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By A. Marek

Reduced downtime and maximized availability of mining machines are the key factors in optimizing production rates. As technology moves industry towards an age of ?intelligent machines? it sets the foundation for change in mining equipment design, functionality, and maintenance. Recent mining machine designs are integrating more and more electronic components into the core functionality of individual piece of equipment due to regulatory and legislative requirements, such as TIER rating, as well as from a safety aspect such as personal detection systems. This paper describes the use of existing on-board electronic equipment to increase machine availability through automated maintenance planning and diagnostic fault-finding capability. The automated maintenance planning takes place on the machine itself in order to be independent of the mine?s infrastructure, while the diagnostic capabilities allow for reduced downtime. The paper also presents a concept to transfer operational data to surface to be able to optimize planning of maintenance and to reduce downtime. Case studies are presented where such systems have been commissioned and are currently in operation, the possibilities of this technology application for the future are examined.

Jan 1, 2012

By M. Klapwijk

Ancient Metallurgy - In addition to the preceding three papers, the following three contributions were received for publication 1. An analysis of Bantu-made iron implements from the Letaba district M. KLAPWIJK INTRODUCTION It has often been stated that it is hardly possible in South Africa to classify Bantumade iron hoes according to their places of origin and manufacture. It is well known that these hoes were used for barter, and as currency in the purchase of livestock or the payment of lobolo, although reports of the value of the hoes differ widely. The difficulty in their classification has arisen from their distribution over a very wide area.

Jan 1, 1974

By C. Botha

The South African democratically-elected government instituted policies and legislation that are intended to bring about change and transformation in the country and have drastically affected different industries, among others, the mining industry. The Mineral and Petroleum Resources Development Act 28 of 2002 (MPRDA) and the Broad-based Socio-economic Empowerment Charter (Mining Charter) were introduced in order to redress previous inequalities and to promote women?s employment in the mining industry. This research evaluates the practical implementation of the MPRDA and the Mining Charter with specific reference to women in underground mining. A quantitative research paradigm was used. Data were collected through a process of a face-to-face interview surveys utilizing a structured questionnaire. The research setting was limited to the platinum-mining industry in the area of Rustenburg in the North West Province. The study population consisted of 195 available women working undergound at the mine. It is clear from the findings that mining companies are facing major challenges in their attempt to incorporate women in underground mining activities. Company policies regarding these issues must be firmly in place; however, the challenge lies in the translation of these policies into practice. In this regard, managers will have to play a leading role in the successful phasing in of women into the underground mining arena.

Jan 1, 2012

By B. Steyn

AngloGold Ashanti?s Obuasi Mine, in Ashanti Province, Ghana, operates a plant utilizing bacterial oxidation technology. The mineralization consists predominately of arsenopyrite with gold associated in the sulphides. The plant was commissioned in 1994, at that time was the largest BIOX® plant in the world, and has been operated by AngloGold Ashanti since 2004. The constraints identified in the plantwere the flotation capacity, BIOX® feed size and disruptions to the BIOX® process. This paper focuses on the practical steps that were followed in order to increase the recovery of the plant. The original plant design was based on a head grade of 10 g/t and an overall recovery of 86.5%. In 2007 the plant was operating at an overall recovery of 79.2% at a head grade of 5.6 g/t and this has improved to 80.6% at a head grade of 5.8 g/t during the last quarter of 2008. Additional improvement to 81.5% are expected by the end of April 2009. Improvements that have been implemented include: ?Replacement of the regrind mill ?Installation of additional float capacity ?Installation of a scavenging circuit ?Installation of a complete water treatment plant. In addition we have focused on the effect of the water circuit on the BIOX® process and the steps involved in correcting this problem. The water circuit was originally designed as an open circuit and through environmental considerations the circuit was closed towards the end of 2007. The bacteria in the BIOX® are particularly sensitive to cyanide and cannot tolerate levels above 0.05ppm (free cyanide), this required extensive modifications to the water circuit. The plant improvement project is ongoing and the current scope of work should be complete by the end of April 2009. Further research and development work is being planned, which includes a ?white rot bacteria? that can consume the preg-robbing carbon and oxidize the sulphide in a single pass.

Jan 1, 2009

By J. Autio, H. Malmlund

The excavation damage zone (EDZ) caused by drill and blast excavation consists of intense damage adjacent to the remaining half barrel of the blast hole, new fracturing induced by the blast and reactivation of pre-existing fractures. The properties of intense microfracturing adjacent to the half barrels of remaining blast holes are based on study of rock samples from ZEDEX-tunnel and it’s extension at Äspö and the Research tunnel at Olkiluoto in Finland. The new and reactivated fracturing is based on results of mapping of slots cut in the rock in ZEDEX-project. The zone of increased porosity adjacent to half barrels, which penetrates to a depth of some centimeters was studied by using novel 14C-PMMA method. The penetration of the radial fractures, which is an order of magnitude deeper, was studied by mapping using color penetrant. Results of EDZ studies carried out at Äspö HRL in Sweden and in Research Tunnel at Olkiluoto show that the properties of EDZ can be affected significantly by engineering design. The result imply that the additional pore volume caused by distinct fracturing around the blast holes caused by state-of-the-art smooth blasting technique could be of the same order as that of the damaged zone adjacent to the surface of the half barrels. Durch Bohr- und Sprengarbeiten entstandene Eigenschaften der Auflockerungszone: Die durch Bohr- und Sprengarbeiten entstandene Auflockerungszone besteht aus einem großen, an das Resthalbrohr des Bohrloches angrenzenden Schaden, neuer Rissbildung und einer Reaktivierung der bereits existierenden Risse. Die Eigenschaften einer immensen, an die Halbrohre der Restbohrlöcher angrenzenden Mikrorissbildung beruhen auf einer Studie über die aus dem ZEDEX-Tunnel gewonnenen Gesteinsproben und deren Fortsetzung im Untertage-Labor Äspö (Hard Rock Laboratory, HRL) in Schweden und im Forschungstunnel bei Olkiluoto in Finnland. Die neue und reaktivierte Rissbildung basiert auf den Auswertungsergebnissen der Schlitze, die bei dem ZEDEX-Projekt ins Gestein geschnitten wurden. Die Zone der vermehrten, an die Halbrohre angrenzenden Porösität, die in ein paar Zentimeter Tiefe eindringt, wurde anhand der neuen 14C-PMMA-Methode untersucht. Das Eindringen der etwas tiefer gelegenen radialen Risse wurde anhand eines farbigen Eindringungsmittels untersucht. Die in Äspö und Olkiluoto durchgeführten EDZ-Studien zeigen, dass die EDZ-Eigenschaften wesentlich durch Engineering Design beeinflusst werden können. Das durch Bohrarbeiten verursachte Hohlraumvolumen der Rissbildung um den Tunnel herum ist ähnlich wie das Hohlraumvolumen der Bruchzone um die Bohrlöcher herum. Caractéristiques d’une zone endommagée par le creusement résultant d’un creusement par forage et dynamitage: La zone endommagée par le creusement résultant d’un creusement par forage et dynamitage présente un endommagement intense adjacent au demi-cylindre restant du trou de mine, une nouvelle fracturation et la réactivation des fractures préexistantes. Les caractéristiques de la microfracturation intense adjacente aux demi-cylindres des trous de mine restants reposent sur l’étude des échantillons de roche en provenance du tunnel ZEDEX et de son extension au laboratoire souterrain d’Äspö (« Hard Rock Laboratory » ou HRL) en Suède et du tunnel de recherche d’Olkiluoto en Finlande. La fracturation nouvelle et réactivée repose sur les résultats du placement des rainures sur la roche dans le projet ZEDEX. La zone de porosité accrue adjacente aux demi-cylindres et pénétrant à une profondeur de quelques centimètres à fait l’objet d’études recourant à la nouvelle méthode 14C-PMMA. La pénétration des fractures radiales, considérablement plus profondes, a été étudiée par placement en utilisant un agent pénétrant coloré. Les résultats des études EDZ effectuées à Äspö et à Olkiluoto montrent que les caractéristiques de l’EDZ sont susceptibles d’être affectées de manière significative par la conception technique. Le volume vide de fracturation résultant du dynamitage autour du tunnel est du même ordre que celui de la zone concassée autour des trous de mine.service.

Jan 1, 2003

By N. Morris

In South Africa, diamonds are mined predominantly in rural areas. Kimberlitic diamonds dominate in Limpopo and the Northern Cape Provinces whereas alluvial diamonds are mainly sourced from the North West and Northern Cape Provinces. There are a few large and medium sized diamond companies (De Beers, Petra Diamonds, Trans Hex, Rockwell and Alexkor). There are many smaller companies involved in diamond mining, mostly alluvial diamond mines. The nature of alluvial diamond deposits is that they are aerially more extensive and thus their surface impacts are more pervasive: The recent downturn in diamond prices has resulted in the sudden cessation of mining at many small and medium diamond operations. The South African Mining Charter requires all mines, including diamond mines, to subscribe' to .the following empowerment objectives:

Jan 1, 2010

By P. Lundman, L. Rosengren, T. Brandshaug

Tunnel 99’, the Swedish National Road Administration´s (SNRA) general technical specification for road tunnels requires that the load bearing system must be designed to withstand specified design explosion loads. This paper presents the results of an SNRA study to provide a perspective of design analyses tailored to comply with the new requirement. The paper provides an example of a design analysis for assumed tunnel conditions using a numerical model. Dynamic predictions of potential damage to the rock mass and ground support system (rock bolts and shotcrete) from the specified explosion loads are presented including the nonlinear response of the shotcrete. “Tunnel 99”, qui fixe les règles de l’Administration nationale suédoise des Routes (Vägverket) concernant la conception et la construction des tunnels routiers, requiert entre autres que la structure porteuse soit capable de résister à certaines contraintes spécifiques résultant d’explosions. Cette étude présente les résultats d’une étude effectuée par Vägverket en vue d’obtenir une perspective d’analyses de conception spécialement élaborées pour se conformer aux nouvelles règles. Il contient un exemple d’analyse en fonction de conditions déterminées, basée sur un modèle numérique. Des prévisions de’études dynamiques de dommages potentiels subis par la masse rocheuse et les systèmes de soutènement (boulons d’ancrage et béton projeté) du fait des contraintes libérées par la force d’explosion spécifiée sont également présentées, y compris dans le cas d’une réaction non linéaire du revêtement en béton. “Tunnel 99” - die Vorschriften für Strassentunnel der Nationalen Schwedischen Behörde für Straßenwesen - (Vägverket) - fordern vom Tragsystem eine Auslegung in der Form, dass es spezifischen Explosionsbelastungen standhält. Dieser Artikel präsentiert die Ergebnisse einer Studie der Behörde, um einen Ausblick über konstruktive Auslegungen zu geben, die geeignet sind, die neuen Vorschriften zu erfüllen. Der Artikel zeigt ein Beispiel einer konstruktive Auslegung für eine angenommene Tunnelsituation mit Hilfe eines numerischen Modells. Voraussagen unter Beachtung der vollständigen dynamischen Berechnung zu potentiellen Schädigungen des Gebirges und des Ausbaus (Anker und Spritzbetonauskleidung) durch die spezifizierten Explosionsbelastungen, einschließlich des nichtlinearen Verhaltens der Spritzbetonauskleidung, werden gezeigt.

Jan 1, 2003

By H. W. Read

Sustained growth in demand for platinum group metals has resulted in an unfettered increase in new projects undertaken by the platinum producers. These projects are outsourced to various degrees. The argument of this paper is that both the client?s project manager and the consultant?s project manager are essential elements in the successful execution of a project. Their roles are not only different but also complementary, and are defined in the paper. Data gained from considerable experience in the South African corporate mining environment underpins the opinions expressed in the paper. Furthermore, an extensive literary search was undertaken to determine trends in related industries outside the South African context. Throughout the argument it is emphasized that project philosophy emanates from a systems view and must be treated as an overarching whole. The elements of project management become meaningless unless they are connected to one another and integrated into a cohesive unit. Advancing the knowledge field of project management is of critical importance to mining investors. There is no doubt that a key determinant of the sustainability of mining business is closely related to the efficiency of all-inclusive project management. From pre-investment and feasibility studies, implementation and execution, to ramping-up to full production, scientific project management has a critical role to play. Guidelines for the pursuance of this ideal are proposed. The introduction contextualizes the subject matter in an overview of the evolution of mine management in South Africa and abroad. Then, brief insight is given into the philosophy of project management. This exposition is confined to the predispositions, presuppositions and defining characteristics of project management. The core of the paper looks at the roles of the client?s project manager in relation to the consultant?s project manager. This section is underpinned by practical experience gained by the author on numerous mining projects. A view is taken of the future of project management in the mining industry. The conclusion summarizes the argument and aligns project management trends in the mining industry.

Jan 1, 2004

Professor G.T. van Rooyen has guided the Department of Materials Science and Metallurgical Engineering at the University of Pretoria in South Africa for the past 32 years. During this period, 54 masters' degrees have been awarded in the Department, and 17 doctorandi promoted. He has also made a major contribution to industry by his incisive analysis, relevant and practical synthesis, and his exceptional talent in innovative design. He is still active in fracture analysis, thermal fatigue, weld simulation transformation kinetics, and fracture mechanics, and continues to contribute to a better understanding of metallurgical problems encountered in practice. He has advised many companies, and has received many honours and recognitions.

Jan 1, 1994

By Y. Mitani, M. Zhang, M. Takeda, T. Esaki

Despite their inadequacy, assumptions of isotropy and homogeneity are often applied to the analyses of in situ permeability tests. In this paper, we exam the effects of anisotropy and heterogeneity on Pressure Pulse Test (PPT) and Constant Head Injection Test (CHIT) by means of numerical approaches. Some new findings that offer practical considerations in properly interpreting in-situ permeability tests are presented. Malgré leur manque de précision inhérente, les assomptions d'anisotropie et d'hétérogénéité sont souvent utilisées lors de test de perméabilité in situ. Dans cette étude, nous examinons les effets de l'anisotropie et de l'hétérogénéité sur le test par pression pulsée et le test à injection à charge hydraulique constante par le biais de la simulation numérique. Des nouveaux résultats permettant une interprétation pratique et claire des test de perméabilité in-situ sont présentés. Trotz ihrer inhärenten Ungeeignetheit werden häufig Annahmen über Isotropie und Homogenität auf Analysen von in-situ Permeabilitätstests angewendet. In diesem Papier untersuchen wir die Effekte von Anisotropie und Heterogenität auf Durck-Impuls-Test (PPT) und Konstant-Kopf-Einspritztest (CHIT) mit numerischen Ansätzen. Einige neue Befunde, die zu praktischen Erwägungen zur richtigen Interpretation von in-situ Permeabilitätstests führen, werden vorgestellt.

Jan 1, 2003

By M. J. U. T. Van Wijngaarden, R. J. Dippenaar, P. M. Van Den Heever

Some of the important properties of solid electrolytes that influence the performance of electrochemical oxygen probes are discussed in the first part of this paper. It is shown that accurate and reproducible measurements of oxygen in liquid steel are possible with commercial oxygen probes only if great care is exercised in the make-up of the chemical and phase composition, as well as the microstructure of the solid electrolytes used in these probes. To prove this premise, the second part of this study reports on the results of an experimental evaluation of new commercially available oxygen probe. An evaluation of the solid electrolytes used in this probe indicated that the consistency of the chemical composition, degree of stabilization, and microstructure of the electrolytes used are well controlled. The successful application of this particular oxygen probe under industrial conditions was confirmed. It was also found possible to make reliable and reproducible measurements, with acceptable uncertainty, by proper handling of the probe. Some recommendations are made as to the interpretation of the e.m.f. response curve and the use of the electronic processing unit supplied with the probe.

Jan 1, 1987

This paper sets out to consider a specific type of corrosion protection system, which is referred to as a Duplex Coating system. The dictionary describes a duplex system as consisting of two elements or a “twofold” composition. The duplex system described in this presentation is such a two-part corrosion protection coating, viz, hot dip galvanizing as the firm base or primer, followed by a top paint system designed to combat severe corrosive environmental conditions encountered in nature. In principle, the duplex system is a cost effective and balanced, designed coating that is suited to the environmental conditions and service life requirements.

Jan 1, 2009

By Maggi Loubser

"In a technologically advanced era, where process control is accurately maintained within the smallest error margins, and accuracy and precision are continuously improved, sampling is still left to the lowest paid employee in the company without training him or her properly to understand the relevance and importance of the task. In this paper the objectives of sampling in a process control environment is discussed and an attempt is made to link it to the theory of sampling. Some of the practical problems associated with implementing theoretical sampling principles in an industrial environment will be elucidated and some examples given on how compromises are made. Different methods of sampling will be discussed with real life examples from different manufacturing industries."

Jan 1, 2014

By SAIMM Journal

COLLOQUIUM ON COMMINUTION The Institute has arranged a colloquium on the above topic to be held on 13th March, 1973, at Kelvin House. The following papers will be presented: 'Peripheral Discharge Mill', by A. H. Mokken, Union Corporation Limited

Jan 1, 1974

'Gee Whiz', a nationwide schools' science and technology competition, is being screened on SATY1 on Friday evenings at 18:00. Twenty-seven schools from al over South Africa are taking part in the series, which will run for 13 weeks. The participating schools were selected on the basis of their performance in the National Youth Science Olympiad and in Mintek's Minsa science competition, which has been organized on an annual basis since 1988.

Jan 1, 1991

What is the role of The South African Institute of Mining and Metallurgy at a time when so much is developing and changing in South Africa? The minerals industry is represented through employers, employees, government, education, research, public views, and even pure political interests. The role of the Institute is to represent the mining and metallurgical profession. In this role the Institute has aligned itself with Associated Scientific and Technical Societies, the Council of Scientific Societies and the Engineering Association. The Institute is also active on the Engineering Council of South Africa.

Jan 1, 1996

ENERGY 73 The First International Energy Exhibition will be held in Brussels from 13th to 18th May, 1973. Vllth INTERNATIONAL CONGRESS ON CYBERNETICS This congress will be held in Namur, Belgium, from 10th to 15th September, 1973. GRADUATE SCHOOL OF BUSINESS ADMINISTRATION University of the Witwatersrand The Sixth Executive Development Programme is scheduled to take place from 2nd July to 24th August, 1973. IFAC SYMPOSIUM 1973 Second announcement ofthe International Symposium on AUTOMATIC CONTROL IN MINING MINERALS AND METAL PROCESSING to be held at the University of New South Wales, Sydney, Australia, from 13th to 17th August, 1973. MINERALS AND ENVIRONMENT The effect ofthe mineral industry on the environment and mineral resources The Institution of Mining and Metallurgy, with the co-operation of the Institute of Quarrying and the Institution of Mining Engineers, will hold a Symposium "Minerals and the environment" at the Institution of Electrical Engineers, Savoy Place, from 4th to 7th June, 1974. COLLOQUIUM ON SLlMES DAMS The South African Institute of Mining and Metallurgy have great pleasure in advising members that a two-day colloquium on slimes dams is bei ng arranged for the 14th and 15th November, 1973. INTERNATIONAL SOLVENT EXTRACTION CONFERENCE This conference will be held at Lyon from 8th to 14th September, 1974. Particulars may be obtained from the Secretary: Dr A. Naylor, British Nuclear Fuels Ltd., Windscale and Carder Works, Seascale, Cumberland CA20 IPG, United Kingdom. Competition for Student Members of the South African Institute of Mining and Metallurgy Each year the Institute offers a prize (or prizes should the entries warrant it) of up to RIOO for the best paper or dissertation on a topic appropriate to the interests of the Institute. Proceedings of the Tenth International Symposium on Application of Computer Methods in the Mineral Industry (APCOM) The Proceedings of the APCOM Symposium held in Johannesburg from April 9th to 14th, 1972, have now been published. Members of the Institute may buy the Proceedings at a special price of R12,50 as against the list price of R25,00 to non-members. Verrigtinge van die Tenth International Symposium on Application of Computer Methods in the Mineral Industry (APCOM) Die Verrigtinge van die APCOM Simposium, wat op April 9 tot 14, 1972, in Johannesburg gehou is, is gepubliseer. Lede van die Instituut kan die Verrigtinge teen 'n spesiale prys van R12,50 koop, in teenstelling met die prys van R25,00 vir nie-Iede.

By T. Ito, N. Tokashiki, H. A. Bilgin, Ö. Aydan, T. Akagi

This study is concerned with the electric potential induced by deformation, fracturing and sliding of non-piezoelectric geomaterials. In experiments, the acoustic emissions together with electric potential measurements were also measured in order to see how the so-called seismic electric signals are associated with new fracture occurrence and propagation and sliding in case of existing discontinuities. The outcomes of this experimental study may explain why such electric potential variations may occur during the fracturing and sliding of non-piezoelectric geomaterials. On the basis of these outcomes, their possible implications in geo-engineering and geo-science are discussed. Ce travail concerne 1e potentiel électrique produit par la déformation, 1a fracture ou le glissement du matériel géologique non-piézoélectrique. Aux expérimentations, on a mesuré 1es émissions acoustiques avec le potentiel électrique afin de constater la relation entre les phénomènes de glissement ou de formation et de propagation de nouvelles fractures le long des discontinuités existantes. Les résultats de ce travail expérimental expliquent pourquoi de telles variations de potentiel electrique se produisent durant le glissement ou la fracture du matériel non-piézoélectrique. Sur la base de ces résultats, on a discuté de leurs éventuelles implications dans les domaines de 1a géologie et de 1a géo-ingénierie. Diese Studie befasst sich mit dem elektrischen Potential das durch die Deformation, Bruchbildung und Gleiten von nicht-piezoelektrischen Geomaterialien induziert wird. In Experimenten wurden die akustischen Emissionen zusammen mit elektrischen Potentialmessungen gemessen, um zu sehen, wie die so genannten seismischen elektrischen Signale mit neuer Bruchbildung und Ausbreitung sowie Gleiten im Falle von vorhandenen Diskontinuitäten verbunden sind. Die Ergebnisse dieser experimentellen Untersuchungen könnten erklären warum elektrische Potential Schwankungen während der Bruchbildung und dem Gleiten von nicht-piezoelektrischen Geomaterialien entstehen. Auf der Basis dieser Ergebnisse, wurden ihre möglichen Implikationen in Geoingenieurwissenschaften behandelt.

Jan 1, 2003

The most important event in the Institute's recent history is the strategic plan, which was drawn up in 1986 during Henry James's term of office. This plan was formulated at a workshop attended by most of the senior members of Council, and from it has flowed a number of changes that have been introduced over the past four years.

Jan 1, 1990

The theory of geostatistics covers a branch of .applied statistics aimed at a mathematical description and analysis of geological observations. Geostatistics can be used in pure geology (for example, for the analysis of trace elements in a metamorphic rock), in mineral exploration (for example, for the analysis of geochemical exploration data), as well as in mine valuation. This book is intended to provide a practical introduction to the theory of geostatistical methods of mineral evaluation. Over the years, various mathematical models have been developed to represent the distribution of values in mineral deposits. The simpler models are based on the assumption that the values are randomly distributed. Classical statistical methods, based on this assumption of the random distribution of values, are used to analyse mineral deposits to which these models apply or are assumed to apply. In all mineral deposits, however, one recognizes the presence of areas where the values are higher or lower than elsewhere. Also, the values of two samples in a mineral deposit are more likely to be similar if the samples are taken close together than if they are taken far apart. This indicates that there exists a degree of correlation between sample values, and that this correlation is a function of the distance between the samples. Models have been developed which take this correlation into consideration, with the degree of correlation between sample values being usually measured by the semivariogram function. In these models the fact that two samples taken next to each other will most probably not have the same value, must also be considered; even for very short distances the correlations are usually not perfect and a purely random component is present in the value distribution. The mathematical models will therefore assume the presence of two sources of variability in the values: a correlated component and a random component. Finally, one must consider the particular and very common case of mineral deposits in which the values present a systematic variation in space. This variation is usually referred to as a drift, or a trend. For example, the grade of an ore body may increase with depth of the ore, or it may decrease when one moves away from a central volcanic pipe. The earlier models did not give a satisfactory representation of drifts, and more complex models have been developed, in which three sources of variation are represented. These models are made up of: a deterministic component, a correlated component, and a random component. The deterministic component is used as a model of the drift. The correlated component explains regular changes in values which are not represented by the drift. The random component represents variations which cannot be explained by any of the above factors. The simpler models, based on the assumption of a single random component, will be described first (Chapter 2). The models based on the hypothesis of the superimposition of a correlated component on a random component, will then be analysed in detail. These models are most commonly used in the analysis of mineral deposits (Chapters 3-1 I). Finally, how to deal with the presence of a drift will be briefly described (Chapter 12). This book has been written essentially for students in mining engineering and for mining engineers who are interested in the background to the theory of geostatistics as well as its practical applications. The assumption is made that the reader has an elementary knowledge of statistics. Some knowledge of linear algebra is useful in part of Chapter 9, and is necessary to read Chapter 12. A proof is given of all the equations related to geostatistics, and which are not usually found in elementary textbooks on statistics. Understanding of these proofs is not necessary for practical application of the theory, and the reader may wish to skip them on a first reading, concentrating attention on the numerous simple practical examples given. Although the theoretical geostatistician will not find much new material in this publication, it is anticipated that he will develop some interest in the practical approach chosen to prove the various geostatistical equations. Many people and institutions contributed to the preparation and completion of this work. I am much indebted to Dr D. G. Krige and the Anglo Transvaal Consolidated In- vestment Company Limited, who gave me the opportunity to spend a considerable amount of time working on geostatistical problems, both theoretical and practical, during the time that I was in their employment. Dr Krige contributed greatly in developing my interest in studying both the theory and practice of geostatistics, always insisting that a correct balance be kept between theory and practice. I am grateful to Professor H. M. Wells and the Mining Department of the University of the Witwatersrand for inviting me to give lectures in a post- graduate course on geostatistics. The notes 1 prepared for that course became the foundation of the present work. I am also indebted to the Centre de Geostatistique of the École des Mines de Paris, where I received my first formal education in geostatistics during a summer course given by Charles Huijbregts, after many years of my lonely plodding through the published literature. Many graphs in the present volume are reproduced with the permission of the Centre de Geostatistique. The Department of Metallurgical and Mineral Engineering of the University of Wisconsin-Madison has also contributed in making this work possible, by allowing me to spend a considerable amount of time and resources in the writing, typing, and correcting of successive drafts. I am thankful to Lynn D. Kendall, who typed the entire manuscript under constant pressure.

Jan 1, 1978