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

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Control Parameters of Magnitude—Seismic Moment Correlation for the Crustal Earthquakes Open Journal of Earthquake Research, 2013, 2, 60-74 http://dx.doi.org/10.4236/ojer.2013.23007 Published Online August 2013 (http://www.scirp.org/journal/ojer) Control Parameters of Magnitude—Seismic Moment Correlation for the Crustal Earthquakes Ernes Mamyrov Institute of Seismology of the National Academy of Sciences of the Kyrgyz Republic, Bishkek, Kyrgyz Republic Email: [email protected]; [email protected] Received June 12, 2013; revised July 19, 2013; accepted August 8, 2013 Copyright © 2013 Ernes Mamyrov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT In connection with conversion from energy class KR (KR log10 ER , 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 stud- ies has been done. It is assumed that in the expression M mMbLs,, M k ii z log10 M 0 , the coefficients ki and zi are 1 controlled by the parameters of ratio logtab01tt log 0 M0 (where tfb 0 ; f0—corner frequency, Brune, 1970, 1971; M0, Nm). According to the new theoretical predictions common functional correlation of the advanced magni- tudes Mm (mbm = mb, MLm = ML, MSm = MS) from log10M0, log10t0 and the elastic properties (Ci) can be presented as M midMlog10 0 2log 10 tC 0 i, where zdii –2 b t, and kCii –2 a t, 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 log10tM 0 0.22log10 0 3.45 , 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. Keywords: Magnitude; Seismic Moment; Energy Class; Earthquakes; Frequency 1. Introduction M0 based on the following findings: 1) proportional magnitudes and the maximum ampli- In world practice, seismological research in assessing the tude of seismic vibrations [1-3]; scale of earthquakes magnitude scale of Gutenberg and Richter [1-3] is fundamental. In the countries of the for- 2) the statistical dependencies of the average magni- mer Soviet Union has been used scale independent en- tude of displacement along the fault u [7-12] and u func- tional relationship with the seismic moment, the shear ergy class KR, defined as the logarithm of the seismic modulus μ and the gap area S [13-14]; energy ER, highlighted by an earthquake, measured in 3) functional relationship corner period tf 1 s joules (KR = log10ER, [4-6]). в 0 For crustal earthquakes Tien Shan when considering with M0, the source radius r0, speed S—wave vS and the transition to magnitude scale was necessary to de- static stress drop Δσ [15,16], as well as the similarity of velop a self-consistent system of quantitative relation- the angular frequency f0 with a fundamental frequency of ships that justify numerous empirical relationships body- the acoustic Debye [17] fD, depending on the amount of source and the elastic properties of the geophysical me- wave magnitude mb, local magnitude on surface waves dium [18]. ML, surface wave magnitude for MS and KR from seismic Our further quantitative construction is based on the moment M0 (N·m), as the reference scale. In connection with the above purpose is to study the quantitative rela- following empirical relationship Gutenberg and Richter [3,12]: tionships mb, ML, MS and energy of seismic radiation ES c Copyright © 2013 SciRes. OJER E. MAMYROV 61 −6 log10 EKGR GR 4.8 1.5 MS (1) (10 м), it is assumed that an upgraded body-wave mag- nitude mbm (equivalent mb, mPV ) is (considering doubling MmSb1.59 3.97 (2) вm on the ground at the focus): logtM 0.32 1.4 (3) 10 0 L mbm log10 вm 6.3 log10 u 6 (12) 2 where EGR—seismic energy according to Getenberg and If Sr 0 in (8) on the basis Equations (9) and (10) Richter, J; t0—fluctuations with a maximum duration of and Equation (12) value mbm equal (M0, N·m; tв, s; µ, Pa; vibration speed А/Т in the near field (А—amplitude, Т— vs, m/s): period), s. mC log M 2log t (13) Use the following generalization of Soviet seismolo- bm 1100100 gists, which were introduced scale energy class K [5], where Cvlog 2 2.34 2 12 6.3 , value С R 110 S 1 the magnitude of surface waves MLH (IC device) and determines the springiness of the geophysical environ- body waves mPV on device SCM [4,9]: ment at mbm. log EK4.0 1.8 M (4) Based on generalizations Christensen [26,27] for the 10 R RLH crust taken: average density 3 log10 tMmL 0.35H 1.4 (5) ρ = 2830 kg/m , vS = 3600 m/s and 2 vS 36.7 GPа in what follows, these quantities mKbR5.53 0.45 14 (6) ρ, vs and μ taken as the standard. mMPV 0.35log10 0 2.75 (7) When these elastic parameters of the geophysical me- dium expression Equation (13) is transformed to the fol- where ER—seismic energy according to [5], in J; KR = lowing form: log10ER; tm—increase the maximum duration of the seis- mMtlog 2log 11.30 mic intensity in the near field, in sec. bm 10 0 10 0 (14) The basis of the theoretical constructs are the follow- 1 3log10M 0 2 3log 10 4.80 ing functional relations [10,13,15,16,19]: Seismic energy radiation ESK by Kanamori [19], based 3 on Equations (8) and (9) and Equation (13) is: M 00Su 16 7 r (8) 16 7 2.35 2 3 Vt3 log EKC2 log M 3log t Sb 10 SK SK 2 10 0 10 0 (15) CC212log mbm 100 t EMSK 2 0 (9) where Cvlog 7 411 2.34 3 3 . 2103 S rvt0 2.34 2 S b (10) Taken for the elastic parameters and subject [19]. 5 EMSK 0 2510 obtain: Δσ = 3.67 MPa and MMW 2 36 log10 0 6.07 (11) 36.7 bar and the expression Equation (8) can be rewritten in a simple form log10t0 = 1/3log10M0 – 5.43, then Equa- where r0—radius of the source, in м; ∆σ—static seismic tion (15) simplifies to: stress drop, in Pа; tb—corner period, s; MW—moment magnitude; (ESK, in J; M0, in N·m; u in m; vS in m/s); for KMt2log 3log 20.61 SK 10 0 10 0 (16) the constructions made t0 = tb = tm. 1.5Mm 4.8 3 3 Many generalizations proved that for a wide range of Wbm changes log10M0 or MW empirical correlations magnitude On the basis of Equations (13)-(16), reflecting the mb, ML and MS from M0 are non-linear, as in Equation (8), functional relationship of ESK from M0, t0, mbm and μ at n E = E introduced upgraded the magnitude of surface as a function of М 00~ f t value of n varies from 3 to GR SK 6, and is increase Δσ [7,12,20-24]. waves MSm (equivalent of MS, MW), while maintaining However, for individual intervals M0 or MW commu- that the formula Equation (1) Gutenberg and Richter nication between magnitudes relationships and depend- [2,3], with Equation (9), Equations (15) and (16) will be: encies of the magnitude log10M0 can be represented as MMt4 3 log 2log C linear relationships. Sm 10 0 10 0 S 2 3 log10MK 0 log 10 10.45 2 3SK 3.2 2. Justification Relations (17) Magnitude—Seismic Moment where CCS 23 2 3.2. Based on the original definition of magnitude on Richter Taken for ρ and vS CS value in Equation (17) is equal [25], under which the numerical value of the earthquake to CS = –16.95, and for the special case of Δσ = 3.67 MPa –5 magnitude is proportional to the logarithm of the maxi- = const and ESK/M0 = 5 × 10 equality: MSm = MW. mum oscillation decimal вm, expressed in microns We also introduce a modernized local magnitude on Copyright © 2013 SciRes. OJER 62 E. MAMYROV surface waves M —equivalent M [18,28], functionally Lm L M Sm 4 3 2вt log10Ma 0 2t 16.95 (26) interconnected with log10M0, logt0, KSK, mbm and MSm: M Lm 7 6 2вt log10Ma 0 2t 14.12 (27) M Lmb0.5mMmSm 7 6 log10M 0 2log tC0 L which provide a self-consistent system of semi empirical 0.5logM 2 3log 7.62 10 0 10 inter magnitude dependencies. For example, the depend- (18) ence of mвm from MSm based on Equations (25) and (26) can be expressed as: where CL = 0.5 (C1 + CS): for standard values ρ and vS value CL is equal: CL = –14.12. 12 в mMaat 2 16.95 2 11.30 (28) Accepted values for ρ and vS by Equation (8) and bm43 2в Sm t t Equation (9) the following relationship: t which is вt = 0.33 and at = −5.43 ransformed into simple log10 log10Mt 0 3log 10 0 9.74 (19) formula Equation (20).
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