Hao, H. Oliveira, C.S. Penzien - 1989 - Multiple-Station Ground Motion Processing and Simu

7/27/2019 Hao, H. Oliveira, C.S. Penzien - 1989 - Multiple-Station Ground Motion Processing and Simu

1/18

Nuclear Engineering and Design 111 (1989) 293-310 293North-Holland, Amsterdam

M U L T I P L E - S T A T I O N G R O U N D M O T I O N P R O C E S S I N G A N D S I M U L A T I O N B A S E DON SMART-1 ARRAY DATAH. HAO 1, C.S. OLIV EIR A 2 and J . PEN ZIE N 3i University of California, Berkeley, US A2 Laboratfrio Nacional de Engenharia Civil, Av. Bra sil, 1799, Lisboa, Portug al-~ University of California, Berkeley, US A and Eastern International Engineers, Inc., Lafayette, California, U SAReceived 21 March 1988

This paper presents a method of generating multiple-support inputs for any given set of n surface locations having spacecoordinates x i and y~ (i = 1, 2 . .. . n) which are compatible with the main wave propagation properties observed in the StrongMotion Array Taiwan (SMART-l). Based on data collected in this array during two different earthquakes, a coherencyfunction "/~j(dL, d~, f) is developed for pairs of stations, i and j, as a continuous function of frequency f and projectedseparation distances d L and d~ in the longitudinal direction of preferential wave propagation and in the transverse direction,respectively. Using three different time windows, changes in apparent wave velocity and power spectral density are determinedas functions of frequency by averaging such functions over the entire array. Finally, an average shape function for the entiretime-history is obtained. This entire process was repeated for each of the three components of motion. Coherency between EWand NS components at each station was also investigated. The results presented herein are essentially valid for stationseparations up to 400m. An algorithm and procedures for generating spatially correlated ground motions which incorporate allof the above mentioned features are presented.

1 . I n t r o d u c t i o n

For some time [1], conside ration has been given towave propagation as it affects the response of structuresto strong ground motions. It has been recognized thatspatial variations induce rotational component s at indi-vidual sites and cause differential movements along thefoundations of extended structures. Until recently, allsuch considerations have been based primarily on theo-retical developments and on observations of overallstructural behavior during earthquakes [2-4]. The studyof strong ground motion characteristics acquired a newdimension, however, when high density arrays of accel-erometers were developed. These arrays permit for thefirst time, among other features, the determination ofspatial correlation of ground accelerations and displace-ments, the evaluation of variability of mot ion within asmall area, the computation of ground strains, and the

determinations of torsional and rotational componentsof ground motion. These characteristics are of impor-tance in the definiti on of ground mot ions to be used forinputs into engineered structures, especially extendedstructures such as buried pipelines, tunnels, bridges, anddams. They are also of importance in defining theinputs to tall structures such as high rise buildings,towers, and chimneys, and the large rigid continuousfoundations such as those of nuclear power plants.

In the present paper, selection and quantificat ion ofthe mai n parameters controlling wave propagation areof primary concern. Using mo dem techniques for wavepropagation detection and the SMART-1 (Strong Mo-tion Array Taiwan) data recorded during two earth-quakes, a methodology is developed for generating timeseries which are compatible with the correlation proper-ties observed in the data. Such time series can be usedas the design multiple input to imp ortant structures.

Graduate Student.2 Researcher.3 Professor Emeritus of University of California and BoardChairman of Eastern International Engineers, Inc.

1 .1 . D e s c r ip t i o n o f S M A R T - 1 a r r a y a n d i t s g e o l o g ic a ls e t t i ng

The SMART-1 array with its 37 stations covering anapproximate area of 12 km2 is located in the northeast

0029-5493/89//$03.50 Elsevier Science Publishers B.V.(North-Holland Physics Publishing Divis ion)


2/18

29 4 H . H a o e t a l. / M u l t i p l e - s t a t i o n g r o u n d m o t i o n p r o c e s s i n gS M A R T t A R R A Y C O N F I G U R R T IO N

A

A

H.-.0t

1-01 7A A A A

E'-01

I I IO. 1 . 2 .

kRE-O~

/ A T , O N

S T A T I O Ni / V o p p

Fig. 1 . Array conf igurat ion and locat ion of recording s ta t ions .

c o r n e r o f T a i w a n n e a r t h e c i t y o f L o t u n g . I t s i n s t a l la -t i o n b e g a n i n S e p t e m b e r 1 9 8 0 a n d w a s c o m p l e t e d b yA u g u s t 19 8 2 . B y D e c e m b e r 1 9 8 7 , 5 1 e a r t h q u a k e e v e n t sh a d b e e n r e c o r d e d i n t h i s a r r a y , p r o d u c i n g r e c o r d s w i t hp e a k g r o u n d a c c e le r a ti o n s , P G A , r a n g i n g f r o m 1 0 t o3 5 0 c m / s 2 . A b o u t 3 5 0 0 re c o r d s a r e n o w a v a i l a b l e . T h ea r r a y c o n s i s t s o f 1 2 t r i a x i a l s t r o n g - m o t i o n a c c e l e r o m e -t e r s l o c a t e d i n e a c h o f t h r e e c o n c e n t r i c c i r c u l a r r i n g sh a v i n g r a d i i o f 2 0 0 m , 1 0 00 m , a n d 2 0 0 0 m , a n d o f o n et r i a x i a l a c c e l e r o m e t e r l o c a t e d a t t h e c e n t e r o f t h e a r r a y ,f i g . 1 . T h e t h r e e r i n g s w i t h 1 2 e q u a l l y s p a c e d s t a t i o n s ,

n u m b e r e d 1 t h r o u g h 1 2 cl o c k w i se a re n a m e d I (i n n e r) ,M ( m i d d l e ) a n d O ( o u t e r ) , r e s p e c t i v e l y ; t h e c e n t e r s t a ti o ni s n a m e d C - 0 0 . T h e d i s t a n c e s b e t w e e n s t a t i o n p a i r s v a r yf r o m a m i n i m u m o f a p p r o x i m a t e l y 1 0 5 m t o a m a x i -m u m o f 4 0 0 0 m . I n J u n e 1 9 8 3 , s t a ti o n s E - 0 1 a n d E - 0 2 ,l o c a t e d r e s p e c t i v e l y a t 2. 8 k m a n d 4 . 8 k m s o u t h o f C - 0 0 ,w e r e a d d e d t o t h e a r r a y . T h e s e l a t t e r t w o s t a t i o n s a r ei n s t a l l e d o n r o c k w h i l e a l l o t h e r s a r e p l a c e d a t t h es u r f a c e o f a r e l a t i v e l y f l a t a l l u v i u m h a v i n g e l e v a t i o n sr a n g i n g f r o m 2 . 4 m t o 1 8 .1 m o v e r t h e a r r a y . F o r a m o r ec o m p l e t e d e s c ri p t i o n o f t h e S M A R T - 1 s i te c o n d i t io n s ,

T ab l e 1Mai n cha r act e ri s ti c s o f SM A R T - 1 even t s unde r s t udyE ven t O r i g i n Mag D i s t . A z i m ut h

t i me ( M s ) ( ki n) ( N - E ) D e p t h P G A ( m a x ) P G D ( m a x )( k m ) ( c m / s 2 ) ( cm ): 2 4 1 9 8 3 . 0 6 . 2 4 6 .7 8 409:06:46.3:~45 1986.11.14 7.8 7921:10:01.2

124

175"48 Hor iz- 65 Hor iz- 2 .0Vert- 15 Vert- 0.5

7 Hor iz-238 Hor iz- 8 .5Ve rt- 104 Vert- 2.5


3/18

H. H ao et al. / Multiple-station ground motion processing 295172t6t'M

t.t.IDI - -F---

M

,o.o T. ,~dfX l ,

121.0 122.0 2 3. 0LONG I UBE

Fig. 2. Epicentral lo catio n of events under study.a n d i t s s t a t i o n c o o r d i n a t e s a n d i n s t r u m e n t a t i o n , s e eB o l t e t a l . [ 5 ] a n d A b r a h a m s o n [ 6 ] .1 .2 . S u m m a r y o f m a i n p r o p e r t i es o f t h e r e c o rd e d e a r t h -q u a k e s

T o s t u d y w a v e p r o p a g a t i o n i n t h e S M A R T - 1 b a s i na n d t o c h a r a c t e r i z e t h e m a i n p r o p e r t i e s a s s o c i a t e d w i t h

i t , t w o e a r t h q u a k e s l i s t e d i n t a b l e 1 a s e v e n t s n u m b e r s2 4 a n d 4 5 w e r e s e l e c t e d. A s i n d i c a t e d , t h e s e e v e n t s h a dm a g n i t u d e s o f 6 . 7 a n d 7 . 8 , e p i c e n t r a l d i s t a n c e s o f 8 4a n d 7 9 k m , f ig . 2 , p e a k g r o u n d a c c e l e r a t i o n s ( P G A ) o f6 5 a n d 2 3 8 c m / s 2, a n d p e a k g r o u n d d i s p l a c e m e n t s( P G D ) o f 2 . 0 a n d 8 . 5 c m . E v e n t N o . 4 5 , t h e l a r g e s te a r t h q u a k e t o t r i g g e r t h e S M A R T - 1 a r r a y s i nc e i t si n s t a l l a t i o n i n 1 98 0, c a u s e d c o n s i d e r a b l e d a m a g e i nn o r t h T a i w a n . T h e e v e n t s r e f e r e n c e d i n t a b l e 1 p r o -d u c e d h i g h ly s p a t i a l l y c o r r e l a t e d g r o u n d m o t i o n d u r i n gt h e a r r i v a l o f t h e S w a v e s [ 7] . F i g . 3 p r e s e n t s t h e N - Sc o m p o n e n t s o f a c c e l e r a t i o n re c o r d e d d u r i n g e v e n t sn u m b e r s 2 4 a n d 4 5 ( F o r m o r e d e t a i l e d i n f o r m a t i o n o nt h e s e e v e n t s s e e O l i v e i r a [ 7] , D a r r a g h [ 8] , a n d B o l t a n dC h i ou [9 ] . )

2 . T h e o r e t i c a l c o n s i d e r a t i o n s o n w a v e p r o p a g a t i o n a n dm u l t i - s i g n a l p r o c e s s i n g

2.1 . Bas ic theory

T h e g e n e r a l r e p r e s e n t a t i o n o f g r o u n d m o t i o n a t a n ys t a t i o n l o c a t i o n P ( x , y , z ) , c o n s i s t s o f t i m e a n d / o rf r e q u e n c y d o m a i n d e s c r i p t i o n s o f th e t h r e e t r a n s l a t i o n a l

~ o

- 1 0 0I 0 0 I

~ o

- 1 0 0

p r o ce s s in g t im e w in d o w s fo r e v e n t 2 4 a n d e v e n t 45e v e n t 24- NS C -00

e v e n t 24. DN C-O 0

w i n d o w I w i n d o w 2 window 3

30 0 -~ o -

- 3 0 0

~ o -- 3 0 0

e v e n t 4.5 NS C-O 0

e v e n t 4.5 DN C-O 0

I 0 20 3O 4ow i nd o w 1 ~ m e ( xc )window 2 w i n d o w 3

Fig. 3. Acceleration traces for events ~24 and ~45. Time windows used.


4/18

296 H. Hap et al. / Mu ltiple-station ground motion processinga n d r o t a t i o n a l c o m p o n e n t s [ 1 0 , 1 1 ] , w i t h r e s p e c t t o a x e s1 , 2 , 3 as g iven byu , , , = d m , z ~ = v m , a n dU r n + 3 = O r e , / ~ r n + 3 = 0 m ,

m = 1 , 2 , 3 ;

/ l m ~ a m ,and / ~ m + 3 = 0 m ,

( 1 )w h e r e d , v , a n d a d e n o t e d i s p l a c e m e n t , v e l o c it y , a n da c c e l e r a t i o n in t h e m d i r e c t i o n , re s p e c t i v el y , a n d w h e r e0 d e n o t e s r o t a t i o n a b o u t t h e m a x i s . B o t h t h e t i m e a n df r e q u e n c y d o m a i n r e p r e s e n t a t i o n s u ( t ) a n d U ( i ~ ) a rec o m p l e t e a n d u n i q u e , w h e r e U ( i ~ ) i s t h e F o u r i e r t r a n s -f o r m o f u ( t ) . D i s p l a c e m e n t i s t r e a t e d h e r e i n a s a f u n c -t i o n o f b o t h t i m e a n d s p a c e , i. e. , u m = u , ~ ( t , x , y , z ) .

O n e d i r e c t w a y o f a n a l y z i n g d a t a o b t a i n e d a t d i s -c r e t e l o c a t i o n s o n t h e g r o u n d s u r f a c e i s t o c o m p u t e t h ec o r re l a t io n f u n c t i o n R , , , , j ( t 0 , r ) b e t w e e n a n y t w oc o m p o n e n t s u m a n d u , a t l o c a t i o n s i a n d j , r e s p e c -t i v e l y , a s d e f i n e d b y

1 r i o + S tR . . . . . j ( t o , ~ ) = ~ T J , o_a ' u m i ( t ) u , j ( t + r ) d t . ( 2 )F o r s i m p l i c i t y , l e t u s f i r s t c o n s i d e r t h e s a m e c o m p o n e n to f g r o u n d m o t i o n a t e a c h o f t h e d i f f e r e n t l o c a t i o n s . I nt h i s c a s e, s u b s c r i p t s m a n d n c a n b e d r o p p e d , l e a v i n gi n d i c e s i a n d j t o c h a r a c t e r i z e t h e p r o c e s s . A s w i l l b es e e n l a t e r, t h i s s i m p l i f i c a t i o n d o e s n o t d i m i n i s h t h eg e n e r a l i t y o f t h e s u b s e q u e n t d e v e l o p m e n t .

F o r a s t a t i o n a r y r a n d o m p r o c e s s , t h e c r o s s c o r r e l a -t i o n f u n c t i o n d e p e n d s o n l y o n t h e t i m e l a g r , a n d i t c a nb e s h o w n t h a t

r ) = ~ S u ( i ~ ) e i ~' d ~ , ( 3)R o (w h e r e t h e f u n c t i o n S , j ( i ~ ) g i v e n b yS o ( i . ) = l i r n 1f ; ~ ~ 2ui(t) - i "d t

f T / 2 . e + i ' ~ t d r ] J _ r / 2 u J I t ) ( 4 )

i s c a l l e d t h e c r o s s p o w e r s p e c t r a l d e n s i t y f u n c t i o n b e -twe en u~ an d u j. I f i = j , e q s . ( 3 ) a n d ( 4 ) e x p r e s s t h ea u t o c o r r e l a t i o n f u n c t i o n R j / ( r ) a n d t h e p o w e r s p e c t r a ld e n s i t y fu n c t i o n ~ j ( ~ ) , r e s p e c ti v e l y . D u e t o th e n a t u r eo f t h e p r o c e s s , S j j (g a ) i s a r e a l q u a n t i t y , a n d S i j ( i ~ ) i s ac o m p l e x q u a n t i t y h a v i n g t h e p r o p e r t i e s R e ( S i j ) =R e ( ~ i ) a n d I m ( S i j ) = - I m ( ~ i ) . T h is m e a n s t h at th em a t r i x c o n t a i n i n g a l l s p e c t r a l d e n s i t y f u n c t i o n s S ~ j ( i ~ )i s H e r m i t i a n .

T h e c r o s s c o r r e l a t i o n f u n c t i o n R u ( 7 ) c a n b e n o r -m a l i z e d a n d b e m a d e d i m e n s i o n l e s s i n a c c o r d a n c e w i tht h e r e l a t i o n

R O ( ~ ' ) ( 5 )p ,.( ~) = v , ~ v ~ . ( 0 )y i e l d i n g a c r o s s c o r r e l a t i o n c o e f f ic i e n t p o ( r ) . T h e c o r -r e s p o n d i n g r e l a t i o n i n t h e f r e q u e n c y d o m a i n i s

S o ( i ; ~ ) ( 6 ), o ( i ~ ) = ~ ~ _

y i e l d i n g a c o h e r e n c y fu n c t i o n y o ( i ~ ) . B o t h p o ( r ) a n dy i } ( i ~ ) - - t h e c o h e r e n c e - a r e g o o d i n d i c a t o r s o f t h ed e g r e e o f c o r r e l a t i o n b e t w e e n t w o d i f f e r e n t s i g na l s . F o rt o t a l l y c o r r e l a t e d s i g n a l s t h e i r c o r r e s p o n d i n g v a l u e s a r e_+ 1 a n d 1 , r e s p e c t i v e l y , w h e r e a s f o r t o t a l l y u n c o r r e l a t e ds i g n a l s , t h e i r v a l u e s a r e , i n e a c h c a s e , z e r o . T h e t i m es h i f t r f o r w h i c h p o ( r ) r e a c h e s i t s m a x i m u m v a l u e i sr e l a t e d t o t h e t i m e t h a t a g i v e n p o r t i o n o f t h e s i g n a lt a k e s t o t r a v e l f r o m s t a t i o n i t o s t a t i o n j . T h i s s h i f t i nt h e t i m e d o m a i n c o r r e s p o n d s t o a c h a n g e e x p ( i ~ r ) i nt h e f r e q u e n c y d o m a i n , w h e r er = d L / v a p p ( 7 )w i t h / 3 a p p d e n o t i n g t h e s u r f a c e a p p a r e n t w a v e v e l o c i t y a tf r e q u e n c y ~ , a n d d L t h e p r o j e c t e d s e p a r a t i o n d i s t a n c ei n t h e l o n g i t u d i n a l d i r e c t io n .T h e c o n c e p t o f c o r r e l a t i o n a n d c o h e r e n c y c a n a l s ob e u s e d t o i d e n t i f y p r i n c i p a l d i r e c t i o n s o f p r o p a g a t i o no f g i v e n w a v e t r a i n s [ 1 2,5 ,6 ], a n d t h e i r d e g r e e o f p o l a r i -z a t i o n [ 8 ] . T o d o s o , t h e t h r e e d i m e n s i o n a l c h a r a c t e r i z a -t i o n o f w a v e p r o p a g a t i o n a s m e a s u r e d a t a g i v e n s i t es h o u l d b e c o n s i d e r e d . T h e r e f e r e n c e C a r t e s i a n s y s t e mf o r t h e t r a v e l l in g w a v e s c a n b e r o t a t e d i n t o a p o s i t i o n t op r o d u c e , f o r a g i v e n t i m e a n d f r e q u e n c y w i n d o w , m a x i -m u m v a l u e s o f p , j a n d % 2j. T h i s i s a c c o m p l i s h e d b yd e t e r m i n i n g t h e e i g e n v e c t o r s o r p r i n c i p a l d i r e c t i o n s o ft h e m a t r i c e s R , j ( r = 0 ) o r R e[S , j ( i ~ ) ] , i , j = 1 , 2 , 3 . I np h y s i c a l t e r m s i t m e a n s t h a t , f o r a g i v e n t i m e i n t e r v a la n d f r e q u e n c y w i n d o w , t h e p a r t i c l e m o t i o n c a n b ed e s c r i b e d b y a s e t o f t h r e e i n d e p e n d e n t o r t h o g o n a lm o t i o n s w i t h p e a k a m p l i t u d e s p r o p o r t i o n a l t o t h e a x e so f a n e l l i p s o i d . T h e r a t i o s b e t w e e n p a i r s o f e i g e n v a lu e sw h i c h p r o v i d e a m e a s u r e o f t h e d e g r e e o f p o l a r i z a t i o no f t h e c o r r e s p o n d i n g m o t i o n s , g i v e t h e d i m e n s i o n s o ft h e e l l i p s o i d a l o n g p r i n c i p a l a x e s. U s i n g t h e c o h e r e n c ym a t r i x a p p r o a c h , i f t h e r a t i o i s c l o s e t o 1 , i t m e a n s t h a tt h e w a v e i s c i r c u l a r l y p o l a r i z e d , a n d i f c l o s e t o z e r o , i t isl i n e a r l y p o l a r i z e d . T h i s t e c h n i q u e h a s b e e n s u c c e s sf u l ly


5/18

H. Ha o et al. / Multiple-station ground mo tion processing 297u s e d i n t h e i d e n t i f i c a t i o n o f p r e d o m i n a n t w a v e t r a i n s ,t h e i r d i r e c t io n s o f p r o p a g a t i o n , t h e i r a p p a r e n t v e l o c i -t i es , a n d t h e i r t y p e s o f p a r t i c l e m o t i o n .

U s i n g t h e s o c a l l e d F - K a p p r o a c h ( f r e q u e n c y - w a v en u m b e r s p e c t r u m ) , B o l t e t a l . [ 5 ] a n d A b r a h a m s o n [ 6 ]i d e n t i f i e d p r e d o m i n a n t w a v e e n e r g y i n t h e b a n d 0 . 5 - 2 . 0H z p r o p a g a t i n g f r o m t h e e p i c e n t e r . E v e n t h o u g h t h i st e c h n i q u e i s n o t a s s e l ec t i ve as t h e o n e d e v e l o p e d b yL o h a n d P e n z i e n [ 1 3 ] , t h e r e s u l t s a r e o f t h e s a m e o r d e ro f m a g n i t u d e .M o r e s o p h i s t i c a t e d t r e a t m e n t s , s u c h a s s t a c k i n g o rt h e s o l u t i o n o f a 3n ( n b e i n g t h e t o t a l n u m b e r o fs t a t i o n s ) e i g e n v a l u e p r o b l e m , w i l l e x t e n d t h e a b o v em e n t i o n e d c o n c e p t s i n v o l v i n g o n e s t a t i o n - p a i r t o a l ls t a t i o n s i n t h e a r r a y . D e t a i l s o f t h is t y p e o f a n a l y s i s c a nb e f o u n d e l s e w h e r e [ 8 ] .2 . 2. D a t a p r o c e s s i n g

1.00

~ 0.50

coherency between i -06ns and i -07nsevent 45

i0 O0 . . . . . . . . . : . . . . . . . . . , . . . . . . . . . ; . . . . . . . . . i . . . . . . . . . i0 .00 500 !000 1 500 20.00 2500frequency(hz)

averaged coherency values for the inner ringEvent 45 NS compon ent1.00

i 0.50~3

I n t h i s i n v e s t ig a t i o n , s y s t e m a t i c u s e h a s b e e n m a d eo f t h e F o u r i e r S p e c t r u m a n d F r e q u e n c y - W a v e N u m b e rS p e c t r u m A n a l y s i s [1 4] , a s t o o l s to d e t e r m i n e t h e m a i nw a v e p r o p a g a t i o n c h a r a c t e r i s ti c s . T h e d i r e c t a n d i n v e r s eF F T a l g o r i t h m s w e r e u s e d , a f t e r a D C r e m o v a l , t o -g e t h e r w i t h a 6 - p o l e B u t t e r w o r t h b a n d - p a s s f i l t e r a p -p l i e d i n b o t h t h e d i r e c t a n d r e v e r s e t i m e a x e s t o r e m o v ep h a s e d i s t o r t i o n s . C o r r e c t e d a c c e l e r a t i o n , v e l o c i t y , a n dd i s p l a c e m e n t t i m e - h i s t o r i e s w e r e o b t a i n e d u s i n g a l o wf r e q u e n c y c u t o f f a t 0.1 H z a n d a h i g h f r e q u e n c y c u t o f fa t 2 5 . 0 H z . A l l i n t e g r a t i o n s w e r e c a r r i e d o u t i n t h ef r e q u e n c y d o m a i n . T h e r e s u l t s o b t a i n e d u s i n g a 6 - p o l eB u t t e r w o r t h f i lt e r w e r e f o u n d t o b e v e r y s i m i l a r t o t h er e s u l t s o b t a i n e d u s i n g t h e s t a n d a r d O r m s b y f i l t e r u s e db y t h e U n i t e d S t a t e s G e o l o g ic a l S u r v ey ( U S G S ) .

T h e F r e q u e n c y - W a v e N u m b e r S p e c t r u m A n a l y s i s( F - K ) , w a s c a r r i e d o u t u s i n g s t a c k i n g h i g h r e s o l u t i o na r r a y t e c h n i q u e s [ 1 4 ] , t o c o m p u t e t h e p o w e r f o r a g i v e nf r e q u e n c y w i n d o w . T h e r e s u l t s p r o v i d e i n f o r m a t i o n o nw a v e n u m b e r s k x a n d k y w h i c h c o n t a i n h i g h p o w e r ,a n d c o n s e q u e n t l y , o n b o t h a p p a r e n t v e l o c i t y a n da z i m u t h o f p r o p a g a t i o n . T h e e x i s t a n c e o f o n l y o n e p e a ki n t h e s p e c t r u m i n d i c a t e s t h e p r e s e n c e o f a p r e d o m i n a n tp l a n e w a v e . I f s e v e r a l p e a k s e x i s t , t h e y i n d i c a t e t h ep r e s e n c e o f m u l t i p l e w a v e s p r o p a g a t i n g s i m u l t a n e o u s l ya t d i f f e r e n t a p p a r e n t v e l o c i ti e s a l o n g d i f f e r e n t a z i m u t h s .

W h i l e c o r r e l a t io n f u n c t i o n s w e r e o b t a i n e d d i r e c t l y i nt h e t i m e d o m a i n , t h e p o w e r s p e c t r a l d e n s i t y a n dc o h e r e n c y f u n c t i o n s w e r e o b t a i n e d i n t h e f r e q u e n c yd o m a i n . T h e c o h e r e n c y c o m p u t a t i o n s r e q u i r e df r e q u e n c y s m o o t h i n g p r i o r t o m u l t i p l i c a t i o n in o r d e r t oo b t a i n v a l u e s d i f f e r e n t f r o m 1 . A t r i a n g u l a r s m o o t h i n gw i n d o w w i t h 9 p o i n t s w a s u s e d f o r t h i s p u r p o s e .

0.00 . . . . . .0,00 5.00 10.00 15.00 20.00 25,00frequency(hz)Fig. 4. Effect of averaging of coherency functions on noisereduction; (a, top) pair of stations, (b, bottom) average of 10pairs o f stations.

A s m e n t i o n e d i n t h e p r e v i o u s s e c t i o n , s p a t i a l w a v ec h a r a c t e r i z a t i o n c a n b e m a d e b y u s i n g e i t h e r t h e c o r r e -l a t i o n c o e f f i c i e n t f u n c t i o n o r t h e c o h e r e n c y f u n c t i o n . I nt h e p r e s e n t i n v e s t i g a t i o n , c o r r e l a t i o n c o e f f i c i e n t f u n c -t i o n s w e r e c o m p u t e d f o r m a n y s t a t i o n p a i r s u s i n g d i f -f e r e n t t i m e a n d f r e q u e n c y w i n d o w s . R e s u l t s p r e v i o u s l yr e p o r t e d b y O l i v e i r a [7 ] s h o w s , a s e x p e c t e d , t h a t t h em a x i m u m v a l u e s o f th e c o r r e l a t i o n c o e f f i c i e n t f u n c t i o n sa r e s t r o n g l y d e p e n d e n t o n t h e t i m e a n d f r e q u e n c yw i n d o w s u s e d . T h e c o r r e l a t i o n c o e f f i c i e n t s , f o r t h en o n - f r e q u e n c y f i l t e re d c a se s , w e re a l m o s t e q u a l t o o n ew h e n t h e w a v e a r r i v a l s w e r e w e l l d e f i n e d , e . g. u s i n gs t a t i o n p a i r s i n t h e i n n e r r i n g i n t h e c a s e o f e v e n t N o .2 4 . E l s e w h e r e i n t h e a r r a y t h e c o r r e l a t i o n c o e f f i c i e n t sd r o p p e d o f f t o v a l u e s o f a p p r o x i m a t e l y 0 .6 . T h i s t e c h -n i q u e p e r m i t s t h e d e t e c t i o n o f t h e t i m e s h i f t ~" f o rm a x i m u m c o r r e l a t i o n a n d , c o n s e q u e n t l y , t h e e v a l u a t i o no f t h e c o r r e s p o n d i n g a p p a r e n t v e l o c i t y . I n t h o s e c a s e so f l o w c o r r e l a t e d s i g n a l s , i t i s d i f f i c u l t b y t h i s t e c h n i q u et o o b t a i n g o o d v a l u e s o f ~ ' ; t h e r e f o r e , o t h e r t e c h n i q u e ss u c h a s t h e a l i g n e d p h a s e s p e c t r u m m e t h o d [1 5], isr e c o m m e n d e d . F r e q u e n c y d e p e n d e n c e s w h i c h h a ve b e e no b s e r v e d b y o t h e r s [ 1 6 , 6 , 1 5 ] , a r e m a j o r f a c t o r s c o n t r o l -l i n g s p a t i a l c o r r e l a t i o n .W o r k i n g i n t h e f r e q u e n c y d o m a i n , c o h e r e n c y i s o b -t a i n e d t h r o u g h c o m p u t a t i o n o f t h e c r o s s s p e c t r a l d e n -


6/18

298 H. H ao et aL / Mu ltiple-station ground motion processings i t y f u n c t i o n s e s t i m a t e s S g j . B a s e d o n h i s s t u d i e s o fS M A R T - 1 d a t a , L o h [ 16 ] s u g g e s te d t h e f o l l o w i n g e q u a -t i o n t o c o m p u t e S , j :S , , ( i ~ ) = S o ( ~ ) 1 3 ' , j ( ~ ) l e x p ( i ~ d ~ / t , ' a p p ) ( S )w h e r e

3 . E s t i m a t i o n o f m o d e l p a r a m e t er s b a s e d on S M A R T - Ia r r a y d a t a

M o d e l p a r a m e t e r s , a s d e s c r i b e d i n s e c t i o n 2 , w e r ee s t i m a t e d u s i n g d a t a r e c o r d e d a t t h e i n n e r - r i n g s t a t i o n so f t h e S M A R T - 1 a r r a y d u r i n g e v e n t s N o . 2 4 a n d N o . 4 5f o l l o w i n g th e p r o c e d u r e g i v e n b e lo w .

I v , j ( ~ ) I = e x p ( - X ~ ] d , ~ l ) , ( 9 )w h e r e ? , i s a f r e q u e n c y d e p e n d e n t p a r a m e t e r h a v i n gun i t s v -1 (v i s ve loc i ty ) , yapp d e n o t e s t h e a p p a r e n tw a v e v e l o c i t y a t f r e q u e n c y ~ , a n d S 0 ( ~ ) is a n a v e r a g e dp o w e r s p e c t r a l d e n s i t y f u n c t i o n f o r t h e e n t i r e a r r a y .S e l e c t in g th i s f o rm o f e q u a t i o n w a s b a s e d o n t h e o b -s e r v a t i o n t h a t t h e c r o s s s p e c t r u m a t t e n u a t e s w i t h d i s -t a n c e a n d e x p e r i e n c e s a p h a s e s h i f t d u e t o t h e t r a v e l li n gw a v e s . A l s o b a s e d o n a s t u d y o f t h e c h a r a c t e r i s t i c s o ff o u r e v e n ts , a n d a s s u m i n g i s o t r o p y i n l o s s o f c o h e r e n c yw i t h d i s t a n c e , H a r i c h a n d r a n a n d V a n m a r c k e [ 1 5 ] s u g -g e s t e d a n e q u a t i o n o f t h e f o r m

y i j ( i ~ ) = A e x p a O ( ~

+ ( 1 - A ) e x p - O - - ~ ( 1 - A + a A )

- u (10 )e x p 1 0 )

w h e r e A a n d a a r e c o n s t a n ts , 0 ( ~ ) r e p r e s e n t s t hes p a t i a l s c a le o f f l u c t u a t i o n a n d d i j i s th e d i s t a n c eb e t w e e n s t a t i o n s i a n d j . T h e s e p a r a m e t e r s w e r e al ld e t e r m i n e d b y t h e m e t h o d o f l e a s t s q u a re s u s i n g m e a -s u r e d g r o u n d m o t i o n d a t a . U s i n g t h e d a t a o f e v e n t N o .2 0 , H a r i c h a n d r a n a n d V a n m a r c k e [ 15 ] o b t a i n e d A , a ,a n d t h e O ( ~ ) f u n c t i o n a s g iv e n b yA = 0 .736 , a = 0 .147 , and

2 1/2

F o r e n g i n e e r i n g p u r p o s e s o n e c a n u s e a s i m p l i f i e d m o d e lf o r r e p r e s e n t i n g t h e t r a v e l l i n g w a v e s w h i c h c o n s i s t s o f as i n g l e m o n o c h r o m a t i c w a v e p r o p a g a t i n g w i t h a s p e c i -f i e d a p p a r e n t v e l o c it y . T h i s a p p r o a c h s e e m s t o g i v eg o o d r e s u l t s w h e n c o n s i d e r i n g d i s p l a c e m e n t s a n d i tp r o v i d e s u p p e r b o u n d r e s u lt s w h e n c o n s i d e r i n g a c c e l e r -a t i o n s .

3 .1 . Co h e re n c yC o h e r e n c i e s f o r a l l p o s s i b l e s t a t i o n p a i r s w e r e o b -

t a i n e d u s i n g t h e f r e q u e n c y d o m a i n a p p r o a c h ( s e e s e c -t i o n 2 .2 ). A s s e e n i n f i g . 4 a , t h e c o h e r e n c y c o m p u t e d f o re a c h i n d i v i d u a l p a i r o f s t a t i o n s s h o w s a l a r g e n o i s ec o n t e n t , w h i c h c a n b e g r e a t l y r e d u c e d b y a v e r a g i n g t h ec o h e r e n c i e s o f s e v e r a l s t a t i o n p a i r s . F i g . 4 b s h o w s h o wt h e a v e r a g i n g p r o c e s s o f th e a m p l i t u d e s o f t h e c o h e r e n c yf u n c t i o n s f o r s t a t i o n p a i r s e l i m i n a t e s t h e p r e s e n c e o fn o i s e . T o i n v e s t i g a t e t h e i n f l u e n c e o f n o i s e i n t h ec o h e r e n c y c o m p u t a t i o n s , w h i t e n o i s e s i g n a ls w e r e g e n e r-a t e d , h a v i n g a b s o l u t e v a l u e s o f c o h e r e n c y f o r i n d i v i d u a lp a i r s o f s i g n a l s e q u a l t o a p p r o x i m a t e l y 0.3 o v e r t h ee n t i r e f r e q u e n c y r a n g e c o n s i d e r e d . A v e r a g i n g o v e r m a n yp a i r s g r e a t l y r e d u c e d t h i s l e ve l o f c o h e r e n c y .

I n o r d e r t o o b t a i n a c o n t i n u o u s f u n c t i o n e x p r e s s i n gl o s s o f c o h e r e n c y w i t h d i s t a n c e , s t a t i o n p a i r s h a v i n ga p p r o x i m a t e l y t h e s a m e p r o j e c t e d s e p a r a t i o n d i s t a n c e s( d ,~ , d ~ ) * w e r e s e le c t e d a n d c a t e g o r i z e d in t o n i n eg r o u p s ( d R , d ~ = 5 0 m , 1 0 0 m , 2 0 0 m ) . V a l u es o fc o h e r e n c y l o s s f o r s t a t i o n p a i r s i n e a c h g r o u p w e r ec a l c u l a t e d a n d a v e r a g e d . T h e a v e r a g e d v a l u es , f o r w h i c ht h e n o i s e h a d b e e n s u b s t a n t i a l l y re m o v e d , w e r e t h e nu s e d a s t h e t r u e v a l u e s e x p r e s s i n g lo s s o f c o h e r e n c yb e t w e e n s t a t i o n p a i r s a t t h e c o r r e s p o n d i n g d i s t a n c e s .

A f t e r c o n s i d e r i n g t h e s e v e r a l m o d e l s a v a i l a b l e in t h el i t e r a t u r e w h i c h r e p r e s e n t l o s s o f c o h e r e n c y w i t h d i s -t a n c e [ 1 5 - 1 9 ] , t h e p r e s e n t a u t h o r s b e l i e v e th a t ac o h e r e n c y f u n c t i o n o f t h e f o r my i j( d /~ , d T , f ) = e x p ( - / 3 , d ~ - / ~ 2 d T )

e x p [ - ( a , d ~ l / 2 + o t2 d l T l / 2 ) f 2 ] e x p ( i 2 " ~ f ~ ) ( 12 ,

a d e q u a t e l y r e p r e s e n t s f i e l d c o n d i t i o n s . I n t h i s e x p r e s -s i o n , f r e p r e s e n t s f r e q u e n c y in H z ( ~ = 2 0 f ) , a n d t h e

* d~ and d are pro jected separat ion d is tances in the longitu-d inal d i rect ion of preferent ial wave propagation and in th etransverse direction, respectively.


7/18

H. Hao e t aL / Mu l t i p l e - s t a ti on g r ound mo t i on p r oc e s s i ngT abl e 2Values of f l l and f12 for d i f ferent components and events

29 9

E v e n t # 2 4 E v e n t # 4 5N S E W D N N S E W D N

fll 4.78 X 10 -4 same 5.1 X 10 - 4 7 .4X 10 - 4 s ame 1 .88 10 -4f12 1.59 x 10 - 4 same 2.64 x 10 - 4 9 .6 10 - s same 3.43 10 - ~

p a r a m e t e r s i l l , f 1 2 , 0 tl , a n d ot 2 m u s t b e e v a l u a t e d b y t h em e t h o d o f l e as t s q u a re f i t ti n g u s i n g S M A R T - 1 d a t a ; f lla n d f12 w e r e o b t a i n e d b y f i t t i n g t h e c o h e r e n c y v a l u e sw h e n f = 0 ; h o w e v e r , a 1 a n d (x2 w e r e c o m p u t e d t o f i tt h e c o h e r e n c y v a l u e s a t e a c h f r e q u e n c y . T a b l e 2 p r e -s e n t s f ll a n d fiE f o r th e d i f f e r e n t c o m p o n e n t s a n dd i f f e r e n t e v e n t s , a n d f i g . 5 p r e s e n t s e q a n d c t 2 a sf u n c t i o n s o f f r e q u e n c y . I n g e n e r a t i n g t h e s e p a r a m e t e r s ,t h e e n t i re t i m e - h i s to r i e s o f g r o u n d m o t i o n w e r e u s e d ,e v e n t h o u g h i t w a s n o t i c e d t h a t t h e f r e q u e n c y d e p e n -d e n t c o h e r e n c y f u n c t i o n s w e r e s o m e w h a t t i m e d e p e n -d e n t . T h i s d e p e n d e n c y w a s j u d g e d t o b e t o o w e ak ,h o w e v e r , to r e q u i r e a d e t a i le d t i m e w i n d o w a n a l y s is . A ss e e n i n f i g . 5 , t h e v a l u e s f o r ch a n d ot 2 s h o w t h ec o h e r e n c y f u n c t i o n t o b e m u c h m o r e d e p e n d e n t o n

l o n g i t u d i n a l d i s t a n c e t h a n o n t r a n s v e r s e d i s t a n c e ; h o w -e v e r , t h i s o b s e r v a t i o n d i d n o t h o l d f o r a l l c o m p o n e n t so f m o t i o n , p a r t i c u l a r l y i n t h e l o w e r f r e q u e n c y r a n g e .F i g . 6 s h o w s t h e c o h e r e n c y v a l u e s c a l c u l a t e d b y e q . ( 1 2 )f o r d i f f e r e n t d i s t a n c e s . A s w i l l b e s h o w n l a t e r i n t h eg e n e r a t i o n o f s u r f a c e g r o u n d m o t i o n s a t th e n s i t el o c a t i o n s ( x i , Y i, i = 1 , 2 . . . . , n ) , t h e d e v e l o p e da l g o r i t h m r e q u i r e s t h e e x i s t e n c e o f b o t h d E a n d d T .

E v e n t h o u g h c o h e r e n c y v a l u e s f o r d i s t a n c e s g r e a t e rt h a n 4 0 0 m w e r e n o t o f m u c h c o n c e r n , t h e i r a v e ra g ev a l u e s w e r e c o m p u t e d f o r a l l s t a t i o n s i n t h e m i d d l e a n do u t e r r i n g s o f t h e S M A R T - 1 a r r a y h a v i n g a v e r a g e s e p -a r a t i o n v a l u e s o f 1 0 00 m a n d 2 0 0 0 m , r e sp e c t i v e l y . E v e nt h e s e r e s u l t s w e r e f o u n d t o b e i n r e a s o n a b l e a g r e e m e n tw i t h t h e c o h e r e n c y m o d e l p r e s e n t e d a b o v e .

0 . 0 1 0

0 . 0 0 5

0 . 0 0 0

ALPHA v a l u e s c a l c u l a t e d b yl e a s t s q u a r e m e t h o dE v e n t 24. N5

A L P H A I

f , ~ " ' ~ w - -ALPHJ~- 0 . 0 0 5

- 0 . 0 1 0 I I I I I I I ] I I I i i i i i i ] i io . o 5 . 0 1 0 . of r e q u e n c y ( h z )

0 . 0 1 0

0 . 0 0 5

0 . 0 0 0

- 0 . 0 0 5

- 0 . 0 1 0

A L P H A v a l u e s c a l c u l a t e d b yl e a s t s q u a r e m e t h o dE v e n t 4 5 N S

/ALPHA2

I I I I I I I I l l l l l l l l l l l l l l l0 . 0 5 . 0 1 0 .0f r e q u s n ~ ( h z )

0 . 0 1 00 . 0 0 5 -

0 . 0 0 0

- 0 . 0 0 5 -

A L P H A v a l u e s c a l c u l a t e d b yl e a s t s q u a r e m e t h o dEvent 24 . DN

ALPHA1

- 0 . 0 1 0 I I I I I I I l l ] l l l l l l l l l [ l l lo . o 5 . 0 l O .Of r e q u s n ~ ( h z )

0 . 0 1 0

0 . 0 0 5

0 . 0 0 0

- 0 . 0 0 5

-0.010

A L P H A v a l u e s c a l c u l a t e d b yl e a s t s q u a r e m e t h o dE v e n t 4 5 D N

ALPHA1

ALPHa.

I I I I I I I I I I I I I I I I I I I ] 1 1 10 . 0 5 . 0 1 0 .0f m q u e n ~ ( h z )

Fig. 5 . Var ia t ion of a I and ~t2 w i t h f r equency f o r d i f f e r en t comp onen t s and even ts .


8/18

300 H. H ao el al. / M ult ipl e-stati on ground motion processing

Table 3Parameter values for the Kanai-Tajimi power spectral density function for different time windows, components and eventsWindow Event #24 Event #45

NS EW DN NS EW DN5, % is % 5, % & % 5, Y k %

1 0.3 1.9 0.95 1.0 0.33 2.0 0.84 0.17 0.80 4.52 0.26 1.1 0.30 1.2 0.83 0.9 0.80 1.2 0.80 4.53 0.58 0.8 0.71 1.0 0.60 0.5 0.95 0.50 0.41 0.6

model coherency Yaiuein a 50m area

0.00 Ll_5mTg-J .000frequency(hz)

0 5 10frequencyjhr)

Fig. 6. Coherency function as derived by eq. (12) for differentseparation distances, d,; and d,:.

3.2. Power spectral density functi onAn averaged power spectral density function for all

inner-ring stations was computed for each time windowusing the Kanai-Tajimi [20] form

So(W) = 0, + 4&?$2(0, - q2 + 4~,2wpzw*. (13)

The numerical values of constant 5, and ws obtained ineach case are shown in table 3. The scaling factor r wasfound to have values as shown in table 4. Fig. 7 showsthe averaged power spectral density functions for theinner-ring stations for different time windows. In only

Table 4Power spectrum scaling factors for different time windows,components and eventsWindow Event #24 Event #45

NS EW DN NS EW DN1 1.0 1.0 1.0 1.0 0.422 1.34 0.94 1.49 1.24 0.423 1.65 1.06 2.07 1.8 1.21

one case, namely the vertical component of event No.24, did this Kanai-Tajimi form not correlate with fieldresults. In this particular case, band-limited white noisewas found to be more representative.

overq ed? auto power apectml densit y functionor ail inner ring rtotions (Event 45)NS tims window 5-10 set

were sdP auto power spectral density functionor 011 inner ring stations (Event 45)t2ooooooo

i

NS time window 9-29 s*c

*ma edP cut0 power spcctml density functionor all inner ring stations (Event 45)NS time window 28-38 set

j,,,h ,,I,,~,, ,,,,,,,, ,,,,I.00 5.60 10.00 15.00 20.00 25Yoofmquscy(hz)Fig. 7. Average power spectral density function for the inner-

ring stations for different time windows.


9/18

H . H ao et al . / M ul t ip le -sta t ion gr ound mot ion processing 3 0 1

q5 N 15-25 5EC 0 .~9 MZ q5 N 15-25 SEC 0 .9 8 HEAZIMUTH OF PEAK = 159 AZIMUTH OF PERK = lq IAPPARENT VELOCITY [ km / lec ] = 2 .3 ~P~ EN T VELOCITY I km/ eec) = 3 ,1HAX lM ~ POWER [cm/ sec}u M2 = q12q MAXI MUM POWER t om/ so t ; Mm2 = 3769

N N1.0 1 .0

0 .6.5

~ -0.6 ~ -0.9- I , 0 . . . . - 1 . 0

- 1 , 0 - 0 , 5 0 , 0 0 . 5 1 , 0 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0S L O W N E S S [ S E C / M H ) S L O W N E S S [ S E C / K M ]

q5 N 15-25 SEE 1 .q6 HE qS N 15-25 6EC i , g 5 HEAZMUTHOF PEAK = 1~6 AZIMUTH OF PERK = 180~P~ EN T VELOCITY ( km/ oec) = 5 .5 APPARENT VELOCITY I km/ oec ] = 6 .7MAXIMUM POWER tom/see) u 2 = 2535 MAXIMUM POWER {cm /ne c}l m2 = 2167

Ni .o _ 1 . oo.6 o.60 . 0 ~

~ -o.s ~-o.s- I . O . . . . . . . 1 . 0

-1,0 - 0 . 6 0 , 0 0 . 5 1 . 0 - I . O - 0 . 5 0 . 0 0 . 5 1 . 0S L Y N E S S [ S E C/ K M ) S L ~ N E 5 5 [ 5 E C /K M )

F i g . 8 . E x a m p l e o f F - K s p e c t r u m t o d e t e r m i n e a p p a r e n t v e l o c -i ty , a z im u t h o f i n c o m i n g w a v e s a n d p o w e r .

1 5 0 . 0 0a v e r a g e d e n v e l o p e f u n c t i o n o f a l l t h e i n n e rr i n g s t a t T o n s E v e n t 4 5 N S c o m p o n e n t

1 0 0 . 0 0 j 11 ! ! ~ i I i ~

5 0 . 0 0 ; ~ 1 ~

0 0 0 . , . . . . . . . . , . . . . . . . . . , . . . . . . . . . i . . . .0 . 0 0 1 0 . 0 0 2 0 . 0 0 30.00 4 0 . 0 0t i m e ( s e c )

F i g . 1 0 . A v e r a g e s h a p e f u n c t i o n f o r t h e i n n e r - r i n g s t a t i o n s .

3.3. A ppar ent veloc i tyA p p a r e n t v e l o c i t y i s t h e m o s t d i f f i c u l t p a r a m e t e r t o

a s s e s s , e v e n t h o u g h p r e f e r e n t i a l w a v e s a r e p r o p a g a t i n gfrom the ep icentra l area to the s i te a t a l l t im es . Thisd i f f i c u l t y i s d u e t o t h e p r e s e n c e o f w a v e s a p p r o a c h i n gf r o m o t h e r d i r e c t i o n s w i t h d i v e r s e a p p a r e n t v e l o c i t i e sw h i c h a r e p a r t i c u l a r l y n o t i c e a b l e f o r f r e q u e n c i e s a b o v ec e r t a i n l i m i t s w i t h i n t h e d i f f e r e n t t i m e i n t e r v a l s . T h i sf e a t u r e , f o u n d t o b e p r e s e n t i n a l l e v e n t s , u n d o u b t e d l yhas charac ter i s t i c s re f l ec t ing the par t icu lar geo log ica le n v i r o n m e n t , w h e r e t h e w a v e s a r e r e f l e c t e d a n d r e -f r a c t e d i n m u l t i p l e f o r m s . T o i l l u s t r a t e t h e d e t e r m i n a -

t im e w i n d ow ( I x c - 6 s e c )141 2 -1 0 -

= . R - -

s -

4 -2 -0 ] I I I I I I I I I I I I I I I I I I0,0 4..0

1' $1 2 -1 0 -

8 -~ _

time window (38QC--81OC)

0 ] l l l l l l l l l l l l l l l l l l2.0 0.0 2.0 4.0f ~ q u o n ~ ( h z ) f ~ q u o n ~ ( h z )

time window (1511ec-2511ec) t ime window (2011o c-30u c)1,4 / 14.1 2 - 1 2 -1 0 - 1 0 -

o . 8 - a . 8 -4 , - 4 . -2 - ~ 2 -0 I I I I I I I I I J I I 1 1 I I I I I 00.0 2.0 .0 0.0frequency(hz) frequeney(hz)

F i g . 9 . A p p a r e n t v e l o c i t ie s a s a f u n c t i o n o f f r e q u e n c y .

I I I I 1 I I I I [ I I I I I I I 1 12,0 4..0


10/18

302 H. H ao et al. / Mu ltiple-station ground motion processingtion of apparent velocity and azimuth of propagation,and F-K diagram of the approaching wave field forevent No. 45, using a frequency window centered onf= 0.49 Hz, is shown in fig. 8. Fig. 9 shows the vari-ations of apparent wave velocities with frequency forthe different time windows.Other techniques to detect apparent velocities andazimuth direct ions were also used such as the ~--x analy-sis [7] and the principle direction analysis [21]. Com-parisons of the current results with those of otherstudies [16,15,21], show similar trends related to ap-parent velocity as a function of frequency.3 .4 . Sh ape func t io n

: , e , 3 g e J c o f c r e - c , . ~ ~ , : $ : e < w e e ~ h : , ~ : : , 'c o ~ ' ) a n e n t " ~ c i i v ' c , ' r q s : a t c ~" q c e n t ~ ,

I' t0 9 8 ~ . . . . . . . . . : , , ~ . . . . . . . , ,. . . . .. . ~ . .. .. .. .O O h 5 0 C i 0 , 0 0 1 5 0 9 ? 0 0 0 2 b O O

' r e q u e n c , ( ! , '!Fig. 11. Average coherency function for different componentsat the same station.

Hilbert transformations of the signals were used todetermine the time envelope functions. For each com-ponent, average envelope functions were obtained forall inner -ring stations. Fig. 10 shows an example of theNS com ponen t for event No. 45. Based on the SMART-1data, a shape function ~(t) of the form~ ( t ) = a t e x p ( - b t 2 ) (14)was found to correlate reasonably well except for thevertical components of motion recorded during eventNo. 24. In this case, a constant envelope intensity wasfound to be more appropriate throughout the entiretime-history. The two parameters a and b in eq. (14)can be determined by normalizing the maximum valueof ~ (t) to one at time t 0, which yields the relationships

1t = 2~- ' a= 2~ e and ~ ' ( t0 )= l .0 , (15)where e is the base of the natural logarithm. Table 5shows the values of a and b for different componentsand different events.3 .5 . C o h e r e n c y a m o n g d i f fe r e n t c o m p o n e n t s a t t h e s a m es ta t ion

Coherency between the horizontal components re-corded at each station was computed station by station,Table 5Shape function paremeter values for different components andevents

#24 #45NS EW DN NS EW DN

tmax(s) 8 11 12 12 8a 0.206 0.15 0.1374 0. 1374 0.206b 0.0078 0.00413 0.00347 0.00347 0.0078

and then, the average was obtained for all stations. Fig.11 shows the results obtained for event No. 45 betweenthe NS and Ew components. The low values shownindicate that correlation among horizontal componentsis very small as one would have expected because theNS direction is nearly the same as epicentral directionwhich normally is close to being the direction of majorprincipal motion.

4 . S i m u l a t i o n o f s t r o n g g r o u n d m o t i o n4 .1 . Gr ound mo t ion gener a t ion in the t ime domain

Assuming that earthquake ground motions are sta-tionary random processes having zero mean values andknown power spectral density functions and crosscoherency functions, a series of spatially correlatedground motions can be generated through the followingmethod [23]:

Assume the ground motion time series have the samepower spectral dens ity funct ion S0(w) (- WN --< ~ --< ~N,where ~N is the Nyquist frequency) at every station,which is a reasonable approach when the focus of theearthquake is at a large distance from the site as com-pared to the site dimension. The cross power spectraldensity functions Su( i~ ) can now be expressed in termsof S0(~ ) and the coherency functions 7/j(i ~) as givenby ]n( ~) -112 i~ ) ... ~q~(i~)S ( ~ ) = "/zl(i~) Vz2(~) ' Y2"(i~) S0(~),

V . l ( i ~ ) V . : ( i ~ ) ... "/,,, (~ )- - ~ N ~ W ~ ( - 0 N . (16)The generated time series u l ( t ) , U 2 ( t ) . . . . . Un( t ) , must


11/18

H . H a o e t a l . / M u l t ip l e - s ta t i on g round m o t i on p roces si ng 303b e c o m p a t i b l e w i th t h e i n d i v i d u a l fu n c t i o n s i n S ( ~ ) .T o d o t h i s , l e t

i N~ , ( t ) = E E A , ~ (~ o )m = l n = lX c o s [ K ~.t + f li ,~ ( m . ) + d0 m. ~ . ) ] , ( 1 7 )

w h e r e a m p l i t u d e s A i m ( ~ . ) a n d p h a s e a n g l e s f l i m ( ~ . )a r e t o b e d e t e r m i n e d t o i n s u r e t h e p r o p e r c o r r e l a t i o nr e l a t i o n s s a t i s fy i n g e q . (1 6 ), a n d ~ , . . ( ~ . ) i s a r a n d o mp h a s e a n g l e u n i f o r m l y d i s t r i b u t e d o v e r t h e r a n g e z e ro t o2 ~ r. N o t e ~ , , , a n d q 'r , s h o u l d b e s t a t i s t i c a l l y i n d e p e n -d e n t u n l e s s m = r , a n d n = s .

T o o b t a i n t h e p r o p e r v a l u e s o f A i m ( ~ ) a n d f l i , , ( - ~ )a t e a c h d i s c r e t e f r e q u e n c y ~ , , t h e c o n t r i b u t i o n t o t h et i m e a v e r a g e ( u i ( t ) u j ( t ) ) , b y u i ( t ) a n d u j ( t ) i n af r e q u e n c y b a n d A m c e n t e r e d o n ~ , i s c o n s i d e r e d . A s -s u m e i > j ; t h e n f o r m o t i o n i n t h i s f r e q u e n c y b a n d , t h et i m e a v e r a g e i s( u i , ( t ) u j , ( t ) )

X A j . c o s ( ~ . t + f l j . + q ~.. , ( 1 8 )

w h e n m > j a n d r n q ~ r , ( u i . ( t ) u j . ( t ) ) = O s i n c e i nt h e s e c a s e s , q ~ , .. a n d q ~ .. a r e s t a t i s t i c a l l y i n d e p e n d e n t .T h e a b o v e e q u a t i o n t h e n r e d u c e s t o( u i . ( t ) u j . ( t ) )

j= Y '~ A i m A j m c o s ( ~ n t + r i m + d P m . )m = l

X C OS(~ . t + f ljm + q~ ,~ . ) )J AimAjm= E ~ ( co s (2 ~ . t + f l , . , + f l j , . + 2q~ , . . )

m = lq- COS(~im -- ~jm)>J AimAjm= E ~ C O S ( ~ i m - - ~ j m ) , n = 1 , 2 . . . . . N .m = l (19)

T h e c r o s s p o w e r s p e c t r a l d e n s i t y f u n c t i o n S i j ( i m )h a s t h e p r o p e r t i e s t h a t i t s r e a l a n d i m a g i n a r y p a r t s a r ee v e n a n d o d d , r e s p e c t i v e l y ; t h u s , t h e c o r r e s p o n d i n ga m p l i t u d e f u n c t i o n A ~ j ( m ) i s e v e n , a n d t h e p h a s e a n g l e

f u n c t i o n f l i j ( ~ ) i s o d d . H e n c e , t h e a b o v e t i m e a v e r a g ef o r d is c r e t e h a r m o n i c m . c a n b e e x p r e s s e d a s

( U i n ( t ) U j n ( t ) ) = Re 2 ei (O 'm-B '~)1

= R e H i m ( i m . ) H ~ . , ( i m . ) ,1n = 1 , 2 . . . . . N ; ( 2 0 )

w h e r e R e d e n o t e s re a l p a r t , H ~ m ( i m , ) = A i m ( m n ) Xe iB ~m (~ -), a n d t h e s u p e r s c r i p t * m e a n s c o m p l e x c o n -j u g a t e .T h e a b o v e t i m e a v e r a g e a t f r e q u e n c y m n o v e r t h eb a n d w i d t h A , -z c a n a l s o b e e x p r e s s e d i n t e rm s o f t h es p e c t r a l d e n s i t y f u n c t i o n s i n t h e f o r m( U i n ( t ) u j n ( t ) )

= S i j ( i m . ) a ~ = s 0 ( m . ) [ y i j ( i m . ) + r i j ( - i m . ) ] A m= R e [ 2 S 0 ( m . ) Y i j ( i m . ) ] a w = R e [ Z S i j ( i ~ . ) ] k ~ ,n = 1 , 2 . . . . . N . ( 2 1 )

S i n c e m a t r i x S ( m ) g i v e n b y e q . ( 1 6 ) i s H e r m i t i a n a n dp o s i t i v e d e f i n i te , it c a n a l w a y s b e d e c o m p o s e d i n t o t h em u l t i p l i c a t i o n o f a c o m p l e x l o w e r t ri a n g u l a r m a t r i xL ( i m ~ ) a n d i t s H e r m i t i a n L H ( i m , ) [ 2 4 ] , a s s h o w n b yS ( m ) = L ( i , 5 ) L n ( i m ) S0 ( o~ ), ( 2 2 )w h e r e

L ( i m ) =[ / n ( ~ ) . 0 .. . 0

/22( ~ )121(i~a) .. . 0I t . ~ ( i ~ ) t . 2 ( i ~ ) . . . l . . ( ~ )

( 2 3 )

a n d w h e r e l~ j ( i = l , 2 , . . . , n , j = l , 2 . . . . . i ) ca n bec a l c u l a t e d b y t h e C h o l e s k y m e t h o d [2 5],

I i- 1 ]1/2l . ( ~ ) = 3 ' i i ( ~ ) - Y'~ l ; k ( i ~ ) l ~ ( i ~ ) ,k = l

i = 1 , 2 . . . . . n ;j - I

7 i j ( i ~ ) - Y ' . l i k ( i ~ ) l T k ( i ~ )l~ j ( i ~ ) = k = ll . ( ~ )

j = l , 2 , . . . , i .

( 2 4 )

( 2 5 )


12/18

3 0 4 H. H ao et al. / Multiple-station gro und motion processingF o r t h e ca s e i > j , t h e i n d i v id u a l f u n c t i o n s in S ( ~ ) c a nbe wr i t t en as

JS , l ( i ~ , ) = 2 S 0 ( ~ , ) ~ L , m ( i ~ , ) L ~ m ( i ~ , ) A W ,m = l

n = 1 , 2 . . . . . N . ( 26 )Equa t ing eqs . (20) and (21) , the fo l lowing expres s ionsa r e ob t a i ne dA ~ j ( ~ ) = ( 4 S0 ( ~0 ) A ~ I / , j ( i ~ ) 1,

= t a n _ a{ I m [ l o ( i ~ ) ]R e [ l ~ j ( i~ ) ] ) 'B,j( ~ )

0 < ~ - < ~ r q ;0 ~ < 0 < 0 N .

( 2 7 )

Making use of eqs . (17) and (27) , a s e t of spa t i a l lycor re la ted t ime se r i es u ~ ( t ) ( i = 1 , 2 . . . . . n ) c a n b e g e n -e r a t e d , a n d t h e c o r r e s p o n d i n g g r o u n d m o t i o n s a ~ ( t )( i = 1 , 2 , . . . , n ) c a n b e o b t a i n e d b y m u l t ip l y i n g e a c ht ime se ri es by a pro per s hap e fun c t ion ~ ' (t ). B y th i sprocedure , one f i r s t genera tes a t ime se r i es for s i t e 1 ,t h e n f o r s i t e 2 b y s u m m i n g u p w a v e c o n t r i b u t i o n s t h a ta re proper ly cor re la ted wi th the f i r s t one . S imi la r ly , thegenera ted t ime se r i es for s i t e 3 wi l l be cor re la ted wi tht hos e p r e v i ous l y ge ne r a t e d f o r bo t h s i t e s 1 a nd 2 , e t c .T he f i r s t t i me s e r i e s c a n be e i t he r a s yn t he t i c mo t i on o ra r e a l mo t i on p r ov i de d i t i s c ompa t i b l e w i t h t he p r e -s c r i be d s pe c t r a l de ns i t y ma t r i x S 0 ( m) .4 ,2. G ro u n d m o t io n g e n e ra t io n i n t h e f re q u e n c y d o m a in

G r o u n d m o t i o n s c a n b e g e n e r a t ed m o r e e a si ly i n t h ef r e q u e n c y d o m a i n t h a n i n t h e t im e d o m a i n b y u s i n g e q.( 17 ) . T o p r oc e e d w i t h t h i s ne w a pp r oa c h , e xp r e s s t heF o u r i e r t r a n s f o r m o f U i ( t ) i n t h e f o r m

iU ~ ( i ~ ) = Y'~ B t , , , ( ~ ) [ c o s a g , ~ ( ~ ) + i s in a ~ , ~ ( ~ ) ] ,m = l

n = 1 , 2 . . . . . N ; ( 28 )w h e r e B , m ( ~ , ) i s t h e a m p l i t u d e a t f r e q u e n c y ~ , a n da , , ( ~ , ) i s t he c o r r e s pond i ng pha s e a ng l e . B y t r a ns f o r -mi ng e q . ( 17 ) i n t o t he f r e que nc y doma i n , i t c a n bes h o w n t h a tB.~ (T.,n) = Agm (~ n ) / 2 ." ,m ( ~ . ) = B , m ( ~ . ) + ~.." (29)T he n , u s i ng e qs . ( 29 ) a nd ( 27 ) , ~ ( i ~ n ) a s e xp r e s s e d i ne q . ( 28 ) be c ome s f u l l y know n; t hus , t he c o r r e s pond i ng

t ime se r i es u ~ ( t ) c a n be ob t a i ne d by i nve r s e t r a ns f o r -m i n g ~ ( i ~ , ) b a c k in t o th e ti m e d o m a i n .4 . 3 . Re sp o n se sp e c t ru m c o m p a t ib l e t im e se r i e s

I t i s c ommon p r a c t i c e i n e ng i ne e r i ng t o s pe c i f y t hed e s ig n e a r t h q u a k e g r o u n d m o t i o n s t h r o u g h t h e s ta n d a r dr e s pons e s pe c t r um , s a y S v ( o~ , ~ ) w h i c h de no t e s t heps e udo ve l oc i t y r e s pons e s pe c t r um f o r t he s i ng l e de g r e eo f f r e e dom s ys t e m ha v i ng a na t u r a l f r e que nc y ~ 0 a nd adamping ra t io ~ . In th i s case , i t i s necessa ry to adjus tt he ge ne r a t e d g r ound a c c e l e r a t i on t i me - h i s t o r y a i ( t ) sot ha t i t i s c ompa t i b l e w i t h t he p r e s c r i be d r e s pons e s pe c -t r um S v( ~ 0 , ~ ). I f A i ( i ~ ) i s t he F ou r i e r t r a n s f o r m o fa i ( t ) , t he f i r s t a d j u s t me n t c a n be ma de by mu l t i p l y i ngA i ( i ~ ) by t he r a t i o S v( O~ = ~ , ~ ) / S v ( ~ = ~ , ~ ), w he r eSv(~0, ~ ) i s the ac tua l response spec t rum for a i ( t ) , a n dt he n i nve r s e F ou r i e r t r a ns f o r mi ng t he p r oduc t ba c k t ot he t i me doma i n t o ge t a n a d j u s t e d a c c e l e r og r a m a ~ ( t ) .T h i s a d j u s t me n t p r oc e du r e c a n be r e pe a t e d a s ne c e s s a r yt o ma ke t he f i na l r e s u l t i ng a c c e l e r og r a m s pe c t r um c om-pa t i b l e t o t he de g r e e r e qu i r e d . T he numbe r o f a d j u s t -me n t s r e qu i r e d i nc r e a s e s w i t h t he da mp i ng r a t i o ~ ;how e ve r f o r t he c om m on l y u s e d r a t i o ~ = 0 . 05 , on l y t w oor t h r e e a d j u s t me n t s a r e ne e de d . N o t e t ha t i n t h i sa d j u s t m e n t p r o c e d u r e , o n l y t h e F o u r i e r a m p l i t u d e s o fa ~ ( t ) a r e c ha nge d , i . e . t he i r pha s e a ng l e s r e ma i n un -c ha nge d . B e c a us e t he pha s e a ng l e s r e ma i n unc ha nge d ,t he c o r r e l a t i on c oe f f i c i e n t be t w e e n a , ( t ) a nd a j ( t ) w i llbe t he s a me a s be t w e e n a ~ ( t ) a nd a j ( t ) . A l s o t h ec ohe r e nc y f unc t i on w i l l no t c ha nge w i t h t h i s a d j u s t me n ti f n o s m o o t h i n g o p e r a t i o n i s a p p l i e d w h e n c o m p u t i n gt he c ohe r e nc y . I f s moo t h i ng i s a pp l i e d s ome c ha nge w i l lb e f o u n d d e p e n d i n g o n t h e s m o o t h i n g w i n d o w l e n g t h .I n one e xa mpl e , t w o t i me s e r i e s w e r e ge ne r a t e d a nda d j us t e d t o be c ompa t i b l e w i t h t he s a me r e s pons e s pe c -t r um. I t w a s f ound t ha t t he c ohe r e nc y , w h i c h w a sc o m p u t e d w i th a s m o o t h i n g w i n d o w o f b a n d w i d t h 0 . 22H z , r e m a i n e d n e a r l y t h e s a m e s h o w i n g a g o o d m a t c hw i t h t he s pe c i f i e d mode l .4 .4. Q u a s i - s ta t i o n a ry p o w e r sp e c t ra l d e n s i t y f u n c t io n

E a r t hqua ke r e c o r ds , s uc h a s t hos e ob t a i ne d a t t heS M A R T - 1 a r r ay , s h o w a t e n d e n c y f o r n o n - s t at i o n a r i tyi n b o t h t h e t im e a n d f r e q u e n c y d o m a i n s . I n r e c o g n i t io no f t he s e t e nde nc i e s , t he non - s t a t i ona r y i n t he t i me do -ma i n c a n be e a s i l y mode l l e d by mu l t i p l y i ng e a c h s t a -t i o n a r y a c c e l e ro g r a m b y a n a p p r o p r i a t e s h a p e f u n c ti o n .T o r e p r e s e n t qu a s i - s t a t i ona r i t y i n the f r e que nc y do -m a i n , t h e t o t a l d u r a t i o n o f g r o u n d m o t i o n i s d i v i d edi n t o s e p a r a t e s e g m e n t s o f s t a t i o n a r y m o t i o n h a v i n g


13/18

H. H ao e t aL / Multiple-station ground motion processing 3051 2 3 4

X-

1000.00500.000.00~ " - 5 0 0 . 0 0o - 1 0 0 0 . 0 0

- 1 5 0 0 . 0 0

g e n e r a t e d r e s p o n s e s p e c t r a l c o m p a t i b l e g r o u n dm o t i o n a c c e l e r a t i o n at = O m (p e a k a c c = O . S g )

0.00 20.00x = l O 0 mlO00.0o500,000.00- 5 0 0 . 0 0- 1 0 0 0 . 0 0- 1 5 0 0 . 0 0 io .o o x = 2 0 0 m 2 0, 00

1000.00500,00 - 0.00~ " - 5 0 0 . 0 0- 1 0 0 0 . 0 0- 1 5 0 0 . 0 0 . Io .o o x = 3 0 0 m 2 0. 00

i ooo.o0 - l500.00,~ 0.00~ -- 5 0 0 . 0 0 ~ 1 . . . . . y- 1 0 0 0 . 0 0

-15 00 .00 F [ ' l - ~ 1 ~ l I " ~ I ' ~ ~ " ' ~ I - i / /- ~ f0.00 5,00 10.00 15,00 20,00time(~e)g e n e r a t e d r e s p o n s e s p e c t r a l c o m p a t i b t e g r o u r / d

m o t i o n d i s p l a c e m e n t a t x = O m ( , p ea k a c c = O . S g )20,000.00

- 2 0 . 0 0- 4 0 .0 0 . . . . . . "10.00 20.00

2 0.0 0 x = ] O 0 mo.00 ~ ~ , ~ - " ~

~ - 2 0 , 0 0- 4 0 . 0 0 0,00 20.00

2 0.0 0 x = 2 @ O m0.o0

~--~" - 20 .0 0--40.00 ]000 20.00

20.00 x = : 5 0 0 mooo-~ " - 2 0 . 0 0

-4,0.00 0.00 5,00 10.00 15.00 20.00t ime( sec )F i g . 1 2 . E x a m p le o f g e n e r a t e d g r o u n d m o t i o n s f o r s e v e r a l s i t e s l o c a t e d a lo n g a s t r a i g h t l i n e ; ( a , t o p ) a c c e l e r a t i o n , ( b , b o t t o m )

d i s p l a c e m e n t .


14/18

3 0 6 H. Ha o e t a l . / Mu l t ip le -s ta t ion grou nd mot ion process ingdifferent power spectral density representations. I n orderto render continuity at the begi nning and end of eachsegment, an overlapping of time windows is used alongwith a specified transitional weighting function.

5. ApplicationsTwo examples were chosen to test the algorithm

developed. The first one deals with the generation ofstationary ground motions for several sites along astraight line. The second example deals with the genera-tion of quasi-stationary ground mo tions for a 2-D net-work of stations.5 .1 . E x a m p l e N o . 1

Spatially correlated ground motion time Series weregenerated for four locations spaced at 100 m intervals(d= 0; 100; 200; 300 m) along the direction of wavepropagation. In this example, the Kanai -Taji mi powerspectral density function with ~g = 0.6 and ~0g --- 2.5 Hz,eq. (13); the Harichandran and Vanmarcke coherencyfunction, eq. (10), with A = 0.736, a = 0.147, and yapp= 2.5 km/s; and the commonly used Jennings shapefunction of the form

( t / t o ) 2 for 0 _< t < to,~( t) = 1 for t o < t < t . ,

e xp [ - 0 . 155 ( t - t , ) ] for t,


15/18

H . Ha o et al. / Multiple-station ground motion processing

1 . 0 0 - -

o 0 . 5 0 -

0.00 0.001.00

o 0.50

a b s o l u t e v a l u e o f c o h e r e n c y f u n c t i o n o f t h eg e n e r a t e d g r o u n d m o t i o n s1 . 0 0 -

I I I I I I I I I I I I I I I I I [5 . 0 0 1 0 . 0 0f r e q u e n c y ( h z )

0.00 I I I I l I I I I I i I I t t l I 0 / O0 . 0 0 1 0 .

0.50

0.00 , , , ,0.00

1 . 0 0

0"50i-

0 , 0 05 . 0 0

f r e q u e n c y ( h z )

i i i 5 , 0 0f r e q u e n c y ( h z )

I = ' ' i , ~ i I10.00

I i i I I i i i i I I I I I I0.00 5.00f r e q u e n c y ( h z )

1i i 110.00

F i g . 14 . C o h e r e n c y f u n c t i o n o f t h e g e n e r a t e d g r o u n d m o t i o n . C o m p a r i s o n w i t h m o d e l .

2 . 5 0a u to p o w e r s p e c t r a l d e n s i ty f u n c t i o n o f t h eg e n e r a t e d g r o u n d m o t i o n s

2.50

2 . o o

1.501 . o o

0 . 5 0

o . o o

2.50

x = O m

I I I I I I i i I [ I I i i i I i i i I} . 0 0 5 . 0 0 1 0 . 0 0f r e q u e n c y ( h z )

2 . 0 0 - -

1 . 5 0 -

1 . 0 0 -

0 . 5 0 -0 . 0 0 0 . 0 0

x = 2 0 0 m

I I I I 5 . 0 0 10.00f r e q u e n c y ( h z )

2 . 0 0

1 , 5 0

1.00

0 . 5 0

0 . 0 0

x = 1 0 0 m

~ A

I l i i ; I i I 1 ] i ; l i i t i I I0 . 0 0 5 . 0 0 I 0 . 0 0I f r e q u e n c y ( h z )

1 .5 o - x = 3 0 0 mA

1 . 0 0 -

0.50

0.00 I I I I i i 1 i I I I l 1 I i i l ; I 10 . 0 0 5 . 0 0 1 0 .0 0f r e q u e n c y ( h z )F i g. 1 5 . P o w e r s p e ct r a l d e n s i t y f u n c t i o n o f g e n e r a t e d g r o u n d m o t i o n . C o m p a r i s o n w i t h m o d e l .

30 7


16/18

3 ~

"su o! lm s jo ~t .~omlau leUO!SUOtu[p-oasl e ao j uoT l o t u puno . t ~ pa l ~a oua~ JO a l d t U ex 3 "L[ -~ . t~(o,~=)o~p, (oo,=)~m!:~

O ~ 0 O f O ~= = _

,..uo=X u.log= x uu o~,'( U~OmXpe~Daeue6 pg%o.~iue6

3i, OZ 0

u~og=X ~uo~;=xpt)~.ojeue6

, O L! l l l l I

= 0 ~] M i l l I

LuOS

( : o = ) o w n3~ O~ 0I I l [ l l l l l l ] l l l l l l l l l l o 0 9 -

,~ 1 ] ~ , ~ r , ~ , ~ ~ o o ~ ' 009

~ O g . X ~ O - Xp e 1 ~ e u e 6

"uo ! l o t u pun o . I ~ pa l R . ] aua~ a iq ! l ~du . ] oo ~ .x laad S "9 [ "~.[~ I(zq), (auanba.~ j

l h l l l l l i i h l l ' , i r i i [ : i I l l l i i L I i i i I ~ ] 1 [

- A / ~ ' /u J 0 0 8 = x

(zq)XouanbaJ~D L ~ L - O L :. OL

] H [ I I ] I I h i [ I l l ] ! h l l l l l i i [

3~

OL . ^ , , ~w O O E = x

OL

~ 0 ~(zq)XouanbaJ~

OL L =_ OL =_ OIl l I I I J l I I I I I I I I I ; J h I l f l J I I L

O L

c O L( z q ) X o u a n b a J }

=O L OL L L-OL ,_OLh l i ] l l i I h l l t l l , i l l l ] iI i i ; I ] I ~ [ T I I i L .

" ~ . S J /

LU 001- = X

O~

u J O = X~ O L

SUO!~.OLU pun o.J6 pa~ .oJeu a6~q~. ~o alqF~oduaoo ID.q.oads a= uods aJ

0L

,Oi.

O~

~ fu! ss a .9 o ad u o ! lo u ~ p u n o a X u o ! lo t s - a ld ! t l n I , g / .1 la on H "H 80~


17/18

H. H ap et al. / Mu ltiple-station ground motion processing 309( 3 ) T h e c o h e r e n c y f u n c t i o n s a l s o a r e i n g o o d a g r e e m e n t

w i t h t h e s p ec i fi e d H a r i c h a n d r a n a n d V a n m a r c k ec o h e r e n c y m o d e l a s c a n b e s e e n i n f i g . 1 4 .

t h i s m o d e l a n d a r e a l s o c o m p a t i b l e w i t h a n y s p e c i fi e dd e s i g n r e s p o n s e s p e c t r u m .

5 . 2 . E x a m p l e N o . 2 AcknowledgementsI n e x a m p l e N o . 2 , g r o u n d m o t i o n s w e r e g e n e r a t e d

f o r s t a t i o n s l o c a t e d i n a 2 - D s i te . F o u r t i m e s e r ie s w e r eg e n e r a t e d a t t h e f o u r c o r n e r s o f a 5 0 m s q u a r e s i t e . T h er e s ul t s o b t a i n e d b y p r o c e s s in g t he S M A R T - 1 d a t a f o re v e n t N o . 4 5 w e r e u s e d . T h e f o u r t i m e s e r i e s o f 4 0 sd u r a t i o n a n d A t = 0 .02 s , f i g . 17 , exh i b i t charac te r i s t i css i m i l a r t o t h e t i m e s e r i e s r e c o r d e d a t t h e S M A R T - 1a r r a y f o r e v e n t N o . 4 5 ( s e c t i o n 3 ) . T h e f o l l o w i n g f u n c -t i o n s w e r e u s e d i n g e n e r a t i n g t h e a c c e l e r a t i o n t i m eh i s t o r i e s :( 1 ) T h e c o h e r e n c y f u n c t i o n s u g g e s t e d b y t h e p r e s e n t

a u t h o r s a s g i v e n b y e q . ( 12 ) w i t h t h e p a r a m e t e r s o ft a b l e 2 w a s u s e d . T h e a p p a r e n t v e l o c i t y w a s a s -s u m e d , f o r s i m p l i c i t y , a s b e i n g c o n s t a n t a n d e q u a lto Yapp = 3 .5 k m / s , a n d t h e w a v e s w e re a s s u m e d t ob e p r o p a g a t i n g i n t h e p o s i t i v e x - d i r e c t i o n .

( 2 ) T h r e e p o w e r s p e c t r a l d e n s i t y f u n c t i o n s o f t h eK a n a i - T a j i m i f o r m w e r e u s e d f o r t h r e e ti m e w i n d o w s( 1 s t w i n d o w 0 - 5 s ; 2 n d w i n d o w 4 - 3 2 s ; 3 rd w i n d o w3 1 - 4 0 s ) . T h e s e s p e c t r a l d e n s i t y f u n c t i o n s h a dp a r a m e t e r s a s g i v e n i n t a b l e s 3 a n d 4 r e p r e s e n t i n gt h e N S c o m p o n e n t s r e c o r d e d d u r i n g e v e n t N o . 4 5 .

( 4 ) T h e s h a p e f u n c t i o n g i v e n b y e q . ( 1 4 ) w i t h t h ep a r a m e t e r s o f t a b l e 5 r e p r e s e n t i n g t h e N S c o m p o -n e n t s o f e v e n t N o . 4 5 w a s u s e d .

A n a n a l y s i s o f t h e f o u r a c c e l e r a t i o n t im e s e r ie s p r e -s e n t e d i n f i g . 1 7 s h o w s t h e f o l l o w i n g :( 1 ) T h e b a s i c f e a t u r e s o f p r o p a g a t i o n a r e o b s e r v e d i n

t h e f o u r t i m e s e r i e s . C o m p u t e d c r o s s c o r r e l a t i o n s ,c o h e r e n c y a n d p o w e r s p e c t r a l d e n s i t y f u n c t i o n s a r ei n g o o d a g r e e m e n t w i t h t h e p r e s c r i b e d m o d e l d e -f i n e d a b o v e .

( 2 ) T h e o v e r a l l f e a tu r e s o f t h e t i m e s e r ie s r e s e m b l eq u i t e w e l l t h o s e o f t h e r e a l t i m e s e ri e s . T h e v a r i a b i l -i t y o f p e a k g r o u n d a c c e l e r a t i o n s a m o n g t h e f o u rr e c o r d s ( a v e r a g e v a l u e o f 0 . 6 g ) i s w i t h i n t h e l i m i t so f m e a s u r e d v a l u e s [ 2 7 ] .

T h i s p a p e r p r e s e n t s t h e r e s u l t s o f r e s e a r c h c a r r i e dd u r i n g 1 9 87 a t t h e S e i s m o g r a p h i c S t a t i o n , U n i v e r s i t y o fC a l i f o r n i a , B e r k e l e y , w h i c h w a s s u p p o r t e d b y t h e N a -t i o n a l S c i e n c e F o u n d a t i o n u n d e r G r a n t N o . E C E -8 4 1 7 8 5 6 . T h e a u t h o r s e x p r e s s t h e i r s i n c e r e t h a n k s a n da p p r e c i a t i o n t o D r . B . A . B o l t , D i r e c t o r o f t h e S e i s m o -g r a p h i c S t a t i o n , fo r h is s u p p o r t a n d e n c o u r a g e m e n td u r i n g t h e e n t i r e s t u d y . T h e F - K c o m p u t e r s o f t w a r em a d e a v a i l a b l e t h r o u g h t h e S e i s m o g r a p h i c S t a t i o n b yD r s . N . A . A b r a h a m s o n a n d R . D . D a r r a g h a n d s p e c i a l -i z e d a s s i s t a n c e p r o v i d e d b y S .J . C h i o u , a r e g r e a t l ya c k n o w l e d g e d .

NomenclatureR ~ j( )S i j ( i w )p ~ j ( z )v , L ( ~ )d ~ jdW

OappF - Kfk xk ySo ( ~, )0)g, ~gF' ~ , "2 , B1 , ~2~ ( t )Sv(,~, ~)

c o r r e l a t i o n f u n c t i o n ,c r o s s s p e c t r a l d e n s i t y f u n c t i o n ,c o r r e l a t i o n c o e f f ic i e n t ,c o h e r e n c y f u n c t i o n ,s e p a r a t i o n d i s t a n c e p r o j e c t e d i n t h el o n g i t u d i n a l d i r e c t i o n o f p r o p a g a t i o n ,s e p a r a t i o n d i s t a n c e p r o j e c t e d i n t h et r a n s v e r s e d i r e c t i o n o f p r o p a g a t i o n ,a p p a r e n t v e l o c i t y ,f r e q u e n c y - w a v e n u m b e r s p e c t r u m ,f r e q u e n c y i n H z ,f r e q ue n c y i n r a d / s ,w a v e n u m b e r o f t h e x - d i r e c t i o n ,w a v e n u m b e r i h t h e y - d i r e c t i o n ,p o w e r s p e c t r a l d e n s i t y f u n c t i o n ,K a n a i - T a j i m i p a r a m e t e r s v a l u e s,s c a l i n g f a c t o r ,c o h e r e n c y p a r a m e t e r s v a l u e s ,s h a p e f u n c t i o n ,p s e u d o - v e l o c i t y r e s p o n s e s p e c t r u m f o rf r e q u e n c y ~ a n d d a m p i n g r a t i o ~ .

6. Concluding statementsA c o h e r e n c y m o d e l o f n o n i s o t r o p i c f o rm f or c o m p o -

n e n t s o f m o t i o n a t m u l t i p l e s t a t i o n s is d e v e l o p e d h e r e i nw h i c h i s b e l i e v e d to h a v e a g e n e r a l a p p l i c a t i o n . F u r t h e r ,a s i m p l e n u m e r i c a l p r o c e d u r e i s p r e s e n t e d w h i c h a l l o w st h e g e n e r a t i o n o f t i m e s e r ie s w h i c h a r e c o m p a t i b l e w i t h

R eferen ces[1] N.M . New mark an d E. Rosenblueth, Fund am entals ofEarthquake Engineering (Prentice Hall, Englewood Cliffs,NJ, 1971).[2] J.P. Wolf, P. Obernhuber and B. Weber, Response of anuclear power p lant on aseismic bear ings to hor izontal


18/18

31 0 H. Ha o e t a l . / Mu l t ip le -s ta t ion groun d mot ion process ingpro pag at in g waves, Ear th q. Engrg. and S t r . Dyn . 11 (1983)4 8 3 - 4 9 9 .

[3] R .T . D ua r t e , S pa t i a l l y va r i ab l e g round m ot io n m ode l s fo rt h e e a r t h q u a k e d e s i g n o f b r i d g e s a n d o t h e r e x t e n d e ds t ruc tu re s , 8 th W or ld C onf . E a r thq . E ngrg . , V o l I I , pp .613-6 20 (P ren t i ce H a l l , E ng l ew o od C l i ff s , N J , 1984) .

[4 ] J . E . L uco , R esponse o f a r i g id founda t ion t o a spa t i a l l yr andom ground m ot ion , E a r thq . E ngrg . and S t r . D yn . 14(1986) 891-908.

[5] B .A. Bol t , C .H. Loh, J . Penzien, Y .B . Tsai and Y,T . Yeh,P r e l i m i n a ry R e p o r t o n t h e S M A R T - 1 s t r o n g m o t i o n a r r a yi n T a iw a n , R e p o r t U B C / E E R C - 8 2 / 1 3 , U n i v e r s i ty ofCal i fornia , Berkeley (1982) .

[6 ] N . A . Abraham son , E s t im a t ion o f s e i sm ic w ave cohe rencya n d r u p t u r e v e l o c i t y u s i n g t h e S M A R T - 1 s t r o n g m o t i o na r ra y r ec o rd i ng s , R ep o r t U C B / E E R C - 8 5 / 2 , U n i v e rs i tyof Cal i fornia , Berkeley (1985) .

[7] C .S . O l ive i r a, V ar i ab i l i t y o f s t rong g rou nd m ot io n cha r -ac t e r i s t ic s ob t a in ed in S M A R T -1 a r r ay , P roc . 12 th R e-g iona l S em ina r on E a r thquake E ng inee r ing , H a lk id ik i ,Greec e (1985).

[8 ] R . B . D ar r agh , Ana lys i s o f t he nea r - source w aves : S ep-a ra t ion o f w ave t ypes us ing s t rong m ot ion a r r ay r eco rd -ings , P hD T hes i s , U n ive r s i t y o f C a l i fo rn i a , B e rke l ey (1987) .

[9] B .A . B o l t and S . J. C h iou , S t rong m ot ion a r r ay a na lys i s o ft he N ovem ber 14 , 1986 T a iw an ea r thquake , P e r sona lC o m m u n i c a t i o n ( 19 87 ).

[10 ] J . P enz ien , L ec tu re no t e s on ea r th quak e eng inee r in g ana l -ys is , Univers i ty of Cal i fornia , Berkeley (1985) .

[11 ] C . S . O l ive i r a , P robab i l i s t i c a s se s sm en t o f s t rong g roundm ot ion , i n : S t rong M ot ion S e i sm ology , E ds . M . O . E rd ikand M . N . T oksoz , Se r i e s C : M a them a t i ca l and P hys i ca lSciences, Vol . 204, pp. 405 -46 0 (D. Rei del Pub l ishi ng Co, ,D ord rech t , H o l l and , 1987) .

[12 ] C .H . L oh , J . P enz ien and Y .B . T sa i , E ng in ee r ing ana lys i so f S M AR T -1 a r r ay acce l e rogram s , E a r thq . E ngrg . and S t r .D yn . 10 (1982) 575-591 .

[13 ] C . H . L oh and J . P enz ien , I den t i f i ca t i on o f w ave t ypes ,d i r ec t ions , and ve loc i t i e s u s ing S M AR T -1 s t rong m ot iona r r ay da t a , 8 th W or ld C onf . E a r thq . E ngrg . , V o l I I , pp .191 -198 (P ren t ice Hal l , Eng lewoo d Cl i ffs , N J , 1984).

[14] N . A . A bra ham son an d B . A . B o l t, Ar r ay ana lys i s andsyn thes i s m app ing o f s t rong se i sm ic m ot ion , i n : S t rong

M ot ion S yn the t i c s , E d . B . A . B o l t , pp . 55 -90 (Academ icPress , Or lando, 1987) .[15 ] R . H ar i chandran and E . V anm arcke , S pace - t im e va r i a t i on

o f e a r t h q u a k e g r o u n d m o t i o n , R e s e a r c h R e p o r t R 8 4 - 1 2 ,D ep t , C iv i l E ngrg . , M assachuse t t s Ins t i t u t e o f T echno logy(1984).

[16 ] C . H . L oh , A na lys i s o f t he spa t i a l va r i a t i on o f s e i sm icw a v e s a n d g r o u n d m o v e m e n t s f o r S M A R T - 1 a r r a y d at a ,E a r thq . E ngrg . and S t r . D yn . 13 (1985) 561-581 .

[17] A . M i t a and J . E . L uko , R espon se o f s t ruc tu re s t o as p a t i a l l y r a n d o m g r o u n d m o t i o n , 3 r d U S N a t i o n a l C o n f .on E a r thq . E ngrg . , pp . 907-918 , C ha r l e s ton , US A (1986) .

[ 1 8 ] N . A . A b r a h a m s o n , C o h e r e n c y a n a l y s i s o f s t r o n g g r o u n dm o t i o n , P e r s o n a l c o m m u n i c a t i o n ( 19 88 ).

[19 ] Y . B . T sa i , E m pi r i ca l cha rac t e r i za t ion o f f r ee - f i e l d g roundm o t i o n f o r s o i l - s t r u c t u r e i n t e r a c t i o n a n a l y s i s , 2 n dW orkshop on S t rong M ot ion Ar rays , T a ipe i ( 1988) .

[20 ] K . K ana i , A n em p i r i ca l f o rm ula fo r t he spec t rum o fs t rong ea r thquake m ot ions , B u l l . E a r thq . R es . I n s t . , Un i -vers i ty of Tokyo, Vol . 39 (1961) .[21 ] J . P enz ien and M . W atabe , C ha rac t e r i s t i c s o f 3 -d im en-s iona l ea r t hquake g round m ot ions , E a r thq . E ngrg . andStr . Dyn. 3 (1975) 365-373.

[22 ] Y . F ukum or i , M . S a to , T . K iku t a , K . Y an abu a nd K .T o k i , D e t e c t i o n o f d i s p e r s i o n c h a r a c t e r i st i c s o f a p p a r e n tw ave ve loc i t y , 8 th W or ld C onf . E a r thq . E ngrg . , V o l I I , pp .223 -230 (P ren t i ce H a l l , E ng l ew o od C l i ff s , N J , 1984).

[23 ] E . S am aras , M . S h ino zuka and A . T su ru i , T im e se r i e sg e n e r a t i o n u s i n g t h e a u t o r eg r e s s iv e m o v i n g a v e r a g e m o d e l ,i n : S tochas t i c M echan ic s , E d . M . S h inozuka , D ep . C iv i lE ngrg . and E ngrg . M echan ic s , C o lum bia Un ive r s i t y , V o lI I , pp . 1 -130 , N ew Y ork (1987) .

[24] F . R . G an tm ac he r , T he T heo ry o f M a t r i ce s (C he l sea P ub-l i sh ing C om pany , N Y , 1977) .[ 2 5] K .E . A t k i n s o n , A n I n t r o d u c t i o n t o N u m e r i c a l A n a l y s i s

( John W i l ey and S ons , N Y , 1978) .[26] N . M . N e w m ark an d W . L H a l l , S e i sm ic des ign c r i te r i a fo r

nuc l ea r r eac to r f ac i l i t ie s , 4 th W o r ld C onf . E a r thq . E ngrg .,V o l I I, pp . B 4 / 37 -B 4 /50 , S an t i ago , C h i l e (1969).

[27 ] C .S . O l ive i r a , F . P im e n te l and J . Azevedo , V a r i ab i l i t y o fs t r o n g g r o u n d m o t i o n o r d e r e d p e a k s a t a s i t e b a s e d o ndense se i sm ic a r r ays , 8 th E urop . C onf . E a r thq . E ngrg , ,Vol . 6 , L isbon, Por tugal (1986) .

Hao, H. Oliveira, C.S. Penzien - 1989 - Multiple-Station Ground Motion Processing and Simu

Documents

Transcript of Hao, H. Oliveira, C.S. Penzien - 1989 - Multiple-Station Ground Motion Processing and Simu