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ISSN 0145 8752, Moscow University Geology Bulletin, 2013, Vol. 68, No. 3, pp. 165­174. © Allerton Press, Inc., 2013. Original Russian Text © V.S. Zakharov, 2013, published in Vestnik Moskovskogo Universiteta. Geologiya, 2013, No. 3, pp. 29­37.

The Dynamic Characteristics of GPS Time Series and their Relation to the Seismotectonic Specific Features of a Region
V. S. Zakharov
Faculty of Geology, Moscow State University, Moscow, Russia e mail: vszakharov@yandex.ru
Received October 23, 2012

Abstract--The noise component in the time series of Earth surface displacements that were obtained with the Global Positioning System (GPS) is analyzed for 19 points. The methods of dynamic system and fractal set theory are applied. The analyzed parameters include the correlation dimension, spectral scaling parame ter, fractal dimension, and the Hurst exponent. We detect that GPS time series demonstrate fractal properties in a range of over one order of magnitude of frequency (flicker noise). The fractal characteristics of the studied series and seismotectonic features of the studied regions are characterized by a relationship that can be explained by the dynamic characteristics of the block models and seismicity. Keywords: GPS, fractal dimension, spectral analysis, Hurst exponent, flicker noise, and seismicity DOI: 10.3103/S0145875213030095

INTRODUCTION Recently, Global Positional System (GPS) data have been widely used in different fields of science and technology, including the Earth sciences. For instance, in geodynamics they make it possible to identify "immediate" relative displacements (averaged for a period of several years) and the speed of the lithospheric plates and blocks, which attain a few tens of millimeters per annum. These results are an instrumental confirma tion of important provisions of the lithospheric plate tectonics. GPS speed values, compared to values based on geological data, are used in many geodynamic reconstructions and models (Khain and Lomize, 2005; Sella et al., 2002). The data on vertical block move ments also carry important information (Freymueller and Fletcher, 2000; Zakharov, 2006). In addition to displacement speeds, which are determined by trends in the GPS time series, the records of these time series can be used as the source of data on the dynamic properties of the Earth's crust system. For this purpose, it is necessary to consider the short period "vibration" overlapping the general trend and seasonal variations and to analyze its properties (Zakharov, 2004). The analysis of time series is widely discussed in the Earth sciences, including analysis from the standpoint of dynamic system and fractal set theory (Goryainov and Ivanyuk, 2001; Lukk et al., 1996; Turcotte, 1997). It has been proven that the "noise" component, which was previously ignored, contains important informa tion on the process, namely, on the way that a complex nonlinear discrete geological medium reacts to exter nal action, i.e., on its dynamic characteristics.

The observed sequences can often be identified with flicker noise (1/f noise), which is relatively widespread in natural systems, including geological and geophysi cal ones, e.g., trends of Benard convection characteris tics (flow rate, etc.), variations in river level, water flow rate, variations in solar activity, and electrometric and seismological characteristics (Lukk et al., 1996; Schro der, 1991; Schuster, 1984; Turcotte, 1997). The "vibration" in the GPS time series is most ade quately described by the combination of white and flicker noises (Prawirodirdjo and Bock, 2004; Wang et al., 2012; etc.). These authors determined the noise component properties largely in order to mark the dis placement trends in the best possible way, in other words, to select the speed model. Our investigation objective is to determine the noise dynamic characteristics in the GPS time series on the basis of fractal analysis and also to compare these characteristics with the geotectonic and geody namic specific features of the studied regions. The self similarity properties of the GPS time series, correlation dimension, spectrum structure, fractal dimension, and Hurst exponent were analyzed (R/S analysis). RESEARCH DATA The daily ground displacement series that were obtained in the course of GPS data processing (http://sideshow.jpl.nasa.gov/post/series.html) in a number of sites were used as initial information. The selection criteria were as follows: the longest analyzed series, the occurrence of analyzed sites in different

165


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60

YELL

KELY STJO WEST METS BOR1 ZECK ANKR BAHR KOUR COCO IPKT JS

30

COSO GOL2 HARV

0

­30
YAR1 YAR1

­60

1

2 3 ­120 ­90 ­60 ­30 0 30 60 90 120 150
MCM4

20 mm/year

­150

Fig. 1. The location of the analyzed GPS points and the speed of their horizontal movement: (1) point name and its speed; (2) earthquake epicenters, according to the PDE catalog data; (3) boundaries of lithosphere plates.

continents and geodynamic settings, the occurrence of observation points in seismic and aseismic regions, and a close location to plate boundaries. Figure 1 demonstrates the horizontal speed values (Sella et al., 2002) that were obtained for nineteen GPS points that were subject to the analysis. Figure 2a demonstrates the time series of the daily displacements of the Earth's surface (latitudinal, lon gitudinal, and vertical components) for the YELL site (Northern America) for 1992­2003. The clearly seen general trend makes it possible to estimate the speed values for geodynamic investigations and seasonal variations. Moreover, significant variations that have a chaotic "noise" form are also observed. According to Figs. 2b and 2c, which demonstrate the three dimen sional pattern of displacements at the same point, the movement of the Earth's surface in space is rather complicated. All the studied time series are characterized by a well defined chaotic component. The NEIC PDE earthquake catalog was used to correlate the position of GPS points and seismicity (http://earthquake. usgs.gov/regional/neic/). ANALYSIS OF THE SELF SIMILAR PROPERTIES OF THE TIME SERIES A time series of the self similar properties of geo physical data were analyzed in detail by Lukk et al. (1996). The initial time series is subject to processing aimed at the removal of regular components (the trend), which are thought to be due to external causes as to the considered system. The time series variations

that remain after this procedure are initiated by the internal dynamics of the studied geodynamic system rather than by measurement errors. The spectral power (SP) of the studied signal is then calculated. The spectrum is analyzed to determine the scaling field, i.e., the area where the dependence of spectral power S on frequency f is of the following form: S(f) = f
­

,

where is a constant; when constructed on a log­log scale: log S(f) = ­ log f + . If is close to zero, then the studied signal has a "flat" (frequency independent) spectrum generated by a random system (so called white or Gaussian noise). White noise means the absence of any relation ship between the system history and its state in the fol lowing moment (no "memory"). If the parameter is close to 1, the studied time series can be considered as flicker noise. The flicker noise occurrence in the sys tem is indicative of its memory, although it is limited in time, in other words, the system "forgets" its past but not at once. The spectral structure of the flicker noise is intermediate between white noise and ordered vari ations. As a whole, what is meant here is generalized noise of the 1/f type, where can take any, including fractional, values. Noise with = 2 is occasionally called brown, noise with = 1 is called pink, while noise with > 2 is called black (Schroeder, 1991). The higher the value is, the greater is the memory of a system that generates such a signal (in other words, it becomes more deterministic, predictable).
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THE DYNAMIC CHARACTERISTICS OF GPS TIME SERIES 20 15 Displacement, mm 10 5 0 ­5 ­10 1992 (b) 2 (a)

167

3

1

1994

1996

1998 Years

2000

2002 (c)

Vertical displacement

4 2 0 ­2 ­4 ­6 ­8 ­10 14 La 12 10 titu 8 din 6 4 al d 2 i sp 0 lac ­2 em en t

Vertical displacement

4 2 0 ­2 ­4 ­6 4 La 3 tit 2 ud in 1 al 0 di ­1 sp lac ­2 em ­3 en t
5

ud git n Lo

t en m e lac sp di al in 0 4 8

­2

t en ce m la isp al d in ud gi t n Lo
4 3 2 1

12 16 20 24

Fig. 2. The displacement of the Earth's surface for the YELL site (Northern America) according to the GPS data: (a) series of daily measurements of three displacement components: (1) vertical, (2) longitudinal, and (3) latitudinal; (b) three dimensional pattern of the YELL site displacements; (c) three dimensional pattern of displacements with a removed trend. Displacement is measured in mm.

The Hurst exponent (H) is one more frequently used characteristic of the self similarity of the time series. It is calculated via the analysis of the R/S rela tionship (Lukk et al., 1996; Feder, 1988; Turcotte, 1997), where R is the range of the time series, i.e., the difference between the highest and lowest accumu lated deviation from the current average value (at the given time interval ) and S is the standard series devi ation at the same interval. It is established that many natural processes can be described by the following relationship: R/S ~ H, (1)
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where H is the Hurst exponent. The method of H cal culation by the dependence of R/S on is analogous to that used above to calculate the scaling parameter for the spectrum. The Hurst exponent values make it pos sible to distinguish time dependences that are charac terized by a stable trend in variation (persistence, at H > 0.5) from those that demonstrate no stability (non persistence, at H < 0.5). The relationships of the scaling parameter values for the time series spectrum with fractal dimension D for the time series itself, as well as with the Hurst expo nent H (Turcotte, 1997), were obtained:
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2

= 0.74

= 1.09 log (spectral power) 1

= 1.03

1000 10
­3

100 10
­2

10

Period, day

10­1 Frequency, day­1

Fig. 3. The dependence of spectral power (SP) on frequency at the log­log scale for the YELL site series demonstrated in Fig. 2: (1) vertical, (2) longitudinal, and (3) latitudinal components. For the demonstration purposes, the spectra diagrams are displaced along the vertical axis. The scaling fields are approximated by the linear dependence; the values are given nearby.

2 H ­ 1 at ­ 1 < < 1 , H 0 at < ­ 1 , H 1 at > 1 ; D (5 ­ )/2 at 1 < < 3.

(2) (3)

RESULTS OF ANALYSIS Figure 3 demonstrates examples of spectra that were calculated by the displacement series for the YELL site at a log­log scale. Two zones are seen on the spectra: a horizontal (high frequency) zone that shows the occurrence of white noise and a linearly dip ping (scaling field) zone (low frequency), which is indicative of the self similarity of the process in a given range. All other analyzed time series were characterized by analogous results. The fractal characteristics of the GPS series, such as spectral scaling parameter , frac tal dimension D, and Hurst exponent H, were calcu lated by independent methods (Table 1). The previous Hurst exponent values were corrected with respect to our previous results (Zakharov, 2004). The correlation ratio R and statistical significance according to the Student criterion t were calculated upon the approxi mation of the line scaling field (log­log scale). The val ues R > 0.8 and t > 10 confirm the statistical reliability of these characteristics. The standard errors of the param eters are as follows: 0.1; H 0.1; and D 0.3.
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Lukk et al. (1996) provide the corrected relationship based on numerical experiments: D = 2.28­0.38 (at 0.5 < < 3). (4) It should be noted that the fractal dimension of the time series is quite a conditional term, because, strictly speaking, the time series differs from an ensemble of points (which the fractal dimension concept was for mulated for) in the inequity of the coordinates in dif ferent axes (in this case, displacement and time). These methods were applied in the analysis of time displacements of the Earth's surface on the basis of the GPS data for different observation points. The spec tral analysis was carried out using the STATISTICA package; the calculation of fractal dimension (D) and Hurst exponent (H), as well as the analysis of the cor relation dimension, were carried out with the author's FraTiS software (Zakharov, 2004).

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THE DYNAMIC CHARACTERISTICS OF GPS TIME SERIES Table 1. The fractal characteristics of the GPS time series of displacements Series length, day Lithospheric plate Anatolian Antarctic Arabian Australian Eurasian Seismicity Rigidness Latitudinal displacement 1.25 0.49 1.12 1.16 0.82 1.25 0.76 1.15 1.41 1.40 0.96 0.79 1.21 1.25 1.03 0.83 1.15 1.02 1.03 D 1.89 1.98 1.93 1.86 1.97 1.92 2.00 1.98 1.80 1.87 1.92 1.97 1.95 1.94 1.93 1.96 1.95 1.94 1.94 H 0.83 0.74 0.85 0.86 0.75 0.88 0.69 0.74 0.89 0.90 0.87 0.69 0.83 0.78 0.78 0.84 0.85 0.87 0.80 Longitudinal displacement 1.05 0.78 1.02 1.04 1.06 1.25 1.09 1.02 1.31 1.39 1.30 0.86 1.00 1.07 1.09 1.12 0.70 1.09 0.82 D 1.89 1.98 1.96 1.92 1.95 1.90 1.99 1.97 1.89 1.84 1.89 1.97 1.96 1.96 1.95 1.91 2.00 1.95 1.96 H 0.84 0.79 0.78 0.79 0.80 0.88 0.73 0.76 0.79 0.93 0.85 0.67 0.80 0.81 0.78 0.87 0.76 0.85 0.77 Vertical displacement 0.97 0.90 1.15 1.05 0.93 0.94 1.09 1.06 0.90 1.36 0.85 1.08 1.02 0.98 0.74 1.31 0.92 1.09 0.99 D 1.94 1.96 1.93 1.94 1.97 1.93 1.96 1.95 1.95 1.86 1.97 1.95 1.96 1.96 1.93 1.98 1.98 1.95 1.95

169

Point

H 0.76 0.79 0.85 0.74 0.76 0.85 0.85 0.78 0.78 0.94 0.79 0.74 0.79 0.74 0.86 0.77 0.76 0.78 0.78

ANKR MCM4 BAHR YAR1 B0R1 IRKT METS ZECK COCO COSO GOL2 KELY STJO WEST YELL HARV KOUR SHAO USUD

1946 3238 2048 3030 2716 2160 2468 1434 1814 2070 2340 2036 3334 1730 3574 2296 2690 2262 3142

Indian North American

Pacific South American Chinese Okhotsk

r n r r r n r n n n n r r r n n r r r

s n n n n s n s n s s n n n n s n s s

As follows from the analysis of the results, the GPS time series demonstrate fractal properties in range of over one order of magnitude of frequency (~ 10­3Â3 â 10­1 day­1). These are similar in their characteristics to generalized Brownian functions rather than to Gauss ian noise (yet, it is also present, as follows from the high frequency horizontal zone of the spectrum). The calculated quantitative characteristics of self similarity (proximity of the self similarity parameter to 1) are indicative of the fact that these series are of the flicker noise type (intermittence noise). This means that the system that generates such signals is not absolutely random, is characterized by some determi nacy, and "remembers" its previous states. The higher or H values (and the lower the fractal dimension D), the greater its memory. Hurst exponent values that sig nificantly exceed 0.5 are indicative of the persistence of the studied movements. Currently there are different models of systems generating flicker noise. For instance, the model of self organized criticality (SOC) (Goryainov and Ivanyuk, 2001; Lukk et al., 1996; Turcotte, 1997). Flicker noise can also occur in a system affected by numerous periodic processes with random amplitudes and phases. One more important class of systems that generate flicker noise is those where intermittence is caused by nonlinear interaction of elements inside the
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system, which complicate its response, even to con stant action and initiate a chaotic distribution of out bursts. Since the , D, and H values are calculated inde pendently, it is possible to confirm, within the error margin, the justification of equations (2)­(4), which were obtained by simulated Brownian functions under the estimation of the fractal characteristics of a natural time series. The relationship between and H for the latitudinal and longitudinal displacement components is in the best agreement with the simulated relation ships. The radial components are characterized by a lower confidence level, but it should be noted that the GPS vertical determination accuracy is also somewhat lower. DISCUSSION As follows from the analysis results, the "noise" "chaotic" component in the studied GPS series is not odd at all. On the contrary, it carries valuable data on the characteristics of the Earth's crust system that gen erates such complicated block movements. We compared the fractal and geodynamic charac teristics of the studied series, such as seismic activity of the region where a GPS point is located, as well as its
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Table 2. The average values of the fractal characteristics by displacement components depending on the studied geody namic parameters Rigidness Characteristics Component r (11) Lat Lon Rad Lat Lon Rad Lat Lon Rad 1.03 0.95 1.01 0.80 0.78 0.78 1.94 1.96 1.95 n (8) 1.10 1.20 1.03 0.83 0.83 0.82 1.92 1.91 1.94 s (10) i.n 1.14 1.04 0.84 0.83 0.80 1.92 1.91 1.94 n (9) 1.00 0.96 0.99 0.79 0.77 0.79 1.94 1.97 1.95 Seismic activity

H

D

confinement to so called rigid (conditional name) blocks. In this case, following Sella et al. (2002), the sets of GPS points are considered to be a rigid block if they belong to stable parts of the plates, move in a consis tent way, and thus are used for the most accurate deter mination of the Euler parameters of plate rotation (the Euler pole and angular speed values). Correspond ingly, the points that are confined to the plate bound aries and internal deformation zones cannot be con sidered as rigid blocks. The analyzed GPS points were considered as seis mically active regions on the basis of the calculation of earthquake epicenter surface density. The density for the time period corresponding to the studied time series (1992­2001) was calculated using the PDE cat alog in a mobile window using the author's FrAnGeO program (Zakharov, 2011b); one event per 104 km2 was regarded as a threshold value. For purposes of simplicity, both characteristics are regarded as binary: r and n are rigid and non rigid blocks, respectively; s and n are seismic and non seis mic regions, respectively; these parameters are given in Table 1. The diagrams for the fractal characteristics of hor izontal components with account for the above indi cated parameters (r and s) are demonstrated in Fig. 4. Initially, it seems that they cannot be used to identify clusters (crowds) of points with similar characteristics accurately. It is possible to roughly distinguish the areas of points that belong to rigid and non rigid blocks using the parameter (Fig. 4, above on the left). Other parameters make it possible to see in general that points corresponding to the studied characteris tics lie higher or lower on a diagram depending on the r and s values. Meanwhile, this grouping (rather con ditional) is systematic in nature. For specification pur poses, we compared the averaged characteristics (while understanding the averaging disadvantages) of

each component depending on the studied geody namic parameters. The results for the longitudinal component are shown in Fig. 5. Other components are characterized by the same results that were reported in Table 2. Such representation of these results makes it possible to establish the systematic nature of the dependence on geodynamic parameters. However, this method is not reliable enough, because the ranges of average values with account for standard errors do not overlap, while those that account for standard deviations overlap. The theory of difference in average values by both grouping (r and s) variables (t criterion) is reliably confirmed (p 0.05) for all characteristics determined by the longitudinal components, while latitudinal and vertical components are distinguished by lower reli ability of differences. The pattern can be described in the following way. The average and H values are lower for the points with a feature r (i.e., those associated with rigid blocks) than for non rigid blocks, while average D is, on the contrary, higher. As mentioned above, the lower and H values and, correspondingly, higher D values are indicative of less certainty and determinacy of the pro cess. Hence, according to the data we obtained, the time series of the displacement of the points associated with the rigid blocks are characterized by less deter minism and by more significant randomness. The sit uation is the opposite for the feature s: the and H val ues are higher, while the D values are lower for the points that occur in the seismically active regions. Hence, the time series of points that are located in seismically active regions, as well as the points that are associated with non rigid blocks, are characterized by a higher degree of determinism than non active regions and rigid blocks. These results seem somewhat paradoxical and need to be explained. First of all, it should be noted that these groups (rigidity and seismic activity) are independent, as a whole: 7 of 19 points are associated simultaneously
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THE DYNAMIC CHARACTERISTICS OF GPS TIME SERIES 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Lat
Lon

171

1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7

Lon

0.6 0.4

0.6

0.8

1.0

1.2

1.4

1.6 Lat

HLon 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.65

HLon 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.65

0.70

0.75

0.80

0.85

0.90

0.95 HLat

0.70

0.75

0.80

0.85

0.90

0.95 HLat

DLon 2.02 2.00 1.98 1.96 1.94 1.92 1.90 1.88 1.86 1.84 1.82 1.82 1.86 1.90 1.94 1.98 2.02 1.78 1.80 1.84 1.88 1.92 1.96 2.00 DLat r n

DLon 2.02 2.00 1.98 1.96 1.94 1.92 1.90 1.88 1.86 1.84 1.82 1.82 1.86 1.90 1.94 1.98 2.02 1.78 1.80 1.84 1.88 1.92 1.96 2.00 DLat s n

Fig. 4. The diagrams of the fractal characteristics for horizontal displacement components. The data on rigid r and non rigid n blocks are given in the left column, while the data on seismic s and non seismic n regions are given in the right column. MOSCOW UNIVERSITY GEOLOGY BULLETIN Vol. 68 No. 3 2013


172 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 r HLon 0.90 0.88 0.86 0.84 0.82 0.80 0.78 0.76 0.74 0.72 0.70 r DLon 2.00 1.98 1.96 1.94 1.92 1.90 1.88 1.86 1.84 r n n n
Lon

ZAKHAROV Rigidity Lon 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 s HLon 0.90 0.88 0.86 0.84 0.82 0.80 0.78 0.76 0.74 0.72 0.70 s DLon 2.00 1.98 1.96 1.94 1.92 1.90 1.88 1.86 1 1.84 s n 2 n n Seismic activity

Fig. 5. A diagram of the average fractal characteristics. The data on rigid r and non rigid n blocks are given in the left column, while the data on seismic s and non seismic n regions are given in the right column. MOSCOW UNIVERSITY GEOLOGY BULLETIN Vol. 68 No. 3 2013


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with rigid blocks and non seismic regions, while 6 points, in contrast, are associated with non rigid blocks and seismic regions; 4 of 6 remaining points are associated with rigid blocks and seismic regions, while 2 points are associated with non rigid blocks and non seismic regions. Such a subdivision of most points into two groups (13 of 19 points, 68%) allows us to suggest that the non rigid blocks correspond to seismic regions, while the rigid blocks correspond to non seis mic regions. This subdivision can be explained in the following way. The non rigid blocks, being heteroge neous, are composed of smaller blocks, which are able to carry out differentiated movements. The interaction at the boundaries of these multiscale blocks manifests itself in the observed seismicity pattern (Zakharov, 2011a). The GPS points that are confined to rigid blocks participate in relatively concurrent joint movement (this very concurrence serves as a criterion for the selection of points to determine the Euler parameters of plate movement). However, due to the fact that we are analyzing time series with a "removed" trend, this general most deterministic component is excluded from consideration. The remaining component con tains a highly chaotic random signal that affects the values of fractal characteristics in a corresponding way. The points that are not confined to the rigid blocks do not have such a significant general component that could be removed along with a trend. However, the smaller blocks could perform complicated movements in these regions. The identification and analysis of such movements were carried out for a number of regions by Zakharov and Simonov (2010). Such com plicated intrablock movements are also characterized by deterministic­chaotic properties. Apparently, its degree of determinism is manifested in the values of the studied fractal characteristics. Possible models of such processes, which are controlled by "dry" friction mechanisms that provide a positive reverse relation ship and their properties, were studied in the works of Zakharov (2004, 2011a). It has been established that the seismically active regions are specified by a certain degree of determin ism. These results are in good agreement with our data (Zakharov, 2010), which were obtained in the course of the analysis of seismic energy time series and dem onstrate that these series are marked by self similarity in the range of over one order of magnitude of frequency and time. The seismic process is very complicated and hardly deterministic, but it is not totally random. How ever, the low seismic and aseismic regions do not have a general process for energy reworking and a key role is played by the action of numerous factors and a stronger influence of random noises. These facts affect the values of the investigated fractal characteristics. Thus, the results we obtained by the comparison of fractal and geodynamic characteristics can be logically explained under the approaches that are applied in the theory of geodynamic systems and our works.
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It should be noted that the investigated dynamic characteristics of the horizontal components are in relatively good agreement, as follows from the "diago nal" distribution of points in the diagrams (Fig. 4). They are closer among themselves than to the analo gous characteristics of vertical components, as follows from the cluster analysis results. As well, it should be noted that in spite of the indi cated differences, the range and dynamic characteris tics of the chaotic variation component are similar for the vertical and horizontal displacements compo nents. The contrasting sign alternating nature of the highly intensive vertical movements was revealed long ago in the course of a geodetic survey (repeated level ing, etc.) (Kuzmin, 1989; Sidorov and Kuzmin, 1989), while the horizontal movements are commonly marked by directed behavior (Khain and Lomize, 2005). Yet, according to the results of our analysis of GPS series, contrasting effects and oscillation are also characteristic of the horizontal component. The occurrence of the flicker noise component in the studied time series of displacements with spectral ratio values in the range of 0.7­2 and a Hurst expo nent in the range of 0.7­0.9 can be explained in two ways. First of all, the identified noise is a generalized Brownian processes involving some displacement of the system at each following stage under the action of numerous external factors, as well as some relaxation (likely, nonlinear) of the previously accumulated noise component. Time series of this type are characterized by the following spectral ratio values: = 1­2 (in gen eral, up to 3). Secondly, the system that generates the observed series is a deterministic chaotic system. Examples of such systems have been offered and studied (Zakharov, 2004, 2011a), viz., models of block dynamics with dry friction playing the role of a positive nonlinear reverse relationship that results in chaotization (in both the model and in nature). These systems have dynamic characteristics that are similar in value to those obtained in the course of our investigation: = 0.9­2 and H = 0.8­0.9 (Table 2). Hence, the discrete dynamic system of the Earth's crust processes the incoming energy in a self similar way. The dynamics of systems with such characteristics assume that rela tively insignificant variations are accompanied by sub stantial outbursts. As well, the scale of these outbursts and the time intervals between them are not regular and cannot be predicted. CONCLUSIONS Hence, it can be concluded that the analysis and forecasting of seismotectonic and geodynamic systems of earthquakes and strong movements should be per formed by adequate methods that were developed using the theory of dynamic systems and fractals.
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It appears that it is impossible to unambiguously select any particular model on the basis of our results; a more detailed analysis is needed. Meanwhile, the established correlation between the dynamic parame ters of investigated displacement series and geody namic characteristics of the regions (rigidity of blocks and seismic activity) is indicative of the fact the deter ministic­chaotic processes (along with random ones) make some contribution to the observed time series of displacements. ACKNOWLEDGMENTS We thank M.V. Rodkin for a productive discussion in the course of the preparation of the paper. REFERENCES
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Translated by E. Maslennikova

MOSCOW UNIVERSITY GEOLOGY BULLETIN

Vol. 68

No. 3

2013