"The book is an excellent resource for students, young scientists, practicing geologists, geophysicists, and geoscientists who want to better understand mathematical methods. ... The handbook is a comprehensive reference that should have a place on the bookshelves of graduate students and researchers in the field of geosciences who are interested in using mathematical methods to analyze data." (Yuriy Kostyuchenko, Mohamed Abioui and Andrea Di Cencio, Mathematical Geosciences, Vol. 52, 2020)

Foreword

Preface

Introduction

B. S. Daya Sagar, Qiuming Cheng, Frits Agterberg

Part I Theory

1. Kriging, Splines, Conditional Simulation, Bayesian In-version and Ensemble Kalman Filtering Olivier Dubrule

1.1 Introduction

1.2 Deterministic Aspects of Geostatistics

1.3 Stochastic Aspects of Geostatistics: Conditional Simulation

1.4 Geostatistical Inversion of Seismic Data

1.5 Kalman Filtering and Ensemble Kalman Filtering

1.6 Beyond the Formal Relationship between Geostatistics and Bayes

1.7 Conclusion

References

2. A Statistical Commentary on Mineral Prospectivity analysis Adrian Baddeley

2.1 Introduction

2.2 Example Data

2.3 Logistic Regression

2.4 Poisson Point Process Models

2.5 Monotone Regression

2.6 Nonparametric Curve Estimation

2.7 ROC curves

2.8 Recursive Partitioning

References

3. Testing joint conditional independence of categorical random variables with a standard log-likelihood ratio test Helmut Schaeben

3.1 Introduction

3.2 From Contingency Tables to Log-Linear Models

3.3 Independence, Conditional Independence of Random variables

3.4 Logistic Regression, and its Special Case of Weightsof Evidence

3.5 Hammersley-Clifford Theorem

3.6 Testing Joint Conditional Independence of Categorical Random Variables

3.7 Conditional Distribution, Logistic Regression

3.8 Practical Applications

3.9 Discussion and Conclusions

References

4. Modelling Compositional Data. The Sample Space Ap-proach Juan José Egozcue and Vera Pawlowsky-Glahn

4.1 Introduction

4.2 Scale Invariance, Key Principle of Compositions

4.3 The Simplex as Sample Space of Compositions

4.4 Perturbation, a Natural Shift Operation on Compositions

4.5 Conditions on Metrics for Compositions

4.6 Consequences of the Aitchison Geometry in the Sample Space of Compositional Data

4.7 Conclusions

References

5. Properties of Sums of Geological Random Variables

G.M. Kaufman

5.1 Introduction

5.2 Preliminaries

5.3 Thumbnail Case Studies

References

6. A Statistical Analysis of the Jacobian in Retrievals of Satellite Data Noel Cressie

6.1 Introduction

6.2 A Statistical Framework for Satellite Retrievals

6.3 The Jacobian Matrix and its Unit-Free Version

6.4 Statistical Significance Filter

6.5 ACOS Retrievals of the Atmospheric State from Japan’s GOSAT Satellite

6.6 Discussion

References

7. All Realizations All the Time

Clayton V. Deutsch

7.1 Introduction

7.2 Simulation

7.3 Decision Making

7.4 Geostatistical Simulation

7.5 Resource Decision Making

7.6 Alternatives to All Realizations

7.7 Concluding Remarks

References

8. Binary Coefficients Redux Michael E. Hohn

8.1 Introduction

8.2 Empirical Comparisons and a Taxonomy

8.3 Effects of Rare and Endemic Taxa

8.4 Adjusting for Poor Sampling

8.5 Metric? Euclidean?

8.6 From Expected Values to Null Association

8.7 Illustrative Example

8.8 Discussion and Conclusions

8.9 Summary

References

9. Tracking Plurigaussian Simulations

M. Armstrong, A. Mondaini and S. Camargo

9.1 Introduction

9.2 Review of Complex Networks

9.3 Network Analysis of Google Citations of Plurigaus sian Simulations

9.4 Diffusion of the New Method into Industry

9.5 Conclusions and Perspectives for Future Work

References

10. Mathematical Geosciences: Local Singularity Analysis of Nonlinear Earth Processes and Extreme Geo-Events Qiuming Cheng

10.1 Introduction

10.2 What is Mathematical Geosciences or Geomathemat ics?

10.3 What contributions has MG made to geosciences?

10.4 Frontiers of Earth science and opportunity of MG

10.5 Fractal density and singularity analysis of nonlinear geo-processes and extreme geo-events

10.6 Fractal Integral and fractal differential operations of nonlinear function

10.7 Earth dynamics processes and extreme events

10.8 Fractal density of continent rheology in phase transition zones and association with earthquakes

10.9 Discussion and Conclusions

References

Part II General Applications

11. Electrofacies in Reservoir Characterization

John Davis

11.1 Introduction

11.2 The Amal Field of Libya

11.3 Electrofacies Analysis

11.4 What Do Amal Electrofacies Mean?

11.5 Conclusions

References

12. Forecast of Shoreline Variations by Means of Median Sets Jean Serra

12.1 Three problems, One Theoretical Tool

12.2 Median Set

12.3 Median and Average for Non-Ordered Sets

12.4 Extrapolations via the Quench Function

12.5 Accretion and Homotopy

12.6 Conclusion

References

13. An Introduction to the Spatio-Temporal Analysis of Sat-ellite Remote Sensing Data for Geostatisticians A. F. Militino, M. D. Ugarte, and U. P´erez-Goya

13.1 Introduction

13.2 Satellite Images

13.3 Derived Variables from Remote Sensing Data

13.4 Pre-processing

13.5 Spatial Interpolation

13.6 Spatio-Temporal Interpolation

13.7 Conclusions

References

14. Flint drinking water crisis: a first attempt to model geo-statistically the space-time distribution of water lead levels Pierre Goovaerts

14.1 Introduction

14.2 Materials and Methods

14.3 Results and Discussion

14.4 Conclusions

References

15. Statistical Parametric Mapping for Geoscience Applica-tions Sean A. McKenna

15.1 Introduction

15.2 Anomaly Detection with Statistical Parametric Mapping

15.3 Example Problems

15.4 Summary

References

16. Water chemistry: are new challenges possible from CoDA (Compositional Data Analysis) point of view? Antonella Buccianti

16.1 Water Chemistry Data as Compositional Data

16.2 Isometric-Log Ratio Transformation: Is this the Key to Decipher the Dynamics of Geochemical Systems?

16.3 Improving CoDA-Dendrogram: Checking for Vari ability, Resilience and Stability

16.4 Conclusions

References

17. Analysis of the United States Portion of the North Ameri-can Soil Geochemical Landscapes Project – A Composi-tional Framework Approach

E. C. Grunsky, L. J. Drew, and D. B. Smith

17.1 Introduction

17.2 Methods

17.3 Results

17.4 Discussion

17.5 Concluding Remarks

References

Part III Exploration and Resource Estimation

18. Quantifying the Impacts of Uncertainty

Peter Dowd

18.1 Introduction

18.2 Sources of In-Situ Uncertainty

18.3 Transfer Uncertainty

18.4 Consequences of In-Situ Uncertainty

18.5 Quantifying Epistemic Uncertainty

18.6 Quantifying the Effects of Transfer Uncertainty

18.7 Conclusion

References

19. Advances in Sensitivity Analysis of Uncertainty due to Sampling Density for Spatially Correlated Attributes Ricardo A. Olea

19.1 Introduction

19.2 Data

19.3 Traditional Uncertainty Assessment

19.4 Kriging

19.5 Stochastic Simulation

19.6 Validation

19.7 Conclusions

References

20. Predicting Molybdenum Deposit Growth

John H. Schuenemeyer, Lawrence J. Drew and James D. Bliss

20.1 Introduction

20.2 Cutoff Grade as a Function of Deposit Grade

20.3 Deposit Growth as a Function of Cutoff Grade

20.4 An Example

20.5 Conclusions

References

21. General Framework of Quantitative Target Selections Guocheng Pan

21.1 Introduction

21.2 Randomness of Mineral Endowment

21.3 Fundamental Geo-Process Relations

21.4 Scarceness, Rareness, and Exceptionalness

21.5 Intrinsic Geological Unit

21.6 Economic Truncation and Translation

21.7 Information Synthesis

21.8 Prediction with Dynamic Control Samples

References

22. Solving the Wrong Resource Assessment Problems Pre-cisely Donald A. Singer

22.1 Introduction

22.2 Target Population

22.3 Examples of Mismatches in Assessments

22.4 How to Correct Type III Errors

22.5 Conclusions

References

23. Two ideas for analysis of multivariate geochemical survey data: proximity regression and principal component re-siduals

G.F. Bonham-Carter and E. C. Grunsky

23.1 Introduction

23.2 Method 1: Direct Prediction of Spatial Proximity

23.3 Method 2: Principal Component Residuals

23.4 Conclusions

References

24. Mathematical minerals: A history of petrophysical pe-trography John H. Doveton

24.1 Pioneering Computer Methods

24.2 Mineralogy of Underdetermined Systems

24.3 Mineralogy of Overdetermined Systems

24.4 Optimization Methods

24.5 Clay Component Estimation

24.6 Normative Estimation by Geochemical Logs

24.7 Conclusion

References

25. Geostatistics for Seismic Characterization of Oil Reser-voirs Amílcar Soares and Leonado Azevedo

25.1 Integration of Geophysical Data for Reservoir Modeling and Characterization

25.2 Iterative Geostatistical Seismic Inversion Methodologies

25.3 Trace-by-Trace Geostatistical Seismic Inversion

25.4 Global Geostatistical Seismic Inversion Methodologies

25.5 Uncertainty and Risk Assessment at early stages of exploration

25.6 Final Remarks

References

26. Statistical Modeling of Regional and Worldwide Size-Frequency Distributions of Metal Deposits Frits Agterberg

26.1 Introduction

26.2 Modified Version of the Model of de Wijs Applied to Worldwide Metal Deposits

26.3 Theory and Applications of the Pareto-Lognormal Model

26.4 Upper Tail Pareto Distribution and its Connection to the Basic Lognormal Distribution

26.5 Prediction of Future Copper Resources

26.6 Concluding Remarks

References

Part IV Reviews

27. Bayesianism in the Geosciences

Jef Caers

27.1 Introduction

27.2 A Historical P

27.3 Science as Knowledge Derived from Facts, Data or Experience

27.4 The Role of Experiments – Data

27.5 Induction vs Deduction

27.6 Falsificationism

27.7 Paradigms

27.8 Bayesianism

27.9 Bayesianism for Subsurface Systems

27.10 Summary

References

28. Geological Objects and Physical Parameter Fields in the Subsurface: A Review Guillaume Caumon

28.1 Introduction

28.2 Motivations for Explicit Geological Parameterizations

28.3 Parameterizations for Physical Models

28.4 Geological Parameterizations

28.5 Conclusions and Challenges

References

29. Fifty Years of Kriging

Jean-Paul Chilès and Nicolas Desassis

29.1 Introduction

29.2 The Origins of Kriging

29.3 Development and Maturity: Trend, Neighborhood Selection

29.4 Iterative Use of Kriging to Handle Inequality Data

29.5 Nonstationary Covariance

29.6 Kriging for Large Data Sets

29.7 Iterative Algorithms for Solving the Kriging System

29.8 Conclusion

References

30. Multiple Point Statistics: A Review

Pejman Tahmasebi

30.1 Introduction

30.2 Two-Point based Stochastic Simulation

30.3 Multiple Point Geostatistics (MPS)

30.4 Simulation Path

30.5 Current Multiple Point Geostatistical Algorithms

30.6 Current Challenges

References

31. When Should We Use Multiple-Point Geostatistics?

Gregoire Mariethoz

31.1 Under-Informed vs Over-Informed Models

31.2 MPS vs Covariance-Based Geostatistics

31.3 Examples for which MPS Works W

31.4 Conclusion

References

32. The Origins of the Multiple-Point Statistics (MPS) Algo-rithm R. Mohan Srivastava

32.1 Introduction

32.2 1970s

32.3 1980s

32.4 1990s

32.5 Concluding Thoughts

References

33. Predictive Geometallurgy: An Interdisciplinary Key Challenge for Mathematical Geosciences? K.G. van den Boogaart and R. Tolosana-Delgado

33.1 Introduction

33.2 Process Modelling

33.3 Ore Characterisation

33.4 Orebody Modelling

33.5 Decision Making

33.6 Conclusions References

34. Data Science for Geoscience: Leveraging Mathematical Geosciences with Semantics and Open Data Xiaogang Ma

34.1 Introduction

34.2 The Intelligent Stage of Mathematical Geosciences

34.3 Case Studies of Data Science in Geoscience

34.4 Concluding Remarks

References

35. Mathematical Morphology in Geosciences and GISci: An Illustrative Review B. S. Daya Sagar

35.1 Introduction

35.2 Terrestrial Pattern Retrieval

35.3 Terrestrial Pattern Analysis

35.4 Geomorphologic Modeling and Simulation

35.5 Geospatial Computing and Visualization

35.6 Conclusions

References

Part V Reminiscences

36. IAMG: Recollections from the Early Years

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

36.1 The Birth of Mathematical Geology and the Origins of the IAMG

36.2 The Role of the Kansas Geological Survey in the origins of the IAMG

36.3 Name and Establishment of the Society

36.4 Foundation of IAMG Publications

36.5 Prague

36.6 Subsequent Events following Prague –

36.7 The Looming Gap

References

37. Forward and Inverse Models over 70 Years

E. H. Timothy Whitten

37.1 Birth of IAMG in 1968

37.2 In the Beginning (one pre-1968 experience)

37.3 Inverse and Forward Geology Problems

37.4 Forward Models in Earth Sciences

37.5 Inverse Models in Earth Sciences

37.6 The Samples Analysed

37.7 The Black Swan Effect

37.8 Concluding Thoughts

References

38. From individual personal contacts 1962–1968 to my 50 years of service Václav Němec

38.1 Introduction

38.2 IAMG Foundation (Prague 1968)

38.3 Activities for the IAMG 1968 – 1993

38.4 Příbram – East - West Gate near the Iron Curtain

38.5 My own professional work

38.6 Two Separate Silver Anniversary Meetings of Mathematical Geologists in Prague (1993)

38.7 From the Silver to the Golden IAMG Jubilee

38.9 Conclusion

References

39. Andrey Borisovich Vistelius

Stephen Henley

39.1 Background

39.2 Scientific Achievements and Insights

39.3 The International Association for Mathematical Geology

39.4 The "Father of Mathematical Geology"?

39.5 Legacy

References

40. Fifty Years’ Experience with Hidden Errors in Applying Classic Mathematical Geology Hannes Thiergärtner

40.1 Introduction and Definitions

40.2 Hidden errors and Case Study Examples

40.3 Conclusion and Suggestions

References

41. Mathematical Geology by Example: Teaching and Learning Perspectives James R. Carr

41.1 Introduction

41.2 Multivariate Analysis of Geochemical Data

41.3 Geostatistics and its Myriad Parameters

41.4 The Variogram as a Stand-Alone Data Analytical Tool

References

42. Linear Unmixing in the Geologic Sciences: More Than a Half of Century of Progress William E. Full

42.1 Introduction

42.2 History of Constant Sum HVA

42.3 Non-Constant Sum Data and Algorithms

42.4 Summary

References

43. Pearce Element Ratio Diagrams and Cumulate Rocks

James Nicholls

43.1 Introduction

43.2 Outline of a Cumulate Rock Paradigm

43.3 Pearce Element Ratio Patterns for Cumulate Rocks

43.4 Compositions of Units of the Skaergaard Intrusion

43.5 Melts of the Skaergaard Intrusion

43.6 Pearce Element Ratios, Cumulate Rocks, and September 11

References

44. Reflections on the Name of IAMG and of the Journal

Donald E. Myers

45. Origin and Early Development of the IAMG

Frits Agterberg

45.1 Introduction

45.2 Pioneers of Mathematical Geology

45.3 Inputs from Mathematical Statisticians

45.4 Concluding Remarks

References

B.S. Daya Sagar is a full Professor at the Systems Science and Informatics Unit (SSIU) at the Indian Statistical Institute in Bangalore, India. Dr. Sagar received his MSc and Ph.D. degrees in geoengineering and remote sensing from the Faculty of Engineering, Andhra University, Visakhapatnam, India, in 1991 and 1994 respectively. He is also first Head of the SSIU. Earlier, he worked at the College of Engineering, Andhra University, and Centre for Remote Imaging Sensing and Processing (CRISP), at the The National University of Singapore in various positions during 1992-2001. He served as Associate Professor and Researcher at the Faculty of Engineering & Technology (FET), Multimedia University, Malaysia, during 2001-2007. Since 2017, he has been a Visiting Professor at the University of Trento, Italy. His research interests include mathematical morphology, GISci, digital image processing, fractals and multifractals, their applications in extraction, analyses, and modelling of geophysical patterns. He has published over 85 papers in scientific journals, and has authored and/or guest edited 11 books and/or special theme issues for journals. He recently authored a book entitled "Mathematical Morphology in Geomorphology and GISci". He recently co-edited two special issues on "Filtering and Segmentation with Mathematical Morphology" for IEEE Journal of Selected Topics in Signal Processing and "Applied Earth Observation and Remote Sensing in India" for IEEE Journal of Selected Topics in Applied Earth Observation and Remote Sensing. He is an elected Fellow of Royal Geographical Society (1999), Indian Geophysical Union (2011), and was a member of the New York Academy of Science during 1995-1996. He received the Dr. Balakrishna Memorial Award from the Andhra Pradesh Academy of Sciences in 1995, the Krishnan Gold Medal from the Indian Geophysical Union in 2002, and the "Georges Matheron Award-2011 (with Lecturership)" of the International Association for Mathematical Geosciences. He is the Founding Chairman of Bangalore Section IEEE GRSS Chapter. He is on the Editorial Boards of Computers & Geosciences, and Frontiers: Environmental Informatics.

Qiuming Cheng did his Ph.D. in Earth Science under supervision of Dr. Frits Agterberg at the University of Ottawa in 1994. Dr. Cheng spent a year at the Geological Survey of Canada as a PDF under the supervision of Dr. Graeme Bonham-Carter, and soon became a faculty member at York University, Toronto, Canada in 1995 with cross appointments in the Department of Earth and Space Science and Engineering and the Department of Geography. He was promoted to associate professor in 1997 and to full professor in 2002. He was awarded a Changjiang Scholar Professorship in China by the China Ministry of Education where he has set up and leads the State Key Lab of Geological Processes and Mineral Resources (GPMR) located on both campuses of China University of Geosciences in Beijing and Wuhan. Currently, he holds a Thousand Talent National Special Professorship of China, serving as the founding director of the GPMR lab. Dr. Cheng has specialized in mathematical geosciences with a research focus on nonlinear mathematical modelling of earth processes and geoinformatics techniques for prediction of mineral resources. He has authored and coauthored more than 300 research articles. He has been awarded several prestigious awards including the Krumbein Medal, the highest award by the International Association for Mathematical Geosciences (IAMG). Dr. Cheng was an elected President of the International Association for Mathematical Geosciences (IAMG) during 2012-16. He is the president of International Union of Geological Sciences (IUGS) for the period between 2016 and 2020. Dr. Cheng is an international leader in the application of nonlinear mathematics and geoinformatics to the analysis, modelling and prediction of a wide range of geological processes and mineral resources quantitative assessment. Dr. Cheng’s primary research interest involves the interdisciplinary study of non-linear properties of the Earth’s systems, as well as quantitative assessment and prediction of natural resources and environmental impacts. His research on fractal density & local singularity analysis theory and geomathematical models has made a major impact in several geoscientific disciplines, including those concerned with ocean ridge heat flow, magmatic flare-up during continent crustal growth and formation of supercontinents, earthquakes, floods, hydrothermal mineralization, and prediction of deeply buried mineral deposits.

Frits Agterberg is a Dutch-born Canadian Mathematical Geologist who served at the Geological Survey of Canada in Ottawa. He attended Utrecht University in The Netherlands from 1954 to 1961. With other founding members, he was instrumental in establishing the International Association for Mathematical Geosciences (IAMG) in 1968. He received the IAMG's William Christian Krumbein Medal in 1978 and he was IAMG Distinguished Lecturer in 2004. In 2017 he was conferred with the Honorary Membership of the IAMG. He has authored or coauthored over 250 scientific papers and five books. He has served the IAMG in many ways, including being its President from 2004 to 2008. After defending his doctoral thesis on structural geology of the Italian Alps at Utrecht University and a one-year fellowship at the University of Wisconsin in Madison, he became "petrological statistician" in his first job at the Geological Survey of Canada (GSC) in 1962. He was asked to create the GSC Geomathematics Section in 1971. He retired from the GSC in 1996 but still has an office at their Ottawa headquarters. In 1968, he became associated with the University of Ottawa where he taught a "Statistics in geology" course for 25 years and has supervised six geomathematical PhD students. From 1978 to 1989, he directed the Quantitative Stratigraphy Project of the International Geological Correlation Program. From 1981 to 2001, Dr. Agterberg was a correspondent of the Royal Netherlands Academy of Arts and Sciences. During the past 20 years, primarily in collaboration with Qiuming Cheng, his colleagues and students at the China University of Geosciences in Wuhan and Beijing and at York University, Toronto, he has worked on applications of multifractals to study the spatial distribution of metals in rocks and ore bodies.

This Open Access handbook published at the IAMG's 50th anniversary, presents a compilation of invited path-breaking research contributions by award-winning geoscientists who have been instrumental in shaping the IAMG. It contains 45 chapters that are categorized broadly into five parts (i) theory, (ii) general applications, (iii) exploration and resource estimation, (iv) reviews, and (v) reminiscences covering related topics like mathematical geosciences, mathematical morphology, geostatistics, fractals and multifractals, spatial statistics, multipoint geostatistics, compositional data analysis, informatics, geocomputation, numerical methods, and chaos theory in the geosciences.

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