Geostatistics

Computers & Geosciences Vol. 17, No. 9, pp. 1345-1349, 1991 Pergamon Press plc. Printed in Great Britain

REVIEWS

Geostafisfics by M. Armstrong, editor, 1989, Kluwer Academic Publishers, Dordrecht, Vol. 1 (xxix p. + p. 1-491) and Vol. 2 (xvii p. + p. 493-1038), ISBN 0-7923-0204-4 (2 volumes), $249.00 (U.S.) These two volumes comprise the Proceedings of the Third International Geostatistics Congress, held in the papal palace, Avignon, France, 5-9 September 1988. The event was attended by 185 scientists and engineers, mainly from France and other European countries. The 78 published papers and 16 abstracts give the reader an excellent idea of the state of the art in geostatistics, a branch of geoscience primarily concerned with the spatial analysis of ore assay values and other data which can be averaged for blocks. The first conference of this series was held in Rome, 1975, the second at Lake Tahoe, California, 1983, and the fourth congress will take place in Troia, Portugal, September, 1992. Geostatistics is becoming used more widely in many subdisciplines of the earth sciences because it provides adequate tools for estimating average values of attributes in blocks and for the contouring of noisy data. The theory and practice of geostatistics remain removed from geology as well as from mathematical statistics. Geologists are accustomed to think in terms of conceptual models that can be used for the interpretation of observations. Geostatisticians are not concerned with the genesis of objects such as ore deposits but with random variables in threedimensional space. It probably is fair to say that such variables have been neglected in mathematical statistics and geostatistics is filling a major gap. The Avignon Proceedings begin with seven geostatistical overviews originally presented in plenary sessions. The first two of these are by the fathers of geostatistics: Daniel Krige, Witwatersrand University, South Africa, and Georges Matheron, Ecole des Mines de Paris. It is useful to begin this review with a summary of the history of geostatistics. The earliest geostatistical methods were developed during the late 1940s by Krige and colleagues in South Africa in order to predict the average gold content of stoping blocks in the large but relatively thin Witwatersrand placer-type gold deposits. Although it is easy to recognize the conglomerate beds ("reefs") and underlying shales ("mats") that are mined, gold concentration per unit of volume within these layers is highly variable and can be determined only by chemical analysis. Krige applied the statistical technique of regression analysis to compensate for the fact that, after mining, average

gold content of a block delineated by drawing a dosed polygon around a set of relatively high assay values, as a rule, turns out to be less than the average of these high assay values. This fact is of crucial significance when the total delineated volume of ore depends on a changing cut-off grade determined by current metal prices. Krige emphasized the economic side of geostatistics, by terming himself a financial engineer throughout his long and distinguished career in the field of ore reserve estimation. During the 1950s, Matheron generalized and greatly expanded the early geostatistical mining applications. His first two volumes on practical geostatistics were published in 1962 (Trait6 de grostatistique appliqure, 1 & 2: Mrmoires du Bureau de Recherches Groiogiques et Minirres, Paris). The so-called variogram was introduced as a basic geostatistical tool. It is a diagram showing the function (h) which represents one-half of the average squared difference between assay values that are a given interval ( = h ) apart. This function provides a measure of spatial continuity. By using the variogram in combination with regression analysis, it is possible to estimate average metal grade values for blocks by weighting the assay values in the vicinity of these blocks. To honor the first practitioner of this approach, Matheron coined the term "krigeage" which was later translated into English as "kriging". In 1965 Matheron published the first comprehensive version of his geostatistical theory in a new book (Les variables rrgionalisres et leur estimation: Masson, Paris). In conceptual approach as well as notation, this book differed from books in the English language concerned with similar topics, During the next 10 years, geostatistics became widely accepted within France but remained relatively unknown elsewhere. Matheron's colleagues including Serra, Journel, Huijbreghts, and David made numerous practical applications in the field of ore reserve estimation. By 1975 when the first international geostatistical congress was held in Rome, Englishspeaking statisticians and geoscientists were impressed by the accomplishments of the French school. Today, at least six good geostatistical texts have been published in English, and Margaret Armstrong has skillfully edited several volumes with geostatistical case-history studies. As illustrated in contributions by Switzer, Cressie, and Myers in the Avignon Proceedings, mathematical statisticians also have helped to reformulate Matheron's geostatistical ideas for wider dissemination.

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In the Foreword, the organizers state that the purpose of the Avignon meeting was to highlight theoretical advances as well as the widening scope of practical applications. The meeting was successful in both respects. Theoretical developments are dealt with in 19 papers. Specific sampling problems are considered in 5 papers. The applications are in the fields of soil science (5 papers), oceanography (5 papers), hydrocarbons (10 papers), hydrology (10 papers), mining (10 papers), and other fields (7 papers), respectively. On the theoretical side, attention is paid to the internal consistency of the geostatistical methods mainly in contributions by Matheron and Rivoirard. In other papers, new models for random functions are presented or adapted from models previously developed in mathematical morphology and random set theory. The relationship between fractal geometry and geostatisticai models is further explored. The objective to achieve internal consistency of geostatistical methods presents interesting paradoxes and mathematical problems. Spherical, exponential, and Gaussian functions are used widely for modeling the variogram. Are these functions compatible with the positively skewed lognormal frequency distribution of measurements which may be approximately satisfied in practice? Matheron provides new answers which are surprising. For example, the spherical variogram is not compatible with the iognormal distribution in three-dimensional (3-D) space. In I-D it is only possible for lognormai distributions with relatively small coefficients of variation. On the other hand, it is well known that models such as the spherical variogram have been applied successfully in numerous situations. It can only be concluded that they are good approximations of as yet unknown models that would be internally consistent. Other interesting new developments treated in detail include isofactorial models and coregionalization analysis. Isofactorial models are based on a relation between concentration values determined for blocks of different sizes. Suppose that X1 and X2 represent concentration values for a large block and a small block, respectively, and that the small block is located at a random point within the large block. It then can be postulated that the conditional expectation of X2 and X~ is equal to Xl. This simple rule establishes a bridge between the frequency distributions of concentration values for blocks of different sizes. Isofactorial models also are used in the nonlinear estimation technique known as disjunctive kriging or indicator cokriging (not to be confused with indicator kriging which is a special type of indicator cokriging). These techniques, which are more demanding computationally because entire frequency distributions are considered instead of means only, occasionally give much better experimental results than ordinary kriging. Coregionalization analysis is a useful form of spatial factor analysis primarily developed by Royer

and Wackernagel who each have papers in the Avignon Proceedings. In their approach, linear multivariate relationships between variables are based on neighborhoods rather than single observation points as in ordinary factor analysis. A characteristic feature of Matheron's early geostatistical approach was the emphasis given to the so-called de Wijsian model. When a logarithmic scale is used for the interval h, the function 7(h) plots as a straightline that dips toward the origin indicating that 7(h) continues to increase when h increases. The attribute of a rock that produces this type of variogram has the property of self-similary. Its pattern of variability remains the same at different scales. In the late 1940s, the Dutch mining engineer Henri de Wijs developed this type of model for a vein-type zinc deposit near Pulacayo, Bolivia. Of course, self-similar features now are well known in many fields of science because of the work by Benoit Mandelbrot who refers to the original application by de Wijs in his book on the fractal geometry of nature. Mathematical statisticians use the autocovariance function a(h) for description and analysis of spatial variabil!ty. Matheron adopted the variogram, which is related to the autocovariance by y (h) = tr (0) - a (h), because y (h) can be used even if the feature of interest does not have a finite variance tr 2 = ~(0). Self-similar features have an infinitely large variance. On the other hand, a feature with a stationary mean and variance has the properties that a(0o) = 0 and 7(0o) = tr(0). In the de Wijsian model this limit for h tending to infinity is never reached because the variogram continues to increase and tr(0) = 0o. During the 1970s, geostatisticians insisted that the variogram should be used instead of the autocovariance function. The variogram remains a characteristic feature of geostatistical textbooks. On the other hand, the Avignon Proceedings contain a convincing article by Srivastava and Parker showing that the autocovariance is to be preferred to the variogram, mainly because as a measure of spatial continuity it is more robust and suffers less from the effects of heteroscedasticity and clustering of observation points. On the practical side, it is obvious that, although mining applications remain important, the emphasis in the Avignon Proceedings is on new applications, especially in hydrology and hydrocarbon reservoir modeling. In these fields, the modeling of fluid flow provides a new challenge, because time is added as an extra dimension. My main criticism of the Avignon Proceedings is that is signifies a continued proliferation of geostatistical terminology which is confusing to the uninitiated. The papers contain discussions and applications of as many as seventeen different types of kriging and three types of cokriging. Fortunately, there is an excellent index at the end which is needed absolutely to keep track of numerous relatively unknown terms. For example, the index contains four

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as Chapters 5, 15, and 16 which now are so comprehensively technically outdated that they must be of dubious value in a collection such as this. These papers cover state-wide mapping using crude raster maps of Maryland (Chaper 5) and digitizing procedures for data capture from paper maps (Chapters 15, 16). Digitizing is rapidly becoming the data capture method of second choice with the arrival of semiautomatic "head's up" vector screen capture of raster scanned maps. However, this book also contains a number of the acknowledged classics of GIS literature dating back to the late 1970s and early 1980s. Amongst this group is the Dangermond paper describing the software components of GIS (Chapter 3), a defining work for GIS design; the description of the Dual Independent Map Encoding (DIME) system by the US Bureau of Geological Survey of Canada F.P. AGTERBERG the Census (Chapter 7), which was the starting point 601 Booth Street for much work on topology; and the anecdotal paper by Douglas entitled "It makes me so CROSS" Ottawa, Ontario, Canada KIA OE8 (Chapter 21), which describes the complexity of the line intersection problem with all the special case Introductory Readings in Geographic Information handling problems. Systems by D. J. Peuquet and D. F. Marble, editors, The core of the book is a selection of papers from 1990, Taylor & Francis, London, 371 p. ISBN journals and conferences held in the late 1980s. A 0-85066-857-3, £40 (cloth), £18 (paper) number of these papers merit particular attention, beginning with the paper by Cowan (Chapter 4) Introductory Readings in Geographic Information which answers the question "GIS vs. CAD vs. Systems is a collection of 26 previously published DBMS: what are the differences?" His definition of papers, selected by the editors to illustrate the devel- GIS as "a decision-support system involving the opment and application of Geographic Information integration of spatially referenced data in a problem Systems (GIS). The rationale for the collection is solving environment" is one of the most accurate and elaborated in the Preface where the editors note that concise I have seen. Another key paper in this collecGIS literature is scattered through journals and tech- tion is the paper by Peuquet (Chapter 19) which connical reports in a number of disciplines, many of siders data models, data structures and their relative which are inaccessible to the general reader although merits in depth in a 35 p. blockbuster. A paper by they cover important basic principles. That a book Vrana (Chapter 20) tackles another thorny problem such as this one is really necessary is a testament to for GIS---that of the handling of historical data. Several papers are worthy of attention as the extremely rapid growth of GIS in the late 1980s and the tremendous interest in what is currently an significant applications of GIS. Hence Tomlin and Johnston (Chapter 11) describe land use allocation in under-published field. The book is divided into five sections covering the the Orpheus project, Bonham-Carter and Agterberg definition of a GIS, examples of practical appli- (Chapter 12) illustrate the use of modeling techniques cations, operations and problems of building a data- for gold exploration using a raster GIS, and Davis, base, GIS internals--data representation and analysis Whigham, and Grant (Chapter 14) show how artifitechniques, and finally GIS design and evaluation. cial intelligence techniques can be applied to environEach of these sections is reviewed and summarized in mental management. There also are a number of short essays by the editors which serve to place the system descriptions which illustrate GIS in practice papers (of widely differing origins) in a modern such as the comprehensive description of the TIGER context. The papers themselves have been refor- system by Marx (Chapter 9) which is used by the US matted to a common style and have been annotated Bureau of the Census to geocode addresses, and the by the editors where changes have taken place since paper by Chrisman (Chapter 18) illustrating the use original publication. of the ODYSSEY system for efficient data capture Inevitably, in a collection such as this the quality procedures. of the individual contributions is variable, in this In summary, this is a valuable collection of papers situation mainly the result of the time that has which will be a useful addition to the library of many elapsed since some of the original publication dates practitioners and students of GIS. Although there is (the earliest is 1970). Although the editors carefully a tangible US bias (no non-N. American papers), and state that they have only included older papers where there are some inconsistencies between the papers, they are of specific merit, there are a few papers such the editors have done a good job in bringing this

cross-references to "atom at origin" which is an expression to describe the occurrence of a peak at zero value in a histogram. It would be too drastic to reformulate all geostatistical results in terms of mathematical statistics. Perhaps a line could be drawn that only the better known terms such as "variogram" and "kriging" are used. Because of the high price and an unavoidable lack of continuity resulting from multiple authorship, these volumes will be owned by relatively few earth scientists. However, they should be widely available for consultation in libraries. Because of the widened scope of practical applications, they contain useful information especially for economic geologists, geochemists, geophysicists, oceanographers, and reservoir geologists.