The Appropriate Statistical Distribution for Residuals in Seismic Attenuation Relationships

Document Type : Review Article

Author

Department of Civil Engineering, Technical University of Buein Zahra, Buein Zahra, Qazvin, 3451745346, Iran.

Abstract

The developed ground motion models are mainly based on the assumption of normality of the residuals. The extreme value distribution is a statistical distribution used in modeling rare events and extreme scenarios. In large earthquakes with a long return period, the recorded peak ground accelerations (PGAs) are large and rare, so the assumption of extreme distributions is not unexpected for these accelerations. The extreme value distribution has two conventional forms: generalized extreme value (GEV) for maximum values of blocks with the same time duration and generalized Pareto distribution (GPD) for values above a determined threshold. Due to the lower recorded numbers of PGAs, using the GPD distribution in examining the extreme values ​​of the PGAs is more appropriate. If the GPD distribution assumption for PGA data be accepted, it is suggested to develop a seismic acceleration attenuation relationship for large or extreme data based on the GPD distribution, and the common assumption of lognormal distribution is discarded. This article reviews the statistical distributions used in ground motion models. The results suggest that in the development of ground motion relationships, the normal distribution for residual should be abandoned with a fundamental revision, and the next generation of these models should be developed based on the GPD distribution.

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Main Subjects

References

[1] W. Milne, A.G. Davenport, Distribution of earthquake risk in Canada, Bulletin of the Seismological Society of America, 59(2) (1969) 729–754.
[2] L. Esteva, E. Rosenblueth, Espectros de temblores a distancias moderadas y grandes, Boletin Sociedad Mexicana de Ingenieria Sesmica, 2(1) (1964) 1–18.
[3] J.A. Blume, The SAM procedure for site-acceleration-magnitude relationships, in:  Proceedings of Sixth World Conference on Earthquake Engineering, 1977, pp. 416–422.
[4] N.C. Donovan, A.E. Bornstein, Uncertainties in seismic risk procedures, Journal of the Geotechnical Engineering Division, 104(7) (1978) 869–887.
[5] L. Esteva, Seismic Risk and Seismic Design Decisions, Massachusetts Inst. of Tech., Cambridge. Univ. of Mexico, Mexico City, 1970.
[6] E. Faccioli, Response spectra for soft soil sites, in:  From Volume I of Earthquake Engineering and Soil Dynamics--Proceedings of the ASCE Geotechnical Engineering Division Specialty Conference, June 19-21, 1978, Pasadena, California. Sponsored by Geotechnical Engineering Division of ASCE in cooperation with: 1978.
[7] W. Milne, Seismic risk maps for Canada, in: Proceedings of Sixth World Conference on Earthquake Engineering, 1977, pp. 2–508.
[8] D. Orphal, J. Lahoud, Prediction of peak ground motion from earthquakes, Bulletin of the Seismological Society of America, 64(5) (1974) 1563–1574.
[9] R.K. McGuire, Seismic design spectra and mapping procedures using hazard analysis based directly on oscillator response, Earthquake Engineering & Structural Dynamics, 5(3) (1977) 211–234.
[10] D. Denham, G. Small, Strong motion data centre: Bureau of Mineral Resources, Canada, Bulletin of the New Zealand Society for Earthquake Engineering, 4(1) (1971) 15–30.
[11] N. Ambraseys, Trends in engineering seismology in Europe, in:  Proceedings of fifth European conference on earthquake engineering, 1975, pp. 39–52.
[12] R.K. McGuire, Seismic ground motion parameter relations, Journal of the Geotechnical Engineering Division, 104(4) (1978) 481–490.
[13] K.W. Campbell, Near-source attenuation of peak horizontal acceleration, Bulletin of the Seismological Society of America, 71(6) (1981) 2039–2070.
[14] M. McCann Jr, H. Echezwia, Investigating the uncertainty in ground motion prediction, in:  of: Proceedings of Eighth World Conference on Earthquake Engineering, 1984, pp. 297–304.
[15] K. Kawashima, K. Aizawa, K. Takahashi, Attenuation of peak ground acceleration, velocity and displacement based on multiple regression analysis of Japanese strong motion records, Earthquake engineering & structural dynamics, 14(2) (1986) 199–215.
[16] K. Campbell, Empirical prediction of near-source soil and soft-rock ground motion for the Diablo Canyon power plant site, San Luis Obispo, California, prepared for Lawrence Livermore National Laboratory, Livermore, cA (1990).
[17] C. Crouse, Ground-motion attenuation equations for earthquakes on the Cascadia subduction zone, Earthquake spectra, 7(2) (1991) 201–236.
[18] C. Crouse, J. McGuire, Site response studies for the purpose of revising NEHRP seismic provisions, Earthquake spectra, 12(3) (1996) 407–439.
[19] N. Theodulidis, B. Papazachos, Dependence of strong ground motion on magnitude-distance, site geology and macroseismic intensity for shallow earthquakes in Greece: I, Peak horizontal acceleration, velocity and displacement, Soil Dynamics and Earthquake Engineering, 11(7) (1992) 387–402.
[20] J.J. Bommer, N.A. Abrahamson, Why do modern probabilistic seismic-hazard analyses often lead to increased hazard estimates?, Bulletin of the Seismological Society of America, 96(6) (2006) 1967–1977.
[21] N. Abrahamson, P. Birkhauser, M. Koller, D. Mayer-Rosa, P. Smit, C. Sprecher, S. Tinic, R. Graf, PEGASOS—A comprehensive probabilistic seismic hazard assessment for nuclear power plants in Switzerland, Proc. of the 12th ECEE,  (2002).
[22] F.O. Strasser, N.A. Abrahamson, J.J. Bommer, Sigma: Issues, insights, and challenges, Seismological Research Letters, 80(1) (2009) 40–56.
[23] S. Ólafsson, S. Remseth, R. Sigbjörnsson, Stochastic models for simulation of strong ground motion in Iceland, Earthquake engineering & structural dynamics, 30(9) (2001) 1305–1331.
[24] J. Douglas, P.M. Smit, How accurate can strong ground motion attenuation relations be?, Bulletin of the Seismological Society of America, 91(6) (2001) 1917–1923.
[25] J.J. Bommer, N.A. Abrahamson, F.O. Strasser, A. Pecker, P.-Y. Bard, H. Bungum, F. Cotton, D. Fäh, F. Sabetta, F. Scherbaum, The challenge of defining upper bounds on earthquake ground motions, Seismological Research Letters, 75(1) (2004) 82–95.
[26] M. Yamada, A.H. Olsen, T.H. Heaton, Statistical features of short-period and long-period near-source ground motions, Bulletin of the Seismological Society of America, 99(6) (2009) 3264–3274.
[27] B. Tavakoli, S. Pezeshk, A new approach to estimate a mixed model–based ground motion prediction equation, Earthquake spectra, 23(3) (2007) 665–684.
[28] V. Graizer, E. Kalkan, Ground motion attenuation model for peak horizontal acceleration from shallow crustal earthquakes, Earthquake Spectra, 23(3) (2007) 585–613.
[29] J.W. Baker, An introduction to probabilistic seismic hazard analysis (PSHA), White paper, version 1(3) (2008).
[30] S. Rezaeian, A. Der Kiureghian, Simulation of synthetic ground motions for specified earthquake and site characteristics, Earthquake Engineering & Structural Dynamics, 39(10) (2010) 1155–1180.
[31] V. Graizer, E. Kalkan, A novel approach to strong ground motion attenuation modeling, in:  Proceedings of Fourteenth World Conference on Earthquake Engineering, 2008, pp. 02–0022.
[32] V. Graizer, Extending and testing Graizer-Kalkan ground motion attenuation model based on Atlas database of shallow crustal events, in:  Proceedings of the 9th US National and 10th Canadian Conference on Earthquake Engineering, 2010, pp. 5525–5534.
[33] D.M. Boore, G.M. Atkinson, Ground-motion prediction equations for the average horizontal component of PGA, PGV, and 5%-damped PSA at spectral periods between 0.01 s and 10.0 s, Earthquake spectra, 24(1) (2008) 99–138.
[34] M. Segou, N. Voulgaris, The use of stochastic optimization in ground motion prediction, Earthquake spectra, 29(1) (2013) 283–308.
[35] M. Soghrat, N. Khaji, H. Zafarani, Simulation of strong ground motion in northern Iran using the specific barrier model, Geophysical Journal International, 188(2) (2012) 645–679.
[36] B. Bradley, M. Hughes, Spatially-distributed ground motion intensity maps: Application for site-specific liquefaction evaluation in Christchurch, in:  2013 NZSEE Conference, 2013.
[37] A. Azarbakht, S. Rahpeyma, M. Mousavi, A new methodology for assessment of the stability of ground‐motion prediction equations, Bulletin of the Seismological Society of America, 104(3) (2014) 1447–1457.
[38] S. Akkar, J.J. Bommer, Empirical equations for the prediction of PGA, PGV, and spectral accelerations in Europe, the Mediterranean region, and the Middle East, Seismological Research Letters, 81(2) (2010) 195–206.
[39] G.M. Atkinson, D.M. Boore, Modifications to existing ground-motion prediction equations in light of new data, Bulletin of the Seismological Society of America, 101(3) (2011) 1121–1135.
[40] C. Cauzzi, E. Faccioli, Broadband (0.05 to 20 s) prediction of displacement response spectra based on worldwide digital records, Journal of Seismology, 12(4) (2008) 453–475.
[41] R. Harris, N.A. Abrahamson, Ground Motions Due to Earthquakes on Creeping Faults, in:  AGU Fall Meeting Abstracts, 2014, pp. S53B–4504.
[42] M.A. Jaimes, J. Lermo, A.D. García‐Soto, Ground‐motion prediction model from local earthquakes of the Mexico basin at the hill zone of Mexico City, Bulletin of the Seismological Society of America, 106(6) (2016) 2532–2544.
[43] M. Kowsari, B. Halldórsson, B. Hrafnkelsson, J. Þór, S.Ó. Snæbjörnsson, S. Jónsson, On the Sensitivity of Ground-Motion Prediction Equations for Earthquake Strong-motions in the South Iceland Seismic Zone, in:  International Workshop on Earthquakes in North Iceland: 31 May–4 June 2016, Húsavík, Iceland, 2016.
[44] D.M. Boore, J.P. Stewart, E. Seyhan, G.M. Atkinson, NGA-West2 equations for predicting PGA, PGV, and 5% damped PSA for shallow crustal earthquakes, Earthquake Spectra, 30(3) (2014) 1057–1085.
[45] M. Mousavi, H. Zafarani, S. Rahpeyma, A. Azarbakht, Test of goodness of the NGA ground‐motion equations to predict the strong motions of the 2012 Ahar–Varzaghan dual earthquakes in northwestern Iran, Bulletin of the Seismological Society of America, 104(5) (2014) 2512–2528.
[46] G. Tusa, H. Langer, Prediction of ground motion parameters for the volcanic area of Mount Etna, Journal of Seismology, 20(1) (2016) 1–42.
[47] M.A. Zanini, C. Vianello, F. Faleschini, L. Hofer, G. Maschio, A framework for probabilistic seismic risk assessment of NG distribution networks, Chemical Engineering Transactions, 53 (2016) 163–168.
[48] D. Bindi, The predictive power of ground‐motion prediction equations, Bulletin of the Seismological Society of America, 107(2) (2017) 1005–1011.
[49] M. Soghrat, M. Ziyaeifar, Ground motion prediction equations for horizontal and vertical components of acceleration in Northern Iran, Journal of Seismology, 21(1) (2017) 99–125.
[50] L. De Haan, A. Ferreira, Extreme value theory: an introduction, Springer, 2006.
[51] N.N. Ambraseys, J. Douglas, S. Sarma, P. Smit, Equations for the estimation of strong ground motions from shallow crustal earthquakes using data from Europe and the Middle East: horizontal peak ground acceleration and spectral acceleration, Bulletin of earthquake engineering, 3(1) (2005) 1–53.
[52] L. Huyse, R. Chen, J. Stamatakos, Application of generalized Pareto distribution to constrain uncertainty in peak ground accelerations, Bulletin of the Seismological Society of America, 100(1) (2010) 87–101.
[53] K.M. McBean, J.G. Anderson, J.N. Brune, R. Anooshehpoor, Statistics of Ground Motions in a Foam Rubber Model of a Strike‐Slip Fault, Bulletin of the Seismological Society of America, 105(3) (2015) 1456–1467.
[54] N. Abrahamson, N. Gregor, K. Addo, BC Hydro ground motion prediction equations for subduction earthquakes, Earthquake Spectra, 32(1) (2016) 23–44.
[55] D.J. Dupuis, J.M. Flemming, Modelling peak accelerations from earthquakes, Earthquake engineering & structural dynamics, 35(8) (2006) 969–987.
[56] M. Raschke, Statistical modeling of ground motion relations for seismic hazard analysis, Journal of seismology, 17(4) (2013) 1157–1182.
[57] V. Pavlenko, Effect of alternative distributions of ground motion variability on results of probabilistic seismic hazard analysis, Natural Hazards, 78(3) (2015) 1917–1930.
[58] V. Pavlenko, Estimation of the upper bound of seismic hazard curve by using the generalised extreme value distribution, Natural Hazards, 89(1) (2017) 19–33.
[59] S. Borzoo, M. Bastami, A. Fallah, Modeling extreme ground-motion intensities using extreme value theory, Pure and Applied Geophysics, 177(10) (2020) 4691–4706.
[60] S. Borzoo, M. Bastami, A. Fallah, Extreme scenarios selection for seismic assessment of expanded lifeline networks, Structure and Infrastructure Engineering, 17(10) (2021) 1386–1403.
[61] S. BORZOO, M. BASTAMI, A. FALLAH, MAGNITUDE SIMULATION USING THE GENERALIZED PARETO DISTRIBUTION.
AUT Journal of Civil Engineering
Volume 10, Issue 1
2026
Pages 91-98

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