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T h e B o d y F o r c e E q u i v a l e n t o a n E a r t h q u a k eA T u t o r ia l

os Pu jo lERI

As noted in the In t roduc t ion to the c lassic paper by Burr idgeand Knopoff (1964) ( th is i ssue) , th is paper p layed a c r i t ica lro l e i n t h e d e v e lo p me n t o f t h e t h e o ry o f t h e e a r th q u a k esource . Because the mathemat ica l deve lopment presupposes alevel of knowledge b eyond tha t o f the se ismologist or se ismol-ogy studen t n ot ac tively engaged in theore t ica l s tudies, i wi l ld iscuss a num ber of basic resul ts tha t wi l l he lp un derstan d thepaper as wel l as to put i t in to a broader perspec t ive . I hopeth is tu tor ia l wi l l en t ice those untheore t icaUy or iented to readthe Burr idge and Knopoff paper , so tha t they can apprec ia tethe essential elegance (as opposed to a brute-force approach)of the a rgumen ts u sed to der ive the expression for the bodyforce equiva lent . To make the tu tor ia l se l f -conta ined a fewexcerpts f rom the In t roduc t ion a re repea ted here .

For a homogeneous iso t ropic e last ic medium the equa-t ion tha t governs the propaga t ion of se ismic waves can bewri t ten as

p i i x , t ) = ; t, + 2 b t )g ra d d iv t t x , t )- / ~ c u r l c u r l u x , t ) + f x , t ) 1 )

where u i s the vec tor tha t descr ibes the d isp lacement of thepar t ic les in the medium, x = x I , x 2 , x 3) denotes the observa-t ion poin t , such as a se ismic s ta t ion , the double dots indica tesecond der iva tive wi th respec t to t ime , ~. and/ , t a re the L amd'sparameters , p i s densi ty , and f i s the body force per uni t vol -ume tha t i s the source of the waves e . g . , A i d and Richards,1980) .

Al though solv ing Equat ion 1 for a g iven f i s a d i ff icu l tproblem, so lu t ions for s imple forces have been known since1849. The problem tha t se ismologists faced when they beganto s tudy the gen era t ion o f waves by an ear thquak e was to f indthe body force to be used in Equat ion 1 . As noted be low, there la t ion be tween faul ts and ear thquakes was wel l establ ishedbefore a model for the ear thquake source was ava i lable andthe problem tha t se ismologists addressed was, br ie f ly s ta ted ,the fo l lowing: W hat i s the equiva lent bod y force tha t in theabsence of the faul t wil l cause exac t ly the sam e disp lacem entfield as slip on the fault?

To see how th is problem was approached f i rs t we mustsu mma r i z e t h e w o rk d o n e to w a rd so lv in g Eq u a t io n 1 . In1849 Stokes presented the so lu t ion for the case of a spa tia l lyconcen tra ted force d i rec ted a long a coordina te axis . Loveconfi rmed th is so lu t ion in 1904 using a d i ffe rent approach

(Miklowitz , 1978) and in addi t ion in t roduced what he ca l leddoub le forces , wh ich are pairs of parallel forces in oppo site

di rec t ions (Love , 1927) . I f the forces have d i ffe rent l ines ofac t ion the pa i r i s known as a couple ; o therwise i t i s known asa d ipole . Assuming tha t the d istance be tween the forces i sinf in i tesimal , the so lu t ion of Equat ion 1 for a doub le forcec a n b e o b ta in e d w i th o u t mu c h e f fo rt w h e n f is a c o n c e n t ra t e d(or point) force. Let f = g(t) 8(x - ~)e., wh ere 6 is Dirac 's delta,~: indicates the locatio n of the :1orce , and e j i s a uni t vec tor inthe : 9 direc t ion , and le t u i j (x , ~ , ~) b e the co rrespon ding i thc o m p o n e n t o f th e v e c to r u . H e re t i s t h e t ime me a su re d f ro mthe t ime a t which the source began to ac t . Expressions for u / jand the i r der iva t ions can be found in Aki and Richards(1 9 8 0 ) , Mik lo w i t z (1 9 7 8 ) , a n d Pu jo l (2 0 0 3 ) , a mo n g o th e r s .For our purposes, however , we need to know only tha t suchexpressions exist .

To f ind the d isp lacement due to a couple consider now aconcentra ted force f having t ime-dependence g t ) / h ac t ing a t

= 0 , O , h / 2 ) in the x 1 d i rec t ion , and anoth er force o f equa lma gni tud e ac t ing a t ~r = (0 , 0 ,- h / 2) in the nega t ive x 1 d i rec-t ion (Figure 1) . Th e sec ond force can be wri t ten as - f ' andassumed to be in the posi t ive x I d irec t ion . T hen the corre -sponding d isp lacements a re g iven by + u i.~ (x, t; ~ ) / h , w h e re U r ]i s the d isp lacement due to the force wi th t ime-dependenceg(t ) . Fina l ly , le t h go to zero . Un der these condi t ions, the d is-p lacement due to th is couple can be wri t ten as

u / , , ; . , , ; o ) - i r a [ u ( ; / ~ 1 7 6 u , , ( , , , ; / o . o . -

0~3where 0 = (0 , 0 , 0) and the superscr ip t s tands for s ingle cou-ple (af te r M iklowitz , 1 978) . The r ig ht-han d side of Equ at ion2 can a lso be obta ined by adding the d isp lacements wri t ten inTaylor expansions to f i rst order (Udfas, 1999) . This der iva tiveoffe rs a prac t ica l and genera l way to obta in the d isp lacementf ie ld due to a couple , but one can ac tua l ly s ta r t wi th the twod i sp l a c e me n t s i n d i c a t e d in Eq u a t io n 2 , i n t ro d u c e a n u mb e ro f a p p ro x ima t io n s , a n d th e n t a k e t h e l im i t. Th i s a p p ro a c hwas used by Lay and Wal lace (1995) to de te rmine the fa rf ield , but i t is much simpler to use the der iva t ive (see Pujo la n d H e r r m a n n , 1 9 9 0 ).

S e i s m o l o g i c a l R e s e a rc h e t te r s M a r c h / A p r il 2 0 0 3 V o l u m e 7 4 ,N u m b e r 2 1 6 3

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A )

_ f ~

~ 3

h / 2

~ 2

f f- h / 2

B )

- h / 2

- f

~ 2

h 2

,~ , F i g u r e 1 . G r a p h i c a l r e p r e s e n t a t io n o f s i n g l e c o u p le s w i th f o rc e s _.f nA ) th e x ~ d i r e c t io n a n d B ) t h e x 3 d i r e c t i o n .

Let us note some features of this s ingle couple . F irs t, theabsolute value of g t ) l h approaches infini ty as h approacheszero. Second, the arm of the couple is in the x 3 direct ion.Thi rd , the couple t ends to p roduce a counterc lockwise ro ta -t ion about the x 2 ax i s . Thus , the couple has a ne t m om ent ,wh ich from mech anics we know i t is equal to force t imes dis -tance, or h x g / h = g . Also note that the three subindices of uon the l e f t -hand s ide o f Equa t ion 2 denote the com pone nt o fthe displace men t vector, the direct ion o f the force, and thedirect ion of the arm of the couple , respectively.

No w cons ide r a p rob lem s im i la r to the p rev ious one b u tle t the forces be in the x 3 direct ion and the arm of the cou plei n th e x 1 d irec t ion , wi th the nega tive fo rce ac t ing a t-h /2 ,0 ,0 ) (F igure 1). In this case the displacemen t isgiven by

= a u , , = , , ; o ) 3 )

Th e effect of this couple is a c lockwise ro ta t ion abo ut the x 2ax is and the s ign o f i ts m om ent i s oppos i t e to tha t o f the p re -vious couple . Therefore , i f the two couples are com bine d in toa s ingle sys tem of forces the resul t is a doub le cou ple w itho utm o m e nt wi th d i sp lacem ent equa l to the sum of the d i sp lace -ments given in Equat ions 2 and 3:

UiSI(X,,;Os o _ U/ 1 S ( X,;Os o ~ _ a U i l ( X t ; O ) _[_ a U iJ (X ; O )a ~ 3 a ~ l

4 )

Th e fi rs t equal i ty in Equa t ion 4 arises because we can changethe o rde r o f the two te rm s on the r igh t -hand s ide . Note tha tthe fac t tha t the dou ble couple does no t have a ne t m echani -ca l m o m e nt does no t m ean tha t i t does no t have s treng th a s aseismic source. For example, the doub le couple eq uivalent toan earthqu ake has zero mechanical mo me nt , b ut i ts s ize isgiven by the scalar se ismic mo me nt .

The deba te ove r the m echanism o f ear thquakes spannedseveral decades . I t began in the 1920 s an d las ted thro ug h th eearly 1960 s a nd is well described by Staude r (19 62). As aresul t of observat ional s tudies that showed a quadrantal pat-tern of P-wave di la ta t ions and co mpress ions an d the recogni-t ion of the c lose re la t ion between earth quakes a nd faul ts , twoleading models , based on the couples described above, wereproposed. O ne of them , favored by California se ismologists ,was a single couple, while Japanese seismologists preferred adouble couple . A problem with the s ingle couple was that i thad a ne t m om ent , w hich was an undes i rab le fea tu re becausei t im pl ied a nonequi l ib r ium condi t ion ins ide the Ea r th . Inprinciple , the observat ions should have a l lowed discrimina-t ion between the two models , but this was not poss ible fortwo reasons. First, the single and double couples have similarP-wave radiat ion pat terns (F igures 2 and 3). Second,al though the corresponding S-wave radiat ion pat terns arequi te different (F igures 2 and 3), the qual i ty of the data wasinadequa te to e s tab l i sh un ique ly which m ode l was m oreappropria te . In addi t ion, even i f the observat ions had a l lowedthe ident if icat ion of the r ight mod el , the fact wo uld haveremaine d that no theo ret ical jus t i f icat ion exis ted for e i ther ofthem .

As thi s deba te con t inued , d ur ing the i950 s ano the r the -oretical tool was broug ht to bear, nam ely dis locat ion theory.This theory o r ig ina ted in the work of a num ber o f I t a l i anmathe matic ian s , part icularly Volterra, w ho used the word d i s -to r s io n e . Dislocat ion is Love s t rans la t ion (Love, 1927). A dis -loca t ion can be v i sua l i zed th rough the fo l lowing thoughtexperiment , based on Steketee (1958). Consider a cut madeover a surface E w ithin an e las tic body. After the cu t has beenmad e there are two surfaces , indicated with s and E-, wh ichwil l be deforme d different ly by appl icat ion of some force dis -t r ibut ion . If the com bine d sys tem of forces is in s ta t ic equi l ib-

1 6 4 S e i s m o l o g i c a l R e s e a rc h L e t t e rs V o lu m e 7 4 . N u m b e r 2 M a r c h / A p r i l 2 0 0 3

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P SV

x 1 x2 x 1 x2

A) B), Figure 2. P- and SV-w ave radiation patterns for the single co uple of Figure lB. In these plots only the orien tation of the observation location with respectto the source location, not the distance, is considered. From P ujol (200 3). This source does not generateSHwaves. For the equations used to generate heplots see also Pujol and Herrmann (1990).

Px3

X x2 x 1 x2

(A ) (B )

x3 x3sv L s _

(C ) (D )i Fi0ure 3. P-, S-, V-, and SH-wave radiation patterns or the double couple obtained by superposition of the two single couples of Figure 1. The S pattern

corresponds to the absolute value of the S-wave displacement. From Pujol (2003). For the equations used to generate the plots see also Pujol and Herrmann(1990). The only d ifference between his P radiation pattern and that in F igure 2 is a factor of 2.

Seismological Research Letters March/April 2003 Volume 74, Number 2 165

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r ium, the body wi l l remain in the or ig ina l equi l ibr ium sta te .Th e resul t of th is opera t ion is a d iscont inui ty in the d isp lace-ment across E, known as a d is loca t ion , which is accommo-da ted by deformat ion wi th in the body. This descr ip t ionshould be compared to our model for a tec tonic ear thquake ,which is represented b y slip on a faul t p lane . Wh en an ear th-quake occurs, the two sides of the faul t suffe r a sudden re la -t ive d isp lacement wi th respec t to each o ther , and th isd iscont inui ty in the d isp lacement across the faul t i s thesource of the d isp lacement e lsewhere in the med ium. A gooddiscussion of the re la tion be tween ear thquakes and d isloca-t ions i s provided by Steke tee (1958) , a l though h is ana lysiswas rest r ic ted to the s ta t ic case . Co ntr ibu t ions f rom otherauthors a re descr ibed by Stauder (1962) .

Th e d e b a t e e n d e d w h e n Ma ru y a ma (1 9 6 3 ) , H a sk e l l( 1 9 6 4 ) , a n d Bu r r id g e a n d K n o p o f f (1 9 6 4 ) d e mo n s t ra t e d th a tthe body force equiva lent was a double couple . In the threecases the der iva t ions w ere based on a num ber of resul tsder ived in the co ntext o f theore tica l e last ic i ty and wave prop-aga t ion . W hile the f i rs t two au thors add ressed the case ofhomogeneous iso t ropic media , what d is t inguishes Burr idgeand Kn opo ff s p aper is i ts generality, as their results apply tohe te rogeneous aniso t ropic media . To achieve th is genera l i tythe authors re l ied heavi ly on the powerful formal manipula-t ion tools provided by D irac s de l ta (see be low).

Because the ana lysis in Burr idge and Knopoff (1964;here inaf te r equa t ion numbers f rom th is paper wi l l be pre-ceded by BK) is comple te ly genera l , the s ta r t ing poin t i s thee q u a t io n

s q X , t ) ) , j- - ~ O X ) U i X , t ) = - - f i x , r 5 )(see B K. 1, page 154), w hich has E qua tion 1 as a special case.Equat ion 5 appl ies to aniso t ropic he te rogeneous media andcan be obta ined from the fo l lowing re la t ions:

~ U i = P U ir i j , j + f - P a t i ( 6 )

where P indica tes densi ty and ~ is the i th co mp one nt of theforce f, and

r / s = c o p q E p q = c , j p q u ~ , q 8 )

og = q ( 9 )Equ at ion 6 is Cauchy s equ a t ion of mot ion , T~ and epq indi-cate the stress and strain tensors, u corresponds to displace-me nt , and the f i rs t equa l i ty in Equat ion 8 is Hook e s law ( e . g . ,

Aki and Richards, 1980) . For readers not famil ia r wi th indi-c ia l nota t ion , when Equat ion 5 is wri t ten in fu l l i t becomes

(10)

To go from th is equa t ion to Equat ion 5 the fo l lowing ru lesare used: Two repea ted indices imply summation and theindices to the r igh t of a comm a indica te der ivat ives wi thre spe c t to t h e c o r re sp o n d in g c o m p o n e n t o f x.

A basic resul t in Burr idge and Knopoff (1964) i s a so-ca l led representa t ion theorem, which is an expression involv-ing in tegra ls tha t re la tes a pa i r of forces and the i r corresp ond-ing d isp lacements , w i th one of the forces be ing a Dirac s de l tain the space and t ime domains. The d isp lacement corre -spon ding to th is concentra ted force is kno wn as Green s func-t ion . Given th is force-disp lacement pa i r , the d isp lacementgenera ted by a second force is g iven by BK.9, p age 156. Thisequa t ion needs some explana t ion . The faul t p lane on whichthe ear thquake occurs i s represented by the surface E in Fig-u re 1 o f Bu r rid g e a n d K n o p o f f (1 9 6 4 ) , w h ic h in p r in c ip l eneed not be p lanar . This surface is embedded in a volume Vof an e lastic med ium bou nded by a surface S. To in t rodu ce Einto the pro blem two steps a re needed. Fi rs t , one of the vol-ume in tegra ls in BK.4 , page 155, i s converted to a surfacein tegra l using Gauss (or d ivergence) theorem . This g ivesB K.5 , page 155, which in turn was wri t ten for the cas e of aforce g i (x , t ) concen tra ted both in space and t ime and assum -ing tha t the d isp lacement a nd G reen s func t ion sa tisfy homo -geneous boun dary c ondi t ions o n the surface S. Th ecorresponding resul t i s BK.8 , page 155, but i t cannot beappl ied d i rec t ly to the ear thquake problem because the d is-p lacement has a d iscont inui ty on ~, which is wi th in the in te -gra t ion volum e V. Th e seco nd step is to accoun t for th issi tua t ion , which requires modif ica t ion of Gauss theorem asfo l lows. Given a vec tor-va lued func t ion b(x , t ) , cont inuouswith in a volume V, the theorem sta tes tha t

~ r V . b d V = ~ s b . n d S (11)Th e in tegrand on the le f t s ide is the d ivergence of b , whichusing indic ial nota t ion is wri t ten as b i , i (a sum ma tion over i i simpl ied) , and n is an outward uni t vec tor normal to S . Inindic ia l nota t ion the dot produ c t b .n i s wri t ten as b i n i .Equ at ion 1 t was used to obta in BK .5 , page 155. W he n b isd iscont inu ous across a surface E wi th in V the theorembecomes

J v V b d V = J s b n d S - f x [ b ] v d E (12)where v i s a uni t vec tor norma l to E an d [b] is the d i ffe rencein the values of b across E ( e . g . , Kraut , 1967; Pujo l , 2003) .Th e p roo f of th is resul t requires the fo l lowing steps: (a ) Sur-

166 Se ismo log ica lResearchL e t t e rs V o l u me7 4 N u m b e r 2 M a r c h / A p r il 2 0 0 3

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c

i F i g u r e 4 . G e o m e t r y f o r G a u s s t h e o r e m i n t h e p re s e n c e o f a s u r f a c e o fd i s c o n t i n u i t y ( E ) . Y _ ., a n d E - i n d i c a t e t h e t w o s i d e s o f Y _ ,,.C + a n d 6 - c o m -b i n e d fo r m a c a v i t y C t h a t s u r r o u n d s Z ; . v + a n d v - a r e u n i t n o r m a l v e c t o r s .T h e v e c t o r v i n E q u a t io n 1 2 i s e q u a l t o v - .r o u n d E b y a c a v i t y h a v i n g s u r f a c e C ( F i g u r e 4 ), ( b ) i n t r o d u c ea v o l u m e V ' h a v i n g e x t e rn a l s u rf a c e S a n d i n t e r n a l s u rf a c eC , i . e ., V i s e q u a l t o V w i t h t h e c a v i t y s u b t r a c t e d , ( c) a p p l yG a u s s ' t h e o r e m t o V ' , w i t h i n w h i c h b is c o n t i n u o u s , a n d ( d )u s e a l i m i t i n g a r g u m e n t t o e x t e n d t h e t h e o r e m t o t h e w h o l eV.

C o m b i n i n g B K . 8 , p a g e 1 5 5 , a n d E q u a t i o n 1 2 g iv e sB K . 9 , p a g e 1 5 6 , w h i c h i n t u r n m u s t b e r e w r i t t e n t o d e r i v et h e e x p r e s s i o n f o r t h e b o d y f o r c e e q u i v a l e n t . T h i s r e q u i r e si n t r o d u c i n g v o l u m e i n t e g r a l s i n t o t h e s e c o n d t e r m o n t h er i g h t - h a n d s i d e o f B K . 9 , p a g e 1 5 6 , w h i c h is d o n e u s i n g~ x - ~) , w i t h ~ : a v e c t o r t h a t i d e n t i f ie s p o i n t s o n s I n B u r r -i d g e a n d K n o p o f f ( i 9 6 4 ) t h i s d e l t a is w r i t t e n a s 8 ( x ;~ ) . T h e n ,f o r a g i v e n f u n c t i o n f ( x ) , t h e f o ll o w i n g ap p l ie s :

/ q > ) = l - v (13)T o i n d i c a t e t h e i n t e g r a t i o n v a r i a b l e t h e s u b s c r i p t x w a s u s e d .T h i s e x p l a i n s t h e f i r s t e q u a t i o n i n t h e E q u i v a l e n t F o r c e s s e c -t i o n o f B u r r i d g e a n d K n o p o f f ( 1 9 6 4 ) , p a g e 1 5 6 . T h e s e c o n de q u a t i o n i s s i m i l a r , b u t n o w a n e x p r e s s i o n i n v o l v i n g t h ed e r i v a t i v e o f th e d e l t a i s u s e d , n a m e l y

] x 7O xq x+ =r

( 1 4 )

(e.g. , P u j o l , 2 0 0 3 ) . U s i n g E q u a t i o n s 1 3 a n d 1 4 , B K . 9 , p a g e1 5 6 , c a n b e w r i t t e n a s i n t h e e q u a t i o n p r e c e d i n g B K . 1 0 , p a g e1 5 6 , w h i c h l e a d s d i r e c t l y t o t h e e x p r e s s i o n f o r t h e b o d y f o r c ee q u i v a l e n t , g i v e n b y B K . 1 0 . F o r a g i v e n v a lu e o f p , t h a t e q u a -t i o n i n v o l v e s u p t o 2 7 t e r m s c o m i n g f r o m t h e s u m s o v e r t h ei n d i c e s i , j , a n d q ( i . e . , 3 3 t e r m s ) .

B K . 1 0 i s c o m p l e t e l y g e n e r a l . T o s p e c i a l i z e i t t o t h e c a s eo f i s o tr o p i c m e d i a w e u s e

( 1 5 )

(e.g. , A k i a n d R i c h a r d s , 1 9 8 0 ) w h e r e 8 /j i s K r o n e c k e r ' s d e l ta ,w h i c h i s e q u a l t o o n e w h e n I = j a n d e q u a l t o z e r o o t h e r w i s e .I f , i n a d d i t i o n , i t i s a s s u m e d t h a t t h e t r a c t i o n v e c t o r h a s n od i s c o n t in u i t y , t h e s e c o n d t e r m i n B K . 1 0 v a n is h e s. C o n t i n u i t yo f t h e t r a c t io n i s a r e a so n a b l e a s s u m p t i o n f o r a n e a r t h q u a k es o u r c e b e c a u s e i t c o r r e s p o n d s t o N e w t o n ' s l a w o f a c t i o n a n dr e a c t i o n a n d i s e x p e c t e d u n l e s s e x t e r n a l f o r c e s a r e a p p l i e d t oE . B u r r i d g e a n d K n o p o f f ( 1 9 6 4 ) d i s c u s s e s s e v e r a l s p e c i f i ce x a m p l e s o f f a u l t i n g . O n e o f t h e m h a s E o n t h e ( x 1 , x 2 ) p l a n e ,s o t h a t t h e n o r m a l v e c t o r v i s i n t h e x 3 d i r e c t i o n , w h i l e t h e d i s -c o n t i n u i t y i n d i s p l a c e m e n t i s n o n z e r o i n t h e x 1 d i r e c t i o n o n l y .In th i s case

( 1 6 )U n d e r t h e s e c o n d i t i o n s E q u a t i o n 1 0 b e c o m e s

C I p q (17)

T h e s u b i n d i ce s 1 a n d 3 c o m e f r o m [ u l ] a n d v3, respectively,w h i c h a r e t h e o n l y c o m p o n e n t s o f u a n d v d i f fe r e n t f r o mz e ro . E q u a t i o n 1 7 s h o w s t h a t t h e p o s s ib l e 2 7 t e r m s i n B K . 1 0r e d u c e t o o n e b e c a u s e t h e o n l y n o n z e r o c o n t r i b u t i o n s c o m ef r o m p = 1 a n d q = 3 a n d p = 3 a n d q = 1. I n b o t h c a se s t h er e s u l t is p . F i n a l ly , u n d e r t h e a s s u m p t i o n t h a t E i s l i m i t e d t ot h e o r i g i n ( p o i n t s o u r c e a p p r o x i m a t i o n ) , t h e t h r e e c o m p o -n e n t s o f t h e b o d y f o r ce a r e g i v e n b y t h e e x p r es s io n s o n p a g e1 5 7 o f B u r r i d g e a n d K n o p o f f ( 1 9 6 4 ) . T o i n t e r p r e t e 1 a p p r o x -i m a t e t h e d e r i v a t i v e i n e , ( i n c l u d i n g t h e m i n u s s i g n ) a s f o l -lows

~3 8(x3 )___ h 8 x 3 - - 8 x 3 + (18)

S i n c e e 1 i s in t h e x 1 d i r e c t i o n , t h e l e f t - h a n d s i d e o f E q u a t i o n1 8 c a n b e r e p r e s e n t e d b y a c o u p l e i n t h e x , d i r e c t i o n w i t h a na r m i n t h e x e d i r e c t i o n , a s s h o w n i n F i g u r e 5 . i n a s i m i l a r w a yi t c a n b e s h o w n t h a t u 3 c o r r e s p o n d s t o a c o u p l e i n t h e x 3 d i r e c -t i o n w i t h a n a r m i n t h e x 1 d i r e c t i o n ( F i g u r e 5 ) . T h e r e f o r e , t h eb o d y f o r c e e q u iv a l e n t is a d o u b l e c o u p l e w i t h o u t m o m e n t .

F i n al ly , it m u s t b e n o t e d t h a t t h e i m p o r t a n c e o f B u r r id g ea n d K n o p o f f ' s p a p e r g o e s m u c h f a r t h e r t h a n h a v i n g d e m o n -s t r a t e d t h a t t h e b o d y f o r c e e q u i v a l e n t is a d o u b l e c o u p l e . I nf a c t, t h e p a p e r w a s u s e d b y A k i ( 1 9 6 6 ) t o d e r i v e th e w e l lk n o w n r e l a t i o n M o = / x ~S , w h e r e M 0 i s th e s c a l a r s e i s m i cm o m e n t , K i s t h e a v e ra g e o f t h e a b s o l u t e v al u e o f [ u ], a n d Si s t h e a r e a o f E . I n a d d i t i o n , B K . 9 , p a g e 1 5 6 , i s e x t r e m e l yi m p o r t a n t i n i t s o w n r i g h t b e c a u s e i t is t h e b a s i s o f t h e

S e i s m o l o g i c a l R e s e a r c h L e t te r s M a r c h / A p r i l 2 0 0 3 V o l u m e 7 4 , N u m b e r 2 1 6 7

8/13/2019 Tutorial Pujol SRL

6/6

A ) X 3

- 5 ( x 3 h / 2 )

h i 2- h / 2

= 6 ( x 3 - h / 2 ). . . . . > X 1

B ) X 3

- h / 2

- ~ x ~ + h / 2 )

( x 3 - h / 2 )

h / 2 -> X 1

, A , F i g u r e 5 . Graphical representationo f ( A ) ~ x 3 i O x 3 a n d ( B )- a ~ X l l a X l .

mom ent tensor density and m ome nt tensor discussed in Akiand R ichards (1980) and because it allows the com putatio nof the displacement caused by an earthquake when Green sfunction is known. For its application to strong motion stud-ies see, e . g . Spudich and Arch uleta (1987). El

ACKNOWLEDGMENTThe material presented here is based on Pujol (2003). CERIcontribut ion No. 458.REFERENCESAki, K. (1966). Generation and propag ation of G waves from the Nii-gata earthquake o f June 16, 1964, Part 2: Estimation o f earth-quake m om ent, released energy, and stress-strain drop from the Gwave spectrum, Bulletin o f the E arthqua ke Research Institute 44,7 3 - 8 8 .Aki, K. and P. Richards (1980). Quantitative Seismolog~ Vol. I, SanFrancisco: W. H . Freeman and Co.Burridge, R. and L. Kno poff (1964). B ody force equivalents for seismicdislocations, Bulletin of the Seismological Society of America 54,1,875-1,888. Reprinted in Seismological Research L etters 74,154-162 .Haskell , N. (1964). Total energy and energy spectral density of elasticwave radiation from p ropagating faults , Bulletin of he SeismologicalSociety of America 54 , 1 ,811-1 ,841 .Kraut, E. (1967). Fundamentals of Mathem atical Physics, New York :McGraw-Hill .Lay, T. an d T. Wa llace (1995). M odern Glob al Seismolog 4 San Diego,

CA: A cadem ic Press.Love, A. (1927). A Treatise on the Mathematical The ory o f Elasticity ,Cambridge: C amb ridge University Press. (Reprinted N ew York:Dover, 1944.)Maruyama. T. (1963). O n the force equivalents of dynamical elastic dis-locations with reference to the earthquake mechanism, Bullet in ofthe Earthquake Research n stitute 4 1 , 4 6 7 - 4 8 6 .Pujol, J . and R. H errm ann (1990). A student s guide to po int sources inhomogeneous med ia , SeismologicalResearch Letters6 1 , 2 0 9 - 2 2 4 .Pujol, J. (2003). Elastic Wave Propagation and Generation in Seismolog~Cam bridge University Press (in press) .Spudich, P. and R. Archuleta (1987). Techniques for earthquakegrou nd-m otion calculation with applications to source parameter-ization o f f inite faults , in B. Bolt (editor) , Seismic Strong MotionSynthetics,Academic P ress , 205-265 .Stauder, W. (1962). The focal mech anism o f earthquakes, in H. Lands-berg and J. Van Mieghem (editors) , Adv ance s in Geophysics9, Aca-demic Press, 1-76.Steketee, J. (195 8). So me geop hysical applications of the elasticity the-ory of dislocations, CanadianJourn al o f Physics36 , 1 ,168-1 ,198 .Udfas, A. (1999). Principles of Seismolog~ Cambr idge : Cambr idge Uni-versity Press.

C E R IT h e U n i v er s it y o f M e m p h i s

M e m p h i s , T N 3 8 1 5 2p u j o l @ c e r i .m e m p h i s .e d u

168 SeismologicalResearchLetters Volum e74 Num ber2 March/April2003