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

Abstract

Fractional calculus (FC) has recently received increasing attention due to its applications in many fields involving complex and nonlinear systems. However, one of the key challenges in using FC to deal with fractal or multifractal phenomena is how to relate functions to geometries with fractal dimensions. The current paper demonstrates how fractal calculus can be used to represent physical properties such as density defined on fractal geometries that no longer have the Lebesgue additive properties required for ordinary calculus. First, it introduces the recently proposed concept of fractal density, that is, densities defined on fractals and multifractals, and then shows how fractal calculus can be used to describe fractal densities. Finally, the singularity analysis based on fractal density calculation is used to analyze the depth clustering distribution of seismic frequencies around the Moho transition zone in the subduction zone of the Pacific plates and the Tethys collision zones. The results show that three solutions (linear, log-linear, and double log-linear) of a unified differential equation can describe the decay rate of the fractal density of depth clusters at the number (frequencies) of earthquakes. The spatial distribution of the three groups of earthquakes is further divided according to the three attenuation relationships. From north latitude to south latitude, from the North Pacific subduction zone to the Tethys collision zone, and then to the South Pacific subduction zone, the attenuation relationships of the earthquake depth distribution are generally from a linear, to log-linear, to double log-linear pattern. This provides insight into the nonlinearity of the depth distribution of earthquake swarms.