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Bulletin of the Seismological Society of America, Vol. 79, No. 1, pp February 1989 SHORT NOTES A NOTE ON THE SELECTION OF MINIMUM MAGNITUDE FOR USE IN SEISMIC HAZARD ANALYSIS BY BERNICE BENDER AND KENNETH W. CAMPBELL Most seismic hazard analyses combine probabilistic ground motions from all earthquakes within some magnitude range, mmin ~ m _-- m... to estimate levels of peak horizontal accleration or peak horizontal velocity that have a specified probability of being exceeded at a site during various time periods (e.g., Algermissen et al., 1982). However, engineers have observed that high peak accelerations from smaller magnitude earthquakes may not cause significant damage to well-engineered structures (e.g., Kennedy, 1986), and therefore, it is arguable whether ground motions from small-magnitude earthquakes should be included in seismic hazard calculations (e.g., EPRI, 1987). The choice of minimum magnitude is particularly important in estimates of hazard at sites in Eastern North America (ENA) because peak accelerations associated with ground motions in this region are typically high in amplitude, even when the motions result from small-magnitude earthquakes. However, the ground motions of ENA are extremely rich in high frequencies because of a relatively high [max in this region (Atkinson, 1984; Boore and Atkinson, 1987; Toro and McGuire, 1987), and may be less hazardous than ground motions of the same amplitude with more low-frequency energy. There is no consensus among seismic hazard analysts as to an appropriate value of minimum magnitude to use in hazard calculations. Reflecting different opinions regarding the choice of a minimum magnitude to use in seismic hazard analyses for sites in ENA, Bernreuter et al. (1985) adopted the value mb = 3.75, for example, whereas EPRI (1986) selected the value mb = 5.0. A comparison of results obtained by the two groups named above for the same sites suggests that differences in the minimum magnitude may significantly affect the ground motion estimates at some probability levels (Bernreuter et al., 1987). In part because a single scalar, such as peak horizontal acceleration, cannot adequately represent the distribution of ground-motion amplitudes with respect to frequency, some recent seismic hazard studies have estimated response spectra in addition to peak horizontal acceleration (e.g., Bernreuter et al., 1985; EPRI, 1986). However, as we shall illustrate, the choice of minimum magnitude is still important, inasmuch as the minimum magnitude used in the calculations can affect both the shape and level of probabilistic response spectra. THE EFFECTS OF MINIMUM MAGNITUDE Table 1 gives peak horizontal acceleration (amax) levels, and Figure 1 shows equalhazard pseudo-velocity response spectra (PSV) calculated as having annual exceedance rates p , 0.001, and 0.01, respectively. Both are calculated for mlg(mln) = 4.0, 4.5, 5.0, and ml~(m~xi = 6.0 and 7.0 at a site at the center of a 200-km by 200- km square source zone. Here a is defined to be the standard deviation of the natural logarithm of the ground motion (a... or PSV) resulting from earthquakes of a given magnitude and distance. Magnitudes are defined in terms of the mlg scale; mlg(min) and ml~(m~x) represent the minimum and maximum magnitudes used in the calcu- 199 200 SHORT NOTES TABLE 1 LEVELS OF PEAK HORIZONTAL ACCELERATION mz~(~): ~: m~cm~n) am~ (% g) p = ! p = p = Levels of peak horizontal acceleration (amax) for exceedance rates of p = , and 0.01 at a site in the center of a source 200 km by 200 km, for various combinations of minimum magnitude mlg(mln), maximum magnitude mlg(ma~,) and a, the standard deviation of the natural logarithm of the attenuation relationship. The attenuation relationship is:that of Boore and Atkinson (1987) for peak horizontal acceleration. The same rate and b-value were used as in Figure 1. lations. The attenuation relationships of Boore and Atkinson (1987) were used to estimate peak horizontal acceleration (amax) and 5 per cent damped pseudo-velocity response (PSV) for hard-rock sites in ENA. Because these relationships are given in terms of moment magnitude M and the seismicity parameters are given in terms of ml~; mlg was converted to M using the moment-magnitude relationship of Boore and Atkinson (1987). Hypocentral distances were calculated assuming a depth of 8 km. Results shown are for b = 1.0 and either a = 0.3 (A) or ~ = 0.6 (B to D). (Boore and Atkinson suggest that their relationship be used with 0.37 _- ~ _ ) Although the source zone used in this example is hypothetical, the earthquake occurrence rate per unit area is comparable to rates used by Algermissen et al. (1982) for moderately active source zones in the Eastern United States. Figure 1 demonstrates that the choice of minimum magnitude appreciably affects the shape of equal-hazard spectra at lower ground-motion levels corresponding to p = 0.01, where, for example, the value of PSV at a 0.1-sec period calculated for mlg(m~,) = 4.0 is approximately a factor 3 higher, and the value at a 1.0-sec period is approximately 40 per cent higher, than the values calculated at the same periods for mlg(~) = 5.0. In contrast, minimum magnitude has virtually no effect on the equal hazard spectra for higher ground-motion values corresponding to p = The differences in the shapes of the equal-hazard spectra for p = 0.01 reflect both the importance of minimum magnitude and the strong influence of magnitude scaling. For example, the greater dependence of PSV on magnitude at long periods tends to reduce the impact of minimum magnitude on these estimates. The lack of any influence of mlg(~) on the equal-hazard spectra for p = reflects the importance of the larger magnitudes in controlling the higher values of PSV. EFFECTS OF MAXIMUM MAGNITUDE AND (~ The effect of maximum magnitude on the shapes of equal-hazard spectra is exactly opposite to the effect of minimum magnitude. Maximum magnitude has virtually no effect on the curves corresponding to p = 0.01, where PSV is controlled SHORT NOTES (A) mlg(max)= (B) mlg(max)=6.0 o = 0.3 o =0.6 1 o 1 1 o I p=o.0ool p = o.o001 EIO p..~o 1 X mlg(rnin ) rnlg(min) '-. : 4.0 ~ I I :' ~H :,::::: : : ::::::: 10-2 i i :;;:::: ; ;:;::: : : :::::: T (sec) T (sec) 10 2! (C) mlg (max)= 7 0 10 2i (D) m Lg(max)=6'0 o'= 0.6 mlg(min ) =4.0 p= ~ 10 1 p=0.o0o~.,~ p=0.001 '~ ~ ~-0- I ~ ~ E 0 oe lo 0 p=o.ol ,.. 1(~ mlg(min) = = I I ::::'::.1 : :::'::: 1 -' : :::'-:: : : ::' .1 : ' :::::: 1 : ::::::: 10 T (sec) T (sec) FIG. 1. Equal-hazard response spectra for five per cent damped pseudo-velocity (PSV) plotted versus period (T), calculated at a site at the center of a hypothetical 200-kin square source, using the relationships of Boore and Atkinson (1987). The curves (A-C) represent pseudo-velocities having annual exceedance rates ofp = 0.01, 0.001, and , calculated for minimum magnitudes, mlg(min) = 4.0, 4.5, and 5.0. Curves (A) and (B) show the sensitivity of the calculations to a when the same maximum magnitude is assumed; curves (B) and (C) show the sensitivity to maximum magnitude when the same a is used, where a is the standard deviation of the natural logarithm of the ground motion. Curve (D) shows the sensitivity of the calculations to ~ when m~g(mln) = 4.0 and mlg(max) = 6.0. Earthquake occurrence rates are arbitrarily set to 0.01 events per yr in the range 5.0 ~mz~ = 5.5, regardless of the value for minimum magnitude, and are consistent with a b-value of 1.0. by small-magnitude earthquakes, whereas it has a noticeable effect on the higher levels of PSV calculated for p = , where PSV is controlled by the largermagnitude earthquakes. In the latter case, the effect is greatest at the longer periods, where the greater dependence of PSV on magnitude increases the influence of the larger magnitudes. 202 SHORT NOTES At some level of exceedance, calculations become more sensitive to assumptions regarding a than to assumptions regarding minimum magnitude. For example, the peak horizontal acceleration levels calculated for p = for a = 0.6 are about 30 to 50 per cent higher than the corresponding levels calculated for a = 0.3 (Table 1); the change in level as a function of a increases as minimum magnitude increases. On the other hand, for a given a, the variation in a~ax due to differences in minimum magnitude (mlg(min) = 4.0 versus 5.0) is only 7 to 20 percent). Similar results are found for equal-hazard spectra (Figure 1). For p = , (and a fixed mlg(~ax)), PSV is found to be 30 to 60 per cent higher for a = 0.6 than for a For a given a, changing the minimum magnitude makes a visible difference only for shortperiod ground motions and for a = 0.6. For lower ground-motion levels, corresponding to p , the effect of a on the equal-hazard PSV spectra depends on mlg(~in) and the period of the ground motion. In this case, changing ~ from 0.3 to 0.6 increases the short-period ground motions by ~20 per cent when mlg(~in) = 4.0 but negligibly when mlg(min) = 5.0. Changing has little effect on the longer period ground motions, for any of the values of mlg(m~). The spectra shown in Figure i can be compared among themselves, but should not be interpreted in terms of absolute amplitudes. For example, if the earthquake occurrence rate were doubled, the PSV levels computed as having an annual rate of exceedance of p = 0.01 would have instead a rate of p = In addition, for the current study a single source of uniform seismicity was assumed; changing the source zonation could affect both the amplitudes and the shape of the spectra. Finally, the spectra shown are for hard-rock sites. Boore and Atkinson (1987) note, There may be significant frequency-dependent amplification by local site conditions that should be addressed on a site-specific basis. Soil amplifications can be very significant, amplifying motions by as much as a factor of 10 or more. In addition, some areas characterized by soft or weathered near-surface rocks may show amplifications of as much as a factor of 2 with respect to hard-rock sites. ACCOUNTING FOR UNCERTAINTY IN DAMAGEABILITY OF SMALLER EARTHQUAKES For the hypothetical site in Table 1, estimated levels of amax increase from 6.3 to 10.9 per cent gravity and from 22.9 to 27.4 per cent gravity when mlg(mln) decreases from 5.0 to 4.0 for the case mlg(max) = 6.0 and ~ = 6.0. If a... is used in decisions relating to the seismic safety of a structure, then only potentially damaging earthquakes should be included in the hazard analysis. We believe that it is unrealistic to assume that damaging earthquakes terminate abruptly at some fixed magnitude retain. We suggest that it may be more realistic to assume a tapered, rather than an abrupt, cutoff of damaging earthquakes at retain, such that the fraction of earthquakes included decreases to zero as magnitude decreases from m~in to some magnitude too. A taper can be constructed to give the same results as those obtained by integrating over a probabilistic minimum magnitude. However, a probabilistic minimum magnitude differs in concept from a tapered minimum magnitude. In the first case, we postulate the existence of a true minimum damaging magnitude, and model our uncertainty regarding what value that magnitude should assume. In the second case, we assert our belief that some, but not all, of the earthquakes at smaller magnitudes could potentially be damaging, and that damaging earthquakes SHORT NOTES 203 do not terminate abruptly at some magnitude Mmin. We believe that the same arguments hold true for PSV. Although PSV is probably a better indicator of damage than a... its relative insensitivity to duration of shaking makes it a poor indicator of the type of damage arising from inelastic deformation. Smaller earthquakes are of shorter duration and not likely to cause damage resulting from inelastic deformation, even though they may contribute significantly to probabilistic PSV spectra. Therefore, a tapered minimum magnitude could be used to select a range of earthquake magnitudes over which damage resulting from inelastic response is believed to become significant enough to be included in the analysis. CONCLUSIONS The choice of minimum magnitude can have a significant effect on calculated probabilistic peak accelerations and equal-hazard response spectra at lower ground motion values. The degree to which minimum magnitude affects the calculated seismic hazard depends on many factors, including the level of seismicity, the type of zonation, the maximum magnitude, the variability in ground motion, the period (in the case of response spectra), and the attenuation relationship. Because the shape of equal-hazard spectra varies with so many different factors, it is not possible to use a single spectra shape to produce probabilistic estimates of response spectra, even if this shape is anchored to a probabilistic estimate of ground motion, such as peak acceleration; in other words, it is not possible to develop equal-hazard response spectra using a normalized spectral shape as is common practice in deterministic analyses. We suggest that in calculating peak horizontal acceleration, peak velocity or response spectra for seismic hazard analyses, a tapered, rather than abrupt, minimum magnitude cutoff be considered. ACKNOWLEDGMENTS We wish to thank Bob Youngs for motivating this note and Dave Perkins and Paul Thenhaus for helpful comments. REFERENCES Algermissen, S. T., D. M. Perkins, P. C. Thenhaus, S. L. Hanson, and B. L. Bender (1982). Probabilistic estimates of maximum acceleration and velocity in rock in the contiguous United States, U.S. Geol. Surv., Open-File Rept Atkinson, G. M. (1984). Attenuation of strong ground motions in Canada from a random vibrations approach. Bull. Seism. Soc. Am. 74, Bernreuter, D. L., J. B. Savy, R. W. Mensing, J. C. Chen, and B. C. Davis (1985). Seismic hazard characterization of the Eastern United States, Vol. 1: Methodology and results for ten sites, Lawrence Livermore National Laboratory Rept UCID-20421, Livermore, Calif. Bernreuter, D. L., J. B. Savy, and R. W. Mensing (1987). Seismic hazard characterization of the Eastern United States: comparative evaluation of the LLNL and EPRI studies. NUREG/CR-4885, UCID Boore, D. M. and G. M. Atkinson {1987). Stochastic prediction of ground motion and spectral response parameters at hard-rock sites in Eastern North America, Bull. Seism. Soc. Am. 77, EPRI (1987). Proceedings of Workshop on Engineering Characterization of Small Magnitude Earthquakes, Palo Alto, Calif., Jan , 1987, Electric Power Research Institute, Palo Alto, Calif. EPRI (1986). Seismic Hazard Methodology for the Central and Eastern United States, Vol. 1: Methodology. Electric Power Research Institute, Rept. EPRI NP-4726, Palo Alto, Calif. 204 SHORT NOTES Kennedy, R. (1987). Engineering characterization of small-magnitude earthquakes, in Proceedings of Workshop on Engineering Characterization of Small Magnitude Earthquakes, Palo Alto, Calif., Jan 28, 1987, Electric Power Research Institute, Palo Alto, Calif., Sect, 2, Toro, G. R. and R. K. McGuire (1987). An investigation into earthquake ground motion characteristics in Eastern North America, Bull. Seism. Sac. Am. 77, UNITED STATES GEOLOGICAL SURVEY BOX M.S. 966 DENVER FEDERAL CENTER DENVER, COLORADO Manuscript received 20 August 1987
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