| Singularity theory states that extreme geo-processes result in anomalous amounts of energy
release or material accumulation within a narrow spatial–temporal interval. The products (e.g.
mass density and energy density) caused by extreme geo-processes depict singularity without
the ordinary derivative and antiderivative (integration) properties. Based on the definition of
fractal density, the density measured in fractal dimensional space, in the current paper the
author is proposing several operations including fractal derivative and fractal integral
to analyze singularity of fractal density. While the ordinary derivative including
fractional derivatives as a fundamental tool measuring the sensitivity of change
of function (quantity as dependent variable) with change of another quantity as
independent variable, the changes are measured in the ordinary space with additive
property, fractal derivative (antiderivative) measures the ratio of changes of two
quantities measured in fractal space-fractal dimensional space. For example, if
the limit of ratio of increment of quantity (Δf) over the associated increment of
time (Δtα) measured in α – dimensional space approaches to a finite value, then
the limit is referred a α-dimensional fractal derivative of function fand denoted
as
f′ = lim Δf--= df-
α Δt→0 Δtα dtα
According to the definition of the fractal derivative the ordinary derivative becomes the
special case if the space becomes non-fractal space with α value as an integer. In the
rest of the paper we demonstrate that fractal density concept and fractal derivative
can be applied in describing singularity property of products caused by extreme
or avalanche events. The extreme earth-thermal processes such as hydrothermal
mineralization occurred in the earth crust, heat flow over ocean ridges, igneous
activities or juvenile crust grows, originated from cascade earth dynamics (mantle
convection, plate tectonics, and continent crust grow etc.) were analyzed by fractal
derivative method according to fractal density of mass accumulation or energy release. |