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ISSN 0145 8752, Moscow University Geology Bulletin, 2012, Vol. 67, No. 2, pp. 133­137. © Allerton Press, Inc., 2012. Original Russian Text © V.S. Zakharov, 2012, published in Vestnik Moskovskogo Universiteta. Geologiya, 2012, No. 2, pp. 52­56.

Preliminary Analysis of the Self Similarity of the Aftershocks of the Japanese Earthquake on March 11, 2011
V. S. Zakharov
Faculty of Geology, Moscow State University, Moscow, 119991 Russia e mail: vszakharov@yandex.ru, zakharov@dynamo.geol.msu.ru
Received October 11, 2011

Abstract--The quantitative parameters of the self similarity of the aftershocks of the Japanese earthquake on March 11, 2011 were obtained. The parameters p in the Omori law (1.06), b in the Gutenberg­Richter law (0.61), and the fractal dimension (D) of the earthquake epicenters (1.52) were determined. Self similarity is manifested in a range of two orders of temporal and spatial scales and four units of magnitude. The stability in time of parameter p and spatial variations in b and p parameters were revealed. Keywords: earthquakes, aftershocks, self similarity, Omori law, Gutenberg­Richter law, fractal dimension. DOI: 10.3103/S0145875212020081

INTRODUCTION An extremely strong earthquake occurred on March 11, 2011 at 5:46 a.m. GMT 129 km east of Honshu Island, Japan. Based on information of the National Earthquake Information Center of the US Geological Survey, its magnitude was 9.0, the coordi nates of the hypocenter were 38.297°N, 142.372°E, and depth was 30 km. The tectonic setting of the
(a) 42 41 40 Depth, m 39 38 37 36 35 34 33 139 140 141 142 143 Pa 144 145
M 8 6 4 2 0

earthquake is the contact zone between the Pacific and North American (Okhotsk) plates. The locations of the main shock and aftershocks with mb 4 from March 11, 2011 until October 17, 2011 based on the worldwide earthquake database (NEIC PDE, http:// earthquake.usgs.gov/regional/neic/index.php) are shown in Fig. 1. A series of works has already been devoted to the tectonic and seismological problems of

Okh

(b) 0 42 41 40 3938 20 37 36 35 34 40 60 80

145 146 143 144

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42 41 40 39 38 37 145 146 36 143 144 35 34 141 142 Longitude Latitude 33 139 140

Fig. 1. The location of the epicenter of the main shock (star) of the earthquake on March 11, 2011 and its aftershocks with mb 4 (circles) from March 11, 2011 until October 17, 2011 based on the NEIC USGS operative earthquake catalogue in plan view (a) and as a 3D image (b). The bold lines are the contours of the coast and the thin line gives the Pacific (Pa) and Okhotsk (Okh) plate boundaries after (Bird, 2003).

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Fig. 2. Aftershocks (mb 4) of the Japanese earthquake from March 11, 2011 until October 17, 2011 on a linear scale (a) and a double logarithmic scale (b) with an estimated p parameter from the Omori law (p = 1.06).

this event (Rogozhin, 2011; Tikhonov and Lomtev, 2011; Tikhonov, 2011). Our work is is focused at the study of the self simi larity of the seismic process. The seismic process is an population of earthquakes, which is considered as energy supplying points in space­time coordinates (Kasahara, 1981). The self similar (fractal) properties of the seismic regime are demonstrated in its spatial­ temporal parameters as distribution power laws. It was established (Kasahara, 1981; Sadovskii and Pisarenko, 1991; Turcotte, 1997) that the seismic process has the characteristic properties of a hierarchical self similar system in the distribution of earthquakes in time, space, and by energy. For the aftershock consistency of strong earthquakes, these properties are manifested in the fractal disposal of their epicenters and are described by the Omori and Gutenberg­Richter laws using the qualitative p and b parameters and the fractal dimension D. The target of the work is the determination of the quantitative properties of the self similarity of the aftershocks of the strongest Japanese earthquake, which occurred on March 11, 2011. DEVELOPMENT OF THE AFTERSHOCK PROCESS IN TIME Based on NEIC USGS data, an aftershock area ~650 km long and ~350 km across extended from Honshu Island up to the deep trench and toward the

east. Most aftershock hypocenters were located at a depth of 20­50 km. The general character of the aftershocks from March 11, 2011 until October 10, 2011 (4459 events) is shown on Fig. 2a. The number of events decreased in time. The bursts of seismic activity at the 12th, 32nd, 61st, and 122nd days are probably related to strong aftershocks (mb = 6.1­6.3). Rogozhin (2011), Tikhonov and Lomtev (2011), and Tikhonov (2011) limit the duration of the after shocks to 30­40 days, which is explained, first of all, by the dates of the writing of these manuscripts (April, 2011). Because a significantly more prolonged series of events is now available, we try to clarify whether these events belong to the aftershocks of this earth quake using the Omori law (Kanamori and Brodsky, 2004), which empirically describes the reduction of aftershock activity: K, (1) n(t) = p (t + c) where n(t) is the number of aftershocks in a unit of time and p is a parameter that characterizes the rate of decrease in seismic activity. The delineation of the aftershocks in the double logarithmic scale allows determination of the p param eter by the slope of the approximate line (p = 1.06 ± 0.04) for the entire data set (Fig. 2b). Here and hereaf ter, we also calculate the accompanied statistical char acteristics for confirmation of the reliability of estima
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Fig. 3. Parameters of the distribution by magnitude for the aftershocks: by MS and the b parameter (b = 0.61) (a) and the calcu lated field of the b parameter (b).

tions. The parameter p of the aftershocks at the 30th, 60th, and 90th days after the major event is 0.95 ± 0.03, 0.96 ± 0.04, and 0.99 ± 0.04, respectively, and correlation coefficient of approximation is no less than 0.9. The stability of the estimations allows us to review the data with the assumption of the entire data set as a single aftershock. THE DISTRIBUTION OF AFTERSHOCKS BY MAGNITUDE AND THE GUTENBERG­ RICHTER LAW A conventional example of proving self similarity of the seismic regime is the Gutenberg Richter law for the distribution of earthquakes in magnitude, which has fundamental significance in seismology (Kasa hara, 1981): log N = ­bM + a, (2) where a and b are empirical constants and N is the number of the earthquake for a certain range with a magnitude over M in a certain region. This ratio is cor rect for the decreasing area of the log N(M) distribu tion plot; the graphic display of this dependence is a recurrence plot (Fig. 3a). The b parameter in the Gutenberg­Richter law for different regions and con ditions varies from 0.5 to 1.2 (Kasahara, 1981). Con sidering that the magnitude is proportional to the log arithm of earthquake energy, then coefficient b is a parameter of the self similarity of the seismic regime. The surface wave (MS) or moment (MW) magni tudes are usually used in recurrence plots. Because most earthquakes are characterized in the NEIC PDE catalogue only by the body wave magnitude (mb), it
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was recalculated into MS using the worldwide average dependence (Fowler, 2005): mb = 2.94 + 0.55MS. (3) After this, the aftershock distribution was plotted. Figure 3a shows the aftershock distribution according to the magnitude MS and the b parameter for all stud ied events (b = 0.61 ± 0.02; the correlation coefficient R = 0.98). The FrAnGeo v.1.0. computer program was cre ated by the author for analysis of the b parameter and the fractal dimension D of the self similarity of the seismic process. The calculation is conducted using a sliding window, whose parameters are specified by the user. The results are visualized in a graphic window and are recorded in a text file which, with the chosen discreteness, records the coordinates of the sliding window (longitude and latitude) and the values of the calculated fractal dimension D, b parameter, and sur face density of point objects. Thus, the program results in a field of studied self similarity characteristics, which distinguishes this program from many others. The FrAnGeo v.1.0 program is also suitable for fractal analysis of point and vector objects with geographical positioning. Figure 3b shows that the calculated field of the b parameter considerably varies within the aftershock area: the maximum (b = 0.9­1.1) is localized at the deep trench and the area of the main event is charac terized by a decreased value. This can probably be explained by the presence of epicenters of strong after shocks in this zone, which leads to the reduction of the slope of the recurrence plot. However, this problem
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Fig. 4. The calculation of the fractal dimension: calculation of the box dimension D for the epicentral field of aftershocks (D = 1.52) (a) and calculated field D (b).

and the variability of the b field in time require further investigation. FRACTAL PROPERTIES OF THE EPICENTRAL AFTERSHOCK FIELD Fractals are objects with scale invariance where each element holds information about the whole. The fractal dimension D occupies an important place in fractal theory, being the index in the following power law: N ~ r ­D, (4) where r is the scale and N is the number of elements. The fractal dimension shows the degree of self sim ilarity of this hierarchic assemblage and degree of complexity of the structure. The degree of self similar ity is estimated by the scale range where the homoge neous power law is observed and the relative degree of complexity is determined by the value of the fractal dimension. The fractal dimension gives a quantitative measure for comparison of objects or processes that are formed, as a rule, at a different time and/or are characterized by various physical values. The spatial structure of the hypo and epicentral field of earthquakes is complicated and heteroge neous, displaying these properties in a broad scale range. Without consideration of the focus size, the population of the focal points has the character of Cantor sets (Turcotte, 1997; Mandelbrot, 1982). To determine the fractal dimension in the FrAnGeo v.1.0 program, we use box dimension method (Turcotte, 1997; Mandelbrot, 1982). To cal

culate the box dimension, the studied object is covered with boxes, each of whose sides is and each cover step changes this value, which is followed by calculation of the number of necessary boxes, N, using different values. This graph is usually plotted in the double log arithmic scale and after this the scaling area, i.e., the interval of values whose dependence is described by equation (4), is approximated by the linear depen dence: log N = ­D log + c, (5) where the fractal dimension D is the angle coefficient and c is a constant. Figure 4a shows the calculation of fractal dimen sion D for the epicentral field of the entire assemblage of studied events (D = 1.52 ± 0.04; the correlation coefficient R is 0.998). Figure 4b shows the calculated field, D. It is obvi ous that parameter D varies significantly within the aftershock area (D = 0.6­1.9). The maximum zone (D = 1.8­1.9) is elongated from southwest to north east; it probably corresponds to the projection of area of the epicenter concentration in the inclined seismo focal zone. 3D analysis, including analysis of the frac tal dimension of the hypocentral field and study of the variation of the D field in time, are necessary for the fine study of this dependence. DISCUSSION AND CONCLUSIONS The main result concerns the quantitative charac teristics of self similarity of the aftershocks of the earthquake on March 11, 2011: the p parameter in the
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Omori law, the b parameter in the Gutenberg­Richter law, and the fractal dimension D of the epicentral field. The self similarity is expressed in a range of at least two orders of the temporal scale, almost two orders of the spatial scale, and four units of magnitude (i.e., six orders of energy). The stability of the p parameter in the Omori law allows us to consider all data sets as a common aftershock sequence. The spatial variations in the b and p parameters indicate the heterogeneity of the studied area and reveal its structure. The properties of self similarity of seismic process, which are manifested as a distribution power law, show that the seismotectonic system is a complicated system with self organized criticality and chaotic behavior (Turcotte, 1997; Kanamori, Brodsky, 2004). Earthquakes not only occur in fault systems but form faults as well. Thus, an interrelationship between the properties of a discrete hierarchical media and a self similar process inside it should exist. This rela tionship has been reviewed by various authors (Kasa hara, 1981; Lukk et al., 1996; Turcotte, 1997; Caneva and Smirnov, 2004). Our previous works (Zakharov, 2008; 2011) showed that the fractal dimensions of the epicentral field D and networks of active faults DF are characterized by similar values. In our opinion, this is evidence of the mutual coordination of seismic pro cesses and fault formation. The fractal structure of fault systems determines the peculiarities of fractal spatial­temporal dynamics of earthquakes and vice versa. This allows the estimation of some topologic features of fault and fracture systems that cannot be studied directly based on similar features of the fields of earthquake epicenters (as in the case where the aftershock area lies in the ocean). REFERENCES
Bird, P., An Updated Digital Model of Plate Boundaries, Geochem., Geophys., Geosyst., 2003, vol. 4, no. 3. doi: 10.1029/2001GC000252 Caneva, A. and Smirnov, V., Using the Fractal Dimension of Earthquake Distributions and the Slope of the

Recurrence Curve To Forecast Earthquakes in Colom bia, Earthquake Sci. Res. J., 2004, vol. 8, no. 1, pp. 3­9. Fowler, C.M.R., The Solid Earth: An Introduction To Global Geophysics, Cambridge: Cambridge Univ. Press, 2005. Kanamori, H. and Brodsky, E.E., The Physics of Earth quakes, Rep. Prog. Phys., 2004, vol. 67, pp. 1429­1496. Kasahara, K., Earthquake Mechanics, Cambridge: Cam bridge Univ. Press, 1981. Lukk, A.A., Deshcherevskii, A.V., Sidorin, A.Ya., and Sidorin, I.A., Variatsii geofizicheskikh polei kak proyav lenie determinirovannogo khaosa vo fraktal'noi srede (Variations in Geophysical Fields as Manifestation of Determinated Chaos in Fractal Medium), Moscow: OIFZ RAN, 1996. Mandelbrot, B., The Fractal Geometry of Nature, New York: W. H. Freeman and Co., 1982. Rogozhin, E.A., March 11, 2011 M 9.0 Tohoku Earthquake in Japan: Tectonic Setting of Source, Macroseismic, Seismological, and Geodynamic Manifestations, Geo tectonics, 2011, vol. 45, no. 5, pp. 337­348. Sadovskii, M.A. and Pisarenko, V.F., Seismicheskii protsess v blokovoi srede (Seismic Process in Block Medium), Moscow: Nauka, 1991. Tikhonov, I.N. and Lomtev, V.L., Tectonic and Seismological Aspects of the Great Earthquake in Japan on March 11, 2011, Geodin. Tektonofiz., 2001, vol. 2, no. 2, pp. 145­ 160. Tikhonov, I.N., Mega Earthquake on 11 March 2011 in Japan and Aftershock Process Dynamics' Develop ment, Rus. J. Earth Sci., 2011, vol. 12, no. ES1003. doi: 10.2205/2011ES000503 Turcotte, D.L., Fractals and Chaos in Geology and Geophys ics, Cambridge: Cambridge Univ. Press, 1997, 2nd ed. Zakharov, V.S., Analysis of the Characteristics of Self Sim ilarity of Seismicity and of the Active Fault Network of Eurasia, Moscow Univ. Geol. Bull., 2011, vol. 66, no. 6, pp. 385­392. Zakharov, V.S., Characteristics of Self Similarity of Seis micity and of the Active Fault Network of Eurasia, GEOrazrez, 2008, no. 1. http://www.georazrez.ru/articles/ 2008/1 1/zakharov kharakteristiki_samopodobiya_ seysmichnosti.pdf. Cited May, 20, 2011.

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