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ISSN 0145 8752, Moscow University Geology Bulletin, 2010, Vol. 65, No. 3, pp. 177184. © Allerton Press, Inc., 2010. Original Russian Text © V.S. Zakharov, D.A. Simonov, 2010, published in Vestnik Moskovskogo Universiteta. Geologiya, 2010, No. 3, pp. 2531.

An Analysis of Modern Discrete Movements of the Earth's Crustal Blocks in Geodynamically Active Regions Based on GPS Data
V. S. Zakharov and D. A. Simonov
Moscow State University, Moscow, 119991 Russia e mail: zakharov@dynamo.geol.msu.ru; simon@geol.msu.ru
Received March 24, 2009

Abstract--In this work original algorithms are proposed that allow us to provide a set of points (clusters) on a surface that are in cooperative motion, on the basis of an analysis of GPS data. A single angular velocity vec tor can describe this movement. The selected clusters are interpreted as rigid blocks. Methods for determining the relative movements of blocks have been developed. The possibilities of the methods are demonstrated using the example of an analysis of blocks in Western Turkey. Key words: geodynamics, kinematics, the Earth's crustal blocks, Euler pole, relative movements. DOI: 10.3103/S0145875210030038

INTRODUCTION High precision measurements using GPS (global positioning system) data are widely used to solve prob lems of modern geodynamics and to determine the current values of the velocities of displacements of the Earth's surface. A GPS system allows us to connect the results of determination of displacement and velocity with a unified coordinate system. This makes it possible to compare and combine the data from dif ferent points of a GPS network if the networks have at least a few common tie points and the time intervals of the measurement are well known. During the last few years many measurements have been made in geody namically active regions, such as the AegeanAnato lian region, Southern California, and Japan, as well as the first attempt to combine data from different GPS networks. As a result, sufficient representative data sets for analysis of the global movements of plates and microplates have been obtained, and the first attempts to define discrete motions of blocks have been made, which represents a new area of research. Such works have not been carried out yet, due to the lack of data and the fact that the techniques for such analysis have still not been developed. One of the earliest works in this area of research is Kahle el al., [2000], in which the displacement points of GPS data in areas at the boundaries of the Eurasian, Arabian and African plates were analyzed, and the values of strain rate in these areas were calculated. In Reilinger et al. [2006] a model of elastic blocks was developed using an analysis of GPS data for the zone of interaction of these plates, which is in agreement with current plate motion (rel ative to Euler poles), the strain values in the interplate zone, and the values of the strain rate along the main faults.

In McClusky et al. [2000] a preliminary analysis and the summation of a set of high quality GPS data from Western Greece to the Aegean arc and Western Turkey was made. These authors used the data to detect hard blocks in the Central Anatolia and South Aegean regions. In particular, they were first to detect the South Aegean microplate and calculate its motion from the east of the Aegean region to Southwestern Peloponnesus. A good summary of the ideas on the geodynamics of the Aegean region is given in Nyst and Thatcher, [2004], in which a quantitative model with four micro plates was proposed, which makes it possible to describe the modern dynamics of the Aegean region quite well. However, the analysis assumes the existence of isolated deformation zones within the nominally rigid microplates. Similar research has been carried out for other regions. There are a number of detailed works devoted to analysis of the dynamics of the joint zone between the North Pacific plate and the North American plate on the basis of GPS data [McCaffrey, 2005; Meade, Hager, 2005; Prawirodirdjo, Bock, 2004; Williams et al., 2006], which taking the quality of the GPS net work in the western United States, particularly in Cal ifornia, into account. The above mentioned works, despite the differences in methods used for determina tion of the quantitative characteristics of the motions of blocks, are characterized by a common approach to the problem of the detection of blocks. Firstly, based on geological, tectonic, geomorpho logic, etc. data relatively rigid (or rigid elastic) blocks and their boundaries are detected, then the GPS data that are obtained in a given block are analyzed, and the kinematic characteristics of motions (the poles of

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rotation, the values of the angular velocity, relative motions) are calculated. A somewhat different approach has been proposed [Zubovitch et al, 2004, 2006, 2007; Kuzikov, 2007; Bogomolov et al., 2007] for the analysis of the current crustal deformation in Central Asia on the basis on GPS data. In these works methods for the detection of lightly deformed sections of crust, which are based on a dis crete set of horizontal velocity values and detection of the domain (block) structure of a field of modern motions for the region of the Central Asian GPS net work, were developed. In addition, the spatial rela tionships between neotectonic discontinuous struc tures, domains, and areas between them were defined, as well as the kinematics of these objects was studied. The detection of blocks is mainly based on the analysis of kinematics, using a complex method based on the method of least squares when the curvature of the Earth's surface is not taken into consideration and the calculations are carried out for the planar case. The aim of this work [bold] is to develop mathe matical methods and software based on GIS, and to create discrete kinematic models for individual regions on a spherical surface based on GPS data. However, we will not develop the complete scheme of the block dynamics of the region, but only demon strate the possibilities of the method proposed. In Simonov et al [2006] the developed methods for the analysis of modern motions of discrete blocks in geodynamically active regions according to GPS data were reported. The method was tested using the exam ple of the AegeanAnatolian region. This is due to the fact that many researchers have studied the geody namics of this region during the last 3035 years, and, as mentioned above, have collected a lot of GPS data. Some results of this analysis are given in this work. In addition, the method we tested using the GPS net works of Southern California demonstrated good results. THE METHOD OF ANALYSIS In order to analyze the GPS data and to develop a kinematic model of blocks we have developed a special method based on the analysis of motions on a spheri cal surface. The GPS velocity data from all sources are given in the so called local Cartesian coordinate sys tem, which has three components: the northern (n), eastern (e) and vertical (d): V = (Vn, Ve, Vd). In this case only horizontal motions are of interest, so Vd = 0. For further analysis the values of velocity are transformed into the global system coordinates of V = (Vx, Vy, Vz), which are associated with the center of the Earth [Cox, Hart, 1989]. Motions of plates, particularly small ones, are quite complicated. They can be represented, firstly, as a rotation around various poles, chosen by any method, and, secondly, as a rotation around an inner axis, and

thirdly, as a combination of both types of motion. Since the direct vectorial operations with the velocity values of the GPS points in the local coordinate sys tem can result in significant errors under the analysis of differential motions one of the first and most funda mental tasks is the detection of individual blocks, which are in differential motion. The standard methods of calculation of Euler poles according to linear velocity data [Zonenshayn, Kuzmin, 1992; Cox, Hart, 1989] were used to analyze the velocity of individual GPS points. This work pro poses original algorithms that allow us to analyze (on the basis of simple enumeration) sets of points ("clus ters") belonging to the same Euler pole and having the same angular velocity (within the specified error). In other words, sets of surface points were selected that are now in coordinated movement, which can be described by a single angular velocity vector . These clusters are interpreted as rigid kinematic units (per haps temporary ones). The consistency of motion was a criterion for interpretation. Hereafter, using the detected "clusters" (blocks) and calculated motion parameters the coordinates (latitude p and longitude p), the pole of rotation P (p and p), the angular velocity , and the relative motions of individual blocks are calculated. METHODS OF CLASSIFICATION OF INPUT DATA AND THE DETECTION OF KINEMATIC BLOCKS (CLUSTERS) The Basic Algorithm (K1) As a basis for the detection of kinematic blocks we have adopted the simplest method, which is based on enumeration. The algorithm of the method is as fol lows: (1) There is a data set of N observation points of the GPS. From the set two points T1 and T2 having GPS velocities of V1 and V2 are selected, whose the Euler pole was not defined; (2) On the basis of the direction of the velocity V1 and V2 the position of the Euler poles of P (p and p) for the points T1 and T2, as described in Zonenshayn and Kuzmin [1992] are calculated; (3) On the basis of the values of linear velocity and calculated position of the Euler pole P for each point of T1 and T2 the values of the instantaneous angular velocity 1 are 2 are calculated. If the values of 1 and 2 are similar (that is, |1 2| , where is the prescribed accuracy (the maximum error) of the value of the angular velocity ), the Euler pole of P is considered as correct and stored, and the points T1 and T2 are attributed to the Euler pole P1. If the difference is |1 2| , then another point T3 is selected from the data set, the Euler pole of which is not still defined; the point T2 is replaced by T3, and finally the operations p. 2 and p. 3 are
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repeated until you have found at least two points belonging to one pole of rotation P1; (4) From the data set an arbitrary point Ti, the Euler pole of which is not still determined, is selected and the azimuth of motion A 'i at this point and the angular velocity i are calculated; these would have been determined during the rotation of this point around the previously defined pole P1. These calcu lated (test) values of azimuth and angular velocity are compared with the actual ones. If the difference between the calculated ( A 'i ) and the actual (Ai) azimuths is less then the previously given azimuth error A(| A 'i Ai|) A, and i and 1 are close to (|1 i| ), then the point Ti is referred to the same pole P (belongs to the same block), if not, then the other points of the whole data set are checked; (5) If as a result of the enumeration of all the points there are those left for which the rotation pole is still not defined, then go to p. 1 to find the Euler poles of P2, P3, P4 etc. for other blocks. We can expect that the main limitations of the method are the definite dependence on the choice of an initial pair of points and the potential possibility to attribute a point to a false pole. However, during the practical testing of the method with "shuffled" data bases such limitations were not revealed. The selected "clusters" were the same regardless of the starting point. The Method of Minimization of Errors (K2) This method is a modification of the method K1 and has the following algorithm: (1) pp. l5 coincide with the relevant paragraphs of the method K1; (6) all the points are checked once again. The errors of determination of azimuth and angu lar velocity for rotation around each of the found poles P1, P2, P3, etc. are calculated. Finally, the point is referred to the pole for which the errors of azimuth and angular velocity are minimal. The advantage of this method is that the points belong to the pole that provides the minimum errors. In this case, the following limitation of the method was revealed: within the selected clusters "foreign" points occur, which formally belong to the same pole, but are actually surrounded by points with another pole. This requires research experts to be attentive. Variational Method (VM) This method is used in addition to the methods of K1 and K2. According to this, it is possible to calculate few poles for each point, meeting the criteria of accu racy; that is, previously, each point can belong to mul
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T1 T2

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Fig. 1. The scheme of calculation of velocity of relative movement using the O1 method. Commentaries are given in the text.

tiple clusters. The final decision is made by a researcher on the basis of additional information obtained. The algorithm of the method is as follows: (1) pp. l5 coincide with the relevant paragraphs of the method K1; The errors in determination of azimuth and angu lar velocity for every pole found, viz., P1, P2, P3, etc. are calculated. The point applies to the pole for which the difference of azimuth and angular velocity is less than the specified error A, , and which meets the additional criteria of the researcher (tectonic, geo morphologic, etc.). Note that all these methods cannot take into account the internal rotation of a block around its own axis, so there is no most correct and best method for classification and definition of the Euler pole on the basis of the GPS data. Therefore, the described meth ods of classification (clusterization) are preliminary and more detailed analysis should be carried out by an expert researcher. The sets of points (clusters) detected by this method can be interpreted as those that belong to separate rigid blocks, but after the confirmation of this fact by geological data. METHODS OF CALCULATION OF THE RELATIVE MOTIONS OF BLOCKS To study the relative motion of plates or blocks it is necessary to calculate the values of the relative veloc ity. We offer four methods of calculating the relative motions, which can be used depending on the task and the quality of the source data. Method O1. In this method, the relative motions of blocks are calculated by subtraction of values of linear velocity of points in the local coordinate system. The method has the following algorithm (Fig. 1): (1) select a point T1, which has a local linear veloc ity VT1 = (VT1n, VT1e) (2) for any other point T2 with a velocity in the local system VT2 = (VT2n, VT2e); the velocity relative to the point T1 (V ' = VT2 VT1) has the next components: V n' = VT2n VT1n and V e' = VT2e VT1e.
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P'

' O

Fig. 2. The scheme of calculation of velocity of relative movement using the O3 method. Commentaries are given in the text.

Fig. 3. The scheme of calculation of velocity of relative movement using the O4 method. Commentaries are given in the text.

Method O2. Firstly, in this method, the values of the velocity are converted from the local coordinate system to the global one. The relative motions of blocks are calculated by subtraction of the values of the linear velocity of points in the global coordinate sys tem. Finally, the relative values of the linear velocities are converted back into the local system. Both these methods (O1 and O2) give acceptable results in the absence of large rotations of blocks and they are reliable for calculating the relative motion of close local points (when the sphericity of the Earth's surface can be neglected). The disadvantage of the method is that one can get the relative motions of the points only in the block itself. For cases where the motion of blocks have a well defined rotation around the Euler pole (that is, at even a rough approximation their movement cannot be regarded as progressive), methods O3 and O4 were developed. Method O3 allows us to determine the relative motion of points, calculating the velocity of their movements in the local coordinate system at rotation around different poles and finding the difference between the values obtained. The method has the fol lowing algorithm (Fig. 2): (1)Select the block that rotates around the pole P (p, p with angular velocity (that is, it is character ized by the angular velocity ); (2) Select a point T(, ), which may belong to another block with the other pole of rotation P2; its actual (measured) velocity in the local coordinate sys tem VT = (Vn, Ve); (3) Calculate the velocity of rotation (VTP) of point T relative to the pole of P (as if it belongs to the pole P) in the global system VTP = [, r], where r = (rx, ry, rz) is a radius vector of the point T (in the global coordi nate system). Components of relative velocity VTPx = yrz zry, VTPy = zrx xrz, VTPz = xry yrz, (4) The rate VTP is converted into a local coordinate system;

(5) Calculate the relative velocity in the local coor dinate system V ' = VTP VTP, which has the compo nents V n' = Vn VTPn, V e' = Ve VTPe. The results obtained using the method O3 are more correct than the methods O1 and O2. The disadvan tage is the possibility of obtaining the motion of a point relative to its block. This is due to the fact that during the clusterization we referred the points to a block with a certain error and therefore only the movements whose quantitative characteristics are more than the defined errors should be considered as significant. Method O4. In this method the determination of relative motions of blocks is made using their angular velocities and the coordinates of Euler poles (that is, the angular velocity vector) in the global coordinate system. The method has the next algorithm (Fig. 3): (1) Choose the block and the angular velocity belong to the block that and the angular velocity with the Euler pole P1(1, 1) vector 1 and the point T that has the Euler pole P2(2, 2) vector 2.

(2) Calculate the angular velocity vector of block 2 relative to block 1: ' = 2 1; (3) Calculate the relative linear velocity of the point T in the global coordinate system V ', where r = (rx, ry, rz) is the radius vector of the point T (in the global system of coordinates). Components of the relative velocity are V 'x = 'y rz 'z ry, V 'y = 'z rx 'x rz, V 'z = 'x ry 'y rx. (4) Calculate the relative velocity in the local coor dinate system. In this method, the points within the block remain stationary in relation to one another within a given error if the block does not undergo internal rotation. The method works in all circumstances, and the most correct results are obtained.
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Fig. 4. The kinematic blocks that were detected using the K2 method for the test region in West Turkey, shown by the frame in the inset map. Points in different blocks are shown by different signs; their velocities are shown by different arrows. The wide dashed bands are the possible boundaries of the blocks detected, taking into account the geological data; the dashed lines are more correct determinations of the boundaries of the blocks, which were detected on the basis of the geomorphologic data.

APPLICATION OF METHODS (FOR EXAMPLE, THE DETECTION OF BLOCKS AND DETERMINATION OF THEIR KINEMATICS IN WESTERN TURKEY) The above described method was tested during the analysis of the data presented in Nyst and Thatcher [2004] for the AegeanAnatolian region. The classifi cation was carried out with the following parameters: the acceptable error in the angular velocity values is ±15% and the acceptable deviation of the motion vec tor is ±5%. We preselected and described only a few clearly detected blocks that have no significant own rotation and are characterized in the first approxima tion by a fairly simple translational motion. The choice of the specific blocks was caused by the representativeness of the initial GPS data and geological data. The analysis carried out made it possible to detect a number of blocks having different poles of rotation and thus, differential motions. The boundaries of blocks are confirmed by geolog ical data. This work does not allow one to represent the results obtained in full volume, so there is only one example in this article showing the relationship of blocks in Western Anatolia. Note, firstly, that during the detection of crustal blocks with different kinemat ics one cannot rely only on formal mathematical
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methods, as a definition can contain significant errors associated with limitations of the methods such as clusterization and the determination of the relative motions. It is necessary to make a thorough analysis of the geological structure and morphology of the relief of the studied areas. However, there are additional complications related to the fact that the boundaries of modern mobile blocks are not always correlated with geological and morphological data. So, linear zones that are well expressed in the relief, which could be interpreted as active block boundaries, may be passive ones at the present stage. The relative motions of points located on different sides of morphologically well expressed and straightened valleys, which are usually interpreted as faults, can be either negligible or absent. In addition, the motion of these points can be described as the rotation of a single block around a sin gle pole. On the other hand, the boundaries of individ ual moving blocks relative to each other may not be clearly manifested in the relief, although some mor phological peculiarities may indicate developing active structures. Here one can talk more about the detection of cer tain regions with different kinematics (Figs. 46), whose boundaries (broad dashed lines) are very diffi cult to accurately determine. In this case, when deter mining the boundaries of blocks it was taken into
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Fig. 5. The velocity of movement of blocks (arrows), which was obtained using the O1 method, relative to the Bakha point. The legend is the same as in Fig. 4.

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Fig. 6. The velocity of movement of blocks (arrows), which was obtained using the O4 method, relative to the Bakha point. The legend is the same as in Fig. 4. MOSCOW UNIVERSITY GEOLOGY BULLETIN Vol. 65 No. 3 2010

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GPS network does not allow us to talk about them with confidence. With regard to the relative movements of the blocks allocated, the motions calculated by method O1 (rela tive to the Baha point) are not significant (see Fig. 5); the velocity values calculated by method O4 (Fig. 6) reach 510 mm/year. This difference is connected with the position of the Euler poles of the blocks detected, and with the relative positions of the blocks themselves (Fig. 7). Under this configuration, method O1, for which the calculation of the relative motions is made in a local Cartesian coordinate sys tem, will certainly give an error in determining the rel ative velocity. Method O4 is more correct. The results obtained can be illustrated by closely located pendu lums, which are fixed in different locations (see the insert map in Fig. 7). During turning of the pendulums the vectorial subtraction of their linear velocity values gives a very small relative velocity, whereas the displace ment of blocks of pendulums will be very noticeable. CONCLUSIONS

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In this work mathematical methods are proposed and developed software based on GIS is demonstrated for the development of discrete kinematic models of individual regions of GPS data, taking into account the sphericity of the Earth's surface. The preliminary kinematic analysis has shown that GPS data can be used not only to determine the motion of plates and microplates, but also for the detection of small mobile blocks, which reflect the internal differential kinemat ics of smaller structural elements. However, along with an analysis of the velocity field, a thorough analysis of geological information is necessary. REFERENCES
1. Bogomolov, L., Bragin, V., Fridman, A. et al., Compar ative analysis of GPS, seismic and electromagnetic data on the central Tien Shan Territory, Tectonophysics, 2007, vol. 431, pp. 143151. 2. Cox, A. and Hart R., Tektonika plit (Plate tectonics: How it Works), Moscow: Mir, 1989. 3. Kahle, H.G., Cocard, M., Peter, Y., et al., GPS derived strain rate field within the boundary zones of the Eur asian, African, and Arabian Plates, J. Geophys. Res., 2000, vol. 105, no. B10, pp. 23 35323 370. 4. Kuzikov, S.I., Structural analysis of horizontal veloci ties by GPS and the nature of current crustal deforma tion in Central Asia, Abstract. of Cand. Sci. Dissertation, Moscow, 2007. 5. McCaffrey, R., Block kinematics of the Pacific North America plate boundary in the southwestern United States from inversion of GPS, seismological, and geologic data, lbid., 2005, vol. 110, no. B07401 (doi:10.1029/2004JB003307). 6. McClusky, S., Balassallian, S., Barka, A. et al., Global Positioning System constraints on plate kinematics and
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Fig. 7. The relative location of Euler poles (stars) of the blocks detected and angular velocities (arrows). In the insert map the model of the relative motions of blocks at the given location of rotation poles is shown.

account that the points attributable to different Euler poles are confined to areas with different geological structures. In this case, when determining the bound aries of blocks it was taken into account that the points attributable to the different Euler poles are confined to areas with different geological structure. For example, the points of Cape Baba (the points are shown with gray triangles and the velocity values with gray arrows) are confined to areas of distribution of MioceneQua ternary sediments, whereas the points of the Baha block with a central point (the points are shown by white triangles and the velocity values by white arrows) are confined to areas of the metamorphosed basement (shaded box in Fig. 46). More accurate detection of block boundaries (dotted lines) is possible on the basis of the geomorphologic data, but the density of the
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ZAKHAROV, SIMONOV dynamics in the eastern Mediterranean and Caucasus, lbid., 2000, vol. 105, no. B3, pp. 56955719. Meade, B.J. and Hager, B.H., Block models of crustal motion in southern California constrained by GPS measurements, lbid., 2005, vol. 110, no. B03403 (doi:10.1029/2004JB003209). Nyst, M. and Thatcher, W., New constraints on the active tectonic deformation of the Aegean, Ibid., 2004, vol. 109, no. B11406 (doi:10.1029/2003JB00283C). Prawirodirdjo, L. and Bock, Y., Instantaneous global plate motion model from 12 ye ars of continuous GPS observations, Ibid., 2004, vol. 109, no. B08405 (doi:10.1029/2003JB002944). Reilinger, R., McClusky, S., Vernant, P., et al., GPS constraints on continental deformation in the Africa Arabia Eurasia continental collision zone and implica tions for the dynamics of plate interactions, Ibid., 2006, vol. 111, no. B05411 (doi:10.1029/2005JB004051). Simonov, D.A., Zakharov, V.S., and Liu S., Methods of analysis of modern discrete block movements in Geo dynamically active regions according to the GPS (for example, the Aegean Anatolian region) in Areas of active tectonogenesis in the modern and ancient history of the Earth, Proceedings of 39th tectonic meeting, Mos cow: GEOS, 2006, issue 2, pp. 215219. 12. Williams, T.B., Kelsey, K.M., and Freymueller, J.T., GPS derived strain in northwestern California: Termi nation of the San Andreas fault system and convergence of the Sierra Nevada Great Valley block contribute to southern Cascadia forearc contraction, Tectonophysics, 2006, vol. 423, pp. 171184. 13. Zonenshayn, L.P. and Kuzmin, M.I., Paleogeodi namika (Paleogeodynamics, Moscow: Nauka, 1992. 14. Zubovitch, A.V., Beisenbayev, R.T., Van Syaochan, et al., Recent Kinematics of the Tarim Tien Shan Altai Region of Central Asia from GPS Measurements, Izvestiya, Physics of the Solid Earth, 2004, no. 9, pp. 3140. 15. Zubovitch, A.V., Makarov, V.I., Kuzikov, S.I., et al., Intracontinental mountain building in Central Asia as inferred from satellite geodetic data, Geotectonics, 2007, no. 1, pp. 1629. 16. Zubovitch, A.V., Mosienko, O.I., Kuzikov, S.I., and Mellors, R., Study of modern tectonics of the Tien Shan on the data of satellite geodesy in Areas of active tectonogenesis in the modern and ancient history of the Earth, Materials of 39 tectonic meeting, issue 1, Mos cow: GEOS, 2006, pp. 243244.

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