The source parameters of earthquakes of Bishkek geodynamic proving ground (Northern Tien Shan)

Based on the method of polarity of signs of P-waves, the focal mechanisms of 1674 earthquakes with M  1.6, which occurred on the territory of the Bishkek geodynamic proving ground (BGPG) from 1994 to 2020, were determined. Some characteristics of the complete catalogue are presented. Quantitative distributions by the type of mechanisms and diagrams of azimuths of the main stress axes are constructed. A variety of focal mechanism of earthquake is observed, most of them are reverse fault, oblique reverse fault, and horizontal strike-slip fault. The compression axis for most of the events has a north-northwest direction and a sub-horizontal position. For 183, dynamic parameters (DP, source parameters) were obtained: spectral density Ω0, corner frequency f0, scalar seismic moment M0, source radius (Brune radius) r, and stress drop Δσ. The correlations between DP and energy characteristic (magnitude) and scalar seismic moment are investigated. The smallest correlation coefficient was obtained for stress drop.


Introduction
The Kyrgyz seismological network, installed in 1991 on the territory of the Northern Tien Shan, consists of 10 digital, broadband stations, each of which is equipped with an STS-2 sensor (Fig. 1). It was found that the stations of the KNET network are low-noise [1] and site under stations does not generate resonance effects [2]. The catalogue of earthquakes created on the basis of the KNET network database is representative [3], which makes it possible to carry out various seismological studies on its basis. The propagation of seismic waves in rocks is a complex process characterized by a number of parameters. It is conventionally accepted to divide the parameters into kinematic and dynamic. On the basis of seismic waves, various source parameters of earthquakes are calculated, which are also usually divided into kinematic and dynamic ones. The focal mechanism is a geomechanical model of an earthquake source based on instrumental data from seismic records. The source model, presented in three-dimensional form, characterizes the orientation of the axes of the main relieved stresses and the position of the nodal planes of displacement along the rupture in the seismic source and is indirectly related to tectonic stresses. The focal mechanism refers to the kinematic characteristic of the source.

Fig. 1. Position of KNET network stations (triangles) and repeaters (squares).
The dynamic characteristics of seismic waves make it possible to determine such source parameters of earthquakes as the source radius r, scalar seismic moment M0, stress drop  etc., which are related to dynamic parameters. The purpose of this work is to calculate the source parameters of earthquakes for their further use in determining the stressed-deformed state of the earth's crust of Northern Tien Shan.

Methodology
Focal mechanisms (FM). The overwhelming amount of experimental information accumulated in modern seismology on the mechanisms of earthquake sources was obtained on the basis of data on the polarity of arrivals of seismic waves. At the present time, in the conditions of equipment of an increasing number of seismic stations with digital equipment, this source continues to be the most important [4][5][6]. When interpreting the picture of the first arrivals, a two-dipole seismic source is considered according to [7]: where t, p are unit direction vectors for the tension and compression axes. The nodal surface for P-waves are two mutually perpendicular planes in the considered case of an isotropic elastic medium. As a rule, the focal mechanism is characterized by the orientation in space of its main axes.
One of the conditions for the reliable determination of the focal mechanism from the signs of the arrival of P-waves is the completeness of the surrounding of the epicenter by observation points. In the conditions of the KNET network, for the calculation of the focal mechanism, only those earthquakes that occurred in the territory bounded by the coordinates of the network edge stations can be considered: 42.0-43.0° N. and 73.75-76.0° E, which is a small area of 100  300 km 2 located on the territory of the Northern Tien Shan (green rectangle in Fig. 2). Potentially, the focal mechanism can be determined for earthquakes marked in black. The earthquakes for which the focal mechanisms were obtained are marked in green. They account for more than 55% of the number of events that occurred in the territory bounded by the green rectangle. To calculate the focal mechanisms, the FPFIT program was used [8][9][10][11][12], which finds solutions for a double pair of rupture planes (initial model) that best suit the given set of recorded values of the first arrivals of an earthquake. Dynamic parameters. To calculate the dynamic parameters, it is necessary to construct a source spectrum and determine its characteristics -spectral density 0 and corner frequency f0 (time of rupture in the source). The method for calculating the source spectrum in a KNET network is presented in detail in [13][14] and is not considered here. The spectral density 0 is used to determine the scalar seismic moment M0. To calculate M0 the expression [15] is used: where R is the distance from the source to the reference point, where the spectrum is determined,  is the density of rocks in the source area, 0 is the directional factor of radiation from the source (for a source described by a displacement jump on the rupture plane), the average value of this factor is taken as 0.64 [16]. The source radius is determined by the formula [17][18][19] r = kVS/f0, (3) where k is a numerical coefficient depending on the source rupture model. The work uses the Brune model [14], for which this coefficient has a value of k = 0.37. To calculate the source radius, it is necessary to know the corner frequency, which is determined from the source spectrum and corresponds to the rupture time in the source point. Fig. 3. Earthquakes epicentral location of (183 events).The minimum circle corresponds to the event with М = 2.7, the maximum -with М = 6.0 Triangles -KNET stations location. Black lines are regional faults.
As the initial data, we used the waveforms of earthquakes that occurred in the territory bounded by the coordinates of the edge stations of the KNET network for 1998-2017 with K = 8.7 -14.8. Geological faults, geographical names of some ridges in the study area and the epicentral location of the seismic events under consideration are shown in Fig. 3. The main part of earthquake epicenters is located on the northern slopes of the Kyrgyz ridge, ridge Karamoynok, mount Sandyk and in the area of the Suusamyr and Kochkor depressions.

Results
Focal mechanisms. As a result of the application of the method for determining the focal mechanisms based on the polarity of the arrival of P-waves at the station, the focal mechanisms of 1674 earthquakes were determined and a catalogue was compiled (Fig. 4). The main part of the catalogue of focal mechanisms is made up of weak events (М = 1.6-2.5) that occurred at a depth of 5-15 km. Most of the earthquakes included in the catalogue occurred from 1999 to 2020. Solutions were received for 7 or more components, 43% of solutions were received for 9-10 components. To build a map of focal mechanisms with a designation of the type of movement by color, samples were formed based on the analysis of the plunge angles of the main stress axes (plunge) Ppl and Tpl. The sampling criterion and quantitative results are presented in table 1. Table 1. Criteria for choosing the type of mechanism, the number of earthquakes according to the criterion for choosing the type of mechanism for the entire catalogue and for individual depths  According to the results obtained, the direction of the compression axes changes from north-north-west to north-north-east (the scatter is limited to the 330° -10° sector) (Fig.  4Bb). For extension axes, the direction of the axes changes from northeast to southeast (the scatter is limited to a sector of 60° -110°) (Fig. 4Bc). This is in good agreement with previously published results [24][25][26]. The sub-horizontal position (up to 30°) has 75% of the compression axes and 53% of the tension axes (Fig. 4Bd).  fig. 5B shows diagrams of the Pand T-axes directions for the reverse fault and normal fault mechanisms. P-axes for reverse fault events have three directions: north-north-west, north and north-north-east, T-axes are mostly oriented sublatitudinally. The P-axes for normal fault mechanisms have many directions, the most representative is the northeast direction, and for the T-axes the western direction is the most representative. It is noted in [27] that, having different rheological properties, the layers of the earth's crust of the Tien Shan react differently to the tectonic compression they experience. According to the study of the azimuths of the compression axes at all considered depths (see Table 1) changes in the 330º -10º sector, however, the maximum peak value of the number of events with depth shifts from 330º (northwest direction) to 360º (north). The direction of the extension axes for all depths varies from northeast to southeast. Dynamic parameters. The dynamic parameters for 183 earthquakes of the Northern Tien Shan have been determined. When calculating the scalar seismic moment M0, source radius r, stress drop , emitted seismic energy ES and reduced seismic energy ePR according to formulas (2) - (6), the values of the rock density in the source and the shear wave velocities were taken to be equal, respectively,  = 2.6 g / cm 3 [28], VS = 3.5 km/s [29]. The boundary values of the obtained DP values are presented in table. 2. In the further, the DP of 182 earthquakes will be considered. The earthquake of December 25, 2006 with K = 14.7, M0 ~ 210 17 Nm and ∆ ~ 214 MPa (which radically differs in value from other events) is excluded from the analysis. The dependences of the dynamic parameters on the scalar seismic moment, the magnitude M, recalculated from the class K, were constructed, and the correlation coefficients between the DP and the magnitude, between the DP and the scalar seismic moment were calculated. Table 3 the obtained correlation coefficients are presented, as well as some regression dependences. The largest correlation coefficient between the seismic moment M0 and the seismic energy ES (k = 0.98), the smallest between the seismic moment M0 and the stress drop  (k = 0.74). Let us consider the features of the distributions of the parameters of the spectral characteristics Ω0, f0 and the dynamic parameters for 182 events. Spectral density and scalar seismic moment. The distribution of the parameter Ω0 (on a logarithmic scale) depending on the magnitude is shown in Fig. 6a, and the distribution of the scalar seismic moment for the same events (M0 ~ Ω0) is shown in Fig. 6b. These logarithmic distributions are well described by the line model. For a scalar seismic moment, the regression dependence on magnitude (best model) can be written in the form:  For the events under consideration, the range of magnitude variation is 2.2  M  6.0, and the values of M0 lie in the range from 6.210 12 to 2.210 17 N·m. It should be noted that, due to the presence of a scatter and a relatively narrow range of magnitudes, the distribution of lg M0 also agrees (with an accuracy of 10%) with the well-known expression lg М0 = 1.6M +8.4 [16].
Taking into account the high value of the correlation coefficient between the magnitudes and the values of the logarithm of the seismic moment, in the future, in a number of cases, we will consider the distribution of DP depending on lg M0 Corner frequency and source radius. In fig. 7a shows the distribution of f0 as a function of lg M0. The corner frequency is characterized by a decrease with increasing lg M0 or magnitude. The source radius was determined from the values of the corner frequency using formula (2). The distribution of r values depending on the logarithm of the seismic moment is shown in Fig. 7b. The Brune radius r various from approximately 210 to 640 m. Seismic energy and reduced seismic energy. The logarithmic distribution of the seismic energy ES and the reduced seismic energy еPR in comparison with the values of lg M0 is shown in Fig. 8. Seismic energy varies in the range from ~1.010 7 to ~8.010 12 J (see Table  2). The linear model describing the ES distribution has the form lg ES = 1.6lg M0 -13.2, and the correlation coefficient between seismic energy and scalar moment is k = 0.98. For the events under consideration, the еPR parameter is on the order of 1.810 -6 to ~0.310 -3 (see Table 2). The distribution of lg еPR is also described by a linear model, although for it the scatter of values relative to the regression line is greater in comparison with the case in Fig.  8a. The expression for the linear model (regression dependence) has the form lg ePR = 0.6lg M0 -13, M0 (Nm), (8) moreover, the correlation coefficient between the values of lg еPR and the logarithm of the scalar seismic moment is 0.92. Fig. 8b shows that the еPR values are localized in a strip, the boundaries of which are parallel to the approximating straight line: about 95% of the imaging points fall inside such strips. The boundaries of the localization band of the lg еPR values in Fig. 8b are determined by the ratio: 0.59lg M0 -13.6  lg еPR  0.59lg M0 -12.9.
Stress drop. The values of the stress drop are in the range from 0.2 to ~ 40 MPa ( Table  2). There is only one earthquake on 25.12.2006, М = 6 with ∆  214, which is excluded from the analysis. The distribution of the stress drop together with the values of lg M0 is shown in Fig. 9a. The best approximation of the ∆ distribution is a power-law dependence on lg M0. Let us explain this circumstance. According to expressions (6), (4), (2), the values of the dropped stresses and reduced seismic energy are proportional: ∆ ~ ePR. Consequently, the distribution lg ∆ will differ from that shown in Fig. 8b of the logarithmic distribution of ePR only by a vertical shift by some constant. For the calculation results for ∆, the following approximation is obtained: ∆ = 3.010 -8 M0 0.59 . The correlation coefficient between drop stress and seismic moment is k = 0.74 (Table 3).   (Table 4), and for each of them the correlation coefficients were determined. The first two samples were made according to the values of lg M0, which characterizes the scale of earthquakes: events with lg M0 < 14.5 constituted the first sample, and events with lg M0  14.5 -the second. Two more samples were formed on the basis of stress drop: the third sample included events with ∆ < 10 MPa, and the fourth -∆  10 MPa. The distinction based on the ∆ feature corresponds with good accuracy to the selection of two clusters in the distribution of significant points in Fig. 9b using the clustering algorithm [30][31]. The distribution of stress drop for the first two samples is shown in Fig. 10a, the latter in Fig. 10b. The color of the icons indicates that the events belong to one of the selections.   Table 4 shows the values of the correlation coefficients between the stress drop and the logarithm of the seismic moment for the above-described samples, as well as for the general sample of 182 events. For samples with lg M0  14.5 and with ∆  10, the correlation coefficients are lower than for the general sample and there is a significant scatter of values relative to the regression line. The high value of the correlation coefficient for events with lg M0 < 14.5 is reflected in the graph by the fact that the points of the values of ∆ in Fig.  10a are more localized near the regression line than points for another sample of events (crimson). For ∆ < 10 MPa, the correlation coefficient is low, which is explained by a small change in the values of the stress drop for the considered sample of events (black dots in Fig.10b). Such formal correlations convey physical meaning to a lesser extent than the correlations between the stress drop and M0, or between the logarithms of these quantities. The obtained regression dependences, in which the exponent of M0 is not less than 0.37, as well as Fig. 10 indicate the absence of self-similarity of earthquake sources on the territory of the BGPG in terms of stress drop and reduced seismic energy (∆ ~ еPR).

Conclusion
For weak earthquakes that occurred on the territory bounded by the coordinates of the edge stations of the KNET network (completely surrounded by stations), the focal mechanisms of the sources were determined by the signs of the arrival of P-waves and a catalogue of focal mechanisms was formed, which includes 1674 earthquakes that occurred from 1994 to 2020 biennium. In the region under consideration, mainly three groups of sources are formed, differing in the nature of the movements occurring in them: reverse fault, oblique reverse fault, and horizontal strike-slip faults. The Normal fault and oblique normal fault events occur less frequently and are localized to a greater extent. A significant part of the events are reverse fault and oblique reverse fault events, most of which have a near-horizontal compression axis. For weak events, the prevailing direction of the azimuths of the compression axes is north-north-west (with depth this direction changes from northnorth-west to north), and for extension axes -east-north-east. A data bank has been formed on the dynamic parameters of earthquakes that occurred on the territory of the Northern Tien Shan (BGPG) in 1998-2017 (183 events). Distributions of dynamic parameters from magnitude M and scalar seismic moment M0 are constructed, regression equations and correlation coefficients of dynamic parameters and scalar seismic moment are obtained. For a scalar seismic moment, the regression dependence on magnitude has the form: lg M0 = 1.1M + 10.1 for BGPG, which is in good agreement with the well-known relationship given by Yu.V. Riznichenko. For earthquake sources on the territory of BGPG, regression was established -a power-law dependence of stress drop on the scalar seismic moment with a correlation coefficient of 0.74. The main contribution to the correlation is made by seismic events with M0 < 3•10 14 N•m (lg M0 <14.5). For stronger earthquakes with lg M0  14.5, the correlation coefficient between the values of  and M0 is no more than 2/3, and this makes it possible to understand the results of works in which no significant correlations were found between the stress drop and the seismic moment (they did not consider relatively weak developments). The presence of an increasing dependence of the stress drop and reduced seismic energy on the seismic moment, established for the BGPG, is consistent with the results on the power-law dependence of ePR(M0) in a number of other regions. The get data on focal mechanisms and dynamic parameters can be used to assess the stress-strain state.