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A critique of the ICOLD method for selecting earthquake ground motions to design large dams
Engineering Geology 80 (2005) 37 – 42 www.elsevier.com/locate/enggeo
Opinion paper
A critique of the ICOLD method for selecting earthquake ground motions to design large dams Manish Shrikhande, Susanta BasuT Department of Earthquake Engineering, Indian Institute of Technology Roorkee, Roorkee-247667, India Received 25 February 2005; accepted 25 February 2005 Available online 12 April 2005
Abstract The practice for selection of earthquake ground motion for design of large dams as recommended by the International Committee on Large Dams (ICOLD) bulletin 72 is examined. It is shown that the recommended practice is flawed and does not bear a scrutiny on the basis of statistical theory. The ground motion attenuation relationships are derived on the basis of different sets of assumptions and different sets of data, and with different standard errors. Estimates from these different attenuation relationships cannot be averaged (weighted, or otherwise) as recommended in the bulletin due to the presence of standard errors for each of these relationships. D 2005 Elsevier B.V. All rights reserved. Keywords: Seismic hazard assessment; Ground motion attenuation; Earthquake resistant design; Regression analysis
1. Introduction The Appendix 2 of the International Committee on Large Dams (ICOLD) Bulletin 72 (ICOLD, 1989), hereafter referred to as the Bulletin, contains a recommendation which states that seismic ground motion can be specified by peak or effective peak values of expected acceleration, velocity, and/or displacement. Therein the attenuation relation is defined as an empirically derived relationship to obtain peak parameter of ground motion at a site in T Corresponding author. E-mail addresses: [email protected] (M. Shrikhande), [email protected] (S. Basu). 0013-7952/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.enggeo.2005.02.005
terms of energy release (magnitude) at source and distance from source to site. These relationships are very sensitive to distance and magnitude. The reasons behind the scatter are also stated in the Bulletin. The peak ground acceleration (PGA) is the most commonly used seismic parameter for a site despite its various shortcomings. The guideline does not recommend the use of any specific attenuation relationship among the various relationships that have been developed in the recent years to estimate PGA. But it recommends that b. . . consideration should be given to using weighted (emphasis added) average of values provided by several of the most accepted and reliable equations for this variableQ. The guideline also provides references for equations
38
M. Shrikhande, S. Basu / Engineering Geology 80 (2005) 37–42
that are deemed to be the most frequently used ones in the United States to estimate PGA. These are (1) Ambraseys, (1973 and 1978 in ICOLD, 1989); (2) Trifunac and Brady (1975); (3) Campbell (1981); (4) Boore and Joyner (1982); (5) Bolt and Abrahamson (1982); (6) Seed and Idriss (1982); and (7) Idriss (1985). This simply implies that weighted average of the derived values of peak horizontal ground acceleration (PGA) from various attenuation relationships (equations) shall be used in arriving at a design parameter for a project. This is a dangerous proposition and cannot be defended by any means. We explore the implications of this proposition and show that the ground motion estimates obtained in this manner lack a physical significance. Prior to that, however, a summary of the basic data provided by the Bulletin (ICOLD, 1989) as several of the most accepted reliable equations need to be presented. A brief discussion on general method of estimation of attenuation law follows and finally we conclude with comments on the said recommendation of the guideline.
2. Summary of the equations Out of all the references for the equations of PGA stated in the Bulletin (ICOLD, 1989), only three (Campbell, 1981; Bolt and Abrahamson, 1982; Idriss, 1985) of them deal with the equation of PGA and it is surprising to note that rest of them do not deal with the attenuation relationships at all! First part of the summary will be with regard to the references cited in the Bulletin that do not directly deal with the subject.
3. Out of context references At serial number 2 in the list of references in the Bulletin figures an entry for bAmbraseys, N.N. (1973): Fifth World Conference on Earthquake EngineeringQ. However, there are no articles by Ambraseys on the subject of attenuation relationship in the entire proceedings of bThe Fifth World Conference on Earthquake EngineeringQ. There are only three discussions by Ambraseys in this proceedings. The first one (Ambraseys, 1973a) is
on response spectra scaling given the information of earthquake source mechanism. The second one (Ambraseys, 1973b) is on cracking of Canyon Dam. The last one (Ambraseys, 1973c) is on the soil–structure interaction effects observed during Caracas Earthquake. The list of reference in the Bulletin does not contain any publication by Ambraseys in the year 1978. An excellent review on the subject of ground motion estimation (Douglas, 2003) is currently available, wherein it is indicated that Ambraseys (1975) proposed an attenuation relation for PGA with Local Magnitude (ML) and hypocentral distance as independent variables based on European strong motion data. Douglas (2003) states that this article provides little information on the data selection. The Richter magnitude data is between 3.5 and 5.0 and the hypocentral distance data between 5 km and 35 km are used in proposing attenuation relation for Europe. The article of Trifunac and Brady (1975) deals with the correlation between Modified Mercalli Intensity (MMI) and peak ground motion. This paper proposes three relationships (i) between MMI and PGA, (ii) between MMI and peak ground velocity (PGV), and (iii) between MMI and peak ground displacement (PGD). The Boore and Joyner (1982) article is a review that deals with the necessary condition to improve the empirical projection (prediction) of strong ground motion. However, it discusses one of their paper (Joyner and Boore, 1981) that is on the subject of discussion. The monograph of Seed and Idriss (1982) shows some illustrative plot of peak horizontal acceleration with closest distance from zone of energy release for some given earthquake surface wave magnitude M S. This monograph does not prescribe any empirical attenuation relation for estimating PGA in terms of earthquake magnitude and distance.
4. References on the subject Campbell (1981) proposed an attenuation relation, using data of western north America and some nearsource earthquakes from other parts of the world, that does not have anelastic attenuation term. It contains a geometric spreading term, exponentially
M. Shrikhande, S. Basu / Engineering Geology 80 (2005) 37–42
dependent on magnitude, that modulates PGA close to the fault. The influence of magnitude and source to site distance is non-separable due to the presence of this term. The proposed relation is for the geometric mean of two horizontal peak acceleration from a record depending on magnitude M and distance to fault rupture from the site. The magnitude M, defined as Richter magnitude, is equal to local magnitude M L for M b 6 and is equal to surface-wave magnitude M S for M z 6. This definition of Richter magnitude will be used henceforth. The magnitude range between 5.0 and 7.7 is considered in the analysis. The analysis is carried out on the logarithm transformed proposed equation with 27 earthquakes that produced 116 records of mean horizontal PGA. Since, the mean PGA is not uniformly sampled over all magnitude and rupture distances in the considered data set, a weighted nonlinear regression analysis using method of least square is performed. The rupture distances (0 to 56.6 km) are distributed in nine-bin to control the influence of well-recorded earthquakes such as San Fernando, 1971 and Imperial Valley, 1979. The weights are assigned according to distances. The weighting scheme is such that if an earthquake has a single recording in each distance classes, then it will carry maximum weight. The weightage scheme is subjective (educated guess) in nature and the chosen scheme influences the coefficients to be estimated. Joyner and Boore (1981) proposed an attenuation relation with all terms as stated above with the coefficient of geometric spreading term being constrained to be unity. The basic feature of this relationship is separability of influence of magnitude and site to source distance on the largest component of the two horizontal PGA. The PGA relationship is defined as a function of moment magnitude M W and closest distance of surface projection of the fault rupture from the site. The data set used for regression analysis is of western north America strong motion that contains a total of 183 largest of the two horizontal component of PGA from accelerograms recorded during 23 earthquakes. One of these earthquakes is an aftershock of the Imperial Valley, 1979 main event. The range of moment magnitude M W is between 5.0 and 7.7 and range of the closest distance of surface projection of fault rupture is [0.5–370] km. They use a two-stage method on the logarithmic
39
transformed multiplicative model for estimation of coefficients to account for (1) record to record variation in the first stage, and (2) earthquake to earthquake variation in the second stage. The accelerograms of well-recorded earthquake are essentially records of strong motion conditional on a given set of source parameters. These data form a cluster of PGA of a single magnitude thus are not independent observations. The Boore and Joyner (1982) article, referred to in the Bulletin, states assumptions that are required to be made for reconciliation of the proposed PGA relationship due to them (Joyner and Boore, 1981) with that of Campbell (1981) as different definitions are used for both dependent and independent variables in the two relationships. Bolt and Abrahamson (1982) proposed equations that prescribe the largest of two horizontal PGA as a function of the closest distance to zone of energy release for different ranges of moment magnitude M W. These relationships are in the form of Pearson family of probability curves. They use data set of Joyner and Boore (1981) and use nonlinear least square estimation on the proposed multiplicative model itself and hence project mean of the largest horizontal PGA. Idriss (1985) proposed relationships for randomly oriented horizontal component of PGA for various Richter magnitude M in the range from 4.5 to 8.5 at an interval of magnitude 0.5. The relation is dependent on closest distance from source for M z 6 and hypocentral distance in km otherwise. These relationships are essentially amplification factor scaled by geometric spreading term.
5. Development of attenuation models The attenuation relationship is a concise abstraction of complex reality. It is, therefore, a mathematical model necessarily limited in scope in comparison with the real system. The attenuation relationships are developed to obtain an interpolation formula as a summary of the data base of PGA, magnitude and distances obtained from various strong motion stations for different seismic events. Regression analysis is used to estimate parameters of the model. A model given by an equation representing dtrueT value of observation or response (dependent variable) is
40
M. Shrikhande, S. Basu / Engineering Geology 80 (2005) 37–42
chosen using some physical basis of seismic ground motion propagation. The following essential assumptions are implicit in every regression analysis: 1. The chosen form of equation is assumed to be approximately correct. 2. The data set is a representative sample of the entire range. 3. The data of responses are statistically uncorrelated. A less restrictive assumption is also made that (1) all observations have same variance (if the variance of the data of dependent variable changes in a known way, a weighted regression is performed), and (2) all the factors or predictors (independent variables) are known without error. The random error in ordinary least squares regression method is assumed to be additive and variance of random error is equal to that of observations. It is necessary to know probability distribution of random error to test hypothesis about coefficients, confidence interval and prediction interval of regression equation. Usually, random error is assumed to be normal and consequently it also becomes statistically independent. One must exercise great care while performing regression analysis for earthquake ground motion parameters as in this case none of the primary assumptions of regression are complied with. First, it is impossible to choose a correct form of equation for earthquake ground motion parameter because all physical causes that affect ground motion are not clearly known. Moreover, the values of the coefficients of the model are dependent on the choice of a data set. This bias in estimate is due to (1) non-uniform data in magnitude– distance space, and (2) the data set being dominated by a few well recorded earthquake. An attenuation relation for PGA is essentially a multiplicative model, which in general is composed of terms related to (1) exponential scaling of magnitude, (2) distance dependent anelastic attenuation, (3) geometric spreading, and (4) site effects with a multiplicative random error. This multiplicative model transforms into a linear (in the coefficients) regression model by taking logarithms. The random error becomes additive after transformation. Moreover, the fitting of multiplicative model directly violates constant variance errors (homoskedasticity) assumption of regression analysis.
6. Discussion The daverageT as recommended in the Bulletin is technically a misnomer. The U.S. Census Bureau defines an average as a number or value that is used to represent a dtypical valueT of a group of numbers (Jaffe and Spirer, 1987). A typical value is defined sometimes as a measure of dlocationT, i.e. where the parameter value belonged on an axis. Most of the times, it is defined as the central tendency of the data. Two of the common measures of central tendency of data are mean and median. The mean is very sensitive to outliers (high and low values of observations). The median estimates are resistant to outliers. Based on linear combination of ordered statistics (L-estimator), various location estimates for symmetric distributions have also been proposed, such as midmean, broadened median, trimean, etc. These estimators represent a compromise between mean and median. In the absence of any qualified statement, it is presumed that arithmetic mean is meant by daverageT in the Bulletin (ICOLD, 1989). The attenuation relations are models of physical phenomena. A model is made by adding idealised assumptions to the data and running them through an assumed equations. These assumptions are necessary to overcome limitations of data. Therefore, all models reveal some part of the real effects of data and others are artifacts of the assumptions made in the process. It can be seen that various authors have proposed models of attenuation relationships of PGA that are different in both independent and dependent variables and are based on some physical basis of strong ground acceleration. Different models are developed with different physical basis and different data set. The obtained equation of the model via regression analysis is different from an ordinary algebraic equation because of the presence of random error term. This equation represents a probabilistic additive model. The weighted mean model that is suggested to be obtained from the various proposed models based on different physical basis is alien to statistical theory and practice. The weightage scheme in regression analysis is primarily performed when the variance of dependent variables change in a known way. It appears that dweighted averageT value is borrowed from weighted regression analysis without dwelling much about its correctness and consequences. The
M. Shrikhande, S. Basu / Engineering Geology 80 (2005) 37–42
choice of weights needs a subjective judgment. Any subjective judgment is a measure of knowledge that is influenced by personal bias and is likely to be questioned. The various models referred to in the Bulletin are for horizontal component of the ground motion that project (1) median of geometric mean of two horizontal PGA (Campbell, 1981), (2) median of the largest PGA (Joyner and Boore, 1981), (3) mean of the largest PGA (Bolt and Abrahamson, 1982), and (4) median of the randomly oriented horizontal PGA (Idriss, 1985). It is obvious that weighted mean of dissimilar object cannot be performed. The data set used in developing the relationships as above are different. Hence, the suggested daverageT is to be performed on non-uniform data set and that is not recommended in statistics. Moreover, Bolt and Abrahamson (1982) have reported standard error of a single observation and not the standard error of regression equation. The covariances between random error of various recommended attenuation relationships of PGA are required to find the variance of the random error of suggested weighted mean model. Unfortunately, the above covariances are not available. Since, the standard error cannot be estimated any confidence intervals and predicted intervals of the weighted mean model cannot be obtained. Thus, it is an estimate without a standard error and hence, according to Sir Jeffreys (Jeffreys, 1967) is a practically meaningless estimator. The only rational way to salvage the situation–in our opinion–is to use a single prediction relationship that is derived from as large and well-represented data set as possible with lower standard error. Glitches in theory cannot be wished away by daverageT. It appears that daverageT is discovered by the advocates of seismic probability as a convenient tool to iron out all problems that are created by uncertainty of earthquake phenomena.
7. Conclusions The ground motion attenuation relationships represent mathematical models for a limited aspect of the phenomenon of ground motion propagation. Each of these models has been derived on the basis of different sets of assumptions and different sets of
41
data. Further, each of these models has different standard error. Estimates from these different attenuation relationships cannot be averaged (weighted, or otherwise) as recommended in the Bulletin due to the presence of standard errors and due to differences in the underlying assumptions for each of these relationships. Moreover, while the independent estimates from each attenuation relationships can be associated with certain confidence limits, no such measure can be ascribed to the average of several attenuation relationships.
References Ambraseys, N., 1973a. Discussion on bCharacterization of response spectra by parameter governing the gross nature of earthquake source mechanismQ by M.D. Trifunac, Session 2C:PD79, pp. 701–704. Proceedings of the Fifth World Conference on Earthquake Engineering, Rome, vol. 1, p. 705. Ambraseys, N., 1973b. Discussion on bEarthquake induced cracking of dry Canyon damQ by Kenneth L. Lee and Henry G. Walters, Session 4D:PD192, pp. 1544–1547. Proceedings of the Fifth World Conference on Earthquake Engineering, Rome, vol. 2, p. 1548. Ambraseys, N., 1973c. Discussion on bSoil–structure interaction effect in the Caracas earth-quake of 1967Q by H. Seed and J. Alonso Session 6A:PD263, pp. 2108–2111. In Proceedings of the Fifth World Conference on Earthquake Engineering, Rome, vol. 2, p. 2112. Ambraseys, N., 1975. Trends in engineering seismology in Europe. Fifth European Conference on Earthquake Engineering, vol. 3, pp. 39 – 52. Bolt, B., Abrahamson, N., 1982. New attenuation relations for peak and expected accelerations of strong ground motion. Bulletin of the Seismological Society of America 72 (6), 2307 – 2322. Boore, D., Joyner, W., 1982. The empirical prediction of ground motion. Bulletin of the Seismological Society of America 72 (6), S43 – S60. Campbell, K., 1981. Near-source attenuation of peak horizontal acceleration. Bulletin of the Seismological Society of America 71 (6), 2039 – 2070. Douglas, J., 2003. Earthquake ground motion estimation using strong-motion records: a review of equations for the estimation of peak ground acceleration and response spectral ordinates. Earth-Sci. Rev. 61, 43 – 104. ICOLD, 1989. Selecting seismic parameters for large dams. Bulletin, vol. 72. International Commission on Large Dams. Idriss, I., 1985. Evaluating seismic risk in engineering practice. Proceeding Eleventh International Conference on Soil Mechanics and Foundation Engineering, vol. 1. A.A. Balkema, San Fransisco, CA, pp. 255 – 320. Jaffe, A., Spirer, H., 1987. Misused Statistics: Straight Talk for Twisted Numbers. Marcel Dekker, Inc., New York.
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Jeffreys, H., 1967. Seismology, statistical methods. In: Runcorn, K. (Ed.), International Dictionary of Geophysics. Pergamon Press, London, pp. 1398 – 1401. Joyner, W., Boore, D., 1981. Peak horizontal acceleration and velocity from strong motion records including records from the 1979 Imperial Valley, California, earthquake. Bulletin of the Seismological Society of America 71 (6), 2011 – 2038.
Seed, H., Idriss, I., 1982. Ground motions and soil liquefaction during earthquakes. Technical report, Earthquake Engineering Research Institute, Berkeley, California, USA. Trifunac, M., Brady, A., 1975. On the correlation of seismic intensity with peaks of recorded strong motion. Bulletin of the Seismological Society of America 65, 139 – 162.
Opinion paper
A critique of the ICOLD method for selecting earthquake ground motions to design large dams Manish Shrikhande, Susanta BasuT Department of Earthquake Engineering, Indian Institute of Technology Roorkee, Roorkee-247667, India Received 25 February 2005; accepted 25 February 2005 Available online 12 April 2005
Abstract The practice for selection of earthquake ground motion for design of large dams as recommended by the International Committee on Large Dams (ICOLD) bulletin 72 is examined. It is shown that the recommended practice is flawed and does not bear a scrutiny on the basis of statistical theory. The ground motion attenuation relationships are derived on the basis of different sets of assumptions and different sets of data, and with different standard errors. Estimates from these different attenuation relationships cannot be averaged (weighted, or otherwise) as recommended in the bulletin due to the presence of standard errors for each of these relationships. D 2005 Elsevier B.V. All rights reserved. Keywords: Seismic hazard assessment; Ground motion attenuation; Earthquake resistant design; Regression analysis
1. Introduction The Appendix 2 of the International Committee on Large Dams (ICOLD) Bulletin 72 (ICOLD, 1989), hereafter referred to as the Bulletin, contains a recommendation which states that seismic ground motion can be specified by peak or effective peak values of expected acceleration, velocity, and/or displacement. Therein the attenuation relation is defined as an empirically derived relationship to obtain peak parameter of ground motion at a site in T Corresponding author. E-mail addresses: [email protected] (M. Shrikhande), [email protected] (S. Basu). 0013-7952/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.enggeo.2005.02.005
terms of energy release (magnitude) at source and distance from source to site. These relationships are very sensitive to distance and magnitude. The reasons behind the scatter are also stated in the Bulletin. The peak ground acceleration (PGA) is the most commonly used seismic parameter for a site despite its various shortcomings. The guideline does not recommend the use of any specific attenuation relationship among the various relationships that have been developed in the recent years to estimate PGA. But it recommends that b. . . consideration should be given to using weighted (emphasis added) average of values provided by several of the most accepted and reliable equations for this variableQ. The guideline also provides references for equations
38
M. Shrikhande, S. Basu / Engineering Geology 80 (2005) 37–42
that are deemed to be the most frequently used ones in the United States to estimate PGA. These are (1) Ambraseys, (1973 and 1978 in ICOLD, 1989); (2) Trifunac and Brady (1975); (3) Campbell (1981); (4) Boore and Joyner (1982); (5) Bolt and Abrahamson (1982); (6) Seed and Idriss (1982); and (7) Idriss (1985). This simply implies that weighted average of the derived values of peak horizontal ground acceleration (PGA) from various attenuation relationships (equations) shall be used in arriving at a design parameter for a project. This is a dangerous proposition and cannot be defended by any means. We explore the implications of this proposition and show that the ground motion estimates obtained in this manner lack a physical significance. Prior to that, however, a summary of the basic data provided by the Bulletin (ICOLD, 1989) as several of the most accepted reliable equations need to be presented. A brief discussion on general method of estimation of attenuation law follows and finally we conclude with comments on the said recommendation of the guideline.
2. Summary of the equations Out of all the references for the equations of PGA stated in the Bulletin (ICOLD, 1989), only three (Campbell, 1981; Bolt and Abrahamson, 1982; Idriss, 1985) of them deal with the equation of PGA and it is surprising to note that rest of them do not deal with the attenuation relationships at all! First part of the summary will be with regard to the references cited in the Bulletin that do not directly deal with the subject.
3. Out of context references At serial number 2 in the list of references in the Bulletin figures an entry for bAmbraseys, N.N. (1973): Fifth World Conference on Earthquake EngineeringQ. However, there are no articles by Ambraseys on the subject of attenuation relationship in the entire proceedings of bThe Fifth World Conference on Earthquake EngineeringQ. There are only three discussions by Ambraseys in this proceedings. The first one (Ambraseys, 1973a) is
on response spectra scaling given the information of earthquake source mechanism. The second one (Ambraseys, 1973b) is on cracking of Canyon Dam. The last one (Ambraseys, 1973c) is on the soil–structure interaction effects observed during Caracas Earthquake. The list of reference in the Bulletin does not contain any publication by Ambraseys in the year 1978. An excellent review on the subject of ground motion estimation (Douglas, 2003) is currently available, wherein it is indicated that Ambraseys (1975) proposed an attenuation relation for PGA with Local Magnitude (ML) and hypocentral distance as independent variables based on European strong motion data. Douglas (2003) states that this article provides little information on the data selection. The Richter magnitude data is between 3.5 and 5.0 and the hypocentral distance data between 5 km and 35 km are used in proposing attenuation relation for Europe. The article of Trifunac and Brady (1975) deals with the correlation between Modified Mercalli Intensity (MMI) and peak ground motion. This paper proposes three relationships (i) between MMI and PGA, (ii) between MMI and peak ground velocity (PGV), and (iii) between MMI and peak ground displacement (PGD). The Boore and Joyner (1982) article is a review that deals with the necessary condition to improve the empirical projection (prediction) of strong ground motion. However, it discusses one of their paper (Joyner and Boore, 1981) that is on the subject of discussion. The monograph of Seed and Idriss (1982) shows some illustrative plot of peak horizontal acceleration with closest distance from zone of energy release for some given earthquake surface wave magnitude M S. This monograph does not prescribe any empirical attenuation relation for estimating PGA in terms of earthquake magnitude and distance.
4. References on the subject Campbell (1981) proposed an attenuation relation, using data of western north America and some nearsource earthquakes from other parts of the world, that does not have anelastic attenuation term. It contains a geometric spreading term, exponentially
M. Shrikhande, S. Basu / Engineering Geology 80 (2005) 37–42
dependent on magnitude, that modulates PGA close to the fault. The influence of magnitude and source to site distance is non-separable due to the presence of this term. The proposed relation is for the geometric mean of two horizontal peak acceleration from a record depending on magnitude M and distance to fault rupture from the site. The magnitude M, defined as Richter magnitude, is equal to local magnitude M L for M b 6 and is equal to surface-wave magnitude M S for M z 6. This definition of Richter magnitude will be used henceforth. The magnitude range between 5.0 and 7.7 is considered in the analysis. The analysis is carried out on the logarithm transformed proposed equation with 27 earthquakes that produced 116 records of mean horizontal PGA. Since, the mean PGA is not uniformly sampled over all magnitude and rupture distances in the considered data set, a weighted nonlinear regression analysis using method of least square is performed. The rupture distances (0 to 56.6 km) are distributed in nine-bin to control the influence of well-recorded earthquakes such as San Fernando, 1971 and Imperial Valley, 1979. The weights are assigned according to distances. The weighting scheme is such that if an earthquake has a single recording in each distance classes, then it will carry maximum weight. The weightage scheme is subjective (educated guess) in nature and the chosen scheme influences the coefficients to be estimated. Joyner and Boore (1981) proposed an attenuation relation with all terms as stated above with the coefficient of geometric spreading term being constrained to be unity. The basic feature of this relationship is separability of influence of magnitude and site to source distance on the largest component of the two horizontal PGA. The PGA relationship is defined as a function of moment magnitude M W and closest distance of surface projection of the fault rupture from the site. The data set used for regression analysis is of western north America strong motion that contains a total of 183 largest of the two horizontal component of PGA from accelerograms recorded during 23 earthquakes. One of these earthquakes is an aftershock of the Imperial Valley, 1979 main event. The range of moment magnitude M W is between 5.0 and 7.7 and range of the closest distance of surface projection of fault rupture is [0.5–370] km. They use a two-stage method on the logarithmic
39
transformed multiplicative model for estimation of coefficients to account for (1) record to record variation in the first stage, and (2) earthquake to earthquake variation in the second stage. The accelerograms of well-recorded earthquake are essentially records of strong motion conditional on a given set of source parameters. These data form a cluster of PGA of a single magnitude thus are not independent observations. The Boore and Joyner (1982) article, referred to in the Bulletin, states assumptions that are required to be made for reconciliation of the proposed PGA relationship due to them (Joyner and Boore, 1981) with that of Campbell (1981) as different definitions are used for both dependent and independent variables in the two relationships. Bolt and Abrahamson (1982) proposed equations that prescribe the largest of two horizontal PGA as a function of the closest distance to zone of energy release for different ranges of moment magnitude M W. These relationships are in the form of Pearson family of probability curves. They use data set of Joyner and Boore (1981) and use nonlinear least square estimation on the proposed multiplicative model itself and hence project mean of the largest horizontal PGA. Idriss (1985) proposed relationships for randomly oriented horizontal component of PGA for various Richter magnitude M in the range from 4.5 to 8.5 at an interval of magnitude 0.5. The relation is dependent on closest distance from source for M z 6 and hypocentral distance in km otherwise. These relationships are essentially amplification factor scaled by geometric spreading term.
5. Development of attenuation models The attenuation relationship is a concise abstraction of complex reality. It is, therefore, a mathematical model necessarily limited in scope in comparison with the real system. The attenuation relationships are developed to obtain an interpolation formula as a summary of the data base of PGA, magnitude and distances obtained from various strong motion stations for different seismic events. Regression analysis is used to estimate parameters of the model. A model given by an equation representing dtrueT value of observation or response (dependent variable) is
40
M. Shrikhande, S. Basu / Engineering Geology 80 (2005) 37–42
chosen using some physical basis of seismic ground motion propagation. The following essential assumptions are implicit in every regression analysis: 1. The chosen form of equation is assumed to be approximately correct. 2. The data set is a representative sample of the entire range. 3. The data of responses are statistically uncorrelated. A less restrictive assumption is also made that (1) all observations have same variance (if the variance of the data of dependent variable changes in a known way, a weighted regression is performed), and (2) all the factors or predictors (independent variables) are known without error. The random error in ordinary least squares regression method is assumed to be additive and variance of random error is equal to that of observations. It is necessary to know probability distribution of random error to test hypothesis about coefficients, confidence interval and prediction interval of regression equation. Usually, random error is assumed to be normal and consequently it also becomes statistically independent. One must exercise great care while performing regression analysis for earthquake ground motion parameters as in this case none of the primary assumptions of regression are complied with. First, it is impossible to choose a correct form of equation for earthquake ground motion parameter because all physical causes that affect ground motion are not clearly known. Moreover, the values of the coefficients of the model are dependent on the choice of a data set. This bias in estimate is due to (1) non-uniform data in magnitude– distance space, and (2) the data set being dominated by a few well recorded earthquake. An attenuation relation for PGA is essentially a multiplicative model, which in general is composed of terms related to (1) exponential scaling of magnitude, (2) distance dependent anelastic attenuation, (3) geometric spreading, and (4) site effects with a multiplicative random error. This multiplicative model transforms into a linear (in the coefficients) regression model by taking logarithms. The random error becomes additive after transformation. Moreover, the fitting of multiplicative model directly violates constant variance errors (homoskedasticity) assumption of regression analysis.
6. Discussion The daverageT as recommended in the Bulletin is technically a misnomer. The U.S. Census Bureau defines an average as a number or value that is used to represent a dtypical valueT of a group of numbers (Jaffe and Spirer, 1987). A typical value is defined sometimes as a measure of dlocationT, i.e. where the parameter value belonged on an axis. Most of the times, it is defined as the central tendency of the data. Two of the common measures of central tendency of data are mean and median. The mean is very sensitive to outliers (high and low values of observations). The median estimates are resistant to outliers. Based on linear combination of ordered statistics (L-estimator), various location estimates for symmetric distributions have also been proposed, such as midmean, broadened median, trimean, etc. These estimators represent a compromise between mean and median. In the absence of any qualified statement, it is presumed that arithmetic mean is meant by daverageT in the Bulletin (ICOLD, 1989). The attenuation relations are models of physical phenomena. A model is made by adding idealised assumptions to the data and running them through an assumed equations. These assumptions are necessary to overcome limitations of data. Therefore, all models reveal some part of the real effects of data and others are artifacts of the assumptions made in the process. It can be seen that various authors have proposed models of attenuation relationships of PGA that are different in both independent and dependent variables and are based on some physical basis of strong ground acceleration. Different models are developed with different physical basis and different data set. The obtained equation of the model via regression analysis is different from an ordinary algebraic equation because of the presence of random error term. This equation represents a probabilistic additive model. The weighted mean model that is suggested to be obtained from the various proposed models based on different physical basis is alien to statistical theory and practice. The weightage scheme in regression analysis is primarily performed when the variance of dependent variables change in a known way. It appears that dweighted averageT value is borrowed from weighted regression analysis without dwelling much about its correctness and consequences. The
M. Shrikhande, S. Basu / Engineering Geology 80 (2005) 37–42
choice of weights needs a subjective judgment. Any subjective judgment is a measure of knowledge that is influenced by personal bias and is likely to be questioned. The various models referred to in the Bulletin are for horizontal component of the ground motion that project (1) median of geometric mean of two horizontal PGA (Campbell, 1981), (2) median of the largest PGA (Joyner and Boore, 1981), (3) mean of the largest PGA (Bolt and Abrahamson, 1982), and (4) median of the randomly oriented horizontal PGA (Idriss, 1985). It is obvious that weighted mean of dissimilar object cannot be performed. The data set used in developing the relationships as above are different. Hence, the suggested daverageT is to be performed on non-uniform data set and that is not recommended in statistics. Moreover, Bolt and Abrahamson (1982) have reported standard error of a single observation and not the standard error of regression equation. The covariances between random error of various recommended attenuation relationships of PGA are required to find the variance of the random error of suggested weighted mean model. Unfortunately, the above covariances are not available. Since, the standard error cannot be estimated any confidence intervals and predicted intervals of the weighted mean model cannot be obtained. Thus, it is an estimate without a standard error and hence, according to Sir Jeffreys (Jeffreys, 1967) is a practically meaningless estimator. The only rational way to salvage the situation–in our opinion–is to use a single prediction relationship that is derived from as large and well-represented data set as possible with lower standard error. Glitches in theory cannot be wished away by daverageT. It appears that daverageT is discovered by the advocates of seismic probability as a convenient tool to iron out all problems that are created by uncertainty of earthquake phenomena.
7. Conclusions The ground motion attenuation relationships represent mathematical models for a limited aspect of the phenomenon of ground motion propagation. Each of these models has been derived on the basis of different sets of assumptions and different sets of
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data. Further, each of these models has different standard error. Estimates from these different attenuation relationships cannot be averaged (weighted, or otherwise) as recommended in the Bulletin due to the presence of standard errors and due to differences in the underlying assumptions for each of these relationships. Moreover, while the independent estimates from each attenuation relationships can be associated with certain confidence limits, no such measure can be ascribed to the average of several attenuation relationships.
References Ambraseys, N., 1973a. Discussion on bCharacterization of response spectra by parameter governing the gross nature of earthquake source mechanismQ by M.D. Trifunac, Session 2C:PD79, pp. 701–704. Proceedings of the Fifth World Conference on Earthquake Engineering, Rome, vol. 1, p. 705. Ambraseys, N., 1973b. Discussion on bEarthquake induced cracking of dry Canyon damQ by Kenneth L. Lee and Henry G. Walters, Session 4D:PD192, pp. 1544–1547. Proceedings of the Fifth World Conference on Earthquake Engineering, Rome, vol. 2, p. 1548. Ambraseys, N., 1973c. Discussion on bSoil–structure interaction effect in the Caracas earth-quake of 1967Q by H. Seed and J. Alonso Session 6A:PD263, pp. 2108–2111. In Proceedings of the Fifth World Conference on Earthquake Engineering, Rome, vol. 2, p. 2112. Ambraseys, N., 1975. Trends in engineering seismology in Europe. Fifth European Conference on Earthquake Engineering, vol. 3, pp. 39 – 52. Bolt, B., Abrahamson, N., 1982. New attenuation relations for peak and expected accelerations of strong ground motion. Bulletin of the Seismological Society of America 72 (6), 2307 – 2322. Boore, D., Joyner, W., 1982. The empirical prediction of ground motion. Bulletin of the Seismological Society of America 72 (6), S43 – S60. Campbell, K., 1981. Near-source attenuation of peak horizontal acceleration. Bulletin of the Seismological Society of America 71 (6), 2039 – 2070. Douglas, J., 2003. Earthquake ground motion estimation using strong-motion records: a review of equations for the estimation of peak ground acceleration and response spectral ordinates. Earth-Sci. Rev. 61, 43 – 104. ICOLD, 1989. Selecting seismic parameters for large dams. Bulletin, vol. 72. International Commission on Large Dams. Idriss, I., 1985. Evaluating seismic risk in engineering practice. Proceeding Eleventh International Conference on Soil Mechanics and Foundation Engineering, vol. 1. A.A. Balkema, San Fransisco, CA, pp. 255 – 320. Jaffe, A., Spirer, H., 1987. Misused Statistics: Straight Talk for Twisted Numbers. Marcel Dekker, Inc., New York.
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Jeffreys, H., 1967. Seismology, statistical methods. In: Runcorn, K. (Ed.), International Dictionary of Geophysics. Pergamon Press, London, pp. 1398 – 1401. Joyner, W., Boore, D., 1981. Peak horizontal acceleration and velocity from strong motion records including records from the 1979 Imperial Valley, California, earthquake. Bulletin of the Seismological Society of America 71 (6), 2011 – 2038.
Seed, H., Idriss, I., 1982. Ground motions and soil liquefaction during earthquakes. Technical report, Earthquake Engineering Research Institute, Berkeley, California, USA. Trifunac, M., Brady, A., 1975. On the correlation of seismic intensity with peaks of recorded strong motion. Bulletin of the Seismological Society of America 65, 139 – 162.