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Bulletinof the SeismologicalSociety of America, Vol. 74, No. 6, pp. 2201-2219, December 1984

T O M O G R A P H I C R E C O N S T R U C T I O N O F V E L O C IT Y A N O M A L I E S

BY JOHN A. FAWCETT AND ROBERT W. CLAYTON

ABSTRACT

A n a p p r o x i m a t e i n v e rs i o n f o r m u l a i s p r o p o s e d f o r t h e r e c o n s tr u c ti o n o f s lo w -

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

s e r v e d t r a v e l - t i m e p e r t u r b a t i o n s f o r r e f l e c t i o n s f r o m a k n o w n p l a n a r r e f l e c t o r .

T h e li m i ta t io n s o f t h e fo r m u l a a r e d i s c u s s e d a n d n u m e r ic a l e x a m p l e s a r e g i v e n .

INTRODUCTION

T o m o g r a p h y r e f e r s t o t h e t e c h n i q u e o f r e c o n s t r u c ti n g a f ie ld f ro m l in e o r s u r fa c e

i n t e g r a ls o f it . I n m e d i c a l X - r a y t o m o g r a p h y , f o r e x a m p l e , t h e t i s s u e d e n s i t y f ie l d

i s d e d u c e d f r o m m e a s u r e m e n t s o f X - r a y a t t e n u a t i o n t h r o u g h t h e p a t i e n t. I n t h i s

c a s e t h e d a t a a r e r e g u l a r l y s a m p l e d l i n e i n t e g r a ls , s o t h e y a r e d i s c r e t e v a lu e s o f a

R a d o n t r an s f o r m .

I n s e i s m o l o gy , t h e d e t e r m i n a t i o n o f s lo w n e s s ( i n v e r s e o f v e l o c it y ) a n d a t t e n u a t i o n

f i el d s c a n a ls o b e v ie w e d in a t o m o g r a p h i c f r a m e w o r k . T h e t r a v e l t im e s o r t h e

a m p l i t u d e d e c a y a l o n g ra y s c o n n e c t i n g t h e s o u r c e s a n d r e c e i v e rs a r e t h e p r o j e c t i o n s

o f t h e f ie l ds . H e r e , t h e p r o b l e m i s c o m p l i c a t e d b y t h e f a c t t h a t t h e r a y s a r e c u r v e d

a n d t h a t t h e r a y p a t h d e p e n d s o n t h e s l o w n e s s f i el d i ts e lf . T h e p r o b l e m i s i n g e n e r a l

n o n l i n e a r . O n e c a n l i n e a r i ze t h e p r o b l e m a b o u t a r e f e r e n c e sl o w n e s s w h i c h e s se n -

t i a l ly d e c o u p l e s t h e r a y p a t h s f r o m t h e u n k n o w n s l o w n e s s fi el d. H o w e v e r , t h i s

l e av e s th e p r o b l e m o f t o m o g r a p h i c r e c o n s t r u c t i o n f r o m l in e i n t e g r a ls a l o n g c u r v e d

ra y s .

O u r m e t h o d s d i s c u s s e d b e lo w w i ll b e t h o u g h t o f i n t e r m s o f a r e f l e c t io n s e is m o l o g y

e x p e r i m e n t . H o w e v e r , t h e r e s u l t s a r e a ls o a p p li c a b le t o t r a n s m i s s i o n p r o b l e m s . T h e

g o al o f o u r t o m o g r a p h i c r e c o n s t r u c t i o n i s t o i d e n t i fy f r o m t r a v e l - t i m e i n f o r m a t i o n

( s o u r ce p o s i ti o n s k n o w n ) a r e a s o f re l a t i v e ly h i g h a n d s lo w v e l o c i ty ( w i t h r e s p e c t t o

a k n o w n b a c k g r o u n d f ie ld ) w i t h i n a l a y e r o f t h e e a r t h .

W e e x a m i n e t h e t h e o r y o f t o m o g r a p h i c r e c o n s t r u c t i o n w h e n t h e r e f e r e n c e s lo w -

n e s s is t a k e n t o b e a k n o w n f u n c t i o n o f d e p t h . A l s o , a l t h o u g h w e s p e c ia l iz e t h e

p r o b l e m t o d e p t h - d e p e n d e n t b a c k g r o u n d v e l o c it ie s a n d f l at r e f le c to r s , w e h o p e t h a t

o u r r e s u l t s f o r t h i s c a s e w i l l i n d i c a t e t h e c o n c e p t s t o a p p l y f o r m o r e g e n e r a ls i tu a t io n s . T h e g e n e r a li z e d I n v e r s e R a d o n T r a n s f o r m w h i c h w e w i ll d e ri v e f o r

c u r v e d r a y p r o j e c t i o n s is s i m i l a r t o t h a t d e r i v e d i n d e p e n d e n t l y b y G . B e y l k i n (1 98 2).

LINEARIZATION OF THE FORWARD PROBLEM

T h e t r a v e l t i m e , b e t w e e n a s o u r c e x s a n d r e c e i v e r Xr, a l o n g t h e r a y p a t h , r(x , , Xr) ,c a n b e w r i t t e n a s

t(Xs, Xr, n) = frra~ n[ r(x , , xr ; n ) ] ds. ( i )

In (1 ) n i s t h e s l o w n e s s , a n d d s i s t h e d i f f e r e n t i a l a r c l e n g t h . T h e e x p r e s s i o n ( 1) i s

n o n l i n e a r i n n . T o d e c o u p le t h e r a y p a t h s r(xs , xr; n) f r o m t h e u n k n o w n s lo w n e ss

f ie ld , n , w e w r i te t h e s l o w n e s s a s a p e r t u r b a t i o n a b o u t a n a s s u m e d r e f e r e n c e f i el d

n0.

n (~ ' ) = n o (~ ') + A n (~ ) . ( 2)

2201



TOMOGRAPHIC RECONSTRUCTION OF VELOCITY ANOMALIES 2203

w h e r e

f ~ p ( h ) c ( y )q(z , h ) = J1 - p ( h ) 2 c ( y ) 2

dy. (7)

I n t h i s n o t a t i o n , p ( h ) , i s t h e r a y p a r a m e t e r f o r a g i v e n o f f s e t h ( p ~ - cos Oo/c(0),

w h e r e Oo i s t h e t a k e - o f f a n g l e o f t h e r a y a t t h e s o u r c e ) .

A n e q u i v a l e n t m o d e l t o t h a t s h o w n i n F i g u r e 1 is o b t a i n e d b y r e f l e c t in g t h e l a y e r

a n d i t s s l o w n e s s f i e ld a b o u t t h e l i n e z = z 0 a s i s s h o w n i n F i g u re 2 . T h i s n e w

g e o m e t r y a ll ow s t h e r a y p a t h t o b e e x p r e s s e d a s

Z z o p ( h ) c ( y )x ( z ) = m - , / { p- c2 dy. ( 8 )

F o r e q u a t i o n (4 ) , w e w r i t e

F 2~o dsA t ( m , h ) = ~ n ( x - q ( y , h) , y ) dzz (y ' h ) d y .

'~0(9 )

mI

I\

i I

n o ( Z )

n o ( Z o - Z )

FIG. 2. Equivalent transmission geometry for symmetrized field.

F o r a c o n s t a n t b a c k g r o u n d v e l o c i t y t h e r a y s a r e s t r a i g h t l i n es . T h u s i n ( 9) , A t i s

s i m p l y t h e s t a n d a r d t w o - d i m e n s i o n a l R a d o n T r a n s f o r m o f A n e x p r e s s e d i n t e r m s

o f m i d - p o i n t a n d o f f se t . H e n c e , a n e x a c t i n v e r s io n f o r m u l a ( a t le a s t i n a d o m a i n o f

F o u r i e r s pa c e ) c a n b e fo u n d . F o r a d e p t h - d e p e n d e n t b a c k g r o u n d f ie l d, t h e r a y s f o r

s m a l l o f fs e t s a r e o n l y s l ig h t l y c u r v e d , a n d u s i n g a n e x p a n s i o n o f t h e i n t e g r a n d o f

(9 ) a b o u t h = 0 , a n d u s i n g a n a p p r o p r i a t e c h a n g e o f v a r i ab l e s, w e c a n o n c e a g a i np u t (9 ) i n t o t h e f o r m o f t h e s t a n d a r d R a d o n T r a n s f o r m . T h e i n v e r si o n f o r m u l a s fo r

t h e a b o v e p r o b l e m s a r e g i v e n in t h e s e c ti o n s o n " S m a l l -O f f s e t A p p r o x i m a t i o n " a n d

" C o n s t a n t B a c k g r o u n d V e l o c i ty " , r e s p e c ti v e l y .

T h e s e t w o c a s e s l e a d o n e t o c o n s i d e r a backprojection a p p r o x i m a t i o n t o A n (x , z ) .

T h a t i s , t o r e c o n s t r u c t t h e s l o w n e s s f i e l d , A n ( x , z ) a t a p o i n t ( x , z ) w e c o m p o s i t e

w e i g h t e d t r a v e l - ti m e p e r t u r b a t i o n s t h a t c o r r e s p o n d t o r a y s w h i c h p a ss t h r o u g h

(x , z ) . A s w e s h a l l n o w s h o w , a g o o d b a c k p r o j e c t i o n f o r m u l a t o c o n s i d e r i s

f h m ~ A t ( 2 + q ( 5 , h ) , h ) I 02q I

n1 (2 , ~ ) = __ - ~ dh .- ~ d s (5, h)d z

(10)

E a c h t r a v e l - t i m e c o n t r i b u t i o n i s d i v id e d b y t h e l o c al a rc l e n g t h , ds / dz ( 5 , h ) , a n d



2204 J O H N A . F A W C E T T A N D R O B E R T W . C L A Y TO N

m u l t i p l ie d b y a r a y f o c u s s in g f a c to r , I (02q/OzOh)(5, h ) I . T h e l im i t s o f i n t e g r a t i o n ,

± h m ,x , e m p h a s i z e t h e f a c t t h a t w e c a n o b s e r v e A t ( m , h ) f o r o n l y a fi n i t e r a n g e o f

o f f s e t s . A l s o , w e a r e o n l y i n t e r e s t e d i n t h o s e r a y s t h a t d o n o t h a v e t u r n i n g p o i n t s

w i t h i n t h e l a y e r.

T o s h o w t h e v a l i d i t y o f e q u a t i o n ( 10 ), w e s u b s t i t u t e (9 ) i n t o (1 0 ) t o o b t a i n

n l ( ~ , 5 ) = A n ( ~ + q ( 5 , h ) - q ( z , h ) , z )-hmax

ds ( z , h )dz l a2q ( 5 , h ) I

ds (5, h ) Ozahd z

dzdh . (11)

W e n o w c o n s i d e r a p o l a r c o o r d i n a t e s y s t e m c e n t e r e d a t ( ~, 5 ), w h i c h w e

s h o w d i a g r a m a t i c a l l y i n F i g u r e 3 . M a t h e m a t i c a l l y , t h e d o m a i n ~2 o f F ig u r e 3 i s

x - ~ = r c os 0

z - ~ = r s in 8

FIG. 3. Ra y/polar coordinatesystem.

((x, Z)l I Z/X [ > t a n O ~ n ). W e n o w c h a n g e v a ri a b l e s f r o m ( h , z ) to ( 0 , r) w h e r e

_ , / O q )0 = -cot ~ ( z , h )

ds 1d z s in O ; x ~ = r cos 0 z 5 r s in O.

(12a)

(12b)

T h e J a c o b i a n o f t h i s a b o v e c h a n g e v a r i ab l e s is

0 ( 0 , 5 = - ~ (5, h ) I-- 1

s in 0 ( 1 3 )

W e c a n n o w w r i t e ( 11 ) a s

~ " A n ( 2 + r c o s 0 + R I (O , r ), 5 + r s in 0 ) ( 1 + R 2 (O , r ) ) rdrd 0r t l ( .~ , ~ )

3, Iz- l( 1 4 )




T h e t e r m s RI(0, r ) a n d R2 (0, r ) a r e r e m a i n d e r t e r m s f r o m t h e f i r s t o r d e r T a y l o r

e x p a n s i o n o f t h e i n t e g r a n d i n ( 14 ) a b o u t z = 2. W e n o w a s s u m e t h a t a g o o d

a p p ro x i m a t i o n t o (1 4) i s a l o c a l a p p ro x i m a t i o n , w h e r e w e l e t t h e z l i m i t s o f f~ g o t o__+oo

An(2 + r cos O, 2 + r s in O)rdrd 0n l ( ~ , 2) J~ I z - ~ l

(15)

W e c a n w r i t e ( 1 5 ) i n t h e f o r m

r e ( Y , 2 ) = 0 ( 2 , 2 ) • ~ n ( ~ , 2 ) (16a)

w h e r e O(x , z ) h a s t h e f o r m

I z l l z ]( x , z ) - fo r Y ~ t a n Omin .

= 0 o t h e rw i s e . (16b)

F o r m a l c a l c u l a t i o n s s h o w t h a t i n t h e F o u r i e r d o m a i n , ~ )(k ~, k z) h a s t h e f o ll o w i n g

f o r m

1 ik z I)(k~, kz) = I~---[ ~ _-. c o t Omm

= 0 o th e rw ise . (16c)

E q u a t i o n s (1 5) a n d ( 16 c ) s u g g e s t t h e f o l l o w i n g i m p r o v e m e n t t o t h e b a c k - p r o j e c t i o n

fo rm u l a (1 0 )

f ~ A t ( ~ + q ( 2 , h ) , h ) I 02q In l (~ , 2 ) - - F (~ ) • _ 0 - ~ dh .

-~= ds (2, h)d z

(17)

H e r e , F ( x ) i s d e f i n e d f o r m a l l y a s F ( x ) = F - I ( I k ~ l ) . I n p r a c t i c e , F (x ) w i l l b e t h e

i n v e r s e F o u r i e r t r a n s f o r m o f s o m e f i n i te a p p r o x i m a t i o n t o I k ~ l. T h e f i l te r F ( x ) is

p r o p o r t i o n a l t o t h e H i l b e r t T r a n s f o r m o f t h e d / d x o p e r a t o r . I n t u i t i v e l y , a s t h e

b a c k p r o j e c t e d f i e ld is a n i n t e g r a l ( a s m o o t h i n g o p e r a t o r ) o v e r a ll tr a v e l - t i m e

p e r t u r b a t i o n s f r o m r a y s p a s s i n g t h r o u g h t h e p o i n t ( ~ , 2 ) , i t i s c l e a r s o m e t y p e o f

h i g h - f r e q u e n c y o p e r a t i o n m u s t b e a p p l ie d t o t h e b a c k - p r o je c t io n a p p r o x i m a t i o n t o

r e c o v e r t h e t r u e f i e ld . T h e n u m e r i c a l i m p l e m e n t a t i o n o f t h e f i l te r is d e s c r i b e d b e l o w

i n " T h e I n v e r s i o n P r o g r a m " [ se e e q u a t i o n ( 26 ) ]. E q u a t i o n ( 17 ) is t h e g e n e r a l iz e d

b a c k - p r o j e c t i o n f o r m u l a w e w i ll e m p l o y . B e l o w , w e g i ve s i m p l i f ic a t i o n s o f (1 7) f o r

v a r i o u s s p e c i a l c a s e s.

SMALL-OFFSET APPROXIMATION

O v e r t h e r a n g e o f i n t e g r a t i o n i n (1 7), h E [ - h . . . . h m ~ ], w e w i ll e x p a n d t h e

i n t e g r a n d i n a f i r s t o r d e r T a y l o r s e r ie s a b o u t h = 0 . T h u s , w e w i ll u s e t h e f o l l o w i n g



2206

expansions

J O H N A . F A W C E T T A N D R O B E R T W . C L A Y T O N

~ ° c ( y ) d y

q ( ~ , h ) = + O(h2)

f o ~° c(y) d y

( 1 8 a )

I 02q [h=O C(Z) + O (h2 )

= i z oc ( y ) a y

( l S b )

d s [ I = 1 + O(h2 ) . (18c)dzz h=O

W e w i l l u s e t h e d e f i n i t i o n s , r -~ ~ c ( y ) d y , ~ o = fr o c ( y ) d y , a n d n o w w e c a n w r it efo r ( 17)

(19)

CONSTAN T BACKGROUND VELOCITY

F o r a c o n s t a n t b a c k g r o u n d v e l o ci ty , w e h a v e t h e f o ll o w in g re l a ti o n s

q(~, h) = (Zo - 5) h (20a)Z0

d s J - ~ + Z02

d z Z o(20b)

c 9 2 qO z O h ~ - Z o . (20c)

T h u s ( 1 1 ) b e c o m e s , e x a c t l y ,

n l (~ , ~ ) A n £ + (y ~ ) h= - - - , y z o d y d h .

-hmax "~0 Zo

(21)

U s i n g ( 20 ) a n d t h e c h a n g e o f v a r i a b l e s t o p o l a r c o o r d i n a t e s ( 12 ), w e o b t a i n e x a c t l y

n l ( e , 2) - - o(~, 2) , An(~, ~) .

H e r e , c o t O m ln= h m a x / Z o .H e n c e , f o r t h i s c a s e , w e c a n w r i t e e x a c t ly ,

(22)

n 1 ( £ , ~ ) =

~ = A t ( £ + h ( z ° - ~ ) ' h ) z oF ( ~ ) , j _ [ " _ _

go - -h m ax ~ _]_ Zo2dh (23)



T O M O G R A P H I C R E C O N S T R U C T I O N O F V E L OC IT Y A N O M A L I E S 2207

w h e r e

51(kx, kz) = AS (kx, k~) fo r 7- - - -Rx z0

= 0 o t h e rw i s e . (24)

T h e s e f o r m u l a s , ( 23 ) a n d ( 24 ), f o r c o n s t a n t b a c k g r o u n d v e l o c it y w e r e d e r i v e d in a

d i f f e r e n t f a s h i o n b y K j a r t a n s s o n (1 98 0) a n d F a w c e t t (1 98 3).

D E P T H A N D T H E R E L A T IV E R E S O L U T I O N

F o r t h e b a c k g r o u n d f i e ld a c o n s t a n t , t h e r a t i o o f m a x i m u m o f f s e t t o t h e r e f le c t o r

d e p t h , h~ax/Zo,d e t e r m i n e s f r o m ( 24 ) h o w w e l l w e c a n r e c o n s t r u c t t h e u n k n o w n f i e ld

i n F o u r i e r s p a c e. F o r a d e p t h v a r y i n g b a c k g r o u n d f i e ld , t h e a n a l o g o f h~ax/Zo i s t h e

m a x i m u m s l o p e , dx/dz(z ) , o f a n y r a y t h a t p a s s e s t h r o u g h a p o i n t (x , z ). F o r a

b a c k g r o u n d f i e ld w h i c h i n c r e a s e s w i t h d e p t h , t h e s l o p e s o f t h e r a y s i n c r e a s e s w i t hd e p t h . H e n c e , i n t u it i v e l y , w e e x p e c t o u r r e c o n s t r u c t i o n t o i m p r o v e w i t h d e p t h f o r

a n i n c r e a si n g b a c k g r o u n d v e lo c it y .

M o r e p h y s i c a l ly , f o r s t r a i g h t r a y s , i t is c l e a r t h a t w i t h o n l y a k n o w l e d g e o f t h e

At(m ,h=O)

/ \An

A t

FI ~ . 4 . E x am p le o f ray s ' re so lu t io n .

n o

m

p ro j e c t i o n s a l o n g r a y s , a ll w i t h t h e s a m e s l op e , i t i s n o t p o s s i b l e t o r e s o l v e v a r i a t i o n s

i n t h e u n k n o w n f ie l d i n t h e d i r e c ti o n o f t h e r a y s. A s a n e x a m p l e o f t h i s s t a t e m e n t ,

c o n s i d er th e g e o m e t r y s h o w n i n F i g u re 4 . W e s e e t h a t f o r t h e a n o m a l y a n d r a y s o f

F i g u r e 4, w e c a n d e t e r m i n e o n l y t h e l a t e r a l e x t e n t o f t h e c i rc l e. F o r s e i sm i c

e x p e r i m e n t s , w h e r e w e h a v e o n l y a f i n it e m a x i m u m o f fs e t, w e d e d u c e t h a t t h e l a c k

o f l a rg e o f f s e t d a t a w i ll c o r r e s p o n d t o p r o b l e m s i n t h e v ~ e ~i ca l r e s o l u t i o n o f t h e

a n o m a l y . F o r a d e p t h i n c r e a s i n g b a c k g r o u n d v e l o c i ty f ie l d, w e e x p e c t t h e v e r t i c a l

r e s o l u t i o n o f t h e a n o m a l y t o i m p r o v e , f o r i n t h i s c a s e t h e e f f e c ti v e s lo p e s o f t h e r a y s

i n c r e a s e w i t h d e p t h .

NUMERICAL EXPERIMENTS

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

t w o c o m p u t e r p r o g ra m s : o n e t o g e n e r a t e s y n t h e t i c t r a v e l- t i m e p e r t u r b a t i o n s f o r

k n o w n a n o m a l y a n d b a c k g r o u n d f i e l d s , a n d s e c o n d , a n i n v e r s i o n p r o g r a m u s i n g

e i t h e r ( 17 ), ( 1 9) , o r ( 23 ) t o i n v e r t t h e t r a v e l - t i m e d a t a . A l l c o m p u t a t i o n s w e r e d o n e

i n si n g l e p r e c i s i o n F o r t r a n - 7 7 o n a V A X c o m p u t e r .

Ge neration o[ synthetic da ta. T h e d a t a a r e g e n e r a t e d f r o m t h e p r o j e c t io n o f t h e

u n p e r t u r b e d r a y t h r o u g h t h e a n o m a l y fi el d. H e n c e , t h e d a t a d o e s n o t c o r r e s p o n d



2208 JOHN A. FAWCETT AND ROBERT W. CLAYTON

e x a c t l y t o t h e p e r t u r b a t i o n s w h i c h w o u l d b e m e a s u r e d i n a t r u e " s e i sm i c " e x p e r i-

m e n t . H o w e v e r , a s d i s c u s s e d a b o v e , t h i s i s a f i r s t - o r d e r a c c u r a t e a p p r o x i m a t i o n .

T h u s , o u r n u m e r i c a l e x a m p l e s b e lo w t e s t o n l y th e t o m o g r a p h i c i n v e r s io n f o r m u l a s

w i t h t h e a s s u m p t i o n t h a t t h e l i n e a r i z e d p r o b l e m i s v a li d .

W e t a k e t h e b a c k g r o u n d v e l o c it y fi el d t o b e o f t h e f o r m c ( z ) = a z + b . T h e v e l o ci ty

a n o m a l i e s a r e t a k e n t o b e d i sk s . T h e d i s k s' r a d i i a n d p o s i t io n a n d t h e v a l u e o f t h e

c o n s t a n t p e r t u r b a t i o n w i t h i n t h e d i s k a r e u s e r i n p u t p a r a m e t e r s . A s m e n t i o n e d

a b ov e , w e a r e a s s u m i n g f o r t h e d a t a g e n e r a t i o n t h a t t h e l i n e a ri z a ti o n a s s u m p t i o n

(3) is va lid . He nc e , we can use s im ple un i t s ( e.g., 1 , 2 , 0 .5 , e tc . ) f o r th e pe r tu rb a t io n

s t r e n g t h w i t h i n t h e d i s k s, a s i t is o n l y t h e r e l a ti v e si ze o f t h e p e r t u r b a t i o n s t h a t i s

r e l e v a n t . O f c o u r s e , in r e a l i ty , t h e l i n e a r i z a t io n i s o n l y v a l id f o r A n s u f f i c ie n t l y

smal l .

T h e r a y s fo r t h e f ie l d c ( z ) = a z + b a r e a r c s o f c i r cl e s . S o m e a l g e b r a a l lo ws o n e t o

d e t e r m i n e t h e e q u a t i o n o f t h e s e c i r c le s fo r a g i v e n o f f se t , h , a n d m i d - p o i n t m . T h e

i n t e r s e c t i o n p o i n t s ( i f t h e r e a r e a n y ) , ~ i ( m , h ) a n d ~ 2~ (m , h ) , o f t h e r a y w i t h t h e i t h

d i sk a r e f o u n d b y a p p l y i n g a q u a d r a t i c f o r m u l a . T h e n , t h e c o n t r i b u t i o n o f t h e i t h

d i s k t o t h e t r a v e l - t i m e p e r t u r b a t i o n , A t ( m , h ) i s n ~ s i , w h e r e n ~ i s t h e c o n s t a n t

s lo w n e s s p e r t u r b a t i o n w i t h i n t h e i t h d is k , a n d si i s t h e a r c l e n g t h o f t h e r a y , in t h e

M m i n =-5

FIG. 5. Geometry for example 1.

M m a x 5 = X

ZO=4

d i s k , b e t we e n £ 1 i ( m , h ) a n d £ 2 i ( m , h ) . T h i s c a l c u l a t i o n i s c a r r i e d o u t f o r a l l t h e

d is k s, a n d f o r b o t h t h e d e s c e n d i n g a n d a s c e n d i n g r a y s e g m e n t s .

T h e p r o g r a m u s e r s p e ci fi es th e m i n i m u m a n d m a x i m u m m i d - p o i n t, m m a~ a n d

m m in, a n d t h e p e r c e n t a g e o f t h e m a x i m u m o f f s e t t o c a l c u l a t e A t ( m , h ) f o r. T h e

m a x i m u m o f fs e t , f o r a v e l o c i ty p r o f i le t h a t i n c r e a s e s w i t h d e p t h , i s t h e o f f s e t o f t h e

r a y t h a t h a s a t u r n i n g p o i n t a t z = Zo.S i x t y - f o u r i n c r e m e n t s i n h a n d m a r e t h e n c a l c u l a te d . W e o n l y c a lc u l a t e h > 0 a s

w e k n o w t h a t A t (m , - h ) = A t ( m , h ) . T h i s d a t a f i l e i s t h e n s t o r e d a s t h e i n p u t f o r

t h e i n v e rs i o n p ro g r a m .

T h e i n v e rs i o n p r o g r a m . T h e t w o b a si c f o r m u l a s w e w i s h to e x a m i n e n u m e r i c a l l y

a r e e q u a t i o n s (1 7) a n d ( 1 9) [ a n d ( 2 3 ) wh i c h is ( 19 ) a n d (1 7) f o r c ( z ) = b ]

( ' ~ x A t ( ~ + q ( 5 ' h ) ' h ) I 02q In1(2 , ~) = F(~)*J_hm .x ~ dh

( ~ , h )d z

(17)

(19)



~mOO

~

oTO~

•o

o

(_Ukp

~

o

'V

o

"~I~uvo

-oO

(ww-

(z

•z+-

(o~

6AOADXNO&DHAO



2210 J O H N A. F A W C E T T A N D R O B E R T W . C L A Y T O N

d e f i n i n g t h e f o l l o w i n g t a p e r e d f u n c t i o n i n d i s c r e t e F o u r i e r s p a c e

2 ~ j ( ~ j ) JP ( J - j ) = F ( j ) = - ~ - c o s - ~ j = 0 , ~ . ( 2 6)

W e n o w t a k e F ( j ) a s t h e i n v e r s e F o u r i e r t r a n s f o r m o f F ( . ] ) . H o w e v e r , w e f i n d i t

I

I | l i I

| , , , , I

I

I

I II

I

II

I I I I l l

I ll

I

X

i i I I |

i I

ii I

I

m

m

m m

m

0 N

FIG. 7. B a c k - p r o j e c t e d f i e l d , c ( z ) = - 1 .

n e c e s s a r y t o p a d i l l ( x / , z ) a n d F ( j ) w i t h z e r o s t o a v o i d t h e w r a p a r o u n d e f f e c t sf r o m t h e s u b s e q u e n t c o n v o l u t i o n .

A s w e s h a l l s e e, t h e d i f f e r e n c e s b e t w e e n t h e r e s u l t s o f u s i n g ( 1 9 ) o r (1 7 ) a r e s li g h t

( a t l e a st , w h e n v i e w e d w i t h o u t p l o t t i n g f o r m a t ) . H o w e v e r , ( 1 9 ) i s m u c h q u i c k e r



T O M O G R A P H I C R E C O N S T R U C T I O N O F V E LO C IT Y A N O M A L I E S 2 2 1 1

c o m p u t a t i o n a l l y , a s t h e a m o u n t o f f u n c t i o n e v a l u a t i o n i n v o l v e d i n t h e i n t e g r a t i o n

i s m u c h l e s s t h a n i n ( 17 ) .

N u m e r i c a l e x a m p l e s

E x a m p l e 1

I n t h i s e x a m p l e , w e c o n s i d e r a d i s k o f r a d i u s 1 , w i t h c o n s t a n t s l o w n e s s p e r t u r -

L

I m , ,I

II I

No

F I O . 8. F i l t e r e d b a c k - p r o j e c t e d f i e ld , c ( z ) ~ 1 .

b a t io n , 1, l o c a te d a t t h e c e n t e r o f t h e f ie ld . O u r m i n i m u m a n d m a x i m u m m i d - p o i n t sa r e f o r t h i s e x a m p l e , mmin ---- - -5 a n d m m ~ = 5 , a n d t h e d e p t h o f t h e r e f l e c t o r is

zo = 4 . T h e g e o m e t r y f o r t h i s e x a m p l e i s s h o w n a b o v e in F i g u r e 5 .

W i l l w i l l v a r y " a " i n c ( z ) = a z + b , a n d t h e m a x i m u m o f f s e t h max , f o r d i f f e r e n t




i n v e r s io n s . I n a l l o u r e x a m p l e s , w e c a l c u l a t e 6 4 m i d - p o i n t p o s it i o n s, w h e r e t h e

d i s c r e t e m i d - p o i n t p o s i t i o n s a r e g i v e n b y m y = m m i n + j ( ( m m = - m m i n ) / 6 3 ) ( j = O,

6 3). F o r o u r f i r s t i n v e r s i o n , we t a k e c ( z ) = 1 . T h e o f f s e t s , h h a r e c a l c u l a t e d f r o m h k

k= ~ - ~ 8 . 9 5 3 ( k = 0 , 6 3 ) . F i g u r e 6 s h o ws t h e t r a v e l - t i m e p e r t u r b a t i o n d a t a f o r t h i s

i i | l

Illi i r a

i i i m

0F I G . 9 . R e c o n s t r u c t e d f ie l d ; h ~ = - -- 4 . 4 8 , c ( z ) m 1 .

m o d e l . W e n o t e t h e t w o " a r m s " o f d a t a . I f w e h a d t a k e n a p o i n t a n o m a l y a t ( fit, ~ )

i n s t e a d o f a fi n i t e t h i c k n e s s d i sk , t h e n t h e a r m s w o u l d b e t w o s t r a i g h t l in e s , a n dt h e s lo p e o f t h e s e l in e s w o u l d g iv e t h e d e p t h t o t h e a n o m a l y . F r o m e q u a t i o n (2 3)

m - f i t= 1 -- -. (27)

h Zo



T O M O G R A P H I C R E C O N S T R U C T I O N O F V E L O C IT Y A N O M A L I E S .2213

To invert the travel-time data, At(m, h), for nl(~, 2), we will numerically

imp leme nt (23). In Figure 7, we sh ow the back-projection app roximation to n l (x, z)

[i.e., F(x) has not yet been applied]. Here, the plot shows 64 mid-point positions

4kand 16 depth incre men ts zk = ~-~(k = 0, 15). Figure 7 agrees qualitatively with the

__......

I IHII

II

I

II

II I

i uH

0 ~ N

FIG. 10. R econ struc ted field from (17); h ~ = 5.98, c(z) = 0.2z + L

concept of convolving the "true" symmetrized field, An(x, z), with the fi lter

O(x, z), described by (16a). We see immediately that a filter F ( x ) which "kills" slow

horizontal variations and amplifies shorter wavelengths will improve the image of

Figure 7. The anomaly field obtained by applying F ( x ) to the back-projected field

is shown in Figure 8. We now h alf the offset coverage u sed above. Th e tomograph-

ically reconstructed field is shown in Figure 9. As we expect from the section on



2 2 1 4 J O H N A. F A W C E T T A N D R O B E R T W . C L A Y T O N

" D e p t h a n d t h e R e l a t i v e R e s o l u ti o n , " t h e a n o m a l y ' s v er t ic a l e x t e n t i s n o w l e s s w e l l

r e s o l v e d .

W e n o w c a l c u la t e A t (m , h ) o r t h e b a c k g r o u n d f i e ld c( z ) = 0 . 2z + 1 . T h e m a x i m u m

o f f s e t w e u s e o f hm a~ = 5 . 9 8 , a n d w e c a l c u l a t e 6 4 i n c r e m e n t s i n m a n d h . F i g u r e 1 0

J

J

I I I I I I I

- - IilllII

J

J

No

F r o . 1 1 . R e c o n s t r u c t e d f i e l d f r o m ( 1 7 ) ; h ~a x = 3 . 9 2 , c ( z ) = z + 1 .

s h o w s t h e r e c o n s t r u c t e d f i e l d u s i n g f o r m u l a ( 1 7 ) . W e n o t e , t h a t a s w e d i s c u s s e d i n" D e p t h a n d t h e R e l a t iv e R e s o l u t i o n ," t h e v e r t i ca l r e s o l u t i o n o f t h e a n o m a l y ,

p a r t i c u l a r l y a t t h e t o p o f t h e f i e ld , h a s d e c r e a s e d . F o r c ( z ) = z ÷ 1 , h m ax = 3 . 9 2 , a n d

t h e r e c o n s t r u c t e d f ie l d , u s i n g ( 1 7 ) , i s s h o w n i n F i g u r e 1 1 , a n d u s i n g ( 1 9 ) , F i g u r e 1 2 .




I n F i g u r e 1 3, w e s h o w t h e r e s u l t o f a n i n v e r s i o n [ d a t a f o r c ( z ) = z + 1 ] u s i n g t h e

b a c k g r o u n d v e l o c i t y f i el d t o b e a c o n s t a n t (i.e ., t h e r a y s a r e s t r a i g h t ) . W e c a n s e e

f r o m th i s e x a m p l e t h a t t h e b e n d i n g o f t h e b a c k g r o u n d r a y s h a s a n i m p o r t a n t e f f e c t

u p o n t h e i n v e r s i o n r e s u lt s .

I

I

I II

: - No

FIG. 12. Reco nst ruct ed field from (19); h ~ = 3.92, c ( z ) = z + 1 .

E x a m p l e 2

A s w e h a v e d i s c u s s ed , f o r a b a c k g r o u n d f i e ld t h a t i n c r e a s e s in d e p t h , t h e r e s o l u t i o n

o f o u r t o m o g r a p h i c r e c o n s t r u c t i o n i m p r o v e s w i t h d e p t h . I n t h i s e x a m p l e , w e c o n s i d e r

t h r e e d i s k s o f v a r y i n g p o s i t io n , r a d ii , a n d s l o w n e s s p e r t u r b a t i o n s t r e n g t h , l o c a t e d




I

L

X

I I I

~ N

F IG . 1 3 . R e c o n s t r u c t e d f ie l d u s i n g c ( z ) c o n s t a n t , h m .x - - 3 . 9 2 , c o ( z ) = z + 1 .

n e a r t h e b o t t o m o f t h e l a y er . S o m e r a y s p a s s t h r o u g h m o r e t h a n o n e d i sk .

O n c e a g a i n , m m i n = - - 5 , m m a x -----5 a n d t h e r e f l e c t o r i s a t Zo = 4 . T h e d i s k s a l l h a v e

t h e i r c e n t e r s a t d e p t h z = 3 , w i t h h o r i z o n t a l c o o r d i n a t e s x ~ = - 3 , x 2 = 0 , x 3 = 2 .

T h e c o n s t a n t s l o w n e s s p e r t u r b a t i o n s i n e a c h a r e 2 , 1 , a n d 2 , r e s p e c t i v e l y . S c h e m a t -i c a ll y , t h e a n o m a l y f i e l d i s s h o w n b e l o w i n F i g u r e 1 4 . T h e t r a v e l - t i m e d a t a ,

At (m, h ) , i s p l o t t e d i n F i g u r e 1 5 . F i n a l l y , i n F i g u r e 1 6 , a a n d b , w e s h o w t h e r e s u l t s

o f t h e i n v e r s i o n u s i n g f o r m u l a s ( 1 7) a n d ( 1 9 ), r e s p e c t i v e ly .



T O M O G R A P H I C R E C O N S T R U C T I O N O F V EL O CI TY A N O M A L I E S 2217

Mmin=-5 Mmax=5

x = - 3

~ / ~ n

~ x = 0 /x=2

:2 ~ A n : 2

FIG. 14. G eom etry or example 2.

Z0= 4- - z = 3

O f f s e t 'h ' I - i

Midpo in t 'm '

Im

m

A t ( r e , h )

~ m

~ mu

~ m m ~ m ~

~ m m m m m m m m ~

~ m m m m ~ m

~ ~ , m m m m m

~ N m m m , , , m m m m

~ m m m m b m , m

~ h

~ h

~ m m ~ ,

~ = ~ - -

~ h

m m m m m m m h . ~

, m m m m . , . - ~ ~

m m m m m . . ~

. m m m m m m , . . ~ ~

• ~ m m . . . - ~ m ~ .

• m m m m - - ~ h

, , , m m m m , . . ~ m h

. ~ m m ~ m . m~

h

h

" ' ~ ' ' ' ~ V4mmm

. ~ . - , . . . m . ~ m m

m ' ' m ~ ' ~ m m m mh

. - - - - - - - - . . m m . . ~ m h

- . . - - - - . . . . m . . . , . ~ h

i~ , m ~

. - . . . . . - - . . , m m m b ~

m ~ .

, m - - . - - . m . , . m m m ,..--

m m ~

m ~ . . . , ~ m m ~ ~ . .

• m ~ m . , ' = ' m m ~ m , ~

. ~ m m m m ~ m m

. . ~ m ~ m m m m m

. ~ . . ~ m m m ~ m

m

~ , m m

~ m m m m

J

. , ~ m m m m m ~

b

m m ~

m m ~ m

m

m

m m m m

m

m

m

m m m

m

m m

m m

M

I

FIG. 15. Travel-Timedata, At(m , h), for Fig ure 16.



2218 J O H N A . F A W C E T T A N D R O B E R T W . C L A Y T O N

a

xT

I I

, . I I |

, , i i I I III

. . . . . , |

i

IL il

~ J

J

, I

P

I

IN

J

0 ~ N 0 ~ N

f

I

IIIII I

lift

im

. . . . . I

. . . . . . . L I m i i i i i

II I

i

I

I I I ~- .

I IIIII

III

IL

m

I I ,,

II I ,'

f "

FIG. 16 . (a) Invers ion from (17) ; c(z) = z + 1, h~ = 3.92. (b) Inv ers io n f rom (19); c(z) = z + 1,

h~a, = 3.92.

C O N C L U S I O N S

A simple generalized inverse radon transform (17) can be used to qualitat ively

reconstruct s lowness anomalies with respect to a depth varying background f ie ld

from observed surface reflection (f lat reflector) travel-t ime data. Mu ch of our

analysis was based upon the assumption that the s lowness anomalies were spat ia l ly

local ized. Thus, we could consider a local coordinate system, centered on ananomaly , and using the constant velocity problem as a model , def ine a local radon

transform. There are certa inly s ituat ions where our a p r i o r i phys ica l a ssumpt ion

may break down [e .g . , the anomalies may not be compact , or the background ray

f ield may have s ingularit ies (caust ics) ] . However, we hope that these ideas derived




f r o m l o c a l a n a l y s is , c a n a l s o b e a p p l i e d to t h e s i t u a t io n o f m o r e g e n e r a l b a c k g r o u n d

m e d i a [ f o r m o r e w o r k o n g e n e r a l i z e d r a d o n t r a n s f o r m s , s e e B e y l k i n ( 1 9 8 2 ) ] . A

s i m p l e r i n v e r s i o n f o r m u l a , { 1 9 ) , b a s e d o n z e r o o f f s e t a p p r o x i m a t i o n s c a n a l s o b e

e f f e c t i v e l y u s e d f o r n o ( ~ ) = n o ( z ) .

T h e q u a l i ty o f t h e a n o m a l y r e c o n s t r u c t io n a t a p o i n t w i t h i n t h e l a y er d e p e n d s

up on th e '"angu lar coverage" ( i .e . , the s i ze o f Omin) o f the rays for the bac kgr oun d

f i e l d . T h i s c o v e r a g e i n c r e a s e s w i t h d e p t h f o r a b a c k ~ o u n d v e l o c i t y f i e l d t h a t

i n c r e a se s w i t h d e p t h . W e h o p e t o a d d r e s s s o m e o f t h e p r o b le m s a n d e x t e n s i o n s o f

o u r m e t h o d s i n f u t u r e w o r k .

ACKNOWLEDGMENTS

This paper is based upon a chapter of the first author's Ph.D. Thesis at the California Institute of

Technology. This chapter was written under the supervision of the second author and Professor H. B.

Keller, whom we would like to thank for his suggestions and encouragement. We would also like tothank Professor J. B. Keller at Stanford for his helpful critiques of earlier versions of this paper.

The first author (J. F.) was supported financially at the California Inst itute of Technology by theU.S. Department of Energy and the Natural Sciences and Engineering Research Council of Canada. At

Stanford, financial support for this research was provided by the Air Force Office of Scientific Research,

the Army Research Office, the Office of Naval Research, and the National Science Foundation.

REFERENCES

Beylkin, G. (1982). Generalized radon transform and its applications, Ph.D. Thesis, New York Universi ty,

New York.

Fawcett, J, (1983). I. Three-dimensional ray-tracing and ray-inversion in layered media. II. Inverse

scattering and curved ray tomography with applications to seismology, Ph.D. Thesis, California

Ins titute of Technology, Pasadena, California.

Kjartansson, E. (1980). Attenuation of seismic waves in rocks, Ph.D. Thesis, Stanford University,

Stanford, California.

DEPARTMENT OF MATHEMATICS

STANFORD UNIVERSITY

STANFORD, CALIFORNIA94305 (J.A.F.)

SEISMOLOGICALLABORATORY

CALIFORNIA NSTITUTEOF TECHNOLOGY

PASADENA, CALIFORNIA91125 (R.W.C.)

Manuscript received 1 March 1984