a comparison and test of various site-response estimation techniques,

7/30/2019 A Comparison and Test of Various Site-Response Estimation Techniques,

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Bulletin of the Seismological Society of America, Vol. 85, No. 4, pp. 1127-1143, August 1995

A Comparison and Tes t of Various Si te-Response Est imat ion Techniques ,Including Three That Are Not Reference-Si te Dependent

b y E d w a r d H . F i e l d a n d K l a u s H . Ja c o b

Abs t r a c t W e c o m p a re v a r i o u s s i t e - r e s p o n s e e s t i m a t i o n t e c h n i q u e s u s i n g a f t e r -s h o c k d a t a o f t h e 1 9 8 9 L o m a P r i e t a e a r t h q u a k e c o l l e c t e d i n Oa k l a n d , C a l i fo rn ia .B e c a u s e o f r e c e n t i n t e r e s t i n c o m p a r i n g r e s u l t s f r o m w e a k a n d s t r o n g m o t i o n ( t oi n f e r a n y n o n l i n e a r i t y ) a n d b e t w e e n d i r e c t S a n d c o d a w a v e s , w e p a y p a r t i c u l a ra t t e n t io n t o t h e u n c e r t a i n t i e s . F ir s t , s e d i m e n t t o b e d ro c k s p e c t r a l - r a t io e s t i m a t e s b e -t we e n p a i r s o f s i t e s a r e c o m p a re d w i t h t h o s e o b t a i n e d f ro m v a r i o u s g e n e ra l i z e d -i n v e r s i o n a p p ro a c h e s wh e re t h e s o u rc e a n d s i t e e f f e c t s o f m u l t i p l y r e c o rd e d e v e n t sa r e s o l v e d fo r s i m u l t a n e o u s l y . W e f i n d t h a t t h e s i t e a m p l i f i c a t i o n f a c t o r s a r e v e rys i m i l a r a m o n g t h e s e a p p ro a c h e s , b u t t h a t t h e u n c e r t a i n t i e s c a n b e s i g n i f i c a n t l y d i f -f e r e n t d e p e n d i n g o n h o w t h e d a t a a r e w e i g h t ed .

W e a l s o e x a m i n e a n d t e s t t h r e e s i t e - r e s p o n s e e s t i m a t i o n t e c h n i q u e s t h a t d o n o tr e l y o n a r e f e r e n c e s it e t o e s t i m a t e s o u rc e a n d p a t h e f f e c t s . T h e f i r s t i n v o l v e s ap a r a m e t e r i z e d s o u r c e - a n d p a t h - e f fe c t s in v e r s io n . E v e n w h e n t h e b e d r o c k d a t a a r ee x c l u d e d f ro m c o n s i d e ra t i o n , th i s a p p ro a c h i s f o u n d t o r e v e a l t h e f r e q u e n c y d e p e n d -e n c e o f s i t e r e s p o n s e a t e a c h o f t h e s e d i m e n t s i t e s . T h e s e c o n d t e c h n i q u e i n v o l v e st a k i n g h o r i z o n t a l - t o v e r t i c a l - c o m p o n e n t s p e c t r a l r a t io s ( r e c e i v e r - fu n c t i o n - t y p e e s -t i m a t e s) o f s h e a r - w a v e a f t e r s h o c k d at a. T h e s e a r e a l s o f o u n d t o r e v e a l t h e f r e q u e n c yd e p e n d e n c e o f s i t e r e s p o n s e a t t h e s e d i m e n t s it e s, a n d t h e r e s u l t s f o r t h e b e d ro c k s i t ea r e r e l a t i v e l y f ia t a n d n e a r u n i t y . T h e t h i rd e s t i m a t e i s f o rm e d b y t a k i n g h o r i z o n t a l -t o v e r t i c a l - c o m p o n e n t r a t i o s o f a m b i e n t s e i s m i c n o is e , a n d t h e s e a r e s h o wn t o r e v e a lt h e f u n d a m e n t a l r e s o n a n t f r e q u e n c y o f t h e s e d i m e n t s i te s . U n f o r t u n a t e l y , d is c r e p -a n c i e s e x i s t a m o n g a l l o f th e s i t e - r e s p o n s e e s t i m a t e s ( a n d w i t h o n e -d i m e n s i o n a l p r e -d i c t i o n s) w i t h r e s p e c t t o a f r e q u e n c y - i n d e p e n d e n t s c a l i n g f a c to r . Ne v e r t h e l e s s , t h eh i g h l y f r e q u e n c y -d e p e n d e n t c h a ra c t e r o f s it e r e s p o n s e i s we l l c o n s t r a i n e d , a n d t h ef a c t t h a t n o n - r e f e r e n c e - s i t e - d e p e n d e n t m e t h o d s a r e c a p a b l e o f r e v e a l i n g th i s is p ro m -i s i n g fo r s i t e - s p e c i fi c h a z a rd a s s e s s m e n t s i n r e g i o n s t h a t l a c k a d e q u a t e r e f e r e n c e s it e s .

In t ro d u c t i o nIt has long been und erstood that earthquake ground mo-tions can b e significantly am plified by sedim entary depositsnea r the Earth 's surfa ce (e.g., Milne, 1898). The potentially

severe consequences of th is phenomenon were recentlydemon strated in the d ama ge patterns of the 1985 Michoacan,Mexico, earthquake (e.g., Singh e t a l . , 1988), the 1988 Ar-menian earthquake (e.g ., B orcherdt e t a l . , 1989), the 1989Lom a Prieta earthquake (e.g ., H ough e t a l . , 1990; Borcherdtand Glassmoyer, 1992), and the recent Northridge earth-quake in Los Angeles , California (EERI Prel iminary Recon-naissance Report, 1994). Numerous other studies have alsodem onstrate d the ability o f surface geologic conditions toalter seismic m otions (e.g., Borcherdt, 197 0; King a ndTucker, 1984; Aki , 1988; Field e t a l . , 1992). Given thegreat ly increased level of dam age that can be produced, i t is

of practical im portance to develop m ethods for assessing thenature of, an d potential for, se dim ent amplification, espe-cial ly when choosing the locat ion and design of cri t ical andessential facilities.The greatest challen ge in estimating site respons e fromearthquake data is rem oving the source and path effects . Themost c omm on procedure, in troduced by B orcherdt (1970),is to divide the spectrum observed at the s ite in quest ion bythat observed at a nearby referen ce si te (preferably on com -petent bedrock). If the two sites have similar source and patheffects, and if the ref eren ce site has a negligible site re-sponse, then the resulting spectral ratio constitutes an esti-mate o f s i te response. And rews (1986) recast the me ,hod ofspectral ratios into a generalized-inverse proble m by solvingthe data o f mult ip ly recorded events for al l source/path ef-

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1128 E.H . Field and K. H. Jacob

fec ts and s i te e ffec ts s imultaneous ly . Others have s ince ap-p l i ed va r ious fo rm s o f th i s gene ra l i z ed - inve rsion approach(B oa twr igh t e t a L , 1991a; Hartze l l , 1992; Seekins and Boat-wright , 1994), and the ques t ion remains as to whether i tp rov ides any advan tage ove r s imply ave rag ing s pec t ra l ra t io sfor each s i te separa te ly ( the t radi t ional approach introducedby B orcherdt , 1970). Th e f i rs t goal of th is a r t ic le , therefore ,i s t o com pare s i t e - re spons e e s t ima te s ob ta ined f rom t rad i-t iona l s pect ra l ra t io s w i th thos e ob ta ined f rom va r ious gen-era l ized-invers ion spectra l -ra t io approaches (a lso de ter-min ed re la t ive to a referen ce s i te ) .

T hes e t e chn iques fo r e s t ima t ing s i te re s pons e depend onthe ava i l ab i l i ty o f an adequa te re fe rence s i t e (w i th a neg l i-g ib le s i t e re s pons e ) . S ince s uch a s i t e may no t a lways beavai lable , as in the Ne w M adrid se ism ic zone , i t is des i rableto deve lop a l t e rna t ive me thods tha t do no t depend on a re f -e rence s it e. T ow ard tha t end , we a l s o examine th ree recen t lydeve loped non- re fe rence -s i te -dependen t t e chn iques by com-pa r ing re s u lt s w i th thos e ob ta ined f rom s ed imen t - to -bedrocks pec tra l r a tio s . T he f i rs t a l te rna t ive me thod emp loys a gen-e ra l i z ed - inve rs ion s cheme in t roduced by B oa twr igh t e t a l .(1991a) , whe re shear-w ave spectra are represented with apa rame te r i zed s ource - and pa th -e f fec t mode l and a f re -quency-dependen t s i t e - re s pons e t e rm fo r e ach s i t e . T hes ource pa rame te rs fo r e a ch even t , t he pa th -e f fec t pa rame te rs ,and the re s pons e fo r e ach s i te a re then s o lved fo r s imul t a -neous ly in the l e a s t - s qua re s s ens e . T he s econd non- re fe r -ence -s i t e -dependen t t e chn ique invo lves d iv id ing the ho r i -zon ta l -componen t s hea r -wave s pec t ra a t e ach s i t e by theve r t i c a l -componen t s pec t rum obs e rved s imul t aneous ly a ttha t s i te (L e rm o e t a l . , 1993). This procedu re , which is anal-ogous to the s o -ca l l ed rece ive r - func t ion t e chn ique app l i edto s tud ie s o f the uppe r man t l e and c rus t f rom te l e s e i s micrecords (e .g . , Langs ton, 1979), assumes tha t the loca l s i tecond i t ions a re re l a t ive ly t r ans pa ren t to the m ot ion tha t ap -pea rs on the ve r t i c a l compon en t . T he l a s t t e chn ique tha t weexam ine i s tha t i n t roduced by N akam ura (1989) fo r ana lyz -ing ambien t s e i s mic no i s e . He hypo thes ized tha t s i t e re -s pons e cou ld be e s t ima ted by d iv id ing hor i zon ta l -compon en tno i s e s pec t ra by ve r t i c a l -componen t no i s e s pec tra . Seve ra ls tud ie s have s ince s how n tha t Nakam ura ' s p rocedure can bes ucces s fu l in iden t i fy ing the fund amen ta l r e s onan t f requencyof s ed im en ta ry depos i t s (Omach i e t a L , 1991 ; L e rm o e t a t . ,1992; Fie ld and Jacob, 1993b; Fie ld e t a l . , 1994a).

W e us e a f t e r s hock da ta ob ta ined in Oak land , C a l i fo rn ia ,fo l lowing the 1989 L o ma P r i e t a e a r thquake to t e s t t he s e s i te -re s pons e e s t ima t ion t e chn iques . De ta i l s r ega rd ing the de -p l o y m e n t c a n b e f o u n d i n H o u g h e t a L (1990) or Fie ld e t a l .(1992). Figure la shows the loca t ion of the f ive s i tes occu-p ied in th i s E as t B ay s tudy : s it e s S1 , $2 , and $5 we re loca tedon B ay M ud depos i t s t ha t cove r Qua te rna ry a l luv ium; s i t e$3 was p laced d i rec t ly on the a l luv ium laye r ; and a re fe rences it e, $4 , was p laced o n the Franc i s can C om plex which con-s t itu t es bedrock in th i s r eg ion . E ach s i te was equ ipped w i thM a rk Produc t s L 22-D geophon es ( th ree -componen t ) , wh ichhave a co rne r f requency a t 2 Hz . T ab le 1 g ive s pe r t inen t

E M E R Y V lL ~ ' . . . : ' ~ l ; , ' ?T R E A S U R E I S . ~ ? / ~ ; ' , ~ , ' . - , : " ~ l l ~ l ~ l l l l J

I l l I I~ ' ~ 2 / . , ' ~ 2 ' - I . j I.I":'..'.::.:';-.':!'.:(:..i::.....'.:.'!::... , ~ \l \ ~ -]

o , z k mE X P L A N A T I O N I

C O LL A P S E D ~ B a y M u d ( H o l o ce n e ) IU N C O L L A P S E D I ': " ~ . " . ~ : . 1 0 - 3 0 m thick I~ l u v i u m ( q u a t e rn a r y) II II W E A K M O T I O N S I T E S t " ~ ~ / 1 0 -1 8 0 m t h i c k I~ ] ~ F ~ ~ C o m p l e x ( M e s o z o i c) lI I I I I I I I I I B a s e m e n t l

1 2 2 " 4 0 ' 1 2 2 " 1 2 1 " 2 2 '38"1 . ~ _.~:. t I

" ~ . . . : . E A R T H Q U A K E S - eW E A K M O T I O N S I T E S - I I

37*

36*45 '

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Fig ure 1. (a) Map of Oakland, California, regionshowing the surface geology and the locations of thefive sites (S1, $2, $3, $4, and $5) occupied duringthe aftershock deploym ent. (b) Regional map sho wingthe epicenter locations (circles) and recording sites(squares).


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V a r i o u s S it e - R e s p o n s e E s t i m a t i o n T e c h n i q u e s 1129

i nf or ma t i on on t he 18 a f t e r shocks used i n t h i s s t udy , andF i gur e l b shows t he i r l oca t i ons r e l a t i ve t o our r ecor d i ngs i t e s . T he magni t udes r ange be t ween 2 .5 and 4 .4 and t hee p i c e n t ra l d i s ta n c e s r a n g e b e t w e e n ~ 7 5 a n d - 1 1 5 k m .

G i v e n t h e l o w a f t e r sh o c k m a g n i t u d e s a n d l a rg e e p i c e n -t r a l d is t ances , t he s i t e - r e sponse e s t i ma t es ob t a i ned h e r e a r ef o r w e a k m o t i o n . H o w e v e r , s o m e e v i d e n c e o f n o n l i n e a r si ter e s p o n s e d u r i n g s t r o n g g r o u n d m o t i o n h a s r e c e n t l y a p p e a r e d( e .g . , Ch i n and Aki , 1991 ; Dar r agh and S haka l , 1991 ; Be-r e s n e v e t a l . , 1994) . T he ex t en t t o whi ch non l i nea r e f f ec t sa r e s i gn i f i can t r ema i ns a con t r ov e r s i a l i s sue . Never t he l e s s ,weak- mot i on s i t e r e sponse w i l l r ema i n a l og i ca l s t a r t i ngp o i n t f o r d e t e r m i n i n g t h e r e s p o n s e o f s e d i m e n t s e v e n i f n o n -l i nea r e f f ec t s a r e i n f l uen ti a l f o r s t r onge r mot i ons t han t hosecons i de r ed i n t h i s s tudy .

Beca use a succes s f u l i den t i f i ca ti on o f non l i nea r i t y w i l lr equ i r e demo ns t r a t i ng t ha t t he s t r ong- m ot i on r e sponse i s s ig -n i f i can t l y d i f f e r en t f r om t he w eak- m ot i on r e sponse , spec i a lca r e mu s t be t aken i n de t e r m i n i ng t he na t u r a l va r iab i l i t y andunce r t a i n t y o f t he e s t i ma t es . T he l eve l o f unce r t a i n t y w i l la l s o d e t e r m i n e h o w m u c h e f f o r t i s w a r r a n te d i n t e s t in g a n dd i s t i ngu i sh i ng be t we en va r i ous t heor e t i ca l mode l s o f s i te r e -sponse . F i na l l y , t he unce r t a i n t i e s a l so need t o be und e r s t oodbef o r e app l y i ng t he r e su l t s to a s ses sment s o f s e i smi c haza rd .F o r t h e s e r e a s o n s , w e d e v o t e c o n s i d e r a b l e a t t e n ti o n t o t h eunce r t a i n t i e s ob t a i ned f r om t he va r i ous s i t e - r e sponse e s t i -ma t i on t echn i ques .

I n t h is s t u d y w e h a v e n o t e x a m i n e d t h e c o d a m e t h o d o fes t i ma t i ng s i t e r e sponse , wh i ch was i n t r oduced by P h i l l i psand Ak i ( 1986) and has be en app l i ed i n s eve r a l si t e - r e sponses t u d ie s ( M a y e d a e t a l . , 1 9 9 1 ; S u e t a l . , 1 9 9 2 ; K o y a n a g i e ta l . , 1992) . Unf or t una t e l y , t he even t - t r i gge r ed w i ndo ws i no u r E a s t B a y s t u d y a r e n o t l o n g e n o u g h t o g e t a n a d e q u a t e

w i n d o w o f c o d a . W e d o , h o w e v e r , d i sc u s s s o m e i m p l i c a ti o n st ha t our r e su l t s have w i t h r e spec t t o t he coda me t hod .

R e p r e s e n ta t io n o f G r o u n d M o t i o nS u p p o s e w e h a v e a n e t w o r k o f I s it es o v e r w h i c h J

e v e n t s h a v e b e e n r e c o r d e d ( e a c h s i te d o e s n o t n e e d t o h a v er ecor ded a l l J even t s ) . T hen t he ampl i t ude spec t r um of t hej t h even t r ecor ded a t t he i t h s it e , O i j ( f ) , can be wr i t t en i n t hef r e q u e n c y d o m a i n a s a p r o d u c t o f a s o u r c e te r m , E j ( f ) , apa t h t e r m, P u ( f ) , and a s i t e - e f f ec t t e r m, S i ( f ) :

O i j f ) " ~ H i ( f ) P i j ( f ) S i ( ~ " ( 1 )T ak i ng t he na t u r a l l oga r i t hm, equa t i on ( 1 ) becomes

I n Oo ( f ) = I n E j ( f ) + I n P o ( f ) + In S , ( f ) . ( 2 )T hi s l i nea r expr es s i on o f t en f o r ms t he bas i s o f a t t empt s t osepa r a t e t he sour ce , pa t h , an d s i t e e f f ec ts .

S p e c t r a l -R a t i o E s t i m a t e sI n t h i s s ec t i on , we compar e t r ad i t i ona l spec t r a l - r a t i o

s i t e - r e sponse e s t i ma t es , i n t r oduced by Bor che r d t ( 1970) ,w i t h t h o s e o b t a i n e d f r o m v a r io u s f o r m s o f t h e g e n e r a li z e di nve r s i on appr oach i n t r oduced by Andr ew s ( 1986) . M ot i -va t ed by r ece n t i n t e re s t in com par i ng s i t e r e sponse be t wee nw e a k a n d s t r o n g m o t i o n , a n d b e t w e e n d i r e c t S a n d c o d awaves ( e .g . , M ar ghe r i t i e t a L , 1994; Ka t o e t a l . , 1995) , ar a t he r de t a i l ed deve l opment o f t he unce r t a i n t y e s t i ma t es i sg i ven . W e f i nd t ha t t he s i t e ampl i f i ca t i on f ac t o r s ob t a i nedf r om t he gene r a l i zed i nve r s i on appr oach a r e ve r y s i mi l a r t o

T ab l e 1Event Information (- - Means the Event Was Not Recorded)

294 00:49 37.038 238.1 41 11.6 4 .1 * * * * - -298 01:27 37.068 238.1 78 9 .4 4 .2 * * * * *297 04:48 37.04 6 238.1 99 4 .2 3 .2 * * * * *298 05:38 37.07 4 238.1 75 9.7 2 .8 * * - - * *297 07:02 37.171 238.0 45 5.0 2 .8 - - * * * *295 08:17 37.171 237.9 29 13.2 2 .6 - - * * * *294 08:32 37.11 2 238.0 06 13.0 2 .7 - - * * * - -297 08 :56 37 . 178 238 . 041 4 . 9 2 . 5 - - - - * * *299 09:01 37.03 9 238.1 18 10.8 3 .6 * * * * - -295 09:44 36.91 6 238.3 28 6. 0 3 .3 * * * * *294 10:57 37.15 8 238.0 17 7.2 2 .6 - - * * * - -294 12:54 37.13 7 238.0 65 5.3 3 .1 * * * * - -295 12 :55 37 . 057 238 . 210 6 . 8 2 . 2 - - * - - * *298 13:00 36.88 9 238.3 63 3.7 3 .7 * * * * *300 13:29 36.95 2 238.2 75 13.9 3 .0 * * - - * - -295 14:24 36.98 4 238.1 90 15.0 3 .7 * * * * *298 22:01 36.98 7 238.1 91 14.7 3 .7 * * * * - -294 22:14 37.05 5 238.1 21 12.7 4 .4 * * * * *

Julian Hour: Latitude Longitude Depth Mag. Site Site Site Site SiteDay Minute (N) (W) (kin) (m0) S1 $2 $3 $4 $5


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1130 E. H . Field and K. H. Jacob

the t rad i t ional spect ral rat ios . The uncer tain ty est imates , onth e o th e r h an d , can b e s i g n i fi can t ly d i f f e r en t d ep en d in g o nth e d a t a we ig h t i n g sch em e u sed . I f a l l d a t a a r e we ig h t edeq u a l l y , t h en t h e u n cer t a in t ie s o b t a in ed f ro m th e g en era l i zedin v er s io n ap p ro ach can b e u p t o a f ac to r o f - -0 .7 l e s s t h anth o se o f t r ad i t i o n a l sp ec t r a l r a t i o s . Th e d a t a we ig h t i n gsch em e in t ro d u ced b y An d rews (1 9 8 6 ) , o n t h e o th e r h an d ,i s sh o wn to p ro d u ce so m ewh at a rb i t r a ry u n cer t a in t i e s an dwe ad v o ca t e an a l t e rn a t i v e ap p ro ach . F in a l l y , t h e ex p res -s i o n s p ro v id ed i n t h i s sec t i o n can b e u sed fo r d ep lo y m en tp l an n in g p u rp o ses ; t h ey in d i ca t e h o w m u ch d a t a n ee d t o b eco l l ec t ed in o rd er t o o b t a in es t im a t es o f a g iv en l ev e l o fp rec i s i o n . Read er s wh o a re n o t i n t e r es t ed i n t h e d e t a i l s o fthe spect ral - rat io technique can easi ly sk ip th is sect ion , toth e d i scu ss io n o f t h e n o n - re fe ren ce - s i t e -d ep en d en t t ech -n iques, wi thout a loss of cont inu i ty .

T rad i t i o n a l S p ec t r a l Ra t i o sS u p p o se we h a v e a r e f e ren ce si t e ( i = R) t h a t we as -

sum e to have a neg l ig ib le s i te respo nse ( In SR = 0) . I f thein t e rs t a t io n sp ac in g i s sm al l co m p ared t o t h e ep i cen t r a l d is -tances , so that P o ~ - P R ; t h en t h e s i te r esp o n se a t eac h s i t ecan b e es t im at ed f ro m

In S R = .= In O R j ~ - " -~ j=l (In O o -- In OR1), (3 )

wh ere J i s t h e n u m b er o f ev en t s r eco rd ed a t b o th s i te i an dth e r e f e ren ce s i t e (wh ich m ay v ary b e tween s i t e s ) , an d weh av e d ro p p ed t h e f r eq u en cy d ep en d en ce fo r n o t a ti o n a l s im -p l i c it y . Eq u a t i o n (3 ) co n s t i tu t es t h e g eo m et r i c - av erag e sp ec-t ral rat io . Using th is es t imate assumes that the observat ionsa re co n s i s t en t wi th a l o g n o rm al d i s t r i b u t i o n , wh ich we ex -p l ici t ly tes t la ter in th is ar t ic le . I f the reference s i te has an o n -n eg l i g ib le s i t e r esp o n se , t h en t h e sp ec t r a l r a t i o s b eco m ere l a t i v e s i t e -r esp o n se es t im at es . T h e s t an d ard d ev i a t i o n o f InSsR, whic h repre sen ts the scat ter o f ind iv idua l spe ct ral rat ios ,i s e s t im a t ed f ro m

asR = [(In 0 o - In O~j) - In SsR] (4 )st d SiTh e s t an d ard d ev i a t i o n o f t h e m ean (o r s t an d ard e r ro r ) ,wh ich r ep resen t s t h e u n cer t a in ty wi th r esp ec t t o t h e t ru em ean , i s g iv en as

1a SR = - a ~ ( 5 )Si ] - ~ s t d Si ", / JNote that s taas , and as f are fo r In S i rather tha n fo r S i i t sel f .

Th e d i s t i n c ti o n b e tween t h e s t an d ard d ev i a t i o n an d t he .s t an d ard d ev i a t i o n o f t h e m ea n i s im p o r t an t fo r h azard an a l -y ses . I f t h e sca t t e r o b se rv ed am o n g t h e i n d iv id u a l sp ec t r a lr a t i o s r esu l t s f ro m so m e k in d o f n o i se i n o u r o b se rv a t i o n s(e.g . , due to scat ter ing f rom lateral heterogenei t ies , which

cau ses r an d o m d i f f e r en ces b e twee n t h e t ru e i n p u t an d t h ere fe ren ce- s i t e o b se rv a t i o n ) , t h en o n e w o u ld w an t t o m in i -m ize t h i s so u rce o f u n cer t a in ty b y u s in g t h e s t an d ard d ev i -a t i o n o f t h e m ea n ( eq u a t i o n 5 ) . On t h e o th e r h an d , i f t h escat ter in spect ral rat ios i s p roduced by an in t r insic var ia-b i l i ty i n t h e s i te r esp o n se i t se l f (b y v i r t u e o f d i f f e r en t i n c i -d en ce an g l es , b ack az im u th s , an d p e rh ap s wav e t y p es f ro mth e d i f f e r en t so u rces ) t h en o n e wo u ld wan t t o acco u n t fo rth is by apply ing the s tandard deviat ion (equat ion 4) to theu n cer t a in ty es t im a t es . F u r th e rm o re , t o co n v in c in g ly i d en t i fya n o n l i n ea r s i t e r esp o n se f ro m sp ec t r a l r a t io s , i t w i l l b e n ec -essa ry t o sh o w th a t t h e r esp o n se fo r t h e ev en t i n q u es t i o n i so u t s i d e t h e r an g e o f sca t t e r r ep resen t ed b y t h e s t an d ard d e -v iat ion .

Th e ab o v e an a ly s i s im p l i c i t l y as su m es t h a t th e l ev e l o fnoise in eac h observa t ion , say Cro0, s the sam e for each e ven t( a o u = a o , ) . I f we fu r t h e r a s su m e th a t t h e l ev e l o f n o i se i st h e sam e a t b o th s i te i an d t h e r e f e ren ce s i t e ( a o , = a o R =a o ) , an d t h a t t h e two a re u n co r re l a t ed , t h en we h av e

~ R = E ~ , + ~ , ]o . ~ f i ~ o ,td Sian d eq u a t i o n (5 ) can b e r ewr i t t en as

a g ~ = ~ a o . ( 6 )

This g ives the uncer tain ty of t rad i t ional spect ral rat ios int e rm s o f t h e n o i se i n t h e d a t a an d t h e n u m b er o f r a t i o s av -e rag ed . We wi l l l a t e r co m p are t h i s wi th t h e u n cer t a in ty o fg en era l i zed i n v er s io n sp ec t r a l r a ti o s .

P lo ts o f t rad i t ional spect ral rat ios for the t ransverseco m p o n en t a t two o f o u r Eas t Bay s i t e s (S 1 an d $ 3 r e l a t i v eto $ 4 ) a r e sh o wn in F ig u re 2 a . Th ese o b se rv a t i o n s , a s we l las t h o se fo r t h e r ad i a l co m p o n en t an d t h e o th e r two s i t e s ($ 2an d $ 5 ) , h av e b een sh o wn an d d i scu ssed p rev io u s ly (F i e lde t a l . , 1992, 1994b) . These est imates , and al l o thers shownin t h i s a r t i c l e , were co m p u ted f ro m 2 0 - sec win d o ws b eg in -n in g a t t h e sh ear -wav e a r r i v a l . On ly n eg l i g ib l e d i f f e r en ceswere o b t a in ed b y u s in g 1 0 - sec win d o ws . Th e sp ec t r a l e s t i-m at es were co m p u ted u s in g t h e h ig h - r eso lu t i o n m eth o d o fTh o m so n (1 9 8 2 ) an d P ark e t a l . (1 9 8 7 ) , wh ich p ro d u ces aneq u iv a l en t sm o o th in g wid th o f 0 .4 Hz . Th e d o t t ed l i n es i nFigure 2a, w hich rep resen t the ___2~tda R limits, defin e theran g e wh ere an ad d i t i o n a l sp ec t r a l r a t i o wo u ld h av e 9 5 %l ik e l i h o o d o f r esid in g . Th ese r ev e a l a l a rg e d eg ree o f v a r i -ab i l i t y am o n g t h e i n d iv id u a l sp ec t r a l r a t i o s . Th e d ash edl i n es , wh ich r ep resen t th e 9 5 % co n f id en ce l im i t s o f t h e m ean(___2asR) , demonst rate that the s i te response can be fai r lywel l co n s t r a in ed wh en sev era l r eco rd in g s a r e av a i l ab l e .Th e s i t e r esp o n se a t $ 3 i s d o m in a t ed b y t h ree p ro m in en tp eak s , wh ich r ep resen t t h e f i rs t th r ee r eso n an t m o d es o f t h ea l l u v iu m l ay er . Th e r esp o n se a t S 1 i s ch arac t e r i zed b y fo u rp ro m in en t p eak s , wh ich r ep resen t co u p l ed r eso n an ces b e -tween t h e m u d an d a l l u v iu m l ay er s a t th i s s i te . Bo th o f t h ese


5/17

Various Site-Response Estimation Techniques 1 1 3 1

a ) t r a d i t io n a l s p e c t r a l r a t i o s

4 o 4 1. , t ; : : : ~ . . . . . : : ~ : . i - / : ' : . ~ . : , i~ ' " . . " / . ' , " / " " " , ' . : : Y ' , : : : " ' . . ~ : , : : : : '. ~ i ' ~ - " < L - q ' : ' ~ i , ~ ' ! , ,! . ~ - - . , ' ' ' , : : : ' , , , " ' , , ~ , ~

2 . 0 -" . -... :.:.:. , ,;,. ,1 . 0 " . . . . . . " . ' " ' " :: . . . . . . ' ~ " ' : " ; ":- : . ~ " ~ . ! ~ , !0 . 4 ~ ' ,' ~ : , 9 i.0 . 2 I , , , "l '"~" '/

4 0 -2 0

" ~ 1 0; ~ 4 . 0

2 . 01 .00 . 40 . 2

b ) A n d r e w s ' w e i g h t s c h e m e e s t i m a t e sS 1

I I I I I I

$3

I I I I I I

c ) u n it w e i g h t s c h e m e e s t i m a t e s4 0 - 1 " " : S 1 -2 0 ~ i : ~ ( : : i ' } -~o--[ , ~ - . ~ : ~ i ~ : ~

4 . 0 - J ~ - . ~ " . . . " ~ . ' . . ' . : . ~ . . . . . . . . : ,2 .0 - ' [~ E N . , ~ : . i : ' , ] : " ! ~ : , , ]0 . 4 " J ~ ' : ' ~ 57~;!0 . 2 I ~ , , , ' " " : ,

$3 ~ ," . . . . " t / ~ " . . . . . ' t ~ " . : ' " " " .. , ' : ' " :~ : ' : . . :~ " :U ]t / ~ i " " " . " . : - t - ; ; 2

; - . , :

I I I I I I

d ) p r e fe r re d w e i g h t s c h e m e e s t i m a t e s

4 . 0

0. 4 . - ,,...,,,0 .2 , , I

0 . 2 0 . 4 1 . 0 2 . 0 4 . 0 1 0 2 0 0 . 2 0 . 4 1 . 0 2 . 0 4 . 0 1 0 2 0frequency (Hz) frequency (Hz)Fi gu re 2 . Spectra l-rat io es t imates o f s i te response for the t ransverse com ponen ts a tS 1 and $3, re lat ive to the reference s i te $4. The dot ted an d dashed l ines represent 2standa rd deviation an d standard devia tion of the mea n limits, respectively. (a) Tradi-tiona l spectral-ratio estimates. (b) Generalized-inversion spectral ratios using An dre w s'data weight ing scheme. ( th is does not have 2 s tandard deviat ion l imits because thedata uncertainties are specified explicitly rather than estimated fro m the data) (c) Ge n-eralized-inversion spectral ratios w here all data have been given unit weight. (d) Gen -eralized-inversion spectral ratios obtaine d using our preferre d we ightin g scheme.


6/17

1132 E .H . F ield and K. H. Jacob

o b s e r v e d r e s p o n s e e s t i m a t e s w e r e f o u n d t o a g r e e w e l l w i t ht h e f r e q u e n c y - d e p e n d e n t c h a r a c t e r o f o n e - d i m e n s i o n a lm o d e l p r e d i c t i o n s b a s e d o n i n d e p e n d e n t g e o t e c h n i c a l d a t a(F i e ld e t a L , 1 9 92 ). H o w ev er , th e sp ec t r a l - r a t i o am p l i t u d esa r e g r e a t e r th a n t h o s e p r e d i c t e d b y a f r e q u e n c y - i n d e p e n d e n ts c a l in g f a c t o r o f a b o u t 2 . S c a t t e ri n g o f e n e r g y f r o m l a t e ra lb o u n d a r i e s o f t h e s e d i m e n t s m a y e x p l a i n t h i s d i s c r e p a n c y .H o w e v e r , t h e r e is n o o b v i o u s e v i d e n c e o f s u c h w a v e p r o p -ag a t io n i n t h e t im e se r i e s o b se rv ed a t t h e sed im en t s i t e s(F i e ld , 1 9 9 4 ) . T h a t i s , ex cep t fo r t h e f ac to r o f 2 d i sc r ep -a n c y , o n e - d i m e n s i o n a l p r e d i c t i o n s r e p r o d u c e t h e o b s e r v e dw a v e f o r m s a n d d o n o t l a c k a n y l a t e r - a r r i v i n g e n e r g y t h a tm ig h t b e a t t r i b u t ed t o l a t e r a l p ro p ag a t io n w i th in t h e sed i -m e n t s .

G e n e r a l i z e d - I n v e r s i o n S p e c t r a l R a t i o sT h e m e t h o d o f s p e c t r a l r a t i o s w a s r e c a s t b y A n d r e w s

( 1 9 86 ) i n to a g e n e r a l i z e d - i n v e r s e ( G I ) p r o b l e m b y s o l v i n geq u a t io n (2 ) fo r a ll t h e s i t e - e f f ec t an d so u rce t e rm s s im u l -t a n e o u s ly . A p a t h e f f e c t r e p re s e n t i n g g e o m e t r i c a t t e n u a ti o nw a s s p e c i f ie d a s

P i j = In = - l n ( ru) ,

w h e r e ri j i s t h e k n o w n h y p o c e n t r a l d i s t a n c e b e t w e e n t h e f l hev en t an d th e i t h s i t e . W i th t h i s ap p l i ed a s a co r r ec t i o n t ot h e d a t a , t h e s y s t e m o f e q u a t i o n s c a n b e r e w r i t t e n a s

l n E j + l n & = l n O k + ln r k ,

w h e re k r ep resen t s t h e k th o b se rv a t io n , an d i an d j a r e n o wim p l i c i t f u n c t io n s o f k. F o l lo w in g th e n o t a t i o n o f M e n k e(1 9 8 9 ) , t h i s can b e w r i t t en i n m a t r ix fo rm as

G m = d , ( 7 )w h e re d i s a d a t a v ec to r co n t a in in g t h e g eo m et r i c - a t t en u a -t i o n - c o r re c t e d o b s e r v a t i o n s , m i s a m o d e l v e c t o r c o n t a i n i n gth e J + I u n k n o w n so u rce an d s i t e t e rm s , an d G i s t h e d a t ak e r n e l m a t r ix , w h i c h h a s J + I c o l u m n s a n d K r o w s ( K c a nb e u p t o J . I t o t h e ex t en t t h a t a l l ev en t s a r e r eco rd ed a t a l ls i t e s ) . G i s a sp a r se m a t r ix , h av in g o n ly tw o n o n ze ro e l e -m e n t s p e r r o w o r c o l u m n .T h e w e ig h ted l eas t - sq u a res so lu t i o n i s o b t a in ed a s t h a tw h i c h m i n i m i z e s t h e d i f f e r e n c e b e t w e e n t h e o b s e r v e d a n dp r e d i c t e d d a t a

Z 2 = ~ 7~ ' ( l n O k + l n r k ) - - ( l n E j + l n S i ) ] , ( 8 )k = 1 O ' o /

w h e r e a o k i s t h e n o i se o r u n ce r t a in ty o f t h e k th o b se rv a t io n( I n O ~ ) .

T o t h e e x t e n t th a t K > J + I , t h e s y s t e m o f e q u a t i o n sa p p e a r s o v e r d e t e r m i n e d . H o w e v e r , t h e r e i s a c t u a l l y o n e u n -

d e t e r m i n e d d e g r e e o f f r e e d o m , w h i c h m e a n s t h a t a ll t h e s i te -r e s p o n s e t e r m s c a n b e m u l t i p l i e d b y s o m e a r b i t r a r y f u n c t io no f f r e q u e n c y , p r o v i d e d a l l t h e s o u r c e t e r m s a r e d i v i d e d b yth e sam e fu n c t io n . T h i s t r ad e -o f f w i l l n o t i n f lu en ce t h e v a lu eo f Z 2 i n eq u a t io n (8 ). A n d rew s (1 9 8 6 ) c o n s t r a in ed t h i s u n -r e s o l v e d d e g r e e o f f r e e d o m b y d e t e r m i n i n g e a c h s i t e - r e -s p o n s e t e r m r e l a t iv e t o t h e n e t w o r k a v e r a g e . B o n a m a s s a a n dM u e l l e r (1 9 8 9 ) , an d o th e r s s i n ce ( e . g . , B o a tw r ig h t e t a l . ,1 9 91 a ; H a r t z e l l, 1 9 92 ) , r e m o v e d t h e d e g r e e o f f r e e d o m b yco n s t r a in in g t h e am p l i f i ca t i o n a t a r e f e r en ce s i t e t o b e ze ro( In S R = 0 ) . T h i s l a t te r ap p ro ach , w h ich w e sh a l l ad o p t h e re ,cau ses t h e r e su l t i n g s i t e - r e sp o n se e s t im a tes t o b e r e l a t i v e t oth a t a t t h e r e f e r en ce s i t e . S in ce t h i s i s an a lo g o u s t o t h e t r a -d i t i o n a l sp ec t r a l - r a t i o e s t im a tes , w e ca l l t h ese g en e ra l i zed -in v e r s io n (G I ) sp ec t r a l r a t i o s .

T h e m o d e l - p a r a m e t e r u n c e r ta i n t ie s a r e o b t a i n e d a s t h es q u a r e r o o t o f t h e d i a g o n a l e l e m e n t s i n t h e m o d e l c o v a r i a n c em at r ix . I f a l l t h e o b se rv a t io n s h av e an eq u a l l ev e l o f n o i se( a o k = a o ) , t h e n t h e m o d e l c o v a r i a n c e m a t r i x i s g i v e n a s

[ c o v m ] = [ G r G ] - ' a g , ( 9 )w h e r e [ G r G ] - ~ i s o f t e n r e f e r r e d t o a s t h e u n i t c o v a r i a n c em a t r i x ( M e n k e , 1 9 89 ) . T h i s e q u a t io n s h o w s t h a t th e m o d e l -p a ra m e te r u n ce r t a in ti e s a r e d i r ec t l y r e l a t ed t o t h e u n ce r t a in -t ies in the data .

O n e m i g h t w o n d e r w h e t h e r t h e g e n e r a l i z e d - i n v e r s i o nap p ro a ch p ro d u ces l e s s u n ce r t a in ty t h an t h e t rad i t i o n a l sp ec -t r a l - r a t i o e s t im a tes . F o r s im p l i c i t y , a s su m e th a t a l l t h e o b -s e r v a t io n s h a v e t h e s a m e l e v e l o f n o i s e ( a o k = ao ) . T h en , i fa l l J ev en t s a r e r eco rd ed a t a l l I s i t e s , t h e so u rce - an d s i t e -e f f ec t u n ce r t a in t i e s a r e

o'GI ~ ~Si 0 (lO)

~ I + I - 1a ~ l = J I a ' ( ] 1 )r e s p e c t i v e l y , d e t e r m i n e d b y e x a m i n i n g t h e u n i t m o d e l c o -v a r i a n c e m a t r i x f o r a w i d e r a n g e o f J / I c o m b i n a t i o n s . C o m -p ar in g eq u a t io n s (1 0 ) an d (6 ) , w e see t h a t t h e s i t e - r e sp o n seu n ce r t a in t i e s fo r t h e G I ap p ro ach a r e i d en t i ca l t o t h o se fo rth e t r ad i t i o n a l sp ec t r a l r a t i o s . O n e a l so f i n d s t h a t t h e s i t e -r e s p o n s e e s t i m a t e s t h e m s e l v e s a r e e q u i v a l e n t , p r o v i d e d t h eg eo m et r i c - a t t en u a t io n co r r ec t i o n i s e i t h e r n eg l ig ib l e , o r h asb een ap p l i ed t o t h e t r ad i t i o n a l sp ec t r a l r a t i o s a s w e l l .

T h e m a i n a d v a n t a g e o f u s in g t h e G I a p p r o a c h a r i s e sw h e n n o t a l l ev en t s a r e r eco rd ed a t a l l s i t es . I n t h i s case , t h eG I e s t i m a t e s a r e m o r e e f f i c ie n t ( h a v e l e s s u n c e r t a i n ty ) t h a nth o se o b t a in ed v i a t r ad i t i o n a l sp ec t r a l r a t i o s . F o r ex am p le ,c o n s i d e r t h e c a s e w h e r e t h e r e f e r e n c e s i t e h a s r e c o r d e d e v e r ye v e n t . T h e s i t u a t i o n w h e r e t h e G I a p p r o a c h w i l l b e m o s tb en e f i c i a l w i l l o ccu r w h en a l l J ev en t s a r e r eco rd ed a t a l l


7/17

V a r i o u s S i t e - R e s p o n s e E s t i m a t i o n T e c h n i q u e s 1133

si tes exc ept one (say the i = 1 s i te , S~) , wh ich record s on lyo n e ev en t (say th e j = 1 ev en t , E0 . F ro m eq u a t i o n (6 ) wek n o w th a t t h e u n cer t a in ty o f In S s R ( the t rad i t ional spect ralrat io for s i te 1 ) wi l l be a sR = ~ a o . By ex am in in g t h e u n i tc o v a r i a n c e m a t ri x o v e r a w i d e r a n g e o f J / 1 , w e f o u n d t h att h e u n cer t a in ty fo r In S ~x ( t h e g e n era l i zed i n v er s io n es t im at efor s i te 1 ) i s g iven by

~ G I ~ ) - - ~s~ a2o + " -_ a . ( 1 2 )

Th i s ex p re ss io n can b e u n d er s to o d i n tu i ti v e ly f ro m th e fo l -lowing . S ince the s i te response at s i te 1 i s ob tained as In$1 = In O11 - In El , the term in brack ets can be understo odas a2o + a z~, where ae~ i s ob tained f rom equat ion (11) wi thI - 1 subst i tu ted for I (s ince on ly I - 1 s i tes rec orde d al lo f t h e ev en t s ) . Wi th eq u a t i o n (1 2) , an d wi th ~R = ~ /2 ao ,t h e r e l a t i v e u n cer t a in t i e s b e tween t h e tw o a p p ro ach es fo r InS~ can b e ex p ressed as

o ' '[~Sl 1 + (13)a s f - J ( I - - 1 ) / J "Ag a in , t h i s ex p resses t h e r e l a t i v e u n cer t a in t i e s i n t h e casewh ere o n e s i t e h as r eco rd ed o n ly o n e ev en t , b u t a l l o th e r( I - 1 ) s it e s h av e r eco rd a l l J ev en t s . F o r a g iv en I an d J ,t h i s r ep resen t s t h e m ax im u m b en ef i t t h e GI ap p ro ach canh av e o v er t h e t r ad i t i o n a l sp ec t ra l r a ti o s . F o r t h e m o re r ea l -i s t i c case wh ere each s i t e r eco rd s so m e n u m b er o f ev en t s ,t h e r e l a t i v e u n cer t a in t i e s wi l l b e so m ewh ere b e tween u n i t yan d t h a t g iv en b y eq u a t i o n (1 3 ) . F o r ex am p le , i n o u r Eas tBay s t u d y we r eco rd ed ev en t s o v er f i v e s i t e s so t h e GI ap -p ro ach u n cer t a in t i e s wi l l b e so m ewh ere i n t h e r an g e 0 .8 as f< Ors, =< aSl . Fu rthe rm ore , equa tion (13) im plies th at fo r ar-b i t r a ry I an d J t h e m a x im u m p o ss ib l e ad v an t ag e ( fo r t h e caseof an infinite nu m be r of ev ent s an d sites) is O~s~ = a s f / ] 2 ~ -0.7asSa . Th ere for e, in gene ral ,

0.7 ~R < Ofs[=< ~ f , ( 1 4 )wh ich sh o ws t h a t t h e g en era l i zed i n v er s io n ap p ro ach t o s i t e-respo nse est im at ion offers a s l igh t , bu t perhap s insign i f ican t ,im p ro v em en t o v er t r ad i t i o n a l sp ec t r a l - r a t i o es t im a t es . Th eim p ro v em en t can b e g rea t e r i f th e r e f e re n ce s i te h as n o t r e -c o r d e d e v e r y e v e n t.

An es t im at e o f th e o b se rv a t i o n a l u n cer t a in t i e s ( ao ) m u s tb e av a i l ab l e t o ap p ly t h e we ig h t ed l eas t - sq u ares i n v er s io n .In g en era l , o n e wo u ld ex p ec t t h e d a t a u n cer t a in ty t o v a rywi th t h e d i s t r i b u t io n o f so u rces an d r ece iv e r s , wi th t h e t y p eo f sp ec t r a l - es t im a t io n t ech n iq u e ( i . e . , sm o o th in g ) ap p l i ed ,an d wi th f r eq u en cy . An d rews (1 9 8 6 ) ap p l i ed th e fo l l o win g :

aok = m ax \Oh '

wh e re N~ i s t h e am p l i t u d e sp ec t ru m o f a sam p le t ak en im -m ed ia t e ly b e fo re t h e a r r i v a l o f t h e wav e o f i n t e r es t . Th i sap p ro ach , wh ich h as su b seq u en t l y b een ap p l i ed b y o th e r s(Bo a twr ig h t e t a l . , 1 9 9 1 a ; S eek in s an d Bo a twr ig h t , 1 9 9 4 ;Marg h er i t i e t a l . , 1 9 9 4 ), i s d es ig n ed t o d o wn -w eig h t r eco rd sth a t h av e a p o o r o b se rv ed s i g n a l - t o -n o i se r a t io , wh i l e a t t h esam e t im e r eq u i r i n g a l l d a ta t o h av e a m in im u m u n cer t a in tyo f 0 .5 . Co n s id e r t h e ca se o f a c l ean d a t a se t i n t h a t N J O k

a comparison and test of various site-response estimation techniques,

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