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ISSN 0145 8752, Moscow University Geology Bulletin, 2011, Vol. 66, No. 6, pp. 385392. © Allerton Press, Inc., 2011. Original Russian Text © V.S. Zakharov, 2011, published in Vestnik Moskovskogo Universiteta. Geologiya, 2011, No. 6, pp. 1017.

Analysis of the Characteristics of Self Similarity of Seismicity and the Active Fault Network of Eurasia
V. S. Zakharov
Faculty of Geology, Moscow State University, Moscow, 119899 Russia e mail: vszakharov@yandex.ru; zakharov@dynamo.geol.msu.ru
Received February 22, 2011

Abstract--An analysis of the possible relationship between fractal dimensions of the active fault network, spatial distribution of earthquake epicenters, and parameter b in the GutenbergRichter law is presented. The quantitative characteristics of self similarity of the seismic process and the active fault network of seis mically active areas of Eurasia are obtained. This self similarity manifests itself over a range of at least two orders of spatial scales and magnitudes. The obtained estimations of the fractal dimensions of the fault net work Df and epicenter field De are close for all the areas analyzed. It is established that the average value con necting values D and b for all the investigated areas is slightly higher than the theoretical value (2.0) and varies within the range of 1.72.4. Keywords: seismotectonics, faults, earthquakes, fractal dimension, GutenbergRichter law. DOI: 10.3103/S0145875211060123

INTRODUCTION The geophysical medium in which a seismic pro cess occurs is strongly non uniform. This non unifor mity manifests itself in geologo geomorphological characteristics (geological faults, thickness of the major crust upper shells of the Earth, folding types, types of endogenous regime, relief, morphostructures, etc.), in various geophysical fields (gravitational, mag netic, thermal, field of hydrogeological characteris tics, field of seismic velocity anomalies, etc.), and in the characteristics of matter and the structure of a medium. There are numerous data on the hierarchical self similar structure of these fields and non unifor mities (Sadovskii and Pisarenko, 1991). The closest connection exists between the structure of a non uniform medium and the seismic process exhibiting the typical properties of a hierarchical self similar system in the spatial distribution of earth quakes. Self similar (fractal) properties of both the seismic process and the medium in which it occurs manifest themselves in the values of fractal dimension and parameter b in the GutenbergRichter law. The possible character of the theoretical connection between these two values was discussed in (Kasahara, 1981; Turcotte, 1997). A rather large number of studies (Volant and Grasso, 1994; Oncel, Alplekin, and Main, 1995; Oncel, Wilson, and Nishizawa, 2001; Caneva and Smirnov, 2004; Zakharov, 2008) have been published where these ratios were studied in practice for different regions of the world. This study is focused on Eurasia. The aim of the present paper is to study the con nection between the fractal dimensions of the active

fault network in different areas of Eurasia, the spatial dis tribution of earthquake epicenters, and parameter b in the GutenbergRichter law for earthquake frequency. The PDE uniform world wide earthquake cata logue (URL: hhtp://earthquake.usgs.gov) for 1973 2010 and maps of active faults of Eurasia (Trifonov, 2004; Soboleva, Trifonov, and Vostrikov, 2002) were used as the source data for this study. Various seismically active regions were included in the analysis (Fig. 1). In addition to the seismotectonic activity, the choice of regions was also conditioned by the completeness of the databases analyzed. Discreteness and self similarity of the block divisi bility of the Earth. The self similar hierarchy of the block divisibility of the Earth has been noted in a num ber of studies (Sadovskii and Pisarenko, 1991; Tur cotte, 1997; Goryainov and Ivanyuk, 2001; Bonnet et al., 2001). The tectonic divisibility of the lithos phere is connected with the manifestation of long range order in the structural organization of the tec tonosphere. The divisibility of rocks (Sadovskii and Pisarenko, 1991) is capable of forming a hierarchical sequence that can be approximately described by a geometrical progression with a power exponent of 2 5 (on average, 3.5 ± 0.9) and is virtually independent of the physicochemical properties of the rock or the method of formation (natural or induced by under ground explosions). This property was accounted for by the automodelity of the formation of lithospheric matter by M.A. Sadovskii. Thus, according to modern conceptions, the lithos phere is a strongly hierarchical (multi level), self simi

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8 9 50° 1 40° 2 3 6 5

11 13 12

10

30° 4 0° 10° 20° 30° 40° 50° 60° 70° 7 80° 90° 100° 110° 120° 130° 140° 150° 160°

Fig. 1. Seismicity according to the PDE earthquake catalogue for 19732010 and the active fault network of Eurasia (Trifonov, 2004; Soboleva, Trifonov, and Vostrikov, 2002). Digits denote the regions: (1) the Alps; (2) the Eastern Mediterranean; (3) the Caucasus; (4) Iran; (5) Afghanistan, Pakistan; (6) the Pamir, Tian Shan; (7) the Himalayas, Tibet; (8) the Altai, Sayan Mountains; (9) Baikal; (10) East China; (11) Sakhalin; (12) Kuril Isles; and (13) Kamchatka Peninsula.

lar, and coarsely discrete medium. New approaches designed in synergetics, theory of dynamic systems, and fractal theory, have been actively used to study it. Fractals and fractal dimension. The fractal theory is a field of knowledge that is actively being developed and possesses powerful tools for the description of complex self similar objects and processes (Mandel brot, 1982). This approach offers a new approximation for the description of complex objects using mathe matical language and allows one to introduce quanti tative characteristics for self similarity, a concept that has been used in various fields of knowledge for a long time, including Earth science at a qualitative level. Fractals are objects that possess scale invariance (scaling), where each element carries information on the entire object. The concept of fractal dimension (D) occupies a significant place in fractal theory; D is the power exponent in the power law N ~ rD, (1 )

where r is the scale under consideration and N is the number of elements. To calculate D in practice, the box counting method is usually used (Mandelbrot, 1982); in this case, the dimension is called box dimension. In order to determine it, the object under study is covered with boxes with a side r; r is varied for each act of covering, and the number of boxes N required to fulfill this pro cedure at various r values is counted. A double loga rithmic scale plot is typically built; the region of scal ing (i.e., the range of r values for which the depen dence appears as (1)), is approximated by the following straight line: log N = D log r + c , (2)

its angular coefficient being the fractal dimension D; c is a constant. The fractal dimension represents the self similarity measure of the hierarchical combination under con sideration and the degree of structure complexity. The similarity measure is estimated by the range of the scale, in which the uniform power law is satisfied, whereas the relative degree of complexity is deter mined by the fractal dimension value. The fractal dimension provides the quantitative measure for com paring objects or processes that are usually character ized by different formation times and/or different physical values. Fractal geometry is a powerful tool for analyzing a vast variety of geological processes and objects, whose characteristics are described by power laws (Goryainov and Ivanyuk, 2001; Lukk et al., 1996; Tur cotte, 1997). Self similarity of a fault network. Among various tectonic structures, it is easiest to represent faults and fracturing using a complex of quantitative parameters and use computer software to perform mathematical processing and numerical simulating (Sherman, 1998). The quantitative parameters of the fractures have been used for a long time for studying fault tectonics. They include the values of the bedding angle, amount of inclination, depth, penetration depth, displace ment amplitude and direction, width of the influence domain, density of the fracture network determining the internal structure of the faults, etc. However, these parameters turn out to be insufficient, both when describing the complexity of the structures that fault tectonics deals with or when comparing these result with those obtained in other fields of geological and geophysical knowledge. Thus, it is rather easy to ascer
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387

4 Df = 1.63 3 40° 2

1

35° 40°

45°

0 50° 1

0

1

2

3 log r

Fig. 2. Determination of the fractal dimension of the Caucasus fault network using the box counting method: (a) covering the fault network with boxes of size r; (b) dependence of the number of boxes N on size r in double logarithmic scale (Df = 1.63).

tain the spatial relationship of the epicenters of certain earthquakes with individual faults; however, there is a need for other methods to compare the fault network of the region with its epicentral field of earthquakes. The use of the approaches of fractal geometry in fault tectonics can considerably enhance the opportunities of the numerical methods employed in it (Turcotte, 1997; Sherman, 1998; Goryainov and Ivanyuk, 2001). Most researchers attribute the emergence of plane tary or local regmatic (lineamentary) networks either to tension of a rotational nature or to the local redistri bution of tensions upon tectonic motion, i.e., a passive response to the external impact is assumed. These net works are frequently bound to two, four (more rare, to six and more) series of systemic directions relative to meridians and parallels. However, the actual fault net work is far from such perfection. It typically is a divis ibility system of different orders with a chaotic or weakly pronounced anisotropic organization. The corresponding apparatus is required to provide an adequate description of this structure. Figure 2 shows an example of determining the fractal dimen sion of the fault network of the Caucasus. The points obtained for different consideration scales fall onto one common straight line corresponding to the uni form power law with power exponent Df = 1.63. This means that the decomposition process in this range of scales occurs in the same manner and is independent of the object size. The estimations of the fractal dimension of the reg matic network of different scaling performed in vari ous studies (Turcotte, 1997; Goryainov and Ivanyuk, 2001; Bonnet et al., 2001; Wilson, 2001; Bour et al., 2002) point to their fractal structure over a wide range of scale with dimension Df = 1.11.9. These estima
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tions quantitatively support the conclusion of a self similar hierarchy of the block divisibility of Earth, which was made by M.A. Sadovskii (Sadovskii and Pisarenko, 1991). Thus, the tectonosphere structure is conditioned by the demand for efficient dissipation of the continu ously supplied endogenic energy. In addition to the reasons mentioned, this fact is indicated by the fractal character of distribution of the hypo and epicenter of earthquakes, which is in general comparable with the character of divisibility structures (Goryainov and Ivanyuk, 2001). Self similarity of the seismic regime. The seismic regime (Kasahara, 1981) is the combination of earth quakes that are considered as points in timespace coordinates and are supplemented by an energy parameter. The seismic regime is a dramatic example of a self similar process. According to the modern conceptions, the seismic process results from the deformation of rock under tectonic forces. Since the seismic process occurs in a hierarchical self similar discrete medium, it also bears the features of hierarchical organization, discreteness, and automodelness (Sadovskii and Pisarenko, 1991). Large partings within a rock system may be repre sented by associations of smaller ones that respond as an entire unity to the external impact. Such a system is capable of exchanging mass and energy both with the external medium and between the partings it consists of. Moreover, the partings may attain the critical energy saturation and lose stability by discharging the excess energy as elastic waves (an earthquake). A sta bility loss may be accompanied by changes in the con figuration of blocks with respect to one another, their separation, or consolidation. However, the properties
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log N 1 4 3

10

(b) r, km 100

1000

N 10000 10000 1000

40°

De = 1.53 2 1 35° 40° 100 10 1 45° 50° 0 1 2 3 4 log r

Fig. 3. Determination of the fractal dimension of distribution of earthquake epicenters of the Caucasus: (a) covering the epicen tral field with boxes of size r; (b) dependence of the number of boxes N on size r in double logarithmic scale (De = 1.53).

of the rock in general remain virtually unchanged; so does the character of energy and mass exchange within the system. In other words, the general character of the seismic process remains unchanged. According to these conceptions, the medium in which a seismic process takes place should be considered as a self organizing system that is in dynamic equilibrium. The moment of an earthquake is an abrupt loss in stability (catastrophe) followed by the rearrangement of the system and adaptation to a new stable state. The specified self similar (fractal) properties of the seismic regime manifest themselves in its spacetime characteristics as power laws of distribution (Main, 1996; Turcotte, 1997). Fractal properties of the spatial distribution of earthquake focuses. The spatial structure of the hypo and epicentral field of earthquakes has an appreciably complex non uniform form; these properties manifest themselves over a wide scale range. Without consider ation of the focus size, the combination of the focal points has the character of Cantor sets (Mandelbrot, 1982). An example of calculation of the dimension of epicenter distribution using the box counting method for the Caucasus is shown in Fig. 3. The dimension value De = 1.67. The relationship between a fault network and an epicenter field confined to them allows one to com pare their fractal dimensions (Volant and Grasso, 1994). According to the studies performed by Sher man (Sherman, 1998), the coincidence of Df (the frac tal dimension of the active fault network) and De (the fractal dimension of the epicentral field) is observed for the Baikal Rift Zone; furthermore, Df = De = 1.68.

Similar values may indicate the similarity between the spatial characteristics of faults and epicenter fields and, therefore, the mutual consistency of the pro cesses of seismicity and fault formation. The fractal structure of fault networks determines the features of the fractal spacetime dynamics of earthquakes, and vice versa. This circumstance make it possible to esti mate certain topological characteristics of the network of faults and fractions, which cannot be directly stud ied, using similar characteristics of the earthquake foci fields. The GutenbergRichter law. The Gutenberg Richter law for the magnitude distribution of earth quakes, which is of fundamental importance in seis mology, is a conventional example that attests to the self similarity of the seismic regime (Kasahara, 1981): log N = bM + a , (3)

where a and b are empirical constants and N is the number of earthquakes with a magnitude higher than M during a certain time in a certain region. This relationship occurs for the region of decrease in the plot of distribution log N(M). The graphic pre sentation of this dependence is called a recurrence plot (Fig. 4). It was ascertained that parameter b in the GutenbergRichter law for different regions and situ ations varies over the range 0.5 < b < 1.2 (Kasahara, 1981). This type of distribution points to the fact that cat astrophic earthquakes occur rarely; however, there is no need for searching for a special mechanism for them, since the mechanism is the same as for lower scale events (Turcotte, 1997; Goryainov and Ivanyuk,
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2011). Taking into account the fact that the magnitude is in proportion to the logarithm of earthquake energy, coefficient b is the self similarity parameter of a seis mic regime. Let us note that earthquakes are not only events that take place in an existing fault structure, they also form these faults. Thus, there should be a mutual rela tionship between the properties of the discrete hierar chical medium and the self similar process that takes place in it. The relationship between parameter b in the Guten bergRichter law and the fractal dimension of the net works of active faults and epicentral fields. The physical mechanisms for earthquakes connected with fracture development give grounds for assuming that the fault length depends on the earthquake energy and, there fore, on magnitude. The power character of the rela tionship between the number of earthquakes (N) and the length of a fault disturbance (l) it is associated with was ascertained empirically (Lukk et al., 1996; Tur cotte, 1997): N = l ,
D

log (Nc)

b = 0.72 100 10 1 0 1 2 3 4 Ms 5 6 7 8

Fig. 4. Magnitude distribution of earthquakes and estima tion of parameter b in the GutenbergRichter law for the Caucasus (b = 0.72).

(4)

where D is the fractal dimension and is an empirical parameter. This law is of a universal character and is satisfied for regions with different tectonic regimes. The self similarity of the seismic regime and fractal characteristics of divisibility of the Earth's crust indi cate that the relationship between the characteristics of the power laws describing these phenomena can be revealed (Lukk et al., 1996; Turcotte, 1997). The seismic moment M0 is related to the deforma tions that occur during an earthquake by the following relationship (Kasahara, 1981; Fowler, 2005): M0 = Au, where is the shear elastic modulus, A is the area involved in a fault, and u is the average displacement along the fault during an earthquake. The following relationship was ascertained experimentally (Kan amori and Anderson, 1975; Kasahara, 1981): log M 0 = cM w + s , (5) which introduces a new magnitude scale Mw, the moment magnitude scale. In Eq. (5), Mw is frequently replaced by magnitude MS determined on the basis of surface waves: log M 0 = 1.5 M S + 9.1 . (6) Moreover, the following relationship takes place for M0 (Kanamori and Anderson, 1975): M0 = A
3/2

.

(7)

By combining (5), (7), and the GutenbergRichter law (3) and taking into account the average value c = 1.5 (Kasahara, 1981; Fowler, 2005), we obtain D = 2b. (8) Thus, this makes it possible to determine the fractal dimension of a fault combination over parameter b in
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the GutenbergRichter law based on the distribution of magnitudes of surface waves MS. Analysis technique. In this paper, the possible rela tionship between parameter b in the Gutenberg Richter law and fractal characteristics of the medium (fault network De and spatial distribution of epicenters Df) for Eurasia was investigated. The procedure com prises three stages. 1. Mapping of active faults for the regions selected using the data (Trifonov, 2004; Soboleva, Trifonov, and Vostrikov, 2002). To obtain the spatial distribution of faults, the fractal dimension Df was determined using the box counting method for a graphic raster image of a fault network using FractalAnalysis freeware. An example of such analysis for the Caucasus is presented in Fig. 2. 2. Mapping earthquake epicenters. The authors' software was used to determine the fractal dimension De of epicenter distribution by the box counting method. An example for the Caucasus is presented in Fig. 3. The focus size was not taken into account, the source was assumed to be a point source. The spatial coordinates were measured in degrees; the Earth's sphericity was not taken into account. 3. Plotting the magnitude distribution of earth quakes and determining parameter b in the Guten bergRichter law. An example for the Caucasus is pre sented in Fig. 4. It was mentioned above that this rela tionship connects the fractal dimension determined using the active fault network (or the epicentral field) and parameter b, which is obtained by analyzing the GutenbergRichter law for the distribution over mag nitudes MS. However, only a few records in PDE cata logue contain determinations of MS. Therefore, in this study, magnitudes mb determined using the spatial waves were recalculated into magnitudes MS using the following equation (Fowler, 2005): m b = 2.94 + 0.55 M S .
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Fractal characteristics of seismicity and the active fault network of Eurasia No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Region The Alps The Eastern Mediterranean The Caucasus Iran Afghanistan, Pakistan The Pamir, the Tian Shan The Himalayas, Tibet The Altai, the Sayan Mountains Baikal East China Sakhalin The Kuril Islands Kamchatka Peninsula Eurasia D
f

D

e

b 0.67 0.73 0.72 0.81 0.71 0.71 0.75 0.66 0.74 0.64 0.61 0.78 0.74 0.79

Df/b 2.44 2.12 2.26 2.11 2.37 2.25 2.14 2.67 1.85 2.34 2.22 1.70 1.88 1.99

De/b 2.63 2.56 2.12 1.99 2.17 2.18 2.16 2.31 1.89 2.42 2.08 2.17 2.18 2.03

1.63 1.55 1.63 1.72 1.69 1.60 1.61 1.75 1.37 1.50 1.35 1.33 1.39 1.57

1.76 1.87 1.53 1.62 1.54 1.55 1.62 1.51 1.40 1.55 1.26 1.69 1.61 1.60

Study results. The procedure given was used for all the regions of Eurasia analyzed (Fig. 1). The table summarizes the values of fractal dimensions Df and De, parameter b determined from the cumulative distribu
D/b 3.0 5 2.5 2.0 1.5 2 1.0 0.5 0 4

tion, and the Df/b and De/b ratios. The graphical rep resentation of these results is given in Fig. 5. The find ings were compared with the theoretical equation (8). During the analysis, the accompanying statistical

(De /b)av = 2.20 (Df /b)av = 2.17 1

3

10. East China

13. Kamchatka Peninsula

1.The Alps

8. The Altai, the Sayan Mountains

Fig. 5. Results of the analysis of fractal dimensions of the active fault network (Df), spatial distribution of the epicenters of earth quakes (De), and parameter b in the GutenbergRichter law for the regions of Eurasia under study: (1) Df; (2) De; (3) b; (4) Df/b; and (5) De/b. The standard errors of parameter estimation and average values (Df/b)av and (De/b)av are also presented. MOSCOW UNIVERSITY GEOLOGY BULLETIN Vol. 66 No. 6 2011

12. The Kuril Islands

3. The Caucasus

2. The Eastern Mediterranean

6. The Pamir, the Tian Shan

7. The Himalayas, Tibet

5. Afghanistan, Pakistan

11. Sakhalin

14. Eurasia

9. Baikal

4. Iran

ANALYSIS OF THE CHARACTERISTICS OF SELF SIMILARITY De, Df 2.0 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.50 0.55 0.60 0.65 0.70 0.75 0.80 1 0.85 2 0.90 b
D 2b D 2b

391

Fig. 6. Dependences of the fractal dimensions of spatial distribution of the epicenters of earthquakes work Df on parameter b in the GutenbergRichter law for the Eurasia regions under study: (1) De; (2) D ~ 2b are shown by dashed line.

De and the active fault net Df. The linear relationships

characteristics were also determined to verify the validity of the resulting estimations. Results and Discussion. The major result of our study was that we obtained the quantitative character istics of self similarity of the seismic process and an active fault network with which its manifestation was associated. This self similarity manifests itself within the range of at least two orders of spatial scale and in magnitude. Let us first note the closeness of the resulting values of the fractal dimension for a fault network Df and those of the epicentral field De. The hierarchical struc ture of the blocks of a geological environment and the spatial structure of the distribution of earthquake hypocenters obey (within an error) equal self similar ity relationships. This allows one to study the structure of the medium using the properties of the seismic pro cess occurring in it. The results given can be regarded as quantitative evidence of the mutual consistency between the seismic process and fault formation. The obtained relationships between b, Df, and De demonstrate that the dependence D = 2b is approxi mately satisfied, although appreciably significant dis crepancies are observed (Fig. 6). The consistency between the fractal distribution of magnitude (and, therefore, the energy) of earthquakes and the fractal distribution of the size of faults quantitatively support
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the hierarchical self similar properties of the seismo tectonic process. However, let us note that the coefficient connecting D and b in most regions, as well as the average coeffi cient, is slightly higher than 2 and varies over the range 1.72.4; the average value (Df/b)av = 2.20, (De/b)av = 2.17. This discrepancy from the theoretical depen dence may be accounted for by the fact that we used recalculation of magnitudes, which is not unambigu ous, as was mentioned above. A rather poor correla tion in certain regions was likely to be obtained because of the insufficiency of the data, mostly the data on faults. Being an open system, the lithosphere adapts to an external impact in such a manner as to efficiently pro cess the energy supplied, which is to a large extent accumulated in the form of elastic potential energy. The system of discretenesses (non uniformities) of the lithosphere selects the structure that would provide the maximum extent of dissipation of the elastic energy; it is a sort of adaptation of the material to the existence conditions. Thus, the formation of structures of part ings (non uniformities) of the geophysical environ ment can be regarded as self organization over a wide scale range.
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ZAKHAROV Determined Chaos in Fractal Environment), Moscow: OIFZRAN, 1996. Main, I., Statistical Physics, Seismogenesis, and Seismic Hazard, Rev. Geophys., 1996, vol. 34, no. 4, pp. 433 462. Mandelbrot, B., The Fractal Geometry of Nature, New York: W. H. Freeman and Co., 1982. Oncel, A.O., Alplekin, O., and Main, I., Temporal Varia tions of the Fractal Properties of Seismicity in the West ern Part of the North Anatolian Fault Zone: Possible Artifacts Due to Improvements in Station Coverage, Nonlinear Proc. Geophys., 1995, vol. 2, pp. 147157. Oncel, A.I., Wilson, T.H., and Nishizawa, O., Size Scaling Relationships in the Active Fault Networks of Japan and Their Correlation with Gutenberg Richter B Val ues, J. Geophys. Res., 2001, vol. 106, pp. 21 82721 841. Sadovskii, M.A. and Pisarenko, V.F., Seismicheskii protsess v blokovoi srede (Seismic Process in Fragmented Medium), Moscow: Nauka, 1991. Soboleva, O.V., Trifonov, R.V., and Vostrikov, G.A., Sovre mennaya geodinamika Al'piisko Gimalaiskogo kollizion nogo poyasa (Recent Geodynamics of the Alpine Himalayan Collisional Belt), Moscow: GEOS, 2002. Sherman, S.I., Fractal Analysis in Fault Tectonics, in Tek tonika i geodinamika: obshchie i regional'nye aspekty (Tectonics and Geodynamics: General and Regional Aspects), Moscow: GEOS, 1998, vol. 2, pp. 274276. Trifonov, V.G., Active Faults in Eurasia: General Remarks, Tectonophysics, 2004, vol. 380, nos. 34, pp. 123130. Turcotte, D.L., Fractals and Chaos in Geology and Geophys ics, Cambridge: Cambridge Univ. Press, 1997, 2nd ed. Volant, P. and Grasso, J.R. The Finite Extension of Fractal Geometry and Power Law Distribution of Shallow Earthquakes: a Geomechanical Effect, J. Geophys. Res., 1994, vol. 99, pp. 21 87921 889. Wilson, T.H. Scale Transitions in Fracture and Active Fault Networks, Math. Geol., 2001, vol. 33, no. 5, pp. 591 613. Zakharov, B.C., Kharakteristiki samopodobiya seismichnosti i setei aktivnykh razlomov Evrazii, (Characteristics of Self Similarity of Seismicity and Active Fault Networks of Eur asia), Electronic Scientific Journal GEOrazrez, 2008, no. 1. http://www.georazrez.ru/articles/2008/1 1/zakharov kharakteristiki_samopodobiya_seysmichnosti.pdf.

CONCLUSIONS From the standpoint of modern physics, an open complex system evolves into a state of self organized criticality; the hierarchical organization of non uni formity (fractality) and power laws of parameter distri bution is typical of this system (Goryainov and Ivanyuk, 2001). The Earth's crust in general and indi vidual components of it belong to the class of such systems. This is the reason for the fact that fractal properties are revealed in various geological structures and processes at different spacetime scales, includ ing those of the seismotectonic process. REFERENCES
Bonnet, E., Bour, O., Odling, N.E., et al., Scaling of Frac ture Systems in Geological Media, Rev. Geophys., 2001, vol. 39, no. 3, pp. 47381. Bour, O., Davy, P., Darcel, C., et al., A Statistical Scaling Model for Fracture Network Geometry, with Validation on a Multiscale Mapping of a Joint Network (Hornelen Basin, Norway), J. Geophys. Res., B, 2002, vol. 107, no. 6. doi 10.1029/2001JB000176. 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, Earth Sci. Res., 2004, vol. 8, no. 1, pp. 39. Fowler, C.M.R., The Solid Earth: an Introduction to Global Geophysics, Cambridge: Cambridge Univ. Press, 2005. Goryainov, P.M. and Ivanyuk, G.Yu., Samoorganizatsiya mineral'nykh sistem. Sinergeticheskie printsipy geolog icheskikh issledovanii (Self Organization in Mineral Systems. Synergetic Principles of Geological Investiga tion), Moscow: GEOS, 2001. Kanamori, H. and Anderson, D.L., Theoretical Basis of Some Empirical Relations in Seismology, Bull. Seismol. Soc. Am., 1975, vol. 65, pp. 10731096. 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

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