statistical uncertainties in seismic hazard evaluations in the united states

7/29/2019 STATISTICAL UNCERTAINTIES IN SEISMIC HAZARD EVALUATIONS IN THE UNITED STATES

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Bulletinof he SeismologicalSocmtyof America, Vol 71, No 4, pp 1287-1308, August1981

STATISTICAL UNCERTAINTIES IN SEISMIC HAZARD EVALUATIONSIN THE UNITED STATES

BY ROBIN K. McGUIRE* AND KAYE M. SHEDLOCKABSTRACT

E f f , c ie n t a n d a c c u r a t e m e t h o d s o f e s t im a t i n g t h e s e n s i t i v it y o f s e i s m i c h a z a r dc a l c u l a ti o n s t o s ta t i s ti c a l u n c e r t a i n t ie s i n m o d e l s a n d p a r a m e t e r s a r e d e m o n -s t r a te d . T h e s e m o d e l s re q u i r e k n o w l e d g e o f t h e e a r t h q u a k e m a g n i t u d e a n dd i s t a n c e t h a t c o n t r i b u t e m o s t t o t h e p r o b a b i l i t y o f e x c e e d e n c e o f a c h o s e na c c e l e r a t io n l e ve l; t h e m e t h o d s e s t i m a t e s e n s i t iv i t ie s u s i n g p o i n t- s o u r c e s e = sm i c -h a z a r d a p p r o x i m a t a o n s f o r w h i c h c l o s e d - f o r m s o l u t i o n s a r e a v a i l a b l e . A n a d d l -t , o n a l r e s u l t i s t h a t t h e u s e o f B a y e s i a n e s t i m a t e s f o r s e i s m i c i t y a n d g r o u n dm o t i o n p a r a m e t e r s in t h e h a z a r d a n a l y s i s p r o d u c e s u n b i a s e d B a y e s i a n e s t i m a t e so f th e s e i s m i c g r o u n d m o t i o n h a z a r d , d u e t o t h e a l m o s t l i n e a r r e l a ti o n s h i pb e t w e e n g r o u n d m o t i o n a m p l i tu d e s a t a g i v e n p ro b a b i l it y l e v e l, a n d p a r a m e t e ru n c e r t a i n t ie s . A p p l i c a t io n o f t h e s e m e t h o d s t o t h e S a n F r a n c i s c o , C a l i fo r n i a,B a y a r e a i n d i c a t e s a c o e f f i c i e n t o f v a r i a t i o n ( c o v ) o f t h e 5 0 0 - y r a c c e l e r a t i o n o fa b o u t 0 . 4 a t s i te s c l o s e to m a j o r f a u lt s , a n d a c o v o f a b o u t 0 . 2 a t s i te s 5 0 k m t ot h e e a s t o f t h e m a j o r e a s t b a y f a u l t s . T h e s e c o v ' s r e s u l t f r o m s t a t i s t ic a l u n c e r -t a i n t y i n t h e d e p t h o f e n e r g y r e l e a s e , t h e a c t i v i t y r a t e a n d R i c h t e r b v a l u e f o re a c h f a u l t , a n d t h e m e a n a c c e l e r a t i o n - a t t e n u a t i o n r e l a t i o n s h i p . A s i m i l a r a n a l y s i si n t h e c e n t r a l M i s s i s s i p p i V a l l e y a r e a i n d i c a t e s a c o v in 5 0 0 - y r a c c e l e r a t i o n o f0 . 4 n e a r t h e m a j o r f a u l ts , w i t h a v a lu e o f a b o u t 0 . 3 a t d i s t a n c e s g r e a t e r t h a n 5 0k m . T h e s o u r c e s o f s t a t is t i c a l u n c e r t a i n t y i n t h i s re g i o n a r e t h e d e p t h o f e n e r g yr e l e a s e a s w e l l a s it s I o c a t m n , t h e a c t i v i t y r a t e a n d R i c h t e r b v a l u e f o r e a c hf a u lt , a n d t h e m e a n a c c e l e r a t io n - a t t e n u a t i o n f u n c t io n .

INTRODUCTIONThe probabilistic assessment of seismic ground motion has gained wide acceptance

as a method of expressing absolute and relative earthquake hazards (e.g., Cornelland Merz, 1975; Kiremidjian and Shah, 1975; Algermissen and Perkins, 1976; VonRolf and Mayer-Rosa, 1978). In these studies a variety of mathematical represen-tations have been used to model the sources of energy release, the seismicityassociated with those sources, and the seismic ground motion. The uncertaintiesinherent in these models, and in the parameters used to define them, have beenrecognized by many investigators as important for the realistic determination ofseismic haza rd (e.g., CorneU and Vanmarcke, 1969; Esteva, 1970; Der Kiureghianand Ang, 1977; McGuire, 1977). The effect of these uncerta int ies has general ly beeninvestigated by sensitivity studies or by explicit inclusion in the risk analysis ofprobability distributions on parameters and models.

In analyzing seismic hazard at many sites for the purpose of drawing a seismic-hazard map, methods of accurately and efficiently including the effects of modeland paramete r uncer tainty are important. Fortunatel y, best (mean-value) estimatesof the seismic hazard can be obtained using the mean value of most parameters,because of the almost linear dependence th at calculated probabilities have on theseparameters. This is demonstrated here. Thus, for best est imates of the seismichazard, explicit inclusion of uncertain ties in most paramete rs is unnecessary.

*Present address. Ertec Rocky Mountain Inc., 1746 Cole Boulevard, Golden, Colorado 804011287


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1288 R O B I N K . M C G U I R E A N D K A Y E M . S H E D L O C KT h e p r i m a r y p u r p o s e o f t h e p r e s e n t s t u d y i s t o d e t e r m i n e t h e s t a t i s t i c a l u n c e r -

t a m t y i n s e i s m i c - h a z a r d e s t i m a t e s r e s u l t i n g f r o m s t a t is t i c a l u n c e r t a i n t i e s i n m o d e l sa n d p a r a m e t e r s u s e d a s i n p u t to t h e a n a l y s is . E x p l i c i t e v a l u a t i o n o f t h i s s t a t is t i ca lu n c e r t a i n ty g i v es a n i n d i c a ti o n o f t h e p r e c is i on o f t h e e s t i m a t e s a n d o f h o w m u c ht h e y a r e l ik e l y t o c h a n g e a s m o r e d a t a a r e c o l le c t e d a n d a s t h e s e i s m i c i t y a n dg e o p h y s i c a l m o d e l s a n d p a r a m e t e r s a r e r e f i n e d . F u r t h e r , e v a l u a t i n g t h e s o u r c e s o fs t a t is t ic a l u n c e r t a i n t i e s i n s e i s m i c h a z a r d w i l l i n d i c a t e t h e m o s t a p p r o p r i a t e a r e a s o fd a t a c o l l e c t i o n a n d r e s e a r c h t o r e d u c e t h a t u n c e r t a i n t y .

TABLE 1SEISMIC I-~AZARD MODELS AND PARAMETERS FOR WHICH STATISTICAL UNCERTAINTY IS EXAMINED1

AND DISCRETE DISTRIBUTIONS USED FOR HYPOTHETICAL EXAMPLE

M o d e l o r P a r a m e t e rD i s t r i b u t i o n U s e d I n E ~ a m p l eValue Probabfllt~

M e a n r u p t ur e l e n g t h [RL (kin)]

M e a n a c c e l e r a t i o n

R i c h t e r b v a l u e

M a x i m u m m a g n i t u d e (M~)

I0- 1085+0 3~9M I10-4 0+o 8M 0

McGuire (1978a) 0.2D o n o v a n a n d B o r n s t e i n (1978) 0 4S c h n a b e l a n d S e e d (1973) 0 4

0.69 0.30 87 0.41.04 0 37 0 1/37 5 1/38.0 1/3

A c tav l ty r a te (v) 0 05 1/3( e v e n t s p e r y e a r ) 0.10 1/3

0.15 1/3F a u l t l o c a t a o n ( i n d l r e c u o n

p e r p e n d i c u l a r t o f a u l t s t r i k e )As shown in Figure I 0,4East 2 km 0 3W e s t ' 2 k i n 0.3

T h i s s t u d y i s c o n d u c t e d i n t h e c o n t e x t o f c u r re n t p r o c ed u r e s u s e d t o e v a l u a t ee a r t h q u a k e h a z a r d s in th i s c ou n t r y; t yp i c a l m a t h e m a t i c a l m o d e l s a n d a s s u m p t i o n sa r e d i s c u s s e d i n t h e n e x t s e c t io n . W i t h t h e a i m o f d e t e r m i n i n g s t a t i s t ic a l u n c e r t a i n -t i e s t h a t w o u l d a p p l y t o m a n y s i te s , si m p l e , e f f i ci e n t m e t h o d s a r e u s e d f o r e s t i m a t i n gt h e s e u n c e r t a i n t i e s g i v e n u n c e r t a i n t i e s i n t h e i n p u t , w i t h o u t e x p l i c i t i n c l u s i o n o fm u l t i - p a r a m e t e r p r o b a b i l i t y d i s t r i b u t i o n s i n t h e a n a l y s i s .

SEISMIC-HAZARD ANALYSIST h e m e t h o d u s e d h e r e f o r a s s e s s i n g s e i s m i c h a z a r d i n c o r p o r a t e s i d e a s p r o p o s e d

b y s e v e r a l i n v e s t i g a t o r s ( C o r n e ll , 1 96 8; D e r K i u r e g h i a n a n d A n g , 1 9 77 ) . T h e p r o b a -b i l it y o f e x c e e d e n c e o f v a r i o u s l e v e l s o f g r o u n d m o t i o n a r e o f i n t er e s t; h e n c e t h ei m p o r t a n t p a r a m e t e r d e s c ri b in g e a r t h q u a k e o c c u r r e n c e s i n t i m e i s t h e e x p e c t e dn u m b e r o f e v e n t s w i t h i n a s p e c if ie d t i m e p e r i od . T h e s iz e a n d l o c a t i o n o f s u c c e s si v ee a r t h q u a k e s a r e i n d e p e n d e n t ; e a r t h q u a k e m a g n i t u d e s a re a s s u m e d t o c o n f o r m to a


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S T A T I S T IC A L U N C E R T A I N T I E S I N S E I S M I C H A Z A R D E V A L U A T I O N S 1289truncated exponential distribution. Each event ruptures a section of a prescribedfault; the length of rupture is lognormally distributed, with the mean value afunction of the earthquake magnitude. The lognormal distribution is truncated toconform to the constraint that the rup ture length is always less than or equal to the

x(16)

x ( ] 2 )x ( 1 5 )X ( 1 4 ) X ( ] ] )X ( ] 3 )

x ( l o )

x(9 )

Eo

FAULTI

MIDPOINTOFFAULT i "-4"~

I

Fic

X(5 )x (6 )

x ( 7 )

X(8 )

(1) (2) (3) (4)X X X X0 2 0 k i n ( ] S I T E N U M B E R

1. Plan view of faults and sit es m example problem.fault length. Characteristics of ground motion at the site of interest are lognormallydistributed; the mean value of a specified ground-motion parameter is a function ofthe earthquake magnitude and the shortest distance between the rupture and thesite.

The seismic hazard is assessed using the total probability theorem. Specifically,the probability that a value of a chosen ground-motion measure is exceeded at a siteduring a specified time is the probabili ty tha t t ha t value is exceeded given a specificseismic event times the probabi lity of that event during the time of interest, summe dover all possible events (all possible magnitudes and locations). A modified comput erprogram (McGuire, 1978b) based on this theorem was used for calculations in thispaper.


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1290 ROBIN K. MCGUIRE AND KAYE M. SHEDLOCKGiven the probabilistic uncertainty in earthquake size, location, and associated

ground motion, there are several models and parameters which are determinedempirically and which therefore have associated statistical uncertainties. Thesestatistical uncertainties result from the finite number of data available on which tobase estimates, and would vanish if an infinite number of appropriate data wereavailable. This is not to say th at probabilistic uncertaint y would vanish. Models andparameters for which uncertainties are examined specifically in this st udy are listedin Table 1, along with discrete distributions used to express that uncertainty in a

E

"7"I . - - -Dz , ,. _ . i

100( I I I I I I I

L S T R I K E - S L I PAULT DATA

101

f " Q

1 0L

I 'I~ /o j /12 3 4 5 6 7 8 9

EARTHQUAKE MAGNITUDEF~G. 2 Length of surface rupture versus magn itud e for strike-sAp faults, and two math emat ica lexpressions of the mean trend (after Bolt, 1978)

hypothetical example described below. It is assumed here that the uncertainties inTable I are statistically independent; this is appropriate if model and p aramet erestimates are based on different, independent data for each model and parameter.For instance, the Richter b value may be estimated by historical seismicity, theactivity rate may be estimated from slip rates observed in fault trenches, and themaximum magnit ude may be esti mated by considering the tectonic regime of thearea. A typical seismic-hazard analysis may not be so simple; if activity rates andRichter b values are determined jointly from historical seismicity, correlation mayresult in these two paramet ers unless care is take n in their det ermi nati on (Veneziano,1975).

Other uncertainties in seismic-hazard analysis are classified here as probab i l i s t i cuncertainties; they are inherent to the method and would not be reduced if aninfinite data set was available. Included are uncertainties in the number of earth-quakes, their locations and magnitudes, in the length of rupture for each event


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S T A T IS T I CA L U N C E R T A I N T I E S I N S E IS M I C H A Z A R D E V A L U A T I O N S 1291( a s s u m i n g a n a c c u r a t e m e a n r u p t u r e l e n g t h ) , a n d i n t h e g r o u n d m o t i o n a t a s i t e f o ra g iv e n e a r t h q u a k e ( u n c e r t a in t y a b o u t t h e e x p e c t e d g r o u n d m o t i o n ). U n t i l m o r es o p h i s t i c a t e d m e t h o d s a n d t e c h n o l o g i e s a r e a v a i l a b l e t o s e i s m i c - h a z a r d a n a l y s i s ,e .g ., t o p r e d i c t t h e s i ze a n d l o c a t i o n o f f u t u r e e a r t h q u a k e s , t h e s e p r o b a b i l i s t i cu n c e r t a i n t i e s w i ll r e m a i n .T h e d i s t r ib u t i o n o f g r o u n d - m o t i o n v a l u e s c a l c u l a t e d b y t h e s e i s m i c - h a z a r d a n a l -y s i s is i n f a c t a d e r i v e d d i s t r i b u t i o n o b t a i n e d f r o m d i s t r i b u t i o n s o n e a r t h q u a k e

700

600

50 0

~-~ 4 0 0 -Z,oI .-.-,..,.. 3 0 0 -

20 0 -

100 -

\ \\ \ \

500 YEARACCELERATION

1 1 l l i

1000 YEAR ACCELERATION

- - Iog lo R L :-1 .085 + O 389 M. . . . IOgloRL= -4 .0+ 0 8 M

_ _ I I [ I i L 1 _ l0 10 20 30 40 50 60 70 80DISTANCE (k m) FROM FAULT

F I a . 3 . 50 0 - a n d 1 0 0 0 - y r a c c e l e r a t i o n a t s i t e s ( 1 ) t h r o u g h ( 4) ( F i g u r e 1 ) , u s i n g t w o r u p t u r e l e n g t hv e r s u s m a g m t u d e r e la t io n s .m a g n i t u d e , l o c a t i o n , a n d s o o n. T h e c o m p l i c a t e d n a t u r e o f t h e p r o b l e m p r o h i b i t s a ne x a c t c l o se d - f o rm d e t e r m i n a t i o n o f t h e e f fe c t o f u n c e r t a i n t y i n th e i n p u t p a r a m e t e r sa n d m o d e l s . T h i s e f f e c t c a n a l w a y s b e d e t e r m i n e d b y e n u m e r a t i o n , i.e ., b y u s in gd i s c re t e d i s tr ib u t i o n s t o r e p r e s e n t p a r a m e t e r a n d m o d e l u n c e r t a in t i e s a n d d o i ngm u l t ip l e - h a z a r d a n a l y s e s f o r a ll c o m b i n a t i o n s o f i n p u t v a l u e s . A p p r o x i m a t e , c l o se d -f o r m s o l u t i o n s a r e p r e s e n t e d i n t h e n e x t s e c t i o n ; t h e i r a c c u r a c y i s v e r i f i e d b ye n u m e r a t i o n w i t h e x a c t h a z a r d a n a l y s i s r e s u l t s .

E S T I M A T I N G E F F E C T S O F S T A T IS T IC A L U N C E R T A I N T I E SHypothetical example. T o e f f ic i e n tl y a n d a c c u r a t e l y d e t e r m i n e t h e e f f e c t s o f

s t a ti s ti c a l u n c e rt a in t ie s , s e v e r a l m e t h o d s a r e r e c o m m e n d e d d e p e n d i n g o n t h e s o u r c eo f u n c e r t a in t y . T h e s e m e t h o d s w e r e v e ri fi ed b y c o m p a r i s o n w i t h e x a c t r e s u lt s f o r ah y p o t h e t i c a l e x a m p le . T h e g e o m e t r y o f t h e f a u l t a n d t h e s i te s e x a m i n e d a r e s h o w ni n F i g u r e 1 ; t h e s t a t is t i c a l d i s t ri b u t i o n s r e p r e s e n t i n g u n c e r t a i n t y i n i n p u t p a r a m e t e r sa n d m o d e l s a r e li s te d i n T a b l e 1. B e c a u s e t h e a p p r o x i m a t e r e s u lt s w e r e t o b e


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1292 ROBIN K. MCGUIRE AND KAYE M. St IEDLOCKchecked by enumeration with multiple hazard analyses, discrete distributions werelimited in number to three values with associated probabilities. These probabilitiesmay represent statistical uncertainties resulting from limited data, or subjectivejudgments on the validity of one model over another. Peak horizontal groundacceleration was used as a ground-motion measure because it has received wideinterest in seismic-hazard studies and because several attenuation functions areavailable to est imate it.

M e an rup ture - l e ng th magn t tude re la t ion . The effect on seismic-hazard estimatesof the relationship between the mean length of fault rupture and earthquakemagni tude was examined by considering the two relat ions shown in Table I. Theserepresent ext reme interpre tations of available dat a (Bolt, 1978) as shown in Figure2. For each relation, the peak accelerations with 0.002 and 0.001 annua l probabilitiesof being exceeded are indicat ed in Figure 3 for the sites located perpendicular to th emid-point of the fault. (The acceleration with 0.002 probability of being exceeded isalso called the "500-yr accelerat ion" and is designated asoo herein.) The sensit ivityof a~0o to the mean rupture-l ength magni tude relation is not large at most sites,leading to t he conclusion tha t uncert ainty in this relation need not be included inseismic-hazard estimates. Uncertainty in rupture length about the mean values h o u l d be included; because this uncertainty is so large [the standard deviation oflog~o (rupture length) is taken to be 0.52], relat ively small changes in the mea nvalues, particular ly around magni tude 7 where the two lines cross in Figure 2, havelittle effect on asoo. As a result of this insensitivity to the mea n value of rupturelength, only the more conservative relationship (the first listed in Table 1) is usedfor further comparisons in this study (thus the second relation listed in Table 1 isassigned zero probability).M e a n ac c e lera tion , b , m ax im um ma gn i tude , ac t i v it y rat e , and fau l t l oca t ion .Model uncertainty in the estimation of mean acceleration for a given magnitudeand distance is represented in this example using three atten uatio n functions whichhave been proposed for peak horizontal ground acceleration as on rock (Schnabeland Seed, 1973; Donovan and Bornstein, 1978; McGuire, 1978a) and which havebeen calibrated with strong mo tion records. The first function listed in Table 1(assigned subjective probabili ty of 0.2) assumes a very simple dependence of a s onearthquake magnitude, M, and source-to-site distance R

as -- exp(cl + c2 M )R c3.Functions of this type are adequate for expressing the dependence of as on magni-tude and distance in the M and R range where data are numerous, but are notappropriate for extrapolation in particular to large magnitudes and small distances.The other two attenuation functions listed in Table 1 assume more realistic butmore complicated dependences of as on M and R; they provide more realisticestimate s of acceleration for large magn itudes at close distances. Un cert aint y in a sa b o u t the mean value is assumed to conform to a lognormal distribution withstandard deviation of In ag equal to 0.62 (McGuire, 1978a) for all three attenuationfunctions.The assumed trun cated exponential distribution on magnitude is specified withtwo parameters: the slope b of the log number versus magnitude relation, and themaximum magnitude M1. To represent statistical uncertainty in b, a mean value of0.87 was chosen (and weighted 0.4) with alt ernate values of -+_20 per cent weigh ted


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S T A T IS T I C A L U N C E R T A I N T I E S I N S E IS M I C H A Z A R D E V A L U A T I O N S 12930.3 each (20 per cent typically corresponds to one standard deviation for historicalseismicity in the United States). For M~, values of 7.0, 7.5, and 8.0 were weightedequally; this represents a large uncert aint y in M~ and is thus an extreme test for theapproximate procedures being developed.

The rate of occurrence of earthquakes is specified by mean activity rate ~.Uncert ainty in ~ is repr esen ted by values of 0.05, 0.1, and 0.15 earthquakes per year,each value weighted equally.

The fault location indicated on Figure 1 is assigned a subjective probabili ty of 0.4,and the major source of energy release is taken to be at a dep th of 5 kin. Alternatefault locations of _+2 km (in the direction perpendicular to the fault) are assigned

(~Pom t-source hazard analys~s, , ~ us ing a l te rna te va lues o f ba r M 14~) Polnt-source \ /hazard analys is 31~ ~ , ~ . ~usin g m ean I ~ / b(,.~.) '50 0 from 41~),~(,2) and

parameter values I \ ~ ~ l . . . . .o " , , , , / . . . . s00T 1 \ ~ a ' s o o ro m % 0 0z , 3 /< ~ and c hanges i n Y

,,x,0 002 - -


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1294 ROBIN K. MCGUIRE AND KAYE M. SHEDLOCKa c c e l e r a t i o n , a ~ oo , a n d w i l l a l s o i n d i c a t e t h e s l o p e o f t h e ( lo g ) p r o b a b i l i t y -v e r s u s - ( l o g ) a c c e l e r a t i o n c u r v e a t a~ oo . A l s o , f o r t h i s v a l u e o f a~ oo , d e t e r m i n ea n e f f e c ti v e m a g n i t u d e M e a n d a n e f f e c t i v e d i s t a n c e R e d e f i n e d a s fo l l o w s

Me = ~ M . P [ A > asoo [ M ] P [ M ] / O . O 0 2M

Re = ~ R P [ A > a so o I R ] P [ R ] / O . O 0 2 .R

(3)M e a n d R e a r e th e m e a n m a g n i t u d e a n d m e a n d i s t a n ce o f s e is m i c e v e n t sc a u s i n g e x c e e d e n c e o f a c c e l e r a t i o n a~ oo a t t h e s it e.

0 . 0 5 I

0 . 0 5 1

0 . 0 5 I

" ~ o . o s la , .

M c G u l r e ( 1 9 7 8 ) A t t e n u a t i o n

i l l l l l l l L I I / l l I 1 1 1J I 1 1 1 1 1 1 I I I I ID o n o v a n ( 1 9 7 , 8 ) A t t e n u a t i o nH l l J I I iS c h n a b e I - S e e d ( 1 9 7 3 ) A t t e n u a t i o no . o s ) l ) l l l l J ) l ) l i l i I l lo . o s l I I ( l ! l l l f I

I I I I0 2 0 0 4 0 0 6 0 0

A c c e l e r a t i o n ( g a l s )Fro. 5. Exact dgstrlbunons (above horizontal hnes) and approxi mate chstnbutlons (below horizontallines) of a~0o for each att enuati on, at si te (2) m Figure 1

2. F o r t h e f ir s t a t t e n u a t i o n f u n c t i o n d e t e r m i n e t h e a c c e l e r a t i o n a l e s t i m a t e d f o rm a g n i t u d e M ~ a n d R e ; d e t e r m i n e a c c e l e r a t i o n s a 2 a n d a 3 s i m i l a r l y f o r t h es e c o n d a n d t h i r d a t t e n u a t i o n s . C o m p u t e t h e w e i g h t e d a v e r a g e a w o f t h e s ea c c e l e ra t i o n s u s i ng t h e a t t e n u a t i o n w e i g h ts . C a l c u l a t e a n a t t e n u a t i o n f a c t o ra l a s t h e r a t i o o f a l t o a w ; c a l c u l a t e f a c t o r s a 2 a n d a ~ s i m i l a r l y .

3 . D e t e r m i n e t h e c l o s e s t d i s t a n c e R e f r o m t h e f a u l t t o t h e s i te ; f o r t h e f i r s ta t t e n u a t i o n f u n c t i o n c a lc u l a te t h e a c c e l e r a t io n a c e s t i m a t e d f o r m a g n i t u d e M ea n d d i s t a n c e R e . F o r e a c h c h a n g e i n f a u l t lo c a t i o n a n d d e p t h , d e t e r m i n e an e w d i s t a n c e R ~', a n e w a c c e l e r a t i o n a 'c, a n d a l o c a t i o n f a c t o r X' a s t h e r a t i o o fa~' to ac .

4 . F o r t h e f i r st a t t e n u a t i o n f u n c t i o n a n d m e a n v a l u e s o f b a n d M 1 , c a l c u l a t e t h ep r o b a b i l i t y t h a t a c c e l e r a t i o n a ~oo is e x c e e d e d a s s u m i n g a n e a r t h q u a k e o fr a n d o m m a g n i t u d e o c c u r s a t d i s ta n c e Re f r o m t h e s it e. T h i s i s a c l o s e d - f o r m


9/22

S T A T I S T I C A L U N C E R T A I N T I E S I N S E I S M I C H A Z A R D E V A L U A T I O N Sc a l c u l a t i o n ( C o r n e l l , 1 9 7 1 )

1295

P [ A > asoo] = (1 - k ) o * ( z / o ) + k , ~ * ( z ' /o )+ k R e p 3 /c 2 e x p ( - f i I n a 5o o /C 2 - f l c l / c 2+ f i m o + B ~ o ~ 1 2 c ~ ) ( ~ * ( z / o - # o l e o ) - ) * ( z ' / , ~ - # , d c ~ ) ) (4)

w h e r e 0 * i s t h e c o m p l e m e n t a r y c u m u l a t i v e f u n c t io n o f t h e s t a n d a r d i z e dn o r m a l d i s t ri b u t i o n , m o i s t h e l o w e r - b o u n d m a g n i t u d e , k = (1 - e x p ( - b ( M 1- mo ))) -1, fi = b In 10 a n d

Z = asoo - c~ - c2M1 - - C3 I n R eZ ' = asoo - c~ - c2 mo - c3 In Re. (5 )

0 5 [

Z0F,-,~04-

L 9 . / ~ L / ( -0 , 3 - - ( 1 5 ) ( 9 ) "~6T }" / 4 , :! .(10)

/,(15)( I ~ / ,(B)

(~.3/

02 03 04

I ) S I T E N U M B E R

ESTIMATED COEFFICIENT OF VARIATIONF IG 6 . C o m p a r i s o n b e t w e e n e x a c t a n d e s t i m a t e d c o v o f a~o o d u e t o s t a h s t ] c a l u n c e r t a i n t y , f o r 1 6 s i t e s

s h o w n m F i g u r e 1

T h e d e r i v a t i o n o f e q u a t i o n (4) is a v a i l a b l e i n M c G u i r e ( 19 76 ).5. F o r t h e f i r s t a t t e n u a t i o n f u n c t i o n a n d a n a l t e r n a t e v a l u e o f b (f r o m it s d is c r e te

d i s t r i b u t i o n ) , o r o f M s , r e c a l c u l a t e t h e p r o b a b i l i t y t h a t a c c e l e r a t i o n a~ 0o i se x c e e d e d u s i n g e q u a t i o n ( 4 ) .

6. F r o m t h e c h a n g e i n p r o b a b i l i t y c a l c u l a t e d f r o m s t e p s 4 a n d 5, a n d t h e s l o p ef r o m s t e p 1 [ i.e ., t h e r a t i o o f c h a n g e s i n ( lo g) p r o b a b i l i t y t o c h a n g e s i n ( lo g)a c c e l e r a t i o n ] , d e t e r m i n e a n e w a~oo a s s o c i a t e d w i t h t h e a l t e r n a t e v a l u e o f b.

7. U s i n g t h e s l op e f r o m s t e p 1, a n d a s s u m i n g t h a t c h a n g e s i n p r o b a b i l i t y a r ep r o p o r t i o n a l t o c h a n g e s i n a c t i v i t y r a t e s , d e t e r m i n e ( f r o m a~ 0o i n st e p 6 ) n e wa c c e l e r a t i o n s as"oo f o r e a c h v a l u e i n t h e d i s c r e t e d i s t r i b u t i o n o f p .

8 . M u l t i p l y a c c e l e r a t i o n s a~'0o b y t h e a t t e n u a t i o n f a c t o r a l ( s t e p 2 ) a n d l o c a t i o nf a c t o r s X ' ( s te p 3 ). F o r e a c h a c c e l e r a t i o n a l ~ ' a ~ o ( a s s o c i a t e d w i t h a s p e c if i c


10/22

1296

.

10.

ROBIN K. MCGUIRE AND KAYE M. SHEDLOCKf a u l t l o c a t i o n a n d s p e c i f ic v a l u e s f o r b , M , , a n d r ) c a l c u l a t e t h e a s s o c i a t e dp r o b a b i l i t y a s a p r o d u c t o f p r o b a b i l i t i e s f o r a t t e n u a t i o n 1, l o c a ti o n , b , M , a n dP.R e p e a t s t e p s 5 th r o u g h 8 f o r a ll c o m b i n a t i o n s o f v a l u e s f o r b a n d M 1 .R e p e a t s t e p s 3 t h r o u g h 9 f o r t h e s e c o n d a n d t h i rd a t t e n u a t i o n . F o r t h ep u r p o s e s o f e q u a t i o n (4 ), c2 c a n b e c a l c u l a t e d a s t h e l o c a l s l o p e o f I n ( a c c e l-e r a t io n ) w i t h r e s p e c t t o m a g n i t u d e a n d c3 i s t h e s l o p e w i t h r e s p e c t t o In( d is t a nc e ) , b o t h e v a l u a t e d a t M e a n d R e ; c l c a n b e o b t a i n e d b y s o lv i n g e q u a t i o n( 3) u s i n g M e , R e , a n d a c c e l e r a t i o n a 2 (o r a 3) c a l c u l a t e d i n s t e p 2 .

500

40 0

Zo~ 3O0Ut , . )

~- 2oo8,x,

100. (3)

(14)e(7)(10)

(15)1 1 1 . 1 %18)., '(16)(12)I10 0

I I

*(1)(5)

"(2)(6)e

o(9}(13)

i) SITE NUMBER

I200 300 400 500ESTIMATED 500 YEAR ACCELERATION (GALS)

FIG. 7. Com par iso n bet wee n exact and e stn nat ed a~00 for 16 sites sho wn m Figure 1.

1 1. F o r t h e c a l c u l a t e d v a l u e s o f a s0 o a n d t h e i r a s s o c i a t e d p r o b a b i l i t i e s ( w h i c h s u mt o u n i ty ) , d e t e r m i n e t h e m e a n v a l u e , s t a n d a r d d e v i a ti o n , a n d c o e f f ic i e n t o fv a r i a t i o n ( c oy ) , t h e r a t i o o f s t a n d a r d d e v i a t i o n to m e a n .T o i l lu s t r a te t h e a c c u r a c y o f th i s m e t h o d , F i g u r e 5 s h o w s d i s t r ib u t i o n s o f aso o a t

t h e s i t e ( 2) l o c a t e d 2 0 k m f r o m t h e m i d - p o i n t o f t h e f a u l t ( F i g u r e 1 ) f o r t h eu n c e r t a i n t i e s i n b , M 1 , a n d f a u l t l o c a t i o n l i s te d i n T a b l e 1, f o r th e t h r e e a t t e n u a t i o nf u n c t io n s . T h e " e x a c t " d i s t r i b u t io n s w e r e d e t e r m i n e d f r o m m u l t i p l e - h a z a r d a n a l y s e s;t h e " a p p r o x i m a t e " d i s t r i b u t i o n s w e r e c a l c u l a t e d u s i n g t h e s t e p s d e s c r i b e d a b o v e .T h e c a l c u l a t e d c o v f o r t h e e x a c t r e s u l t s ( i n c lu d i n g u n c e r t a i n t y i n r ) i s 0 .3 0; f o r t h ea p p r o x i m a t e r e s u l t s i t i s 0 .3 2. T h i s a g r e e m e n t i s r e m a r k a b l e g i v e n t h e s i m p l i c it y o fm a k i n g t h e s e a p p r o x i m a t e c a l c u l a t i o n s i n c o n t r a s t t o r e p e a t i n g t h e s e i s m i c - h a z a r da n a l y s i s w h i c h i n v o l v e s n u m e r i c a l i n t e g r a t i o n o v e r m u l t i p l e v a r ia b l e s . A g r e e m e n tb e t w e e n e x a c t a n d a p p r o x i m a t e c o v ' s w a s g e n e r a l l y c l o s e a t a l l 1 6 s i t e s s h o w n i nF i g u r e 1 ; t h i s a g r e e m e n t i s i l l u s t r a t e d i n F i g u r e 6 .


11/22

S T A T I S T I C A L U N C E R T A I N T I E S I N S E I S M I C H A Z A R D E V A L U A T I O N S 1297A r e s u l t e q u a l l y a s im p o r t a n t i s t h a t t h e u s e o f m e a n v a l u e s f o r fa u l t lo c a t i o n ,

a t t e n u a t i o n , b , M 1 , a n d v a s i n p u t t o t h e h a z a r d a n a l y s i s p r o v i d e d a c c u r a t e e s t z m a t e so f a ~ oo . T h i s i s i ll u s t r a t e d i n F i g u r e 7 , w h i c h s h o w s a so o f r o m t h e h a z a r d a n a l y s i su s in g m e a n - v a l u e p o i n t e s t i m a t e s v e r s u s t h e m e a n o f t h e v a l u e s o f a so o o b t a i n e d

NORTHERN ~ODGERSSAN ANDREAS CREEKGREENVALLEY

XCONCORDHAYWARD \.- ' ~ X CALAVERAS, ~ ~ SUNOL

PENINSULARSAN ANDREAS

' ID W A Y,O,T ERN

~ SAN JOAQUINOREE~V,LLE~SOUTHER~~ , SAN

MIDDLE ~.JOA QU ,iSAN JOAQUIN%O'NEILLCALAVERAS\ \ X .

|IILm

SAN GREGORIO" ' " :~Ok,n

FIG 8 Fau lts m t h e S a n F r a n c i s c o B a y a r e a.

f r o m m u l t i p le - h a z a r d a n a l y s i s ( t h e " e x a ct " m e a n a s o o) . T h e a g r e e m e n t i s e x c e ll e n t ;t h i s m e a n s t h a t p a r a m e t e r u n c e r t a in t i e s o f t h e t y p e e x a m i n e d h e r e n e e d n o t b ec o n s i d e r e d e x p l i c i t l y in h a z a r d a n a l y s i s f o r b e s t e s t im a t e s o f th e s e i s m i c h a z a r d . T oe n s u r e t h e a p p l i c a b i li t y o f t h i s re s u l t, i t w a s e x a m i n e d a n d f o u n d t o b e a c c u r a t e o na p a r a m e t e r - b y - p a r a m e t e r b a s is , a s w e l l a s in t h e c a s e w h e n t h e r e a r e u n c e r t a i n t i e si n s e v e r a l p a r a m e t e r s ( a s il l u s tr a t e d h e r e ) .

F o r s i t e s a f f e c t e d b y m o r e t h a n o n e f a u l t , t h e s t e p s a b o v e s h o u l d b e r e p e a t e d f o re a c h f a u lt , a n d t h e p r o b a b i l i t ie s a s s o c i a t e d w i t h e a c h a c c e l e r a t i o n a so o s h o u l d b e


12/22

1298 R O B I N K . M C G U I R E A N D K A Y E M . S H E D L O C Km u l t i p l i e d b y a f a u l t - w e i g h t i n g f a c t o r . T h i s f a c t o r c a n b e c o m p u t e d f r o m t h e " b e s te s t i m a t e " h a z a r d a n a l y s i s a s t h e c o n t r i b u t i o n e a c h f a u l t m a k e s t o t h e t o t a l p r o b a -b i l i t y o f exceede nce a t a~00 ( 0.002 pe r yea r ) . The accu r a cy o f t h i s m e t hod ha s b eenv e r i f i e d w i t h a s i m p l e t w o - f a u l t e x a m p l e ; t h e s e r e s u l t s a r e n o t p r e s e n t e d h e r e .

TABLE 2F A U L T S Y S T E M S I N T H E S A N F R A N C I S C O BAY AREA, A N D MEAN V A L U E S O F A S S O C I A T E D

P A R A M E T E R S ( A F T E R S H E D L O C K E T A L , 1 9 8 0 )Seismic TotalR~chter CreepR i c h t e r L e n g t h S h p S A pNo Fau lt* Ma gmtude b ( k in ) ( cm/ ( cm/ ( c r n / E ( M0)R a n g e y r )yr) yr)

Activi tyRate~+( even t s /yr)

1 N or the rn San Andr eas 7.6-8.2 0 72 430 1.90 0 -- 6.89 1027 0.0036Peninsular San Andreas

2 Nort he rn San Andre as 7.5-8.2 0.72 410 0.95 0 -- 5.82 1027 0 0020San Gregorio3 N or the rn San And reas 7 6-7.9 0.7 2 270 0.15 0 3 4.09 1027 0.00034 Sa n Gregorio 5-7.5 0.75 140 0.05 0 1 2.29 102~ 0 00925 Pen ins ula r San Andre as 5-7.6 0 72 160 0.10 0 2 3.03 1025 0.01596 Rodgers Creek 5-6.9 0.75 50 0.15 0.6 0.75 8.15 1024 0.02767 Maacam a 5-7.5 0.75 140 0.15 0.6 0 75 2.29 1025 0.02758 Cal ave ras 5-7.7 0.9 170 0.15 - - I[ - - [[ 186 x 1025 0.0411

Calaveras-Sunol9 Calaveras-Hayward 5-7.7 0.9 190 0.15 0 6 0 75 1.86 1025 0.045910 Calaveras- Sunol 5-7.1 0 9 70 0 15 - - [[ - - II 7 95 1024 0 039611 Green Valley 5-7.3 0 75 90 0 15 0 6 0 75 1.62 1025 0.024912 Concord 5-6.4 0.9 20 0 15 0 6 0.75 2 86 1024 0 031513 Sar gen t 5-6.4 0.72 20 0.1 0.3 0.4 3.57 x 1024 0.016814 Qulen Sabe 5-6 4 0.9 20 0.1 0.0 0 1 2.86 x 1024 0 021015 Greenville 5-7.0 0.9 50 0.02 0.0 0.02 6.89 1024 0 004416 Las Posl tas 5-6.2 0 9 14 0.02 0 0.02 2.12 1024 0 004017 Monte Vis ta 5-7 2 0 75 20 0.02 0 0.02 1 37 1025 0.000918 Ever green 5-7.1 0.75 14 0.02 0 0 02 1.15 102~ 0 000719 Midway 5-7 0 0.9 11 0 005 0 0.005 6.89 1024 0.000220 Ve rona 5-7.0 0.9 10 0.02 0 0,02 6.89 x 1024 0 000921 O'Nei ll 5-7.3 0.9 24 0.01 0 0,01 1 06 1025 0 000722 Ortlgalita 5-7.6 0 9 60 0.01 0 0.01 1.62 x 1025 0.001123 Nort h San Joaquin 5-7.3 0.9 60 0.02 0 0.02 1.06 x 102~ 0.003424 Midd le San Joaquln 5-6.6 0.9 14 0.02 0 0.02 2 85 x 1024 0.002225 Sout h San Joaqul n 5-6.8 0.9 20 0.02 0 0.02 5 16 1024 0.0023

* All faults were given a depth of energy release of 7.5 km with a probability of 1/3; alternate valuesof 5 and 10 km were assigned probabilities of 1/3 each

t Ri cht er b values shown were assigned probab ilities of 0 4; values differing by __10 per cent wereassigned probab ilitie s of 0.2, and va lues differing by +20 pe r cen t were asmgned probabiliti es of 0 1

Rat es show n were assigned probab ilities of 0.2; values differing by ___25 per cent were asmgnedprobab flties of 0.2, and values differing by +_50 per cent were asmgned probabflitms of 0.2

Total sAp is counted in faults 3 to 5 Modified from Shedlock e t a l . (1980) to yaeld activity rates cons istent with h istorical selsmlclty.[[ On the Calave ras-Sunol section, seismic slip -- 0.3 cm/ yr (total) and creep = 0.45 cm /y r On the

Calaveras-Calaveras-Sunol section, seismic s h p = 0.15 cm/ yr and creep = 0.6 cm/yr. This preserves 0.75crn/yr total sllp for these sections.


13/22

S T A T IS T IC A L U N C E R T A I N T I E S I N S E I SM I C H A Z A R D E V A L U A T I O N S 1299APPLICATIONS IN THEUNITED STATES

San Francisco Bay area. The procedures described in the previous section wereused to determine the effects of statistical uncertainties on peak horizontal groundacceleration with a 500-yr return period in the San Francisco Bay area. This is anarea in which a large amount of geologic investigation has been done to understandfaulting and its relationship to seismicity.

1000

50C

(Dz- ~00

50

5 2 5 6 6 0 6 6 7 0 7 6 8 0 8 5M AGN I T U D E

FIG. 9 M e a n a c c e l e r a t i o n v e r s u s m a g m t u d e f o r s e v e r a l s o u r c e - t o - s i t e d i s t a n c e s , u s i n g S c h n a b e l a n dS e e d ( 19 73 ) a t t e n u a t i o n ( s o li d h n e s) . D a s h e d l m e s i n d m a t e m o d i f i c a n o n m a d e f o r a p p h c a t l o n t o S a nF r a n c is c o B a y a r e a

M a j o r f a u l ts i n t h e a r e a a r e s h o w n i n F i g u r e 8; t h e s e w e r e t a k e n f r o m S h e d l o c ke t a l . ( 1 9 80 ) . S e i s m i c i t y i n t h e a r e a w a s m o d e l e d c o n s i d e r i n g f a u l t s s i n g l y a n d i nc o m b i n a t i o n ; t h e f a u l t s y s t e m s a n d m e a n v a l u e s o f a s s o c i a te d p a r a m e t e r s a r e l is t e di n T a b l e 2 { a f t e r S h e d l o c k e t a l . , 1 9 8 0 ) . S t a t i s t i c a l u n c e r t a i n t i e s i n t h e m e a n r a t e o fe a r t h q u a k e o c c u r r e n c e , i n t h e R i c h t e r b v a l u e , a n d i n t h e d e p t h o f e n e r g y r e le a s e,w e r e m o d e l e d a n d a r e e x p l a in e d i n f o o t n o t e s t o T a b l e 2 . T h e s e r e fl e ct o u r j u d g m e n t so n t h e u n c e r t a i n t y o f t h e s e p a r a m e t e r s f o r t h i s a r e a . G e o l o g i s t s f a m i l i a r w i t h t h ea r ea e x p r e ss e d n o u n c e r t a i n t y a b o u t m a x i m u m m a g n i t u d e e a r t h q u a k e s o r a b o u tf a u lt lo c a t i o n s , s o n e i th e r o f t h e s e u n c e r t a i n t i e s w a s m o d e l e d m t h i s a p p l ic a t io n .

T o e s t i m a t e g r o u n d a c c e le r a ti o n , t h e t h r e e a t t e n u a t i o n f u n c t i o n s g i v e n i n T a b l e


14/22

1300 ROBIN K. MCGUIRE AND KAYE M. SHEDLOCK1 were used, with weights as shown, except that a modified form of the Schnabeland Seed (1973) function was used. This modification is shown as dashed lines inFigure 9, and represents the observation that small-magnitude earthquakes, on theaverage, generate larger ground accelerations than those estimated by the original

60

20

400kin 50kin

FIG. 10 Peak horizont al ground acceleration (in per cen t of gravity) with 0.002 annual exceedenceprol~abihty, San Francisco Bay area.

Schnabel and Seed curves. This has been substantiated by recent data, at least atclose distances (Seekins and Hanks, 1978). A lognormal distribution of accelerationvalues (given the mean) was used, w ith uncer ta in ty specified by a~n% = 0.62.It is impor tant to note tha t these seismic activity rates were calculated based onRicht er magnitudes. The Richt er magni tude scale sat urates for M _-> 7.5 and maynot be a good measure of earthquake source strength for faults on which largerevents may originate (e.g., the San Andreas and Calaveras systems). A nonsat urat ingscale (Hanks and Kanamori, 1979) may be a be tter choice for use in calculating theactivity rates for long faults. However, these activity rates may be several times the


15/22

STATISTICAL UNCERTAINTIES IN SEISMIC HAZARD EVALUATIONS 1301activity rates observed historically (Shedlock e t a l . , 1980) due to uncertainty inparameters such as upper-bound magnitude and rate of fault creep. F urthermore, ithas been shown that activity rates which are consistent with observations in therecent past are most appropriate for estimating seismic hazard in the near future

0.3

0.2

\

\

0.3

v . '~0kin 50kinFro. 11. Cov of accelerat ion with 0 002 annual exceede nce probability, San Francisco Bay area

(McGuire and Barnhard, 1981). A second consideration for seismic-hazard analysesis th at the magni tude scale used for fault activity calculations should be consistentwith that used for ground motion estimates. For these reasons we adopt estimatesof seismic activi ty for the San Francisco Bay region based on Rich ter magnit ude forthis example application.

Figure 10 shows mean value estimate s of as00 for the area, using mean values ofthe seismicity parameter s (Table 2) and the weighted attenu ation functions {Table


16/22

1 3 0 2 ROBIN K. MCGUIRE AND KAYE M. SHEDLOCK1 ). I t i s i m p o r t a n t t o i n t e r p r e t F i g u r e 1 0 in t e r m s o f t h e f a u l t s y s t e m s a n d t h e i rp a r a m e t e r s ( T a b l e 2 ) . F o r e x a m p l e , s e i s m i c s l ip o n t h e s e c t i o n o f t h e S a n A n d r e a sn o r t h o f t h e G o l d e n G a t e i s a t t r i b u t e d e n t i r e l y t o la r g e e a r t h q u a k e s ( m a g n i t u d e s 7 .6t o 8 .3 ) w h i c h a r e r a r e ( r e c u r r en c e i n t e r v a l o f a b o u t 1 70 y r ). B y c o n t r a s t , s e i s m i c s l i p

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o n f a u l ts in t h e e a s t b a y a r e a ( C a l a v e ra s , H a y w a r d , a n d S u n o l ) i s a t t r ib u t e d t o s m a l la n d m o d e r a t e s h o c k s ( m a g n i t u d e s 5 . 0 t o 7 . 7 ) w h i c h a r e m u c h m o r e f r e q u e n t( r e c u r r e n ce i n t e r v a l o f a b o u t 2 5 t o 4 0 y r ).C o n t o u r s o f t h e c o v o f a ~oo a r e s h o w n i n F i g u r e 1 1 . V a l u e s a r e l a r g e s t ( s o m e w h a tg r e a t e r t h a n 0 .4 ) a l o n g t h e m a j o r f a u l ts , d e c r e a s e t o t h e l o w e s t v a l u e s ( l e s s t h a n 0 . 2)a t s i te s a b o u t 5 0 k m f r o m t h e m a j o r s o u r c e s o f s e i s m i c h a z a r d , a n d in c r e a s e a t s i t e s


17/22

S T A T I S T I C A L U N C E R T A I N T I E S I N S E I S M I C H A Z A R D E V A L U A T I O N S 1303T A B L E 3

S E ISM O G E N IC Z O N E S A N D T H E IR P A R A M E T E R S IN T H E C E N T R A L M IS S IS S IP P I V A L L E Y A R E ALocat ion Ac t iv i ty Rmhter t Depth ofSelsmogemcZone Dtstnbutton Magm tude Rate* Energy(probability) Ran ge b(events/yr) Release:~

S t G e n e v m v e f a u l t A s m F i g u r e 1 2 ( 0 8 ) 4 . 5 - 7 5 0 0 3 1 0 . 87 1 0N E 5 k i n ( 0.1 )S W : 2 5 k m ( 0.1 )

C e n t r a l G r a b e n Z o n e A s m F i g u r e 1 2 ( 0.8 ) 4 5 - 7 . 5 0 .1 5 6 0 . 87 1 0S E 2 2 . 5 k m ( 0 .1 )N W : 2 2 .5 k m ( 0. 1)

C o t t o n w o o d G r o v e f a u l t A s m F i g u r e 1 2 ( 0. 8)S E 1 .5 k m ( 0 .1 )N W 3 k m ( 0.1 )

4 . 5 - 6 . 0 0 . 0 2 3 0 . 8 7 1 0

R e e l f o o t f a u l t A s m F i g u r e 1 2 ( 0 8 ) 4 . 5 - 5 .5 0 0 0 8E 1 k m ( 0 .1 )W 1 k m ( 0 1 )

C r o s s G r a b e n A s i n F i g u r e 1 2 ( 0 8 ) 4 5 - 7 5 0 . 15 6E : 5 k m ( 0 .1 )W 1 5 k m ( 0 .1 )

W e s t B r a n c h Z o n e A s m F i g u r e 1 2 ( 0.8 ) 4 . 5 - 5 . 0 0 .0 0 3N . 1 0 k m { 0 .1 )S 1 0 k m ( 0 . 1)

N o r t h e a s t B r a n c h Z o n e A s m F i g u r e 1 2 (0 .8 ) 4 5 - 7 5 0 1 56S E : 1 0 k m ( 0 .1 )N W " 1 0 k m ( 0 1 )

0 . 8 7 1 0

0 . 8 7 1 0

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C o t t a g e G r o v e f a u l t A s i n F i g u r e 1 2 ( 0.8 ) 4 . 5 - 7 .5 0 0 3 1 0 8 7 1 0S W 5 k m ( 0. 1)N E : 5 k m ( 0.1 )

R o u g h C r e e k f a u l t A s m F i g u r e 1 2 ( 0 8 ) 4 . 5 - 7 5 0 0 3 1 0 8 7 1 0S W 5 k m ( 0.1 )N E - 5 k m ( 0 .1 )

R l d g e l y f a u l t A s m F i g u r e 1 2 ( 0 8 ) 4 . 5 - 5 .5 0 .0 0 8 0 . 87 1 0S E 1 5 k m ( 0 .1 )N W 1 5 k m ( 0.1 )

S h a w n e e T o w n f a u l t A s m F i g u r e 1 2 ( 0.8 ) 4 5 - 6 . 0 0 . 00 2S W 5 k m ( 0 1 )N E 5 k m ( 0 .1 )

0 . 8 7 1 0

* R a t e s s h o w n w e r e a s s i g n e d p r o b a b f l i t m s o f 0 .2 v a l u e s d l f f e n n g b y _ +2 5 p e r c e n t w e r e a s s i g n e dp r o b a b i l i t ie s o f 0 2 , a n d v a l u e s d T f f e nn g b y _ + 50 p e r c e n t w e r e a s m g n e d p r o b a b i l i t ie s o f 0 2 .

t R i c h t e r b v a l u e s s h o w n w e r e a s s i g n e d p r o b a b i l i t m s o f 0 .2 9 ; t h e o t h e r a s s i g n a t i o n s w e r e a s f o ll o w s b+ 5 p e r c e n t - - 0 . 1 2; b + 1 6 p e r c e n t = 0 12 ; b + 2 6 p e r c e n t - - 0 . 06 ; b - 5 p e r c e n t = 0 1 8; b - i 6 p e r c e n t= 0 . 23 .

T h e d e p t h s h o w n w a s a s s i g n e d a p r o b a b i l i ty o f 1 /3 ; a l te r n a t e v a l u e s o f 5 a n d 1 5 k m w e r e a s s i g n e dp r o b a b d m e s o f 1 /3 e a c h .


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1304 R O B I N K . M C G U I R E A N D K A Y E M . S H E D L O C Kf a r t h e r t o t h e e a s t . C l o s e t o t h e f a u l t s , t h e m a j o r e f fe c t s a r e t h e u n c e r t a i n t i e s in t h em e a n a c c e l e r a ti o n a n d d e p t h o f e n e r g y r e le a s e. A t 5 0 k m d i s ta n c e , t h e t h r e ea t t e n u a t i o n f u n c t i o n s e s t im a t e a b o u t t h e s a m e a c c e l e r a ti o n . T h e r e f o r e u n c e r t a i n t i e si n a ~0 0 a t s i t e s 5 0 k m f r o m m a j o r f a u l t s r e s u l t p r e d o m i n a n t l y f r o m u n c e r t a i n t i e s i n

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t h e a c t i v i t y r a t e a n d R i c h t e r b v a l u e . A t p r o g r e s s i v e l y f a r th e r s i te s , t h e a c c e l e r a t i o ne s t i m a t e s a g a i n d i v er g e f o r d if fe r e n t a t t e n u a t i o n s b e c a u s e o f t h e a l t e r n a t e f u n c t i o n a lf o r m s a d o p t e d , r e s u l t i n g i n p r o g r e s s i v e l y l a r g e r c o v 's .M i s s i s s i p p ~ V a l l e y a r e a . T h e p r o c e d u r e s d e v e l o p e d i n t h i s s t u d y w e r e a p p l i e d t ot h e c e n t r a l M i s s i s s i p p i V a l l e y a r e a t o d e t e r m i n e s t a t i s t i c a l u n c e r t a i n t i e s i n s e i s m i ch a z a r d f o r t h a t r e g i o n . T h e a r ea a n d i t s f a u l ts a n d s e i s m o g e n i c z o n e s ( w h e r e c u r r e n t


19/22

STATISTICAL UNCERTAINTIES IN SEISMIC HAZARDEVALUATIONS 1305t e c t o n i c p r o c e s s e s a r e s ti l l p o o r l y u n d e r s t o o d ) a r e s h o w n i n F i g u r e 1 2 . I t is r e c o g n i z e dt h a t s o m e d i s a g r e e m e n t e x i s t s a m o n g i n v e s t i g a t o r s i n t h e M i s s i s s i p p i V a l l e y a r e ao v e r w h i c h f a u l ts a r e a c t i v e a n d h o w t o d e l i m i t s e i s m o g e n i c z o n e s . T h e f a u lt s a n dz o n e s s h o w n i n F i g u r e 1 2 r e p r e s e n t o n l y o n e o f s e v e r a l p o s s i b l e i n t e r p r e t a t i o n s o f

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t h e a r e a ( M c K e o w n , p e r s o n a l c o m m u n i c a t i o n , 1 97 9) a n d a r e u s e d h e r e o n l y a s a ne x a m p l e t o i ll u st r a te t h e m e t h o d o l o g y . P a r a m e t e r s a s s o c i a t e d w i t h t h e s e f a u l t s a n dz o n e s a r e l is t e d i n T a b l e 3 . N o u n c e r t a i n t y i n m a x i m u m m a g n i t u d e w a s u s e d b e c a u s eg e o l o g i s ts f a m i li a r w i t h t h e a r e a w e r e i n a g r e e m e n t o n t h e v a l u e s s h o w n i n T a b l e 3 .A l o w e r - b o u n d m a g n i t u d e o f 4.5 w a s a d o p t e d b e c a u s e o f th e l o w a t t e n u a t i o n o fg r o u n d a c c e l e r a t i o n i n t h e c e n t r a l U n i t e d S t a t e s : s m a l l m a g n i t u d e e a r t h q u a k e s ,


20/22

1306 ROBIN K. MCGUIRE AND KAYE M. SHEDLOCKbecause of their large relative number and the low attenuati on of ground motion,may contribute significantly to ground-acceleration hazard.

The mea n act ivity rates shown in Table 3 were det ermi ned using geologic evidenceto estima te recurrence rates for earthquakes of the largest Modified Mercalli (MM)intensity. Assuming that magnitude is proportional to ma ximum MM intensity andthat the constant of proportionality is 2/3, these recurrence rates were assumed tobe accurate for magnit udes between M1 - 2/3 and M1, and consistent rates formagnitudes greater than 4.5 were then calculated (Table 3). In this case the strictstatistical independence between activity rate a nd b value does not hold, as it did inthe San Francisco example, but any correlation between these two parameters isignored in this study. In any case, the total activity rate for the area is consistentwith historical seismicity as reported by Algermissen and Perkins (1976).

The mean acceleration attenuat ion function used was tha t proposed by Algermis-sen and Perkins {1976) for the Central an d East ern United States. Accelerationsestimated by this function for magnitudes up to about 6.6 are close to thoseestimated by the attenuation equation of Nuttli (personal communication, 1979),which became available after this study was initiated. For higher magnitudes, in therange 7.0 to 7.5, the Nutt li equation predicts larger accelerations by a factor of about2. If the Nuttli curves had been used in the Mississippi Valley region they wouldhave produced higher values of a~0o than those report ed in this study but about thesame values for the cov of asoo. To model un cert ain ty in the mea n acceleration, auniform distribution was used, with values ranging from 0.5 to 1.5 of the mean.Uncertainty in peak acceleration about the mean value was assumed lognormallydistributed, with ( ~ l n a g = 0.62.

The calcula ted values of a~0o are shown in Figure 13. The largest values occurnear the "central graben" and "cross graben" zones, where large earthquakes(magni tude ~ 7.5) are es timated to occur once every 600 yr, on the average. Figure14 shows the cov of these accelerations due to the statistical uncer tainti es indicatedin Table 3 and due to u ncertai nty in the m ean acceleration. The largest uncertaintiesoccur in the area of highest acceleration risk because of uncertainties in faultlocation and depth of energy release; at locations farther from faults, smalleruncertainties are obtained. The uncert ainty in mean acceleration has been assumedconstant with distance, so there is no increase in coy at large distances. If, insteadof the assumed range of mean accelerations (0.5 to 1.5 times th e mean), a range of0.75 to 1.25 was used, the values shown in Figure 14 would decrease by abou t 0.1.

CONCLUSIONSThe procedure developed in this study provides a computationaUy efficient

metho d of determining the uncert ainty in seismic-hazard calculations which resultfrom statistical uncertainty in the assumed models and their parameters. "Bestestimates" of seismic hazard at most sites can be acurately obtained using meanvalues of statistically uncertain parameters, and uncertainties in the seismic hazardcan be dete rmined subsequent ly in a postprocessing operation.Uncertainties in seismic hazard in the San Francisco Bay area, as expressed bythe cov of the 500-yr acceleration, range from greater than 0.4 to less than 0.2. Largeuncertainties near the major faults result from uncertainties in mean accelerationgiven an event and from uncertainties in the depth of energy release. The lowestuncertainty is calculated about 50 km east of the major faults. At farther distances,values of acceleration are low but un cer tai nty is high due to differences in predicted


21/22

S T A T IS T I C A L U N C E R T A I N T I E S I N S E IS M I C H A Z A R D E V A L U A T I O N S 1 3 0 7a c c e le r a ti o n s a t th e s e l o n g d i s ta n c e s a m o n g v a r i o u s a t t e n u a t i o n e q u a t i o n s.

I n t h e c e n t r a l M i s s i s si p p i V a l le y , u n c e r t a i n t y i n t h e 5 0 0 - y r a c c e l e r a t io n i s l a r g e stn e a r t h e m a j o r fa u l t s ( c o v gr e a t er t h a n 0 .4 ) a n d l o w e r a t f a r th e r d i s t a n c e s ( a r o u n d0 .3 a t d i s t a n c e s g r e a t er t h a n 1 0 0 k i n ). T h e s e v a l u e s d e p e n d h e a v i l y o n th e a s s u m e du n c e r t a i n t y i n m e a n a c c e l e r a t i o n ; h a l v i n g t h e r a n g e u s e d h e r e w o u l d d e c r e a s e c o v ' sb y a b o u t 0 . 1 .

A l t h o u g h n o t d o n e h e r e , i t w o u l d b e p r o p e r a n d l o g i c a l t o i n t e r p r e t t h e l a r g e s ts o u r c e s o f s ta t i s ti c a l u n c e r t a i n t y a s a r e a s i n w h i c h r e s e a r c h a n d d a t a c o l l e c t i o ns h o u l d b e c o n d u c t e d t o r e d u c e u n c e r t a in t i e s i n se i s m i c -h a z a r d e v a l u a t io n s . T h er e l a ti v e u n c e r t a i n t i e s a l s o g i v e a n i n d i c a t i o n o f h o w h a z a r d c a l c u l a t i o n s w i l l c h a n g ei n t i m e , w i t h a d d i t i o n a l r e s e a r c h a n d d a t a . T h e m e t h o d e x p l a i n e d a n d i ll u s t r a t e dh e r e p r o v i d e s a c o m p u t a t i o n a l l y e f f ic i e n t fo r m a t fo r d e t e r m i n i n g t h e s e u n c e r t a i n t ie s .

A C K N O W L E D G M E N T ST h e a u t h o r s a c k n o w l e d g e t h e h e l p o f E . E B r a b b , A . J . C r o n e , D . G . H e r d , F . A . M c K e o w n , a n d D .

P . R u s s i n p r o v i d i n g m f o r m a t l o n o n f a u l t s a n d s e l s m o g e n i c z o n e s a n d t h e i r p a r a m e t e r s i n t h e S a nF r a n c i s c o B a y a r e a a n d m t h e c e n t r a l M i s m s m p p l V a l l e y a r e a . W e a p p r e c i a t e a l s o t h e r e v m w s o f e a r l yd r a f t s o f t h i s w o r k b y C A C o r n e l l, R . M H a m i l t o n , O W N u t t l i , D . M . P e r k i n s , a n d A . M R o g e r s , J r . ,w h o o f f e re d m a n y u s e f u l s u g g e s ti o n s .

R E F E R E N C E SA l g e r m l s s e n , S T a n d D M P e r k i n s { 1 97 6) A p r o b a b i h s t m e s t u n a t e o f m a x u n u m a c c e l e r a t i o n m ro c k

m t h e c o n t i g u o u s U m t e d S t a t es , U S G e o l . S u r v O p e n - F ~ l e R e p t . 7 6 -4 1 6, 4 5 p pB o l t , B A . ( 19 78 ). I n c o m p l e t e f o r m u l a t i o n s o f t h e r e g r e s s i o n o f e a r t h q u a k e m a g n i t u d e w i t h s u rf a c e f a u l t

r u p t u r e l e n g t h , G e o l o g y 6 , 233-235C o r n e l l, C A { 19 68 ) E n g m e e n n g s e ls m m r is k a n a ly s i s , B u l l . S e t s m . S o c . A m . 5 8 , 1 5 8 3 - 1 6 0 6C o r n e l l , C . A ( 19 71 ). P r o b a b f l l s t l c a n a l y s i s o f d a m a g e t o s t r u c t u r e s u n d e r s e l s m m l o a d , i n D y n a m i c

W a v e s m C w t l E n g i n e e r i n g , D . A H o w e l ls , I P . H a l g h , a n d C T a y l o r , E ( f it o r s , W i l e y - I n t e r s c m n c e ,L o n d o n , 4 7 3 - 4 88

C o r n e l l , C A . a n d H . A . M e r z ( 1 97 5 ) S e i s m i c r i s k a n a l y s i s o f B o s t o n , J S t r u c. D w . , A m S o c . C w dE n g r s . 1 0 1 , 2 0 2 7 - 2 0 4 3

C o r n e ll , C A a n d E . H . V a n m a r c k e { 1 9 6 9 ) T h e m a j o r m f l u e n c e s o n s e is m m ri s k , P r o c T h t r d W o r mC o n f o n E a r t h q u a k e E n g , S a n t m g o , C h i l e , A - l , 6 9 - 8 3 .

D e r K m r e g h m n , A . a n d A H - S . A n g (1 97 7) . A f a u l t - r u p t u r e m o d e l f o r s m s m m r i s k a n al ys ~ s, B u l l S e t s mS o c . A m . 67 , 1173-1194

D o n o v a n , N C a n d A . E B o r n s t e i n ( 19 78 ) U n c e r t a m t l e s i n s e i s m m r i s k p r o c e d u r e s , J G e o t e c h E n g .D w , A m S o c C t v d E n g r s 1 6 4 , 8 6 9 - 8 8 7 .

E s t e v a , L . ( 19 7 0) . S m s m l c r is k a n d s m s m m d e s i g n d e c l s m n s , m S ez sm ~ c D e s i g n f o r N u c l e a r P o w e r P l a n t s ,R . J H a n s e n , E d i to r , M . I T P r e s s , C a m b r i d g e , 1 42 - 18 2

E s t e v a , L ( 19 76 ) S e l s m m l t y , m S e t s m w R ~ s k a n d E n g i n e e r i n g D e c z s m n s , E R o s e n b l u e t h a n d C .L o m m t z , E d i to r s , E l s e v i e r S c m n t i f i c P u b h s h m g C o ., A m s t e r d a m , 1 7 9 - 2 2 4

H a n k s , T C a n d H K a n a m o n ( 19 79 ) A m o m e n t m a g n i t u d e s ca le , J G e o p h y s R e s 8 4 , 2 3 4 8 - 2 3 5 0 .K l r e m l d p a n , A S . a n d H C . S h a h ( 19 75 ). S e i s m i c h a z a r d m a p p i n g o f C a l i fo r n i a , J o h n A B l u m e

E a r t h q u a k e E n g i n e e r i n g C e n t e r R e p o r t N o . 2 1 , S t a n f o r d U n i v e rs i ty , N o v e m b e r , 9 8 p p.M c G m r e , R . K . (1 97 6) . F o r t r a n c o m p u t e r p r o g r a m f o r s e is m a c r i s k an a l ym s , U S . G e o l . S u r v . O p e n - F d e

R e p t 7 6 - 6 7 , 9O ppM c G u i r e , R K (1 97 7) E f f e c ts o f u n c e r t a i n t y i n s em m i c lt y o n e s t i m a t e s o f s m s m l c h a z a r d f o r t h e e a s t

c o a s t o f t h e U n i t e d S t a t e s , B u l l . S e t s m . S o c . A m . 6 7 , 8 2 7 - 8 4 8M c G m r e , R K . (1 9 78 a ) S m s m i c g r o u n d m o t i o n p a r a m e t e r r e l a t m n s , J G e o t e c h E n g . D w . , A m . S o c

C w d E n g rs 1 6 4 , 4 8 1 - 4 9 0M c G m r e , R K . ( 1 97 8 b ) F R I S K - c o m p u t e r p r o g r a m f o r s e m m i c r i s k a n a l y s i s u s i n g f a u l t s a s e a r t h q u a k e

s o u r c e s , U S G e o l S u r v O p e n - F d e R e p t . 7 8- 10 07 , 71 pp .M c G m r e , R . K a n d T P B a r n h a r d { 19 81 ) E ff e c ts o f t e m p o r a l v a r l a t m n s in sm s m i c l ty o n s e ls m m h a z ar d ,

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statistical uncertainties in seismic hazard evaluations in the united states

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