Mathematical Geology, Vol. 2, No. 4, 1970

Reviews Geological Data Processing: Using FORTRANIV by F. G. Smith, Professor of Geology, University of Toronto Harper & Row, New York, 1966 284 p., $14.00 The title is misleading; except for a few exercises with casual mention of grain sizes or melting temperatures of minerals, the book contains but one treatment of specific geological relevance, the computation of a norm. As the editor himself puts it " . . . it would not be much of a misnomer if the book were to be entitled simply Data ProcesshTg." Constant reference is made to FORTRAN; a thorough prior knowledge of the language is assumed. A short presentation of FORTRAN IV is the object of Chap. 12, but its careful reading would not in itself give a sufficient preparation to understand the rest of the book. Incidentally, one may wonder why this chapter was not placed at the beginning instead of at the end. The book, then, is intended to help the FORTRAN initiate who realizes that mastering a programming language, while making possible the use of mathematics, does not replace the knowledge of this science. The author presents in simple terms those parts of mathematics most likely to be of use to process data by FORTRANprograms; nowhere has any attempt been made at mathematical rigor in definitions. Rather, the concepts are presented intuitively, with FORTRANserving as reference. Beginning by symbolic logic and Boolean algebra, elements, expressions, Venn diagrams, are swiftly explained and their use illustrated by FORTRANstatements. The 3rd and 4th chapters, "Arithmetic" and "Algebra," present the binary and octal numbers, integer arithmetic, real numbers, BCD, and the use of the DO loop, all subjects being strictly oriented to computer use. In the next chapter, "Vectors and Matrices," the notions of unit, diagonal, triangular and symmetrical matrices are illustrated, and an attempt is made at explaining what the cofactors, the determinant and the inverse of a matrix amount to. The recurring chain of explanations is not completely avoided however, the notion of cofactors being explained by the notion of determinants, and vice versa. The clarity of the chapter would have been improved by a longer and more thorough treatment of what really is one of the most important stumbling blocks in programming for beginners. In particular, words of caution for the manipulation of subscripts with two- or three-dimensional matrices would have been welcome, because errors on these subscripts are one of the plagues of debugging. The chapters on "Calculus of Discrete and Continuous Functions," although 415

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short, are probably the best of the work. Useful rules for checking tables of data and for programming methods, such as Newton's iterative method, are clearly explained. The chapters dealing with "Probability" and "Statistics" are well exemplified with FORTRANstatements to compute such commodities as regression and correlation coeMcients, and contain useful reminders of what these and other statistics are to serve. These notions are applied in the next chapter to a practical discussion on "Error, Precision, and Accuracy," which will probably prove to be the most useful part of the book to most readers. It contains, among others, an excellent charge against the bad habit of rounding final results. A small chapter on flow diagrams introduces the 12th chapter which gives a brief introduction to FORTRAN. With so many excellent manuals, courses, and books on the subject presently in commerce, one may wonder if this chapter, which is necessarily incomplete, was really worth its space. This presentation of the grammar rules with examples adapted to the IBM 7090, not to the 360 series, is as good as the other presentations published, and does not avoid frequent use of "as explained below;" I have found this practice of referring to notions not yet explained in nearly all the texts and manuals on programming languages. They probably are unavoidable. But as a complementary reading for beginners, the chapter contains useful hints, and a list of well chosen exercises (no answers are provided). The book ends by different "Examples of Data Processing," with corresponding FORTRANstatements and flow diagrams; most examples refer to statistical problems; only one is purely geological. The examples are clearly commented upon, but it is unfortunate that the arithmetic IF is used whereas in many places a logical IF would have made the program much easier to read. The book could do more harm than good if it is considered as a mathematical textbook. But, taken for what it is, and is intended to be, the first pier of a bridge from FORTRAN to mathematics, it is undoubtedly a useful book. A. Hubaux

Cetis Euratom, Ispra (Italy) Scientific Method in Analysis of Sediments

by John C. Griffiths with 108 figures, 136 tables, vii+508 pages McGraw-Hill, Inc., New York, 1967, $17.50 The aim of the book, stated in the preface, i s " . . , to outline a [scientific] method, as an algorithm, along with a basis for critical appraisal and testing of the information gathered according to the plan, so that the problem-solving procedure can be improved and some workable solutions achieved." The algorithmic representation of method is given in Chap. 1, together with a discussion of the concepts' purpose, value, accuracy, and precision, particularly as applied to the evaluation of variation in properties of sedimentary rocks. The unique nature of the scientific method provides the main theme, as successive steps in the plan of experimental investigation are elaborated in the succeeding chapters: (1) Sampling design (Chap. 2) for target populations in sedimentary sequences is

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treated by means of a three-stage sampling model (Figs. 2-3, p. 20), use of which nonetheless necessitates preliminary experimentation. Difficulties inherent to random sampling of geological populations are indicated and tests for randomness given. The need for specific experimentation, directed towards solutions to geologic sampling problems, is stressed. (2) A model for the measurement of properties of sedimentary rocks (Chap. 3) entails initial characterization of the population (rock) as a function of five fundamental properties of elements (mineral grains)--types and proportions of elements, as well as their size, shape, orientation and packing. Considerations of interrelationships of the fundamental properties lead to the view that for adequate definition of a rock specimen each property of an aggregate should be evaluated simultaneously, prior to further investigation by multivariate analysis. The more complex, derived properties, such as porosity and permeability, usually may be evaluated only on the basis of the interdependent fundamental properties, considered together with variables introduced during experiments. (3) Measurement techniques and procedures, yielding mainly numerical data and applicable to both fundamental (Chaps. 4-10) and derived (Chap. 11) properties of sandstones, are described in 183 pages. Relatively simple procedures, involving direct measurement and giving large amounts of data, amenable to expression as summary statistics for frequency distributions are recommended. The need for a definite sampling plan, selected to fit the variability pattern of a given population, is repeatedly stressed. Measurement of bulk density, porosity, and permeability is only briefly discussed and a test, based on variation in mean value of fundamental properties from one sample to another, is proposed as a quantitative measure of layering (see Fig. 7.2, p. 149, and Fig. 11.3, p. 234). (4) Data processing and selection of summary statistics are treated in Chap. 12. Procedures for quantification of data and measurement of information content are discussed in terms of scale theory. Considerations of summary statistics as estimators of their corresponding parameters in terms of unbiasedness, consistency, efficiency, and sufficiency indicate that knowledge of the frequency function of a variate is essential and that parametric analysis is necessary for meaningful solutions. (5) Statisticalanalysis of data is covered in 222 pages (Chaps. 13-21) in the logical sequence : (a) types of model frequency distribution, their properties, and experimental situations giving rise to them (Chaps. 13 and 14); (b) model and actual relationships of mean and dispersion, application of the concept of control to detect systematic bias (Chap. 15); (c) types of significance test and their applications (Chaps. 16 and 17); (d) one-, two-, and three-way-classification analyses of variance (Chaps. 18, 19, and 20); (e) regression and correlation (Chap. 21). (6) The final chapter (Chap. 22) has to do with decision-makhTg aspects of hypothesis testing and the status of experimental inquiry in the geosciences.

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The importance of the two main sections of the book (3 and 5 above) lies in the fact that measurement procedures and methods of statistical analysis are critically discussed and the weight in geologic interpretation of alternative approaches given. Though emphasis is placed on parametric statistical analysis for reduction of data, reference also is made in several chapters to nonparametric statistical tests, for example, in Sees. 5.3, 15.4, 16.6, and 22.3. The list of references occupies 17 pages. Literature published up to 1964 is cited and mostly relates to geologic studies, but numerous source references for formal mathematical proofs are also given. The subject index is six pages long; unfortunately, there is no author index. The style is lucid and the text well organized, in accordance with the outline of scientific method given in the first chapter. Extensive use is made of figures and summary tables. Printing errors, which include the title to Chap. 21 and absence of headings for some of the early tables, are relatively scarce. This important book will be extensively used, not only as a reference text on both textural and compositional properties of the main classes of sandstones and statistical analysis of sedimentological data, but also as a systematic approach to experimental investigation in the geosciences generally. Frank Simpson

Province of Saskatchewan Subsurface GeologicalLaboratory Regina, Saskatchewan (Canada) Geological Survey Computer Contributions available free on application to the Chief, Computer Science Division U.S. Geological Survey Washington, D.C. 20242 The U.S. Geological Survey has started a series of open-file reports describing computing applications to problems of interest to earth scientists. The first five numbers, reviewed here, indicate that this series is similar in format to the Kansas Geological Survey's Computer Contribution series. Most reports contain a listing of a computer program, examples of input and output, a description of the program, and a brief introduction to the relevant mathematical method. One report contains the proceedings of a symposium meeting. Because of the heavy demand for the first five reports, future issues will be available from the Clearinghouse for Scientific and Technical Information, Department of Commerce, Springfield, Virginia, either as hard copy or in microfiche. Of the programs described so far, all have been written for IBM 360/65 computers, three of them in FORTRAN,one in r'L/l. The programs are available as card decks or in card-image format on magnetic tape. Broadly speaking, the subject matter treated so far covers applications to cartography, multivariate statistical procedures, and simulation modeling. These reports will be of interest to a wide variety of computer-oriented persons in the earth sciences, computer science, engineering, and other fields.

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Number 1. Weighted Triangulation Adjustment, by Walker L. Anderson, 1969. Program Number W8250, written in FORTRAN IV (H) for IBM 360/65 This program cleans up geodetic survey measurements obtained in the field, to make them internally consistent. Given a few 'fixed' stations whose map positions are known exactly, and 'adjustable' stations whose positions are estimated approximately, observed distances and directions between stations are used to compute new positions for the adjustable stations using a least squares method. The program is long, nearly 1500 cards, appears to be well documented with comment cards, and contains a number of built-in error checks. The NAMELISTstatement is used in the main program, which may not be available on some nonlBM compilers.

Number 2. Perspective Center Determination, by John D. McLaurin, May t969. Program Number W5344, written in FORTRAN IV (G) for IBM 360/65 From the introduction to this paper: "Certain methods of independent-model aerotriangulation . . . require the coordinates of the perspective center of each projector so that the models can be joined in a strip. This documentation describes a computer program for determining the three-dimensional coordinates of these perspective centers." The mathematical method for this photogrammetric application is described in some detail. The program is compact, less than 300 statements long, and is well documented with flow charts, detailed input instructions, and a sample of output.

Number 3. Nonconstant Variance Regression Analysis Program, by Marshall Strong Hellman, 1970. Program Number W8251 written in FORTRAN IV (H) for IBM 360/65 Classical multivariate regression models assume that the standard deviations of the dependent variable are constant for each independent variable, or that they differ from one another by known weight factors. This program makes the less demanding assumption that the standard deviations of the independent variables are proportional to a function of the expected value o f the dependent variable. The mathematical theory and method of calculation is described in some detail. The input and output of the program is thoroughly documented, with a long list of diagnostic messages that may be generated. The program itself is long, containing nearly 1500 statements, distributed between a long main program and 20 subroutines. There are no comment cards, so the program is rather difficult to dissect. It would have been helpful to see a trial run using a geologic example to illustrate better the use of what appears to be a valuable program.

Number 4. Hot Pipe, by Patrick C. Doherty, 1970. Program Number C102, written in PL/I for IBM 360/65 This report describes a program that was written in response to a request for information on the possible effects that construction of the proposed pipeline across Alaska would have on the natural environment. The program computes the temperature

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history of a cross section of permafrost in which an oil pipe, whose surface is maintained at 80°C, is buried in a 10-ft-deep trench. The surface of the permafrost is subject to seasonal variation in temperature, and the thermal properties of the frozen and thawed material are treated, as well as the latent heat of melting. Finite-difference methods are used to solve the equation of heat conduction, and output is in the form of printer plots of isothermal lines on vertical cross sections of permafrost in the region of the pipe. This is an interesting application of a deterministic simulation model to a problem in environmental analysis. The numerical method is clearly described and the input and output well documented. The program contains less than 750 statements, interspersed with sufficient comments to aid program dissection.

Number 5. Proceedings of the Symposium on Map and Chart Digitizing, sponsored by Topographic Division, U.S.G.S., and Environmental Cartography Unit, Royal College of Art, London, October 7-8, 1969 This appears to be a transcript of the oral presentations and discussions of a symposium held to exchange information on methods and equipment for converting linear and point data from graphic to digital form. The document may be useful for those seeking the technical state-of-the-art as of 1969. It is not recommended for general reading, however, as it is long (81 pages of small type), poorly organized, and verbose, as one would expect of a transcript of oral discussions. Graeme Bonham-Carter

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