Tomography Reconstruction of Velocity Animalies by Fawcett and Clayton 1984

7/28/2019 Tomography Reconstruction of Velocity Animalies by Fawcett and Clayton 1984

http://slidepdf.com/reader/full/tomography-reconstruction-of-velocity-animalies-by-fawcett-and-clayton-1984 1/19

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

Documents

Tomography Reconstruction of Velocity Animalies by Fawcett and Clayton 1984