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Multiple-point geostatistics should be used when there is either too little or too much information available for other types of geostatistics.

First proposed in the early 1990s, the geostatistical algorithm known as multiple-point statistics (MPS) now enjoys widespread use, particularly in petroleum studies. It has become part of the toolkit that new practitioners are trained to use in several oil companies; it has been incorporated into commercial software; and research programs in many universities continue to tap into the central MPS idea of extracting statistical information directly from a training image. The inspiration for the development of a proof-of-concept MPS prototype code owes much to several different researchers and research programs in the late 1980s and early 1990s: the sequential algorithms pioneered at Stanford University, the work of Chris Farmer, then at UK Atomic Energy, and the growing use of outcrop studies by several oil companies. This largely accidental confluence of divergent theoretical perspectives, and of distinct practical workflows, serves as an example of how science often advances through the intersection of ideas that are not only disparate but even contradictory.

Predictive geometallurgy tries to optimize the mineral value chain based on a precise and quantitative understanding of: the geology and mineralogy of the ores, the minerals processing, and the economics of mineral commodities. This chapter describes the state of the art and the mathematical building blocks of a possible solution to this problem. This solution heavily relies on all classical fields of mathematical geosciences and geoinformatics, but requires new mathematical and computational developments. Geometallurgy can thus become a new defining challenge for mathematical geosciences, in the same fashion as geostatistics has been in the first 50 years of the IAMG.

Mathematical geosciences are now in an intelligent stage. The freshly new data environment enabled by the Semantic Web and Open Data poses both new challenges and opportunities for the conduction of geomathematical research. As an interdisciplinary domain, mathematical geosciences share many topics in common with data science. Facing the new data environment, will data science inject new blood into mathematical geosciences, and can data science benefit from the achievements and experiences of mathematical geosciences? This chapter presents a perspective on these questions and introduces a few recent case studies on data management and data analysis in the geosciences.

Georges Matheron and Jean Serra of the Centre of Mathematical Morphology, Fontainebleau founded Mathematical Morphology (MM). Since the birth of MM in the mid 1960s, its applications in a wide ranging disciplines have illustrated that intuitive researchers can find varied application-domains to extend the applications of MM. This chapter provides a concise review of application of Mathematical Morphology in Geosciences and Geographical Information Science (GISci). The motivation for this chapter stems from the fact that Mathematical Morphology is one of the better choices to deal with highly intertwined topics such as retrieval, analysis, reasoning, and simulation and modeling of terrestrial phenomena and processes. This chapter provides an illustrative review of various studies carried out by the author over a period of 25 years—related to applications of Mathematical Morphology and Fractal Geometry—in the contexts of Geosciences and Geographical Information Science (GISci). However, the reader is encouraged to refer to the cited publications to gather more details on the review provided in an abstract manner.

John Cubitt and Stephen Henley, with contributions from T. Victor (Vic) Loudon, EHT (Tim) Whitten, John Gower, Daniel (Dan) Merriam, Thomas (Tom) Jones, and Hannes Thiergärtner.

The transition over 70 years from qualitative rock description to attempted quantitative description of rocks and rock bodies (inverse modelling) and testing of process models with observation data (forward models) are outlined. Dramatic increases of readily measured variables, combined with almost unlimited computing power, yielded a plethora of varied inverse models, but limited attention has been given to critical sampling, variance, closure, ‘black swan’, and nonlinear issues; recent approaches to closure problems hold promise. Especially for plutonic rocks, paucity of quantitative process modelling left exciting forward-modelling opportunities neglected. Resulting challenges ahead are anticipated.

The author’s initial personal random contacts with pioneers in introducing mathematics and computers to geology in Russia, USA and France evolved thanks to the 23rd International Geological Congress and the foundation of the IAMG in Prague 1968. An incredibly large set of colleagues from all over the world have continuously contributed to a long series of regular international sessions at the Mining Příbram Symposia—a unique East–West gateway for the IAMG during the period 1968–1989. Very intensive work has been continuing until 2000 with several new peaks. The author has used many positive international organizational experiences from the work for the IAMG in developing geoethics, where many experts of mathematical geology have brought a considerable contribution to this new field.

This chapter provides a glimpse of the legacy of Professor Andrey Borisovich Vistelius, who served as the first President of the International Association for Mathematical Geoscientists (IAMG) during 1968–1972. Professor Andrey Borisovich Vistelius (1915–1995) was arguably the founder of the field of mathematical geology, and he was the first President of the International Association for Mathematical Geology. As a 1982 recipient of the President’s Prize (later renamed the Andrey Borisovich Vistelius Research Award) I consider it a great privilege to have been invited to contribute this chapter in his honour. The scientific heritage of Professor Vistelius is extremely rich. His active work on fundamental and applied problems of geology, and especially mathematical geology, continued to the last days of his life. He was responsible for more than 200 published works, each representing a significant contribution to science. His works cover a wide range of subjects, with contributions to the development of stratigraphy, mineralogy, petrography, petrology and geochemistry. The mathematical approach to geoscientific research, pioneered by Vistelius, has gained recognition worldwide. As applied in practice, these works also represent building blocks to more effective methods of search for minerals. There have been a number of publications about Vistelius, and in attempting to present a rounded view of his life and works, this chapter quotes from them extensively: particularly Dvali et al. (), Romanova and Sarmanov (), Dech and Glebovitsky (), Merriam (), Henley (), Dech and Henley (), and Whitten (). I also wish to acknowledge unpublished sources including Whitten, the late Merriam, Pshenichny, and Dech.

Classical mathematical geology is a branch of mathematical geosciences in which mathematical methods and models—not specifically developed for and not exclusive to specific geosciences—are applied to describe, to model and to analyse quantitatively geoscientific subjects and processes. It was the dominant approach in the 1960s to 1980s and it is still used today to solve numerous, mostly limited and less complex problems. The methods have been implemented in the form of algorithms in commercial software packages that are widely used in geological practice. Their application frequently assumes specific pre-conditions, which are often difficult, if not impossible, to verify. This situation can result in significantly spurious output and errors that are often not recognised (hidden errors). In this paper five case studies are used to demonstrate these errors. In particular, they demonstrate that small mistakes can lead to serious, but often unrecognised, misinterpretations. The main conclusion is that there is a need to improve education and training in classical mathematical geology especially for engineering sections of consulting firms, governmental agencies and individual consultants.