Mathematical Geology, Vol. 6, No. 3, 1974

Reviews Proceedings of a Symposium on Decision-Making in Mineral Exploration H A. M. Kelly and A. J. Sinclair, editors University of British Columbia, Center for Continuing Education, Vancouver, B.C., Canada, 6-7 February, 1969, 279 p., no price stated. At this symposium, the second of a series, fourteen papers were presented on several subjects. Several authors treat data selection, storage, and retrieval-of particular importance in Canada, the world's second largest country. C. F. Burk, Jr. points out that the supply of data is much smaller than the demand (a problem that has now been at least partly solved). A. D. Drummond describes the information system of Placer Development, Ltd., and E. T. Lonergan outlines some methods to code diamond-drillhole data. P. G. Sutterlin and J. De Planke develop a storage and retrieval system with the significant advantages that the format is "free-form" and data-controlled and user-oriented rather than being designed for the convenience of computers and computer-oriented people. Techniques of data plotting are developed by W. J. Coulthard and J. L. Leigh. Trend-surface analysis, a familiar and controversial subject, is introduced by A. J. Sinclair, using regional variations of minor elements in sulfides at Slocan, British Columbia, as example data. F. P. Agterberg continues with a discussion of several new concepts applied to copper values from the Whalesback mine, Springdale, Newfoundland. Using geochemical and geophysical data from a one-square-mile area in British Columbia, D. R. Cochrane fits fourth-order trend surfaces. Multivariate procedures other than trend-surface analysis are treated by J. E. Klovan in a lucid and exceptionally well-illustrated account of classification methods. One of these methods, Q-mode factor analysis, is applied by J. D. S. Wilson and A. J. Sinclair to geochemical data from Park City, Utah, U.S.A. Two papers concern geophysics. T. J. Ulrych writes on wavenumber domain analysis and design of potential field filters, and G. K. C. Clarke discusses application of optimum filters. 273 9 1974 Plenum Publishing Corporation, 227 West 17th Street, N e w York, N.Y. 10011. N o part o f this publication m a y be reproduced, stored in a retrieval system, o r transmitted, in any f o r m or by an.y means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission o f the publisher.

274

Reviews

Finally, L. E. Borgman examines the precision of ore-grade estimates through a simulation study. In summary, the papers treat selected topics from the broad field of mineral exploration. The editors and the University of British Columbia are to be thanked for making the work available to a wide audience. George S. Koch, Jr.

University of Georgia Athens, Georgia (USA)

Topics in Mathematical Geology M. A. Romanova and O. V. Sarmanov, editors (Russian translations by J. P. Fitzsimmons) Consultants Bureau, New York, New York, 1970, 281 p., $37.50. This is a collection of 25 papers originally published in Russian by Nauka Press in 1968. According to the foreword, the book displays the general level of mathematization of geology in 1966. Fifteen articles were translated from the Russian. Contributions by G. V. Middleton, F. Chayes, D. M. Shaw, W. Schwarzacher, W. T. Fox, F. I-Iori, E. H. T. Whitten, D. G. Krige, and D. F. Merriam were printed from the original manuscripts. An article by B. A. Choubert on geochemical behavior of elements in the lithosphere was translated from the French. The volume begins with a description of the life and work of A. B. Vistelius. The editors and several Russian contributors are from the Laboratory of Mathematical Geology of the Steklov Mathematical Institute of the Academy of Sciences of the USSR in Leningrad. Most Russian papers are developments of concepts originally proposed by Vistelius in earlier publications of which a complete list is provided. A set of papers initially prepared almost a decade ago is bound to contain parts which have become obsolete. In several papers detailed discussions are given of propagation of errors, e.g., in matrix inversion, caused by lack of precision on the digital computer. Problems of this type should be less serious today. One of the purposes of the original edition was to acquaint Russian readers with methods of trend-surface analysis practiced outside the USSR before 1966. This translation is of current interest because it shows the standard usage of mathematics by geologists in the USSR. A characteristic feature of the Russian contributions is that most problems are firmly based in theoretical geology. L. D. Knoring presents a statistical analysis of over 45,000 orientaion measurements on joints from six regions. In this paper and that by

Reviews

275

M. E. Demina and O. M. Kalinin, a number of useful rapid statistical tests are documented and exemplified. The relationship between mathematical applications and geological concepts is particularly close in the section on paragenetic analysis. E. P. Kalinin, M. V. Fishman, and B. A. Goldin used multivariate statistics to study the concentration of niobium in a suite of metasomatically transformed granites in the polar Urals. Y. U. Nagaitsev and Y. V. Podol'skii studied gradational changes in the compositions of whole rock, garnets, and biotites in micaschists and gneisses of the Ladoga Formation with metamorphic zones ranging from the epidote-amphibolite facies to the granulite facies. M. A. Romanova presents a detailed study on the eolian sorting of grains of different minerals in recent sands of the Kara Kum Desert. These sands were transported by more or less steady winds and Romanova's exponential models are well suited to describe the sand composition trends. K. I. Kheiskanen studied the process of differential subsidence and sedimentation in a Proterozoic basin by fitting harmonic functions to vertical variations in the mean grain size of clastic sediments. D. N. Ivanov derived a sequence of crystallization for mineral phases in granites of central Kazakhstan. Markov-chain models were used to distinguish between textures arising from random crystallization in a magma and more regular textures superimposed during metasomatic transformations. Two papers are more theoretical. A. V. Faas and O. V. Sarmanov related sets of measurable percentage values which form a closed-number system to several theoretical random variables in the open system. T. S. Rivlina has developed a stochastic process model for stratigraphic sequences with interstratal erosion defined as a separate state. Perhaps not all methods of approach displayed in this volume should be accepted without further questions. In the papers by E. P. Kalinin and others, and Y. V. Nagaitsev and Y. V. Podol'skii, weight-percent values for major oxides in rock samples are divided by total weight-percent oxygen before correlation coefficients are computed. Reference is made to a method by Vistelius and Sarmanov who, in 1961, had applied the method to chemical data on micas where weight-percent oxygen is relatively more variable than in sets of whole-rock analyses from similar rock samples. The authors claim to have avoided false correlations as may arise between percentage values in a closednumber system. However, when weight-percent oxygen is approximately constant, the transformation may not have a significant effect. In the paper by M. D. Belonin and I. M. Zhukov and that by M. A. Romanova, exponential functions are used for trend fitting. Placing the polynomial in the exponent can be more advantageous than performing an ordinary polynomial trend-surface analysis. However, the resulting exponential model is nonlinear. The authors reduce it to a linear model by

276

Reviews

applying a logarithmic transformation. It is not mentioned that this procedure provides unbiased results only if the observed values are relatively closely approximated by the calculated values of the exponential function. When the frequency distribution of the residuals is positively skew, the sum of the calculated values will be less than the sum of the observed values. This specific type of bias does not occur when linear models are used and may constitute a severe drawback of the exponential model. The quality of the Russian translations is excellent. Locally, some reinterpretation may be needed. For "'value" in the expression "single independent random value" (p. V), I would read variable. The words "matrix equalities corresponding to systems of linear equations are invariant, relatively arbitrary, linear transformations" (p. 189) may puzzle the reader. However, from the context it may become clear that the solutions of these systems of linear equations should be independent of linear transformations applied to the geographical coordinates. The preceding more critical remarks should not detract from the value of this collection which provides a unique insight into mathematical geology as practiced in the USSR. The volume should have its place on the shelves of every geological library. F. P. Agterberg

Geological Survey of Canada Ottawa, Ontario (Canada)

Elektronische Datenverarbeitung in der Tektonik (Electronic Data Processing in Structural Geology) by R. E. Adler, F. Krueckeberg, W. Pfisterer, A. Pilger, and W. M. Schmidt. Clausthaler Tektonische Itefte 8, Verlag Ellen Pilger, Clausthal-Zellerfeld, West Germany,1969, ~57 p. Adler (p. 79-90, p. 93-149) discusses the aspects of modern statistical analysis and synthesis based on geologic-tectonic mapping and provides a system of coding structural features and parameters. The data are standardized for punching on 80-column data cards. Krueckeberg (p. 7-53) describes a FORTRAN program for directional data, which are punched on data cards or written on magnetic tapes according to the coding system of Adler. The program control cards are written in a type of computer language with list structure. This language is named GELI. For most procedures in structural geology there is a corresponding instruction in GELI, for example storage and retrieval of data, plotting fabric diagrams, rotating diagrams. GELI is an

Reviews

277

easy to handle language which may be used by geologists who are not familiar with FORTRAN or another language. Adler, Pfisterer, and Schmidt (p. 55-78) present extensive examples making use of all features of the GELI program from coal mines of the Ruhr area and the Harz region of Germany. Pilger (p. 1--4, 153-157) outlines the work already done and the future work which is necessary to solve problems in structural geology by means of computers. Heinrich Siemes

hlstitute fiir Mineralogie und Lagerstiittenlehre Rhein.-Westf. Technische Hochschule Aachen Aachen (West Germany)

Quantitative Geomorphology: Some Aspects and Applications Marie Morisawa, editor Proceedings of the 2nd Annual Geomorphology Symposia Series SUN-Y, Binghamton, New York, 315 p., softbound, $4.00 This softbound book is the outcome of a symposium held in Binghamton in the fall of 1971, where a large and successful gathering of geologists and geomorphologists heard and discussed many of the papers presented now in this volume. As one who attended and read a paper at the meeting I think it is slightly disappointing that the repartee of the question and answer period could not be carried forward into the volume. In many ways these exchanges provide the real meat and benefit of such meetings. Quantitative geomorphology is discussed in opening remarks by both the editor and N. E. Salisbury. Both are primarily fluvial geomorphologists by interest and hence, (and by-and-large quite rightly) attach considerable significance to the developments in this field in tracing the changing nature of "Quantitative Geomorphology." Salisbury states that the "quantitative revolution is over" (p. 9) and this is borne out by the majority of papers which are using techniques for a specific purpose and not examining the general relevance of a particular method of statistical manipulation. There is also an underlying sense that we need to understand the deterministic aspects of a problem, not just the probabilistic. The volume is divided into three parts: Part 1 is entitled "Theoretical, Historical and Educational Aspects"; Part 2 is called "Some Typical Studies and Techniques"; and Part 3 is labelled "Some Practical Applications." It should be noted that these meetings are annual events and that this volume

278

Reviews

forms part of a series. All in all, the papers make useful reading for geomorphologists. J. T. Andrews Institute of Arctic and Alpine Research University of Colorado, Boulder, Colorado (USA)

GEOCOM Programs

R. J. Howarth, editor Geosystems, P.O. Box 1024, Westminster, London SW1P 2JL, England Issued about 5 times/year, individually priced, or s for five issues GEOCOM Programs is a worthy successor to the Kansas Geological Survey Computer Contributions. The series is edited by R. J. Howarth of Imperial College. The intent is to publish about five issues per year; the programs also included in the GEOCOM Bulletin. Started in 1971, nine programs are now available. (1) Spatial Filters and FORTRAN IV Program for Filtering Geologic Maps, by J. E. Robinson and M. J. Ellis, 1971, 21 p. (2) A First Generation Simulation Model for Selecting Amongst Exploration Programs with Special Application to the Search for Uranium Ore Bodies, by J. C. Griffiths and D. A. Singer, 1971, 42 p. (3) FORTRAN IV Programs for Computation and Printer Display of Mathematically Defined Surfaces, by M. V. Kirk and D. A. Preston, 1972, 25 p. (4) ELIPGRID, A FORTRAN IV Program for Calculating the Probability of Success in Locating Elliptical Targets with Square, Rectangular and Hexagonal Grids, by D. A. Singer, 1972, 16 p. (5) MAXLIKE: FORTRAN IV Program for Maximum Likelihood Estimation, by T. A. Jones and W. R. James, 1972, 16 p. (6) Geochemical Similarity Analysis, by G. A. Borchardt, 1972, 31 p. (7) FORTRAN IV Programs for Empirical Discriminant Classification of Spatial Data, by R. J. Howarth, 1973, 31 p. (8) COSLOP-2: A comprehensive Model Program for Simulating Slope Profile Development, by F. Ahnert, 1973, 24 p. (9) FORTRAN IV Program for Spline-Surface Interpolation and Contour Map Production, by M. E. V. Koelling and E. H. T. Whitten, 1973, 12 p. The spatial filtering program by Robinson and Ellis was long awaited in geological circles. Robinson and his colleagues have been publishing results

Reviews

279

using the program since 1968. Now, the detailed explanation of how the program works is available along with examples. The simulation model program by Griffiths and Singer is a sophisticated approach to exploration for nonrenewable natural resources. "'The Engel Model, considered as a first generation simulation model, i s . . . described" ; economic geologists will find this program of special interest. The program for computation and display of maps of mathematically defined surfaces by Kirk and Preston is a valuable addition to graphic programs presently available. The program will be welcome by all those without special equipment as it requires only a line printer for output. Singer's program ELIPGRID is a program for determining " . . . the probability for finding a randomly oriented target or a target with a specific orientation with respect to the grid." It will be of use to geologists (and other scientists) with sampling problems where the target is defined. MAXLIKE is a program for statistical estimation. The Jones-James program will allow " . . . one to estimate the parameters in two complex distributions: the mixed normal and mixed circular normal." The procedure should be helpful for those working with complex distributions such as in sedimentology with grain-size distributions. The geochemical similarity analysis by Borchardt will find use by all those working with classifications. The program classifies into " . . . groups on the basis of multivariate data by means of a similarity coefficient and comparisons among replicated samples." It can handle as many as 135 samples with 25 variables. The seventh publication in the series by Howarth is also a classification program. "Two programs are presented for the implementation of Specht's non-linear empirical discriminant function in the context of spatially distributed data." As pointed out by the author, the algorithm is generalized and could be used easily with other data sets. Examples are utilizing stratigraphic and regional geochemical data. COSLOP-2 by Ahnert simulates slope profile development under a wide variety of conditions, and it is of use to quantitative geomorphologists as a slope model. The ninth program by Koelling and Whitten presents a new method for contouring gridded data by a spline-interpolation procedure. The reproduction of the series is attractive--a double-column, rightjustified format, 888 by 1188 inches in overall size, and well illustrated. Some of the computer printout is not especially good but it is legible (this is a general printing problem involving computer printout). Thus, the format is not only pleasant in appearance but practical as well. The series, as the subtitle "Computer programs for geoscientists" suggests, is just that. The series is definitely needed and from the first 9 issues, it is obvious that the standards are high and that the presentations will equal or surpass the old, now discontinued, Kansas series. Undoubtedly, every geol-

280

Reviews

ogist working with computers will want to obtain copies of these programs (if they have not already). GEOCOM Programs along with the regular GEOCOM Bulletin, are indispensibte for serious quantitatively oriented earth scientists. D. F. Merriam

Department of Geology Syracuse University Syracuse, New York (USA)

Computer Programs in Oceanography Compiled by C. Dinger, NODC Publication C-5 (second revision) 1970, 143 p. available free on application to the National Oceanographic Data Center, Washington, D.C. As in all social movements, the cybernetic revolution has fostered its own ancillary folklore. One aspect of the folklore which has come to haunt computer science is the quixotic notion that software is transportable. This nemesis attacks from two fronts. It, on the one hand, cajoles the uninitiated user into buying or borrowing software from others which, with probability equal to about 0.97, will not run on his machine without a programming effort which he is typically not only unable to provide, but may be unable to understand. On the other hand, the beast rears its ugly head in meetings of those who fund computer-oriented activities, draws attention to the "proliferation" of hardware systems, and suggests that great economies could be realized if all the transportable software was run on a single, powerful central installation. The history of computer applications in the sciences provides little support for this myth, but it persists and spawns catalogues of contributed programs. The publication, "Computer Programs in Oceanography" is a wellintended compilation of applications programs, but in its catholic zeal it indiscriminantly catalogues together programs that range from those potentially interesting to a general readership to those which were interesting to the person who wrote the code. The catalogue seemingly contains an entry for every program ever described to the National Oceanographic Data Center (NODC) (since incorporated in the Environmental Data Service (EDS)), on one of the Center's forms. The form is similar to that which appears as the last page of the Kansas Geological Survey Computer Contribution series, and the NODC publication lists only this information which includes the

Reviews

281

program name, language, hardware for which it was written, and a brief description of the purpose of the program. The format is essentially that used in the GOSSIP List compiled by T. Victor London and E. P. Adams at Reading University (England). Many of the programs described also have been catalogued elsewhere and a useful list of other applications software catalogues is provided in the NODC publication. It is impossible to determine the relationship between the NODC catalogue and these others. The NODC catalogue is not a superset including these others as proper subsets, but some intersection with their union. Judging, at least from the list of Kansas Survey publications and the GOSSIP list, the nature of the criterion defining the intersections is most elusive. The catalogue of "Computer Programs in Oceanography" is particularly noteworthy for its internal variance. It contains scattered entries for programs which are both of sufficient generality and sufficient complexity to warrant investigation. Software packages are described for presentation of data from productivity stations, STD casts, spectral analysis, sediment textural analysis, and others which are of general interest. These are interspersed with programs for calculating salinity and sigma-t in various manners. For many of these programs, which properly fall into the class of FORTRAN SUBROUTINES or FUNCTIONS, more time has been expended compiling the catalogue entry than is required to write the code from scratch. There are also a number of programs described which are of limited interest such as one which provides "summaries of results of f o u l i n g . . , on exposed and control panels," but if the reader is willing to take the time to sift through the cataIogue he may well be rewarded with a program of interest. From a technical point of view the compilation is reasonably well done. There are three indices provided. The first, which lists programs by contributing institution, is of primarily political interest within the oceanographic community. The second, which is indexed by language, and the third, which is indexed by machine type, are of genuine value to the reader, although it is a bit of a shock to find programs written in FORTRAN VI or programs written for the IBM 7030 computer. Some of the cross-referencing is incorrect between indices and the body of the catalogue. Within the body, programs are listed by category. The category list itself mixes applications categories, e.g., "Biological Oceanography," with data systems categories, e.g., "Editing" and " R e t r i e v a l . . . " and some of the assignations of programs to categories show a good deal of imagination. There are interesting revelations throughout the publication such as programs that compute "various oceanographic parameters" and, of interest to statisticians, a program that computes "concentrations... from approximate raw data." Overall the publication is useful despite the promiscuity of its

282

Reviews

contents, but a paraphase of the classical advice, caveat lector, is recommended. James C. Kelley

Department of Oceanography University of WashhTgton Seattle, Washington (USA)

A Collection of Computer Programs in BASIC for Geology and Geophysics by Roger Till, David T. Hopkins, and Clive McCann Reading University Geological Reports, Number 5. Geology Department, University of Reading (England), 1972 The authors present 31 computer programs which have been developed in the Geology Department of Reading University. Their use of conversational computing methods and of the BASIC language is particularly appropriate to the teaching of computer applications in geology, and the programs were developed mainly for this purpose. Four groups of programs are described, under the headings of "Geophysics," "Statistics," "Larger programs," and "Miscellanea." The geophysical programs include programs for gravity and magnetic data processing, and for generation of gravity profiles using geometric models, as well as a simple plotting routine of general application, and Fourier transform routines. The 19 statistical programs include the usual range of basic statistics, correlation, regression (including reduced major axis), Student t test, and moving average smoothing, among other commonly used methods; there are programs also for random-walk experiments, for generation of binomial and Poisson distributions, for a runs test, and an interesting program for identification of discontinuities in a sequence of numbers. Four larger programs comprise the next section: to one who is more familiar with F O R T R A N or ALGOL programming, these will not seem particularly large, but in the context in which they were designed, they do seem to be useful. They include simple analysis of variance, computation of a correlation matrix, another, more comprehensive linear regression program, and a program for computing frequency distribution moments from phi-scale grain-size data. Two small simulation programs complete the collection in this book: the first computes density/depth relationships within the earth, and the second provides a simple model of erosion and isostatic adjustment. The collection of programs as a whole forcibly illustrates that interactive computing has a part to play in the earth sciences, and illustrates how the

Reviews

283

BASIC language bridges the gap between calculating machine and largescale computing methods. Although these programs were written within an educational context, and will probably be of great use in this field, they should not be dismissed as merely trivial exercises in elementary statistics: for many purposes it may be more appropriate (and cost-effective) to compute in conversational mode using a language such as BASIC or APL, than to use a large-scale batch-oriented language like F O R T R A N or A L G O L . Two situations other than education which spring to mind are the testing of new statistical or mathematical methods, and simulation. In each of these applications, fast turnround is important, and the volume of data to be processed may be relatively small, as it must be in a conversational system when all data must be entered through the keyboard. It is unfortunate, perhaps, then, that the authors restrict themselves in their introduction to a single page, and do not discuss the possible use of their programs in a wider context. A more serious omission, however, is an account (rather than mere reference to) the machine-dependent features of their programs : a prospective user of these programs may be frustrated early if his installation or his version of BASIC do not allow the use of disc files. He also will be frustrated in any situation if he tries to obtain the confidence interval on the intercept with program RMA, because the line which computes this value is omitted from the listing. It should perhaps be 150 L E T D = SQR(TI*(1 - R ' R ) * ( 1 + S * S * ( N - 1)/(N*N*S))/S1)/(N- 1). A minor error in the same program is also the inclusion of an irrelevant remark in line 11. None of these few omissions or errors, however, obscure the fact that this collection of programs is potentially of great value not only in geological education, but also to the greater number of geologists who use numerical methods but feel the need for an alternative to large-scale computer power for small-scale jobs. S. Henley

Computer Unit Institute of Geological Sciences Exhibition Road London (England) Quantitative Geology Peter Fenner, editor Geological Society of America, Special Paper 146, 1972, 101 p., paperbound, $5.25 This publication contains seven papers and one abstract, based on talks given

284

Reviews

at a G.S.A. symposium held in Atlantic City during November 1969. Problems associated with quantification of geological information are discussed by W. C. Krumbein. In particular, he notes that the choice of operational definitions may have severe effects on the shape and characteristics of observed frequency distributions. This is an important phenomenon that could confuse the validation of theoretical models. The estimation of lithologic composition of sedimentary rocks, based on geophysical well logs, is a problem that has faced oil companies for some time. Using linear relationships between porosity and well-log response (either neutron, acoustic, or density) it is possible to determine uniquely the relative proportions of only two lithologies from a rock. A unique solution for cases with several lithologies can be obtained, however, if some independent function of the variables is being maximized or minimized. R. B. McCammon proposes that Shannon's entropy measure be used as the objective function. The problem then is reduced to nonlinear optimization, with linear constraints, for which a solution is obtained using Lagrangian multipliers. The author shows that the method works reasonably well on a 7-component Pennsylvanian cyclothem. This seems to be an important, albeit rather empirical, tool for oil exploration. Anybody who has measured stratigraphic sections, and attempted to determine areal or linear trends in isopach thicknesses, grain-size variation, and the like, has faced the problem of uncertainties due to measurement error. J. M. Dennison points out that isopach maps should somehow reflect levels of confidence in the data, and this is not possible without a knowledge of the measurement error in the data itself. A test for determining the least contour interval that can be detected with 95 percent certainty, borrowed from the biologists, can be applied (assuming known measurement error), and the author suggests that a variable contour interval be used, so that the uncertainty is reflected in the map. The resulting map is more difficult to read, but certainly contains more information. Using a similar line of reasoning, Kelley and McManus (1969) introduce the cost of sampling and measurement into this problem. They determine the sampling density required to minimize cost, yet achieve a known level of precision. G. S. Watson brings the skill and geologically unbiased eye of a professional statistician to review the methods of trend-surface analysis and spatial correlation as applied by geologists. Many people have been confused as to the relation between trend-surface analysis as developed in geology by W. C. Krumbein, J. W. Harbaugh, D. F. Merriam, and others, and kriging as developed by G. Matheron and his coworkers. Watson bridges this gap in a easily read style, and makes some interesting comments about types of stochastic models that may be interesting to develop in a geological context. R. A. Park and J. W. Wilkinson provide us with an abstract about their

Reviews

285

ecosystem modeling of Lake George. This is a project of considerable interest to those involved with the chemical and biological dynamics of lakes, and they have published several technical reports since I969, for example Park and Wilkinson (1971). The powerful techniques of optical processing are discussed by J. C. Davis and F. W. Preston. There is no doubt that photographic storage of pictorial information is more efficient in terms of space than magnetic tape storage of digitized data. Furthermore the spectral characteristics of such information can be obtained optically at a fraction of the cost required to analyze digital data using Fourier techniques. The authors discuss the application of optical techniques to the analysis of pore-grain geometry in sedimentary rocks. For several years, Louis Briggs and Henry Pollack at the University of Michigan ran a summer conference on computer techniques for petroleum geologists, at which Peter Buttner presented material on systems analysis and model building. Buttner now provides us with a summary of this material, and with some intriguing diagrams and suggested applications of systems techniques. For example, the application of queuing theory and linear programming to sedimentologic problems is briefly introduced. Unfortunately the treatment is too cryptic to appreciate the validity of the applications, and I fear that the average geological reader may be put off by the jargon. Let us hope that this material will be presented in greater depth elsewhere. Those who have listened to Prof. J. C. Griffiths give one of his stimulating and amusing lectures will enjoy his short article on cybernetics-mathematics interaction. Griffiths touches on problems of applying Markov chains in stratigraphy, the Goedel Incompleteness Theorem, the large number of possible alternatives due to the interactions in real-world systems, and the likely growth of geomathematics in the 1970s. Thought-provoking stuff. It is a pity that this Special Paper appeared three years after the conference at which the material was first presented. This review will appear in print five years after that initial presentation, and readers who now pick up Special Paper 146 may find it rather "old hat." Much of the material has appeared since 1969 in various forms elsewhere. Although the papers are generally well written and well edited, the overall impression is that the publication is "thin" on content. Nevertheless, geologists interested in mathematical applications should be aware of this Special Paper, which is likely to be in most geology libraries.

References Park, R. A., and Wilkinson,J. W., 1971, Lake George modellingphilosophy: Rensselaer PolytechnicInst., Intern. Biol. Prog., Rept. 71-19, 60 p.

286

Reviews

Kelley, J. C., and McManus, D. A., 1969, Optimizing sediment sampling plans: Marine Geology, v. 7, no. 5, p. 465-471. Graeme Bonham-Carter

Department of Geological Sciences University of Rochester Rochester, New York (USA)

Mathematical Models of Sedimentary Processes

D. F. Merriam, editor 3rd in a series, Computer Applications in the Earth Sciences Plenum Press, New York, 1972, 271 p. $t6.50 Modelling in geology has retained its popularity. Most likely this will continue so long as the mechanisms which underlie geologic processes remain inadequately understood. An exact quantitative description of past geologic events in other words is not imminent. One reason for this is an imperfect geologic record. This is particularly true in sedimentary geology where the stratigraphic record is filled with examples of unconformities, diastrophism, and periods of nondeposition. What makes it even more difficult is that the rock record is not always directly observable. It is not surprising therefore that many geologists have turned to modelling as a means of better understanding the past. The thirteen essays which comprise this book were part of a symposium held in Heidelberg during the International Sedimentological Congress meetings in 1971, cosponsored by the International Association of Mathematical Geology and, as one might suspect, focused on the subject, "Mathematical Models of Sedimentary Processes." More or less by design the essays are exploratory in their treatment of various topics related to mathematical modelling of sedimentary processes. Certainly none qualify as treatises; however, many hold promise of being a precursor to a future major work. There is for instance the paper by J. Jacod and P. Joathon in which a random-genetic model for stratigraphic succession is put forward. Its future value may well lie in the interesting blend of stratigraphy and probability which the authors have put together. There is another paper by G. de Marsily in a similar vein in which a mathematical model is used to predict groundwater conditions in a sedimentary basin in an area where data are lacking. It is for a worthy cause as well, for what is at stake is the future water supply of a wine growing region in France. More along the lines of sedimentary geology, K. I. Heiskanen in a paper

Reviews

287

makes use of a diffusion model to account for the size-frequency distributions of particles in sedimentation from turbulent flows. His paper may have important implications in reconstructing past depositional environments. In another paper by D. Marsal, a more or less quantitative theory is developed for the mechanism of the dissolution of grain aggregates in liquids. Marsal takes up a number of special examples in detail. There is a group of papers which treat various aspects of vertical successions. These include a paper by J. C. Davis and J. M. Cocke who propose a substitutability measure for interpreting complex lithotogic successions and a paper by W. A. Read and D. F. Merriam who give a simple quantitative technique for comparing cyclically deposited sedimentary successions. In another paper, W. Schwarzacher elegantly puts forward the argument that an understanding of the mechanism of bed formation is necessary before satisfactory probabilistic models of stratigraphic succession can be developed. A final group of papers deal mainly with statistical applications. These include a paper by G. F. Bonham-Carter who discusses optimization criteria for sampling in geology, a paper by F. Demirmen outlining a search procedure for reducing the complexity of petrographically derived data for the purpose of constructing facies models, a paper by R. A. Reyment in which canonical correlation is used to analyze the correlation structure of trace element data collected for deltaic sediments and a paper by W. C. Krumbein who considers the troublesome problem of the effect of point correlations on spatially distributed variables. The reader who makes his way through this book may well find that there are few if any final answers to problems considered by the different authors. For a book of this type however, that is as it should be. R. B. McCammon

Department of Geological Sciences University of Illinois at Chicago Circle Chicago, Illinois (USA)