Bulletin of the Seismolo gical Society of Ameri ca, Vol. 74 , No. 6, pp. 2201-2219, December 198 4 TOMOGRAPHIC RECONSTRUCTION OF VELOCITY ANOMALIES BY JOHN A. FAWCETT AND ROBERT W. CLAYTON ABSTRACT An approximate inversion formula is proposed for the reconstruction of slow- ness anomalies in a known depth varying background field. The data are ob- served travel-time perturbations for reflections from a known planar reflector. The limitations of the formula are discussed and numerical examples are given. INTRODUCTION Tomography refers to the technique of reconstructing a field from line or surface integrals of it. In medical X-ray tomography, for example, the tissue density field is deduced from measurements of X-ray attenuation through the patient. In this case the data are regularly sampled line integrals, so they are discrete values of a Radon transform. In seismology, the determination of slowness (inverse of velocity) and attenuation fields can also be viewed in a tomographic framework. The travel times or the amplitude decay along rays connecting the sources and receivers are the projections of the fields. Here, the problem is complicated by the fact that the rays are curved and that the ray path depends on the slowness field itself. The problem is in general nonlinear. One can linearize the problem about a reference slowness which essen- tially decouples the ray paths from the unknown slowness field. However, this leaves the problem of tomographic reconstruction from line integrals along curved rays. Our methods discussed below will be thought of in terms of a reflection seismology experiment. However, the results are also applicable to transmission problems. The goal of our tomographic reconstruction is to identify from travel-time information (source positions known) areas of relatively high and slow velocity (with respect to a known background field) within a layer of the earth. We examine the theory of tomographic reconstruction when the reference slow- ness is taken to be a known function of depth. Also, although we specialize the problem to depth-dependent background velocities and flat reflect ors, we hope that our results for this case will indicate the concepts to apply for more general situations. The generalized Inverse Radon Transform which we will derive for curved ray projections is similar to that derived independently by G. Beylkin (1982). LINEARIZATION OF THE FORWARD PROBLEM The travel time, between a source xs and receiver X r, along the ray path, r(x,, Xr), can be written as t(Xs, X r , n) = frra~ n[r(x,, xr; n ) ] d s . ( i ) In (1) n is the slowness, and d s is the differential arclength. The expression (1) is nonlinear in n. To decouple the ray paths r(xs, xr; n) from the unknown slowness field, n, we write the slowness as a perturbation about an assumed reference field n0 . n(~') = no(~') + An(~). (2) 2201
20
Embed
Tomography Reconstruction of Velocity Animalies by Fawcett and Clayton 1984
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
7/28/2019 Tomography Reconstruction of Velocity Animalies by Fawcett and Clayton 1984
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
7/28/2019 Tomography Reconstruction of Velocity Animalies by Fawcett and Clayton 1984
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
7/28/2019 Tomography Reconstruction of Velocity Animalies by Fawcett and Clayton 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 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
7/28/2019 Tomography Reconstruction of Velocity Animalies by Fawcett and Clayton 1984
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)
7/28/2019 Tomography Reconstruction of Velocity Animalies by Fawcett and Clayton 1984
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
7/28/2019 Tomography Reconstruction of Velocity Animalies by Fawcett and Clayton 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 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
7/28/2019 Tomography Reconstruction of Velocity Animalies by Fawcett and Clayton 1984
2212 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
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
7/28/2019 Tomography Reconstruction of Velocity Animalies by Fawcett and Clayton 1984
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 .
7/28/2019 Tomography Reconstruction of Velocity Animalies by Fawcett and Clayton 1984
2216 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
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 .
7/28/2019 Tomography Reconstruction of Velocity Animalies by Fawcett and Clayton 1984
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
7/28/2019 Tomography Reconstruction of Velocity Animalies by Fawcett and Clayton 1984
TOMOGRAPHIC RECONSTRUCTION OF VELOCITY ANOMALIES 2219
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,