Probabilistic Seismic Hazard Analysis and Design Earthquakes: Closing the Loop

Transcription

1 Bulletin of the Seismological Society of America, Vol. 85, No. 5, pp , October 1995 Probabilistic Seismic Hazard Analysis and Design Earthquakes: Closing the Loop by Robin K. McGuire Abstract Probabilistic seismic hazard analysis (PSHA) is conducted because there is a perceived earthquake threat: active seismic sources in the region may produce a moderate-to-large earthquake. The analysis considers a multitude of earthquake occurrences and ground motions, and produces an integrated description of seismic hazard representing all events. For design, analysis, retrofit, or other seismic risk decisions a single "design earthquake" is often desired wherein the earthquake threat is characterized by a single magnitude, distance, and perhaps other parameters. This allows additional characteristics of the ground shaking to be modeled, such as duration, nonstationarity of motion, and critical pulses. This study describes a method wherein a design earthquake can be obtained that accurately represents the uniform hazard spectrum from a PSHA. There are two key steps in the derivation. First, the contribution to hazard by magnitude M, distance R, and e must be maintained separately for each attenuation equation used in the analysis. Here, e is the number of standard deviations that the target ground motion is above or below the median predicted motion for that equation. Second, the hazard for two natural frequencies (herein taken to be 10 and 1 Hz) must be examined by seismic source to see if one source dominates the hazard at both frequencies. This allows us to determine whether it is reasonable to represent the hazard with a single design earthquake, and if so to select the most-likely combination of M, R, and e (herein called the "beta earthquake") to accurately replicate the uniform hazard spectrum. This closes the loop between the original perception of the earthquake threat, the consideration of all possible seismic events that might contribute to that threat, and the representation of the threat with a single (or few) set of parameters for design or analysis. Introduction The primary advantage of probabilistic seismic hazard analysis (PSHA) over alternative representations of the earthquake threat is that PSHA integrates over all possible earthquake occurrences and ground motions to calculate a combined probability of exceedence that incorporates the relative frequencies of occurrence of different earthquakes and ground-motion characteristics. Modern PSHA also considers multiple hypotheses on input assumptions and thereby reflects the relative credibilities of competing scientific hypotheses. These features of PSHA allow the ground-motion hazard to be expressed at multiple sites consistently in terms of earthquake sizes, frequencies of occurrence, attenuation, and associated ground motion. As a result, consistent decisions can be made to choose seismic design or retrofit levels, to make insurance and demolition decisions, and to optimize resources to reduce earthquake risk vis-a-vis other causes of loss. A disadvantage of PSHA is that the concept of a "design earthquake" is lost; i.e., there is no single event (specified, in simplest terms, by a magnitude and distance) that represents the earthquake threat at, for example, the 10,000-yr ground-motion level (which we call the "target ground motion"). This disadvantage results directly from the integrative nature of PSHA, and it means that other characteristics of the ground motion (e.g., the duration or nonstationarity) must be estimated in an ad hoc fashion if these characteristics are important for analysis or design. This disadvantage was recognized by the Aki committee (NRC, 1988), which recommended that a "recursive" PSHA be performed to determine the dominant earthquake at any particular hazard level. The earthquake and attenuation models can then be revised to more accurately reflect the characteristics of this dominant earthquake, and the hazard analysis can be repeated using these more accurate models. The dominant earthquake was 1275

2 1276 R.K. McGuire recommended to be the mean magnitude/f/and distance/~ of the seismic event that caused a ground-motion exceedence at the chosen return period. The concept of M and k was introduced by McGuire and Shedlock (1981) in the context of evaluating uncertainties in hazard. Independently, Kameda and his co-workers in Japan have examined the concept of M and/~ for single-hypothesis PSHA, i.e, without uncertainties in inputs (Ishikawa and Kameda, 1988, 1991, 1993). They have recommended that 114 and/t be determined separately for each natural frequency of interest and for each seismic zone that contributes to hazard. These magnitudes and distances can then be used with the relevant attenuation equation to calculate ground motion, but this motion must be scaled to give the same amplitude as the target ground motion. Also, several studies (e.g., Stepp et ai., 1993; Chapman, 1995) have de-aggregated the seismic hazard into its contribution by magnitude and distance (and, for the former reference, e) to show which events dominate the hazard at the site. These studies have not addressed the issue of uncertainty in ground-motion equations, which is a feature of many current seismic hazard analyses, and have not considered how to derive a single dominant earthquake for a range of response spectrum periods. There are distinct advantages to developing one or a few "design earthquakes" that can be used for detailed analysis and decision making. One obtains an intuitive feel for the earthquake that is dominating the hazard at the chosen probability level. This earthquake can be associated with a known fault or seismic zone and can be ascribed familiar characteristics such as a magnitude, stress drop, azimuth, depth, and distance. Also, more detailed analyses can be performed for the design earthquake, for example by generating durations and time histories of motion for nonlinear structural analysis. What has been lacking in the definition of design earthquakes to date is an overall format within which to derive a design earthquake, by which we mean a magnitude, distance, and perhaps other parameters that are consistent with the target ground motion. The problem with the use of)f/and/~ or with a "modal" earthquake (the most likely values of M and R) is that the resulting ground motion must be scaled in a heuristic way to equal the target ground motion. Also, previous recommendations have not addressed multiple hypotheses explicitly and have only used one structural period. The latter characteristic means that different/17/'s and/~'s (and different ground motions) might be calculated for 10 and I Hz; in some cases this would not be practical or realistic for the analysis of a building with two natural frequencies in the range 10 to 1 Hz. Developing such a format is the purpose of this article, so that we might understand design earthquakes over a range of natural frequencies and for multiple input hypotheses with uncertainties. A hypothetical but realistic example is used, in order to investigate what drives seismic hazard when high frequencies are controlled by one source of earthquakes but low frequencies are controlled by another. Developing workable recommendations for this case will ensure that the conclusions are applicable over a wide range of contributions from multiple faults and zones. Site Application We demonstrate the contribution to seismic hazard by examining the PSHA results at multiple periods T. (We use "period" instead of "frequency" to avoid confusion in terminology with "annual frequency.") The site chosen is Watkins, Colorado, latitude N and longitude W; a map showing the site and historical seismicity in the region is presented in Figure 1. A PSHA requires a definition of seismic zones. For this example the seismic zones developed by the U.S. Geological Survey (Algermissen et al., 1982) for the national seismic hazard map are adopted. These zones are also illustrated in Figure 1. In addition, a hypothetical fanlt is added: fault i lies west of the site about 20 km and is used to represent the high seismic activity indicated in Figure 1. Much of this seismicity was induced by deep fluid injection at the Rocky Mountain Arsenal during the 1960s, and the alignment of recorded epicenters at the time suggested that a fault might lie in the area. A detailed, nonhypothetical application of PSHA likely would disregard fault 1 or would assign it a much lower activity, as most of the associated earthquakes were not naturally occurring. The inclusion of this fault here allows us to demonstrate the applicability of the recommended procedures when a nearby fault dominates the short periods but distant diffuse zones of seismicity dominate the long periods (as will be seen). Seismicity parameters for the source areas and for fault 1 are shown in Table 1, for the two examples considered here. For the source areas the rate of seismic activity is derived from the results presented by Algermissen et al. (1982); for the fault the activity rate is purely hypothetical. All b values are assumed to be 0.9. The maximum magnitudes listed in Table 1 are chosen to accentuate the contribution of small magnitudes at close distances (for the fault) and large magnitudes at large distances (for the seismic zones). For assumptions on ground-motion attenuation, two empirically derived equations were used. These were taken from the work by Boore et al. (1994) and Campbell (1993). The form of these equations is substantially different; the Boore et al. form (herein called "attenuation 1") is lny= a + b(m- 6) + c(m- 6) 2 + din (R1) + ~R,, (1) where Y is a ground-motion measure, a through d are empirically derived coefficients, M is moment magnitude, and R1 is calculated using the distance to the surface projection of the fault surface Rs for faults: R1 = (R~ + h2) m. In the

3 Probabilistic Seismic Hazard Analysis and Design Earthquakes: Closing the Loop / oi 41" b t ee e o Magnitude 40" o e ~ / > e 45 o/ e " ~ _ USeS SouSe e,,~'''~jl Fault 1 e ~'~~o o eoo, ~ - <3.o :,,,,,"'~% 8, e SITE e O ~,~eeo I e e ol 39". g ~ ' ~ " ~ USGS Source 7,1 o "X k k 38" o ~_ ",Z\ ~\, o o.. o\ \ o o * / -109" -108" -107" " -104" -103" -102" Figure 1. Historical seismicity in Colorado, USGS sources 71 and 45, and hypothetical fault 1. Table 1 Seismicity Parameters Used in Examples Seismic Magnitude Activity Maximum Example Source Distribution Rate* b Value Magnitude 1 fault 1 exponential 0.00l USGS source 71 exponential tjsgs source 45 exponential fault 1 exponential LrSGS source 71 exponential USGS source 45 exponential * Annual rate of earthquakes above M = 5. latter expression, h is a depth specified by the authors. The Campbell form (herein called "attenuation 2") is In Y = a +bm + c tanh (d(m - 4.7)) - lnr- er2 + (f- glnrz)s, (2) where a through g are empirical coefficients, R 2 is distance to the rupture surface, r = f(r2,m), and S indicates soil type. These equations are used herein to predict spectral acceleration (SA) at T = 0.04, 0.1, 0.2, 0.4, 1, and 2 sec, for 5% of critical damping. The above two equations were developed by the respective authors based on empirical data from California, and were adopted for use in this example assuming rock conditions. In a nonhypothetical calculation of seismic haz- ard, the equations might be changed to account for regional attenuation, rock properties, and local soil response. The equations are directly applicable to earthquakes occurring on faults in this example. For earthquakes in source areas, we modify the application of these equations by subtracting a distance equal to one-quarter the earthquake's estimated rupture length, as an approximate correction. The randomness term for each period T is used as reported by the authors. Seismic Hazard Calculations The standard formulation of probabilistic seismic hazard is to calculate a frequency of exceedence 7 of a groundmotion amplitude y as 7(y) = ~ v~ ( f fv(m)fr(r)p[y > ylm, r]dm dr, (3) where v~ is the activity rate for source i. The probability in the integrand of equation (3) can be expressed explicitly as a function of the ground-motion randomness e, giving 7(y)= ~ v~ f ~.( f~(m)fr(r)f~(e)p[y > ylm, r, e]dmdrde, (4)

4 1278 R.K. McGuire where e is defined as the number of logarithmic standard deviations by which the logarithmic ground motion deviates from the median. In this formulation, the probability in the integrand is simply the Heaviside step function: P[Y > ylm, r, e] = H[ln Y(m, r, e) - In y], (5) which is zero if In g(m, r, e) from equation (1) or (2) is less than In y, and 1 otherwise. To derive contributions to 7(y) by M, R, and e, we use instead P[Y > ylm, r, el = J[ln Y(m, r, e) - In y], (6) where 6 is the Dirac delta function, which gives a probability of 1 at In Y(m, r, e) = In y and zero otherwise. (J is used because we wish to obtain M-R-e sets that equal the target ground motion, not that exceed it.) The formulations in equations (4) and (6) are used below to derive the contributions to ~(y) by m, r, and e. For example 1, Figures 2 and 3 illustrate the total hazard curves for T = 0.1 and 1 sec, respectively, and the contribution by fault and source areas. The site is in an area of low seismicity; for T = 0,1 sec fault 1 contributes most to the seismic hazard because of its proximity, and for T = 1 sec USGS source 45 contributes most to the seismic hazard because of its high rate of activity and high maximum magnitude. Thus the response spectrum is dominated by different earthquakes at different periods, which makes the goal of deriving a single earthquake that will reasonably represent the uniform hazard spectrum particularly challenging. For this example, we assume that the design motion should correspond to an annual frequency of exceedence of To provide further insight into the contribution to hazard, we deaggregate the hazard results by magnitude M, distance R, and ground-motion deviation e. This is achieved by accumulating (by magnitude, distance, and e) the annual frequencies of exceedence of the target ground-motion amplitude for each period T separately. Dividing these annual frequencies by the total hazard (the total annual frequency) gives the probability that, given an exceedence of that amplitude, it has been caused by a certain combination of M, R, and e. Figure 4 (top) illustrates the joint distribution of M, R, and e for combinations causing the exceedence of the 10,000-yr ground motion for T sec. The major contribution comes from small magnitudes at close distance (fault 1) with 0 < e < 1. The distribution for T = 1 sec, Figure 4 (bottom), shows the contribution from large M, large R, and epsilon values generally between 1 and 2 (from source 45). The four-dimensional plot necessitates wide intervals of M, R, and e, which hides some of the distributions' features. To provide more detail, the joint distribution is represented as marginal distributions on M, R, and e in Figures 5 (for r = 0.1 sec) and 6 (for T = 1 sec). Mathematically, these distributions are defined as e~ o o o g,,q O O O -d Contribution to T=O.1 s hazard by source ~ All Sources \\... Fault 1 \ \... USGS Source 71 '\ " " "k \ \'. "\ \ ", \ ", "\ \ ",.\\ ", \ "x,~ \ \ \ \ i0 0.i5 0.i0 0.25" 0.30 Spectral acceleration (g) Figure 2. Contribution to hazard by source for T = 0.1 sec, example 1. tm Q L' Contribution to T=I s hazard by source \ "\\, \ "?\ - - All Sources Fault 1 USGS Source 71 USGS Source 45 " q-", ','-,\ \ "x Spectral acceleration (g) Figure 3. Contribution to hazard by source for T = 1 sec, example 1. N

5 Probabilistic Seismic Hazard Analysis and Design Earthquakes: Closing the Loop 1279 T=0.1s l e : 2+ me: lto e:0to I e: -1 to m e: -2 to J T=ls o.29 Figure 4. = 1 sec Contribution to 10-4 hazard by M, R, and e for T = 0.1 sec 1. (bottom), example (top) and T

6 1280 R.K. McGuire [.. 0.o8 ~ HAZA!~ CON~. BUT ION BY.MA,.GN!,.T~. E I 0.4 ~_ 0.04 T=0.1 s, All sources 0 ' ~... "~ ' ' 6.fi 7.0 ' MAGNITUDE HAZARD CONTRIBUTION BY DISTANCE ,..,... '..,... T=0.1 s, All sources HAZARD CONTRIBUTION BY MAGNITUDE 2.0/...., ' " " ,...., ' I 0"~.0' 5'5... b'0... 6'5... 7'0... 7'5... S.0 MAGNITUDE b* HAZARD CONTRIBUTION BY DISTANCE ,......,...,.....,.. T=I s, All sources ~ 0.08 ~0.06 ~0,04 ~0.02 ~0.02 ~0.00 DISTANCE (kin) ~0.o~q SO DISTANCE (kin) 1.0 ~ HAZARD CONTRIBUTION BY EPSILON " r=0.i' [~ t.0 ~ 0.8.~ 0.6 HAZARD CONTRIBUTION BY EPSILON.... L i....,.... [.. ',.,.... i.... T=I s, All sources ~0.4 ~ 0.2 ~0.0 ~..~ ~ , l EPSILON 0.4 ~0.2 m ~ 0.o T :i.5 -i.0'' :d.$' ' g.o... b'.~ " ' o ' 13 EPSILON / Figure 5. Marginal contributions to hazard by M, R, and e for 10 4 hazard, T = 0.1 sec, example 1. f' m) = f f ld de = lo v, f dmde f;(e) = 104~viffldmdr, where the integrand I in all three equations is (7) (8) (9) I = fm(m)fr(r)f+(e)j[ln Y(m, r, e) - In y], (10) and the constant 104 is used to normalize the total annual frequency of 10-4 to obtain a proper probability density function. The magnitude distribution in Figure 5 (top) was drawn from hazard contributions calculated at 0.1 magnitude intervals. There are contributions from all magnitudes from 5.0 to 7.5, reflecting earthquakes on fault 1 and USGS sources 71 and 45, but the largest contribution comes from fault 1, confirming Figure 2. Figure 6. Marginal contributions to hazard by M, R, and e for 10-4 hazar d, T = 1 sec, example 1. The distribution of distance (which is epicentral distance for the areal sources and rupture distance for the fault) is illustrated in Figure 5 (middle). These results were calculated at 5-kin intervals. The large contribution from 20 to 27.5 km reflects the location of fault 1. Figure 5 (bottom) illustrates the marginal distribution of values. These results were obtained using bins that are 0.2~ units wide; the values at + 2 include all contributions below -2 and above + 2, respectively. The distribution of e has the largest contributions between 0.5 and 1.0. This reflects the contribution of fault 1, with small magnitudes (less than 5.2) requiring higher than median ground motions (positive e values) to equal the 10,000-yr target ground motion. Note that e is defined as the number of standard deviations of the ground-motion distribution such that the 10,000-yr ground motion is equaled, even though the probability of exceedence of that ground motion is calculated by the PSHA. e is defined in this way because we wish to replicate the 10,000- yr amplitude with the design earthquake. Figure 6 is similar to Figure 5 except that it shows the contribution by M, R, and e for 1 Hz. Here the contribution

7 Probabilistic Seismic Hazard Analysis and Design Earthquakes: Closing the Loop 1281 to hazard of the larger magnitude, more distant earthquakes is evident. Table 2 presents values of mean magnitude/f/, mean distance/~, and mean deviation g for T = 0.1 and 1 sec calculated from the joint distribution shown in Figure 4 (these are also mean values from the marginal distributions of Figs. 5 and 6). This table also presents the most likely (modal) values of magnitude, distance, and e from the joint distribution, which are labeled h)/,/~, and g. These modal values from the joint distribution do not necessarily correspond to the mode of the marginal distributions. Also shown are predicted ground motions using the mean and modal values of M, R, and e. These predicted ground motions weight each attenuation equation equally, as the hazard analysis does. The mean values do not guarantee a combination of ~Q,/~, and g that is consistent with the 10,000-yr motion (there is no theoretical reason why they should). The modal values do not replicate the target ground motions, because different attenuation equations may require different values of e to equal the target ground motion and this is not taken into account by a single g. Determination of Design Earthquakes To derive a single or several design earthquakes that can replicate the uniform hazard spectrum over all natural periods, we examine a "composite seismic hazard analysis" that combines the contributions by M, R, and e at the two periods. The key is to keep the hazard results deaggregated by M, R, and e, for each attenuation equation separately, and for each natural period. Then we can guarantee mathematically (using equation 6) that the specific combinations of M, R, and e will reproduce the target ground motion (at least within the precision associated with a certain M, R, and e bin), and we can combine the distributions at multiple periods to obtain a joint distribution that will reproduce the target ground motion. To conduct a composite seismic hazard analysis, we first form the distribution of contributions to hazard by M, R, and e as previously described, for T = 0.1 and 1 sec at the target annual probability. Separate distributions are formed for each attenuation equation used in the analysis. The sum of contributions in each M, R, and e interval will equal the target annual probability times the weight assigned to that attenuation equation. Next, for each attenuation equation we form a composite distribution from the two natural periods distributions by (a) considering only intervals in which there is a nonzero contribution at both periods, (b) adding the annual frequency in the corresponding intervals for the two periods, and (c) assigning zero to the remaining intervals (for which one or both periods have a zero annual frequency). A simple explanation of this concept is worthwhile. We have two distributions (for T = 0.1 and 1 sec) of contributions to hazard by M, R, and e. If they overlap closely (in the limit, if they are identical), this means that similar earthquakes drive the hazard for T = 0.1 and 1 sec. If the distri- Table 2 Spectral Accelerations Estimated for Mean and Modal Earthquakes, Example 1 Predicted Ground Motion* Period Description Values 0A sec 1 sec 0.1 sec mean /17/= g(-25%) 0.030g(-39%) earthquake R = 28.9 km g = sec mean M = g (-47%) g (-30%) earthquake /? = 66.5 km g= sec modal h~/ = g (-25%) g (43%) earthquake /? = 22.5 km g = sec modal M = g (+ 15%) g (- 15%) earthquake R = 17.5 km ~= 1.1 * 10 4 target motions are 0.16 g for 0.1 sec and g for 1 sec. Values in parentheses are deviations from target motions. butions are different (in the limit, if they are completely separate), different earthquakes drive the hazard. The choice of interval size is important: small intervals will result in little apparent overlap, and large intervals will result in large apparent overlap, for the same distributions. The interval sizes used here (0.1 on M, 5 km on R, and 0.2 on e) appear to work well. This composite seismic hazard analysis is the tool used to determine a single design earthquake, if appropriate. Using the composite distribution (or the individual distribution for each period), we can determine the most likely earthquake Mp, Rp, and ep. This is herein called the "beta earthquake" after the beta point in structural reliability analysis, the most likely point causing failure. The term beta earthquake herein designates the most likely combination of M, R, and e determined separately by attenuation equation, to distinguish it from the modal earthquake derived earlier for the distribution with all attenuation equations combined. In most applications, the following steps can be used to derive one or more design earthquakes: (1) Determine the contribution to hazard at the target ground motion level separately by source for T = 0.1 and 1 sec. (2) If one source is the dominant contributor at both 0.1 and 1 sec, then one design earthquake generally can be used to represent the hazard. In this case, (2A) Perform a composite seismic hazard analysis for the dominant source and for each attenuation equation i separately, to determine Mpcl, R~ci, and eflci. (2B) Calculate a weighted M,~ c, Rpc, and epc by weighting the M~ci'S, Rpci' s, and e~c~'s using the weights assigned to each attenuation equation. (2C) Adjust e~c so that the final predicted ground mo-

8 1282 R.K. McGuire Table 3 Spectral Accelerations Estimated for Design Earthquakes, Example 1 Period T Description Values Predicted Ground Motion* 0.1 sec beta earthquake for attenuation sec beta earthquake for attenuation sec average 0.1 sec weighted beta earthquake 0.1 sec adjusted weighted beta earthquake = "beta earthquake" 1 sec beta earthquake for attenuation 1 1 sec beta earthquake for attenuation 2 1 sec average 1 sec weighted beta earthquake 1 sec adjusted weighted beta earthquake = "beta earthquake" M~ = 5.15, R~ = 17.5 kin, ea~ = 1.5 m~2 = 5.15, R~z = 22.5 km, e~l = 0.90 M~ = 5.15, Ra~ = 20 km, ea~ = 1.2 M w= 5.15, Ra~ = 20 km, ~ = 1.12 M~ = 7.35, Re, = kna, e~ = 0.5 M~ = 6.75, Ra~ = 72.5 km, e~ = 0.7 M~ = 7.05, R ~ = 87.5 kin, ea~ = 0.6 M~,~ = 7.05, R~ = 87.5 km, eaw= g g 0.16 g (0) g (+4%) 0.16 g (0) g g g (+ 2%) g (-4%) g (0) * 10-4 target ground motions are 0.16 g at 0.1 sec and g at 1 sec. Values in parentheses are deviations from target values. tions at both T = 0.1 and 1 sec equal or exceed the target motions. (3) If different sources are the dominant contributors for T = 0.1 and 1 sec, we generally should not use one earthquake to represent the hazard. In this case, (3A) Perform a seismic hazard analysis for the dominant source at T = 0.1 sec, and for each attenuation equation separately determine Mp~, Rp~, and ep~. (3B) Calculate a weighted M~w, Rew, and epw by weighting the Ma{s Rpi' s, and ep~'s using the weights assigned to each attenuation equation. (3C) Adjust the weighted e~w so that the final predicted ground motion (weighted by attenuation equation) equals the 10-Hz target motion. (3D) Repeat steps (3A) through (3C) for T = 1 sec. The important trait of the composite beta values (M~c ~, Rt~c~, and epci) is that they guarantee that the target ground motion will be replicated by the attenuation equation (to the degree of precision associated with an M-R-e bin). This is the case because each M-R-e combination just causes the target ground motion to be equaled in equation (6). If the attenuation equations have similar characteristics (e.g., similar magnitude and distance dependencies), the values of Mpci, R~ci, and epc~ will be similar. Estimates Using Design Earthquakes values of Mpi and R~i should make it clear whether the design earthquake corresponds to a fault or a source area; in the case of example 1, the values of M~g and R~j at T = 0.1 sec clearly correspond to an earthquake on fault 1, whereas they correspond to an earthquake in source 45 for T = 1 sec. The ground motions shown in Table 3 closely match the target ground motions when averaged over the attenuation equations, even though the estimates for each equation separately may differ from the target ground motion. These differences result from the finite sizes of the M- R-e bins used to calculate the fi point; our experience is that the average ground motion (over the attenuation equations) generally matches the target ground motion, even if the individual estimates differ. Also, the "beta earthquake" matches the target ground motion exactly, because the e value has been chosen to achieve that match. Figure 7 shows the "Design Spectra" obtained from the z o d el DESIGN SPECTRA, EXAMPLE ,., ,,.,, Uniform hazard spectrum s beta spectrum... 1 s beta spectrum / " / *~-. ~?, For example 1 the seismic hazard analysis indicates that different sources contribute most to the hazard for T = 0.1 and 1 sec (see Figs. 2 and 3). As a result, steps (3A) through (3D) are used to derive two design earthquakes for example 1. Table 3 shows the values of M~i, Rpj, and e~ derived from the analysis and the associated ground motions. The weighted values M~w, Rpw, and eb~ provide a close estimate of the target ground motion (within 4%) for both T = 0.1 and 1 sec. The r~ e,i,,,i r i i, illl,,,t r, i i, i i i PERIOD (s) Figure 7. Uniform hazard spectrum and beta earthquake spectra for 10-4 hazard, example 1.

9 Probabilistic Seismic Hazard Analysis and Design Earthquakes: Closing the Loop 1283 Table 4 Spectral Accelerations Estimated for Design Earthquakes, Example 2 Predicted SA* Period T Description Values 0.1 and 1 sec Composite beta earthquake for attenuation and 1 sec composite beta earthquake for attenuation and 1 sec average 0.1 and 1 sec weighted composite beta earthquake 0.1 and 1 sec adjusted weighted composite beta earthquake = "composite beta earthquake" M~cl = 5.95, Rpc 1 = 12.5 km, ~/~cl = 0.1 Mac2 = 5,35, R cj = 12.5 km, eacl = 0.3 M~c = 5.65, Rac = 12.5 km, ece = 0.2 M c = 5.65, R~c = 12.5 km, epc = sec 1 sec g (+18%) g (+ 12%) g ( + 15%) g ( + 14%) g (+ 11%) g (-21%) g (- 10%) g (- 16%) g (+4%) g (0) * 10-4 target motion is g at 0.1 sec and g at 1 sec. Values in parentheses are deviations from target values. uniform hazard spectrum, the T = 0.1 sec adjusted beta earthquake, and the T = 1 sec adjusted beta earthquake. The differences in frequency content are clear, as is the need for,~, two design earthquakes to represent the hazard. Z The parameters selected for example 2 (see Table 1) mean that USGS source 71 dominates the hazard at both T,~ = 0.1 and 1 sec. As a result, steps (2A) through (2C) apply and the use of one design earthquake (for each attenuation equation) is justified. Table 4 shows the values of M~c~, R~ci,,~ and e~c ~ for this example, and illustrates that the "adjusted weighted composite beta earthquake" (or "composite beta earthquake" for short) gives reasonable values of M, R, and rj and results in ground motions close to the target levels. Figure 8 compares the uniform hazard spectrum for ex- o~ ample 2 with the spectrum from the composite beta earthquake and shows that the two match reasonably well for all frequencies. The composite beta earthquakes for each attenuation equation would match the uniform hazard spectrum also, but these individual earthquakes have the disadvantage that they do not provide a single M, R, and e (the values are different for the two attenuation equations). el el Ir DESIGN SPECTRA, EXAMPLE Uniform hazard spectrum,,,~ i i i i i i i i,,,i, r i i r i l l,,,~ PERIOD (s) Figure 8. Uniform hazard spectrum and composite beta earthquake spectrum for 10.4 hazard, example 2. Conclusions Seismic hazard can be deaggregated to show the contribution by magnitude, distance, and ground-motion deviation e, where e is the number of standard deviations that the target ground motion is above or below the median for a given M and R (in logarithmic units). Any uncertainties in seismicity parameters can be incorporated and represented in this deaggregation. The decision on whether a single design earthquake can represent the entire response spectrum hinges on whether one seismic source dominates the hazard at all frequencies. If it does, the preferred design earthquake is obtained from a "composite seismic hazard analysis" wherein a joint distribution of contributions by M, R, and e is obtained by combining the deaggregated hazard results for multiple natural periods (1 and 0.1 sec), keeping separate distributions by ground-motion attenuation equation. From the composite analysis, a composite beta earthquake (M~ci, Rpci, and epci) is derived for each ground-motion attenuation equation i as the most likely combination of M, R, and e that will generate the target ground motions. The final design spectra or other characteristics of the ground motion can be obtained using the individual values of Mpci, Rm~, and e~c ~ for each attenuation equation or by weighting them to obtain a single composite beta earthquake Mnc, R~c, and epc. If different seismic sources dominate the hazard at different frequencies, then contributions to the hazard come from significantly different magnitudes, distances, and ds, so that a single design earthquake is not appropriate. In this case, values of Mew, R~w, and epw derived at T = 0.1 and 1 sec by weighting beta earthquakes for each attenuation equation can be used to represent design earthquakes for short and long periods, respectively.

10 1284 R.K. McGuire The guidelines offered here are meant to be just that. They may not be appropriate in all situations. Ultimately, the judgment of the analyst in selecting one or a few design earthquakes is critical. Deaggregating a seismic hazard analysis in this way achieves two important goals. First, the cause of the seismic hazard in terms of M, R, and e is better understood. There is a relationship between the specified sources of seismicity and the calculated hazard. Second, we can close the loop between the multitude of earthquakes considered by the hazard analysis and the requirement for one or a few design earthquakes by users of hazard studies. The result should be better understood seismic hazards and better decisions on seismic design, analysis, and retrofit. Acknowledgments The author appreciates comments from two anonymous reviewers that were incisive, helpful, and prompt. This work was funded in part by the Electric Power Research Institute, Contract Number RP This support is gratefully acknowledged. References Algermissen, S. T., D. M. Perkins, P. C. Thenhans, 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 , 99 pp. Boore, D. M., W. B. Joyner, and T. E. Fumal (1994). Estimation of response spectra and peak accelerations from western North American earthquakes: an interim report part 2, U.S. Geol. Surv. Open-File Rept , 40 pp. Campbell, K. W. (1993). Empirical prediction of near-source ground motion from large earthquakes, in Proc., Int. Workshop on Earthquake Hazard and Large Dams in the Himalaya, INTACH, New Delhi. Chapman, M. C. (1995). A probabilistic approach to ground motion selection for engineering design, Bull. Seism. Soc. Am. 85, no. 3, Ishikawa, Y. and H. Kameda (1988). Hazard-consistent magnitude and distance for extended seismic risk analysis, in Proc. 9th WCEE, Vol. II, Ishikawa, Y. and H. Kameda (1991). Probability-based determination of specific scenario earthquakes, in Proc. 4th International Conference on Seismic Zonation, Vol. II, Ishikawa, Y. and H. Kameda (1993). Scenario earthquakes vs. probabilistic seismic hazard, in Proc., Int. Conf on Structural Safety and Reliability, lnnsbruck. McGuire, R. K. and K. M. Shedlock (1981). Statistical uncertainties in seismic hazard evaluations in the United States, Bull. Seism. Soc. Am. 71, no. 4, National Research Council (1988). Probabilistic seismic hazard analysis, Rept. of the Panel on Seismic Hazard Analysis, National Academy Press, Washington, D.C., 97. Stepp, J. C., W. J. Silva, R. K. McGuire, and R. T. Sewell (1993). Determination of earthquake design loads for a high level nuclear waste repository facility, in Proc., 4th DOE Natural Phen. Haz. Mitig. Conf., Atlanta, Vol. II, Risk Engineering, Inc Darley Ave., Suite A Boulder, Colorado Manuscript received 21 November 1994.