^{*}

In connection with conversion from energy class *K*_{R} (*K*_{R }= log_{10}*E*_{ R}, where *E*_{R }— seismic energy, J) to the universal magnitude estimation of the Tien Shan crustal earthquakes the development of the self-coordinated correlation of the magnitudes (*m*_{b }, *M*_{L}, *Ms *) and *K*_{R} with the seismic moment *M*_{0} 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 *M *(*m*_{b }*, M*_{L }*, Ms )* =

*K*

_{i }+

*z*

_{i }log

_{10}

*M*

_{0 }, the coefficients

*k*

_{i}and

*z*

_{i}are controlled by the parameters of ratio (where ;

*f*

_{0 }—corner frequency, Brune, 1970, 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 , where , and , for the averaged elastic properties of the Earth’s crust for thembmthe coefficients

*C*

_{i}= –11.30 and

*d*

_{i }= 1.0, for

*M*

_{Lm}:

*C*

_{i }= –14.12,

*di*= 7/6; for

*M*

_{Sm }

*: C*

_{i}= –16.95 and

*d*

_{i }= 4/3. For theTien Shan earthquakes (1960-2012 years) it was obtained that , and on the basis of the above expressions we received that

*M*

_{Sm }= 1.59

*m*

_{bm }– 3.06. According to the instrumental data the correlation

*M*

_{s }= 1.57

*m*

_{b }– 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.

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}, [4-6]).

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 [7-12] and u functional relationship with the seismic moment, the shear modulus μ and the gap area S [13-14];

3) functional relationship corner period 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 [_{D}, depending on the amount of source and the elastic properties of the geophysical medium [

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, Т— period), s.

Use the following generalization of Soviet seismologists, which were introduced scale energy class K_{R} [_{LH} (IC device) and body waves m_{PV} on device SCM [4,9]:

where E_{R}—seismic energy according to [_{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]:

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 [7,12,20-24].

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.

Based on the original definition of magnitude on Richter [_{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):

If 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_{b}_{m}.

Based on generalizations Christensen [26,27] for the crust taken: average density

ρ = 2830 kg/m^{3}, v_{S} = 3600 m/s and 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 [

where.

Taken for the elastic parameters and subject [_{10}t_{0} = 1/3log_{10}M_{0} – 5.43, then Equation (15) simplifies to:

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: M_{Sm} = M_{W}.

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} = 0.5 (C_{1} + C_{S}): for standard values ρ and v_{S} value C_{L} is equal: C_{L} = –14.12.

Accepted values for ρ and v_{S} by Equation (8) and Equation (9) the following relationship:

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 of M_{W} for large earthquakes [19,29] (m_{b} ≥ 6):

which were used 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 or , that closely coincides with on [19,29].

In the sequel will be shown.

Equation (20) agrees satisfactorily with other empirical relationship [

(m_{PV} = m_{b} + 0.18):

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:

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_{в}_{m} from M_{Sm} based on Equations (25) and (26) can be expressed as:

which is в_{t} = 0.33 and a_{t} = −5.43 ransformed into simple formula Equation (20).

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} –consider according to Thatcher and Hanks [_{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: С_{1}= −11.09, С_{S} = −16.5, С_{L} = −13.72. With known ρ, v_{s}, Δσ/2μ = 5 × 10^{−5} between log_{10}t_{0} and log_{10}M_{0} would expect the following relationship:, but the instrumental data obtained (

i.e. in accordance with (19) with increasing values of M_{0}log_{10}Δσ increases:

. Therefore, for the considered data characteristic dependence, said Nuttli [

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:

which is in good agreement with the expression (3) Gutenberg and Richter [_{0}_{ }= t_{в}_{ }= t_{m}_{. }

In _{10}t_{0} and M_{L} according to Thatcher [_{L} = M_{Lm} based on Equation (3) Gutenberg and Richter [

which is in satisfactory agreement with the expression (29).

In _{L} ≤ 6.8 shows the correlation ratio M_{Lm} of M_{L} for Southern California earthquakes [_{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} (

From numerous publications on nonlinear relations log_{10}M_{0} – M_{L} acceptability of new assumptions considered on the basis of Hasegawa [_{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} [

For the first group of small earthquakes characterized by the following relationship (10^{5 }< ∆σ < 10^{6} Pа):, but for another group (10^{6 }≤ ∆σ < 5 × 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 [

Finally, for the Southern California Earthquake Equation (18) and Equation (29) we can obtain the following relationship:, which coincides with the ratio of [

According to Equations (23) and (24) and Equation (27) if в_{t} = 0.25 we get, 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 NorthWest Europe [

As in the case of search based M_{L} − log_{10}M_{0}, for bodywave magnitude m_{b} consider empirical relationships According to Zapolsky [

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 [

A little-known empirical formula Equation (32) Gutenberg [_{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:

Substitution in Equation (31) into (13) leads to the following formula:

which is in good agreement with (33) provided m_{b}_{ }= m_{bm}_{.}

Graphic expressions Equations (33)-(35) are shown in _{b} = m_{bm} (_{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:

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 [

In _{10}t_{0} from log_{10}M_{0} for earthquakes in the world (1981-1991) by the Catalogue Choy [_{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 (

for which the range 17 ≤ log_{10}M_{0} ≤ 21 value of log_{10}Δσ by Equation (19) increases from 6.60 to 7.10.

Substituting (38) in (13) leads to (С_{1} = –11.30):

which agrees closely with the empirical formula:

shown on

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 (

_{bm} (

When log_{10}Δσ ≥ 7.1 value of is close to the m_{bm} same as for Great Chilean earthquake and m_{bm} = 7.71, for Tangshan (1976) and m_{bm}_{ }= 6.92, Yanyuan (1976)., m_{bm} = 6.18, and if 6.36 ≤ log_{10}∆σ < 7.0 value of more then m_{bm} (

_{bm} and (21) for 80 major earthquakes of the world for 2000- 2012 for calculations m_{bm}, and M_{Sm} used data from Global CMT Catalogue (_{10}Δσ from _{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, and for values M_{Sm} characterized by inequality: M_{Sm}_{ }> M_{W} (_{bm} −—for earthquakes in Japan and the Kuril Islands (φ = 30˚ + 40˚, λ = 140˚ + 150˚) for the 1993-2012 shown in

Thus for large earthquakes 1960-1984 and 1993-2012 at logΔσ > 7.1 m_{bm} values coincide closely with the magnitude calculated from the true maximum amplitude (A_{g}) of seismic vibrations, the magnitude of which is proportional to the seismic moment: to Houston [_{bm}

value is proportional to the log_{10}А_{g}.

The ratio of M_{S} – log_{10}M_{0}. In Mamyrov’s papers [_{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

In _{S} from log_{10}M_{0} for earthquakes of the world for 1981-1991 according to the Catalog Chou et al. [

which is in satisfactory agreement with the dependence (_{S} = M_{Sm} with log_{10}M_{0} are in good agreement with the generalization of Perez [

In _{S} with log_{10}M_{0} (solid line) for the earthquakes in Japan and the Kuril Islands in 1993-2012:

, here, we show the same relationship M_{Sm} from log_{10}M_{0} (

, obtained with (N = 521, r = 0.99):

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 (_{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 < 7.10.

Ratio m_{b }– M_{S} и m_{bm}_{ }– M_{Sm}. In _{b }– M_{S} for crustal earthquakes of the Kuril Islands and Japan for 1993-2011:

which is in good agreement with the expression:

derived from (42) and (28) for в_{t}_{ }= 0.32 и a_{t} = −5,43 (

_{b} – M_{S}_{ } for crustal earthquakes in South America for the years 1993- 2012, (φ = −40˚ − 0˚, λ = −85˚ − 65˚) by Global CMT Catalogue:

for this region was obtained (N = 576, r = 0.99):

the substitution of which in (26), в_{t} = 0.32 and a_{t} = −5.48 leads to the formula

Equations (43)-(46) are in good agreement with Equations (21) and (22).

_{10}t_{0} of log_{10}M_{0} for earthquakes of the Tien Shan (φ = 38.5˚ − 45˚, λ = 63˚ − 96˚) for 1960-2012 in interval 13.0 ≤ log_{10}M_{0} ≤ 21.5 (N = 684, r = 0.85):

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 _{S} from m_{b} for Tien Shan’s earthquakes is expressed by the following relation (N = 1183, r = 0.95,

Calculated dependence of M_{Sm} from m_{bm} based on Equations (25), (26) and (47) for the elastic parameters of the standard as follows:

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 _{bm}, M_{Lm} and M_{Sm} from M_{W} to (11) for crustal earthquakes. From

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.

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, , 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.