability laws or interval-valued tools). Though the fuzzy-probabilistic framework for uncertainty propagation is computationally more expensive, it presents the major advantage to highlight situations of high degree of epistemic uncertainty: this enables nuancing a too-optimistic decision-making only supported by a single probabilistic curve (i.e. using the Monte-Carlo results). On this basis, we demonstrate how fuzzy-based sensitivity analysis can help identifying how to reduce the imprecision in an effective way, which has useful applications for additional studies. This study highlights the importance of choices in the mathematical tools for representing the lack of knowledge especially in the early phases of the project, where data is scarce, incomplete and imprecise.     

Bioturbation and erosion rates along the soil-hillslope conveyor belt,part 1: Insights from single-grain feldspar luminescence

The interplay of bioturbation, soil production and long-term erosion–deposition in soil and landscape co-evolution is poorly understood. Single-grain post-infrared infrared stimulated luminescence (post-IR IRSL) measurements on sand-sized grains of feldspar from the soil matrix can provide direct information on all three processes. To explore the potential of this novel method, we propose a conceptual model of how post-IR IRSL-derived burial age and fraction of surface-visiting grains change with soil depth and along a hillslope catena. We then tested this conceptual model by comparison with post-IR IRSL results for 15 samples taken at different depths within four soil profiles along a hillslope catena in the Santa Clotilde Critical Zone Observatory (southern Spain). In our work, we observed clear differences in apparent post-IR IRSL burial age distributions with depth along the catena, with younger ages and more linear age–depth structure for the hill-base profile, indicating the influence of lateral deposition processes. We noted shallower soils and truncated burial age–depth functions for the two erosional mid-slope profiles, and an exponential decline of burial age with depth for the hill-top profile. We suggest that the downslope increase in the fraction of surface-visiting grains at intermediate depths (20 cm) indicates creep to be the dominant erosion process. Our study demonstrates that single-grain feldspar luminescence signature-depth profiles provide a new way of tracing vertical and lateral soil mixing and transport processes. In addition, we propose a new objective luminescence-based criterion for mapping the soil-bedrock boundary, thus producing soil depths in better agreement with geomorphological process considerations. Our work highlights the possibilities of feldspar single grain techniques to provide quantitative insights into soil production, bioturbation and erosion–deposition. © 2019 The Authors. Earth Surface Processes and Landforms Published by John Wiley & Sons Ltd.