Automatic Identification of Rock Fracture Sets Using Finite Mixture Models

The clustering and classification of fracture orientation data are crucial tasks in geotechnical engineering and rock engineering design. The explicit simulation of fracture orientations is always applied to compensate for the lack of direct measurements over the entire rock mass. In this study, a single step approach based on the theory of finite mixture models, where the component distributions are Fisher distributions, is proposed for automatic clustering and simulation of fracture orientation data. In the proposed workflow, the spherical K-means algorithm is applied to select the initial cluster centers, and the component-wise expectation–maximization algorithm using the minimum message length criterion is used to automatically determine the optimal number of fracture sets. An additional advantage of the proposed method is the representation of orientation data using a full sphere, instead of the conventional hemispherical characterization. The use of a full spherical representation effectively solves the issue of clustering for fractures with high dip angles. In addition, the calculation process of the mean direction is also simplified. The effectiveness of the model-based clustering method is tested with a complicated artificial data set and two real world data sets. Cluster validity is introduced to evaluate the clustering results. In addition, two other clustering algorithms are also presented for comparison. The results demonstrate that the proposed method can successfully detect the optimal number of clusters, and the parameters of the distributions are well estimated. In addition, the proposed method also exhibits good computational performance.     

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.     

Utilization of self-organizing map and fuzzy clustering for site characterization using piezocone data

In this paper soft computing techniques, self-organizing maps and fuzzy clustering techniques have been proposed to isolate different layers in stratified soil based on available cone penetration test results. The results have been compared with that obtained from cone classification chart, hierarchical and K-mean clustering techniques. It was observed that variation in result with self-organizing map (SOM) and fuzzy clustering for isolating soil layers is marginal. These techniques are found to be efficient compared to hierarchical clustering technique. The results of K-mean clustering show that the identified soil strata are similar to that obtained from cone classification chart, SOM and fuzzy clustering.     

An example of cluster analysis applied to a large geologic data set: Aerial radiometric data from Copper Mountain, Wyoming

One objective of the aerial radiometric surveys flown as part of the U.S. Department of Energy's National Uranium Resource Evaluation (NURE) program was to ascertain the spatial distribution of near-surface radioelement abundances on a regional scale. Some method for identifying groups of observations with similar -ray spectral signatures and radioelement concentration values was therefore required. It is shown in this paper that cluster analysis can identify such groups with or without a priori knowledge of the geology of an area. An approach that combines principal components analysis with convergentk-means cluster analysis is used to classify 6991 observations (each observation comprising three radiometric variables) from the Precambrian rocks of the Copper Mountain, Wyoming area. This method is compared with a convergentk-means analysis that utilizes available geologic knowledge. Both methods identify four clusters. Three of the clusters represent background values for the Precambrian rocks of the area, and the fourth represents outliers (anomalously high²¹⁴Bi). A segmentation of the data corresponding to geologic reality as interpreted by other methods has been achieved by perceptive quantitative analysis of aerial radiometric data. The techniques employed are composites of classical clustering methods designed to handle the special problems presented by large data sets.     

A Methodology for the Definition of Geotechnical Mine Sectors Based on Multivariate Cluster Analysis

This paper offers a new method for the definition of geotechnical sectors in open pit mines based on multivariate cluster analysis. A geological-geotechnical data set of a manganese open pit mine was used to demonstrate the methodology. The data set consists of a survey of geological and geotechnical parameters of the rock mass, measured directly in several points of the mine, structured initially in twenty-eight variables. After the preprocessing of the data set, the clustering technique was applied using the k-Prototype algorithm. The squared Euclidean distance was used to quantify the proximity between numerical variables, and the Jaccard's coefficient of similarity was used to quantify the proximity between the nominal variables. The different cluster results obtained were validated by the multivariate analysis of variance. The identification of cluster structures was achieved by plotting them on the mine map for spatial visualization and definition of geotechnical sectors. These sectors are spatially contiguous and relatively homogeneous regarding their geological–geotechnical properties, indicated by a high density of points of the same group. It was possible to observe a great adherence of the proposed sectors to the mine geology, demonstrating the practical representativeness of the clustering results and the proposed sectors.

     

Evaluation of graphical and multivariate statistical methods for classification of water chemistry data

A robust classification scheme for partitioning water chemistry samples into homogeneous groups is an important tool for the characterization of hydrologic systems. In this paper we test the performance of the many available graphical and statistical methodologies used to classify water samples including: Collins bar diagram, pie diagram, Stiff pattern diagram, Schoeller plot, Piper diagram, Q-mode hierarchical cluster analysis, K-means clustering, principal components analysis, and fuzzy k-means clustering. All the methods are discussed and compared as to their ability to cluster, ease of use, and ease of interpretation. In addition, several issues related to data preparation, database editing, data-gap filling, data screening, and data quality assurance are discussed and a database construction methodology is presented. The use of graphical techniques proved to have limitations compared with the multivariate methods for large data sets. Principal components analysis is useful for data reduction and to assess the continuity/overlap of clusters or clustering/similarities in the data. The most efficient grouping was achieved by statistical clustering techniques. However, these techniques do not provide information on the chemistry of the statistical groups. The combination of graphical and statistical techniques provides a consistent and objective means to classify large numbers of samples while retaining the ease of classic graphical presentations. Electronic Publication     

Magnetic studies and elemental analysis of river sediments: a case study from the Ponnaiyar River (Southeastern India)

The Ponnaiyar River is one of the largest rivers of the Tamil Nadu state (India), flowing a distance of 430 km from its point of origin to the sea. This work contributes with new data of magnetic and elemental composition of river sediments, and improves the knowledge obtained by preliminary and previous studies of rivers from Southeastern India. Magnetic susceptibility, anhysteretic and isothermal remanent magnetization and chemical determinations (major and trace metals) were measured. Magnetic results reveal the predominance of magnetite-like mineral with magnetic grain size variations along the river and in depth. Most of the uppermost samples have the major presence of trace metals and higher values of magnetic concentration. Magnetic and chemical variables were also analysed as potential pollution indicators using multivariate statistical techniques: canonical correlation and fuzzy c-means clustering analyses, which confirmed the existence of relationships, but not in a simple way, between magnetic and chemical variables. Furthermore, fuzzy analysis allows classifying the data in different well-differentiated groups regarding the trace metal load, concentration and feature-dependent parameters. The most polluted samples show high concentration of trace elements and magnetic carriers, softer and coarser magnetic minerals; on the contrary, the unpolluted samples (from the deepest sediments) have the opposite characteristics.     

The application of fuzzy c-means cluster analysis and non-linear mapping to geochemical datasets: examples from Portugal

In the interpretation of relatively small multivariate datasets, deviations from homogeneity may cause severe problems. In these cases fuzzy c-means cluster analysis (FCM) and non-linear mapping (NLM) are conceptionally suited to discern structure in the datasets. Particularly, the combined use of FCM and NLM furnishes a powerful method to find meaningful data groupings within a dataset. This is illustrated with two case studies, for water and combined water and stream sediment analyses, respectively, where FCM and NLM were applied. The results are easily related to geology, mineral occurrences and environmental factors.     

Application of limited fuzzy clusters to anomaly recognition in complex geological environments

In stream sediment and soil surveys, samples represent mixtures of components from different geological environments. Such mixed samples are misclassified when using conventional “hard” cluster methods. In fuzzy clustering, each sample is allowed to belong to several clusters. Similar to element concentrations, these cluster contributions can be displayed in contour maps (e.g. kriging maps). The amount of an element that is explained by the cluster contribution and element residuals can be calculated. The modified fuzzy clustering algorithm called “limited fuzzy clusters” used in this paper avoids negative residuals.Stream sediment data of Sierra de San Carlos, Tamaulipas, Mexico are used to demonstrate the possibilities of limited fuzzy clustering in geochemical exploration and mapping. From the different drainage systems, 681 stream sediment samples were taken and analyzed for 24 elements. A nineteen-element data set was used to calculate limited fuzzy clusters and element residuals. The contribution values for the clusters and element residuals are displayed in contour maps. All geological units were outlined by the cluster contributions. Extended anomalies are characterized by their own cluster. Small anomalies are clearly identified from the element residuals.     

Stratigraphic delineation by three-dimensional clustering of piezocone data

Piezocone soundings are a fast and economical approach for geotechnical site characterization, providing three separate and continuous channels of data with depth, including: tip resistance q _T, porewater pressure u ₂ and sleeve friction f _s. Literally hundreds to thousands of data points are collected by a single sounding. Since these readings are functions of both soil type and soil behaviour, they can be used for the delineation of soil stratigraphy.

One way to process large amounts of data involves clustering. Cluster analysis is an efficient statistical way to analyse the stratigraphic vertical profiling of geomaterials and means to detect the inherent similarity between data sets and group them together. Clustering in previous geotechnical research was based on only two channels of piezocone data (q _T and u₂). The method works well for soils that are under the groundwater table and was applied to soundings in clay deposits.

In the present paper, a new cluster analysis approach is developed based on all three channels of data, thus extending the method to soils above the water table and applicable to sands, silts, and clays. Example soil profiles derived by three-channel cluster analysis are presented herein and compared with conventional soil boring and sampling data.     

Prediction of the Rock Mass Diggability Index by Using Fuzzy Clustering-Based,ANN and Multiple Regression Methods

Rock mass classification systems are one of the most common ways of determining rock mass excavatability and related equipment assessment. However, the strength and weak points of such rating-based classifications have always been questionable. Such classification systems assign quantifiable values to predefined classified geotechnical parameters of rock mass. This causes particular ambiguities, leading to the misuse of such classifications in practical applications. Recently, intelligence system approaches such as artificial neural networks (ANNs) and neuro-fuzzy methods, along with multiple regression models, have been used successfully to overcome such uncertainties. The purpose of the present study is the construction of several models by using an adaptive neuro-fuzzy inference system (ANFIS) method with two data clustering approaches, including fuzzy c-means (FCM) clustering and subtractive clustering, an ANN and non-linear multiple regression to estimate the basic rock mass diggability index. A set of data from several case studies was used to obtain the real rock mass diggability index and compared to the predicted values by the constructed models. In conclusion, it was observed that ANFIS based on the FCM model shows higher accuracy and correlation with actual data compared to that of the ANN and multiple regression. As a result, one can use the assimilation of ANNs with fuzzy clustering-based models to construct such rigorous predictor tools.     

A statistical approach to evaluating distance metrics and analog assignments for pollen records

The modern analog technique typically uses a distance metric to determine the dissimilarity between fossil and modern biological assemblages. Despite this quantitative approach, interpretation of distance metrics is usually qualitative and rules for selection of analogs tend to be ad hoc. We present a statistical tool, the receiver operating characteristic (ROC) curve, which provides a framework for identifying analogs from distance metrics. If modern assemblages are placed into groups (e.g., biomes), this method can (1) evaluate the ability of different distance metrics to distinguish among groups, (2) objectively identify thresholds of the distance metric for determining analogs, and (3) compute a likelihood ratio and a Bayesian probability that a modern group is an analog for an unknown (fossil) assemblage. Applied to a set of 1689 modern pollen assemblages from eastern North America classified into eight biomes, ROC analysis confirmed that the squared-chord distance (SCD) outperforms most other distance metrics. The optimal threshold increased when more dissimilar biomes were compared. The probability of an analog vs no-analog result (a likelihood ratio) increased sharply when SCD decreased below the optimal threshold, indicating a nonlinear relationship between SCD and the probability of analog. Probabilities of analog computed for a postglacial pollen record at Tannersville Bog (Pennsylvania, USA) identified transitions between biomes and periods of no analog.     

Wind erosion hazard assessment of the Mongolian Plateau using FCM and GIS techniques

The most serious environmental problems of the Mongolian Plateau are land degradation and sand storms caused by wind erosion, but the evaluation of wind erosion at regional scales has been a difficult process in wind erosion research. In this study, fuzzy c-means clustering (FCM) was used to assess the spatial pattern of wind erosion hazard on the Mongolian Plateau. By fuzzy clustering four main wind erosion factors (vegetation cover, average degree of land surface relief, degree of soil dryness and intensity of wind energy), wind erosion hazard was classified into six grades. Results show that FCM can effectively integrate related information between wind erosion and environmental factors, which provides the basis for predictive mapping of wind erosion hazard. Spatial patterns of wind erosion hazard indicate a gradual trend of increasing hazard in the Mongolian Plateau from east to west. Similar patterns were also found in NDVI and soil dryness, indicating that soil moisture and vegetation are the most important factors in the formation of wind erosion hazard. In addition, the distribution of different levels of wind erosion hazard is basically consistent with the regional distribution of landscape vegetation types in the Mongolian Plateau.     

Susceptibility mapping of landslides in Beichuan County using cluster and MLC methods

Cluster analysis and maximum likelihood classification (MLC) are exploited to map the post-earthquake landslide susceptibility in Beichuan County that was affected by the Ms 8.0 Wenchuan earthquake. The methodology is applicable even if there is short of training data. Six effective factors are chosen for mapping the susceptibility, including land use, seismic intensity, average annual rainfall, relative relief, slop gradient and lithology. Four clusters are grouped from sampling grid cells by k-means clustering approach. MLC classifies all the cells in the study area into the four clusters according to their statistical characteristics. Four susceptibility classes (extreme low, low, moderate and high) are assigned to these clusters applying expert experience and hazard density. The final map gives a reasonable assessment of post-earthquake landslide susceptibility in Beichuan County. Comparing with the pre-earthquake susceptibility map made in Beichuan County geological disaster survey project, the result t using cluster and MLC classification has a better agreement with the dot density value of post-earthquake landslides in Beichuan County. The susceptibility map can be used to identify safety spots within the high danger area, which are suitable for habitations and facilities. It is also found that more landslides are densely concentrated at the boundary between high and moderate regions, and between high and extreme low regions.     

地质分类中一种新的模糊聚类方法

没有分类,就没有鉴别,也就无法认识客观事物.因此,分类是各个领域中经常遇到的基本问题之一.以定量分类为特征的聚类分析法,和一般的定性分析方法相比,失误的可能性小得多.因此,在分类问题上,该方法已经起了很大作用.但是,和其它任何方法一样,聚类分析法也有局限性.1.聚类分析法是建立在普通集合论基础之上的.任一集合A的特征函数C_Λ(X)只能用二值变量来描述.即:     

基于混合聚类的岩体结构面优势分组方法

提出一种基于凝聚层次法和模糊C均值法的混合聚类法,用于对岩体结构面的优势组划分。该方法将结构面投放到在单位球面上,并使用欧式距离作为极点的相似性度量准则。先剔除结构面数据中的孤值产状,然后用凝聚层次法得到初步聚类结果,并将其作为FCM法的初始聚类中心,最后用FCM法划分优势组。通对人工生成产状样本的分组,验证了该法的正确性。将该方法应用于大藤峡坝址区实测的结构面数据的划分。在实测数据中寻找到两个孤值产状,成功将大藤峡D1y^1-3地层岩体结构面划分为两组,得到了符合实际的分组结果。     

Ore Grade Prediction Using a Genetic Algorithm and Clustering Based Ensemble Neural Network Model

Accurate prediction of ore grade is essential for many basic mine operations, including mine planning and design, pit optimization, and ore grade control. Preference is given to the neural network over other interpolation techniques for ore grade estimation because of its ability to learn any linear or non-linear relationship between inputs and outputs. In many cases, ensembles of neural networks have been shown, both theoretically and empirically, to outperform a single network. The performance of an ensemble model largely depends on the accuracy and diversity of member networks. In this study, techniques of a genetic algorithm (GA) and k-means clustering are used for the ensemble neural network modeling of a lead–zinc deposit. Two types of ensemble neural network modeling are investigated, a resampling-based neural ensemble and a parameter-based neural ensemble. The k-means clustering is used for selecting diversified ensemble members. The GA is used for improving accuracy by calculating ensemble weights. Results are compared with average ensemble, weighted ensemble, best individual networks, and ordinary kriging models. It is observed that the developed method works fairly well for predicting zinc grades, but shows no significant improvement in predicting lead grades. It is also observed that, while a resampling-based neural ensemble model performs better than the parameter-based neural ensemble model for predicting lead grades, the parameter-based ensemble model performs better for predicting zinc grades.     

Numerical evaluation of multicomponent cation exchange reactive transport in physically and geochemically heterogeneous porous media

Experimental evidence and stochastic studies strongly show that the transport of reactive solutes in porous media is significantly influenced by heterogeneities in hydraulic conductivity, porosity, and sorption parameters. In this paper, we present Monte Carlo numerical simulations of multicomponent reactive transport involving competitive cation exchange reactions in a two-dimensional vertical physically and geochemically heterogeneous medium. Log hydraulic conductivity, log K, and log cation exchange capacity (log CEC) are assumed to be random Gaussian functions with spherical semivariograms. Random realizations of log K and log CEC are used as input data for the numerical simulation of multicomponent reactive transport with CORE^2D, a general purpose reactive transport code. Longitudinal features of the fronts of reactive and conservative species are computed from the temporal and spatial moments of depth-averaged concentrations. Monte Carlo simulations show that: (1) the displacement of reactive fronts increases with increasing variance of log K, while it decreases with the variance of log CEC; (2) second-order spatial moments increase with increasing variances of log K and log CEC; (3) uncertainties in the mean arrival time are largest (smallest) for negatively (positively) correlated log K and Log CEC; (4) cations undergoing competitive cation exchange exhibit different apparent velocities and retardation factors due to both physical and geochemical heterogeneities; and (5) the correlation between log K and log CEC affects significantly apparent cation retardation factors in heterogeneous aquifers.     

Proof of multiplicity of solutions for groundwater seepage in recharged heterogeneous unconfined aquifers

A new analytical proof is presented for steady‐state seepage in recharged heterogeneous unconfined aquifers. The paper also presents a detailed procedure and important rules for performing correctly numerical studies of unsaturated seepage. Once a numerical solution is calibrated with field data, using a set of spatially distributed values for hydraulic conductivity K and effective infiltration EI, any new numerical analysis with a set of αK and αEI values, where α is a constant, yields an equally good calibration. However, if the effective porosities of each layer are unchanged, the groundwater velocities are multiplied by α, whereas the travel times are divided by α, which may help to select α in order to match known travel time data. This is a clear example of multiple solutions to an inverse problem. The paper underlines the role and the need to finely mesh unsaturated zones and also contacts between layers to reach the asymptotic convergence range, as it was carried out to verify the proof and as it should be completed to study any seepage problem. A few consequences of the new analytical proof and the rigorous procedure are shown with examples. Copyright © 2016 John Wiley & Sons, Ltd.     

节理岩体结构面优势产状确定方法研究

节理岩体结构面产状的分析在岩体的力学和水力学分析中都是极为重要的基础性工作.本文分析了传统的结构面产状图形分析法、模糊等价聚类方法和模糊C均值聚类方法(FCM算法)在节理结构面产状分析中的优缺点,针对上述3种方法各自单独使用时的利弊将这3种方法有效结合起来,得出了一种更为准确合理的产状统计分析的综合性方法.应用此方法对...