Geometrical Fractal Analysis of Malard Earthquake (20 November, 2017)

Document Type : Applied Article


Associate Professor, Payam Noor University, Iran


A moderate earthquake (M: 5.1) impacted the urban regions of Malard-Meshkindasht in Alborz province (far west of Tehran) with noticeable crashing of indoor objects that injured people around main shock epicenter in 20 December 2017. In this research, I have used Fibonacci numbers to set 53 years epicenters (IIEES, 1964-2017) based on Perimeter-Area fractal model containing foreshocks, main event and aftershocks onsets. Malard spiral revealed a meaningful geometrical relationship between the recent and recorded earthquakes in catalogue. Several epicenters have been patterned by geometrical angles in self-organized features. Also a set of foreshocks illustrate spiral distributions as well as in aftershocks due to a coherent association with seismic fault systems. Also an integrative spiral has been illustrated in the end of research to realize the hazard of Malard earthquake on NTF. For the time being, there is no inductive case from Malard into NTF because of represented Fibonacci and fractal evidences for this research.    
Fibonacci numbers are famous series in mathematics for introducing golden sequences in natural creatures [1]. In this sequence, each number is found by adding up two numbers before it. Starting with 0 and 1, the sequence goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so forth. Written as a rule, the expression is:

Fn = Fn-1 + Fxn-2


Where, Fn is obtained Fibo-number, and Fn-1, Fn-2 are two sequences before Fn respectively [1].
Both real sets and integer sequences impressed many of natural phenomena, and therefore known as the mathematical keys for terrestrial and infra-terrestrial solutions. For instance, earthquakes have close and meaningful relations with rectangular spiral distributions as a result of spatio-temporal evolution in nature.
A simple sequence of Fibo-numbers is shown as below:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …, Fn
In above sequence, a ratio of (Fn/Fn-1) gives constant value (golden ratio, Ҩ) equal with 1.618.

Ҩ=Fn/F(n-1) = 1.618


This ratio has important roles in geosciences such as a main role in earthquake distribution geometrically. From geodynamical points, crustal motions regulate themselves by Ҩ to originate self-organized patterns on the basis of chaos theory [2]. Two types of golden ratios (1/618 and 1/133) are involved in golden shapes such as triangles, golden circles and golden rectangles [3,4]. Also a simple distribution of earthquakes can be seen in Gnomons and circles around the main shock events. Malard circle has two unequal arcs which one of them has golden angle equal with 137.5 degree of angle. The third is golden rectangle which has spiral function to correlate with geological patterns [3]. For example, earthquakes and their focal distributions are relevant process to golden rectangles with affinity to appear in spiral distributions as is it shown in Figure 1.
Fig. 1. Spiral function of earthquakes based on a golden rectangular distribution.
AGFD (3-a) is a rectangle that is reproduced by a simple ABCD square (3-b) within a multiple statement as below:

(0.5b)*(51/2) = FD   [Where: FD/CD = 1.618


Here, a golden geometrical distribution of half century earthquakes is well done by applying Fibonacci numbers into IIEES catalogue. Natural earthquakes usually array in spirals and therefore geoscientists are interested to start a seismic spatial interpretation according to Fractals and Fibo-sequences [2,4,5]. At least, 53 years backgrounds of Malard seismic events (IIEES, since 1964) facilitate this opportunity to answer the question of “where is the next destructive earthquake in regional local scale?”
It means, with a dense and accurate catalogue, scientists will be able to locate the future earthquakes based on geometrical precursors. Also we know that Fibonacci numbers have close impressions to nature as a key for earthquake prediction [3,4]. According to post prediction algorithm, a main shock record such as Malard event (20 Dec., 2017) , not only initialized post seismic processes, but is relevant to long term catalogues as a regional Fibonacci variable. In practice, Alborz seismic databases including location of epicenters [6,7] and structural lineaments [8], have been gridded by GIS facilities to reveal geometrical relationships of the epicenters to illustrate golden peculiarities of Malard earthquakes.            
Malard earthquake (2017-12-20) seems to be initial point for a short range of post seismic events, which many of them should be considered as aftershocks activities. In Figure 2-a, meaningful triangular distribution can be seen in north side of Malard main shock epicenter. Also, an obvious golden circle (within radius lesser than 2 Km from main shock event) can be seen in Figure 2-b as below.
Fig. 2, a) two kinds of golden triangles (ordinary and gnomons) in Malard seismic pattern.  b) A symmetric golden circle with approximate radius =1.6 near Malard main shock region.
As a primary result, above mentioned facts indicate to natural seismic resources of Malard-Meshkindasht activities, and as second, a rectangular distribution of magnitudes (M>2, since 1964) give rises to spiral function, which contain two types of post seismic potentials (High and Low PSP) shown in Figure 3.
Malard spiral is a dependent geometrical variable to post seismic events as well as its dependency to foreshocks base on Perimeter-Area fractal applied in catalogue. This spiral is Meridian type with west seen affinity that is centralized by Malard seismic events in Dec. 2017.
Fig. 3, Spiral distribution of Malard earthquakes (Ref. Database: IIEES, 1964-2017)
-This research introduced geometrical fractal analysis of 53 years catalogue (Malard region) as recently active zone in eastern part of Alborz province.
-Rectangular distribution of Malard earthquakes, make an easier and accurate forecasting of future events (usually aftershocks or other seismic activities) that is originated from main resources maybe encircled by golden circles and limited by fractals. 
Figure 4. A Fractal separation in random - regular Fibonacci sequences in Malard seismic events (20 Dec. 2017)
In figure 4, forecasted areas (red hatched), have been determined by cross-cutting the isosceles of golden rectangles (Fibo-grids). Therefore, many of Fibo-epicenters are shown on this figure, but few of them are involved with PSP.
- Rectangular distribution of Malard earthquakes makes an accurate way to forecast the future events (aftershocks) originate from main seismic resources by both fractal and Fibo analysis of catalogue.
-For the time being, Malard aftershocks maybe continued toward the west to complete seismic gaps in this cycle (high PSP, Figure 3). Also a rare scenario maybe occurred in west or east directions due to triggering Eshtehard fault or NTF system respectively. Moreover, from geometrical points of view, this scenario is temporary with no longer effects after a monthly reducing in post seismic activities.    


[1] Livio, M. (2002), The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books, p. 85.
[2] Mandelbrot B.B., (2003).  “The Fractal Geometry of  Nature”  W.H. Freeman and Company Press, YALE University, New York, USA, 466P.
[3] Pappas, T. (1989). "The Golden Rectangle." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 102-106.
[4] Kagan, Y.Y. (2002). Aftershock Zone Scaling, Bull. of American Seismological Society, Volume 92, 641-655
[5] Turcotte, D.D. (1997). Fractals and Chaos in Geology and Geophysics, New York, Cambridge University, Cambridge University Press, p397, 2nd Edition.
[6] Turcotte, D.D. (1997). Fractals and Chaos in Geology and Geophysics, New York, Cambridge University, Cambridge University Press, p397, 2nd Edition.
[7] Iranian Seismological Center, Institute of Geophysics, University of Tehran. (2017). Official Report on A magnitude 5.1 earthquake struck Alborz Province near Malard district, December, 20, 2017at 23:27 pm local time,
[8] Berberian, M. (2014). Earthquakes and Coseismic Surface Faulting on the Iranian Plateau, Elsevier, 978-0-444-63297-5, Volume 17 - 1st Edition.