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

Abstract

In the first part of the chapter, the status of the discipline of mathematical geosciences (MG) is reviewed and a new definition of MG as an interdisciplinary field of science is suggested. Similar to other disciplines such as geochemistry and geophysics, mathematical geosciences or geomathematics is the science of studying mathematical properties and processes of the Earth (and other planets) with prediction of its resources and changing environments. In the second part of the chapter, original research results are presented. The new concepts of fractal density and local singularity are introduced. In the context of fractal density and singularity a new power-law model is proposed to associate differential stress with depth increments at the phase transition zone in the Earth's lithosphere. A case study is utilized to demonstrate the application of local singularity analysis for modeling the clustering frequency-depth distribution of earthquakes from the Pacific subduction zones. Datasets of earthquakes with magnitudes of at least 3 were selected from the Ring of Fire, subduction zones of Pacific plates. The results show that datasets from the Pacific subduction zones except from northeastern zones depict a profound frequency -depth cluster around the Moho. Further it is demonstrated that the clusters of frequency-depth distributions of earthquakes in the colder and older southwestern boundaries of the Pacific plates generally depict stronger singularity than those obtained from the earthquakes in their hotter and younger eastern boundaries.

Fractal Derivatives and Singularity Analysis of Frequency—Depth Clusters of Earthquakes along Converging Plate Boundaries

“…Fractals can be considered geometries with fractal dimensions (e.g., noninteger) that no longer possess the Lebesgue additive property [26]. Accordingly, the ordinary integral and derivative operations no longer apply to the fractal geometry with singularity [25]. Before introducing an extension of classical calculus to fractal calculus, there is a need to briefly introduce the common notions used for the representation of the Lebesgue integral and derivative operations and FC.…”

“…Studies also indicated that earthquakes with magnitudes greater than 5.0 worldwide recorded from January 1991 to 31 December 1996 are generally shallow, with depths less than 70 km [17]. The author's earlier research on the distribution of earthquakes along the subduction zones of Pacific plates indicated that earthquakes are clustered around 34-44 km, and the depth distribution of earthquakes can be described by power law function with singularity [25]. There is a lack of systemic and comparative studies about the nonlinear decay of clusters of earthquakes around the Moho and their associations with global plate tectonics.…”

“…An alternative fractal integral and fractal differentiation operations were proposed by the current author [25], which can be expressed as follows:…”

Unveiling Temporal Cyclicities in Seismic b-Values and Major Earthquake Events in Japan by Local Singularity Analysis and Wavelet Methods

“…The local singularity analysis have been successfully utilized to explore complex and nonlinear extreme geological events and geological processes, especially in the field of geochemistry [20,21]. For instance, local singularity analysis method has been applied to the analysis of global zircon U-Pb ages which uncovers the evolutionary process of the continental crust [18].…”