Skip to main content
SpringerLink
Log in

Reviews of seismicity around Taiwan: Weibull distribution

  • Original Paper
  • Published:
Natural Hazards Aims and scope Submit manuscript

Abstract

Statistical studies and empirical models play an important role in earthquake research. In this paper, a new statistical study was presented, evaluating if earthquake magnitude probability functions could be modeled by the Weibull distribution that is commonly used in many areas. On the basis of more than 50,000 earthquake data around Taiwan, the statistical analyses show that the hypothesis examined was not rejected by the statistics. That is, the earthquake magnitude probability function around Taiwan could be modeled by the Weibull distribution, with a substantial statistical significance.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  • Ahmadi MV, Doostparast M, Ahmadi J (2013) Estimating the lifetime performance index with Weibull distribution based on progressive first-failure censoring scheme. J Comput Appl Math 239:93–102

    Article  Google Scholar 

  • Allen RM, Kanamori H (2003) The potential for earthquake early warning in Southern California. Science 300:685–848

    Article  Google Scholar 

  • Ang A, Tang W (2007) Probability concepts in engineering: emphasis on applications to civil and environmental engineering, 2nd edn. Wiley, New Jersey

    Google Scholar 

  • Ashtari Jafari M (2010) Statistical prediction of the next great earthquake around Tehran. J Geodyn 49:14–18

    Article  Google Scholar 

  • Cao ZJ, Wang Y (2013) Bayesian approach for probabilistic site characterization using cone penetration tests. J Geotech Geoenvironmental Eng ASCE 139:267–276

    Article  Google Scholar 

  • Cao ZJ, Wang Y (2014) Bayesian model comparison and characterization of undrained shear strength. J Geotech Geoenvironmental Eng ASCE 140:1–9

    Article  Google Scholar 

  • Chen Q, Gerlach RH (2013) The two-sided Weibull distribution and forecasting financial tail risk. Int J Forecast 29:527–540

    Article  Google Scholar 

  • Chen CH, Wang JP, Wu YM, Chan CH, Chang CH (2013) A study of earthquake inter-occurrence times distribution models in Taiwan. Nat Hazards 69:1335–1350

    Article  Google Scholar 

  • Cheng CT, Chiou SJ, Lee CT, Tsai YB (2007) Study on probabilistic seismic hazard maps of Taiwan after Chi-Chi earthquake. J GeoEngin 2:19–28

    Google Scholar 

  • Corral A (2004) Long-term clustering, scaling, and universality in the temporal occurrence of earthquakes. Phys Rev Lett 92:108501

    Article  Google Scholar 

  • Devore JL (2008) Probability and statistics for engineering and the sciences. Brooks/Cole, Boston

    Google Scholar 

  • Erdik M, Aydinoglu N, Fahjan Y, Sesetyan K, Demircioglu M, Siyahi B, Durukal E, Ozbey C, Biro Y, Akman H, Yuzugullu O (2003) Earthquake risk assessment for Istanbul metropolitan area. Earthq Eng Eng Vib 2:1–23

    Article  Google Scholar 

  • Gan ZJ, Tung CC (1983) Extreme value distribution of earthquake magnitude. Phys Earth Planet Inter 32:325–330

    Article  Google Scholar 

  • Gardner JK, Knopoff L (1974) Is the sequence of earthquakes in Southern California, with aftershocks removed, poissonian? Bull Seismol Soc Am 64:1363–1367

    Google Scholar 

  • Geller RJ, Jackson DD, Kagan YY, Mulargia F (1997) Earthquake cannot be predicted. Science 275:1616–1617

    Article  Google Scholar 

  • Goebel THW, Schorlemmer D, Becker TW, Dresen G, Sammis CG (2013) Acoustic emissions document stress change over many seismic cycles in stick-slip experiments. Geophys Res Lett 40:1–6

    Article  Google Scholar 

  • Gualtieri G, Secci S (2014) Extrapolating wind speed time series versus Weibull distribution to assess wind resource to the turbine hub height: a case study on coastal location in Southern Italy. Renew Energy 62:164–176

    Article  Google Scholar 

  • Gutenberg B, Richter CF (1944) Frequency of Earthquake in California. Bull Seismol Soc Am 34:1985-1988

    Google Scholar 

  • Haktanir T, Elcuman H, Cobaner M (2012) Frequency analysis of annual maximum earthquakes within a geographical region. Soil Dyn Earthq Eng 43:323–328

    Article  Google Scholar 

  • Hasumi T, Akimoto T, Aizawa Y (2009) The Weibull-log Weibull distribution for interoccurrence times of earthquakes. Phys A Stat Mech Appl 388:491–498

    Article  Google Scholar 

  • Islam MR, Saidura R, Rahim NA (2011) Assessment of wind energy potentiality at Kudat and Labuan, Malaysia using Weibull distribution function. Energy 36:985–992

    Article  Google Scholar 

  • Joyner WB, Boore DM (1981) Peak horizontal acceleration and velocity from strong-motion records including records from the 1979 Imperial Valley, California, earthquake. Bull Seismol Soc Am 71:2011–2038

    Google Scholar 

  • Kanamori H, Hauksson E, Heaton T (1997) Real-time seismology and earthquake hazard mitigation. Nature 390:461–464

    Article  Google Scholar 

  • Kramer SL (1996) Geotechnical earthquake engineering. Prentice Hall Inc., Hoboken

    Google Scholar 

  • Kun F, Varga I, Lennartz-Sassinek S, Main IG (2013) Approach to failure in porous granular materials under compression. Phys Rev E 88:062207

    Article  Google Scholar 

  • Lanceiri M, Zollo AA (2008) A Bayesian approach to the real-time estimation of magnitude from the early P and S wave displacement peaks. J Geophy Res Solid Earth 113:17

    Google Scholar 

  • Lin PS, Lee CT, Cheng CT, Sung CH (2011) Response spectral attenuation relations for shallow crustal earthquakes in Taiwan. Eng Geol 121:150–164

    Article  Google Scholar 

  • Martinez EZ, Achcar JA, Jacome AA, Santos JS (2013) Mixture and non-mixture cure fraction models based on the generalized modified Weibull distribution with an application to gastric cancer data. Comput Methods Programs Biomed 112:343–355

    Article  Google Scholar 

  • McGuire RK, Arabasz WJ (1990) An introduction to probabilistic seismic hazard analysis. Geotech Environ Geophys 1:333–353

    Google Scholar 

  • Mualchin L (2005) Seismic hazard analysis for critical infrastructures in California. Eng Geol 79:177–184

    Article  Google Scholar 

  • Nanjo KZ, Hirata N, Obara K, Kasahara K (2012) Decade-scale decrease in b value prior to the M9-class 2011 Tohoku and 2004 Sumatra quakes. Geophys Res Lett 39:L20304

    Article  Google Scholar 

  • Tormann T, Enescu B, Woessner J, Wiemer S (2015) Randomness of megathrust earthquakes implied by rapid stress recovery after the Japan earthquake. Nat Geosci 8:152–158

    Article  Google Scholar 

  • Wang JP, Chan CH, Wu YM (2011) The distribution of annual maximum earthquake magnitude around Taiwan and its application in the estimation of catastrophic earthquake recurrence probability. Nat Hazards 59:553–570

    Article  Google Scholar 

  • Wang JP, Lin CW, Taheri H, Chen WS (2012) Impact of fault parameter uncertainties on earthquake recurrence probability by Monte Carlo simulation—an example in central Taiwan. Eng Geol 126:67–74

    Article  Google Scholar 

  • Wang JP, Huang D, Cheng CT, Shao KS, Wu YC, Chang CW (2013) Seismic hazard analysis for Taipei City including deaggregation, design spectra, and time history with excel applications. Comput Geosci 52:146–154

    Article  Google Scholar 

  • Wang JP, Huang D, Chang SC, Wu YM (2014) New evidence and perspective to the Poisson process and earthquake temporal distribution from 55,000 events around Taiwan since 1900. Nat Hazards Rev ASCE 15:38–47

    Article  Google Scholar 

  • Wang JP, Chang SC, Wu YM, Xu Y (2015) Bayesian analysis on earthquake magnitude related to an active fault in Taiwan. Soil Dyn Earthq Eng 75:18–26

    Article  Google Scholar 

  • Weibull W (1939) The phenomenon of rupture in solids. In: Proceedings of the Royal Swedish Institute for Engineering Research 153. Stockholm, 1–55

  • Weibull W (1951) A statistical distribution function of wide applicability. J Appl Mech 18:293–297

    Google Scholar 

  • Weichert DH (1980) Estimation of the earthquake recurrence parameters for unequal observation periods for different magnitudes. Bull Seismol Soc Am 70:1337–1346

    Google Scholar 

  • Wells DL, Coppersmith KJ (1994) New empirical relationships among magnitude, rupture length, rupture width, rupture area, and surface displacement. Bull Seismol Soc Am 84:974–1002

    Google Scholar 

  • Wu YM, Chiao LY (2006) Seismic quiescence before the 1999 Chi-Chi, Taiwan Mw7.6 earthquake. Bull Seismol Soc Am 96:321–327

    Article  Google Scholar 

Download references

Acknowledgments

The author appreciates the Editor and Reviewers for their comments on the submission, making it much improved in so many aspects. The author also appreciates the financial support of HKUST with the project DAG09/10.EG03 for the study.

Author information

Authors and Affiliations

  1. Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong

    J. P. Wang

Authors
  1. J. P. Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. P. Wang.

Ethics declarations

Humans and animal rights

The research does not involve experiments on humans and animals.

Conflict of interest

None.

Appendix: The Chi-square test

Appendix: The Chi-square test

The first step to conduct a Chi-square test is to compute the total difference between theoretical and observational frequencies in a histogram, and the difference is referred to as the Chi-square value (χ 2):

$$\chi^{2} = \sum\limits_{i = 1}^{n} {\frac{{(e_{i} - o_{i} )^{2} }}{{e_{i} }}}$$
(5)

where o i and e i denote observational and theoretical frequencies; n is the number of data bins in the histogram. Then, according to probability and statistics, the total difference (or the Chi-square value) is considered a random variable following the Chi-square distribution with a degree of freedom of (n – 1 − k), where k is the number of parameters of the statistical model examined (Ang and Tang 2007). For example, k is equal to 2 for the Weibull distribution given model parameters α and β present.

The next step of the Chi-square test is to calculate the critical value at a given level of significance (usually 5 %). As the schematic diagram shown in Fig. 9, the level of significance is actually the right-tail probability against the critical value. As a result, if the hypothesis is reasonable and acceptable, there should be a relatively large probability (i.e., 95 %) for the Chi-square value calculated less than the critical value. By contrast, if the Chi-square value is still greater than the critical value even though the probability is rather small (i.e., 5 %), the model used for the simulation might not be acceptable, or the hypothesis should be rejected from a statistical point of view.

Fig. 9
figure 9

Schematic diagram illustrating the basics of the Chi-square test; if the Chi-square value or the total difference is greater than the critical value, which is a small probability event, then the hypothesis examined will be rejected

Full size image

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, J.P. Reviews of seismicity around Taiwan: Weibull distribution . Nat Hazards 80, 1651–1668 (2016). https://doi.org/10.1007/s11069-015-2045-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11069-015-2045-7

Keywords

Search

Navigation