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

When this handbook is published, the International Association for Mathematical Geosciences (IAMG) is celebrating its 50th anniversary. Mathematical geosciences as a scientific discipline has become mature after half a century of development since the IAMG was established in 1968 at the 23rd International Geological Congress (IGC) in Prague. It had grown from mathematical geology to mathematical geosciences by the time its name was changed at the 32th IGC held in Oslo in 2008. Not only has the subject been accepted widely within the geoscience community but the association has also been recognized for its reputation and significant influence on the earth sciences in general. IAMG has affiliations with several major geoscience organizations including the International Union of Geological Sciences (IUGS), International Statistical Institute (ISI), and the International Union of Geodesy and Geophysics (IUGG). Diverse earth science topics have been published in IAMG conference proceedings and the IAMG journals (Mathematical Geosciences, Computers & Geosciences and Natural Resources Research). However, we have to realize that as a relatively young discipline, MG still has not been very widely accepted and is often ignored by main stream geoscientists. While several definitions and terminologies were proposed to describe mathematical geology, there have been few attempts to define mathematical geosciences. For example, mathematical geosciences have often simply been referred as applications of mathematical and statistical methods for the analysis of geological (earth science) data and the development of quantitative predictive models (Howarth 2017). The mission of the IAMG as shown on the IAMG website was defined as promoting the development and application of mathematics, statistics and informatics in the geosciences. Whether MG should be defined as a formal discipline of science or simply as applications of mathematics in the geosciences is a fundamental question with critical impact on the development of the subject. In this chapter, I will review the status of the discipline and suggests a new definition for MG followed by examples to demonstrate what contributions of MG have been made to earth science and what the current developments in the field are. For the first part I will elaborate on MG on the basis of literature review and for the second on my own research in nonlinear MG as an example of a new field of MG.

One of the original definitions of mathematical geology was given by Vistelius (1962) and used in the name of the association: International Association for Mathematical Geology (IAMG) when it was first established in 1968. Geostatistics is one of the successful fields of IAMG, which originally was developed by MG scientists within the IAMG community. It has been used not only in the geosciences but later in many other fields of science as well. Geostatistics focuses on application of statistical methods in the earth sciences (e.g. Merriam 1970; McCammon 1975a, b) and still appears to be used by many in that sense. The term geomathematics was also used by several authors including Agterberg (1974) who used the term as the title of his two books (Agterberg 1974, 2014). After the name of the association was changed from mathematical geology to mathematical geosciences in 2008, the term mathematical geosciences more often appears in the literature of the IAMG and also in the titles of conferences, as well as in the name of its journal Mathematical Geosciences. When the author of the current chapter served as president of IAMG (2012–2016), dedication to IAMG was given by promoting the discipline of mathematical geosciences. Several notes on this were published in the President’s Forums in IAMG newsletters (Issues 76–79th). The distinction between mathematical geology and mathematical geosciences is not simply in terminology but also in the scope of the discipline. While mathematical geology refers to a branch of geology, mathematical geosciences must be a subdiscipline of the geosciences which includes geology as one of its subfields. Other relevant subjects covered in the geosciences include but are not limited to geochemistry, geophysics, geobiology, and hydrology. Mathematical geosciences should be a discipline parallel to other subdisciplines in the geosciences such as geochemistry, geophysics and geobiology rather than a branch of geology. In the author’s personal view this distinction is critical for the development of the discipline. Under the concept of mathematical geology, the subject is limited to the application of mathematics in geology but as mathematical geosciences just like geochemistry and geophysics, it serves the entire earth science. So, what should be the definition of mathematical geosciences or geomathematics and what are the roles mathematical geosciences should play in the family of geosciences? Here I will briefly elaborate on these questions and introduce several major contributions of MG to earth science. In order to provide a proper definition of mathematical geosciences, we should look at the definitions of other relevant disciplines such as geochemistry, geophysics and geobiology:

The definitions of the preceding relevant disciplines share the common concept of an interdisciplinary geoscience field. A similar definition was proposed by the author in 2014 with consultation of the IAMG Executive Committee Members and published in the President’s Forum of IAMG newsletter (Issue No. 79).

The ultimate question arising from this definition is what are the mathematical properties and processes of the Earth, with prediction and assessment of its resources and environments which have to be dealt with by mathematical geoscience for integration with other geoscience subdisciplines. Similar to other interdisciplinary fields including geochemistry and geophysics, mathematical subjects such as geometry, calculus, functional analysis, morphology, probability and mathematical statistics provide essential theory and methods for quantitative study of the Earth ranging from geometry and dynamics of the Earth, uncertainties of measurements, and observations for the prediction of Earth events.

There are many examples demonstrating that MG has made indispensable contributions to the geosciences. For example, the mathematical model of the Earth’s shape (e.g. Clark ellipsoid, and Hayford ellipsoid) which serves as the foundation of geodesy, navigation systems (e.g. GPS), remote sensing technology (RS) and geographical information systems (GIS), and the fast growing field of geomatics; the mathematical model of mantle convection and models for plate motions (McKenzie and Parker 1967) serve as foundation of plate tectonics, the most notable development of earth science in the last century; mathematical symmetry and symmetry operations as principles of crystallography and optical mineralogy (e.g. in 1830, Hessel proved the existence of the 32 groups of crystal symmetry) which constitute a foundation of solid earth science; the mathematical topological model as foundation of geographical information systems (e.g. as basis of spatial data modeling in ArcGIS), one of the most useful technologies in geoscience; mathematical and statistical theories providing foundations for describing the spatial distribution and correlation of elements, uncertainty and error bars in geochemistry including isotope geochemistry and geochronology as are also used for the geological time scale; mathematical modeling and uncertainty of prediction of climate change, a pressing issue of the geosciences; probability theory and stochastic models for prediction of energy and mineral resources, highly demanded by many nations for economic and societal development; geo-complexity theory such as fractals, multifractals, chaos and self-organized criticality for modeling and predicting singular events and extreme phenomenal issues; and information extraction (big data mining, machine learning, geo-intelligence) in the geosciences, just to name a few.

As the International Association for Mathematical Geosciences, IAMG has earned its reputation by promoting and fostering its members to make contributions to science. Original and significant studies have been published in IAMG journals, books and conference proceedings. However, a large amount of work is documented elsewhere in publications which cover almost every mathematical subject and aspects of geosciences ranging from statistical data analysis, geometrical modeling, dynamics and processes simulation, to prediction and assessment of Earth system. MG theories and methods have been applied not only in tackling conventional solid earth issues such as assessment of mineral and energy resources, but also in other fields including hydrology, climate change, water resources, alternative energy resources and environmental issues. While the importance of MG in the geosciences has been increasingly demonstrated, the discipline of MG has not yet been fully recognized and, to some extent, buried in oblivion. There is hardly any hiring of highly qualified personal (HQP) in academic institutions or industry with as job title Mathematical Geoscientist or Geomathematician. As a matter of fact, most of our IAMG members are employed with job titles such as geologists, geophysicists, geochemists, geodesists, computer scientists, mathematicians and geoinformatical specialists instead of MG or GM. University students who are talented in mathematics and geosciences wanting to pursue mathematical geoscience have to enroll in geophysics or other fields simply because MG does not exist as such in university programs, at least in most of the programs in developed nations. There are very few interdisciplinary university programs except actuarial science, mathematical physics and mathematics for business, which have mathematics as integral part of their subject. A common misconception is that learning mathematics either can only result in kinds of two jobs: pure mathematician or mathematics teacher, or as a prerequisite for other careers in engineering, science or business. This might be one of the reasons there are not so many students wanting to pursue mathematics related subjects in their choice of career. Thus, MG faces significant challenges when promoting MG as a discipline and for facilitating training and education of future generations. This presents the bottleneck for the IAMG to grow further and to become a more successful and influential association.

The International Year of Mathematics of Planet Earth (MPE) celebrated in 2013 generated a much needed publicity of mathematics in geoscience. Mathematical courses are offered in all schools from primary to high school to university. Earth science is also a common choice of topic in essays by students. Integration of math and earth subjects must provide proper and interesting topics for students’ math or science projects. The mathematical and geoinformatical techniques learned by students early on are already powerful tools for exploring the Earth. An excellent example is the work headlined in the media with publication by a high school student Alice R. Zhai who has analyzed 73 tropical cyclones that made landfall in US and used multivariate regression to examine the dependence of hurricane economic loss on maximum wind speed and storm size. This study (Zhai and Jiang 2014) not only proposes a new model by which hurricane damage might be predicted but also provides new evidence showing the area-density power law property of extreme events which, as is to be introduced in the remainder of this chapter, has deep origins in nonlinear dynamics.

The development of modern information technology enables everyone to easily retrieve big data to support their studies via internet and web services in a cloud environment. To access and process huge amounts of data is no longer only for paid professionals. More and more specialized software packages and multi-media teaching and training materials or online courses available in the public domain with Twitter, Facebook and You Tube, provide new ways for self-learning. Online communication, discussion and consultation through the internet in and out of the classroom have become common for students. It should encourage middle school, high school and university students to develop their curiosity in, passion for, and dedication to mathematical geosciences.

IAMG has been rapidly expanding its scope from traditional geostatistics or statistical geology to more comprehensive interdisciplinary sciences for mathematically studying properties and processes of the Earth with prediction and assessment of its resources and environments. What are the current trends of MG and how are they associated with the Earth Science frontiers? It is impossible to create an accurate list of frontiers for MG. Of course, there exist several previous publications by IAMG members that have discussed past, current and future trends for the IAMG (Agterberg 2003). Here I will just share some thoughts based on my personal observations of several recent events and activities. Several international organizations have developed and published white papers illustrating prospective review on trends of scientific research within their organizations and strategic plans for the next 5–10 years; for example, the International Council for Science Union (ICSU) published its strategic research agenda for Future Earth 2025 Vision (http://​www.​futureearth.​org/​sites/​default/​files/​future-earth_​10-year-vision_​web.​pdf); the International Union of Geological Sciences (IUGS) is jointly with UNESCO offering the International Geological Correlation Program (IGCP) in addition to various other big science programs and new initiatives such as the Resourcing Future Generations (RFG), an international collaborative program (http://​iugs.​org/​uploads/​RFG.​pdf); the US National Science Foundation (NSF) has published a strategic plan for 2014–2018 (https://​www.​nsf.​gov/​publications/​pub_​summ.​jsp?​ods_​key=​nsf14043); the American Geophysics Union (AGU) produced a scientific trends report (https://​about.​agu.​org/​trends-earth-space-science/​); the American Natural Science Foundation published its strategic plan for tectonics (https://​docgo.​net/​national-science-foundation-nsf-strategic-plan-fy-2006-2011-nsf-06-48); a white paper resulting from NSF sponsored workshops on “mathematics in geosciences” was published by a group of geoscientists in 2012 (https://​cpb-us-e1.​wpmucdn.​com/​sites.​northwestern.​edu/​dist/​8/​1676/​files/​2017/​10/​agenda-xwphux.​pdf), just to name a few. Relevant publications resulting from international conferences such as the International Geological Congress (IGCs), AGU, EGU, GSA as well as special articles in several journals such as Nature and Science have also been concerned with these issues. The following summary of key topics can be extracted from the preceding sources of information to reflect current trends and frontiers of the earth sciences. These key topics include but are not limited to data science, data analysis, big data and geo-intelligence, computation, inter-/multi-/cross-/transdisciplinary science, integrated models, uncertainty relative to observations and predictions, properties and dynamics of the planet, climate change, disruptive processes such as earthquakes and storms, and special studies of the Arctic, Antarctic and Tibet Plateau. The fundamental issues are for understanding Earth and environmental systems and their interactions with human activities, and for developing reliable monitoring systems, models, and information technologies for predictions and early warnings of large-scale and rapid change. The current challenges facing earth scientists are understanding and modeling the geo-complexity of the Earth and environmental systems with their interactions, chaotic nature and predictability of geo-processes, Earth singularity and human mitigation and adaptation to extreme events, plus observation and monitoring multiple-scale mixing nonlinear processes. Although most organizations neither recognize nor explicitly mention this, the majority of these frontiers are fundamentally related to MG. A long period of incremental advances of new mathematical theories and models in conjunction with modern technologies for solving these earth science problems may lead to creative leaps of innovation. MG has huge challenges and responsibilities facing the earth science frontiers. MG scientists are indeed at the frontier of earth science tackling fundamental problems of the Earth as can be evidenced by the recent advancements reflected in the topics of plenary presentations at IAMG conferences and in the best papers published in IAMG journals; for example, on multi-point geostatistics—a new field of spatial-temporal modeling (Mariethoz and Caers 2014); compositional data analysis—a new way to explore the composites of the Earth (Pawlowsky-Glahn et al. 2015); singularity analysis and singularity physics—new theory and methods of studying geodynamics and geo-complexity (Cheng 2007, 2017a; Agterberg 2017); big data visual analytics for exploratory data analysis; semantic web technology for geoinformation; uncertainty in ecosystem mapping by remote sensing; integrating structural geological data into inverse modeling frameworks; stationary and isotropic vector random fields on spheres; and mathematical morphology modeling, just to name a few.

For the past several decades nonlinear theory and geocomplexity marked an era of new geoscience that deals with nonlinear processes and extreme phenomena which occurred in the evolution of earth systems. Irregular geometry was not popularized in the past until the term “fractal” was coined by Mandelbrot in the 1970s. Fractal geometry rapidly became a new field of mathematics dealing with roughness and irregularity of geometries. For example, fractals have been used for modeling complex and self-similar patterns generated by nonlinear processes (Mandelbrot 1972; Feder 1988). The concept of fractals and fractal dimension was further extended to multifractals involving self-similar measures defined on support which can be fractal itself (Mandelbrot 1972; Meakin 1987; Schertzer and Lovejoy 1987). Multifractal measures have been further extended to fractal density in local singularity analysis (Cheng 1999a, 2001). In the following sections the concept of fractal density will be introduced and followed by discussion and application of new methods for fractal differential operation and fractal integration (Cheng 2017a).

As mentioned in Eq. (10.2) for heterogenetic matter or substances, the derivative of mass over scale can be used for defining localized density of substance. Accordingly, the mass or volume of a heterogenetic substance can be calculated using integration. Obviously, integration and differentiation are two fundamental operations in calculus and used for many mathematical and physical subjects. The traditional integral and differential operations are defined on the basis of additive property of Lebesgue measure. When the measure no longer possesses additive property, then the classical integral and differential may not exist. Therefore, the ordinary integral and differential operations are not applicable to fractal density with singularity. The author has proposed the following fractal integral and differential (Cheng 2017a)

where Δf(x) and Δx represent the increments of a function f(x) for an increment of x. The convergence of the limit in Eq. (10.12) can be defined as the α-fractal derivative of the function f(x). Similarly, we can define the fractal integral of the function f(x) as follows

where f(x_i) is the magnitude of the function f(x) over the small range [x_i, x_i + Δx]. If the limit of Eq. (10.13) converges, then it can be named the α-fractal integral of the function f(x). It must be kept in mind that the fractal derivative defined in this paper is different from the fractional derivative (fractional order) known in the literature as f^(v)(x), where v can be a non-integer order. The fractional derivative assumes that the normal integer order derivative f⁽ⁿ⁾(x) does exist. The fractal derivative is based on fractal dimension of the measure whereas the fractional derivative is based on fractional order of derivative defined on normal measure. As an example, let us take a power-law function to demonstrate the fractal derivative. Assume a power law function, f(x) = c(x − x₀)^b, with ordinary derivative of the function f^′(x) = cb(x − x₀)^b−1, which does not exist at x = x₀ if 0 < b < 1. The integral of the function then is \( \int {f({\text{x)}}} \,{\text{dx}}\,{\text{ = }}\,{{\text{c}} \mathord{\left/ {\vphantom {{\text{c}} {({\text{b}} + }}} \right. \kern-\nulldelimiterspace} {({\text{b}} + }}1)({\text{x}}\, - {\text{x}}_{0} )^{{{\text{b}} + 1}}\), which does not converge if b < −1 at x = x₀. According to Eq. (10.12), the fractal derivative at x = x₀ exists and f_α′(x) = c, if α = b, or f_α′(x) = 0, if α < b and f_α′(x) = ∞ if α > b.

A new concept of Hausdorff derivative underlying the Hausdorff dimension of metric space/time was proposed by Chen (2006) who introduced the systematic mathematical operation of Hausdorff derivative with applications to derive a linear anomalous transport–diffusion equation underlying an anomalous diffusion process. The Hausdorff derivative operation proposed by Chen (2006) is expressed as follows

This formalism was termed the Hausdorff derivative of a function f(x) with respect to fractal measure x^α.

It has to be pointed out that the fractal derivation defined in Eq. (10.12) is different from that defined in Eq. (10.14) considering that, in general, if x₀ ≠ 0, then

The two sides in Eq. (10.15) become equal only if x₀ = 0. Otherwise, according to Taylor expansion, we can obtain \( \Delta x^{\alpha } = x^{\alpha } - x_{0}^{\alpha } = \alpha x_{0}^{\alpha - 1} \Delta x + o\left( {\Delta x} \right), \) so substitution into Eq. (10.14) gives

which implies that the derivative of f(x) defined in Eq. (10.14) is indeed corresponding to the ordinary derivative except for the factor \( \frac{1}{{\alpha x_{0}^{\alpha - 1} }} \). Reconsidering the example used previously with f(x) = c(x − x₀)^b, the derivative of Eq. (10.14) at x = x₀ does not exist if b < 1.

In the remainder of this chapter I demonstrate that fractal density (Δα ≠ 0) characterizes anomalous mass accumulation or energy release caused by extreme geo-processes, which occurred in the Earth’s lithosphere originated from cascade earth dynamics (plumes, mantle convection and plate tectonics) and self-organized criticality involved in phase transitions (avalanches of slab breakoffs, faults, and volcanic eruptions).

Mantle convection at high Rayleigh number generates thermal plumes episodically which upon arrival in the crust could cause major flood basalt events, igneous provinces as well as spreading of continents and mid ocean ridges (Richards et al. 1989; White and McKenzie 1989). On a larger scale, Wilson cycles (Wilson 1966) corresponding to the periodic fragmentation and reformation of supercontinents could be linked to temporal variability in plate tectonics. Numerous studies have revealed that mantle convections can induce exchange of mass between upper and lower mantle across the endothermic phase transition zone at about 660 km. The cold downwelling material penetrates into the lower layer and, simultaneously, the hot upwelling fluid is pushed into the upper layer. The exchange of mass between the upper and lower mantle layers can occur in short bursts (often described with superlatives such as “catastrophic”, overturn, “avalanche” subduction, or “superplumes”) (Zhong and Gurnis 1994). The quick injection of lower mantle hot fluid into the upper mantle can cause not only mantle heterogeneity but also anomalous thermal distribution near the surface (Le Bars and Davaille 2004). This has been considered to be the first order cause of vigorous magmatism. Deep subductions of continental crust into the deep earth interior and rebounded back to the surface of the Earth have been ascertained by the discoveries of regional metamorphic coesite (Chopin 1984; Smith 1984), and subsequently by unusual ultrahigh pressure (UHP) terranes (Hacker and Gerya 2013).

Within the lithosphere there are various types of “catastrophic” events occurring during plate subduction. Formation of magmatic arc can be caused by subduction in which the subducting or subducted oceanic crust material releases volatiles (e.g. H₂O and CO₂) which cause partial melting of the mantle and form magma at depth under the overriding plate. Earthquakes occur at certain depths at the edges of three types of plate boundaries: convergent (subductions and collisions), divergent, and transformative.

In the first part of the chapter, the purpose of including suggestions about mathematical geosciences or geomathematics as a discipline and introduction to examples of significant contributions of mathematical geoscience scientists to science was to appeal to the public and geoscientists to appreciate the indispensable role that MG can play in the family of geosciences. In the second part of the chapter, the fractal density model was introduced and used for characterizing the power-law rheology of phase transition, and singularity analysis of earthquakes from subduction zones of Pacific plates was demonstrated to be a new and promising nonlinear MG method for modeling extreme and “avalanche” geo-events. Examples of application of singularity analysis not only include earthquakes as introduced in the current chapter but also other types of extreme events such as magmatic flare ups (Cheng 2017a), mid ocean ridge anomalous heat flow (Cheng 2016), flooding caused by tropic storms (Cheng 2008), and mineral deposits as well as ore-caused anomalies in surface media (Cheng 2007). Further comprehensive analysis of earthquakes from other regions and clustering depths will be published in separate papers.