V

PREFACE

Most scientific data are of a multivariate nature, and petrology is no exception. The more we delve into the details of petrological processes the more it becomes obvious that there are many factors involved which have not been taken into account. This is particularly true of petrology and geochemistry as more and more elements are commonly being analysed on a routine basis - aided and abetted by sophisticated analytical equipment which makes the task that much easier. As a result there has been an enormous explosion of petrological data over the last two decades, in both the number of analyses and in the number of elements determined.

In order to process this data efficiently, petrologists will have to become more numerate. Just as they had to master the complicated electronic equipment with the advent of, for example, the electron microprobe in the early 60 Ts, so they will have to learn statistical and computing techniques in the next decade, if they wish to keep in the forefront of research.

The main objective of this book, therefore, is to try to present under one cover some of the techniques that are likely to be useful to the average petrologist, together with some of the thoughts that have occurred during many years of research and teaching in the field of what I have loosely called "numerical petrology". Although most of the contents of the book can be found in a variety of statistical publications, some is new material. In particular the sections on generalized petrological mixing models and their errors in Chapter 6; on dealing with the problems of closed data and singular matrices in multiple discriminant analysis and classification procedure in Chapters 9 and 10, respectively; and on the propagation of errors in mineral and normative recalculations in Chapter 11. Several of the Tables in the Appendix are also new and have been compiled with the numerate petrologist in mind. In particular the theoretical chemical compositions of most of the common rock-forming minerals given in Table 13 which, among other things, can be useful for mixing model calculations, and Table 14 which gives the common silicates in sorted order of their chemical composition.

As I have assumed that the reader has a background in petrology rather than statistics and mathematics, I have included many diagrams and a large number of worked numerical examples all from the field of petrology, as I believe that this is what petrologists need and require to become familiar with some of the techniques that they will have to learn. For similar reasons, most of the complicated mathematics, that is not absolutely essential to the understanding; of the problem in hand, has been omitted and references given to where it can be obtained, if required. This includes the theory of matrices which, of course, is given in innumerable texts.

Throughout the book much emphasis has been placed upon the

VI

interpretation of the results, as most petrologists do not write their own computer programs to process their data, but rely upon other peoples. It is, however, essential that they have a clear understanding of how their results were obtained, so that they can interpret them correctly. While this does not mean that all petrologists must be fully conversant with computer programming, it is highly recommended that they have a working knowledge of it so that, at least, they can make a few minor modifications to existing programs. It is then but a short step to writing a program of their own!

The reader may also be interested in the production of this book, as without computers parts of it would never have been produced. The book was typed directly into a word processor, using the WORD 11 system on the University of Melbourne PDP 11/70. All the data for the numerical examples and Tables were prepared in the required format on the University of Melbourne CDC Cyber 73, transferred to the PDP 11/70 and put directly into WORD 11, so that nearly all of the Tables are "untouched by human hand", so to speak. This greatly minimized the problem of typographical errors in the Tables, which are always difficult to detect. However, as a result of the calculations being performed with far more significant figures than are printed, some of the tabulated data will not reproduce exactly the results shown, due to rounding-off errors. The photo-ready text was then printed on a Diablo 630 printer, using a program that enables up to four different print-wheels to be used for each page of text. However, due to the lack of print-wheels with suitable typefaces, the Greek letters, bold-faced characters and summation signs had to be inserted by hand. This is also why the super- and sub-scripted numerals are the same size as the normal numerals.

Finally, I would like to extend my sincere thanks to all those who have assisted and put up with me during the production of this book. In particular to my wife for constant encouragement; to Carol Sugden for secretarial assistance and advice; to Don Campbell and Mary Hutchison for drafting the diagrams; to Margaret Mitchell and Peter Hylands for invaluable advice on editorial and artistic matters; to the staff of the Computer Centre, especially Geoffrey Hudson and Jefferey McDonell, for much help in pushing WORD 11 to its absolute limits - and at times beyond; to F.H.C. Marriott for kind permission to use one of his diagrams, which is presented in a slightly modified form as Fig. 10.2; and to Felix Chayes, Aldo Cundari, Max Hey, Geoffrey Hill, Mike Horder, Bob Howie, Al Miesch, Felix Mutschler and Evan Williams for critically reviewing various parts of the book - their suggestions were most valuable. However, I must apologize to one or two of the reviewers for not incorporating all of their suggestions, as there are a few points on which we must agree to differ.

Melbourne, Australia. April, 1982

R.W. LE MAITRE