Modern Approaches Approaches in Solid Earth Sciences
Marat Abzalov
Applied Mining Geology
32
3
of the mine faces. If safety and logistic permits it is a good good prac practi tice ce to mark mark and and surv survey ey the control points which will be used for verification the surface model generated by photogrammetric technology. The control points should be chosen in such such mann manner er that that ensu ensure ress thei theirr loca locati tion on as clos closee as possible to the centre of the overlapping overlapping pair of stereo images. Camera positions and physical layout of the data acquisition process need to be determined and ensured that it is feasible for a given camera and lens configuration. As a rule of thumb
Separation between cameras Distance from cameras to mine face
the ratio should be within the range from 1/6 to 1/8. Camera should be used with tripod equipped with spirit level for ensuring the horizontal and non-tilted position. Prior Prior to taking taking photos photos the follo followin wing g measur measureements need to be obtained: • The 3D coordina coordinates tes (absol (absolute ute or relativ relative) e) of the camera at the time each image is acquired need to be accurately determined; • Thre Threee orie orient ntat atio ions ns of the the came camera ra need need to be meas measur ured ed each each time time when when imag imagee was acquired: (a) the azimuth of the optical axis of the camera. Stereo photographs should be taken with as little convergence as possible, ideally with the photos of all images made by cameras cameras aligned aligned in parallel. parallel. Conver Convergence gence in ı the order of 10 is likely to result in large error
Mine Mapping
in 3D map caused by poor matching of the stereo pairs; (b) the dip of the axial plan of the camera camera (zero-de (zero-degree gree elevat elevation ion is horizonta horizontal, l, ninety degrees elevation is vertically up); (c) the tilt of the camera. The tilt of the camera should be zero if possible; • The focal focal length length of the camera camera lens usual usually ly determined during lens calibration; • From camera camera specific specification ationss it is necessary necessary to to obtain the image format including number of rows and columns of pixels and the pixel size. The obtained data are processed using commercially available software, which conventionally used for construction digital surfaces. File structure and processing procedures vary depending on chosen software however their basic principles and main steps are the same: firstly, the pair of stereo images needs to be matched and then built-in algorithm will calculate coordinates of each pixel using above mentioned geometric principles (Fig. 3.15 (Fig. 3.15). ). As an outcome of this process each pixel will be assigned a 3D coordinates and then the surface can be presented as wireframe frame (triangula (triangulated ted mesh) mesh) model model (Fig. (Fig. 3.17a). 3.17a). Infilling the wireframe with the pixels recorded by digital photography produces the textured 3D photographic image of the surface closely reproducing the actual look of that face at the time when photo was made (Fig. 3.17b (Fig. 3.17b)).
Fig. 3.17 3D photographic photographic image of the pit wall obtained obtained using SIROVISIO SIROVISION N technology technology (Courtesy (Courtesy of Datamine). Datamine). Bench height 10 m: (a) wireframe; (b) wireframe model infilled by pixels of the photographic image
Mod Modern ern Appr pproach oaches es in Soli Solid d Earth arth Sciences Volume 12 Series Editors Yildirim Dilek, Department of Geology and Environmental Earth Sciences, Miami University, Oxford, OH, U.S.A Franco Pirajno, Geological Survey of Western Australia, and The University of Western Australia, Perth, Australia Brian Windley, Department of Geology, The University of Leicester, UK
More information about this series at http://www.springer.com/series/7377
Marat Abzalov
Applied Mining Geology
1 3
Marat Abzalov MASSA geoservices Mount Claremont WA, Australia Centre for Exploration Targeting (CET) University of Western Australia Crawley, WA, Australia
Responsible Series Editor : F. Pirajno
Additional material to this book can be downloaded from http://extras.springer.com. ISSN 1876-1682 ISSN 1876-1690 (electronic) Modern Approaches in Solid Earth Sciences ISBN ISBN 978978-33-31 3199-39 3926 2633-9 9 ISBN ISBN 978978-33-31 3199-39 3926 2644-6 6 (eBo (eBook ok)) DOI 10.1007/978-3-319-39264-6 Library of Congress Control Number: 2016943190 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland
To my family, Svetlana, Aygul and Shamil, for their help, patience and love
Abou Ab outt th the e Aut utho horr
Dr Abza Abzalo lov v is a geol geolog ogis istt with with 35 year yearss of expe experi rien ence ce.. He obta obtain ined ed a PhD in geology studying nickel deposits in Russia and Fennoscandia and undertook additional postgraduate studies in applied mathematics at Murdoch University, Australia, and geostatistics in Fontainebleau, France. In his long and diverse diverse geological geological career career,, he has fulfilled fulfilled differen differentt roles roles in research, research, exploration and mining geology, including senior management positions at WMC Resources, Rio Tinto and BOSS Resources. With diverse commodity and geographic experience, with the projects encomp encompass assing ing five five contin continent entss and differ different ent deposi depositt types types shown shown on the map, Dr Abzalov has demonstrated skills in greenfields, brownfields and mathematical geological modelling. Using an innovative approach of geostatistically assisted 3D structural modelling, he has led WMC Resources to the successful resource growth at Olympic Dam and Cliffs deposits. He was also instrumental in the discovery of the uranium resources in Jordan. Dr Abzalo Abzalov v is an adjunc adjunctt geosci geoscient entist ist at the Centr Centree for Explor Explorati ation on Targeting, the University of Western Australia, and is successfully sharing his practical work in the mining industry with academic research. In 2015, he was awarded the Dani Krige’s Gold Medal by SAIMM for his novel geostatistical method LUC (localised uniform conditioning).
vii
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World map showing location of the geological projects studied by Dr M. Abzalov
About the Author
Acknowledgements
I express my sincere gratitude to the copyright holders of the originals of many figures and tables reproduced in this book for the permission to use them. In particular, I would like to acknowledge the following organisations: The Australasian Institute of Mining and Metallurgy (AusIMM) for permission to reprint in the book the figures and tables earlier published by the book’s author in the AusIMM Monograph 23 (2014) and the proceedings of the AusIMM conferences, including ‘Ore body knowledge and strategic mine planning’ (2004) and ‘Heavy minerals conference’ (2011). The Applied Earth Science (AES) journal and the publisher Taylor and Francis (www.tandfonline.com) for permission to reprint in the book, a material earlier published by the book’s author in the journal’s several issues, particularly AES (2014) 123/2, AES (2013) 122/1 and AES (2010) 119/3. The Canadian Institute of Mining, Metallurgy and Petroleum (CIM) for permission to reprint some materials published by the book’s author in the issues of Exploration and Mining Geology journal (2009) 18/1–4 and (2008) 17/3–4. The Mathematical Geology ( MG) journal and the publisher Springer for permission to reproduce the description of the LUC technique and reprint the diagrams, published by the book’s author in MG (2006) 38/4. The Society for Mining Metallurgy and Exploration (SME) for permission to reprint the diagrams published in the monograph Underground Mining Methods: Engineering Fundamentals and International Case Studies . The book has benefited from numerous discussions with my friends and colleagues in the mining industry, including geologists and mining engineers of WMC Resources, Rio Tinto, BOSS Resources, BHP Billiton, Newcrest, Harmony, Vale, Jordanian Uranium Mining Company and Anglo Gold Ashanti and also geoscientists from the different universities and research institutes. I am also grateful to R. Minnitt, R. Reid, S. Masters and anonymous reviewers of Springer for their critical reading of the manuscript and many useful comments. I am in particular indebted to my family, Svetlana, Aygul and Shamil, for their enormous support through all my career.
ix
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Part I
1 2
Mine Design, Mine Mapping and Sampling
2
Mining Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Open Pit Mines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Underground Mines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Underground Selective Mining Methods . . . . . . 2.2.2 Underground Bulk Mining Methods . . . . . . . . . . 2.2.3 Mining of the Gently Dipping Ore Bodies. . . . . 2.3 Unconventional Mining. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 In situ Leach (ISL) Technique . . . . . . . . . . . . . . . 2.3.2 Dredging of the Mineral Sands . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 6 7 9 10 14 15 16 16 18
3
Mine Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Mine Mapping Principles. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Mapping Open Pit Mines . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Mapping of Underground Mines . . . . . . . . . . . . . . . . . . . . . 3.4 Mapping Using Digital Photogrammetry and Laser Technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Mapping Mining Faces Using Photogrammetry 3.4.2 Remote Mapping of the Mines Using Laser . . . 3.5 Optimisation of the Mine Mapping Procedures . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19 19 20 23
4
Drilling Techniques and Drill Holes Logging . . . . . . . . . . . . . . . 4.1 Drilling Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Diamond Core Drilling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Core Quality and Representativeness . . . . . . . . . 4.2.2 Orientated Core. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Logging Diamond Core Holes . . . . . . . . . . . . . . 4.2.4 Sampling Diamond Core. . . . . . . . . . . . . . . . . . . 4.3 Open Hole Percussion Drilling . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Sampling Blastholes for Grade Control Purpose in the Open Pits . . . . . . . . . . . . . . . . . . .
30 30 33 34 37 39 39 41 45 49 54 58 59 60
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4.3.2
Use of ‘Jumbo’ Drilling for Delineation of Underground Stopes . . . . . . . . . . . . . . . . . . . . 4.4 Reverse Circulation (RC) Percussion Drilling . . . . . . . . . . 4.4.1 Logging RC Holes . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Sampling RC Holes . . . . . . . . . . . . . . . . . . . . . . . 4.5 Sonic Drilling Technologies . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Strength and Weakness of the Sonic Drilling . . 4.5.2 Logging and Sampling Sonic Drill Holes . . . . . 4.6 Auger Drilling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Rotary Drilling Using Tricone Bit . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64 65 67 69 69 71 73 74 76 76
5
Sampling of the Mine Workings . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Sampling Rock Faces in the Underground Mines . . . . . . . 5.1.1 Channel Sampling. . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Rock Chip Sampling. . . . . . . . . . . . . . . . . . . . . . 5.2 Sampling of the Broken Ore . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Trenching and Winzing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79 79 80 80 82 84 85
6
Geotechnical Logging and Mapping . . . . . . . . . . . . . . . . . . . . . . . 6.1 Geotechnical Logging of the Drill Core . . . . . . . . . . . . . . . 6.1.1 Drilling Parameters and Core Recovery. . . . . . . 6.1.2 Rock Weathering . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Rock Strength. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4 Rock Quality Designation Index (RQD) . . . . . . 6.1.5 Natural Breaks. . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Geotechnical Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Geotechnical Applications of Rock Mass Classification Schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87 87 88 88 89 89 90 91 92 95
7
Dry Bulk Density (DBD) of Rocks . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Types of the Rock Densities Used in the Mining Industry 7.2 Dry Bulk Density Measurement Techniques . . . . . . . . . . . 7.2.1 Competent Non-porous Rocks . . . . . . . . . . . . . . 7.2.2 Porous and Weathered Rocks . . . . . . . . . . . . . . . 7.2.3 Non-consolidated Sediments. . . . . . . . . . . . . . . . 7.3 Spatial Distribution of the Rock Density Measurements. . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97 98 98 98 100 104 104 110
8
Data Points Location (Surveying) . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Surface Points Location. . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Down-Hole Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
111 112 112 115
Part II 9
Sampling Errors
Introduction to the Theory of Sampling . . . . . . . . . . . . . . . . . . . . 119 9.1 Types of Sampling Errors. . . . . . . . . . . . . . . . . . . . . . . . . . . 119
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9.2
Fundamental Sampling Error . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Theoretical Background. . . . . . . . . . . . . . . . . . . . 9.2.2 Experimental Calibration of the Sampling Constants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Sampling Nomogram. . . . . . . . . . . . . . . . . . . . . . 9.3 Grouping – Segregation Error . . . . . . . . . . . . . . . . . . . . . . . 9.4 Errors Related to the Sampling Practices . . . . . . . . . . . . . . 9.5 Instrumental Errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
11
Quality Control and Assurance (QAQC) . . . . . . . . . . . . . . . . . . . 10.1 Accuracy Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Statistical Tests for Assessing Performance of the Standard Samples . . . . . . . . . . . . . . . . . . . 10.1.2 Statistical Tests for Assessing the Data Bias Using the Duplicate Samples . . . . . . . . . . . . . . . 10.1.3 Diagnostic Diagram: Pattern Recognition Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Precision Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Matching Pairs of Data . . . . . . . . . . . . . . . . . . . . 10.2.2 Processing and Interpretation of Duplicate Samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Comparative Analysis of the Statistical Estimation Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Guidelines for Optimisation of the Sampling Programmes 10.4.1 Planning and Implementation of the Sampling Programmes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Frequency of Inserting QAQC Material to Assay Batches . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.3 Distribution of the Reference Materials. . . . . . . 10.4.4 Distribution of the Duplicate Samples . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Twin Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Method Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Objectives of the Twinned Holes Study. . . . . . . 11.1.2 Statistical Treatment of the Results . . . . . . . . . . 11.1.3 Distance Between Twinned Holes . . . . . . . . . . . 11.1.4 Drilling Quality and Quantity . . . . . . . . . . . . . . . 11.1.5 Comparison of Studied Variables . . . . . . . . . . . . 11.1.6 Practice of Drilling Twinned Holes for Mining Geology Applications . . . . . . . . . . . . . . . . . . . . . 11.2 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Gold Deposits: Confirmation of High-Grade Intersections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Twin Holes Studies in Iron Ore Deposits . . . . . . 11.2.3 Mineral Sands Deposits: Validation of Historic Drilling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
121 121 123 128 129 131 132 133 135 135 136 140 140 142 142 143 150 154 154 155 156 156 158 161 162 162 163 163 163 165 166 167 168 169 171
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Contents
11.2.4
Bauxites: Use of Twin Holes as a Routine Control of Drilling Quality . . . . . . . . . . . . . . . . . 171 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 12
Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Construction of the Database . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Data Entry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Electronic Data Transfer. . . . . . . . . . . . . . . . . . . 12.2.2 Keyboard Data Entry . . . . . . . . . . . . . . . . . . . . . . 12.2.3 Special Values. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Management of the Data Flow . . . . . . . . . . . . . . . . . . . . . . . 12.4 Database Safety and Security . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Part III
177 178 180 180 180 181 182 183 183
Mineral Resources
13
Data Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Data Compositing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.1 Data Coding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.2 Compositing Algorithms . . . . . . . . . . . . . . . . . . . 13.1.3 Choice of the Optimal Compositing Intervals . . 13.1.4 Validating of the Composited Assays . . . . . . . . . 13.2 High Grade Cut-Off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
187 187 187 188 188 190 191 192
14
Geological Constraints of Mineralisation . . . . . . . . . . . . . . . . . . . 14.1 Introduction to Wireframing . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Characterisation of the Mineralisation Contacts . . . . . . . . 14.2.1 Contact Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2 Determining of the Cut-Off Value for Constraining Mineralisation . . . . . . . . . . . . . . . . 14.2.3 Contact Topography. . . . . . . . . . . . . . . . . . . . . . . 14.2.4 Uncertainty of the Contacts . . . . . . . . . . . . . . . . . 14.3 Geometry and Internal Structure of the Mineralised Domains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 14.3.1 Unfolding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
193 193 195 195
202 202 205
Exploratory Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Objective of the EDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Overview of the EDA Techniques . . . . . . . . . . . . . . . . . . . . 15.2.1 Spider Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.2 Data Declustering. . . . . . . . . . . . . . . . . . . . . . . . . 15.2.3 Q-Q Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.4 Box-and-Whisker Plot (Box Plot) . . . . . . . . . . . . 15.3 Grouping and Analysis of the Data . . . . . . . . . . . . . . . . . . . 15.3.1 Data Types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.2 Data Generations . . . . . . . . . . . . . . . . . . . . . . . . .
207 207 208 208 208 213 213 214 214 216
15
198 199 200
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15.3.3
Grouping Samples by Geological Characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 15.4 Statistical Analysis of the Resource Domains . . . . . . . . . . 217 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 16
Resource Estimation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Polygonal Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Estimation by Triangulation. . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Cross-Sectional Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.1 Extrapolation of the Cross-Sections. . . . . . . . . . 16.3.2 Interpolation Between Cross-Sections . . . . . . . . 16.4 Estimation by Panels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5 Inverse Distance Weighting Method . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Part IV
221 222 223 224 224 226 228 228 230
Applied Mining Geostatistics
17
Introduction to Geostatistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 17.1 Regionalised Variable and Random Function . . . . . . . . . . . 234 17.2 Stationarity and Intrinsic Hypothesis . . . . . . . . . . . . . . . . . 235 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
18
Variography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1 Quantitative Analysis of the Spatial Continuity . . . . . . . . . 18.2 Intuitive Look at Variogram . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Geostatistical Definition of Variogram . . . . . . . . . . . . . . . . 18.4 Directional, Omnidirectional and Average Variograms . . . 18.5 Properties of the Variograms. . . . . . . . . . . . . . . . . . . . . . . . 18.5.1 Behaviour Near Origin. . . . . . . . . . . . . . . . . . . . . 18.5.2 Anisotropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.6 Analysis of the Data Continuity Using a Variogram Map . 18.7 Presence of Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.8 Proportional Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.9 Variogram Sill and the Sample Variance . . . . . . . . . . . . . . . 18.10 Impact of the Different Support . . . . . . . . . . . . . . . . . . . . . . 18.11 Variogram Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.11.1 Common Variogram Models . . . . . . . . . . . . . . . . 18.11.2 Modelling Geometric Anisotropy . . . . . . . . . . . . 18.11.3 Nested Structures . . . . . . . . . . . . . . . . . . . . . . . . . 18.11.4 Modelling Zonal Anisotropy . . . . . . . . . . . . . . . . 18.12 Troublesome Variograms . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.12.1 Hole Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.12.2 Saw-Tooth Shaped and Erratic Variograms . . . . 18.13 Alternative Measures of a Spatial Continuity . . . . . . . . . . . 18.13.1 Variograms of the Gaussian Transformed Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.13.2 Relative (Normalised) Variograms . . . . . . . . . . . 18.13.3 Different Structural Tools . . . . . . . . . . . . . . . . . .
239 239 240 241 242 242 243 244 245 247 247 248 249 249 249 251 252 252 253 254 254 255 256 257 258
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Contents
18.14 Indicator Variograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.15 Variograms in the Multivariate Environment . . . . . . . . . . . 18.15.1 Multivariate Geostatistical Functions. . . . . . . . . 18.15.2 Linear Model of Coregionalisation . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
259 259 260 260 261
