Modeling Spatial Variability with One and Multidimensional Continuous-Lag Markov Chains

The continuous-lag Markov chain provides a conceptually simple, mathematically compact, and theoretically powerful model of spatial variability for categorical variables. Markov chains have a long-standing record of applicability to one-dimensional (1-D) geologic data, but 2- and 3-D applications are rare. Theoretically, a multidimensional Markov chain may assume that 1-D Markov chains characterize spatial variability in all directions. Given that a 1-D continuous Markov chain can be described concisely by a transition rate matrix, this paper develops 3-D continuous-lag Markov chain models by interpolating transition rate matrices established for three principal directions, say strike, dip, and vertical. The transition rate matrix for each principal direction can be developed directly from data or indirectly by conceptual approaches. Application of Sylvester's theorem facilitates establishment of the transition rate matrix, as well as calculation of transition probabilities. The resulting 3-D continuous-lag Markov chain models then can be applied to geo-statistical estimation and simulation techniques, such as indicator cokriging, disjunctive kriging, sequential indicator simulation, and simulated annealing.     

Bayesian Gaussian Mixture Linear Inversion for Geophysical Inverse Problems

A Bayesian linear inversion methodology based on Gaussian mixture models and its application to geophysical inverse problems are presented in this paper. The proposed inverse method is based on a Bayesian approach under the assumptions of a Gaussian mixture random field for the prior model and a Gaussian linear likelihood function. The model for the latent discrete variable is defined to be a stationary first-order Markov chain. In this approach, a recursive exact solution to an approximation of the posterior distribution of the inverse problem is proposed. A Markov chain Monte Carlo algorithm can be used to efficiently simulate realizations from the correct posterior model. Two inversion studies based on real well log data are presented, and the main results are the posterior distributions of the reservoir properties of interest, the corresponding predictions and prediction intervals, and a set of conditional realizations. The first application is a seismic inversion study for the prediction of lithological facies, P- and S-impedance, where an improvement of 30% in the root-mean-square error of the predictions compared to the traditional Gaussian inversion is obtained. The second application is a rock physics inversion study for the prediction of lithological facies, porosity, and clay volume, where predictions slightly improve compared to the Gaussian inversion approach.     

History matching of approximated lithofacies models under uncertainty

In history matching of lithofacies reservoir model, we attempt to find multiple realizations of lithofacies configuration that are conditional to dynamic data and representative of the model uncertainty space. This problem can be formalized in the Bayesian framework. Given a truncated Gaussian model as a prior and the dynamic data with its associated measurement error, we want to sample from the conditional distribution of the facies given the data. A relevant way to generate conditioned realizations is to use Markov chains Monte Carlo (MCMC). However, the dimensions of the model and the computational cost of each iteration are two important pitfalls for the use of MCMC. Furthermore, classical MCMC algorithms mix slowly, that is, they will not explore the whole support of the posterior in the time of the simulation. In this paper, we extend the methodology already described in a previous work to the problem of history matching of a Gaussian-related lithofacies reservoir model. We first show how to drastically reduce the dimension of the problem by using a truncated Karhunen-Loève expansion of the Gaussian random field underlying the lithofacies model. Moreover, we propose an innovative criterion of the choice of the number of components based on the connexity function. Then, we show how we improve the mixing properties of classical single MCMC, without increasing the global computational cost, by the use of parallel interacting Markov chains. Applying the dimension reduction and this innovative sampling method drastically lowers the number of iterations needed to sample efficiently from the posterior. We show the encouraging results obtained when applying the methodology to a synthetic history-matching case.     

Direct Pattern-Based Simulation of Non-stationary Geostatistical Models

Non-stationary models often capture better spatial variation of real world spatial phenomena than stationary ones. However, the construction of such models can be tedious as it requires modeling both statistical trend and stationary stochastic component. Non-stationary models are an important issue in the recent development of multiple-point geostatistical models. This new modeling paradigm, with its reliance on the training image as the source for spatial statistics or patterns, has had considerable practical appeal. However, the role and construction of the training image in the non-stationary case remains a problematic issue from both a modeling and practical point of view. In this paper, we provide an easy to use, computationally efficient methodology for creating non-stationary multiple-point geostatistical models, for both discrete and continuous variables, based on a distance-based modeling and simulation of patterns. In that regard, the paper builds on pattern-based modeling previously published by the authors, whereby a geostatistical realization is created by laying down patterns as puzzle pieces on the simulation grid, such that the simulated patterns are consistent (in terms of a similarity definition) with any previously simulated ones. In this paper we add the spatial coordinate to the pattern similarity calculation, thereby only borrowing patterns locally from the training image instead of globally. The latter would entail a stationary assumption. Two ways of adding the geographical coordinate are presented, (1) based on a functional that decreases gradually away from the location where the pattern is simulated and (2) based on an automatic segmentation of the training image into stationary regions. Using ample two-dimensional and three-dimensional case studies we study the behavior in terms of spatial and ensemble uncertainty of the generated realizations.     

Bayesian Modelling of Spatial Data Using Markov Random Fields, With Application to Elemental Composition of Forest Soil

Spatial datasets are common in the environmental sciences. In this study we suggest a hierarchical model for a spatial stochastic field. The main focus of this article is to approximate a stochastic field with a Gaussian Markov Random Field (GMRF) to exploit computational advantages of the Markov field, concerning predictions, etc. The variation of the stochastic field is modelled as a linear trend plus microvariation in the form of a GMRF defined on a lattice. To estimate model parameters we adopt a Bayesian perspective, and use Monte Carlo integration with samples from Markov Chain simulations. Our methods does not demand lattice, or near-lattice data, but are developed for a general spatial data-set, leaving the lattice to be specified by the modeller. The model selection problem that comes with the artificial grid is in this article addressed with cross-validation, but we also suggest other alternatives. From the application of the methods to a data set of elemental composition of forest soil, we obtained predictive distributions at arbitrary locations as well as estimates of model parameters.     

Block Kriging for Lognormal Spatial Processes

Lognormal spatial data are common in mining and soil-science applications. Modeling the underlying spatial process as normal on the log scale is sensible; point kriging allows the whole region of interest to be mapped. However, mining and precision agriculture is carried out selectively and is based on block averages of the process on the original scale. Finding spatial predictions of the blocks assuming a lognormal spatial process has a long history in geostatistics. In this article, we make the case that a particular method for block prediction, overlooked in past times of low computing power, deserves to be reconsidered. In fact, for known mean, it is optimal. We also consider the predictor based on the “law” of permanence of lognormality. Mean squared prediction errors of both are derived and compared both theoretically and via simulation; the predictor based on the permanence-of-lognormality assumption is seen to be less efficient. Our methodology is applied to block kriging of phosphorus to guide precision-agriculture treatment of soil on Broom's Barn Farm, UK.     

Application of Dempster-Shafer theory of evidence to GIS-based landslide susceptibility analysis

GIS-based spatial data integration tasks for predictive geological applications, such as landslide susceptibility analysis, have been regarded as one of the primary geological application issues of GIS. An efficient framework for proper representation and integration is required for this kind of application. This paper presents a data integration framework based on the Dempster-Shafer theory of evidence for landslide susceptibility mapping with multiple geospatial data. A data-driven information representation approach based on spatial association between known landslide occurrences and input geospatial data layers is used to assign mass functions. After defining mass functions for multiple geospatial data layers, Dempster’s rule of combination is applied to obtain a series of combined mass functions. Landslide susceptibility mapping using multiple geospatial data sets from Jangheung in Korea was conducted to illustrate the application of this methodology. The results of the case study indicated that the proposed methodology efficiently represented and integrated multiple data sets and showed better prediction capability than that of a traditional logistic regression model.     

A Markov Chain Model for Subsurface Characterization: Theory and Applications

This paper proposes an extension of a single coupled Markov chain model to characterize heterogeneity of geological formations, and to make conditioning on any number of well data possible. The methodology is based on the concept of conditioning a Markov chain on the future states. Because the conditioning is performed in an explicit way, the methodology is efficient in terms of computer time and storage. Applications to synthetic and field data show good results.     

A Fixed-Path Markov Chain Algorithm for Conditional Simulation of Discrete Spatial Variables

The Markov chain random field (MCRF) theory provided the theoretical foundation for a nonlinear Markov chain geostatistics. In a MCRF, the single Markov chain is also called a “spatial Markov chain” (SMC). This paper introduces an efficient fixed-path SMC algorithm for conditional simulation of discrete spatial variables (i.e., multinomial classes) on point samples with incorporation of interclass dependencies. The algorithm considers four nearest known neighbors in orthogonal directions. Transiograms are estimated from samples and are model-fitted to provide parameter input to the simulation algorithm. Results from a simulation example show that this efficient method can effectively capture the spatial patterns of the target variable and fairly generate all classes. Because of the incorporation of interclass dependencies in the simulation algorithm, simulated realizations are relatively imitative of each other in patterns. Large-scale patterns are well produced in realizations. Spatial uncertainty is visualized as occurrence probability maps, and transition zones between classes are demonstrated by maximum occurrence probability maps. Transiogram analysis shows that the algorithm can reproduce the spatial structure of multinomial classes described by transiograms with some ergodic fluctuations. A special characteristic of the method is that when simulation is conditioned on a number of sample points, simulated transiograms have the tendency to follow the experimental ones, which implies that conditioning sample data play a crucial role in determining spatial patterns of multinomial classes. The efficient algorithm may provide a powerful tool for large-scale structure simulation and spatial uncertainty analysis of discrete spatial variables.     

Conditioning Surface-Based Geological Models to Well and Thickness Data

Geostatistical simulation methods aim to represent spatial uncertainty through realizations that reflect a certain geological concept by means of a spatial continuity model. Most common spatial continuity models are either variogram, training image, or Boolean based. In this paper, a more recent spatial model of geological continuity is developed, termed the event, or surface-based model, which is specifically applicable to modeling cases with complex stratigraphy, such as in sedimentary systems. These methods rely on a rule-based stacking of events, which are mathematically represented by two-dimensional thickness variations over the domain, where positive thickness is associated with deposition and negative thickness with erosion. Although it has been demonstrated that the surface-based models accurately represent the geological variation present in complex layered systems, they are more difficult to constrain to hard and soft data as is typically required of practical geostatistical techniques. In this paper, we develop a practical methodology for constraining such models to hard data from wells and thickness data interpreted from geophysics, such as seismic data. Our iterative methodology relies on a decomposition of the parameter optimization problem into smaller, manageable problems that are solved sequentially. We demonstrate this method on a real case study of a turbidite sedimentary basin.     

Prioritization of disaster risk in a community using GIS

Prioritization of disaster risk was carried out for a community in Toronto, Canada. Geographic information systems (GIS) were used for spatial analysis, including spatial overlays and clipping for extracting spatial and attribute information related to people’s vulnerability, critical infrastructure and landuse. In order to determine disaster risk, the overall community vulnerability was evaluated by combining social, economic, physical and environmental vulnerabilities. This paper uses the propane explosion incident as the case in point to demonstrate the methodology and procedure used to evaluate risk using GIS techniques. City of Toronto spatial data have been integrated with the study area to gather landuse information, identify risk zones based on the propane storage facility location and evaluate risks. Statistics Canada 2006 census data have been used for area demographics and people’s social and economic status. Vulnerability indicators were determined based on the GIS-derived spatial and attribute data for the hazard and evacuation zones followed by a quantitative spatial risk estimation and ranking. The methodology of this study, based on the risk evaluation and prioritization conducted, can be applied to future decision making in effective landuse planning and the development of risk management strategies.     

Integration of Markov mesh models and data assimilation techniques in complex reservoirs

We present a methodology that allows conditioning the spatial distribution of geological and petrophysical properties of reservoir model realizations on available production data. The approach is fully consistent with modern concepts depicting natural reservoirs as composite media where the distribution of both lithological units (or facies) and associated attributes are modeled as stochastic processes of space. We represent the uncertain spatial distribution of the facies through a Markov mesh (MM) model, which allows describing complex and detailed facies geometries in a rigorous Bayesian framework. The latter is then embedded within a history matching workflow based on an iterative form of the ensemble Kalman filter (EnKF). We test the proposed methodology by way of a synthetic study characterized by the presence of two distinct facies. We analyze the accuracy and computational efficiency of our algorithm and its ability with respect to the standard EnKF to properly estimate model parameters and assess future reservoir production. We show the feasibility of integrating MM in a data assimilation scheme. Our methodology is conducive to a set of updated model realizations characterized by a realistic spatial distribution of facies and their log permeabilities. Model realizations updated through our proposed algorithm correctly capture the production dynamics.     

Spatial estimation of geotechnical parameters for numerical tunneling simulations and TBM performance models

In this paper, we further elaborate on a methodology dedicated to the modeling of geotechnical data to be used as input in numerical simulation and TBM performance codes. The expression “geotechnical data” refers collectively to the spatial variability and uncertainty exhibited by the boundaries and the mechanical or other parameters of each geological formation filling a prescribed 3D domain. Apart from commercial design and visualization software such as AutoCAD Land Desktop^® software and 3D solid modelling and meshing pre-processors, the new tools that are employed in this methodology include relational databases of soil and rock test data, Kriging estimation and simulation methods, and a fast algorithm for forward or backward analysis of TBM logged data. The latter refers to the continuous upgrade of the soil or rock mass geotechnical model during underground construction based on feedback from excavation machines for a continuous reduction of the uncertainty of predictions in unsampled areas. The approach presented here is non-intrusive since it may be used in conjunction with a commercial or any other available numerical tunneling simulation code. The application of these tools is demonstrated in Mas-Blau section of L9 tunnel in Barcelona.     

Probabilistic analysis of responses of cantilever wall-supported excavations in sands considering vertical spatial variability

The influence of vertical spatial variability of sands on the excavation-induced lateral wall deflection and bending moment of excavations supported by cantilever retaining walls is investigated in this paper. Herein, the random finite element method (RFEM) is adopted to explicitly study the effect of one-dimensional spatial variability of internal friction angle of sands on the predicted wall and ground responses. The RFEM analysis consists of three components: (1) finite element method for analyzing lateral wall deflection and bending moment, (2) random field theory implemented with Monte Carlo simulation (MCS), and (3) statistical interpretation of MCS results through confidence intervals. This study reveals the importance of random field modeling in coping with the spatial variability of sands in the problem of supported excavations: (1) neglecting spatial variability of soil property will cause an overestimation of the variation in the predicted wall deflection and bending moment; (2) the estimated probability of failure based on a well-established serviceability limit state may be overestimated or underestimated depending on the chosen limiting lateral wall deflection. This study further investigates the effect of the number of MCS on the confidence intervals of the predicted statistics of the maximum lateral wall deflection and the maximum bending moment. The results also demonstrate that the confidence interval analysis of the predicted statistics of the maximum lateral wall deflection and the maximum bending moment provides a rational tool for interpreting the statistical data from RFEM.     

Accounting for spatial patterns of multiple geological data sets in geological thematic mapping using GIS-based spatial analysis

This paper presents a Geographic Information System (GIS)-based spatial analysis scheme to account for spatial patterns and association in geological thematic mapping with multiple geological data sets. The multi-buffer zone analysis, the main part of the present study, was addressed to reveal the spatial pattern around geological source primitives and statistical analysis based on a contingency table was performed to extract information for the assessment of an integrated layer. Mineral potential mapping using multiple geological data sets from Ogdong in Korea was carried out to illustrate application of this methodology. The results obtained from the case study indicated that some geochemical elements and residual magnetic anomaly dominantly affected spatial patterns of the mineral potential map in the study area and the dominant classes of input data layers were also extracted. This information on spatial patterns of multiple geological data sets around mines could be used as effective evidences for the interpretation of the integrated layer within GIS.     

Stochastic simulation of time-series models combined with geostatistics to predict water-table scenarios in a Guarani Aquifer System outcrop area, Brazil

Stochastic methods based on time-series modeling combined with geostatistics can be useful tools to describe the variability of water-table levels in time and space and to account for uncertainty. Monitoring water-level networks can give information about the dynamic of the aquifer domain in both dimensions. Time-series modeling is an elegant way to treat monitoring data without the complexity of physical mechanistic models. Time-series model predictions can be interpolated spatially, with the spatial differences in water-table dynamics determined by the spatial variation in the system properties and the temporal variation driven by the dynamics of the inputs into the system. An integration of stochastic methods is presented, based on time-series modeling and geostatistics as a framework to predict water levels for decision making in groundwater management and land-use planning. The methodology is applied in a case study in a Guarani Aquifer System (GAS) outcrop area located in the southeastern part of Brazil. Communication of results in a clear and understandable form, via simulated scenarios, is discussed as an alternative, when translating scientific knowledge into applications of stochastic hydrogeology in large aquifers with limited monitoring network coverage like the GAS.     

Reducing geological uncertainty by conditioning on boreholes: the coupled Markov chain approach

The CMC (coupled Markov chain) model, which is based on the extension of Markov chains in two-dimensions, is used in the reduction of uncertainty in geological structures when conditioned (i.e., honours the data and their location) on a number of boreholes. The model has been applied to an unconsolidated aquifer deposit located in the central Rhine-Meuse delta (the Gorkum study area) in the Netherlands. A comparison is also made between the CMC and the SIS (sequential indicator simulation) model, which is based on Kriging and co-Kriging theories on the same deposit. The results show the potential applicability of the CMC model in reducing the uncertainty in geological configurations when a sufficient number of boreholes is available. Reproduction of the global geological features requires relatively few boreholes (in this case study, nine boreholes with 30-m spacing over a distance of 240 m). However, reproduction of the proportion of each state requires a relatively large number of boreholes (in this case study 31 boreholes with 8-m spacing over a distance of 240 m). It has been shown that variograms can be deceptive in modeling the spatial pattern and that they reflect only part of the complete spatial structure in the field. The use of transition probabilities via the CMC model provides a better alternative approach, because it uses multiple point information. Amro M. M. Elfeki on leave from Department of Irrigation and Hydraulics, Faculty of Engineering, Mansoura University, Mansoura, Egypt     

Characterising geotechnical model uncertainty by hybrid Markov Chain Monte Carlo simulation

Geotechnical models are usually associated with considerable amounts of model uncertainty. In this study, the model uncertainty of a geotechnical model is characterised through a systematic comparison between model predictions and past performance data. During such a comparison, model input parameters (such as soil properties) may also be uncertain, and the observed performance may be subjected to measurement errors. To consider these uncertainties, the model uncertainty parameters, uncertain model input parameters and actual performance variables are modelled as random variables, and their distributions are updated simultaneously using Bayes’ theorem. When the number of variables to update is large, solving the Bayesian updating problem is computationally challenging. A hybrid Markov Chain Monte Carlo simulation is employed in this paper to decompose the high-dimensional Bayesian updating problem into a series of updating problems in lower dimensions. To increase the efficiency of the Markov chain, the model uncertainty is first characterised with a first order second moment method approximately, and the knowledge learned from the approximate solution is then used to design key parameters in the Markov chain. Two examples are used to illustrate the proposed methodology for model uncertainty characterisation, with insights, discussions, and comparison with previous methods.