Introduction

The contents of this chapter constitute a tool-kit for use in the subsequent chapters on data analysis. §3.2 deals with some basic mathematical methods for vectors and matrices; §3.3 and §3.4 are concerned with methods of data display, and qualitative (or descriptive) features of spherical data sets. In particular, the basic method we have adopted for displaying vectorial data, which may cover both hemispheres, is explained in §3.3.1. §3.5 describes some standard statistical methods for deciding whether a given random sample of observations is adequately fitted by some specified probability distribution, and whether two independent samples have been drawn from the same (unspecified) distribution; §3.6 describes the use of simulation as an aid in complicated analyses; §3.7 describes jackknife procedures and permutation tests; and §3.8 is a brief discourse on problems of data collection.

The mathematical results presented in §3.2 are purely for reference purposes, and no derivations are given; most, if not all, of the results are available in standard texts.

Mathematical methods for unit vectors and axes in three dimensions

Mean direction, resultant length and centre of mass

Consider a collection of points P1, …,Pn on the surface of the unit sphere centred at O, with Pi corresponding to a unit vector with polar coordinates (θi, φi) and direction cosines xi, = sin θi, cos φi, yi = sin θi, sin φi, zi, = cos θi, i = 1,…, n.

The purpose of the book

The analysis of data in the form of directions in space, or equivalently of positions of points on a spherical surface, is required in many contexts in the Earth Sciences, Astrophysics and other fields. While the contexts vary, the statistical methodology required is common to most of these data situations. Some of the methods date back to the beginning of the century, but the main developments have been from about 1950 onwards. A large body of results and techniques is now disseminated throughout the literature. This book aims to present a unified and up-to-date account of these methods for practical use.

It is directed to several categories of reader:

1. to the working scientist dealing with spherical data;

2. to undergraduate or graduate students whose taught courses or research require an understanding of aspects of spherical data analysis, for whom it would be a useful supporting text or working manual;

3. to statistical research workers, for reference with regard to current solved and unsolved problems in the field.

Because of the range of readership, priority has been given to providing a manual for the working scientist. Statistical notions are spelt out in some detail, whereas the statistical theory underlying the methods is, by and large, not included; for the statistician, references are given to this theory, and to related work. In particular, only one procedure is given for any specific problem. In some cases, the choice of procedure is dictated by certain optimality considerations, whereas in other cases a somewhat arbitrary choice has been made from several essentially equivalent procedures.

Introduction

Many different ways of representing a three-dimensional unit vector or axis have been developed over the centuries, due not only to the requirements of different disciplines (Astronomy, Geodesy, Geology, Geophysics, Mathematics, …) but also to diverse needs within a discipline: in Geology, for example, there appear to be five or six systems in current use. In this book, we shall use either polar coordinates or the corresponding direction cosines for all purposes of statistical analysis. The following sub-section (§2.2) defines several of the coordinate systems and gives the mathematical relationship of each to polar coordinates.

Later chapters of this book, concerned with statistical analysis, abound with words and phrases which have particular meanings in Statistics, and, possibly, rather different meanings in other areas. A good example of this is the word “sample”, which for our purposes is loosely taken to mean a collection of measurements of a particular characteristic, but which has a general scientific meaning of an observational or sampling unit (e.g. a drill-core specimen on which a single measurement may be made). §2.3 gives definitions of a number of such words and phrases.

Spherical coordinate systems

The type of data we shall be dealing with will be either directed lines or undirected lines. For the former, the measurements will be unit vectors, such as the direction of magnetisation of a rock specimen, or the direction of palaeocurrent flow. For the latter, which we shall term axes (cf. §2.3), the line measured might be the normal to a fracture plane, and so have no sense (direction) unless this is ascribed on some other basis.