Evaluating Thickness of Bauxite Deposit Using Indicator Geostatistics and Fuzzy Estimation

Evaluating the geological properties of a mineral deposit is a fundamental task for mine planning and it requires an assessment of reserve parameters such as thickness and grade. This paper presents a linguistic model for estimating bauxite thickness within a deposit in a fuzzy environment using indicator geostatistics and fuzzy modeling. The proposed model consists of two main stages: determining the orebody boundary and estimating the thickness. In order to estimate the thickness, a rule‐based fuzzy inference mechanism has been developed based on data statistics. Results and performance of the model have been compared with that of a well‐known conventional technique, geostatistics (kriging), and it is shown that the proposed model has bigger estimation power. In addition, the fuzzy approach is more flexible than the kriging approach. The fuzzy methodology used in the present paper is convenient for modeling reserve parameters.     

Probability of serviceability failure in a braced excavation in a spatially random field: Fuzzy finite element approach

A simplified framework is proposed for evaluating the probability of “serviceability failure” in a braced excavation in a spatially random field. Here, the “serviceability failure” is said to occur when the excavation-induced wall or ground movement exceeds specified limiting values. Knowledge of this probability can aid in engineering decision-making to prevent damage to adjacent infrastructures. The proposed framework consists of five elements: (1) finite element method (FEM) for analyzing wall and ground responses in a braced excavation, (2) fuzzy set modeling of parameter uncertainty, (3) spatial averaging technique for handling spatial variability, (4) vertex method for processing fuzzy input through FEM model, and (5) interpretation of fuzzy output. The proposed framework is demonstrated through a well-documented case history. The results show the proposed framework is simple and effective for assessing the probability of serviceability failure in a braced excavation in a spatially random field. To focus on the proposed fuzzy FEM approach, the scope of this paper is limited to one-dimensional modeling of spatial variability with an assumed exponential autocorrelation function.     

A Multiple Training Image Approach for Spatial Modeling of Geologic Domains

Characterization of complex geological features and patterns remains one of the most challenging tasks in geostatistics. Multiple point statistics (MPS) simulation offers an alternative to accomplish this aim by going beyond classical two-point statistics. Reproduction of features in the final realizations is achieved by borrowing high-order spatial statistics from a training image. Most MPS algorithms use one training image at a time chosen by the geomodeler. This paper proposes the use of multiple training images simultaneously for spatial modeling through a scheme of data integration for conditional probabilities known as a linear opinion pool. The training images (TIs) are based on the available information and not on conceptual geological models; one image comes from modeling the categories by a deterministic approach and another comes from the application of conventional sequential indicator simulation. The first is too continuous and the second too random. The mixing of TIs requires weights for each of them. A methodology for calibrating the weights based on the available drillholes is proposed. A measure of multipoint entropy along the drillholes is matched by the combination of the two TIs. The proposed methodology reproduces geologic features from both TIs with the correct amount of continuity and variability. There is no need for a conceptual training image from another modeling technique; the data-driven TIs permit a robust inference of spatial structure from reasonably spaced drillhole data.     

Characteristics of agro-meteorological disasters and their risk in Gansu Province against the background of climate change

A formal Bayesian approach that uses the Markov chain Monte Carlo method to estimate the uncertainties of natural hazards has attracted significant attention in recent years, and a fuzzy graph can be considered an estimation of the relationship that we want to know in risk systems. However, the challenge with such approaches is to sufficiently consider uncertainty without much prior knowledge and adequate measurement. This paper proposes a new adaptive Bayesian framework that is based on the conventional Bayesian scheme and the optimal information diffusion model to more precisely calculate the conditional probabilities in the fuzzy graph for recognizing relationships and estimating uncertainty in natural disasters with scant data. This methodology is applied to study the relationship between the earthquake’s magnitude and the isoseismal area with strong-motion earthquake observations. It is also compared with other techniques, including classic Bayesian regression and artificial neural networks. The results show that the new method achieves better performance than do the main existing methods with incomplete data.     

Modelling Short‐Scale Variability and Uncertainty During Mineral Resource Estimation Using a Novel Fuzzy Estimation Technique

For mineral resource assessment, techniques based on fuzzy logic are attractive because they are capable of incorporating uncertainty associated with measured variables and can also quantify the uncertainty of the estimated grade, tonnage etc. The fuzzy grade estimation model is independent of the distribution of data, avoiding assumptions and constraints made during advanced geostatistical simulation, e.g., the turning bands method. Initially, fuzzy modelling classifies the data using all the component variables in the data set. We adopt a novel approach by taking into account the spatial irregularity of mineralisation patterns using the Gustafson–Kessel classification algorithm. The uncertainty at the point of estimation was derived through antecedent memberships in the input space (i.e., spatial coordinates) and transformed onto the output space (i.e., grades) through consequent membership at the point of estimation. Rather than probabilistic confidence intervals, this uncertainty was expressed in terms of fuzzy memberships, which indicated the occurrence of mixtures of different mineralogical phases at the point of estimation. Data from different sources (other than grades) could also be utilised during estimation. Application of the proposed technique on a real data set gave results that were comparable to those obtained from a turning bands simulation.     

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.     

Imprecise (fuzzy) information in geostatistics

A methodology based on fuzzy set theory for the utilization of imprecise data in geostatistics is presented. A common problem preventing a broader use of geostatistics has been the insufficient amount of accurate measurement data. In certain cases, additional but uncertain (soft) information is available and can be encoded as subjective probabilities, and then the soft kriging method can be applied (Journel, 1986). In other cases, a fuzzy encoding of soft information may be more realistic and simplify the numerical calculations. Imprecise (fuzzy) spatial information on the possible variogram is integrated into a single variogram which is used in a fuzzy kriging procedure. The overall uncertainty of prediction is represented by the estimation variance and the calculated membership function for each kriged point. The methodology is applied to the permeability prediction of a soil liner for hazardous waste containment. The available number of hard measurement data (20) was not enough for a classical geostatistical analysis. An additional 20 soft data made it possible to prepare kriged contour maps using the fuzzy geostatistical procedure.This paper was presented at MGUS 87 Conference, Redwood City, California, 14 April 1987.     

Fuzzy modeling of holding capacity of offshore suction caisson anchors

A fuzzy algorithm, the Takagi–Sugeno model, is implemented to develop a fuzzy inference system for predicting the holding capacity of suction caisson foundations for offshore platforms. The premise parameters of the fuzzy model are optimized by using a subtractive clustering algorithm. The consequent parameters are optimally determined via a weighted least square estimation. The input variables used for training the fuzzy model include the aspect ratio of the caisson, the undrained shear strength of the clay, and the angle that the chain force forms with the horizontal. The output of the proposed fuzzy model is the capacity of the suction caisson anchor. To demonstrate the effectiveness of the fuzzy modeling framework, the results of extensive finite element analyses are investigated. Comparisons of the trained fuzzy model with the data demonstrate that the proposed modeling framework is an effective method to estimate the holding capacity of offshore suction caisson systems. Moreover, the performance of the fuzzy model is robust against higher levels of input data uncertainties. Copyright © 2017 John Wiley & Sons, Ltd.     

中国降水混沌识别及空间聚类

0-1混沌测试方法是一种新的可直接作用于时间序列的混沌识别方法,不需要进行相空间重构,通过对Chebyshev映射的检验验证了有效性。运用0-1方法对中国640个气象站1960-2011年月降水序列进行混沌识别,并运用基于密度的顾及非空间属性的空间聚类方法（DBSC）对计算结果进行空间聚类。结果表明,各气象站月降水序列均表现出明显的混沌特性,且K值的空间分布具有明显的分区特征：从大尺度上看,全国可分为西北高值区、南方次高区、华北-东北中值区和青藏低值区;从小尺度上看,全国分出了29个空间簇。中国降水混沌空间聚类结果不仅与大尺度的气候类型分布相吻合,而且也反映了局部降水动力系统变化特征,这就为降水系统的时空规律研究提供了一条新的途径和方法。     

3D矿床建模技术在数字矿产勘查中的应用

为弥补现有数字矿床建模技术在地质矿产勘查处理和应用中的不足,从原始勘查数据建库及标准化、多指标单工程矿体自动圈定、基于语义识别的剖面矿体的连接与外推、矿体表面和品位建模、基于剖面矿体线框模型构建矿体表面模型及基于TIN+Octree数据结构和地质统计学理论建立矿体的空间属性模型等5个方面总结提出一套面向地质矿产勘查业务处理的矿床建模流程和总体技术解决方案.提高了地质矿产勘查研究精度,为进一步的矿山开采提供可靠的数据模型.      

An interval fuzzy model for magnetic monitoring: estimation of a pollution index

In this contribution, a methodology is reported in order to build an interval fuzzy model for the pollution index PLI (a composite index using relevant heavy metal concentration) with magnetic parameters as input variables. In general, modelling based on fuzzy set theory is designed to mimic how the human brain tends to classify imprecise information or data. The “interval fuzzy model” reported here, based on fuzzy logic and arithmetic of fuzzy numbers, calculates an “estimation interval” and seems to be an adequate mathematical tool for this nonlinear problem. For this model, fuzzy c-means clustering is used to partition data, hence the membership functions and rules are built. In addition, interval arithmetic is used to obtain the fuzzy intervals. The studied sets are different examples of pollution by different anthropogenic sources, in two different study areas: (a) soil samples collected in Antarctica and (b) road-deposited sediments collected in Argentina. The datasets comprise magnetic and chemical variables, and for both cases, relevant variables were selected: magnetic concentration-dependent variables, magnetic features-dependent variables and one chemical variable. The model output gives an estimation interval; its width depends on the data density, for the measured values. The results show not only satisfactory agreement between the estimation interval and data, but also provide valued information from the rules analysis that allows understanding the magnetic behaviour of the studied variables under different conditions.     

A Bayesian/maximum-entropy view to the spatial estimation problem

The purpose of this paper is to stress the importance of a Bayesian/maximum-entropy view toward the spatial estimation problem. According to this view, the estimation equations emerge through a process that balances two requirements: High prior information about the spatial variability and high posterior probability about the estimated map. The first requirement uses a variety of sources of prior information and involves the maximization of an entropy function. The second requirement leads to the maximization of a so-called Bayes function. Certain fundamental results and attractive features of the proposed approach in the context of the random field theory are discussed, and a systematic spatial estimation scheme is presented. The latter satisfies a variety of useful properties beyond those implied by the traditional stochastic estimation methods.     

Teacher''s Aide Variogram Interpretation and Modeling

The variogram is a critical input to geostatistical studies: (1) it is a tool to investigate and quantify the spatial variability of the phenomenon under study, and (2) most geostatistical estimation or simulation algorithms require an analytical variogram model, which they will reproduce with statistical fluctuations. In the construction of numerical models, the variogram reflects some of our understanding of the geometry and continuity of the variable, and can have a very important impact on predictions from such numerical models. The principles of variogram modeling are developed and illustrated with a number of practical examples. A three-dimensional interpretation of the variogram is necessary to fully describe geologic continuity. Directional continuity must be described simultaneously to be consistent with principles of geological deposition and for a legitimate measure of spatial variability for geostatistical modeling algorithms. Interpretation principles are discussed in detail. Variograms are modeled with particular functions for reasons of mathematical consistency. Used correctly, such variogram models account for the experimental data, geological interpretation, and analogue information. The steps in this essential data integration exercise are described in detail through the introduction of a rigorous methodology.     

Teacher's Aide Variogram Interpretation and Modeling

The variogram is a critical input to geostatistical studies: (1) it is a tool to investigate and quantify the spatial variability of the phenomenon under study, and (2) most geostatistical estimation or simulation algorithms require an analytical variogram model, which they will reproduce with statistical fluctuations. In the construction of numerical models, the variogram reflects some of our understanding of the geometry and continuity of the variable, and can have a very important impact on predictions from such numerical models. The principles of variogram modeling are developed and illustrated with a number of practical examples. A three-dimensional interpretation of the variogram is necessary to fully describe geologic continuity. Directional continuity must be described simultaneously to be consistent with principles of geological deposition and for a legitimate measure of spatial variability for geostatistical modeling algorithms. Interpretation principles are discussed in detail. Variograms are modeled with particular functions for reasons of mathematical consistency. Used correctly, such variogram models account for the experimental data, geological interpretation, and analogue information. The steps in this essential data integration exercise are described in detail through the introduction of a rigorous methodology.     

Three-dimensional slope reliability and risk assessment using auxiliary random finite element method

This paper aims to propose an auxiliary random finite element method (ARFEM) for efficient three-dimensional (3-D) slope reliability analysis and risk assessment considering spatial variability of soil properties. The ARFEM mainly consists of two steps: (1) preliminary analysis using a relatively coarse finite-element model and Subset Simulation, and (2) target analysis using a detailed finite-element model and response conditioning method. The 3-D spatial variability of soil properties is explicitly modeled using the expansion optimal linear estimation approach. A 3-D soil slope example is presented to demonstrate the validity of ARFEM. Finally, a sensitivity study is carried out to explore the effect of horizontal spatial variability. The results indicate that the proposed ARFEM not only provides reasonably accurate estimates of slope failure probability and risk, but also significantly reduces the computational effort at small probability levels. 3-D slope probabilistic analysis (including both 3-D slope stability analysis and 3-D spatial variability modeling) can reflect slope failure mechanism more realistically in terms of the shape, location and length of slip surface. Horizontal spatial variability can significantly influence the failure mode, reliability and risk of 3-D slopes, especially for long slopes with relatively strong horizontal spatial variability. These effects can be properly incorporated into 3-D slope reliability analysis and risk assessment using ARFEM.     

Conditioning generative adversarial networks on nonlinear data for subsurface flow model calibration and uncertainty quantification

Conditioning complex subsurface flow models on nonlinear data is complicated by the need to preserve the expected geological connectivity patterns to maintain solution plausibility. Generative adversarial networks (GANs) have recently been proposed as a promising approach for low-dimensional representation of complex high-dimensional images. The method has also been adopted for low-rank parameterization of complex geologic models to facilitate uncertainty quantification workflows. A difficulty in adopting these methods for subsurface flow modeling is the complexity associated with nonlinear flow data conditioning. While conditional GAN (CGAN) can condition simulated images on labels, application to subsurface problems requires efficient conditioning workflows for nonlinear data, which is far more complex. We present two approaches for generating flow-conditioned models with complex spatial patterns using GAN. The first method is through conditional GAN, whereby a production response label is used as an auxiliary input during the training stage of GAN. The production label is derived from clustering of the flow responses of the prior model realizations (i.e., training data). The underlying assumption of this approach is that GAN can learn the association between the spatial features corresponding to the production responses within each cluster. An alternative method is to use a subset of samples from the training data that are within a certain distance from the observed flow responses and use them as training data within GAN to generate new model realizations. In this case, GAN is not required to learn the nonlinear relation between production responses and spatial patterns. Instead, it is tasked to learn the patterns in the selected realizations that provide a close match to the observed data. The conditional low-dimensional parameterization for complex geologic models with diverse spatial features (i.e., when multiple geologic scenarios are plausible) performed by GAN allows for exploring the spatial variability in the conditional realizations, which can be critical for decision-making. We present and discuss the important properties of GAN for data conditioning using several examples with increasing complexity.

     

Applicability of Statistical Learning Algorithms for Spatial Variability of Rock Depth

Two algorithms are outlined, each of which has interesting features for modeling of spatial variability of rock depth. In this paper, reduced level of rock at Bangalore, India, is arrived from the 652 boreholes data in the area covering 220 sq⋅km. Support vector machine (SVM) and relevance vector machine (RVM) have been utilized to predict the reduced level of rock in the subsurface of Bangalore and to study the spatial variability of the rock depth. The support vector machine (SVM) that is firmly based on the theory of statistical learning theory uses regression technique by introducing ε-insensitive loss function has been adopted. RVM is a probabilistic model similar to the widespread SVM, but where the training takes place in a Bayesian framework. Prediction results show the ability of learning machine to build accurate models for spatial variability of rock depth with strong predictive capabilities. The paper also highlights the capability of RVM over the SVM model.     

Settlement prediction by spatial-temporal random process using Asaoka's method

A methodology was presented for observation-based settlement prediction with consideration of the spatial correlation structure of soil. The spatial correlation is introduced among the settlement model parameters and the settlements at various points are spatially correlated through these geotechnical parameters, which naturally describe the phenomenon. The method is based on Bayesian estimation by considering both prior information, including spatial correlation and observed settlement, to search for the best estimates of the parameters at any arbitrary points on the ground. Within the Bayesian framework, the optimised selection of auto-correlation distance by Akaike's Bayesian Information Criterion (ABIC) is also proposed. The application of the proposed approach in consolidation settlement prediction using Asaoka's method is presented in this paper. Several case studies were carried out using simulated settlement data to investigate the performance the proposed approach. It is concluded that the accuracy of the settlement prediction can be improved by taking into account the spatial correlation structure and the proposed approach gives the rational prediction of the settlement at any location at any time with quantified uncertainty.     

Weighted fuzzy kernel-clustering algorithm with adaptive differential evolution and its application on flood classification

Flood classification is the fundamental problem of flood risk analysis and plays an important role in flood disaster risk management. Considering the fact that flood classification is a problem of multi-attribute and multi-stage fuzzy synthetically evaluation, this paper mainly proposed the weighted fuzzy kernel-clustering algorithm (WFKCA) with adaptive differential evolution algorithm (ADE) to solve this problem. Firstly, WFKCA is detailed introduced, and then the differential evolution algorithm (DE) is applied for the fuzzy clustering, thus to obtain the better results. Taking into consideration the disadvantage of DE, ADE is present after the introduction of DE. Finally, the combination of WFKCA and ADE is applied for flood classification, and the results demonstrated the methodology is reasonable and reliable, thus provide a new effective approach for flood classification.     

A proposed new approach for groundwater resources assessment in India

Assessment of groundwater resources in India is guided by National Water Policy (1987, 2002) which states that groundwater resources can be exploited only up to its recharge limit. The methodology for groundwater resources assessment in India is broadly based on Ground Water Resources Estimation Methodology, 1997 and it involves assessment of annual replenishable groundwater resources (recharge), annual groundwater draft (utilization) and the percentage of utilization with respect to recharge (stage of development). The assessment units (blocks/watersheds) are categorized based on stage of groundwater development (utilization) and the long term water level trend. The present methodology though useful in identification and prioritization of areas for groundwater management, falls short of addressing several critical issues like spatial and temporal variation of groundwater availability within the aquifer, accessibility of groundwater resources and quality of groundwater. This paper introduces a new categorisation scheme considering the above issues. The proposed scheme takes into account four criteria, viz. (i) stage of exploitation, (ii) extractability factor, (iii) temporal availability factor and (iv) quality factor. In comparison to the existing method used for categorisation, the proposed approach is more inclusive. The methodology is also equally suitable for both alluvial and hard rock terrain since it takes into consideration the variable characteristics of different types of aquifers and convergence of quantitative and qualitative assessment. The categorisation proposed here involves GIS based integration of different parameters/ themes. This allows better representation of spatial variability. The proposed methodology is demonstrated in this paper taking a case study from a hard rock terrain in central India.