Verifying the Dependence of Fractal Coefficients on Different Spatial Distributions

Abstract

A fractal distribution requires that the number of objects larger than a specific size r has a power-law dependence on the size N(r) =C/r D ∝ r-D where D is the fractal dimension. Usually the correlation integral is calculated to estimate the correlation fractal dimension of epicentres. A 'box-counting' procedure could also be applied giving the 'capacity' fractal dimension. The fractal dimension can be an integer and then it is equivalent to a Euclidean dimension (it is zero of a point, one of a segment, of a square is two and of a cube is three). In general the fractal dimension is not an integer but a fractional dimension and there comes the origin of the term 'fractal'. The use of a power-law to statistically describe a set of events or phenomena reveals the lack of a characteristic length scale, that is fractal objects are scale invariant. Scaling invariance and chaotic behavior constitute the base of a lot of natural hazards phenomena. Many studies of earthquakes reveal that their occurrence exhibits scale-invariant properties, so the fractal dimension can characterize them. It has first been confirmed that both aftershock rate decay in time and earthquake size distribution follow a power law. Recently many other earthquake distributions have been found to be scale-invariant. The spatial distribution of both regional seismicity and aftershocks show some fractal features. Earthquake spatial distributions are considered fractal, but indirectly. There are two possible models, which result in fractal earthquake distributions. The first model considers that a fractal distribution of faults leads to a fractal distribution of earthquakes, because each earthquake is characteristic of the fault on which it occurs. The second assumes that each fault has a fractal distribution of earthquakes. Observations strongly favour the first hypothesis. The fractal coefficients analysis provides some important advantages in examining earthquake spatial distribution, which are:-Simple way to quantify scale-invariant distributions of complex objects or phenomena by a small number of parameters.-It is becoming evident that the applicability of fractal distributions to geological problems could have a more fundamental basis. Chaotic behaviour could underlay the geotectonic processes and the applicable statistics could often be fractal. The application of fractal distribution analysis has, however, some specific aspects. It is usually difficult to present an adequate interpretation of the obtained values of fractal coefficients for earthquake epicenter or hypocenter distributions. That is why in this paper we aimed at other goals-to verify how a fractal coefficient depends on different spatial distributions. We simulated earthquake spatial data by generating randomly points first in a 3D space-cube, then in a parallelepiped, diminishing one of its sides. We then continued this procedure in 2D and 1D space. For each simulated data set we calculated the points' fractal coefficient (correlation fractal dimension of epicentres) and then ...

Box Counting Fractal Dimension and Frequency Size Distributon of Earthquakes in the Central Himalaya Region

“…With the advantage of simple mathematical formulation and empirical estimation, box-counting dimension becomes the widely used dimension (Falconer, 1997). The box counting technique estimates the capacity dimension by measuring the space-filling characters of a fracture set with the change in grid of the scale (Gospodinov et al, 2010;Mandal & Rodkin, 2011). In the box-counting method, the epicenter distribution is covered by squares or cubes of decreasing size (r).…”