Absorbing Acoustic Conditions

7/30/2019 Absorbing Acoustic Conditions

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Bulletin of the SeismologicalSocietyof America. Vol. 67, No. 6, pp. 1529-1540.December 977

A B S O R B I N G B O U N D A R Y C O N D I T I O N S F O R A C O U S T I C A N DE L A S T I C W A V E E Q U A T I O N S

B y ROBERT CLAYTON AND B~ 6R~ ENGQUISTABSTRACT

B o u n d a r y c o n d i t i o n s a r e d e r i v e d f o r n u m e r i c a l w a v e s i m u l a t i o n t h a t m i n i m i z ear t i f ic i a l r e fl ec ti o n s f ro m t h e ed g e s o f t h e d o m ai n o f co mp u t a t io n . In t h i s w a ya c o u s ti c a n d e la s ti c w a v e p r o p a g a t i o n in a l im i te d a r e a c a n b e e f f ic i e n t l y u s edt o d e s c r i b e p h y s i c a l b e h a v i o r i n a n u n b o u n d e d d o m a i n . T h e b o u n d a r y c o n d i t i o n sa r e b a s e d o n p a r a x i a l a p p r o x i m a t i o n s o f th e s c a la r a n d e la s tic w a v e e q u a ti o n s .T h e y a r e c o m p u t a t i o n a l l y i n e x p e n s i v e a n d s i m p l e t o a p p l y , a n d t h e y r e d u c e r e -f l e c t i o n s o v e r a w i d e r a n g e o f i n c i d e n t a n g l e s .

INTRODUCTIONO n e o f th e p e r s i s t e n t p r o b l e m s in t h e n u m e r i c a l si m u l a t i o n o f w a v e p h e n o m e n a i s

t h e a r ti fi c ia l r e fl e ct io n s t h a t a r e i n t r o d u c e d b y t h e e d g e o f t h e c o m p u t a t i o n a l g r i d.T h e s e r e f le c ti o n s, w h i c h e v e n t u a l l y p r o p a g a t e i n w a r d , m a s k t h e t r u e s o l u t io n o f t h ep r o b l e m i n a n i n f in i te m e d i u m . H e n c e , i t is o f i n t e r e s t t o d e v e l o p b o u n d a r y c o n d i t io n st h a t m a k e t h e p e r i m e t e r o f t h e g r id " t r a n s p a r e n t " t o o u t w a r d - m o v i n g w a v e s ; o t h e r-w i se a m u c h g r e a t e r n u m b e r o f m e s h p o i n t s w o u l d b e r e q u ir e d .

O n e s o l u t i o n t o t h e p r o b l e m t h a t h a s b e e n p r o p o s e d ( L y s m e r a n d K u h l e m e y e r ,] 9 6 9 ) is t h e v i s c o u s d a m p i n g o f n o r m a l a n d s h e a r s t re s s c o m p o n e n t s a l o n g t h e b o u n d -a r y . T h i s m e t h o d a p p r o x i m a t e l y a t t e n u a t e s t h e r e f l e c t e d c o m p r e s s i o n a l w a v e s o v e ra w i d e r a n g e o f i n c i d e n t a n g l e s t o t h e b o u n d a r y , b u t i t d o e s n o t d i m i n is h r e f l e c te ds h e a r w a v e s a s c o m p l e t e l y . A n o t h e r m e t h o d t h a t c a n b e m a d e t o w o r k p e r f e c t l y f o ra ll in c i d e n t a n g l e s h a s b e e n p r o p o s e d b y S m i t h ( 19 74 ). W i t h t h i s a p p r o a c h , t h e s i m u -l a t i o n i s d o n e t w i c e f o r e a c h a b s o r b i n g b o u n d a r y : o n c e w i t h D i r i c h l e t b o u n d a r y c o n -d i t i o n s , a n d o n c e w i t h N e t u n a n n b o u n d a r y c o n d i t i o n s . S i n c e t h e s e t w o b o u n d a r yc o n d i t io n s p r o d u c e r e fl e ct io n s t h a t a r e o p p o s i t e i n s ig n , t h e s u m o f t h e t w o c a s es w il lc a n c e l t h e r e fl e ct io n s . T h e c h ie f s h o r t c o m i n g o f t h i s m e t h o d i s t h a t t h e e n t i re s e t o fc o m p u t a t i o n s h a s t o b e r e p e a t e d m a n y t i m e s .

I n t h i s p a p e r w e p r e s e n t a s e t o f a b s o r b i n g b o u n d a r y c o n d i t i o n s t h a t a r e b a s e d o np a r a x i a l a p p r o x i m a t i o n s ( P A ) o f th e s c a l a r a n d e l as t ic w a v e e q u a t i o n s . A d i sc u s s io no f t h e s e t y p e s o f b o u n d a r y c o n d i t i o n s b a s e d o n p s e u d o - d i f f e r e n t i a l o p e r a t o r s , f o r ag e n e r a l c l a s s o f d i f f e r e n t i a l e q u a t i o n s , c a n b e f o u n d i n E n g q u i s t a n d M a j d a ( 1 9 7 7 ) .T h e c h i ef f e a t u r e o f t h e P A t h a t w e w il l e x p l o it i s t h a t t h e o u t w a r d - m o v i n g w a v e f ie ldc a n b e s e p a r a t e d f r o m t h e i n w a r d - m o v i n g o n e . A l o n g t h e b o u n d a r y , t h e n , t h e P Ac a n b e u s e d t o m o d e l o n l y t h e o u t w a r d - m o v i n g e n e r g y a n d h e n c e r e d u c e t h e r e f l e c -t io n s . T h e b o u n d a r y c o n d i ti o n s t h a t w e p re s e n t a r e s t a b le a n d c o m p u t a t io n a l l ve ff ic ie n t i n t h a t t h e y r e q u ir e a b o u t t h e s a m e a m o u n t o f w o r k p e r m e s h p o i n t f o rf i n i t e d i f f e r ence app l i ca t i ons a s does t he fu l l wave equa t i on .

I n t h e f i rs t p a r t o f t h i s p a p e r , s o m e p a r a x i a l a p p r o x i m a t i o n s t o t h e s c a la r a n de l a st ic w a v e e q u a t i o n s a r e p r e s e n t e d . I n t h e s e c o n d p a r t , t h e P A a r e u s e d a s a b s o r b i n gbo un da ry condi t i ons and express ions fo r t he e f f ec t i ve r e f l ec ti on coe f fi c ien t s a long t heb o u n d a r y a r e g i v e n . S o m e n u m e r i c a l e x a m p l e s a r e p r e s e n t e d i n t h e f i n a l s e c t i o n .

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1530 ROBERT_ CLAYTON AND BJORN ENGQUIST

PARAXIAL APPROXIMATIONS OF TI-IE WAV E EQUATIONParaxiM approximat ions of the scMar wave equat ion have been extensively de-

veloped by Claerbout (1970, 1976), and Claerbout and Johnson (1971), and in thefi rs t part of th is sect ion we present a brief review of that work. In the second partwe develop paraxial approximations for the elastic wave equation.The two-dimensionM scMar wave equat ion

P ~ + P = = v - ~ P t t , (1)is usually considered for modeling purposes to b e init ial value d in t ime. T he stabil i tyof the equat ion for t ime extrapolat ion is ensured by the fact tha t in i t s dispersion re-lation :-

o = v( k~ 2 + k,2')~/2, (2)the freq uen cy 0 is a real func tion of the spatial w ave num bers k~ and k, .

If we now consider spatial extrapola tion of (1) (say, in the z-direction), th e ap-propriate form of the dispersion relation would be

k~ = : l :(~/v)[1 - ( ' / 2 2 k : ~ 2 / 5 0 2 ) ] 1 1 2 , , (3)Clearly there are stabil i ty problems wl~en I vk~/~ I > 1 (evanescent wave s) , becausek~ becomes imaginary. I t i s therefore necessary to modify the wave equat ion in sucha way as to eliminate the evanescent components of the solution. A relatively simplewa y of accomplishing this i s to res t r ict th e range of solut ions to those waves tha t aretraveling within a cone of the z-axis (paraxial waves). Note that the : t : sign in equa-tion (3) is for wav e fields moving in opposite directions in z, and we will mod el thesefields separately.

To form the paraxial approximat ion of (1) , we expand the square-root operator of(3) as a rational approximation about small v k ~ / z . Three such approximations [forthe -4- sign of equation (3)] are

h l : v k J ~ = 1 + O(]vk~/~12) , (4)1 2A2: vk~/~ = 1 - -~ (v k J ~ ) + o(IvkJ ,14), (5)

A3: vk~ = 1 :~ v, ~/ ~/~-- I ~ ( v k ~ / . , ) ~ + n c ~ ~ / ~ . (6)

A ge neral order expansion, which leads to stable differencing schemes, can be fo undby th e recursion relation for a Pa d~ series approxim ation to a sq uare ro ot (F rancisMuir , personal communicat ion)

at 1 (v = /~) + 0 ( i 2jv k ~ / ,~ I ) , a l = 1 , ( 7 )1 Jr" a~-iwhere the j th paraxial approximat ion is given by vk~/~ = a j . Th e dispersion relationsA1, A2, A3, and the dispersion relation of the full wav e equation are shown in Fig ure 1.The error term in the expansions indicates that the approximat ions are val id for


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ABSORBING BOUNDARY CONDITIONS FOR ACOUSTIC AND ELASTIC WAVES 1531wa ves t r ave l i ng wi th in a cone o f t he z -ax is . A s imi l a r s e t o f equ a t i ons can be de r ivedf o r t h e m i n u s s ig n o f e q u a t i o n ( 3 ) . I n t h i s w a y , t h e in c o m i n g a n d o u t g o i n g w a v ef ie ld s a r e s e p a r a t e d b y t h e p a r a x i a l a p p r o x i m a t i o n . I t i s i n t e r e s ti n g t o n o t e t h a th i g h e r -o r d e r a p p r o x i m a t i o n s b a s e d o n T a y l o r s e r ie s e x p a n s io n s o f th e s q u a r e r o o t ine q u a t i o n ( 3 ) l e a d t o u n s t a b l e d i ff e re n c in g s c h e m e s ( E n g q u i s t a n d M a j d a , 1 9 7 7 ).In t he i r d i f f e r en t i a l f o rm, equa t i ons (4 t o 6 ) appear a s

A I : P ~ + ( 1 / v ) P t = O , (8 )A 2 : P , t + ( 1 / v ) P , - - ( v / 2 ) P ~ = O , ( 9 )A 3 : P ~ tt - ( v 2 /4 ) P ~ .~ + ( 1 / v ) P , , - ( 3 v / 4 ) P t ~ , = O . (10)

L k

"kx

FIG. 1. Dispersion relations for the scalar case. The curves A1, A2, and A3 are the dispersionrelations of the paraxial approximations of the scalar wave equation (the circle).N o t e t h a t e q u a t i o n s ( 8 to 1 0 ) a r e s o m e w h a t d if f e re n t f r o m th o s e o f C l a e r b o u t ( 1 9 7 6 )b e c a u s e h e u s e s a r e t a r d e d t i m e c o o r d i n a t e s y s t e m w h i c h c a n n o t b e u s e d h e r e.

P a r a x i a l a p p r o x i m a t i o n s f o r t h e e l a st ic w a v e e q u a t i o n a n a l o g o u s t o t h o s e o f t h es c a l a r w a v e e q u a t i o n c a n a l s o b e f o u n d . W e c a n n o t , h o w e v e r , p e r f o r m t h e a n a l y s i sby cons ide r ing expans ions o f t he d i spe r s ion r e l a t i on because t he d i f f e r en t i a l equa t i onsfor vec to r f i e lds a r e no t un ique ly spec i f i ed f rom the i r d i spe r s ion r e l a t i ons . Ins t ead ,w e u s e t h e s c a l a r c a s e t o p r o v i d e a h i n t a s t o t h e g e n e r a l fo r m o f t h e p a r a x i a l a p -p r o x i m a t i o n a n d f it t h e c o e ff ic ie n ts b y m a t c h i n g t o t h e f u ll e la s ti c w a v e e q u a t i o n .

W e w r i t e t h e e l a s t i c w a v e e q u a t i o n f o r a h o m o g e n e o u s , i s o t r o p i c m e d i u m i n t h eform

u_tt = D,u~ + H u ~ + D 2 u ~ , ( 1 1 )w h e r e

u = \ v e r t i c a l d i s p l a c e m e n t f ie ld ] '( o 0 ) 0 )


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1 5 3 2 R O B E R T C L A Y T O N A N D B J O R N E N G Q U I ST

an d a an d f~ a r e the comp ress iona l an d sh ea r ve loc it i es , r e spec t ive ly . Th e F our ie rt r a n s f o r m o f t h i s e q u a t i o n i s[ I - D ~ ( k , / ~ ) 2 - H ( k , / ) ( k ~ / ~ ) - D 2 ( k ~ / ~ ) 2 l f i ( ~ , k , , k~) = 0 . ( 12)

W e n o w c o n s i d e r t wo f o r m s o f t h e p a r a x i a l a p p r o x i m a t i o n

a n dA I : u__,, t- Bl U _ t = O ,

A 2 : u_t~ + Clu .~ t "4 - C2 u _t~: - t - C 3 u .~ = O .

(13)

(14)I n A2 t h e C 2 t e r m d e s c ri b e s t h e c o u p l in g o f u a n d w . T h e F o u r i e r t r a n s f o r m s o f t h e s ea p p r o x i m a t i o n s a r e

a n dA I : [ t ( k J ~ ) - B , ] ~ ( ~ , k ~ , ko ) = 0 , ( 1 5 )

A 2 : [ I ( k J ~ ) - C 1 -4- C 2 ( k x / ~ ) - C 3 ( k ~ / ~ ) 2 1 ~ . ( ~ , k~ , k~) = 0. (16 )I n e q u a t i o n s ( 1 5 ) a n d ( 1 6 ) , t h e ( k J ~ ) t e r m m a y b e i s o la t ed a n d s u b s t i t u t e d i n( 1 2 ) t o d e t e r m i n e t h e c o e ff ic i en t m a t r i c e s in ( 1 3 ) a n d ( 1 4 ) . T h e r e s u l t s a r e

1 / ~ '(0c ~ = ( ~ - ~ ) 1 / ~ 'o )~ = ~ 0 ~ - 2 ~ "

T h e e r r o r t e r m s f o r t h e A1 a n d A2 a p p r o x i m a t i o n s a r e 0 ( [ k ~ / ~ I ) and 0( I k = / ~ I~),r e s p e c t i v e l y .I n F i g u r e 2 t h e d i s p e r si o n r e l a ti o n s fo r ( 1 3 ) a n d ( 1 4 ) a r e p r e s e n t e d . T h e d i s p e rs i o n

r e l a ti o n s i n t h i s c a s e a r e t h e l o ci o f z e ro s o f t h e d e t e r m i n a n t o f t h e s q u a r e - b r a c k e t e dq u a n t i t i e s i n ( 1 5 ) a n d ( 1 6 ) . T h e f a c t t h a t t h e r e a r e t wo c u r v e s f o r e a c h a p p r o x i m a -t i o n i n d i c a t e s t h a t t h e y d e c o u p l e i n t o c o m p r e s s i o n a l a n d s h e a r m o t i o n s , a s d o e s t h ef u ll wa v e e q u a t i o n . F o r t h e a p p r o x i m a t i o n A2 , t h e s h a p e o f t h e d i s p e rs i o n c u r v e sdep end s on the r a t io o f a and f~, and in F igu re 2 an a / ~ r a t io o f ~ i s used . I n genera l ,t h e l a r g e r t h e v e l o c i t y r a t i o b e c o m e s , t h e p o o r e r t h e a p p r o x i m a t i o n f o r s h e a r wa v e s .

I n F i g u r e 2 , we h a v e a ls o p r e s e n t e d t h e d i sp e r si o n re l a t io n f o r t h e v i sc o u s b o u n d -a ri es o f L y s m e r a n d K u h l e m e y e r ( 1 9 6 9 ). T h e c u r v e is a h y p e r b o l a t h a t i s i n d e p e n d e n to f v e l o c i t y ra t i o , a n d i t i n d i c a t e s t h a t t h e v i s c o u s b o u n d a r i e s ~ il l m o d e l sh e a r wa v e sl es s a c c u r a t e l y t h a n e i t h e r t h e A1 o r A2 a p p r o x i m a t i o n s .ABSORBING BOUNDARY CONDITIONS

T h e d i s p e r s i o n c u r v e s p r e s e n t e d i n t h e p r e v i o u s s e c t i o n i n d i c a t e t h a t t h e p a r a x i a la p p r o x i m a t i o n s c a n b e u s e d t o m o d e l b o t h e l a s t i c a n d s c a l a r wa v e s m o v i n g i n o n e


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A B S O R B I N G B O U N D A R Y C O N D I T I O N S F O R A C O U S T I C A N D E L A S T I C W A V E S 1533genera l d i rec tion and to d i sc r imina te aga ins t waves mo ving in the oppos i t e d i rec t ion .T o a bs o r b i nc i de n t e ne r gy a l ong a g i ve n bounda r y , t he n , w e c a n u s e t he pa r a x i a la pp r ox i m a t i on t ha t m ode l s on l y e ne r gy m ov i ng ou t w a r d f r om t he i n t e ri o r o f t he g r i d

J

I - K

k k z IJ

> A 1

7"'kx

F I G . 2. D i s p e r s i o n r e l a t i o n s f o r t h e e l a s t ic c a s e . T h e c u r v e s A 1 a n d A 2 a r e t h e d i s p e r s i o n r e -l a t i o n s o f t h e p a r a x i a l a p p r o x i m a t i o n s o f t h e e l a s t i c w a v e e q u a t i o n ( t h e c i r c l e s) . F o r e a c h a p -p r o x i m a t i o n t h e r e a r e t w o c u r v e s : t h a t a p p r o x i m a t i n g t h e la r g e r c i r c le i s f o r s h e a r w a v e s w h i l et h e o t h e r i s f o r c o m p r e s s i o n a l w a v e s . T h e d a s h e d c u r v e s ( l a b e l e d L K ) a r e t h e d i s p e r s i o n c u r v e so f t h e v i sc o u s b o u n d a r y c o n d i t i o n s o f L y s m e r a n d K u h l e m e y e r ( 19 6 9) .0 1 * N

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

i X

F I G . 3 . Computational grid.t ow a r d t ha t bounda r y . A l l o f t he a pp r ox i m a t i ons p r e s e n t e d i n t he p r e v i ous s e c t i onm o de l e d w a ve s m o v i ng i n t he pos i ti ve z- d ir e ct ion . H e nc e , on t h e g r i d s how n i n F i gu r e3 , t he y w ou l d be s u it a b le f o r a bs o r b ing bou nda r y c ond i ti ons a l ong t h e bo t t o m e dge .F o r t he t op e dge , w e w ou l d u s e t he pa r a x i a l a pp r ox i m a t i ons w h i c h c o r r e s pond t otak ing the nega t ive s ign on the rad ica l o f equa t ion (3) . In th i s case , the d i spe rs ion


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1534< ROBERT CLAYTON AND BJORN ENGQUIST

r e l a t i o n s w o u l d b e a p p r o x i m a t i n g t h e l o w e r h a l f o f t h e s e m i c i r c l e . T h e b o u n d a r yc o n d i t i o n s f o r t h e s i d e s a r e f o u n d b y i n t e r c h a n g i n g x a n d z i n t h e t w o c a s e s a b o v e .

T o c l a ri fy t h e m e t h o d w e sh a ll p r e s e n t t h e s c h e m e u s e d f o r t h e n u m e r i c a l e x a m p l e si n s o m e d e t ai l. F o r % a c h t i m e s t e p :

1 . W e so lved a l l i n t e r io r po in t s (ma rked " o " i n F igure 3 ) wi th an exp l i c i t f i n i t ed i f fe rence fo rm of t he fu l l w av e equ a t ion [ equa t ions (1) o r (11)] . See K e l ly e t a l . (1976)for t he d i f f e rence fo rmulas .2 . W e u s e d t h e a p p r o p r i a t e p a r a x i a l a p p r o x i m a t i o n t o e x t r a p o l a t e s p a t i a l ly t h ei n t e r i o r s o l u t i o n o n e m e s h r o w o u t w a r d t o f i l l i n t h e b o u n d a r y r o w ( m a r k e d " x " inF igure 3 ) . S ince a ll o f t he pa rax i a l appro x ima t ions a re fi r st o rde r i n t he spa t i a l ex t r ap-o l a t io n d i r e ct io n , o n l y t h e n e a r e s t in t e r io r r o w i s n e e d e d f o r e a c h b o u n d a r y . T h ea c t u a l d i f fe r e n ce sc h e m e s a r e p r e s e n t e d i n t h e A p p e n d i x .

T h e e f f e c t i v e n e s s o f t h e v a r i o u s a b s o r b i n g b o u n d a r y c o n d i t i o n s g i v e n h e r e c a n b eg a u g e d b y c o m p a r i n g t h e i r e f f ec t iv e r e fl e c ti o n c o e ff ic ie n ts a t t h e b o u n d a r i e s . F o r t h es c a l a r w a v e e q u a t i o n , c o n s i d e r a n i n c i d e n t p l a n e w a v e t r a v e l i n g i n t h e p o s i t i v e z - d i -r e c t i o n

P~ = exp ( i k ~ x + i k ~ z - i ~ t ) ,w h i c h i n i ti a t e s a r e fl e ct io n f r o m t h e b o t t o m b o u n d a r y o f t h e f o r m

PR = r exp ( i k ~ x - i k ~ z - i ~ t ) ,wh ere r i s t he e f f ec t ive r e f lec t ion coe f f ic i en t . Loc a l ly nea r t he b ou nd ary , t h e w av ef ie ld P~ - 5 P R w il l s a t i s fy b o t h t h e b o u n d a r y c o n d i t io n a n d t h e i n t e ri o r e q u a t io n .A p p l y i n g t h e b o u n d a r y c o n d i t io n A 2 , fo r ex a m p l e , w e o b t a i n

( v v k 2 ) ( P z -}- r P R ) = 0 .

S o l v i n g f o r r a n d e v a l u a t i n g a t t h e b o u n d a r y , w e o b t a i n1 21 - ~ ( v k ~ / ~ ) - ( v k o / ~ )r = 1 - l ( v k J ~ ) ~ + ( vk ~ /o , ) "

B y n o t i n g t h a t f r o m t h e i n t e r i o r e q u a t i o n ( v k ~ / ~ ) 2 = 1 - ( v k / o ) 2 a n d i d e n t i f y i n gv k ~ / ~ as cos Ow h e r e 0 i s t h e a n g l e o f i n c id e n c e m e a s u r e d f r o m t h e n o r m a l t o t h e b o u n d -a ry , we can wr i t e t h e r e f lec t i on coe f fi c ien t a s

r ( O ) = - [ ( 1 - c o s 0 ) / ( 1 + c o s 0 ) ] 2 .Thi s express ion ma y b e genera l i zed fo r t he A ~. bo un da ry cond i t i on t o

r(O) -= - [ ( 1 - c o s O ) / (1 + cos O)]~.In F igu re 4 , t he r e f l ec ti on coe f fi c ien t s fo r A1 , A2 , and A3 a re p lo t t ed . I t shou ld ben o t e d t h a t a n g l e s w h e r e r ( O ) i s l a r g e c o r r e s p o n d t o w a v e s t r a v e l i n g a l m o s t p a r a l l e lt o t h e b o u n d a r y . T h e y w o u l d t h e r e f o r e l i k e l y s t r i k e a n o t h e r a b s o r b i n g b o u n d a r yb e f o r e i n t e r a c ti n g w i t h t h e s o l u t io n a t t h e c e n t e r o f t h e g r id . A m e a s u r e o f t h e e n e r g y


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ABSO RBING BOUNDARY CONDITIONS FOR ACOUSTIC AND ELASTIC WAV ES 15 35

r ad i a t e d to ward 't he cen ter o f t he g r id would be[r~-(O) sin O]2.

To find the effect ive reflect ion coefficients for elast icwave p o ten t i a l so lu t ions o f t he fo rm waves , we assume p l ane-

= 0 exp i (~o /a ) (1 ,x + n ,z ) + re exp i ( c o / a ) ( l . x - - n , z ) ,= ~0 exp i (w /~) ( l~ x + n~a) + rs exp i (w /#) ( l~x - - n~z) ,

l . - - o~_UJ ot .~ ,=I . U ~ _

A!

I I I I I I I I IIO,O 3 0,0 ~Q.O 40,0 tO.Q GQ.O 70, 0 ~EiO,O lOt OINCIDENT ANGLE

F IG. 4 . R e f l e c t i on c oe f f i c i e n ts fo r t he s c a l a r e a se . The c u rv e s a re t he e f fe c t i ve r e f l e c t i on c oe f f i c i en t sf o r t h e s c a l a r a b s o r b i n g b o u n d a r y c o n d i t i o n s A 1 , A 2 , a n d A 3 .

where l~ , n~ , l~ , and n~ are t he d i rec tion cos ines o f t he wave f ron t . For i nc iden t com-pression al wave s, 0 = 1 an d ~I'0 = 0, and for incid en t she ar wa ves 0 = 0 and ~0- - 1 . The d i sp l acement f ie lds a re found by the t r ansfo rma t ion

u = ~ -t- ~I,:,

The express ions fo r t he hor i zon ta l an d ver t i ca l d i sp l acement s a re subs t i t u t ed in tothe t ime- t r ansfo rmed fo rm of t he boundary cond i t i ons A1 and A2 , and the r e f l ec t ioncoefficients rp and r~ are d eterm ined. In Figures 5 a nd 6 , we show th e ref lect ion co-ef fi c ien ts fo r t he b ou nd ary cond i t i ons A1 and A2 , r espec tive ly . The a brup t changes inref lect ion coeff icients (m ar ke d . ,by ar rows) for incident s hear w aves a re d ue to thefo rmat ion o f a pseudo-head wave a long the boundary . These waves do no t i n p rac t i cei n t e rf e r e w i t h t h e co m p u t a t i o n a t t h e cen t e r o f t h e g r id .

In F igure 7 , t he e f fec t ive r e f lec t ion coef f ic i en ts fo r t he v i scous boundar i es (Ly sm erand Kuhlemeyer , 1969) a re shown.

In th e d i scussion above on absorb ing bound ary cond i t i ons , we have assum ed tha tthere i s no a priori i n fo rmat ion on wave d i rec t ion a t t he boundar i es . In some s tud ies ,su ch as mono chrom at i c su r face wa ve s tud ies , t h is i n fo rm at ion does ex i st and can beu sed t o " t u n e" t h e b o u n d a r y co n d i ti o n . I n t h i s ca se t h e p a r ax ia l ap p r o x i m a t i o n s a reexpanded abou t t he p refer red wave d i rec t ion ra ther t han normal i nc idence . In F igures1 and 2 , t h i s am oun t s t o a ro t a t ion o f t he d i sper s ion curves abou t t he o r ig in , so th a t


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1536 R O B E R T C L A Y T O N A N D B J O R N E N G Q U I S Tt h e b e s t f i t t o t h e c i r c l e s o c c u r s a t t h e p r e f e r r e d p r o p a g a t i o n a n g l e . I n t h e n u m e r i c a le x a m p l e s t h a t f o ll o w , t h e c o r n e r p o i n t s o f t h e m e s h w e r e m a d e t r a n s p a r e n t b y a4 5 r o t a t i o n o f t h e A 1 b o u n d a r y c o n d i ti o n s . T h u s a ll w a v e s t r a v e l i n g d i r e c t l y in t ot h e c o r n e r s a r e p e r f e c t l y a b s o r b e d .

N U M E R I C A L E X A M P L E ST o i l lu s t r a t e t h e e f fe c t iv e n e s s o f t h e b o u n d a r y c o n d i t io n s , w e p r e s e n t t h r e e e x -

a m p l e s . I n e a c h c a s e , t h e c o m p u t a t i o n s w e r e d o n e w i t h t h e u s u a l N e u m a n n ( z e r o -

o ~.,j a'--q~)m

(~E D0__"d 'Q,O LO.O aa,O 8Q.O 4Q.O SO,O GO.O 700 80 . 10.0

S INCIOENT ANGL E

~ o ez c a ~

W~- J d "W d "0g ~

, ~ _

P

So . ,'.o ".o ~'.o .o ".0 ". ".o ". ~.0P INCIDENT ANGLE

F I Q . 5. R e f l e c t i o n c o e f f ic i e n t s f o r t h e A 1 e l a s t i c b o u n d a r y c o n d i t i o n . I n t h e t o p p a n e l s h e a r( S ) a n d c o m p r e s s i o n a l ( P ) r e f l e c t io n s a r e s h o w n f o r a n i n c i d e n t S w a v e o f u n i t s t r e n g t h , w h i l ei n t h e b o t t o m p a n e l P a n d S r e f le c t i o n s a r e s h o w n f o r i n c i d e n t P w a v e s . T h i s w a v e d o e s n o tp e n e t r a t e i n t o t h e r e g i o n o f c o m p u t a t i o n .

s l o p e ) b o u n d a r y c o n d i t i o n s a n d w i t h t h e A 2 a b s o r b i n g b o u n d a r y c o n d i t i o n s . A 4 0b y 4 0 g r i d w a s u s e d w i t h t h e m e s h s p a c i n g c h o s e n t o m a k e t h e e x p l i c i t d i f f e r e n c i n gs c h e m e f o r t h e i n t e ri o r s o l u t io n s t a b l e .

T h e f i r s t e x a m p l e , F i g u r e 8 , i s a c i r c u l a r l y s p r e a d i n g s c a l a r w a v e . T h e N e u / n a n nc o n d i t i o n s m a k e t h e b o u n d a r y a c t a s a p e r f e c t r e f l e c t o r , b u t t h e a b s o r b i n g b o u n d a r yc o n d i t i o n s a l l o w t h e w a v e t o p a s s t h r o u g h t h e g r i d p e r i m e t e r .

T h e s e c o n d e x a m p l e , F i g u r e 9 , i s a c i r c u l a r c o m p r e s s i o n a l e l a s t i c w a v e . B o t h t h eh o r i z o n t a l a n d v e r t i c a l w a v e f i e l d s a r e d e p i c t e d w i t h " b l a c k " r e p r e s e n t i n g p o s i t i v ea m p l i t u d e s a n d " w h i t e " r e p r e s e n ti n g n e g a t i v e a m p l i t u d e s . A v e l o c i t y r a t i o o f a / f l- % / ~ w a s u s e d .

T h e f in a l e x a m p l e , F i g u r e 1 0, i s a c ir c u la r s h e a r w a v e , a n d a g a i n a / ~ = % / 3 w a su s e d .


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(/')aZ4-

C..)~". - ld"IJ-w

d -

o

S

o., ,'.o - ~ -.o -.o ".o ;.o ".o -.o -.oS INCIDENT ANGLE

~. ".o ~'.o ".o ;.o ".o ".o ".oP INCIDENT ANGLE

o

( ./') eZ,~-a ~~ -LLh,m...1 d"b- mLd~-~ "

~ . d "Q

FIG. 6. Reflection coefficients for the A2 elastic boundary condition. See Figure 5 for details.In this case, the S reflection for incident P is much more diminished than in the A1 case, whichis important because it reflects more closely to the normal than does the P wave. For incident Swaves, both P and S reflections are smaller than with the A1 ease.o

(A oZ4-

(._)~'.-.._I.~"

. , '.o ". ; . o g.o ".o ~'. , ' .o ,LoS INCIDENT ANGLE

~ i~o "

%. (J 0~

o.o ,;.o .Lo ~i~ . . . . . . :P ~NCIDENT ANGLE

FIG. 7. Reflection coefficients of the viscous boundary condition for the elastic equation.See Figure 5 for details. For both P and S incident waves, the ampl itude of the reflected waves islarger than with the A2 boundary conditions.1537


10/12

1538 R O B E R T C L A Y T O N A N D B J O R N E N G Q U I S T

llllllillll....

6 0

/ i /R e f l e c t i n g

llllIIllllIIOllll[IIIIIl

))))))))))))))))))liliii)))'"'"'" lluumuul

(lll)lllllll,H,,,H,H mHIUUNii UIIIIIIIUI,,m,,,,,,,,illlllllllllllllllll,0ol;llillll:lllll;llll;;lllllt;A b s o r b i n g

U W

1 U

fill

U w

o tllllillIIllH~l)j)IuLL"I~III'I~ilnl"""""~lil)))ll(ll'"l " '

li,([li,,,,,,,,,,,,,,,z,,Ill~lilllmkmll~llmk~li~llilil!l~I b u

Ref lect ing AbsorbingFze. 9

U w U w~m,...,.,,~Itllli1~n~ IIII,,,.,.,,,IIIIltw~Imli(N,,Ill I,,, 'il(,l,,)l,,,,,,j i i),,

li'il .a

II Ill,')II,l:l,I,eflect ing AbsorbingFIG. 8 Fm. 10FIG. 8. Reflecting and absorbing boundary conditions for an expand ing circular scalar wave.The absorbing bounda ry condition was applied with the A2 scalar differencing scheme given inthe Appendix. The numbers refer to the time step index.FIG. 9. Reflecting and absorbing boundary conditions for an expand ing compressional wave.The absorb ing boundary condition was applied with the A2 elastic difference scheme given inthe Appendix. The horizontal and vertical displacements are denoted by U and W, respectively.

The numbers are the time index.FIG. 10. Reflecting and absorbing boundary conditions for an expanding shear wave.Refer to Figure 9.


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ABSORBING BOUNDARY CONDITIONS FOR ACOUSTIC AND ELASTIC WAVES 1539CONCLUSIONS

A s e t o f bo un da r y c ond i t i ons f o r s c a l a r a nd e l a s t ic wa ve s t h a t r e duc e a r t if i c ia lr e f le c t ions ha s be e n p r e s e n t e d . The ' bo un da r y c ond i t i ons a r e e s s e n t i a l l y pa r a x i a la p p r o x i m a t i o n s o f t h e s c a l a r a n d e l a s ti c w a v e e q u a t io n s . T h e a d v a n t a g e s o f th e s eb o u n d a r y c o n d it i o ns a r e , f ir st , t h a t t h e y a b s o r b e n e r g y o v e r a w i d e r a n g e o f i n c i d e n ta n g l e s , a n d , s e c o n d , t h a t t h e y a r e c o m p u t a t i o n a l l y i n e x p e n s i v e a n d s i m p l e t o a p p l y l

B y u s i n g t h e p a r a x i a l a p p r o x i m a t i o n t o d e r i v e a b s o r b i n g b o u n d a r y c o n d i t i o n s ,w e h a v e p r e s e n t e d a g e n e r a l a p p r o a c h f o r d i ff e r en t ia l e q u a t io n s o f v a r i o u s t y p e s .T h e a c c u r a c y o f t h e s e b o u n d a r y c o n d i ti o n s c a n b e i n cr e a s ed b y s i m p l y t a k i n g h i g h e r -o r d e r p a r a x i a l a p p r o x i m a t i o n s .

ACKNOWLEDGMENTSWe than k Professor Jon Claerbout for his valuable suggestions, and the sponsors of the Stan-ford Explo ration Projec t for their financial suppo rt. ~,

REFERENCESClaerbout, J. F. (1970). Coarse grid calculations of waves in inhomogeneous media with appli-cation to delineation of complicated seismic structure, Geophysics 35,407~i18.Claerbout, J. F. (1976). Fundamentals of Geophysical Data Processing, McGraw-Hil l, New York,163-226.Claerbout, J. F. and A. G. Johnson (1971). Extra polatio n of t ime dependent w aveforms alongtheir path of propagat ion, Geophys. J. 25,285-295.Engqu ist , B. and A. Ma jda (1977). Absorbing boundary conditions for the n umerical simulationof waves, Math. Comp. (in press).Kelly, K. R., R. W. Ward, S. Treitel , and R. M. Alford (1976). Synthetic seismograms: a finite-difference approach, Geophysics 41, 2-27.Lysm er, J. and R. L. Ku hlemey er (1969). Finite dynamic model for infinite media, J. Eng. Mech.Div. , ASCE 95 EM4,859-877.Smith, W. D . (1974). A non-reflecting plane boun dary for wave prop agation problem s, J. Comp.Phys. 15,492-503.DEPARTMENT OF GEOPHYSICSSTANFORD UNIVERSITYSTANFORD, CALIFORNIA (R.C.)

Manuscript received May 11, 1977

DEPARTMENT OF COMPUTER SCIENCESUPPSALA UNIVERSITYUPPSALA, SWEDEN (B.E .)

APPENDIXT h e f i n i t e d i f f e r e n c e f o r m u l a s f o r t h e A 2 s c a l a r a b s o r b i n g b o u n d a r y c o n d i t i o n

( e q u a t i o n 9 ) a r e g i v e n b e lo w . R e f e r t o F i g u r e 3 f o r t h e c o m p u t a t i o n a l g r id .T o p ( k = 0 )

D zr~ tD~ __ 1 ~ t = n x n ,,+ 1-'o . , j,o (~v) D+ D _ (P:,o -k P~a) "-k (v /4 )D + D _ (P~+I,1 -k Pj -l , 0) = 0.B o t t o m ( k = K )

D Z~ tDn n x n n- ' - ' 0 - ' ~ " - k ( v ) D+t D- t ( P~ - - k P:- ,~-I) - - ( v / 4 ) D + D _ ( P j + I .K- I - k P j - I ,~ ) = 0 .Le f t s i de ( n = 0 )

1 z z I 0'+~~r~0tD0~s,~ -- (v)D+ tD- t (P~,k + Pj , k ) --k (v /4) D + D_ (Fj+l ,k q- Pj - i , k ) --- 0R i g h t s id e ( n = N )

D _ - n tD,~~,o~:,k -k (v)D+ tD- ' (P~,k -[- p# - i~ , , N- I p ~y,k : - - ( v / 4)D + D_ ( P j + I , ~ - J- j - l , k ) - - O.


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1540 ROBERT CLAYTON AND BJORN ENGQUISTHere

Pi;~ ~ P ( t j , zk , xn),and D+ q, D_ q, and Doq are, respectively, the forward, backward, and center difference

z n ~ n ~Zoperators with respect to the variable q; i.e., D+ Pj,k = (Pj,k+l -- P~,k) / For the second-order elastic boundary conditions (equation 14), the differenceschemes are:Top (k -- 0)

z t n __ 1 t t 0 n 1 t x n nD + Do u_i ,o (~ )C 1D + D _ (u_~,o + u_i ,1) - (: )C 2D + Do (u_i-l,O + u_j a)- (~) C3 D+ D- (u_ ~-l ,O + u_+1,1) = O.

Bottom (k = K)- uo _uj,,~ (~)C~D+ D _ (_u;,,: u_:,~_l) (~)C2D+ Do (u_;-1,~ _uj,~_~)

1 x x n n+ (~)C3D+ 1) _ (u_j-l,~: + Ui+~,K--1) = O.Lef t side (n = 0) :

D x~ . t 0 _ 1 t t 0 1 1 t z 0 1+ /Jo Uj,k (~)CI D+ D _ (u_i,k -4- u_ik) - (~) C2 D+ Do ( U _ i - - 1 , / : . A[- U i , k )1 Z Z 0 1-- (~ )C 3D + D _ (u_i-l,k + u_i+x,~) = O.

Right side (n = N)D x~. t N 1 t t hr N- -1 1 t z h r N- -1- 1)o u_:,k + (~ )C ~D + 1) - (u_j,k + u_s,k ) + (~)C2D+ Do (u_~-l,~ + u_i,~ )1 z z N N- -I+ (~ )C aD + D _ (u_i-~,~ + u+~,~) = O.

Here,

The three points along the boundary in each corner are computed with a 45 ro-tation of the A1 boundary condition. For example, the differential and differenceboundary condition formulas for the lower right-hand corner are:scalar waves

P~ + P~ -4- (V '2 /v )P t = O,- - t p~[D- ~ -4- D Z + (~ / 2/v )D _ ] ,k = O,

elastic wavesu_z + u__x + M u t = O,

t n( D Z + D_" + M D _ )u_i,k = 0

where1 / ' (1 / / 9 ) + ( I / a )M = ~ ix(i/B ) _ ( l /a)

(k,n) = (K, N- l) , (K--1, N), (K, N);

(k,n) = (K,N--1) , (K- 1, N), (K,N );

(1/~) - (1/~)~(1/~) + (1/~)]"

Absorbing Acoustic Conditions

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