Control Parameters of Magnitude — Seismic Moment Correlation for the Crustal Earthquakes

In connection with conversion from energy class KR ( 10 log R R K E  , where ER—seismic energy, J) to the universal magnitude estimation of the Tien Shan crustal earthquakes the development of the self-coordinated correlation of the magnitudes (mb, ML, Ms) and KR with the seismic moment M0 as the base scale became necessary. To this purpose, the first attempt to develop functional correlations in the magnitude—seismic moment system subject to the previous studies has been done. It is assumed that in the expression   10 0 , , log b L s i i M m M M k z M   , the coefficients ki and zi are controlled by the parameters of ratio 0 10 0 log log t t t a b M   (where 1 0 b t f   ; f0—corner frequency, Brune, 1970, 1971; M0, Nm). According to the new theoretical predictions common functional correlation of the advanced magnitudes Mm (mbm = mb, MLm = ML, MSm = MS) from log10M0, log10t0 and the elastic properties (Ci) can be presented as 10 0 10 0 log 2log m i i M d M t C    , where – 2 i i t z d b  , and – 2 i i t k C a  , for the averaged elastic properties of the Earth’s crust for the mbm the coefficients Ci = –11.30 and di = 1.0, for MLm: Ci = –14.12, di = 7/6; for MSm: Ci = –16.95 and di = 4/3. For the Tien Shan earthquakes (1960-2012 years) it was obtained that 10 0 10 0 log 0.22 log 3.45 t M   , and on the basis of the above expressions we received that MSm = 1.59mbm – 3.06. According to the instrumental data the correlation Ms = 1.57mb – 3.05 was determined. Some other examples of comparison of the calculated and observed magnitude—seismic moment ratios for earthquakes of California, the Kuril Islands, Japan, Sumatra and South America are presented.

, the coefficients k i and z i are controlled by the parameters of ratio 0 1 0 0 log log (where ; f 0 -corner frequency, Brune, 1970Brune, , 1971; M 0, Nm).According to the new theoretical predictions common functional correlation of the advanced magnitudes M m (m bm = m b , M Lm = M L , M Sm = M S ) from log 10 M 0 , log 10 t 0 and the elastic properties (C i ) can be presented as 10 0 10 0 log 2 log , and -2 , for the averaged elastic properties of the Earth's crust for the m bm the coefficients C i = -11.30and d i = 1.0, for M Lm : C i = -14.12,d i = 7/6; for M Sm : C i = -16.95and d i = 4/3.For the Tien Shan earthquakes (1960-2012 years) it was obtained that

Introduction
In world practice, seismological research in assessing the scale of earthquakes magnitude scale of Gutenberg and Richter [1-3] is fundamental.In the countries of the former Soviet Union has been used scale independent energy class K R , defined as the logarithm of the seismic energy E R , highlighted by an earthquake, measured in joules (K R = log 10 E R , [4][5][6]).
For crustal earthquakes Tien Shan when considering the transition to magnitude scale was necessary to develop a self-consistent system of quantitative relationships that justify numerous empirical relationships bodywave magnitude m b , local magnitude on surface waves M L , surface wave magnitude for M S and K R from seismic moment M 0 (N•m), as the reference scale.In connection with the above purpose is to study the quantitative relationships m b , M L , M S and energy of seismic radiation E S c M 0 based on the following findings: 1) proportional magnitudes and the maximum amplitude of seismic vibrations [1-3]; 2) the statistical dependencies of the average magnitude of displacement along the fault u [7][8][9][10][11][12] and u functional relationship with the seismic moment, the shear modulus μ and the gap area S [13][14]; 3) functional relationship corner period 1 0 s в t f   with M 0 , the source radius r 0 , speed S-wave v S and static stress drop Δσ [15,16], as well as the similarity of the angular frequency f 0 with a fundamental frequency of the acoustic Debye [17] f D , depending on the amount of source and the elastic properties of the geophysical medium [18].
Our further quantitative construction is based on the following empirical relationship Gutenberg and Richter [3,12] where E GR -seismic energy according to Getenberg and Richter, J; t 0 -fluctuations with a maximum duration of vibration speed А/Т in the near field (А-amplitude, Тperiod), s.
Use the following generalization of Soviet seismologists, which were introduced scale energy class K R [5], the magnitude of surface waves M LH (IC device) and body waves m PV on device SCM [4,9]: where E R -seismic energy according to [5], in J; K R = log 10 E R ; t m -increase the maximum duration of the seismic intensity in the near field, in sec.The basis of the theoretical constructs are the following functional relations [10,13,15,16,19]:   10 0 2 36 log 6.07 where r 0 -radius of the source, in м; ∆σ-static seismic stress drop, in Pа; t b -corner period, s; M W -moment magnitude; (E SK , in J; M 0 , in N•m; u in m; v S in m/s); for the constructions made t 0 = t b = t m .Many generalizations proved that for a wide range of changes log 10 M 0 or M W empirical correlations magnitude m b , M L and M S from M 0 are non-linear, as in Equation ( 8), as a function of value of n varies from 3 to 6, and is increase Δσ [7,12,[20][21][22][23][24].
However, for individual intervals M 0 or M W communication between magnitudes relationships and dependencies of the magnitude log 10 M 0 can be represented as linear relationships.

Justification Relations Magnitude-Seismic Moment
Based on the original definition of magnitude on Richter [25], under which the numerical value of the earthquake magnitude is proportional to the logarithm of the maximum oscillation decimal в m , expressed in microns (10 −6 м), it is assumed that an upgraded body-wave magnitude m bm (equivalent m b , m PV ) is (considering doubling в m on the ground at the focus): in (8) on the basis Equations ( 9) and (10) and Equation ( 12) value m bm equal (M 0 , N•m; t в , s; µ, Pa; v s , m/s): where , value С 1 determines the springiness of the geophysical environment at m bm .
Based on generalizations Christensen [26,27] for the crust taken: average density ρ = 2830 kg/m 3 , v S = 3600 m/s and 2 36.7 GPа in what follows, these quantities ρ, v s and μ taken as the standard.
When these elastic parameters of the geophysical medium expression Equation ( 13) is transformed to the following form: Seismic energy radiation E SK by Kanamori [19], based on Equations ( 8) and ( 9) and Equation ( 13) is: where Taken for the elastic parameters and subject [19].
On the basis of Equations ( 13)-( 16), reflecting the functional relationship of E SK from M 0 , t 0 , m bm and μ at E GR = E SK introduced upgraded the magnitude of surface waves M Sm (equivalent of M S , M W ), while maintaining that the formula Equation (1) Gutenberg and Richter [2,3], with Equation (9), Equations ( 15) and ( 16) will be: where Taken for ρ and v S C S value in Equation ( 17) is equal to C S = -16.95,and for the special case of Δσ = 3.67 MPa = const and E SK /M 0 = 5 × 10 -5 equality: We also introduce a modernized local magnitude on surface waves M Lm -equivalent M L [18,28], functionally interconnected with log 10 M 0 , logt 0 , K SK , m bm and M Sm : where C L = 0.
With the standard values ρ, v S and Δσ = 3.67 MPa, based on Equation (14) and Equation ( 17) we obtain the following theoretical relation: which is within the accuracy of the definitions of the same magnitude satisfactory empirical relation refined body wave magnitude ˆb m of M W for large earthquakes [19,29] which were used ˆb m to calculate the true maximum oscillation amplitude A g , taken from seismograms; . Here it should be emphasized that at a constant value of Δσ Equation (12) and Equation (14) the value of the maximum amplitude в m is proportional to Equation (20) agrees satisfactorily with other empirical relationship [9] (m PV = m b + 0.18): 2.86 0.525 The above quantitative ratios indicate that between modernized magnitudes M m (m bm , M Lm , M Sm ) and log 10 M 0 may exist linear functional relationship of the form: in which the coefficients k i and z i at the control parameter a t and в t in the ratio: 10 0 10 0 log log where ∆σ = const = 3.67 МPa в t = 1/3 = const and a t = -5.43,but for other cases в t is not a constant.
In view of Equations ( 23) and ( 24) correlations Equations ( 14), (17) and Equation (18) for m bm , M Sm and M Lm (standard values ρ and v S ) can be written as follows: which provide a self-consistent system of semi empirical inter magnitude dependencies.For example, the dependence of m вm from M Sm based on Equations ( 25) and ( 26) can be expressed as: which is в t = 0.33 and a t = −5.43ransformed into simple formula Equation (20).

Discussion of Empirical and Theoretical Relations Magnitude-Seismic Moment
Local magnitude-seismic moment.Since the value of the local magnitude is directly related to the maximum oscillation amplitude of the surface waves and the first inter magnitude connections [2,3] have been developed for California earthquakes, relations M L -0 0 log t M consider according to Thatcher and Hanks [30] in this region (2 ≤ M L ≤ 6.8).
For this region, the authors have taken ρ = 2700 kg/m 3 and v S = 3200 m/s, and by ( 13) and ( 17) a constant values will be: i.e. in accordance with (19) with increasing values of M 0 log 10 Δσ increases: M f t , said Nuttli [12] for mid-plate earthquakes.
If true theoretical Equations ( 13), ( 17) and ( 18), then Equation ( 29) and the relationship between M Lm and log 10 t 0 is given by: 10 0 log 0.37 1.68 which is in good agreement with the expression (3) Gutenberg and Richter [2] and Equation ( 5) Soviet seismologists [31] which allows to consider t 0 = t в = t m.
In Figure 2 shows the correlation log 10 t 0 and M L according to Thatcher [30], which also shows the relationship Equation (3) and Equation (30).The presented data show that the semi-empirical formula Equation ( 30) is in good agreement with generalizations instrumental data (Figure 2).It should also be noted that the M L = M Lm based on Equation (3) Gutenberg and Richter [2], and Equation ( 18) can be obtained which is in satisfactory agreement with the expression (29).
In Figure 3 in the range of 0.5 ≤ M L ≤ 6.8 shows the correlation ratio M Lm of M L for Southern California earthquakes [30], South-West Germany [32] and Central Japan [33].In calculations M Lm by Equation ( 18) for the earthquakes in these regions were considered elastic parameters of the geophysical medium according to these authors.The statistical data confirm the validity of our assumptions on the possible equality M L and M Lm (Figure 3).From numerous publications on nonlinear relations log 10 M 0 -M L acceptability of new assumptions considered on the basis of Hasegawa [34] for earthquakes in Eastern Canada.In the range 0 < M L ≤ 6.3 are two of the interval 0 < M L ≤ 3.9 and 3.9 ≤ M L ≤ 6.3, which have different dependencies on log 10 t 0 of M L and log 10 M 0 from M L [34].
For the first group of small earthquakes characterized by the following relationship (10 5 < ∆σ < 10 6 Pа): .On the basis of these empirical formulas for Equation (18) and Equation (24) with C L = −14.21(ρ = 2800 kg/m 3 and v s = 3800 m/s) Figures 4 and 5 shows the calculated dependences of log 10 t 0 from M Lm and log 10 M 0 from M Lm , which in satisfactory agreement with the relations log 10 t 0 − M L and log 10 M 0 − M L (Figures 4 and 5) by Hasegawa [34].
Finally, for the Southern California Earthquake Equation (18) and Equation ( 29) we can obtain the following relationship: According to Equations ( 23) and (24) and Equation ( 27) if в t = 0.25 we get 7 6 2 0.67 , which indicates the acceptability of the proposed relations.
From Equation (32) it follows that b t = 0.25 in Equation (24) the values of M L and M Lm magnitude M W corresponds to Equation (11).Probably, the presence of the form Equation ( 29) between log 10 t 0 and log 10 M 0 explains equality M L = M W for earthquakes with M W ≤ 7.0 North-West Europe [35], New Zealand [36], western Canada [37] and about Taiwan [38].   the observations in the epicentral area, showed that the duration t 0 , determine the energy of the oscillations with the maximum intensity depends strongly on the magnitude and 2.5-fold increases with increasing magnitude of m b on unit [31].
A little-known empirical formula Equation (32) Gutenberg [1] is a key for further generalizations of our constructions on relations m b − log 10 M 0, and m b − M S.
On the basis of ( 13) and ( 29) with С 1 = −11.09,we can get: which is in good agreement with (33) provided m b = m bm.
Graphic expressions Equations ( 33)-( 35) are shown in Figure 6, from which it can be assumed about the close convergence of these relations and the possible equality m b = m bm (Figure 6).At equality m b = m b m-based Equations ( 13) and ( 33) for the standard ρ and v S can obtain the expression: which is in good agreement with Equations ( 29) and ( 31), which may indicate the consistency of our constructions relating m b, m bm, M L, M Lm and log 10 t 0 with log 10 M 0 for earthquakes in California, despite the fact that the conclusions are based on statistical formulas in which the correlation coefficients are not equal to unity (r = 0.75 -0.90)If we use the Equation (36), on the basis of Equation ( 24) with в t = 0.22 and Equations ( 25) and ( 26) for the standard values ρ and v S , M Sm dependence on m bm can be expressed as: 1.59 3.20 which almost corresponds to the classical formula Equation (2) Gutenberg and Richter (1956в) and for which the equality M Sm = m bm complied with M sm = 5.40, which coincides closely with generalizations Chen [7], Gusev [9], Nuttli [12] and Utsu [24].In Figure 7 shows the correlation of log 10 t 0 from log 10 M 0 for earthquakes in the world (1981)(1982)(1983)(1984)(1985)(1986)(1987)(1988)(1989)(1990)(1991) by the Catalogue Choy [39], for which the value of t 0 was taken from the Global CMT Catalogue.The ratio of log 10 t 0 from log 10 M 0 for these data is given by (Figure 7   Equation ( 39) is in good agreement with the dependence on m b from log 10 M 0 for Sumatra island earthquake (φ = -10˚ + 10˚, λ = +90˚ + 100˚) for 1993-2012 (Figure 9).
Table 1 shows a comparison of the magnitude ˆb m obtained by the true maximum amplitude [19,29,40] and the calculated value m bm (Table 1)  , m bm = 6.18, and if 6.36 ≤ log 10 ∆σ < 7.0 value of ˆb m more then m bm (Table 1).Table 2 presents a comparison of calculated m bm and ˆb m (21) for 80 major earthquakes of the world for 2000-2012 for calculations m bm , ˆb m and M Sm used data from Global CMT Catalogue (Table 2).When comparing log 10 Δσ from Table 1 to Table 2 shows that with increasing log 10 M 0 from 19.15 to 22.72 for the 2000-2012 earthquakes log 10 Δσ value ranges from 6.75 -7.58 with an average of 7.16, that is, much higher than for earthquakes 1960-1984 (Tables 1 and 2) and higher than the standard logΔσ = 6.56.
For such high values Δσ values m bm closely coincide with the design ˆb m , and for values M Sm characterized by inequality: M Sm > M W (Table 2) confirmed that conclusion is the relation m bm − ˆb m -for earthquakes in Japan and the Kuril Islands (φ = 30˚ + 40˚, λ = 140˚ + 150˚) for the 1993-2012 shown in Figure 10.
Thus for large earthquakes 1960-1984 and 1993-2012 at logΔσ > 7.1 m bm values coincide closely with the magnitude ˆb m calculated from the true maximum amplitude (A g ) of seismic vibrations, the magnitude of which is proportional to the seismic moment: to Houston [29] and Kanamori [19].Consequently, the m bm  value is proportional to the log 10 А g .
The ratio of M S -log 10 M 0 .In Mamyrov's papers [18], [28] have shown that in the range of 16 ≤ log 10 M 0 < 21.0 if log 10 Δσ ≤ 7.0 at the rated M Sm closely coincides with M S and M W , and for high Δσ ≥ 10 7 Pa following inequality M Sm > M S , as shown in Table 2.
In Figure 11 shows the correlation of M S from log 10 M 0 for earthquakes of the world for 1981-1991 according to the Catalog Chou et al. [39]:  38) and ( 26).These relations with M S = M Sm with log 10 M 0 are in good agreement with the generalization of Perez [41] for crustal earthquakes of the world for the years 1950-1997:     From the data that the value M Sm an average of 0.5 more than the M S , because according to the relation log 10 t 0 with log 10 M 0 (from 42) with growth log 10 M 0 from 16 to 22 on the basis of ( 19), the value increases from 7.19 logΔσ to 7.43 (Figure 12), and using equation (38) in the same size ranges of log 10 M 0 the value of log 10 ∆σ increases from 6.5 to 7.10.It is likely that for most crustal earthquakes before 1993 was characterized by the above limits to growth log 10 ∆σ < 7. derived from ( 42) and ( 28) for в t = 0.32 и a t = −5,43 (Figure 13).which closely coincides with Equations ( 29), ( 31) and (36) typical for earthquakes in California (Figures 1 and  15).Therefore, we can expect that the relationship between magnitudes m b -M S for earthquakes of the two regions may be similar in this range of seismic moment.Indeed, the data in   Calculated dependence of M Sm from m bm based on Equations ( 25), ( 26) and (47) for the elastic parameters of the standard as follows: 1.59 3.06 which is in good agreement with Equations ( 2), ( 37) and (49).Therefore, we have adopted model of the relationship of linear relations between M (m b , M L , M S ) and log 10 t 0 with log 10 M 0 explains many existing empirical formulas.
For a wide range 6 ≤ log 10 M 0 ≤ 23 changing log 10 t 0 , to a first approximation, can be described by a nonlinear dependence of (A 0 = log 10 M 0 ):   in which the first two terms describes the linear growth log 10 t 0 in the range 6 ≤ A 0 ≤ 15.On the basis of Equations ( 25)-( 27) and (51) in Figure 17 shows estimates nonlinear dependence m bm , M Lm and M Sm from M W to (11) for crustal earthquakes.From Figure 17 shows that in the

Conclusions
1) A broad range of local Richter magnitude M L , m b, and M S crustal earthquakes in different regions shows a possible functional relationship with the seismic moment magnitude, corner frequency, voltage and depressurized seismic elastic parameters of the geophysical environment.These links justify numerous empirical relationships with magnitudes of seismic moment.
2) It is assumed that an upgraded body-wave magnitude m bm for large earthquakes is proportional to the logarithm of the average displacement along the fault log 10 u, ˆb m , the true magnitude and the maximum amplitude of seismic vibrations A g ; magnitude M Sm is proportional to the logarithm of the square average displacement along the fault (2log 10 u) and local magnitude proportional 1.5log 10 u.
3) Control parameters of the quantitative relations with seismic moment magnitudes are coefficients depending on the change in corner period of seismic stress drop or discharged from the seismic moment, which provide a self-consistent system of equations between the main source parameters of crustal earthquakes.
Δσ = 3.67 MPa and 36.7 bar and the expression Equation (8) can be rewritten in a simple form log 10 t 0 = 1/3log 10 M 0 -5.43, then Equation (15) simplifies to: −5 between log 10 t 0 and log 10 M 0 would expect the following relationship: instrumental data obtained (Figure1, N-the number of data, r-correlation coefficient):

Figure 3 .
Figure 3.The ratio of calculated M Lm and instrumental M L for Southern California earthquakes by Thatcher and Hanks [30], South-West Germany (Scherbaum et al. 1983) and Cental Japan (Jin et al., 2000).    Lm L M M 0.9 0.03 0.28 0.05     , N =

3. 1 .
Ratio m b − log 10 M 0 : Design and Data Tools As in the case of search based M L − log 10 M 0 , for bodywave magnitude m b consider empirical relationships According to Zapolsky [31], Gutenberg [1], specifically examining the relationship between the energy of focal radiation and earthquake magnitude according to

Figure 4 .
Figure 4. Correlation log 10 t 0 from M L (full line-Hasegawa [34] and from calculated M Lm (dashed line, see the text) for East Canada earthquakes.

Figure 5 .
Figure 5. Correlation log 10 M 0 from M L (full line-Hasegawa [34] and from calculated M Lm (dashed line, see the text) for East Canada earthquakes.

Figure 6 .
Figure 6.Correlation log 10 t 0 and m b (full line), log 10 t 0 and m bm (dashed line, see the text).

Figure 8 .
Figure 8. Correlation dependence m b from log 10 M 0 (full line) for major earthquakes of the world (1981-1991) by Choy's Catalogue et al. (1995) for 1981-1991.    for a number of large earthquakes in 1960-1984.The presented data suggest that for most of the earthquakes characterized by the following inequality: ˆb Δσ ≥ 7.1 value of ˆb m is close to the m bm same as for Great Chilean earthquake ˆ7.57b m  and m bm = 7.71, for Tangshan (1976) ˆ6.9 b m  and m bm = 6.92,Yanyuan (1976).ˆ6.5 b m 

Figure 10 .
Figure 10.Correlation calculated magnitudes m bm and b m  for the earthquakes in Japan and Kuril Islands for 1992-2012 years.    1.0 0.02 0.14 0.003 bm b m m      , N = 521, r = 0.97.Values of b m  were calculated according to the

11 ,
dashed line), derived from Equations ( shows the correlation M S with log 10 M 0 (solid line) for the earthquakes in Japan and the Kuril Islands in 1993-2012: show the same relationship M Sm from log 10 M 0 (Figure 12, dashed line): with (N = 521, r = 0.99): the earthquakes N1-10 log 10 t 0 calculated according to ∆σ Kasachara (1984), Purcaru and Berkhemer (1982); for the rest earthquakes N11-52 all data have been taken according to Global CMT Cataloge, for the earthquakes N53-data have been taken from USSR's catalogue, 1984.

Figure 11 .Figure 12 .N
Figure 11.Correlation of magnitudes M S and log 10 M 0 (full line) by Catalogue of major earthquakes of the world Choy [39]:    

10 .
Ratio m b -M S и m bm -M Sm .In Figure 13 shows the correlation ratio m b -M S for crustal earthquakes of the Kuril Islands and Japan for 1993-2011:

Figure 16
confirmed these assumptions and empirical relationship of M S from m b for Tien Shan's earthquakes is expressed by the following relation (N = 1183, r = 0.95, Figure16):

Figure 13 .
Figure 13.Correlation of magnitudes m b and M S for the earthquakes in Japan and Kuril Islands for 1993-2012 years.    b S m M 0.52 0.03 2.78 0.02     , N = 514, r = 0.84.Calculated dependence bm

Figure 14 .
Figure 14.Correlation of magnitudes m b and M S for the earthquakes in South America 1993-2012 years.    b S m M 0.52 0.03 2.64 0.02     , N = 547, r = 0.82.Calculated dependence bm

Figure 17 .
Figure 17.Averaged according M Sm M Lm and m bm from M W for crustal earthquakes (see text), the dashed line represents the intersection of the curves M Sm  M Lm  m bm  M W  5.26 − 5.50.interval 4 ≤ M W ≤ 6,5 numerical values of magnitudes m bm  m b , M Lm  M L , M Sm  M S and M W within the accuracy of these parameters are close.In accordance with Equations (19) and (51) in the interval 6.0 < A ≤ 23.0 log 10 ∆σ value increases from 1.75 to 7.53, and the most intense increase in this parameter is in the range 6.0 ≤ A 0 ≤ 15.0. :