Celebrating Frits Agterberg’s half-century of publication activity in geomathematics, this volume’s 28 timely papers, written by his friends and colleagues, treat a variety of subjects of current interest, many of them also studied by Frits, including: spatial analysis in mineral resource assessment, quantitative stratigraphy, nonlinear multifractal models, compositional data analysis, time series analysis, image analysis, and geostatistics. Professor Agterberg published his first paper as a graduate student in 1958 and has since produced (and continues to publish) a steady stream of research papers on a wide variety of subjects of interest to geomathematical practitioners. Most of the papers included here address methodology and feature practical case studies, so that the book likely has broad appeal to those interested in mathematical geosciences, both to academic researchers seeking a comprehensive overview and also to practitioners of geomathematical approaches in industry.

The authors introduce geomathematics as an active research area to a wider audience. Chapter 1 presents an introduction to the Earth as a system to apply scientific methods. Emphasis is laid on transfers from virtual models to reality and vice versa. In the second chapter geomathematics is introduced as a new scientific area which nevertheless has its roots in antiquity. The modern conception of geomathematics is outlined from different points of view and its challenging nature is described as well as its interdisciplinarity. Geomathematics is shown as the bridge between the real world and the virtual world. The complex mathematical tools are shown from a variety of fields necessary to tackle geoscientific problems in the mathematical language. Chapter 3 contains some exemplary applications as novel exploration methods. Particular importance is laid on the change of language when it comes to translate measurements to mathematical models. New solution methods like the multiscale mollifier technique are presented. Further applications discussed are aspects of reflection seismics. Chapter 4 is devoted to the short description of recent activities in geomathematics. The Appendix (Chapter 5) is devoted to the GEM – International Journal on Geomathematics founded ten years ago. Besides a detailed structural analysis of the editorial goals an index of all papers published in former issues is given.

Geomathematics provides a comprehensive summary of the mathematical principles behind key topics in geophysics and geodesy, covering the foundations of gravimetry, geomagnetics and seismology. Theorems and their proofs explain why physical realities in geoscience are the logical mathematical consequences of basic laws. The book also derives and analyzes the theory and numerical aspects of established systems of basis functions; and presents an algorithm for combining different types of trial functions. Topics cover inverse problems and their regularization, the Laplace/Poisson equation, boundary-value problems, foundations of potential theory, the Poisson integral formula, spherical harmonics, Legendre polynomials and functions, radial basis functions, the Biot-Savart law, decomposition theorems (orthogonal, Helmholtz, and Mie), basics of continuum mechanics, conservation laws, modelling of seismic waves, the Cauchy-Navier equation, seismic rays, and travel-time tomography. Each chapter ends with review questions, with solutions for instructors available online, providing a valuable reference for graduate students and researchers.

During the last three decades geosciences and geo-engineering were influenced by two essential scenarios: First, the technological progress has changed completely the observational and measurement techniques. Modern high speed computers and satellite based techniques are entering more and more all geodisciplines. Second, there is a growing public concern about the future of our planet, its climate, its environment, and about an expected shortage of natural resources. Obviously, both aspects, viz. efficient strategies of protection against threats of a changing Earth and the exceptional situation of getting terrestrial, airborne as well as spaceborne data of better and better quality explain the strong need of new mathematical structures, tools, and methods. Mathematics concerned with geoscientific problems, i.e., Geomathematics, is becoming increasingly important. The ‘Handbook Geomathematics’ as a central reference work in this area comprises the following scientific fields: (I) observational and measurement key technologies (II) modelling of the system Earth (geosphere, cryosphere, hydrosphere, atmosphere, biosphere) (III) analytic, algebraic, and operator-theoretic methods (IV) statistical and stochastic methods (V) computational and numerical analysis methods (VI) historical background and future perspectives.

This book provides a wealth of geomathematical case history studies performed by the author during his career at the Ministry of Natural Resources Canada, Geological Survey of Canada (NRCan-GSC). Several of the techniques newly developed by the author and colleagues that are described in this book have become widely adopted, not only for further research by geomathematical colleagues, but by government organizations and industry worldwide. These include Weights-of-Evidence modelling, mineral resource estimation technology, trend surface analysis, automatic stratigraphic correlation and nonlinear geochemical exploration methods. The author has developed maximum likelihood methodology and spline-fitting techniques for the construction of the international numerical geologic timescale. He has introduced the application of new theory of fractals and multi fractals in the geostatistical evaluation of regional mineral resources and ore reserves and to study the spatial distribution of metals in rocks. The book also contains sections deemed important by the author but that have not been widely adopted because they require further research. These include the geometry of preferred orientations of contours and edge effects on maps, time series analysis of Quaternary retreating ice sheet related sedimentary data, estimation of first and last appearances of fossil taxa from frequency distributions of their observed first and last occurrences, tectonic reactivation along pre-existing schistosity planes in fold belts, use of the grouped jackknife method for bias reduction in geometrical extrapolations and new applications of the theory of permanent, volume-independent frequency distributions.

Since founding at the 23rd International Geological Congress in Prague in 1968, the International Association for Mathematical Geology has organized sessions in conjunction with the Congress. The 27th IGC in Moscow was no exception and the IAMG again held sessions and assisted the Congress in organizing Section 20 -Mathematical Geology and Geological Information ( D. F. Merriam, D. A. Rodionov, and R. Sinding-Larsen, conveners). All together 128 abstracts were published in the technical proceedings. Several of the papers were published prior to the Congress, others were not available, and others deemed not appropriate for publication in this volume. This collection then contains those papers aVailable and representative of the sessions. The collection is truly international with contributions from Canada, China, France, Poland, the UK, USA, and USSR. They are representative of the state-of-the-art as of the early 1980s in a variety of fields. The application of geomathematics/geostatistics to geological problems has been hastened by the availability of computers. These papers reflect that orientation -most of the results would not have been possible without the use of computers. Most of the approaches utilize techniques readily aVailable and adapted to solving geological problems -simulation, image analysis, decision theory, fuzzy sets, etc. However, one area, that of geostatistiques which includes Kriging, has been designed especially for use by earth scientists of the French school to solve geological problems.