20:
357:
341:-value decrease observed prior to the failure of samples deformed in the laboratory has led to the suggestion that this is a precursor to major macro-failure. Statistical physics provides a theoretical framework for explaining both the steadiness of the Gutenberg–Richter law for large catalogs and its evolution when the macro-failure is approached, but application to earthquake forecasting is currently out of reach. Alternatively, a b-value significantly different from 1.0 may suggest a problem with the data set; e.g. it is incomplete or contains errors in calculating magnitude.
321:
345:
369:
This may in large part be caused by incompleteness of any data set due to the inability to detect and characterize small events. That is, many low-magnitude earthquakes are not catalogued because fewer stations detect and record them due to decreasing instrumental signal to noise levels. Some modern models of earthquake dynamics, however, predict a physical roll-off in the earthquake size distribution.
27:, during the Aug 22 - Sep 1 period. Notice that the linear fit fails at the upper and lower end, due to lack of registered events. Since the recording period is only 10 days, events of magnitude greater than 6 has not yet appeared. Since the recording devices are unable to detect earthquake events near or below the background noise level, most of the events with magnitude lower than 1.5 are not detected.
368:
There is an apparent b-value decrease for smaller magnitude event ranges in all empirical catalogues of earthquakes. This effect is described as "roll-off" of the b-value, a description due to the plot of the logarithmic version of the GR law becoming flatter at the low magnitude end of the plot.
328:
The parameter b (commonly referred to as the "b-value") is commonly close to 1.0 in seismically active regions. This means that for a given frequency of magnitude 4.0 or larger events there will be 10 times as many magnitude 3.0 or larger quakes and 100 times as many magnitude 2.0 or larger quakes.
336:
There is debate concerning the interpretation of some observed spatial and temporal variations of b-values. The most frequently cited factors to explain these variations are: the stress applied to the material, the depth, the focal mechanism, the strength heterogeneity of the material, and the
316:
in a 1944 paper studying earthquakes in
California, and generalised in a worldwide study in 1949. This relationship between event magnitude and frequency of occurrence is remarkably common, although the values of a and b may vary significantly from region to region or over time.
776:
1079:
Burud, Nitin B; Kishen, J M Chandra. "Application of generalized logistic equation for b-value analysis in fracture of plain concrete beams under flexure", Engineering
Fracture Mechanics Vol 210, 2019, pp. 228–246.
996:
Lockner, D. A., et J. D. Byerlee (1991), Precursory AE patterns leading to rock fracture, in Vth Conf. AE/MS Geol. Str. and Mat., édité par Hardy, pp. 45–58, Trans Tech
Publication, Germany, The pennsylvania State
969:
Mori, J., et R. E. Abercombie (1997), Depth dependence of earthquake frequency-magnitude distributions in
California: Implication for rupture initiation, Journal of Geophysical Research, 102(B7), 15081–15090.
987:
Mogi, K. (1962), Magnitude frequency relations for elastic shocks accompanying fractures of various materials and some related problems in earthquakes, Bull. Earthquake Res. Inst. Univ. Tokyo, 40, 831–853.
868:
in concrete by N. Burud and J. M. Chandra Kishen. Burud showed the b-value obtained from generalized logistic equation monotonically increases with damage and referred it as a damage compliant b-value.
796:
It is possible to see in an article published by N. V. Sarlis, E. S. Skordas, and P. A. Varotsos, that above some magnitude threshold this equation reduces to original
Gutenberg–Richter form with
1019:
Amitrano, D. (2012), Variability in the power-law distributions of rupture events, how and why does b-value change, Eur. Phys. J.-Spec. Top., 205(1), 199–215, doi:10.1140/epjst/e2012-01571-9.
1070:
Lev A. Maslov and
Vladimir M. Anokhin, "Derivation of the Gutenberg-Richter empirical formula from the solution of the generalized logistic equation", Natural Science, 04, 08, (648), (2012).
437:
793:
represents the non-extensivity parameter introduced by
Constantino Tsallis to characterize systems not explained by the Boltzmann–Gibbs statistical form for equilibrium physical systems.
497:
855:
1096:
Sanchez E; Vega-Jorquera P. "New
Bayesian frequency–magnitude distribution model for earthquakes applied in Chile", Physica A: Stat. Mech. and its Appl. Vol 508, 2018, pp. 305–312.
104:
587:
New models show a generalization of the original
Gutenberg–Richter model. Among these is the one released by Oscar Sotolongo-Costa and A. Posadas in 2004, of which R. Silva
978:
Schorlemmer, D., S. Wiemer, et M. Wyss (2005), Variations in earthquake-size distribution across different stress regimes, Nature, 437, 539–542, doi: 10.1038/nature04094.
152:
566:
530:
201:
329:
There is some variation of b-values in the approximate range of 0.5 to 2 depending on the source environment of the region. A notable example of this is during
283:
263:
243:
223:
178:
864:
were found for events recorded in
Central Atlantic, Canary Islands, Magellan Mountains and the Sea of Japan. The generalized logistic equation is applied to
1155:
Jon D. Pelletier, "Spring-block models of seismicity: review and analysis of a structurally heterogeneous model coupled to the viscous asthenosphere"
597:
376:
represents the total seismicity rate of the region. This is more easily seen when the GR law is expressed in terms of the total number of events:
1061:
N. V. Sarlis, E. S. Skordas, and P. A. Varotsos, "Nonextensivity and natural time: The case of seismicity", Physical Review E 82 (2010), 021110.
860:
In addition, another generalization was obtained from the solution of the generalized logistic equation. In this model, values of parameter
960:
Scholz, C. H. (1968), the frequency-magnitude relation of microfracturing in rock and its relation to earthquakes, BSSA, 58(1), 399–415.
876:
of Gutenberg–Richter is presented. The model was applied to intense earthquakes occurred in Chile, from the year 2010 to the year 2016.
1185:
19:
361:
24:
1043:
Sotolongo-Costa O., Posadas A., "Fragment-Asperity Interaction Model for Earthquakes", Phys. Rev. Lett. 92 (2004) 048501.
872:
A new generalization was published using Bayesian statistical techniques, from which an alternative form for parameter
1052:
Silva R., Franca G.S., Vilar C.S., Alcaniz J.S., "Nonextensive models for earthquakes", Phys. Rev. E 73 (2006) 026102.
1164:
1132:
382:
448:
909:
1190:
802:
333:
when b can become as high as 2.5, thus indicating a very high proportion of small earthquakes to large ones.
1139:
572:
61:
1119:, Kamal, and Debashis Samanta, "Fractal models of earthquake dynamics", Heinz Georg Schuster (ed),
309:
115:
44:
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505:
356:
183:
922:
8:
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268:
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208:
163:
1085:
300:
analysis due to a close resemblance of acoustic emission phenomenon to seismogenesis.
1160:
1145:
1128:
905:
865:
297:
1097:
1081:
330:
308:
The relationship between earthquake magnitude and frequency was first proposed by
576:
1101:
364:(red dot) and aftershocks (which continued to occur after the period shown here)
771:{\displaystyle \log N_{>m}=\log N+\left({\frac {2-q}{1-q}}\right)\log \left}
313:
1010:-value as an earthquake precursor, Nature, 289, 136–139; doi:10.1038/289136a0.
1174:
1149:
320:
48:
32:
902:
Understanding Systems: A Grand Challenge For 21st Century Engineering
344:
16:
Law in seismology describing earthquake frequency and magnitude
289:
Since magnitude is logarithmic, this is an instance of the
571:
Modern attempts to understand the law involve theories of
245:
are constants, i.e. they are the same for all values of
23:
Gutenberg–Richter law fitted to the aftershocks of the
805:
600:
538:
508:
451:
385:
271:
251:
231:
211:
186:
166:
118:
64:
849:
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491:
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296:The Gutenberg–Richter law is also widely used for
277:
257:
237:
217:
195:
172:
146:
98:
1172:
1141:Seismicity of the Earth and Associated Phenomena
927:Bulletin of the Seismological Society of America
591:presented the following modified form in 2006,
502:the total number of events (above M=0). Since
432:{\displaystyle N=N_{\mathrm {TOT} }10^{-bM}\ }
1157:Geocomplexity and the Physics of Earthquakes
1121:Reviews of Nonlinear Dynamics and Complexity
492:{\displaystyle N_{\mathrm {TOT} }=10^{a},\ }
180:is the number of events having a magnitude
568:must be the probability of those events.
43:) expresses the relationship between the
923:"Frequency of Earthquakes in California"
355:
343:
319:
18:
850:{\displaystyle b={\frac {2(2-q)}{q-1}}}
348:Roll-off compared to ideal GR law with
51:in any given region and time period of
1173:
938:Gutenberg & Richter (1949), p. 17
900:Jamshid Ghaboussi, Michael F Insana,
929:, vol. 34, iss. 4, pp. 185–188, 1944
891:Gutenberg and Richter (1949), p. 17.
362:August 2016 Central Italy earthquake
25:August 2016 Central Italy earthquake
1159:, American Geophysical Union, 2000
1144:, Princeton University Press, 1949
324:GR law plotted for various b-values
13:
789:is a proportionality constant and
464:
461:
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404:
401:
398:
14:
1202:
1186:Independence (probability theory)
1086:10.1016/j.engfracmech.2018.09.011
904:, p. 255, World Scientific, 2017
582:
1138:B. Gutenberg and C. F. Richter,
337:proximity of macro-failure. The
99:{\displaystyle \log _{10}N=a-bM}
1109:
1090:
1073:
1064:
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785:is the total number of events,
532:is the total number of events,
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1:
921:B. Gutenberg, C. F. Richter,
879:
303:
7:
1102:10.1016/j.physa.2018.05.119
147:{\displaystyle N=10^{a-bM}}
10:
1207:
573:self-organized criticality
561:{\displaystyle 10^{-bM}\ }
1006:Smith, W. D. (1981), The
1115:Pathikrit Bhattacharya,
525:{\displaystyle 10^{a}\ }
310:Charles Francis Richter
851:
772:
562:
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353:
325:
279:
259:
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196:{\displaystyle \geq M}
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28:
1034:Pelletier, pp. 34–36.
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37:Gutenberg–Richter law
22:
1191:Probabilistic models
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47:and total number of
1123:, pp. 107–150
1117:Bikas K Chakrabarti
291:Pareto distribution
1127:, Wiley-VCH, 2009
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866:acoustic emission
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360:Magnitude of the
331:earthquake swarms
298:acoustic emission
278:{\displaystyle M}
258:{\displaystyle N}
238:{\displaystyle b}
218:{\displaystyle a}
173:{\displaystyle N}
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55:that magnitude.
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577:self similarity
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583:Generalization
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314:Beno Gutenberg
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1165:0-87590-978-7
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1133:3-527-40850-9
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1114:
1113:
1103:
1099:
1093:
1087:
1083:
1076:
1067:
1058:
1049:
1040:
1032:, pp. 119–121
1031:
1028:Bhattacharya
1025:
1016:
1009:
1003:
993:
984:
975:
966:
957:
950:
947:Bhattacharya
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935:
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66:
58:
57:
56:
54:
50:
46:
42:
38:
34:
26:
21:
1156:
1140:
1124:
1120:
1110:Bibliography
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1075:
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1029:
1024:
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1007:
1002:
992:
983:
974:
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156:
108:
52:
40:
36:
30:
997:University.
49:earthquakes
1181:Seismology
1175:Categories
1150:1323229850
925:, p. 186,
910:9813225971
880:References
304:Background
33:seismology
839:−
825:−
708:−
697:−
684:−
673:
657:−
646:−
627:
605:
545:−
416:−
188:≥
134:−
88:−
76:
45:magnitude
951:, p. 120
53:at least
374:a-value
1163:
1148:
1131:
1030:et al.
949:et al.
908:
781:where
589:et al.
556:
520:
487:
442:where
427:
157:where
41:GR law
35:, the
1161:ISBN
1146:OCLC
1129:ISBN
906:ISBN
613:>
372:The
312:and
265:and
225:and
1125:V.2
1098:doi
1082:doi
670:log
624:log
602:log
575:or
109:or
67:log
31:In
1177::
728:10
579:.
541:10
511:10
475:10
412:10
352:=1
293:.
127:10
71:10
1167:.
1152:.
1135:.
1100::
1084::
1008:b
912:.
874:b
862:b
842:1
836:q
831:)
828:q
822:2
819:(
816:2
810:=
807:b
791:q
787:a
783:N
765:]
760:)
753:3
749:/
745:2
741:a
735:m
732:2
722:(
717:)
711:q
705:2
700:q
694:1
688:(
681:1
677:[
666:)
660:q
654:1
649:q
643:2
637:(
633:+
630:N
621:=
616:m
609:N
551:M
548:b
515:a
484:,
479:a
471:=
465:T
462:O
459:T
454:N
422:M
419:b
405:T
402:O
399:T
394:N
390:=
387:N
350:b
339:b
285:.
273:M
253:N
233:b
213:a
203:,
191:M
168:N
140:M
137:b
131:a
123:=
120:N
94:M
91:b
85:a
82:=
79:N
39:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
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