From Wide Angle Rpf lection to Leaking Mode...,1 Elastic Wave Propagation in an Infinite Medium Due 11 to a Line Source.2 Generalized Ray Theory 16.3 The Solution of Equation (£.27)

F r o m W i d e A n g l e R p f l e c t i o n t o L e a k i n g M o d e

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t h e r e q u i r e m e n t s f o r t h e d e g r e e o f

D o c t o r o f P h i l o s o p h y

i n t h e U n i v e r s i t y o f T o r o n t o

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M a r c h , T 9 7 0

© C . N . G . D a m p n p y , 1 9 7 0

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

A b s t r a c t

S e i s m o g r a m s o b s e r v e d by a d e t e c t o r i n t h e w i d e a n g l e r e f l e c t i o n

t o l e a k i n g mode r e g i o n , ( t h e i n t e r m e d i a t e z o n e ) , e x h i b i t a p l e t h o r a

o f i n t e r e s t i n g f e a t u r e s . Head w a ve s f o r m a n d i n t e r f e r e w i t h d i r e c t l y

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

l a r g e a m p l i t u d e s a n d t h e n o r m a l a n d l e a k i n g modes g r a d u a l l y becom e

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

t r a n s i t i o n r e g i o n f r o m s e i s m o g r a m s c o n s i s t i n g m o s t l y o f d i r e c t l y

r e f l e c t e d w a v e s t o s p i s m o g r a m s w h e r e t h e t o t a l c h a r a c t e r i s t i c s o f t h e

medium a r e i n t e g r a t e d by t h e n o r m a l a n d l e a k i n g modes i s t h e r e f o r e

an i m p o r t a n t a r e a ' o f s t u d y .

The e i g e n f u n c t i o n e x p a n s i o n o f an i n f i n i t e med ium e l a s t i c w ave

f i e l d i n c a r t e s i a n c o - o r d i n a t e s i s an i n f i n i t e s e r i e s o f p l a n e w aves

i J e s c h 'i 'b i 'n g ’ • g e n e ra T i-z e d T ay ff” w t t h te - f f l an d c o m p le x a n g l e s o f p r o p

a g a t i o n . The e x p a n s i o n c a n be m o d i f i e d f o r a p l a n e l a y e r e d medium by

a s s i g n i n g t r a n s m i s s i o n f a c t o r s t o t h e r a y s a s t h e y e n c o u n t e r e a c h

i n t e r f a c e p r o d u c i n g f u r t h e r r a y s .4

The f S h e r w o o d - C a g n i a r d " t e c h n i q u e , ( S h e r w o o d (1 9 5 8 and I 9 6 0 ) ) ,

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

i m p u l s i v e l i n e s o u r c e . I n t h i s t h e s i s i t h a s b e e n u s e d i n a m o d i f i e d

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

s e q u e n c e o f t r a n s m i s s i o n s a n d v e l o c i t i e s i n a two d i m e n s i o n a l med ium.

F u r t h e r m o r e , t h i s s o l u t i o n i s shown t o b e e a s i l y r e l a t e d t o c y l i n d -

r i c a l l y s y m m e t r i c t h r e ^ d i m e n s i o n a l wave p r o p a g a t i o n . The m a t h e m a t i c a l

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

i n t e r m e d i a t e z o n e . The t e c h n i q u e i s d e v e l o p e d and made s y s t e m a t i c by


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

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

s e i s m o g r a m s o b s e r v e d o n m o d e l s .

T h e t r a n s m i s s i o n f a c t o r a s s i g n e d t o r a y s w i t h a g i v e n a n g l e

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

f r o n t a n d t h e i n t e r f a c e . Thu s t h e m e t h o d i s e x t e n d e d t o p l a n e d i p p i n g

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

F i n a l l y an a t t e m p t i s made t o g e n e r a l i z e t h e t e c h n i q u e t o c u r v e d

i n t e r f a c e s . E v e n h e r ® some a g r e e m e n t i s o b t a i n e d w i t h t h e e x p e r i m e n t a l

s e i s m o g r a m s .

The e x p e r i m e n t a l s e i s m o g r a m s a r e m e a s u r e d o n a c a l i b r a t e d tw o

d i m e n s i o n a l s e i s m i c m o d e l s y s t e m . The s y s t e m f e a t u r e s a new

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

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


Ac know]odgpmonts

I n 1 9 6 3 I w r o t e t o Dr* J . T * W i l s o n r e q u e s t i n g i n f o r m a t i o n

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

o f T o r o n t o * H i s r e p l y , e n t h u s i a s t i c a l l y d e s c r i b i n g t h e a d v a n t a g e s

o f t h e G e o p h y s i c a l L a b o r a t o r i e s , l e d me t o a p p l y i n 1 9 6 6 t o e n t e r»

T o r o n t o * :

I w i s h t o a c k n o w l e d g e t h a t Dr* W i l s o n ' s e n t h u s i a s m i s

j u s t i f i e d * I am g r a t e f u l t o Dr* W e s t , my s u p e r v i s o r , f o r h i s

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

n o t h i n g a p p e a r s t o w o r k , a n d t o Dr* G r a n t f o r h i s a d v i c e w h i c h

h e l p e d me f o r m u l a t e some o f t h e t h e o r e t i c a l c o n c e p t s u s e d * I w i s h t o

t h a n k D r s * S a v a g e a n d C e r v e n y f o r t h e u s e f u l d i s c u s s i o n s I was

a b l e t o h a v e w i t h t h e m i n t r y i n g t o u n t i e some o f t h e k n o t t i e r

i d e a s i n S e i s m o l o g y *

I am a p r e c i a t i v e o f t h e f r e e u s e o f t h e U n i v e r s i t y o f T o r o n t o

IBM 3 6 0 / 6 5 c o m p u t e r * Ted C l e e ' s a d v i c e a n d h e l p w i t h t h i s s y s t e m

i s p a r t i c u l a r l y a c k n o w l e d g e d .

Th e r e s e a r c h w a s c a r r i e d o u t w h i l r I h e l d a B l y t h e F e l l o w s h i p

( 1 9 6 6 - 6 7 ) a n d a F r o v i n c e o f O n t a r i o G o v e r n m e n t F e l l o w s h i p ( 1 9 6 7 - 7 0 ) .


T a b le o f C o n t e n t s

pa geA b s t r a c t

A c k n o w l e d g e m e n t s

1 . I n t r o d u c t i o n 1

2 . T h e S h e r w o o d - C a g n i a r d T e c h n i q u e 10

, 1 E l a s t i c Wave P r o p a g a t i o n i n a n I n f i n i t e Medium Due 11

t o a L i n e S o u r c e

. 2 G e n e r a l i z e d Ray T h e o r y 16

. 3 The S o l u t i o n o f E q u a t i o n ( £ . 2 7 ) a n d i t s P h y s i c a l 2 0

I n t e r p r e t a t i o n

. 3 1 Head W a v e s / D i r e c t A r r i v a l s a n d R a y l e i g h / S t o n e l e y 2 7

Wave s

. 4 The ' R e l a t i o n s h i p 'B e tw e e n Two "an d T h r e e 'D in ro n rs io n a 1 28

• S c -a l-a r Wave P r o p a g a t i o n

•5 The R e f l e c t i o n / R e f r a c t i o n C o e f f i c i e n t s 32

•51 The He ad Wave C o e f f i c i e n t s 4 0i ■ *

3 . S e i s m o g r a m s f o r P l a n e L a y e r e d M o d e l s 44

. 1 L a m p ' s P r o b l p m 4 4

. 2 S o u r c e / R e c e i v e r o n t h e I n t e r f a c e B e t w e e n Two S e m i - 5 0

I n f i n i t e Med ia i n W e l d e d C o n t a c t

. 3 L a y e r O v e r a H a l f S p a c e C a s e 57

. 3 1 S o l u t i o n t o t h e S i n g l e L a y e r O v e r a H a l f S p a c e 58

P r o b l e m

. 3 2 D y n a m i c a l l y E q u i v a l e n t R a y s 61

. 4 M u l t i - L a y e r e d M e d i a 62


pa ge

. 5 P r o g r a m m i n g t h e G e n e r a l i z e d R a y T h e o r y 6 3

. 6 . P r o g r a m S t r u c t u r e 6 6

. 7 S u b r o u t i n e T h e o r y 6 7

, 7 1 L e v e l I S u b r o u t i n e s . 6 7

, 7 2 L e v e l 2 S u b r o u t i n e s 6 8

, 7 3 L e v e l 3 S u b r o u t i n e s 7 0

. 7 4 L e v e l 4 a n d 5 S u b r o u t i n e s * 7 0

. 7 4 1 S u b r o u t i n e TIMEV 7 0

. 7 4 2 S u b r o u t i n e s 1 D I F F a n d RTCOEF 7 3

. 8 M o d i f i c a t i o n t o A c c o u n t f o r D y n a m i c a n d K i n e m a t i c 7 3

E q u i v a l e n c e

. 9 M o d i f i c a t i o n s t o A c c o u n t f o r R e a l S o u r c e 7 4

, ( 1 0 ) P r o g r a m T ime E s t i m a t e 8 0

, (*11 ) T he T h e o r e t i c a l a n d ■ E x p e r i m e n t a l - S e i s m o g r a m s 81

4 . The D i p p i n g a n d C u r v e d L a y e r C a s e s 8 4

. 1 T h e S o l u t i o n t o t h p D i p p i n g L a y e r C a s e 8 4

• 1 1 B e h a v i o r o f t h e S o l u t i o n ’ 9 0

, 2 M o d i f i c a t i o n s o f t h e P r o g r a m m e f o r t h e D i p p i n g 91

L a y e r C a s e

, 3 T h e o r e t i c a l a n d E x p e r i m e n t a l S e i s m o g r a m s f o r t h e 9 2

D i p p i n g L a y e r C a s e

, 4 T h e C u r v e d L a y e r C a s e 9 2

, 5 S t u d y o f a Head Wave 9 4

. 6 T h e M u l t i v a l u e d n e s s o f t h e O p t i c a l T r a v e l T im e 9 6

F u n c t i o n

. 7 T h e o r e t i c a l a n d E x p e r i m e n t a l S e i s m o g r a m s f o r t h e 1 0 0

C u r v e d L a y e r C a s e


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5* -M odel S e i s m o l o g y 1 0 2

•1 T h e S e i s m i c M o d e l S y s t e m 1 0 2

• 1 1 P r o b e A s s e m b l y 105

• 2 S y s t e m R e s p o n s e 1 0 7

• 3 C h e c k i n g t h e S y s t e m U s i n g L a m b ' s P r o b l e m 1 1 2

• 4 T he C o n s t r u c t i o n o f L a y e r e d M o d e l s 11 5

• 4 1 T h e P o l y m e r i z a t i o n o f E p o x y R e s i n s 1 1 8

• 4 2 C o u p l i n g A g e n t s 1 2 0

^ 4*3 M e c h a n i c a l C o n s t r u c t i o n 1 2 2

A p p e n d i x A T h e S h e r w o o d - C a g n i a r d T e c h n i q u e

A p p e n d i x B T h e L o c u s o f T im e i n t h e C o m p l e x ® P l a n e

R e f e r e n c e s


1 . INTRODUCTION

T h e e a r t h ’ s c r u s t h a s b e e n d e f i n e d b y H o l m e s ( 1 9 6 6

p a g e 1 7 ) a s " c o n s i s t i n g o f a v a r i e d a s s e m b l a g e o f r o c k s o v e r -

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

s u r r o u n d i n g t h e m a n t l e " . T h i s u p p e r m o s t l a y e r o f t h e e a r t h ♦

c a n b e e x a m i n e d b y t h e s e i s m i c t e c h n i q u e w h i c h s t u d i e s t h e

d i s p l a c e m e n t g e n e r a t e d o n t h e e a r t h ' s s u r f a c e a s a r e s u l t o f

t r a n s i e n t f o r c e s a c t i n g a t o t h e r p o i n t s o f t h e c r u s t .

T h e s e t r a n s i e n t f o r c e s a r e p r o d u c e d e i t h e r n a t u r a l l y

b y e a r t h q u a k e s o r a r t i f i c i a l l y b y e x p l o s i o n s n e a r o r o n t h e

e a r t h ' s s u r f a c e . S h o c k w a v e s a r e c r e a t e d w h i c h r a p i d l y

r e d u c e t o s m a l l a m p l i t u d e d i s t u r b a n c e s i n t h e r e l a t i v e

v a s t n e s s o f t h e c r u s t . To a n a l y s e t h e s e d i s t u r b a n c e s m any

a s s u m p t i o n s m u s t b e m ad e a b o u t t h e c r u s t ' s p r o p e r t i e s .

T h e e q u a t i o n o f m o t i o n o f a s m a l l a m p l i t u d e d i s

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

m e d i u m

( 1. 1)

w a s d i s c o v e r e d b y C a u c h y a n d P o i s s o n i n 1 8 2 8 . F u r t h e r w o r k

d u r i n g t h e n i n e t e e n t h c e n t u r y d e m o n s t r a t e d t h a t t h i s d i s

t u r b a n c e c a n b e d e s c r i b e d b y t h e p r o p a g a t i o n o f t w o w a v e s

- t h e P ( p r i m a r y ) a n d S ( s e c o n d a r y ) w a v e s - w h i c h h a v e

d i f f e r e n t v e l o c i t i e s . T h e s e w a v e s a r e now r e f e r r e d t o a $


. t h e p r e s s u r e a n d s .h e a r w a v e s r e s p e c t i v e l y * F r o m e q u a t i o n ( l )

m u l t i p l y b y t h e o p e r a t i o n s a n d V * V t o o b t a i n

(1 . 2 )

( 1 . 3 )

w h e r e *44- — '-P — \ / X yyi /f< v '■ v ' . /V ( 1 . 4 )

Up t o t h e t i m e o f Lamb ( 1 9 0 4 ) c o n s i d e r a b l e c o n t r o

v e r s y r a g e d b e t w e e n t h e t h e o r i s t s who p r e d i c t e d t h e s e t w ot’

d o m i n a n t w a v e s t o b e e x p e c t e d i n t h e s e i s m o g z a m a n d t h e s e i s

m o l o g i s t who f o u n d l i t t l e c o r r e l a t i o n w i t h t h i s t h e o r y *

•w av es f ro rm a s e i s i n o g r a m - t h e P , S , a n d s u r f a c e w a v e s * T h i s

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

p a r t o f a s e i s i n o g r a m * L a m b ’ s m o n u m e n t a l c o n t r i b u t i o n t o

t h e o r e t i c a l s e i s m o l o g y l i e s i n h i s d e m o n s t r a t i n g t h e e x i s t e n c e

o f s u r f a c e ( R a y l e i g h ) w a v e s on- a s e m i - i n f i n i t e e l a s t i c m ed iu m *

T h i s b r o u g h t t h e o r y a n d p r a c t i c e i n t o a s e m b l a n c e o f a g r e e m e n t *

T he s u r f a c e w a v e s h a v e t h e i r o r i g i n m a t h e m a t i c a l l y i n t h e

b o u n d a r y c o n d i t i o n s e x i s t i n g i n t h e m e d i u m , r a t h e r t h a n

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

e x t e n t * L a m b ’ s w o r k was f o l l o w e d b y t h e s t u d y o f m o r e

c o m p l i c a t e d m o d e l s o f t h e e a r t h ’ s c r u s t a n d t h e t h e o r y t o

p r e d i c t f e a t u r e s o f s e i s m o g r a m s e x p e c t e d f r o m t h e s e m o d e l s *


O l d h a m , ( 1 9 0 0 ) h a d i d e n t i f i e d t h r e e m a j o r t y p e s o f

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

' m e d i u m f o r t h e p u r p o s e s o f a n a l y s i s * E x p e r i e n c e h a s s h o w n t h a t

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

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

n o t . a l w a y s ).. . i n c r e a s e s . w i t h d e p t h * E a c h l a y e r i s a s s u m e d

h o m o g e n e o u s . a n d . . i s o t r o p i c •

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

t h e t h e o r y . i s d i f f i c u l t e n o u g h w i t h o u t i n t r o d u c i n g f u r t h e r

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

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

th < ! . c r u s t h a v i n g . . p r o p e r t i e s o u t s i d e t h e f r a m e w o r k o f o u r

a s s u m p t i o n s * T h e m e d i u m m a y b e l o s s y . o r t h e i n t e r f a c e s may

b e . . . g r a d u a l r a t h e r t h a n a b r u p t * T h e s e c o m p l i c a t i o n s may s o m e

t i m e s . b e i n t r o . d u c . e d a t a . L a t e r - s t a g e a s ~ > p e r t u r b a t i . o n s .o n t h e

t h e b r y *

T h e - d e t a i l e d . c a l c u l a t i o n o f some t h e o r e t i c a l

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

a c c o m p l i s h e d w i t h i n . t h e l a s t t e n y e a r s * H i g h s p e e d d i g i t a l

c o m p u t e r s c a p a b l e o f t h e v a s t n u m b e r o f n u m e r i c a l o p e r a t i o n s

n e c e s s a r y - h a v e , b e e n p r i m a r i l y r e s p o n s i b l e *

E v e n t h i s . r e c e n t w o r k i s l i m i t e d t o r a n g e s o f t h e

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

a n d t o s o m e p a r t s o f t h e s e i s m o g r a m . W u e n s c h e l ( i 9 6 0 ) h a s

g e n e r a t e d s e i s m o g r a m s . f o r v e r t i c a l l y r e f l e c t e d w a v e s i n

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


- 4 -

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

v e r t i c a l - r e f l e e t i o n . . s e i s m o g r a m s t o c o m p a r e w i t h p r a c t i c a l

s e l s m o g r .a m s . a a d e . I n t h e f i e l d ^

On t h e . o t b e i h a n d f a . n a o b e s o f a u t h o r s ( e . g « E w i n g ,

J a r d e t z k y a n d P r e s s ( 1 9 5 7 ) ) h a v e i n t e r p r e t e d c r u s t a l a n d

u p p e r . m a n t l e s t r u c t u r e f r o m . s e i s m o g r a m s a t l a r g e e p i c e n t r a l

d i s t a n c e s . f r o m t h e . d i s p e r s i o n o f t h e s u r f a c e w a v e s . T h e

t h e o r e t i c a l . . t e c h n i q u e i n v o l v e s . a m o d a l o r e i g e n f u n c t i o n

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

m e d iu m .,

T h e . . . r e m a i n i n g , a r e a o f . i n t e r e s t l i e s b e t w e e n t h e s e

t w o - e x t r e m e . . . c a s .e s .o - S e l s m o g r a m s . a t " i n t e r m e d i a t e " d i s t a n c e s

d i s p l a y a . . p l e t h o r a a f _ . . i n t e r e a t i n g , p h e n o m e n a o H e a d w a v e s

b e c a m e , v i s i b l e a n d . . i n t e r f e r e n c e b e t w e e n h e a d w a v e s a n d

d i r e c t . a r r i v a l s , o c c u r « S u p e r b - c r i t i c a l r e f l e c t i o n r e s u l t s

i n u n e x p e c t e d l y . . l a r g e . . . a m p l i t u d e a r r i v a l s - * " L e a k i n g m o d e s "

d e v e l o p , a n d h a v e , b e e n . e x p l o i t e d . ( S u a n d D o r m a n ( 1 9 6 5 ) ) t o

i n t e r p r e t , t h e . s t r u c t u r e . O f . t h e . c r u s t o L e a k i n g m o d e a n a l y s i s

d e m o n s t r a t e s t h e w a y t h e o r e t i c a l s e i s m o l o g i s t s h a v e e x t e n d e d

t h e . r a n g e , o f v a l i d i t y , o f ...mode t h e o r y , t o w a r d s t h e i n t e r m e d i a t e

z o n e s . A& w e l l 9.. c o m m e r c i a l , e x p l o r a t i o n s e i s m o l o g y h a s

t e n d e d o u t :t o w a r d s w id e , - a n g l e . , r e f l e c t i o n m e t h o d s o

T h e - g e n e r a t i o n o f t h e o r e t i c a l s e i s m o g r a m s i n t h e

i n t e r m e d i a t e . z o n e . i s . t h e e n d r e s u l t o f t h i s t h e s i s » T h e

t h e o r e t i c a l t e c h n i q u e u s e d , t h e " g e n e r a l i z e d r a y t h e o r y


m e t h o d " , i s a c o n c e p t u a l e x t e n s i o n o f b o t h g e o m e t r i c a l o p t i c s

and t h e C a g n i a r d ( 1 9 6 2 ) " T r a n s m i s s i o n f a c t o r ” a p p r o a c h . A p a r t

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

s e i s m o g r a m s c a n be c o m p u t e d , a s e c o n d r e s u l t o f t h i s w o rk i s

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

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

g r a m s •

i n t h e s e two e x t r e m e c a s e s h a s b e e n r e c o g n i z e d b y a n u m b er

o f a u t h o r s . K l i n e a n d Kay ( 1 9 6 5 ) d e v o t e t h e i r b o o k t o t h e

a n a l o g o u s e l e c t r o m a g n e t i c p r o b l e m s “ We p r o p o s e t o e x p l o r e

and e x p l o i t t h e r e l a t i o n s h i p b e t w e e n M a x w e l l ' s e l e c t r o

m a g n e t i c t h e o r y an d g e o m e t r i c a l o p t i c s " . I n s e i s m o l o g y

Ben Menahem ( 1 9 6 4 ) d i s c u s s e s t h e way r a y s b u i l d up t o f o r m

t h e f u n d a m e n t a l modes o f t h e e a r t h . K n o p c f f e t a l ( i 9 6 0 )

b u i l d u p t h e l e a k i n g and n o r m a l m o de s i n a o n e - l a y e r o v e r a

h a l f - s p a c e m o d e l by summing u p t h e c o m p o n e n t m u l t i p l y

r e f l e c t e d a n d r e f r a c t e d b o d y w a v e s .

I n t e g r a l f o r m i f t h e b o u n d a r y c o n d i t i o n s o f t h e med ium a r e

k n o w n . S u p p o s e t h a t t h e s o u r c e i s s p e c i f i e d a s a p o i n t s o u r c e

and t h a t t h e med ium i s i n f i n i t e . T h e n v i a W e y l ' s s o l u t i o n

( 1 9 1 9 ) we o b t a i n

The i m p o r t a n c e o f r e c o n c i l i n g t h e t h e o r y r e q u i r e d

E q u a t i o n s ( 1 . 2 ) and ( 1 . 3 ) c a n b e e x p r e s s e d I n

( 1 . 5 )


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

I f t h i s m e d i u m i s n o t i n f i n i t e a n d b o u n d a r i e s e x i s t ,

a d d i t i o n a l w a v e s a r e g e n e r a t e d a s a c o n s e q u e n c e o f t h e b o u n d a r y

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

( 1 9 6 5 ) , p . 1 6 6 ) .

T h u s t o o b t a i n a c o m p l e t e s o l u t i o n o f t h e w a v e s

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

a n d r e f r a c t e d w a v e s ^ i n t e g r a t e t e r m b y t e r m ^ t h e n s u m . T h i s

i s t h e g e r e r a l a p p r o a c h b e h i n d t h e " g e n e r a l i z e d r a y t h e o r y " . .

I n c i d e n t a l l y t h i s s u m m a t i o n c a n b e c a r r i e d o u t a l g e b r a i c a l l y

b y f i r s t s u m m i n g a n d t h e n i n t e g r a t i n g fcy the H a s k e l l ( 1 9 5 3 ) , -

T h o m s o n ( 1 9 5 0 ) m e t h o d ( s e e H a n n o n ( 1 9 6 4 ) f o r e x a m p l e ) t o

o b t a i n t h e i n t e g r a l f r o m w h i c h t h e m o d a l e x p a n s i o n i s m a d e .

w a v e p r o p a g a t i o n ’ h a s n o t b e e n o b t a i n e d i n c l o s e d f o r m . U s i n g

C a g n i a r d ' s ( 1 9 6 2 ) t e c h n i q u e , t r i p l e i n t e g r a l s o f t h e f o r m

o f e q u a t i o n (1 .5 ) c a n b e r e d u c e d t o s i n g l e i n t e g r a l s . S h e r w o o d

( 1 9 5 8 ) w a s a b l e t o s i m i l a r l y r e d u c e t w o - d i m e n s i o n a l w a v e*

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

we c a n a r g u e t h a t t h e r e i s a d i r e c t l i n k b e t w e e n t h e t w o -

d i m e n s i o n a l a n d t h r e e - d i m e n s i o n a l p r o p a g a t i o n a s u n d e r

c e r t a i n s y m m e t r y c o n d i t i o n s n e i t h e r d e p e n d s o n a t h i r d

v a r i a b l e ( s e e F i g u r e l . l ) . P r a g m a t i c a l l y , d e s p i t e t h e

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

e n t a t i o n i n t h e s p i r i t o f W e y l * s s o l u t i o n ( s e e G r a n t a n d W e s t

T h e c o m p l e t e s o l u t i o n o f t h r e e d i m e n s i o n a l e l a s t i c


z

>XTWO-DIMENSIONAL

* k

+ rTHREE-DIMENSIONAL

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

w a v e p r o p a g a t i o n .


m a t h e m a t i c a l d i f f e r e n c e .. i n . . t h e t w o a n d t h r e e d i m e n s i o n a l i n t e g r a l

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

t w o - d i m e n s i o n a l r e s u l t s t o t h r e e - d i m e n s i o n a l p r o p a g a t i o n * I n

t h i s t h e s i s o n l y t w o d i m e n s i o n a l w a v e p r o p a g a t i o n h a s b e e n

c o n s i d e r e d .

U s i n g t h e g e n e r a l i z e d r a y t h e o r y e x t e n s i o n o f

S h e r w o o d ’ s w o r k ( i 9 6 0 ) i t i s p o s s i b l e t o o b t a i n a c l o s e d f o r m

s e r r a t i o n " f o r t h e d i s p l a c e m e n t d u e t o a n y m u l t i p l y - r e f l e c t e d

r a y i n a n i n f i n i t e l a y e r e d h a l f s p a c e . T h i s " S h e r w o o d - C a g n i a r d

t e c h n i q u e " h a s b e e n p u t i n t o p r a c t i c e t o . c o m p u t e i n t e r m e d i a t e

z o n e t h e o r e t i c a l s e i s m o g r a m s . T h i s r e q u i r e d t h e d e v e l o p m e n t

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

f a c i l i t a t e s c o m p u t a t i o n .

. . . I n t e r m e d i a t e z o n e • s e ,i s B io g ^ a m s *0'an *be -S 'tu d - le d

c l o s e l y t o d e t e r m i n e t h e m a n n e r i n w h i c h h e a d w a v e s d e v e l o p ^

I n t e r f e r e n c e o c c u r s , l e a k i n g m o d e s a n d n o r m a l m o d e s a r i s e .

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

a s s o c i a t e s d i r e c t . a n d . h e a d w a.ve a r r i v a l s w i t h w a v e s h a v i n g

a r e a l . , a n g l e o f i n c i d e n c e a n d R a y l e i g h / S t o n e l e y w a v e s w i t h

c o m p l e x a n g l e s .

T h e S h e r w o o d - C a g n i a r d t e c h n i q u e w a s t h e n e x t e n d e d

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

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

o f g e n t l y c u r v e d i n t e r f a c e s .


A c o n t i n u i n g s t i m u l u s t o t h e p r o g r e s s o f s c i e n c e

i s t h e r e a c t i o n b e t v / e e n t h e o r y a n d e x p e r i m e n t t o e x p l a i n

o b s e r v e d p h e n o m e n a . T h e o r y w i t h o u t e x p e r i m e n t i s a r e m n a n t

o f t h e G r e e k s c i e n t i f i c m e t h o d o l o g y . I n s e i s m o l o g y t h e

d i f f i c u l t i e s o f c a r r y i n g o u t m a c r o —e x p e r i m e n t s w i t h a n

u n k n o w n e n v i r o n m e n t ( t h e e a r t h ) h a v e b e e n r e d u c e d b y t h e

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

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

d i c t e d . T h e m o d e l s e i s m o l o g y i s a t l e a s t a n i n t e r m e d i a t e

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

We i n v e s t i g a t e t h e s e i s r a o .g x . a a . f r o j a .±.w.* . p o i o t s .» .f v i e w -

t h e o r y a n d m o d e l s e i s m o l o g y . A c o m p l e t e i n v e s t i g a t i o n

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

b u t t h i s i s b e y o n d t h e s c o p e o f t h i s t h e s i s .

M o d e l s e i s m o l o g y i s a n a n a l o g u e t o o l w h i c h h a s

b e e n u s e d i n c r e a s i n g l y t o s o l v e e l a s t i c w a v e p r o p a g a t i o n .

T h e a p p a r a t u s u s e d i s a n i m p r o v e d v e r s i o n o f M o h a n t y ’ s

( 1 9 6 5 ) w o r k . C a l i b r a t i o n w a s c a r r i e d o u t o n t h e n e w

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

a n a n a l o g u e s o l u t i o n t o L a m b * s p r o b l e m . A m a j o r d i f f i c u l t y


9

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

u n d e r l i n e d b y t h e w o r k o f S c h w a b ( l 9 6 7 ) a n d S c h w a b a n d B u r r i d g e ( 1 9 6 8 ) .

By u s i n g n e w l y a v a i l a b l e m a t e r i a l s t h e m o d e l l i n g m e t h o d o f H e a l y a n d

P r e s s ( i 9 6 0 ) w a s s u c c e s s f u l l y m o d i f i e d t o o b t a i n a w e l d e d c h e m i c a l

c o n t a c t *


- 1 0 -

2 . THE SHERWOOD CAGNIARD TECHNIQUE

" I t i s n o t g e n e r a l l y k n o w n t h a t L a m b u s e d t h e

e s s e n t i a l , t r a n s f o r m a t i o n i n h i s 1 9 0 4 . p a p e r t o d e r i v e t h e

e x a c t s o l u t i o n o f t h e i m p u l s i v e l i n e s o u r c e p r o b l e m f o r t h e

r e s t r i c t e d c a s e w h e r e b a t h s o u r c e a n d r e c e i v e r a r e o n t h e

s u r f a c e o f . a h a l f —s p a c e " . ( S u b s t a n c e o f c o m m u n i c a t i o n f r o m

F * . G i l b e r t t o E . A . F l i n n a n d C • H . D i x a s q u o t e d i n t h e i r

t r a n s l a t o r 1 s p r e f i x t o C a g n i a r d . _ . ( 1 9 6 2 ) ) •

T h e ^ C a g n i a r d . . T e c h n i q u e " h a s b e e n w i d e l y u s e d b y

a n u m b e r o f . i n v e s t i g a t o r s ( S m i r n o v a n d S o b o l e v ( 1 9 3 3 ) , a n d

P e k e r i s ( 1 9 4 1 , 1 9 5 5 ) , b o t h i n d e p e n d e n t l y o f C a g n i a r d ' s w o r k ,

G a r v i n ( 1 9 5 6 ) , d e H o o p ( i 9 6 0 ) f o r e x a m p l e ) . T h e p o w e r o f

' t h e ' t e c h n i q u e i s ' t h a t i t c a n r e d u c e d o u b l e i n t e g r a l s t h a t

a r e . g e n e r a t e d i n t h e o r e t i c a l s e i s m o l o g y t o c l o s e d e x p r e s s i o n s .

( A . b r i e f d e s c r i p t i o n o f t h e C a g n i a r d t e c h n i q u e i s g i v e n i n

A p p e n d i x A ) •

M o s t . i n v e s t i g a t o r s h a v e b e e n a b l e t o a p p l y t h e

t e c h n i q u e o n l y , a f t e r . . . l a b o r i o u s m a n i p u l a t i o n o f t h e c o n t o u r

o f . i n t e g r a t i o n i n t h e c o m p l e x p l a n e . S h e r w o o d ' s ( 1 9 5 8 )

c o n t r i b u t i o n i s t h a t h e m a n i p u l a t e s t h e c o n t o u r p a t h o f

i n t e g r a t i o n i . n „ a s i m p l e w a y t o o b t a i n a n e x a c t s o l u t i o n . f o r

2 - 0 s e i s m i c p r o p a g a t i o n . T h e a p p r o a c h h a s p h y s i c a l m e a n i n g

a n d . i s e x t e n d e d i n C h a p t e r s 3 a n d 4 t o m o r e c o m p l i c a t e d

l a y e r e d m e d i a h a v i n g d i p p i n g a n d c u r v e d l a y e r s .


- 11 -

2 , 1 E l a s t i c Wave P r o p a g a t i o n i n a n I n f i n i t e M e d i u m d u e t o

a L i n e S o u r c e

__ C o n s i d e r t h e . h o m o g e n e o u s e q u a t i o n o f e l a s t i c w av e

p r o p a g a t i o n

/O y - y « . V x 'U (8.1)

- a n d t a k e i t s F o u r i e r t r a n s f o r m w i t h r e s p e c t t o t

( Z + Z s O ? ( ? ■ $ - / * 2 * z x

*hexe~U(c*>) =

.-AJL&o rp o - t .e n t ia .1 s 3 a n d ^ may b e d e f i n e d s u c h t h a t«'v

y = y f - y * * z > v - f = ° ( 8 . 3 )

U . 4 )i t*

o

a n d F * F * ^ = f o ' f <*-s >O' P O'

T he. f o l l o w i n g . . d e r i v a t i o n c l o s e l y f o l l o w s S h e r w o o d

( 1 9 5 8 ) , S u p p o s e . . . t h a t a . f o r c e p e r u n i t v o l u m e / = z f x y

a c t s o n t h e m e d i u m ; t h e n e q u a t i o n ( 2 , 2 ) b e c o m e s

(* + 2 / $ j ( 7 . y ) - / t y x y x u*;F <*.•>= - O

* •*Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

- 12 -

a nd t h u s

= U ( 2 . 7 >^ fOco*

I f f . i s an i m p u l s i v e l i n e f o r c e l y i n g a l o n g t h e y

a x i s . w i t h . s t r e n g t h Z p e r u n i t l e n g t h , and d i r e c t e d i n t h e Jc d i r e c t i o n

Then / = Z f& )

so t h a t f = £ S (~ ) S ( z ) A ( 2 . 8 )

From . s y m m e t r y . c o n s i d e r a t i o n s

i ( 2 *9 )* -v * v

-T h u s - t a k i n g t h e a n d *o f - e q u a t i o n ( 2 . 7 ) -we ' h a v e

- ■ v 9§ - * $^ (2 . 10)

Y *1? = _ f * ' ? T a k i n g . F o u r i e r t r a n s f o r m s w i t h r e s p e c t t o x and z

Joo -Zco y o t x c t *

g i v e s f r o m e q u a t i o n s ( 2 . 1 0 )

- ( F ^ x 3) $ ( F , % ^ x / f ^

- $ ( & X < * 0


- 13 -

T h e r e f o r e

+ 2 j E{&****(§*+ ?£s — £<*9/ f S 9J

P e r f o r m i n g 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 f o r t h e

x , z c o o r d i n a t e s

£ » ) = . < ■ ? r r r < ^ ^ rr ( ' J f p > L L

( 2 . 14 )

^ i / r r

E a c h i n t e g r a n d h a s a p o l e a t

r — -/- /CO3’ Tr ySz-' - ( e » ~ V (8*l s >

w h e r e ^ o r

When ^ ^ ^ t h e i n t e g r a l w i t h r e s p e c t t o

o v e r t h e c o n t o u r f r o m . 0 0 t o . t o - 00 i s z e r o a n d w h e n Z ^ CD

t h e I n t e g r a l . . . f r o m . 0 0 t o — «®o t o - o© i s z e r o .

T h e r e f o r e ^ -A#/Sj

/ - ± j l r * ' < * s ) ‘ î « , « > * ( 2 . 16)

$ ^ £ 2 r U l E

4 r r f r ) ,/s*££


t h e a l t e r n a t i v e s i g n s d e n o t i n g p o s i t i v e o r n e g a t i v e i?

H e n c e

f = u r - t & r * ' - * <***■8l?*P -**> -<*> s * \v ( 2 - 1 7 )

^ 8 r r ^ L ^ L -

We c a n now r e c o g n i z e ^ a n d a s t h e i n t e g r a l

o v e r p l a n e w a v e s p r o p a g a t i n g a t a n a n g l e S t o t h e z - a x i s

s u c h t h a t

t a n S> = - I * ) 41 ( 2 . 1 8 )

B e r r y a n d W e s t ( 1 9 6 6 ) g i v e a more c o m p l e t e d e s c r i p t i o n o f

t h i s i n t e g r a t i o n f o r t h e t h r e e d i m e n s i o n a l c a s e *

P u t t i n g

F =t - ^ 2 s/+? <2oe, *c-

( 2 . 1 9 )

su n S io r - " f * " ’ 7*

t h e p a t h o f i n t e g r a t i o n i s o v e r t h e c o m p l e x & p l a n e a n d

we h a v e

f = C C fa fc c e s g (2

y * = c o /° p j t s"* * ^ c t ~(x 7 ^

20 )

\*/A-cr C - 2?


w h e r e , a l t e r n a t i v e s i g n s - a r e f o r p o s i t i v e o r n e g a t i v e i ? a n d

O n i s t h e o p e r a t o r

i— / o o / ‘-t'*o + r r /z

< * * [ ] - / & [ / a * ? “ • “ »* “ v / - o o ' t o o - 7 T /- J ^

.N o te t h a t . . t h e p a t h o f i n t e g r a t i o n p a s s e s b r a n c h

p o i n t s . g i v e n b y COS ^ a n d we c h o o s e b r a n c h c u t s0

- a s _ L l l u s t r a t e d i n F i g u r e . ( 2 * 1 ) .

.T h u s , we s e e t h a t a . p o i n t f o r c e r a d i a t e s . p l a n e w a v e s ,

w i t h a m p l i t u d e s g i v e n f o r r e a l a n g l e s a s i n e q u a t i o n ( 2 . 2 0 ) .

T h e t e r m s ~f~ COS d ? d , a n d . S'*7 d. i s t h e " s o u r c e

f u n c t i o n ” ( d e n o t e d , b y ) f o r . a n i m p u l s i v e l i n e f o r c e .

T h i s s o l u t i o n i s n o w .• g e n e r a l i z e d b y t h e c o n c e p t

o f " ^ g e n e r a l i z e d . r a y " . t h e o r y - . a . t e r m u s e d . b y , £ p e n . c e r . ( i 9 6 0 . ) .

T he p l a n e w a v e s a r e g i v e n p h y s i c a l i d e n t i t y a n d a r e i n d i v i d

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

a m u l t i —l a y e r e d m e d i u m a c c o r d i n g t o S n e l l ? s l a w * S h e r w o o d

( i 9 6 0 ) u s e d t h i s t e c h n i q u e i n e x t e n d i n g t h e w o r k o f h i s

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

a t h r e e d i m e n s i o n a l m u l t i - l a y e r e d m e d i u m .

T h e c o n c e p t h a s l o n g b e e n i m p l i c i t i n t h e l i t e r a t u r e ;

f o r e x a m p l e t h e ” t r a n s m i s s i o n c o e f f i c i e n t ” o f t h e C a g n i a r d

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

b y E w i n g , J a r d e t s k y a n d P r e s s ( 1 9 5 7 ) .


F i g u r e 2

V>>f>V

*

b r a n c h p o i n t s

Tf/tL^ fkC®)

_1 P a t h o f . i n t e g r a t i o i I n t h e c o m p l e x <9 p l a n e

( e q u a t i o n ( 2 . 2 1 ) 3


- 16 -

2 , 2 G e n e r a l i z e d R a y T h e o r y

C o n s i d e r . t h e . d i s p l a c e m e n t d u e t o t h e r a y f o l l o w i n g

t h e p a t h a s i n F i g u r e 2 . 2 . T h i s s a m e . r a y i s c o n s i d e r e d b y

B e r r y a n d W e s t ( 1 9 6 6 ) * A t 0 j u s t a b o v e t h e i n t e r f a c e

“ T ‘ p ' f e s e n t s : t h e - s o u r c e , f u n c t i o n . , w h i c h we a s s u m e t o b e

i m p u l s i v e , a n d - t h e r e f o r e n o t d e p e n d e n t u p o n CO , S n e l l ' s

l a w t h e n r e q u i r e s

S/*> - s t r , <§.t< * , c**.

(2 . 22)

•a s ( £ . - c o T i c s y o t i c l s - t o t h e a n g l e o f i n c i d e n c e o f t K i s r a y o n

th i* s i n t e r f a c e

C°s Q , ( 2 . 2 3 )

By c o n t i n u a t i o n

2 4 )

w h e r e

** - O £ £ £ . 4 2 c o s & x ( 2 . 2 5 )W ^ / w A #


F i g u r e 2 « 2 T h e p a t h o f t h e r e y F e o n e l d e r e d

t h e t e x t


- 17 -

S ~ ("X- s/n Q+ A COS. jg , ^ ( 8 . 2 6 )

M r a j«*/ '

c i s . . t h e m a x i m u m v e l o c i t y o v e r t h e p a t h o f S , r e p r e s e n t s

t h e t h i c k n e s s o f t h e f i r s t , l a y e r . . a n d i s t h e

r e f r a c t i o n o r . r e f l e c t i o n c o e f f i c i e n t *

. T h u s t h e c o m p l e t e e x p r e s s i o n f o r t h i s r a y i s

f = r ° 7 So / ( s '* t f ) ( 2 . 2 7 )

w h e r e . T » 77" ( t r a n s m i s s i o n c o e f f i c i e n t s o f t h e w a v e•»» - /

a s . i t p a s s e s t h r o u g h e a c h o f t h e N i n t e r f a c e s ) * ( 2 * 2 8 )

( 2 . 2 9 )

f o r e x a m p l e f o r t h e w a v e i n f i g u r e 2 * 2

A / r ~(?C-3roJ ( 2 . 3 0 )

♦1- / C J Cy*

w h e r e y i s a r b i t r a r y , ( s e e B e r r y a n d W e s t ( 1 9 6 6 ) e q u a t i o n 7 . 1 )

a n d i s t h e t h i c k n e s s o f t h e n**1 l a y e r * E q u a t i o n s

( 2 . 2 7 ) c a n t h e n b e r e w r i t t e n


- 18 -

f f ^ ^ s *•* & ) ( 2 . 3 1 )' / o o CA / « o C

I n t e r m s a f g e o m e t r i c o p t i c s i f & — ^ i s

r e a l , . S . r e p r e s e n t s , t h e t i m e i t . t a k e s t o f o l l o w t h e r a y p a t h

a n d i s k n o w n a s t h e o p t i c a l t r a v e l t i m e ( K e l l e r ( 1 9 5 8 ) ) .

T h i s t h e o r y r e p r e s e n t s t h e g e n e r a l i z a t i o n o f g e o m e t r i c o p t i c s

f o r c o m p l e x a n g l e s o f p r o p a g a t i o n *

N o t e t h a t t h e f u n c t i o n S. i s . a m b i g u o u s t o t h e e x t e n t

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

a v a r i e t y o f . p a t h s f o r a g i v e n i n i t i a l a n g l e . S u p p o s e a r a y

f o l l o w s a p a t h a s i n d i c a t e d i n F i g u r e 2 . 3 . T h e n t h e t o t a l ,

t r a v e l t i m e S i s g i v e n b y

S • * ( -* , - ( z - 2 \ c o s j s t '• t Cl

- h / c c , a r ) sj*2 j9^ ( * / - 2 / ) c o s

+- c o s Q

( 2 . 3 2 )

. JB u .t._ .as- t h e _ i n t e r f a c e . i s . a . h o r i z o n t a l p l a n e ^ 'js.£i

a n d f u r t h e r m o r e a s a r e s u l t o f S n e l l ' s L a w

sty> = S**o =■ S/+7 Q ~ S/r> ( 2 . 3 3 )1 ev K 1 ■ ■ 1 ■ ll—1 / ■Cj ct c

T h e r e f e r e


F.l_gure 2 . 3 T h e p a t h o f a g c n e x a l l z e d r a y m o v i n g a n a r b i t r a r y

d i s t a n c e a l o n g t h e b o u n d a r y

t

F i g u r e 2 . 4 T h e m u l t i p l e g e n e r a l i z e d r a y p a t h s d e f i n e d b y a

s i n g l e f u n c t i o n 3..


- 19 -

E x te n d i n g . t h i s t h e o r y t h r o u g h t w o more i n t e r f a c e s

so- t h a t S- c o r r e s p o n d s - t o t h e . i n f i n i t e n um b er , o f p a t h s sh o w n

i n F i g u r e 2 * 4 . F rom a n o t h e r p o i n t o f v i e w , t h i s a m b i g u i t y

o f S i s . c a u s e d b y r a y s t r a v e l l i n g a n a r b i t r a r y d i s t a n c e a l o n g

a n y i n i e r f a c e . t h e y e n c o u n t e r . N o t e t h a t i n t h e d e r i v a t i o n

o f t h e r e f l e c t i o n c o e f f i c i e n t s n o t h i n g p r e v e n t s an e l e m e n t

o f a p l a n e w av e f r o n t b e i n g a r b i t r a r i l y d i s p l a c e d a l o n g an

i n t e r f a c e . B r e k h o v s k i k h ( i 9 6 0 ) h a s shown t h a t b o u n d e d

p l a n e b e a m s a r e d i s p l a c e d a l o n g a n i n t e r f a c e on r e f r a c t i o n and t h a t

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

o f - h e a d w a v e s .

. W h i le —F e rm a t .1 s P r i n c i p l e o f L e a s t Time i m p l i e s

t h a t : r a y s w i lX . J n X lo w a . s t r a i g h t l i n e i n u n i f o r m m e d i a ,

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

a l t e r t h i s . I n p a r t i c u l a r , a c c o r d i n g t o F e r m a t ' s P r i n c i p l e

c u r v e d b o u n d a r i e s w i l l g e n e r a t e r a y s f o l l o w i n g a c i r c u l a r

p a t h ( s e e . F i g u r e . 2 .5 .) .

T h u s t h e g e n e r a l i z e d r a y t h e o r y w i l l b e f u r t h e r

e x t e n d e d i n C h a p t e r 4 t o t h e s o l u t i o n o f a c u r v e d i n t e r f a c e .


Sou r c e R e c e i v e r

LOW VELOCITY MEDIUM

HIGH VELOCITY MEDIUM

F i g u r e 2 . 5 A r a y p a t h a c c o n l i r . g t o F e r m a t * 6 P r i n c i p l e


- 20 -

2 . 3 T h e S o l u t l - o n o f E q u a t i o n ( 2 . 2 7 ) a n d I t s P h y s i c a l

I n t e r p r e t a t i o n

c O - ^ 4 ~T( s , „s ) c S /s j j^ 0 c \S 7 _ o o Z o o K. C. S ( 2 . 3 6 )

We f i r s t n o t e t h a t m u s t b e r e a l . I n e q u a t i o n ( 2 . 1 4 )

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

= ' £ &> F ) ( 2 . 3 7 )

t a k i n g c o n j u g a t e s a n d t h e n p u t t i n g JF = *F a n d ^ ^ .

T a k i n g 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 s w i t h r e s p e c t t o f r e q u e n c y - o f

. ^ . . a n d f^ .v /h J L c h . .a re e.v.en . f u n c t i o n s o f

0 & F ( 8 . 3 8 )

w h e r e F = ^ ( 2 . 3 9 )

p u t t i n g c o - - - ^ i n F •

T h u s a s *P i s r e a l i n i t i a l l y i t w i l l r e m a i n r e a l a f t e r

t r a n s m i s s i o n . T h e r e f o r e e q u a t i o n ( 2 . 3 6 ) b e c o m e s ^

<j) = f k . [ £ 2 ,C4 F S -Z * ^ ~ ^

( a ^ r r / j z ) ( 2 * 0 )

The c o n t o u r o f i n t e g r a t i o n w i t h r e s p e c t t o u c a n b e d e f o r m e d

t o t h a t i l l u s t r a t e d f o r e x a m p l e i n f i g u r e 2 . 6 . F i g u r e 2 . 6 a l s o s h o w s

t h e l o c a t i o n o f b r a n c h p o i n t s , b r a n c h l i n e s a n d p o l e s t h a t o c c u r


- r r / z

0

*Trr,

/y/

b r a n c h p o i n t s l i e a l o n g r e a l a x i s

s , ' ' ' ' * ' > - “ T*&C6> fT t/2

P o l e s l i e on l i n e

F i g u r e 2 . 6 A p p r o x i m a t e c o n t o u r o f i n t e g r a t i o n ( e q u a t i o n ( 2 . 4 0 ) )


21 -

in the integrand o f equation (2*36)•The branch points are located on the rea l

axis as a r e s u l t o f the. factor*. o o » ^ • which occur in the

transm ission fu n ction ./ . Poles corresponding to .s ingul a r i t i e s in the r e f l e s t io n /r e f r a s t io n c o e f f i c i e n t s of the transm ission function l i e along the l in e 5 •

. \rThesd p o les are p h y s ic a l ly expressed by Raylelgty^Stoneley waves•

Sherwood's (1958) contribution i s his recognition that sn equation o f t h i s type (2 .4 0 ) could be e a s i l y evaluated i f the orders of in tegration could be changed.To interchange the in tegration order i t i s necessary to a l t e r the contour o f the in tegra tion so th a t theintegrand remains f i n i t e for a l l ^ and CO • This cond i t io n i s f u l f i l l e d i f the exponent i s purely imaginary for a l l &c • That i s , i f

^ *9*1 ~ ^ (2.41)

which g iv es the form of the new contour o f ^ in tegration

Therefore


- zz -

T h e l i m i t s o f e a s s h o w n i n a p p e n d i x B a r e — < oo ex. ——» c’©o_ ^

w h e r e ^ ^ ^ • T h i s l a s t s t e p i s v a l i d a s

t h e i n t e g r a t i o n f r o m **<’oo a to - c « o -/• Try Q.

i s z e r o a n d i t w i l l b e s h o w n b e l o w t h a t n o b r a n c h p o i n t s

o r p o l e s a r e c r o s s e d a l t e r i n g t h e c o n t o u r o f (Pc i n t e g r a t i o n .

I n t e r m s o f 'X* - , ( f b e c o m e s

a s

[ w h e r e H i s t h e H e a v i s i d e o r s t e p f u n c t i o n .

( 2 . 4 3 )

( 2 . 4 4 )

T h e l a s t s t e p i n t h e S h e r w o o d ” t e c h n i q u e i n v o l v e s

' i n d i n g t h e c o m p o n e n t s o f d i s p l a c e m e n t ^ d u e t o ^ a n d

‘h u s , f o r e x a m p l e ,

<P -=» ( 2 . 4 5 )

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

r e s p e c t t o t h e c o o r d i n a t e s OC o r JSr i s t o p r o d u c e a d e l t a

f u n c t i o n i n t h e i n t e g r a n d a s

( 2 . 4 6 )

<5*


- 23 -

T h e r e f o r e

' < < / * - -Sia. fe n c f S 6 * ) 7&>)s/r>Qj. <f(£ -£)e6> e ts ( 2 . 4 7 ) 6 J US

The s o l u t i o n o f e q u a t i o n ( 2 . 3 6 ) c a n now b e e x p r e s s e d i n

c l o s e d f o r m *

* $ f6 0 - & *> f f f a ) T fi> ) 2 (2.48)A C * )

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

£ S f i > ) J •=; f ( 2 . 4 9 )

a s

O ( 2 . 5 0 )

F r o m e q u a t i o n ( 2 . 3 5 ) i s g i v e n b y

S - - x s > ” & + 2 2 * y c o s < i l i / cj ( 2 . 5 1 )

c

w h e r e i t i s f o u n d c o n v e n i e n t t o p u t

C ■= MGLX £ CjJ

. „ d <92 G>c J O j = Q . ( 2 . 5 2 )


- 2 4 -

s o t h a t ( P i s t h e a n g l e ©F t h e r a y i n t h e h i g h e s t v e l o c i t y

l a y e r t h a t i t t r a v e r s e s .

AS i S 9 f t n e r a l l Y c o m p l e x a n d d e f i n e d i n t h e

f o u r t h a n d s e c o n d q u a d r a n t s * p u t

( 2 ' 5 3 )

F r o m e q u a t i o n ( 2 . 4 9 ) t h e r e l a t i o n s h i p b e t w e e n

a n d i s g i v e n b y

A'x s m j o cos A * -r h , c osi> c e sA <?_ - t (2.54)

C j * 1 V

a n d

*■sc £ O S f S s t n A o _ 2 2 —J. S ' ” / 0, S / n A o .s O (2 .55)

C J S,Cj / */\ < t

An e x a m i n a t i o n o f t h e s e t w o e q u a t i o n s r e v e a l s t h a t t h e

o r i g i o n o f t h e c o m p l e x ^ p l a n e c o r r e s p o n d s t o t h e t i m e i t t a k e s a

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

c o n s i d e r o n l y t h e f o u r t h q u a d r a n t o f t h e ^ p l a n e a s t h e s e c o n d

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

c a u s a l i t y .

E q u a t i o n s ( 2 . 4 8 ) t o ( 2 . 5 5 ) t h u s g i v e t h e s o l u t i o n . T h e


i n t e g r a t i o n o v e r t h e c o n t o u r o f i n t e g r a t i o n h a s b e e n r e d u c e d

t o t h e e v a l u a t i o n a t o n e p o i n t o f & r e l a t e d t o t i m e ZT

T h u s t h e c o n t o u r o f i n t e g r a t i o n b e c o m e s s y n o n o m o u s

w i t h t h e l o c u s o f t i m e w i t h ( P • T h e d i s p l a c e m e n t

i s r e l a t e d t o t i m e C b y t h e p a r a m e t r i c e q u a t i o n s ( £ . 4 8 ) a n d

( 2 . 4 9 ) i n (9 .T h e m a j o r t a s k i n u t i l i z i n g t h e s o l u t i o n ( 2 . 4 8 )

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

(9) g i v e n b y e q u a t i o n s ( 2 . 5 4 ) a n d ( 2 . 5 5 ) . T h i s a n a l y s i s - i s

e x t r e m e l y l a b o r i o u s a l g e b r a i c a l l y a n d i s g i v e n i n A p p e n d i x 8 .

T h e r e s u l t s a r e a s f o l l o w s a n d a r e i l l u s t r a t e d i n

F i g u r e , ( 2 . 7 ) : -

a ) T h e p a t h may b e d i v i d e d i n t o t w o p a r t s j o i n e d

a t a c u s p p o i n t d e p e n d i n g o n w h e t h e r ^ o r

n o t . ( W h e n £ -- o . 1 1 ? = O ) .

T h e p a t h s t a r t s a t t h e o r i g i n t h e n f o l l o w s t h e

r e a l a x i s o f dP to the c u s p p o i n t 0 w h e r e i t d e p a r t s*

a l o n g t h e c o m p l e x p a r t o f t h e p a t h .

b ) T h e p o s i t i o n o f d j - i s ' f o u n d

b y c o n s i d e r i n g t h e l o c u s a s g ^ <2 • T h e n

\ ^ S / _ / ? • - C ( 2 . 5 6 )

j „ J i -

c ) T h e a s y m p t o t e o f t h e c o n t o u r i s f o u n d b y c o n

s i d e r i n g t h e l o c u s a s ^ ^ 00 • T h e n


_ E r . r c h p o i n t c o r r e s p o n d i n g t o h ( sc1 w t v e ~

Cusp cc r : :e s j or t o d i r e « ; l >

P o l e c o r r e s p o n d i n g t o R a y l e i g h o r S t o n e l e y wave

3 1 ( f o r e x a m p l e

F i g u r e 2 . 7 C o n t o u r p a t h s c l i r t f t r a i l on f o r t h e r a y s

a n d f ° r eyi " p i e


- 26 -

K

S " 7 fU ( 2 . 5 7 )- /

N o t e t h a t w h e n a l l / ? . “ / , t h e n t h i s r e s u l t\J

i s t h e s a m e a s e q u a t i o n ( 2 . 5 6 ) . T h u s a r a y w i t h a

c o n s t a n t v e l o c i t y t h r o u g h o u t i t s p a t .h h a s a l o c u s

o f t i m e w i t h $ a s i l l u s t r a t e d i n F i g u r e ( 2 . 7 ) .

d ) T h e v a l i d i t y o f o u r s o l u t i o n i s j u s t i f i e d f r o m

t h e a b o v e r e s u l t s . T h e c o n t o u r o f i n t e g r a t i o n i n

f i g u r e ( 2 . ' l ) c a n b e d e f o r m e d t o t h e c o n t o u r i l l u s t r a t e d

i n f i g u r e ( 2 . 7 ) w i t h o u t c r o s s i n g a n y b r a n c h p o i n t s o r

l i n e s o r p o l e s .

e ) The f a c t o r /A i n e q u a t i o n C 2 . 4 8 ) i s s h o w n

t o b e c o m e z e r o a t t h e c u s p p o i n t •

' A‘I t in / a ( 2 . 5 8 )

• - O ' O or f £~Orrt ( 2 . 5 6 )

F r o m t h e v a r i a t i o n o f t w i t h q a s ^ ^ O

g i v e n i n A p p e n d i x &

/ / B - & A/ ?/ coV° / K


- 27 -

A (l>) - < B ( t*- ( 2 . 6 0 )

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

e q u a t i o n ( 2 . 4 8 ) i s

= C C Met- 4)

C - - < ■ . « >

T h e d i r e c t w a v e f r o m a n i m p u l s i v e f o r c e s o u r c e

t h u s h a s a n i m p u l s i v e d i s p l a c e m e n t .

2 . 3 1 H e a d w a v e s j D i r e c t A r r i v a l s a n d R a y l e i q h / S t o n e l e y W av e s

— - - A p a r t f r o m t h e d i r e c t a r r i v a l , - o t h e r f e a t u r e s o f

t h e s o l u t i o n c a n b e s e e n f r o m a g e n e r a l e x a m i n a t i o n o f e q u a t i o n

( 2 . 4 8 ) .

I n o r d e r f o r t o b e n o n - z e r o , o n e o f t h e

t h r e e f u n c t i o n s ^ , T ”- a n d m u s t b e c o m e n o n - r e a l .

P r o g r e s s i n g a l o n g t h e r e a l (P a x i s f r o m t h e o r i g i n a l l

t h e s e f u n c t i o n s a r e r e a l u n t i l t h e b r a n c h p o i n t B , c o r r e s -

p o n d i n g t o COS (P i s r e a c h e d . N o t e t h a t ^ C

A t t h i s p o i n t b e c o m e s s u d d e n l y c o m p l e x c o r r e s p o n d i n g t o

t h e a b r u p t a r r i v a l o f a h e a d w a v e . N o t e t h a t w h i l e t h e h e a d

w a v e a r r i v a l i s a b r u p t , i n c o n t r a s t t o t h e d i r e c t a r r i v a l ,


- 28 -

i t i s n o t s i n g u l a r >• R e f e r r i n g b a c k t o t h e r a y p a t h

d i s c u s s i o n ( s e c t i o n ( S . R ) ) y we c a n now s e e t h a t w h i l e a

g e n e r a l i z e d r a y may move a n a r b i t r a r y d i s t a n c e a l o n g i n t e r

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

a s h e a d w a v e s e x i s t s . A l o n g t h e i n t e r f a c e t h e r a y h a s a

c o m p l e x a n g l e o f p r o p a g a t i o n

6 > _ s , ~ 6 >) = r r / a - < e (g.62)I

w i t h r e a l p a r t - 7 ^ 2

A t t h e c u s p b e c o m e s c o m p l e x b e g i n n i n g w i t h t h e

a r r i v a l o f t h e d i r e c t w a v e . F r o m t h i s p o i n t o n

n o n - z e r o •

T h e p o l e s a l s o h a v e a p r o f o u n d " I n f l u e n c e o n t h e

s o l u t i o n . When t h e p a t h &ay i s i n t h e v i c i n i t y o f a

p o l e t h e m a g n i t u d e o f ~ r& ) b e c o m e s l a r g e c o r r e s p o n d i n g t o

t h e e x i s t e n c e o f a R a y l e i g h / S t o n e l e y w a v e . As SC b e c o m e s

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

m o r e c l o s e l y . T h i s s h o w s m a t h e m a t i c a l l y t h e p h y s i c a l f a c t

t h a t t h e r e l a t i v e a m p l i t u d e o f R a y l e i g h / S t o n e l e y w a v e s

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

2 . 4 T h e R e l a t i o n s h i p b e t w e e n Two a n d T h r e e D i m e n s i o n a l

S c a l a r W ave P r o p a g a t i o n

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

a g r e a t d e a l o f a t t e n t i o n h a s b e e n p a i d t o t h r e e d i m e n s i o n a l


- 29 -

c y l in d r ie a l ly symmetric wave propagation. Tho main incentive for developing two dimensional theory ia that closed form so lut ions aro found and that two dinensional model seismograms can bo r e la t iv e ly e a s i ly measured for comparison purposes. I t would be idea l i f a simple re la t io n sh ip existed between the two typos of seismegranps.

represents the so lu t ion at (r , z , t ) to a point source in a cy l in d r ie a l ly symmetric three dimensional medium at the

from using descent of dimensions (Duff and Naylor,(1966), p. 387). The response of the medium to a lino source i s given by (see Figure (2 .8 ) )

Suppose that

o r ig in . Consider r\^ Z , C J as the so lu t ion of a l ine source in tho throe dimensional medium. We can find / ( ,

(2 .63)

(by rec ip roc ity )


2*

F i g u r e 2 . 8 A l i n e s o u r e e I n t ; h c e e d i m e n s i o n a l m e d i u m

( e q u a t i o n ( 2 * 6 3 ) )


30

w h e r e / ? * ~ * * + J

■ • t ) s. f }*3 (R j Zj ( 8 . 6 6 )

* ( I - * ) * ’

putting J"* ~ /?* and S - X *Put / ^ ( s ) = KA (x, *, ^ «*✓ a; ( 'f) = K3 (R t ^ t )In a process analegeus te inverting Abel's equatien,

eperate on beth sides ef equatien (2.66) by

o (s ( 2 . 6 7 )

' s ' ( s - s ' ) ' 4 1

F i g u r e ( 2 . 9 ) i l l u s t r a t e s t h e ^ J d # » a i n e f i n t e g r a t i e n

e f K3 (F )

K ( s ) ~ H cC s r K ( £ ) c t s

' r (*~ ’ J s ' 4s ( S - S ') 1*-

- r r / s T ^ ( F ) * S £

A p p l y i n g t h e e p e r a t e r e > b e t h s i d e s a n d d r e p p i n g

p r i m e s

K*® * 7F i l I " K* <2*69>TT O f Jg £S _ ^y/k,

K ( * , g, $ s ~ £ rTT.n On J^* ^


AREA OF INTEGRATION

F i g u r e 2 , 9 ,, T h e ^ ■$ d o m a i n o f i n t e g r a t i o n , , S e e e q u a t i o n ( 2 . 6 8 )


31 -

The s o l u t i o n i s c o m p l e t e l y g e n e r a l . C o n s i d e r , f o r e x a m p l e ,

t h e f o r m o f t h e a r r i v a l s i n g u l a r i t i e s d u e t o a d i r e c t wave an d

t o a wave d i r e c t l y r e f l e c t e d f r om a n I n t e r f a c e a s i l l u s t r a t e d i n

f i g u r e ( 2 . 1 0 ) . I n tw o d i m e n s i o n s t h e a r r i v a l s a r e g i v e n by

f/x y ( 2 . 7 0 )C t z - 1 , 2) ^ ( c * -

o t <? - t h a t i - ^ 3 n d ~ -b- tj-h ^

I n a t h r e e d i m e n s i o n a l c y l i n d r i c a l l y s y m m e t r i c medium t h e

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

~ TT Ccz - x Y t / iy* - J„z.

, 3. W*< + tr (t*-&g>\ ~ JT £-7^ ^

T

- £ - t < L V 1 ) ( 2 - 71

T h e r e f o r e i n t h r e e d i m e n s i o n s an i m p u l s i v e p o i n t f o r c e

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


S o u r c e R e c e i v e r* ^^ s

v\

\

INTERFACE

F i g u r e 2 . 1 0 , The t w o w a v e s , r e p r e s e n t e d b y t h e i r r a y p a t h s ,

c o n s i d e r e d i n e q u a t i o n s ( 2 * 7 0 ) a n d ( 2 . 7 l ) .


- 3 2 -

2 * 5 T h e R e f l e c t i o n / R e f r a c t i o n C o e f f i c i e n t s

W h i l e t h e g e n e r a l p r o p e r t i e s o f t h e s o l u t i o n h a v e

a l r e a d y b e e n e x t e n s i v e l y d i s c u s s e d , a m a j o r i m p e d i m e n t t o

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

r e f l e c t i o n / r e f r a c t i o n c o e f f i c i e n t s . A l a r g e n u m b e r o f a u t h o r s

g i v e t h * e x p r e s s i o n s f o r t h e v a r i o u s c o e f f i c i e n t s . G r a n t a n d

W e s t ( 1 9 6 5 ) , Y se o iu b . ( 1 9 6 1 ) . H o w e v e r e r r o r s i n t h e s e f o r m u l a s

a r e f r e q u e n t . I h a v e g e n e r a t e d niy o w n s o l u t i o n t o t h e ^ ^ 4*

s y s t e m o f e q u a t i o n s a r i s i n g f r o m t h e b o u n d a r y c o n d i t i o n s .

G l e e - ^ p e e a t n a i c o m m u n i c a t i o n ) h o o t o o t e d ' t h e s e f o r mu l e s

e x t e n s i v e l y a n d m a d e s o m e c o r r e c t i o n s . T h e n e e d f o r p r o p e r

q u a l i t y C o n t r o l i n t h e a l g e b r a a t t h i s l e v e l i s a b s o l u t e l y

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

o f t h e s e c o e f f i c i e n t s .

L e t t e r m s s u c h a s /R <c> / ? d e n o t e t h e K n o t tn * /

( 1 5 9 9 ) r e f l e c t i o n c o e f f i c i e n t f o r t h e Pt r e f l e c t e d w a v e . ^

C o n v e n t i o n s T h e p o s i t i v e d i r e c t i o n w i l l b e t r&ken a s d o w n .

C a l c u l a t i o n s T h e r e f l e c t i o n / r e f r a c t i o n c o e f f i c i e n t s a r e

c a l c u l a t e d f r o m t h e b o u n d a r y c o n d i t i o n s f o r e l a s t i c w a v e

p r o p a g a t i o n b e t w e e n t w o a e m i - i n f i n i t e m e d i q i n w e l d e d c o n t a c t

a t a p l a n e i n t e r f a c e a s t h e y a f f e c t a p l a n e i n c i d e n t w a v e .

E x a m p l e s T h u s c o n s i d e r t h e r e f l e c t e d a n d r e f r a c t e d w a v e s

g e n e r a t e d b y a n i n c i d e n t p l a n e P h a r m o n i c w av e i n t h e u p p e r

m e d i u m o n t o t h e i n t e r f a c e .


- 33 -

----------------- T h i s d e r i v a t i o n f o l l o w s t h e p r o c e s s

• f ; t h e t r a n s m i s s i o n f a c t o r t e c h n i q u e a s o u t l i n e d b y E w i n g ,

J a r d e t s k y , a n d P r e s s ( 1 9 5 7 ) . T h e P a n d S w a v e v e l o c i t i e s i n

m e d i a 1 a n d 2 a r e g i v e n b y ce. } cC a n d } . N o te

t h a t t h e i n t e r f a c e i s c h a r a c t e r i z e d b y 7* ~ O mt CO Z~

N e g l e c t i n g t h e f a c t o r - c

<0c) = -*XP l~ 0* s'” ^ + * Cos

f c Z ) n y - * n c o s ® }

' & % ) p p , s & s , n % ~ z c o * < % ) j

t - i i ) = s " ’ % * 2 “ * $ ) ]

T h e b o u n d a r y c o n d i t i o n s a r e g i v e n b y t h e c o n t i n u i t y o f s t r e s s

a n d d i s p l a c e m e n t e q u a t i o n s # H e n c e a s 2 - 0 o n t h e i n t e r f a c e

( 2 . 7 2 )

t h e f i r s t b o u n d a r y c o n d i t i o n g i v e s

^ ^

- c U , / ? „ *r t [*■

jQ rr * € ^ i L S ,~ - s* * %t * ~ ’

( 2 . 7 3 )

Ur


- 34 -

This condition must apply for a l l values of af • Thus wo get S n e l l 's Law

(8.74)

PS* (2 .75)

~ S ir) ^ a s i r ) £>* — s Q ~■*. P> f l r

Applying a l l the boundary conditions we get

R r ,r , * /** - ( £ * - ( f t * - * ) * * 4 *

R p,Pl

- 2/ i ■” (-£* - = - ( 3s “ ~ z- / i )

(2 .76)Putting s\J - - S ih 4 A

Put ut - cos **> ' & ■

. .=

< = r c 5J&*•(t

-

%r

r *

a

% -■ e° s<& . tH)'4 ^

(2 .77)


- 35 —

f a

f a

S f i

f a7 4 .

* . ? .

* J 5

“ ^

r A

(2.78)

« / / * f* t

f a = ^ < - v * -

(2 .79)

He nee

V*

1*1a1£

f l ' i-

• e -e. <i> - v* v ° i 3 * /

■fi £ . ~Sfi, - / /

f a A / , - / * * /!% A

(2 .80)

Continuing t h i s process for incident P waves in the 2 medium and S waves in the 1 and 2 media we have

*l>' f a *%p* t

<e ftppn \*L ' U

/ , “ i * - f a - f a fipsns* f a s , ' k

a A I - j f c ^ s

- ' I *

A

/ *

A

> »

- / /

(2 .81 )

‘ A

« £ > / ? = J 9 'Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

An in sp ec t io n of tho sbove equation revea ls that

the four colutans of £ are operators r e s p e c t iv e ly connected

with the r e f l e c t e d /r e f r a c t e d ^ j and ^ waves.

The four rows are operators connected with the in c id en t

P % j j and 5 waves.In terms o f tho c o - f actors • o f we have

equations (2 .8 2 ) (see next page).

Carrying out tho required c a lc u la t io n g ives putting

Q s ( f^ r O .

Dx - O , + f>% ~ 2 -+ f> ,

q - p, * 3 P*equations ( 2 . 8 3 ) and ( 2 . 8 4 ) written out on page 3 8 .


- 37 -

R f? r , 1 - J ? ty o t„ - S J ^ . 3

- ~ 3 c£„ - ^ f a <*t,3

V .

\ r > , - - 2 -e+ <?t„ - * f a «*, s

.<=£

* * *S . - 2

f y s , -

f t * * , = ^ f a ° * 3 i

* * *_ £5^1 - 2 f t e£JZ

- 2 * ^ , 3 - ' Z f a ^ J Z

± -f + ^ f i, e^ i Z

- ~ 2 * + < z - s f a ^ * z

'l V- - 9 - 0 - - Z f a e t i y

- - a J 5 * ^ v

R fi s<ST S? <S ,M -* * . £ £

# S C =* - 2 - ?-*v - z f a <** y

( 2»82 )


- 30

HI HIiO - K t ^ U l ) = 1 . 0 - < H 2 * ( Hl *H2 * H3 +6 '» * n : ' * l )2 J+VPS ' -n lvOn/ in .Ui j . ' t

g u r u 2HI SI

1 K E F U 1 ) = +V*H1*( 2 . 0 * 0 1 * u 2 * 6 ' t * l ) 2 + l ) i *U3 ) / 0 6 n Ui-iGU TU 2

P1H212 K t r U ( I ) = + »: l * P l * < 6 4 * | ) 2 - t 3*1)3 ) /Ur:iMtli*n

GU TU 2H1S2

.3'* KEf -U I ) = - E l * V # P l # U > l + 2 . 0 # O l # f c : 2 * « E 3 ) / l > E N U i - l .GU TU 2

IC S1H1I ) = — 6 3-^/-* ( 0 2 *2-. 0 * U l aai-2 *6*v+l) 1 s.m)3 ) /D60Ui*i

GU T(J ? r

(C SI SIF15 REFI.C I )

GU TiJ 2iC S1H216 * 6 F i _ ( i )

= - 1 . 0 + 6 3 * (!: 2* (<f .O*U2*V2 *{£ i #hA + Hl^ H2 ) + 6 1 * 0 3 * 0 3 ) /l)6l\'UM

S + v * P l * f = 3 * ( 2 . 0 * 0 1 * 6 l * 6 * t + U l ) / O t UUt tGU T U 2

C S 1 S 217 * £ F L ( I )

GO TU 2C P2H118 K t r L ( I )

( 2 . 8 3 )= - 6 3 * R l * ( 6 1 * 0 3 - 6 2 * 0 2 ) / 0 6 N 0 f i

= -+62 * »*2 * ( F4-*D2 - 6 3* 03U- / 1 > 6 NU wGU TU 2

C P2S119 K ^ F U I )

GU TU 2C P2P220 K6FI. ( I )

= - E 2 * H 2 * V * < 0 1 + 2 . 0 * 0 1 * 6 !*£*» ) /IJENUrt

= 1 , 0 - ( 6 1*( H1*P2*64- + 6 3 » 0 3 * Q 3 ) + V 2 * U l * 0 1 ) /06NUHGU TU 2

C P 2 S 22 1 * r t E F L( I ) = - 6 2 * V * ( 2 . 0 * 6 1 * E 3 * 0 l * l ) 3 + D l * l ' 2 ) /OErtUi'i

GU TO 2 C S2H1.22 rCfFl. ( I ) = + V*P2*EA*{ ? .0*<>l*£2*63+l .>) ) /D6UUi*l

GU TU 20 S2S123 K c F L ( I ) = + E ^ * H 2 * ( 6 2 * l ) 2 - f c l * U 3 ) / Urr i\iUt*i

GU TU 2C S2H224 i < c r 1 _ ( I ) = + V * h ^ * ( 2 . 0 * 6 l * n 3 * O l * l ) 3 + U l i-i i» n / U E imUH

ITuTTu ?C S 2S 22 6 KdFl . ( i ) = - 1 . n + E 4 * ( E l * ( 4 . 0 * U ? v V 2 * 6 2 * 6 3 + H l * H 2 l + 6 2 * t > 2 * 0 2 ) / l > 6 ‘i>iUM

GU r u 2I K E F L ( I ) = k HFI. ( I l /DErtUH

U ~ DENIM = ( V2*l ) I. *IJ I, +1:l * H3 * D 3 * l ) 3 + 6 2 * 6 4 * i ' 2 *1)2 + ( 6 1 * 6 4 + 6 2 * 6 3 ) * H l * H 2L + ' t . O * V 2 * 6 1 * 6 2 * 1 : 3 * 6 4 * 0 2 ) / 2 . 0 .

( 2 . 8 4 )

Reproduced with permission of the copyright owner. Further reproduction prohibited withoutpermission.

- 39 -

These c o e f f ic ie n ts wore checked by Clee for con

servation of energy. Further checks were possible by

comparing waves which are supposed to be dynamically equi

valent (Cerveny (1962)). For exampley the displacements

dtoe to the ^ and ^ waves In a layerever e ha lf space must be dynamically equivalent (see section

S . 3 ) .

In addition we note as a consequence of the direction

of p o s i t iv e Z that i f we interchange subscripts 1 and 2

denoting the medium we must also change the sign of the

c o e f f ic ie n t s A? i f the wave i s of type R PS »x /?S(D .

Thus wo have, i f we interchange subscripts but

keep 2 p o s i t iv e in the original d irection

R p ,f i — * ' W

* s s , -* “ St5*.

(2.85)

RP. S. * *4

RSP

— " A * .


- 4 0 -

T h e r e f l e c t i o n c o e f f i c i e n t s w h e n o n e o f t h e m e d i a

i s r e p l a c e d b y f r e e s p a c e a r e f o u n d s i m i l a r l y * R e p l a c i n g

t h e p a r a m e t e r s o f o n e m e d i u m b y z e r o d o e s n o t g i v e t h e f r e e

s u r f a c e r e f l e c t i o n c o e f f i c i e n t s a s t h e r e s u l t o f t h i s

o p e r a t i o n i s i n d e t e r m i n a t e .

T h e f r e e s u r f a c e r e f l e c t i o n c o e f f i c i e n t s a r e

R r,P , = = ( 3 , % - + * * )

R s,p, = £ / ( ? , % ■ * & )

' E x a m p l e s o f r e f l e c t i o n / r e f r a c t i o n c o e f f i c i e n t s f o r«

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

( 2 * 1 1 ) ( c o u r t e s y T . C l e e ) . T h e o u t s t a n d i n g f e a t u r e o f t h e s e

c u r v e s i s t h a t t h e y a r e c o m p l i c a t e d f u n c t i o n s o f (9 • I n

p a r t i c u l a r t h e z e r o p o i n t s o f t h e r e f l e c t i o n / r e f r a c t i o n

c o e f f i c i e n t s f o r (9 9^ O or TT^Z c o r r e s p o n d t o c r i t i c a l a n g l e s

T h i s a p p a r e n t c o n f l i c t w i t h t h e e x i s t e n c e o f h e a d w a v e s i s

e x p l a i n e d a s f o l l o w s .

2 . 5 1 T h e H e a d V/ave C o e f f i c i e n t s

C o n s i d e r t h e c a s e i n 2 - D s e i s m i c w a v e p r o p a g a t i o n

w h e n t h e b r a n c h p o i n t g i v e n b y COS Q s — O i s

e n c o u n t e r e d b y t h e c o n t o u r o f i n t e g r a t i o n a t t h e p o i n t P .


AM

PLIT

UD

E1.0 r—

0 . 5

The P . P , r e f l e c t i o n c o e f f i c i e n t

o ,

1 . 0

o.s —

The P j S ^ r e f l e c t i o n c o e f f i c i e n t s

F i g u r e 2 . 1 1 . K n o t t r e f l e c t i o n c o e f f i c i e n t s f o r an i n c i d e n t P wave .

V e l o c i t i e s and d e n s i t i e s a r e g i v e n i n f i g u r e ( 3 . 2 k ) .


- 4 1 -

S u p p o s e t h a t t h e t e r m COS o c c u r s i n a

r e f l e c t i o n c o e f f i c i e n t / R • T h u s b e f o r e P

5 ^ 7 ( /? } - O a s (c o s = o ( 2 . 8 7 )

a f t e r

H e n c e i n r e f e r e n c e t o e q u a t i o n ( 2 . 4 8 ) t h e f u n c t i o n

i n s i d e t h e b r a c k e t s b e c o m e s c o m p l e x a n d t h u s b e c o m e s n o n —

z e r o . C o n s i d e r t h e j u m p i n f R a s i t p a s s e s t h e b r a n c h p o i n t .

The j u m p c o r r e s p o n d s t o t h e f a c t t h a t ^COS

' c h a n g r e s ‘s * i g n o v e r t h e b r a n c h ' l i n e ( B e r r y a n d W e s t ( T 9 6 6 ) )

A R . = - # P_ <8 - 8 8 ) - ,

= x / / « f t ? # -

- c 2 ^ Hr V

j

( 2 . 8 9 )

( 2 . 9 0 )

w h e r e / / / ? i s t h e h e a d w a v e c o e f f i c i e n t f o r t h e r e f l e c t i o n

d e s c r i b e d b y /R •


- 42

' / «cos - ft — y?1 *****

C* ( 2 . 9 1 )

p u t 9 ^ ~ / V

A t f - /?+—who r e Str7 - C y A ^

Cos <£>, =: ( 2 . 9 2 )

= < S/*t ^ x s

As &-*- <9pii —>& + ( 2 . 9 3 )

T h u s we s e e t h a t t h e h e a d wa ve r e p r e s e n t s a d i s c o n

t i n u o u s c h a n g e o f s l o p e i n a s •£»£ ^ ^ ^

T h e c o n t r i b u t i o n o f a p a r t i c u l a r r e f l e c t i o n

c o e f f i c i e n t d u e t o a h e a d w a v e h a s an i n t e r e s t i n g c o n n e c t i o n

w i t h o t h e r c o e f f i c i e n t s .

C o n s i d e r t h e c o e f f i c i e n t of. COS ^

r e f l e c t i o n c o e f f i c i e n t Rp, /= " h i c h i s k n o w n a s t h e

h e a d w a v e c o e f f i c i e n t R p .P .P ( B e r r y a n d W e s t ,* t

1 9 6 6 ) .

T h e n c a r r y i n g t h e r e q u i r e d a l g e b r a we f i n d t h a t

/ / - R # <2,94)Hp ,r ^ ~ a ' \ r ,


- 4 3 -

T h i s r e l a t i o n s h i p e a n a l s o b e a n t i e i p a t e d b y

s u p p o s i n g t h a t t h e r a d i u s o f o u r v a t u r e o f t h e i n t e r f a o e

e n c o u n t e r e d b y t h e i n o i d e n t w a v e i n F i g u r e ( 2 . 1 2 ) g o e s t o

i n f i n i t y . C e r v e n y a n d R a v i n d r a ( 1 9 7 0 ) h a v e a l s o f o u n d

o t h e r I n t e r e s t i n g a l g e b r a i o r e l a t i o n s h i p s b e t w e e n t h e

v a r i o u s r e f l e c t i o n a n d r e f r a c t i o n c o e f f i c i e n t s .


F i g u r e 2 . 1 2 T he r a y P j P j t h e h e a d wa ve P 1p gP 1

a s t h e r a d i u s o l a v i r v a t u r e b e c o m e s i n f i n i t e

( e q u a t i o n £ . 9 4 ) <


T h e t h e o r y o f C h a p t e r 2 c a n b e u s e d d i r e c t l y t o

c a l c u l a t e s e i s m o g r a m s f o r i m p u l s i v e p o i n t 9 » u r c e s i n a m u l t i

l a y e r e d m e d i u m . T h e o n l y s i t u a t i o n n o t c o v e r e d i s w h e n t h e

s o u r c e o r r e c e i v e r a r e l o c a t e d o n an i n t e r f a c e o r f r e e b o u n d a r y .

I n e a r t h q u a k e s e i s m o l o g y t h e s e i s m o m e t e r ( r e c e i v e r )

i s a l m o s t a l w a y s o n t h e e a r t h ' s s u r f a c e , a n d i n e x p l o r a t i o n

s e i s m o l o g y t h e s o u r c e i s a l s o e i t h e r o n o r j u s t b e l o w t h e

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

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

T h u s i t i s i m p o r t a n t t o s o l v e t h i s s i t u a t i o n .

' 3 . 1 L a m b ' s P r o b l e mT h i s p r o b l e m h a s b e e n u s o l v e d " b y a l a r g e n u m b e r o f

i n v e s t i g a t o r s i n c l u d i n g L a p w o o d , ( 1 9 4 9 ) , H o n d a , N a k a m u r a a n d T a k a g ' i " ' ( ' T 9 5 6 ) , G a r v i n ('19-55*)* -Fek e rd - s ’( 1 3 5 5 ) - a n d o t h e r s ,

a p a r t f r o m Lamb ( 1 9 0 4 ) h i m s e l f .A s o l u t i o n f o r t v/o d i m e n s i o n s i s d u e t o S h e r w o o d ( 1 9 5 8 )

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

r e a l i t y t o t h e g e n e r a l i z e d r a y t h e o r y a p p r o a c h .

______________________ SURFACE__________________________________

SEMI I N F I N I T E

MEDIUM

S o u r c e


- 45 -

C o n s i d e r a P wave o r i g i n a t i n g a t ( x , ; z , ) t r a v e l l i n g i n t h e n e g a t i v e x and z d i r e c t i o n a n d i n c i d e n t o n a f r e e s u r f a c e 2o x &

. T h e n t h e d i s p l a c e m e n t d u e t o t h e p o t e n t i a l wave ^* n s u r f a c e i s g i v e n by

( s e e B u l l e n ( 1 9 6 3 , p . 1 2 9 ) )

- ^ £ . < 8 - ^ * £ & ] <3 - 8 >

w h e r e f r o m e q u a t i o n ( 2 „ 4 7 )

° / ° [ ] ~~ f t - [ * f f & s ( t - s) [ ] d v ’

5 ^ - O a ht¥t - ( r * z , ) c * L & J

5 * ^ -- / t ~ . f - ( * - * , ) s a t * - ( K + * t e s i & ( 3 ' 3)

+ ( Z . ^ Z a) fcos <g . etc

j ^ f - f e ' - (Z0 ¥■ Z,) C0S <Q + ^ C*>S (9/5

a»e/ d ( t - S * ' - y / i a = cesQ/ct.; 2 ( t - S * ‘ -cosQ/c*.We now h a v e a n e x p r e s s i o n f o r a s u r f a c e r e c e i v e r and b u r i e d s o u r c e * To o b t a i n t h e s u r f a c e r e c e i v e r , s u r f a c e s o u r c e e x p r e s s i o n , we c o u l d t a k e t h e l i m i t o f ( 3 * 2 ) ( 3 * 3 ) , a s Ân a l t e r n a t i v e a p p r o a c h i s t o u s e t h e r e c i p r o c i t y r e l a t i o n s h i p .


- 46 -

By r e c i p r o c i t y , t h e d i s p l a c e m e n t a t ^ Z f ) d u e t o a

s o u r c e a t ( O ) i s g i v e r r by t h e s a m e e x p r e s s i o n .

T h e r e f o r e

* ') J ° /° [ 0 ~ Rp,p)c-g & - Rp,s, } ( 3 *4 )

T h i s e q u a t i o n c a n b e i n t e r p r e t e d a s t h e d i s p l a c e m e n t

d u e t o t h e p o t e n t i a l 2 ,^ w h e r e f r o m e q u a t i o n ( 2 . 4 3 )

f f a i * '') = -& *■ [C £ 0 ° - ) ^ - p j ( 3 . 5 )

/ ? t h e b o u n d a r y t e r r a i s g i v e n b yP ---------------------

P p - (3-6)

~ ~ 3 (* • £ / ( 3 ,f a * f > 3)( 3 . 7 )

w h i c h a r e d e f i n e d i n e q u a t i o n ( 2 . 7 8 ) . S u b s t i t u t i n g t h e

s o u r c e t e r m e q u a t i o n ( 2 . 2 1 ) ,

= & c j W i / Bp c o s ju / / 4 ( s t* &/&■)? ( 3 . 8 )j f p ^ f %c

w h e r e S . - 2T S ■/ £ cos 6LaC ot


- 47 -

— Oc.( 3 . 9 )

T h i s e x p r e s s i o n w a s f o u n d t o a g r e e w i t h S h e r w o o d

( 1 9 5 8 ) . S i m i l a r l y c a n b e f o u n d *

Bs - 4 (3 . 10)

a n d

t t j f = 2 $ B

TT(°p ^ t3 ^ aJ t* S i( 3 . 1 1 )

An o u t s t a n d i n g f e a t u r e o f t h i s a p p r o a c h i s t h a t t h e

b o u n d a r y c o n d i t i o n s - f o r a f r e e s u r f a c e a r e o b e y e d * e x c e p t a t

OC - & a n d "C — O w h e r e t h e b o u n d a r y c o n d i t i o n s

a r e a l t e r e d b y t h e p r e s e n c e o f t h e s o u r c e *

T h e s o l u t i o n f o r a s u r f a c e r e c e i v e r c a n b e f o u n d

b y t a k i n g j£t - O

T h e e q u a t i o n s £ s

s a m e r e s u l t f o r £

a n d & - g i v e t h e

£ = OC S'/r7 ^ /oC =■ OC S/fl ^ ( 3 . 1 2 )

F o x t h i s s i m p l e m o d e l t h e p a r a m e t r i c e q u a t i o n s o f

an d £ i n t e r m s o f & a r e s u f f i c i e n t l y s i m p l e t h a t


- 48 -

i s in e x p l i c i t funetien ef t~

Thusc o s « .

. (3 .13 )

C O S Sj, o 0 - («rF/)Hence we see that and axe zere whiler • <

t < j r as n e terms in equations (3 .8) and (3 .11) have imaginary eempenents.

Put equatiens (3 .8 ) and ( 3 .1 l ) in terms e f X <****{ &

*** ~ - ^ ( p> / £ " )T 1 ^ ( 3 . 1 4 )

« y - a? 4** ( % /* 1Tt F acwhere

then

s - 0 - & T ) ' A ( ' - * ( » ( ) ) V

e - « ( S f / O - b f f T O - O O T

+ ( £ ) ( ' - H U ffT


- 49 -

w h e r e

f * + g ( ' - ( $ ? ) * ( * • ” >

S u m m a r i z i n g t h e b e h a v i e u r e f a s 'C l n e r e a i e i s -

w h e n 3 T ^ C ^ 2£ P i » i m a g i n a r y* P

and£ = A + < ' S ( 3 . 1 9 )

where

A “ C C c ) ( ^ ^ ( $ 0 * ) ^ ( 3 - 2 0 )

b - * ( n fM & T - T O - W O T

Henee= Z * £ ~ A _ <3 * 8 1 > ^

rr(a^ x S3)Thus wa hsvs an abrupt arriva l aarraspanding ta ? a zx r/i* -

When/

7 C ( 3 - 2 2 ) Xsc £


- 50 -

w h e r e

t - ( 3 . 2 3 )

; - i m m ' - r n r - rI n p a r t i c u l a r t h e r e i s 4 v a l u e e f 'C w h e n ^

b e c o m e s i n f i n i t e a s E = 0 . 0 . T h i s e e r r e s p e n d s t e t h e a r r i v a l

e f t h e R a y l e i g h w a v e . A l s e a t C = t h e f u n e t i e n a l

f e r n e f c h a n g e s d i s e e n t i n u e u s l y . A l t h e u g h

^ ( * - f ~ ) s ( £ * j f + ) ( 3 - M )

a s B = 0 . 0 ( e q u a t i e n ( 3 . 2 0 ) ) a t £ - > t h e g r a d i e n t e f <6^

i s d i s a e n t i n u e u s . T h u s t h e a r r i v a l e f t h e s h e a r w a v e i s

s e e n e n l y a s a n a b r u p t c h a n g e e f s l e p e i n • A s y n t h e t i c

La m b ’ s p r e b l e m s e l s m e g r a m i s g i v e n i n F i g u r e ( 3 . 1 ) .

A l l t h e s e c h a r a c t e r i s t i c s a r e s i m i l a r t c t h e t h r e e

d i m e n s i e n a l c a s e s a l v e d b y P e k e r i s ( 1 9 5 5 ) .X

3 . 2 S e u r c e / R e c e l v e r a n t h e I n t e r f a c e b e t w e e n Twe S e m i -

I n f i n i t e M e d i a I n W e l d e d C c n t a c t

T h i s p a r t i c u l a r p r e b l e m , i l l u s t r a t e d w i t h p c s l t i v e

d i r e c t i o n i n F i g u r e ( 3 . 2 ) , r e p r e s e n t s a g e n e r a l i z a t i e n o f

La m b ’ s p r e b l e m w h e r e t h e v a c u u m a b e v e t h e s e m i - i n f i n i t e med ium

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

s e c t i o n i s a s m a l l d i v e r s i o n i n t h e t h e m # e f t h i s c h a p t e r>»

and t h e r e a d e r may w i s h t e go d i r e c t l y t e s e c t i o n 3 . 3 .


TIME IN MICROSECONDS 88 103 133118

Figure 3 ,1 , Lamb's problem d e l ta function response on the surface of p lex ig lass .

P wave ve loc ity = 2.355 mm./microsec.

S wave ve loc i ty = 1.379 mm./microsec.

*Y

R e c e i v e rM e d i u m ( l . ‘

- Sou - r ce

M e d i u m . (?.

F i c r u r e 3 „ 2 The s o u r c e o n th*i i f . " t i f f t c o b e t w e e n t w o s e r a i —i n f i n i t e

m e d i a i n w e l c U c e c r t i c t p r o b l e m


»r lu tU n t» th l i . pjohlia*.. fa t . n s f th«c t a p l t x int«xa«tL*n that t t i i t f i e u s bttHtcn d ir e s t , head and

in terface waves I t i s an. in terest in g . study•

ths in tsr fa e s we. generalize the appreash e f s s s t l s n 3 .1 . Ths algsbsa in v a lv a d . is lang and ted ieu s . I t saeas sa s is r

t s uss Sherweed** (1958) appreaeh — f i t a s s lu t i s n t s ths prebleu and ass i f i t i s s e r r e c t .

Thus we have far a nspual fares sn ths in tsr faes

(sss squatisn (2 .3 6 ) )

whsrs /ja i s ths amplitude s f th is gsnspalizsd P wave dus t s a naraal fares acting sn ths in tsr fa es and i t insludes

(in rsfspsnss t s 3 .2 ) ths ssures and bsundary tern s .

Ts find the., displacement due t s a l in e ssures sn

Frau the paint s f view s f r e e ip r e s i ty and symmetryws see that ths P wavs received, in medium @ f r e m t h e s o u r c e

sn t h e in tevfase w i l l depend sn ths s s s f f i e i s n t s / ( L .

Thus ws put

(3.27)


- 5 2 *-

= [ 0 - * n ) * * * * 1 ,, • * * * ,

Fr om e q u a t i o n ( 2 . 8 2 ) we c a n e x p r e s s t h e f o u r c o e f f i c i e n t s a b o v e

i n t e r r a s o f t h e c o - f a c t o r s o f 3

Thus

^ = d r f * ( - * * ^

- * 9 1 %

- 2 tf, (ft * fe)c{i3 / l l ’&ll

( 3 . 2 8 )

I n a s i m i l a r f a s h i o n we f i n d

F < " \ -- - 2 y , ( f , + / > ) ct32/( /£ > ll

.p & ) - 2 .4 U' ( 3 . 2 9 )

F#> - - 2 ^ (p. ///•*//As i n , s e c t i o n ( 3 . 1 ) p u t t i n g t ~ ^ a n d

4 t * 3 5 / t ? ^

t h e r e f o r e

a f = o r-g ^ j - l z l ^ ( 3 . 3 0 )

T h u s a t 2* - 0 t h e a n d Z c o m p o n e n t s d u e t o

e a c h o f t h e ^ a n d ^ p o t e n t i a l c o n t r i b u t i o n s f r o m a p o i n t

f o r c e l o c a t e d o n t h e i n t e r f a c e a r e g i v e n b y


- 5 3 -

•Of0 = c h *A) et*J* £ / / .£ > / / a t ,

* Z fl£>//t>, ?*>**/»- £■ f 2 //g, * / O V /» ? (3.31)

d ( j ig ft a ,

* C j-'tn f g (p ,4 f£l'**& e£n ?

L (/J}f/ 6 , JvC /bcTh«st re la t ien sh ip s have b»tn dtrlved threugh a

large ameunt af algabra in a nen-rigereus faehian. Ta prave that these expreesiens are a s e lu t ie n te the preblem we ensure that the beundary eendltiens and wave equatiens are ebeyed.I t i s seen that the individual p e te n t ia ls de ebay the wave

equatien.Te eheck the beundary eend it iens i t i s neeessary te

find the displacement in medium due te the impulsive l ine

seuree•Carrying eut the derivatien as befere we find that

II

§- 3 - K Cf>, « y a ) /H & H

II

r\ 4

l4«* (p , < 3 / / / ■ * /(3.32)

(/*> +

F€ ' 1 -- - 2 - % ( p , +■ fo )


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

a n d ^ * T h *V t h e n a g r e e d w i t h t h e k e r n e l i n L a m b ' s

p r o b l e m .

As b e f o r e , we f i n d t h e d i s p l a c e m e n t s i n m e d i u m J2 -

a l o n g 2* * O j c o r r e s p o n d i n g t o t h e a m p l i t u d e s o f t h e p l a n e

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

p o i n t f o r c e o n t h e i n t e r f a c e .

~ f - 3 (p, -t-Pl) ?

V "4) S ' J r J s 1 C * S ^ ( 3 . 3 3 )1 ~ C f - 2 ( /* ,* f i . ) - e l A * ?x . L A t - i / x

— C 4'h*' f -2 (/=*< »/3) j. ^ .* ; /)<&// V<*A *

- c ^ C v r - -2 ^ -f /%.).&■ e&. ^ 3 ? ^ a_ * • I t w f \

We a r e n o w i n a p o s i t i o n t o c h e b k t h a t t h e b o u n d a r y

c o n d i t i o n s a r e o b e y e d . T h i s a l g e b r a h a s b e e h . c a r r i e d o u t a n d

t h e s o l u t i o n f o u n d t o b e s a t i s f a c t o r y .

3 . 2 1 T h e B e h a v i o u r o f t h e D i s p l a c e m e n t d u e t o a F o r c e o n a na

I n t e r f a c e b e t w e e n Two S e m i - I n f i n i t e M e d i a

T h e d i s p l a c e m e n t h a s b e e n s h o w n t o b e m a d e u p o f t w o

p a r t s c o r r e s p o n d i n g t o t h e r o t a t i o n a l a n d d i l a t i o n a l c o m p o n e n t s

o f t h e e l a s t i c w a v e f i e l d . C o n s i d e r t h e b e h a v i o u r o f t h e

t e r r a s i n t h e e q u a t i o n s ( 3 , 3 1 ) a s ®° a l o n g a s &

a s t i m e b e c o m e s i n d e f i n i t e l y l a r g e .

C o n s i d e r

* //<& // *■*


- 55 -

( 3 . 3 4 )

By inspeetien the varieus terms are seen te have the asymptetie

values as 00

% - & (■ * '* ) j> a i - ° 0 / * * )

» 0 ( i o * ) , d a , O f r * ) ( 3 - 3 S )

/ / - * > / / - < ^ 9

The la s t r e s u l t i s seen frem the expansien e f In equatien (2*84)« The terms in eaneel.

There fere as 't**' - ^

(< ’'> )) - ( 2 ^ ) ( 3 - 3 6 >

Henae 4 ^ tends te inorease*Hewever i f we eensider the t e t a l dlsplaeement

r «* ^ 7** ^^ ^ * (3 .37)

= C 4 '* ~ f + f O O * < ^ r v ?t //<£>// 'x *t >

f r e m eqtfati'enV - (3 .31) •

^ < 3 - y ^ ~

Thu. 4 t Cl* *. o (3 .38)


T h u s w h i l e t h e d l l a t i o n a l a n d r o t a t i o n a l c o m p o n e n t s

do n o t . I n d i v i d u a l l y b e h a v e p h y s i c a l l y , t h e i r s u m d o e s . T he

r e a s o n f o r t h i s i s ( M o r s e & F e s h b a c h ( 1 9 5 3 ) ) b e c a u s e we h a v e

t a k e n t h e d l l a t i o n a l a n d r o t a t i o n a l c o m p o n e n t s o f t h e d e l t a

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

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

c a s e t h e y d o n o t .*

T h e f o l l o w i n g f i g u r e s ( 3 . 3^4 ) s h o w t h e b e h a v i o u r o f t h e

f o u r f u n c t i o n s

* 4 ^ ( 3 - 3 9 )s x j % j £

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

T h e s e f u n c t i o n s a r e t h e f o u r n o n - z e r o e l e m e n t s o f t h e G r e e n ' s

dyad ic ( " £ / j f , £ ( ° )

A l o n g t h e i n t e r f a c e t h e d i s p l a c e m e n t h a v e a s t r i k i n g

r e s e m b l a n c e t o t h e f o r r a o f L a m b * s p r o b l e m . C o m p a r e F i g u r e s

( 3 . 3 a ) a n d ( 3 . 1 ) .

T h e d o m i n a n t d i s t u r b a n c e , w h i c h e f f e c t i v e l y sw am p s

a l l t h e d i r e c t a r r i v a l s i s t h e p s e u d o - S t o n e l e y w a v e , G i l b e r t

& L a s t e r ( 1 9 6 2 ) . T h i s wave h a s a l s o b e e n o b s e r v e d t h e o r e t i c a l l y

b y A l t e r r a a n & K a r a l ( 1 9 6 8 ) . T h e r e a r e a c t u a l l y no r o o t s o f

1 1 -9 1 1 c o r r e s p o n d i n g t o t h e S t o n e l e y w a v e a r r i v a l f o r t h e

v e l o c i t i e s u s e d , b u t a w a v e i s g e n e r a t e d i n a n y c a s e . U n l i k e

t h e R a y . l e i g h w a v e i n s e c t i o n 3 . 1 , w h o s e a m p l i t u d e r e m a i n s

Reproduced with permission o f* , copyright owner Further reproduction prohibited without permission.

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

SXDN ** deno t e*- . sc c o m p o n e n t o f d i r . p l a c r e m e n t d u e t o d l l a t i o n a l w a v e s

SXWN - d e n o t e s £ c o m p o n e n t o f d i s p i a c f ra»>nt d u e t o r o t a t i o n a l w a v ^ s ,

SXH - d e n o t e t o t a l sc c o m p o n e n t o f d i r p i a c e m e n t .

SZDN, SZ'.V*l, SZN •• a*; a b o v r e x c r p t 7_ c o m p o n e n t * .

V e l o c i t i e s :

P j s 2 . 0 mm. / m i c r o s e c . , = 1 . 2 mm. / m i c r o b e , ,

^ 2 =■ 2 . 5 mm. / m i c r o s e c . , = 3 . 0 mm. / m i c r o s e c ,

• •

S o u r c e i s l o c a t e d o n t h r o i i c j i o n ; a n d t h e r e c e i v e r a t v a r y i n g

v e r t i c a l d i s t a n c e Z a w a y f r o m t h e i n t e r f a c e .

F i g u r e s 3 . 4 a - d . As f i g u r e s ( 3 , 3 a - d ) e x c e p t t h e i m p u l s e r e s p o n d

s e i s m o g r a m s d u e t o a t a n g e n t ! a 1 l i n e f o r c e s o u r c e .

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o o n s t a n t f o r a l l ( i . e . I n f i n i t e s ) , t h i s w a v e a l o w l y •

• a i t e n u a t e s w i t h i n c r e a s i n g d i s t a n c e a w a y f r o m t h e s o u r c e a l o n g

t h e i n t e r f a c e .

T h e s e q u e n c e o f d i a g r a m s f o r v a l u e s o f tET i n c r e a s i n g l y

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

I n t e r f e r e n c e z o n e s b e t w e e n h e a d w a v e s a n d d i r e c t a r r i v a l s a r e

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

o n t h e s l o w l y v a r y i n g f o r m o f t h e S t o n e l e y w a v e . A l l t h e

s c t i s mo g r a m s f o r t h i s a p p a r e n t l y s i m p l e c a s e a r e s e e n t o b e a

c o m p l i c a t e d i n t e r p l a y b e t w e e n d i r e c t a r r i v a l s a n d h e a d w a v e s

s u p e r i m p o s e d o n t h e l e a k i n g , o r p s e u d o - S t o n e l e y w a v e .

3 . 3 L a v e r o v e r H a l f - S o n s e C a s e

T h i s c a s e i s a p r i m e e x a m p l e o f t h e w a y i n t e r f e r e n c e

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

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

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

As w e l l t h e m a n n e r i n w h i c h n o r m a l m o d e s J j u i l d u p

i n a l a y e r e d m e d i u m c a n b o i n v e s t i g a t e d .

T h i s c a s e h a s b e e n e x t e n s i v e l y c o n s i d e r e d b y s u c h

a u t h o r s a s K n o p o f f e t a l ( i 9 6 0 ) , N e w l a n d s ( 1 9 5 3 ) a n d P e k e r i s

e t a l ( 1 9 5 9 ) . K n o p o f f a t t e m p t e d t o b u i l d u p t h e m o d a l

s o l u t i o n t o t h e c a s e b y a d d i n g t o g e t h e r t h e c o n t r i b u t i o n s d u e

t o t h e f i r s t 8 4 a r r i v a l s . He d e m o n s t r a t e d t h a t t h e e r r o r i n

c o n s i d e r i n g a s m a l l n u m b e r o f t e r m s c a n b e v e r y l a r g e .


• •

- 5 6 -

T h i s c a s e i s s o l v e d u s i n g t h e g e n e r a l i z e d r a y t h e o r y

- a p p r o a c h * An a d v a n t a g e o f t h e s o l u t i o n i s t h a t i t i s f a s t

i n t e r m s o f c o m p u t e r t i m e . I t r e p r e s e n t s a n e x c e l l e n t c a s e

t o c o m p a r e w i t h m o d e l s e i s m o l o g y .

3 . 3 1 S o l u t i o n t o t h e s i n g l e l a y e r o v e r a H a l f - S p a c e P r o b l e m

C o n s i d e r t h e v a r i o u s r a y s a r r i v i n g a t t h e r e c e i v e r

f r o m t h e s o u r c e i n F i g u r e ( 3 . 5 ) .

1 ) T h e d i r e c t a r r i v a l a l o n g t h e s u r f a c e w h i c h

c o r r e s p o n d s t o t h e s o l u t i o n o f L a m b ' s p r o b l e m .

2 ) T h e f o u r w a v e s P P , P S , S P , SS w h i c h f o l l o w

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

f r o m t h e b o t t o m o f t h e l a y e r .

3 ) T h e s i x t e e n w a v e s P P P P , ........................... w h i c h h a v e• * *

f o u r s e g m e n t s .

4 ) e t c .

E a c h o f t h e s e r a y g r o u p s } w h i c h a r e c h a r a c t e r i z e d

fey h a v i n g a n e q u a l n u m b e r o f s e g m e n t s , a r e c a l l e d a n s u i t e u

o f r a y s .

T h u s c o n s i d e r t h e c o n t r i b u t i o n o f t h e f o u r r a y s

P P , P S , S P , a n d S S . As d e s c r i b e d i n C h a p t e r 2 , we may

g e n e r a t e a s o l u t i o n u s i n g t h e g e n e r a l i z e d r a y t h e o r y s o t h a t

t h e p o t e n t i a l s d u e t o t h e f o u r , w a v e s a r es s . *


s u r f a c e a r r i v a l

c e l v e r

F i g u r e 3 . 5 S u i t e s o f o r d e r 0 a n d 2 I n t h e s i n g l e l a y e r o v e r

h a l f s p a a e p r o b l e m


t o . r ^ r ^ ^ Rs>^ ' / a l= C I eLfiM _ ^ ' t x p m r t - j ) d ^%*, J — 03 f,, ^ ^ (3.40)

t o 4 5 ^ ‘ .

Thus ths v s r t i s s l displacements arising fr tn the p s te n t i s l s are

= C

a C £'**> Jf ' -T *

- C £**1

u *s, ~ C Jthe abeve derivatien that

(3.41)

receiver has a een p le te ly f la t frequency r e s p e n s e . Physically we nets that the Bedel i s eempletely syaaetrie with respect te seuree and r e c e iv e r . By res ipree ity the ray SP frem seurse tp receiver sheuld give the same dlsplaeeaent as the ray PS tra v e l l in g frem rece iver te seurce. T i m s a s the s e u r c e a n d

receiver can b e ' interchanged by symmetry, therefere the d i s placement at the receiver due te PS sheuld equal the d i s p l a c e -


- 6 0 -

■int dut te SP* As

S = 5 1 . (3*42>“S.p, P.s, >

t h a t i s t h e t r a v e l t i m e f u n c t i o n s a r e e q u a l , t h e rays a r e

kinematically equivalent*In additien th e ir amplitudes a r e e q u a l s o t h a t t h e

PS a n d SP waves are dynamically e q u i v a l e n t . Frem equatien

(3*41) the respense due te the SP a n d PS w a v e s a r e

v , ' < ■ *-{- % \ % * /3 ,) }

«,s - c f .*V |

L ' ' ' * ' ' ( 3 . 4 3 )

S . !. 4,

Frem equatien ( 2 . 8 3 ) we nete t h a t

R s.r, ~ ~ /% J/ /% ( 8 ’ 44>

and also that .w 9/ * *

Thus subst itu t ing far the functions that ^

and depend upon

Bp, BSt FPSi ^ '» /* ? J

c d S / * > J* * ' ( 3 . 4 5 )


The t e t a l e e n t r i b u t i e n e f t h i s s u i t s s f r a y s

j a s y m p t o t i c a l l y s p p r s a c h z e r o f a r l a r g e v a l u e e f t i n e 't 9

a l t h e u g h a s i n t h e t w e s e m i - i n f i n i t e m e d i a c a s e t h e

d i s p l a c e m e n t s d u e t e e a a h r a y d e n e t .

Te e a l a u l a t e t h e s e l s m e g r a m d u e t e t h e s u m e f a

l a r g e n u m b e r e f r a y e e n t r i b u t i e n s we t a k e a d v a n t a g e e f

k i n e m a t i c a n d d y n a m i e e q u i v a l e n c e .

K i n e m a t i c a l l y e q u i v a l e n t r a y s h a v e t h e s a m e

v e r s u s t i m e r e l a t i e n s h i p a s t h e t r a v e l t i m e £ u a * t i e n s

a r e J e q u a l . K i n e m a t i c a l l y e q u i v a l e n t r a y s p e s s e s s t h e

s a m e n u m b e r e f P s e g m e n t s a n d S s e g m e n t s i n t h e i r p a t h .

I t w a s f e u n d t h a t m e s t e f t h e e e m p u t i n g t i m e r e q u i r e d

wa s t a k e n u p i n t h e t i m e v e r s u s V c a l c u l a t i o n . H e n c e

c e n s i d e r a b l e t i m e i s s a v e d b y g r e u p i n g r a y s i n a p a r t i c u l a r

s u i t e i n t h i s w a y . F u r t h e r t i m e i s s a v e d b y g r e u p i n g t h e

d y n a m i c a l l y e q u i v a l e n t r a y s w i t h i n e a c h k i n e m a t i c a l l y

e q u i v a l e n t g r o u p .

3 . 3 2 D y n a m i c a l l y E q u i v a l e n t R a y s

T h i s d i s c u s s i o n i s r e s t r i c t e d t o r a y s f a r m e d w i t h i n«

a s i n g l e l a y e r .

R a y s w h i c h a r e d y n a m i c a l l y e q u i v a l e n t h a v e t h e

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

d i f f e r e n t e r d e r s . I n p a r t i c u l a r r a y s e f t h e t y p e


- 6 2 -

x y / = ? y x (3*47)

by x t i i p r M l t y and i*u r«*-xt«a ivtr • yam*try. * , > r OL**dt

r*pr*s*nt greups *r s in g le P «nd S rays .In p a r t ic u la r the fe l lew in g rays drawn schematically

in Figure (9 .6 ) are *quival*nt. Dynamically equivalent rays art characterized by th* fallowing fa s te r s s

l ) Kinematic equivalence3) Sam* number e f PP9 (PS cr SP) and SS r e f le c t ie n s

en upper i n t e r f a c e .3) Same number ef PP9 (PS er SP) and SS r e f l e c t ie n s

en lewer i n t e r f a c e .These fa c te r s are u t i l i z e d la te r where pregramming

theery i s d iscu ssed .%

Figures (3 .7 ) fe llew ing shew th e e r e t ic a l seismegrams generated f e r a layer ever a half space fer varieus seurce- reeeiver d is ta n c e / la y e r thickness r a t i e s .

3.4 Multi-Levered MediaThe theery e f sec t ien 3 .3 i s very e a s i ly extended

te m ult i- layered media and has already been generally cen- sidered in Chapter 2 . Th* majer cemplicatien i s the increased pregramming cemplexity. The fe l lew ing sec t ien describes th* leg ic e f a pregram fer a m ulti-layered medium.


T a b l e b e l o w g i v e s t h e k i n e m a t i c a l l y a n d d y n a m i c a l l y e q u i v a l e n t r a y s w i t h 3 P a n d 3 S w a v e s e g m e n t s t h a t f o l l o w t h i s p a t h .

pIs Is IpI s IpP I S i P i S I S I P

P I ^ I ^ I ^ I P I P

pi pi si si si p si pi si pi si pPISIPI SIPI S S j S j P i S t P i P

SIPXSI SIPI P s Is IpIpI s Ip PIPISI SIPI S PISIPIPXSI Spi pi si pi si ss i p i p i s i s x p PISISTPTPI S SI S1SIPIPIP PIPIPISI SIS SIPISIPIPI S SIPIPISIPI S SISIPIPI PI S SIPIPI PISI S

F i g u r e 3 . 6 . D y n a m i c a l l y a n d k i n e m a t i c a l l y e q u i v a l e n t r a y s

D y n a m i c a l l y e q u i v a l e n t r a y s a r e b r a c k e t e d - a l l r a y s a r e k i n e m a t i c a l l y

e q u i v a l e n t


Figure 3.7 Impulse respose selsmogram for a layer over a half spaces h denotes layer thickness. Velocities and densities are given in figure (3.22). Compare with figures

(3.22) and (3.23)

300.0

20.0

L J- 30.0

1 0 0 . o

o 20.0

25.0

r \ i- 35.0

r w- 40.0

50.0 100.0TIMP I IJ HICROGrCOIJDS

i r, o . o


- 6 3 -

3 . 5 P r o g r a m m i n g t h e g e n e r a l i z e d r a y t h e o r y

P r o g r a m m i n g t h e o r y t o c a l c u l a t e t h e s e l s m o g r a r a s

g e n e r a t e d i n t h i s c h a p t e r i s b e s t d e s c r i b e d u n d e r t w o

h e a d i n g s :

l ) P r o g r a m S t r u c t u r e - - t h e o v e r a l l l o g i c s t r u c t u r e

o f t h e p r o g r a m a n d s u b r o u t i n e o r g a n i z a t i o n w i t h t h e p u r p o s e

o f e a c h s u b r o u t i n e d e s c r i b e d , a n d

—2-)— I - n d i v i d u a l S u b r o u t i n e T h e o r y — t h e wa y i n w h i c h

e a c h s u b r o u t i n e * s p u r p o s e i s c a r r i e d o u t .

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

t h e f o r m

= 5C f j fe ) T fo c ) £(*«+*) ? (3.48)

w h e r e — S/*i .6^

a n d = a n g l e b e t w e e n r a y a n d n o r m a l t o i i n t e r f a c e /

b o u n d a r y e n c o u n t e r e d' KiJ.

T ^ ,) = 7 7 / ( r e f l e c t i o n / r e f r a c t i o n /c ” ^ “7 ■

b o u n d a r y / i n t e r f a c e c o e f f i c i e n t s o f r a y p a t h ) j

s o u r c e f u n c t i o n

£ (* * •* ) - r e c e i v e r f u n c t i o n

(^ x ) ■=* P 5 - SL=5 3 ^3^>

w h e r e -?> - w h e r e


- 6 4 -

w h e r e .S ' i s t h e ' o p t i c a l t r a v e l t i m e * ( e i c k o n a l ) f o r t h e

p a r t i c u l a r r a y f r o m s o u r c e t o r e c e i v e r .

i s g i v e n b y t h d b c j u a t i e n s

(s') * o <3,49)

I n t h e p l a n e p a r a l l e l l a y e r c a s e - "L? •£f e r a l l

As w i l l b e s e e n i n f o l l o w i n g c h a p t e r s , t h i s e x p r e s s i o n

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

c u r v e d l a y e r e a s e s .y^ i s t h e •<( * i n d i ’C e t o rr ^ ) 'SC - o r -5 - c o m p o n e n t

v/ . Tf to f t h e d i s p l a c e m e n t d u e t o t h e ^ r a y w i t h /*C s e g m e n t s

d e s c r i b e d b y t h e X ~ ^ o r p o t e n t i a l .

• * »

T h e p r o g r a m m i n g t h e o r y i s r e s t r i c t e d t o t h e e a s e

w h e r e b o t h s o u r c e a n d r e c e i v e r a r e o n a n i n t e r f a c e o r t h e

s u r f a c e . T h u s a r a y w i t h /C s e g m e n t s e n c o u n t e r s f t ¥■ 2

i n t e r f a c e s o r b o u n d a r i e s .

A r a y c a n t r a v e r s e o n e s e g m e n t w i t h e i t h e r t h e P

o r S v e l o c i t y . T h u s c o n s i d e r a l l t h e r a y s w i t h K s e g m e n t s

( d e f i n e d a s a . s u i t e ) w h i c h f o l l o w s a p a r t i c u l a r p a t h £ , t h e n

t h e n u m b e r e f d i f f e r e n t r a y s A / i n a s u i t e i s

/sj = 2* ( 3 - 5 0 >


- 6 5 -

. tifThe 1 eempenent e f the t e t a l displaoement due the s u i t e e f erder fK e f rays i s given by

a K

"Prea Figure ( 9 .8 ) i t i s seen that there may be a number e f s u i t e s e f erder K •

Thus the ^ eempenent ef the displacement due te a l l L su i te s e f erder K i s g iven by

F in a lly the J eempenent e f the displacementdue te a l l s u i t e s i s g iven by

4C. - ‘&V (3 .53)K *

where & (P i s the minimum number e f segments eennecting^f/^r

seurse te r e c e iv e r .i s \ O (3 .5 4 )

where Ki*» 9 O r e fe r s t e the ease where beth seuree and rece iver are en the same in te r fa c e er beundary and the ray neves aleng the beundary Lamb's preblem.

While the abeve d e s e r ip t ie n e f equatien (3 .48 ) and


V{>p**r f u n d a m e n t a l s u i t a

S o u r c e

l o w e r f u n d n m e n t a l s u i t e

F i g u r e 3 . 8 I l l u s t r a t i o n o f t h e I. s u i t e s o f o r d e r K


- 6 6

the methed t« ••■put* th« t e t a l displacement a p p t u t • •■ p l ic a te d , i t i t id e a l ly su ited t# a eemputer. In a la te r se« t i«n t in e • s t in a ta s are given and are seen te be very lew. In f a e t , the t in e eensuning part e f the pregran i s the deuble i t e r a t iv e technique required te sa lve the twe equatiens (3 .49) inand $'***('& ) *•* • given t in e C* •

3*6 Prearan StnuetureThe prdgran s e n s i s t s e f abeut 45 subreutines whieh

in past l e g i e a l i y fe l lew the sequence e f eperatiens given fren equatiens (3*48) te (3*53) in the previeus seetien*

While the structure e f the pregraas far the P a r a l le l ,Dipping and Curved Layer nedels are s im ilar , indiv idualpregrams were eenstrusted te deal with each ef these three cases*

As an input/eutput device the pregram i s fed m e d e l

parameter infermatien as shewn in Figure (3*9) and eutputs the eerrespendlng t e t a l seismcgrems and ( i f required) se isn e - grams ef eaeh s u i t e e f trays, es even each ray. T h e r a y ; n a m e s

and th e ir in d iv id u a l arr iva l times are a lse eutput as shewn in Figure ( 3 .1 0 ) . Varieus a n c i l la r y subreutines which carry eut simple a lgebraic eperatiens such as in te r p e la t ie n , grid

generating e t c . are emitted.The ca lcu la t io n e f the t e t a l seismegram i s breken

dawn in te the hierarchy e f eperatiens es l e v e l s shewn in


I N P U T m o d e l p a r a m e t e r s

an*J t i m e a n d s u i t e r a n g e

r e q u i r e d

C a l c u l a t e t o t a l s e i $ r a o -

j g r a r a b y s u m m i n g a l l s u i t e sIi! * f o r d e r K

O U T P U T S v e r t i c a l d i s p l a c e m e n t

s e i s i n o g r a m s a s a f u n c t i o n o f

t i m e w i t h d i s t a n c e a s a p a ram ete r

F i g u r e 3 . 9 . T h e p r o g r a m a s a n i n p u t o u t p u t d e v i


s u i t LF (.nC£i<

F i t i r i M P 1 S 1F i P 1

»■*r i v , m . n v rI • v v 2 L J 2 1 . 2 ' W mF C2

A NFL I f t^ .F1 , o 3 1 c r - - F 31 . 2 3 r •: f c r

F ‘i I- l b I F 1 i • c •» 7 h i * t £. 1 . 2 3 * o 71 0i 11: i f i f i 1 • •'» / *\ i: C 2 * l . c i t s r —C2f 1P 1 F I £ 1 1 . 2 A/ AC C2 1 . C 1 0 E F - C 2P U 1 S I F 1 i •' ) 0 3 i? C. 32 - 1 . : o c 7 c - < ; iS i l - l S l P l 1 . 4«- . :SF 32 - E . 6 2 M K - C . 1P J 5 1F 1 6 1 I . m c 7. *i F ■; 2 - F . c 2 3 5 F - 0 1:> i S 1 F i V 1 P ) F 1 i 1 S 1 S IF 1F1S1 b l & i S l F l P 1 5 1 S 1 £ 1 S 1 S 1 F 1 S 1

T m n r r r rS1S1SLS1

t ' l i , ; UT TZ~ . ' . C i t f C2 . m ~ t 02 • ; / u *> r c 2• 22. i: 02

- L . C o . \ i; — i 1 - 1 . 3 t * C S E - r 1

1 .KC Cf en -O l 6 . ! i 7 l t f - - a 2 6 . £ 7 1 5 * = - C 2 3 . 2 C 3 6 E - 0 1 i t i t >. I

- 6 . ’5A2 iIE OC. - / i r i. . t 2 S 11 C 2

Fi g u r ^ 3 . 1 0 . P r o g r a m p r o d u c p s t a b l p o f r a y s w i t h t h o i r

• a r r i v a l t i r j p s a n d a n p l i t u d p s •

Reproduced w,,d permission o „h e copyright owner. Fodder reproduCion prohibited whhou, permissio^.

- 6 7 -

F i g u r e ( 3 . 1 l ) , A s u b r o u t i n e a t a p a r t i c u l a r l e v e l c o n t r o l s

s u b r o u t i n e s a t t h e n e x t l o w e r l e v e l ,

A s u b r o u t i n e c a n b e c o n s i d e r e d a s a d e v i c e t h a t

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

1 , G e n e r a t e s i n f o r m a t i o n f r o m i t s o w n l o g i c a n d f r o m

m e r e b a s i c i n f o r m a t i o n s u p p l i e d ( I N P U T ) f r o m t h e l e v e l a b o v e ,

2 , I n p u t s t h i s i n f o r m a t i o n t o s u b r o u t i n e s a t t h e

l e v e l b e l o w a n d c o l l e c t s t h e i r o u t p u t . T h e s e l o w e r l e v e l

s u b r o u t i n e s a r e d e a l t w i t h s e q u e n t i a l l y i n t h i s w a y , (A

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

p r o g r a m i n t h e d o u b l e i t e r a t i o n o p e r a t i o n ) ,

3 , O p e r a t e s o n t h e i n f o r m a t i o n r e t u r n e d f r o m b e l o w

a n d f i n a l l y OUTPUTS ( o r r e t u r n s ) i t s i n f o r m a t i o n t o t h e

l e v e l a b o v e .

T h e v a r i o u s l e v e l s o f s u b r o u t i n e s m ay b e f u r t h e r

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

t o b e c a l l e d .

3 , 7 S u b r o u t i n e T h e o r y

T h e l o g i c o f e a c h l e v e l o f s u b r o u t i n e s f o l l o w s :

3 , 7 1 L e v e l 1 S u b r o u t i n e s ( s e e F i g u r e ( 3 , 1 2 ) )

. T h i s - l e v e l c o n s i s t s o f s u b r o u t i n e MAIN w h i c h r e a d s

I n t h e m o d e l p a r a m e t e r i n f o r m a t i o n i n c l u d i n g P a n d S w a v e

v e l o c i t i e s , t h i c k n e s s a n d d e n s i t y o f e a c h l a y e r f o r t h e


,EVEL 1

LEVEL 2

IjEVEL 3

LEVEL 4

LEVEL 5 Ci^wUti ref lest ien seefficlente

Calculate All Suites - Sua eve? K

If K= O CaleulateL a m b ' s P r e b l e a

V far each say gene*Caleulate tlae vexeua

a t e d t

Deuble lteratien te •elveJt* (s) = t

(S ) - O

Caleulate funetian » preduet af

xefleetlen eeeffa. eagrespending ta eaeh t l f e

Generate a ll xayt fax eaeh suite i f O and ee l-eulate amplitude vaxiatien with tlae

Generate a l l suites ef erde* K by calculating path Paxaaetegs* Adjustand s u a e v e r L

Figure 3.11 The hierarchy ef subreutines


LEVEL I LEVEL 2

Inpijj’*. a l l m o d e l s o u r c e ,

r e c e i v e r , t i m e r a n g e a n d

i n t e r v a l a n d s u i t e r a n g e

parame t e r s

G e n e r a t e t i m e g r i d j i n i t i a l i z e

Sum ^ •

wher e i s r e t u r n e d f r o m

l e v e l 2 s u b r o u t i n e S U I T E S

O u t p u t g r a p h s o f d i s p l a c e

ment d a t a , p u n c h e d c a r d s

as r e q u i r e d .

S U IT E S - s e e f i g u r e ( 3 . 1 3 )

F i g u r e 3 . 1 2 The l e v e l 1 s u b r o u t i n e s


- 68 -

Paralle l Layer Caaa as an example. I t generates a time grid as sp ss if ia d fxsa th« sang* and interval input, e a l l s Level 2 subreutine SUITES t ines e e l le e t in g and sunningi t s eutput t e feris the t e t a l displasenentequatien ( 3 .5 3 ) . F in a l ly , MAIN eutputs required graphs and tables ef • Figure (3 .9) shews the subreutine as aninput-eutput d e v ise .

3.72 Level 2 Subreutines (see Figure (3 .13 ))This le v e l e e n s is t s ef ene subreutine SUITES. I t

hat twe nain funetienss1) Te generate path indleaters fer the L su ites

ef erder K whieh give the behavieur ef the su ite at eaeh in ter faee , and

2) Sert nedel layer paraneters Intesuite segnent paraneters. Ne r e s tr ie t the theery te the sate when beth seuree and reeeiver are en an Interfaee er beundary.

The generfcl preeedure fer ( l ) i s te traee the lew envelepe e f a l l the su ite s e f erder K end then by an i te ra t iv e preeedure ea leu late up te the upper envelepe the remaining s u i t e s . These twe envelepes are referred te as the lewer and upper fundamental su ites resp ective ly -

Figure ( 3 .8 ) .Five su ite path indieaters are required fer each


LEVEL 1 LEVEL 2 LEVEL 3

id* 1 l a y e r j iraaetexs tia# g r id

Outputu :o

Generate auita aaguea t parametera by*1) Interchanging aaurea and raaalvar i f n a a e s s a r y

2 ) Chaak If auita ia paaaibla .3) Ganarata fundamental auita indiaatara Ts and aall apprapslate level 3 aabaautin*4) Itarativaly ganarata remaining, auitaata the upper laval fundamental auita calling

( K \Klaval 3 aubrautinaa ta find .aaah auita ia generated5) N. B. in 3) and 4) CONVERT MODEL LAYER PARAMETERS TO SUITE SEGMENT PARAMETERS.

Sum a l l Lm auitaa af erder P(

* - Z M€*1

SUITES

Floura 3.13. Laval 2 aubrautinaa


- 69 -

• f t h e l n t « r f a i « i s — v i z .2 $ ( j j') i n d i c a t e s z d i r e c t i o n o f i n c o m i n g r a y

i n d i c a t e * z d i r e c t i o n o f o u t g o i n g r a yr s /T J 3J i n d i c a t e s s o u r c e o n i n t e r f a c eXS?XJ 4 ) i n d i c a t e s f r e e s u r f a c e o r i n t e r f a c e ( 3 . 5 5 )

i n d i c a t e s l a y e r n u m b e r b e l o w i n t e r f a e e

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

p e s i t i e n s i f t h e s e u r c e i s a b o v e t h e r e c e i v e r . T h i s i s

j u s t i f i e d b y t h e s y m m e t r y e f G r e e n ’ s F u n c t i o n .

As F i g u r e ( 3 . 1 4 ) s h e w s , i t may b e i m p o s s i b l e f e r a

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

r e t u r n s i e n t r e l t e L e v e l 1 . When t h e f u n d a m e n t a l s u i t e

e n c o u n t e r s t h e l o w e s t i n t e r f a e e a b o v e t h e i n f i n i t e h a l f -

s p a c e , t h e f u n d a m e n t a l s u i t e b e c o m e s - as- s h e w n i n F i g u r e ( 3 . 1 5 ) .

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

l a y e r p a r a m e t e r s a r e r e - s e r t e d t e s u i t e s e g m e n t p a r a m e t e r s .

T h i s i n f o r m a t i o n i s t h e n f e d t e t h e a p p r o p r i a t e L e v e l 3

w is u b r e u t i n e t e c a l c u l a t e t h e t e t a l d i s p l a c e m e n t s f Ja f # r

a g i v e n s u i t e e f r a y s .

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

b y " f i l l i n g i n c o r n e r s " a s i n d i c a t e d i n F i g u r e ( 3 . 1 6 ) .

P r o g r a m c h e c k s l a s t s u i t e g e n e r a t e d u n t i l a c o r n e r l i k e A

i s e n c o u n t e r e d — i t t h e n f o r m s t h e n e x t s u i t e by c h a n g i n g

t h e s u i t e p a t h i n d i c a t e d a t B 9 C a n d O a p p r o p r i a t e l y .

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

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

r e s t r i c t e d , ( F i g u r e ( 8 H 7 ) ) .


S o u r c e * ----------

R e c e i v e r7*----------------

F i g u r e 3 » 1 4 No s u i t e s o f o r d e r s 1 , 3 , 5 , 7 . „ „ e x i s t

R e c e i v e rS o u r c e

F i g u r e 3 . 1 5 T h e l o w e r f u n d a m e n t a l s u i t e i s r e s t r i c t e d b y t h e

b a s e m e n t


F i g u r e 3 . 1 6 The m e t h o d o f i t e r a t i v e l y g e n e r a t i n g s u i t e s b y " f i l l i n g i n c o r n e r s *

S o u r c e

f i g u r e 3 „ I 7 U p p e r f u n d a m e n t a l s u i t e r e s t r i c t e d b y f r e e s u r f a c e


- 7 0 -

3.73 L m l 3 Subreutines ( tee Figure (3 .1 8 ))

This l e v e l c a lc u la te s the t e t a l displacement fer a

Preble* subreutine LAMB ie c a l le d . Otherwise subreutine

are generated by permutating a l l eembinatlens e f the P and

S v e l e e l t i e a e f eaeh segment ef the su ite path. V e lee ity

indicatere (be nete whether P er S wave) fer eaeh ray are

else generated ee th a t eutput as in Figure (3 .10) shewing

rays with arr iv a l time can be made.The remaining funetiens ef subreutine RAYS i s

indioatedj. in the f ig u re ( 3 .1 8 ) .

3.74 Level 4 and Level 5 Subreutines (aee Figure (3 .19 )This l e v e l e e n s is t s ef three majer subreutines —

TIMEV, DIFF and RTCOEF which are a l l ca l led by Level 3

subreutine RAYS.

3.741 Subreutine TIMEVThis subreutine tegether with the Level 5 subreutine

PATH se lv e s the twe equatiens (3 .4 9 ) . Frem equatien ( E . E 4 )

and (2 .55 ) we have expressed in terms e f and ^ the rea l

and imaginary parts e f ^

s u i t e e f erder K then Lamb's

RAYS Is c a l l e d , and the varieus rays e f the s u ite

(3 .56)


LEVEL 3 LEVEL 4

uitff path ndloators ltd segment irarac t e r s

K-O

C a l e u l a t e L a m b ' s p r o b l e m ( 4 . 2 ) -

M o d i f i e d t e s u b r o u t i n e ZERO f e r

r e a l s e u r e e t i m e c o n v o l u t i o n

'KfO

ptput t o t a l

^ p l a c e m e n t

|»r s u i t e

G e n e r a t e a l l r a y s i:oj ; a p a r t i c u l a r s u i t e

b y g e n e r a t i n g a l l c u n o i n a t i o n s e f v e l o c i t y

a n d v e l o o i t y i n d i .

C a l l c a l c u l a t i o n f i r u a l u t i o n o f e q u a t i o n*

( 2 ) i n e l u d i n g d i r o t t a r r i v a l t i m e f o r e a e h

r a y .

G e n e r a t e s u c c e s s i v e v e l o c i t i e s a n d . d e n

s i t i e s a b o u t e a c h i n t e r f a c e t o e a l e u l a t o

T O > - t h e p r o d u t t or t h e r e f l o s t i e n /

t r a n s m i s s i o n c o e f f i c i e n t s f o u n d b y o a l l i ' n g

RTCOEF.

C a l l c a l c u l a t i o n f o r f u n c t i o n A H -

C a l e u l a t e J a n d sum f o r a l l r a y s

a f t e r s h i f t i n g t i n e g r i d t o r e f o r o n o o t i m e

g r i d .

TIMEV

1

RTCOEF

DI F F

F i g u r e 3 . 1 8 . L e v e l 3 s u b r o u t i n e s


E V C L 3

l o t r a y1th p a r a m e t e r s |ae i n t e r v a l Id naxiraum

Inui r e d .

I tpu t t i m e | r s u s

a t i n e|id

If-

LEVEL 4

C a l l a r r i v a l t i m e s u b r o u t i n e

TARR

I t e r a t e

C a l c u l a t e 'IS' f r e a n q u a t Len (340) u s i n g d e u b l e

i t e r a t i o n * T r y a p p r o p r i a t e c a s h ^ f a r "C

a n d a a l l l e v e l 5 j u h r . a j - f c i n • PATH t a f i n d

c ' = / ( / 5 « h ‘ » r e ^ i s f o u n d f r a n

s o l v i n g J (f* /

u n t i l £ — £■ K. ^ i s v e r y s m a l l

c a l l DPATH ( d o u b l e p . r > » a i s i o n ) i n s t e a d a f

PATH. T h e s u b r o u t i n e f o l l o w s t h e p a t h i n

t h e c o m p l e x ® ~ p l a n e f r o m TjYAX

t o TARR a t i n t o i ” a. ' ; i o f D17

I f ^ TsA &R, ^ -■* c2 . - j i n g l e i t e r a t i o n

v e r s u s jp r e q u i r e d u n t i l • • 2yy/,v

I s m i n i m u m t i m e r e q t i r u d *

>f t

'Tihev

LEVEL

TRAVEL

S s PATH

a r

/ DPATH

~ ^ «S j fF i n d

DI F I-

C a l c u l a t e r e f l e c t i n r . / t r a r t E n i s s i o n o a -

e f f i c i e n t sRTCOEI*

F i g u r e 3 . 1 9 L e v e l 4 ant? 5 s u b r o u t i n e s


OC COSfO Sfn h p - £ £ s ,n / j s**iA VThu aubrautina flaw chart la givan in Plgura (3.2^) and la aaan ta ba a daubla i ta ra t lan algarithm. Having aalaulatad

tha valua af and £ aarraapanding ta >*4 * thaaa valuaa aan ba uaad aa a f l r a t appraximatlan far tha tina — & T ,Thua tha pragran falluwa tha laaua af t laa in tha aamplax plana (Flgura ( 2 .7 ) ) aa tima daareasaa. k%£ + O tha alaglu praalaian arithmetic uaad in aubrautina PATH braaka dawn. Cantral la tharafara diverted ta DPATH which ia written in daubla praaialan.

Subrautina PATH and DPATH aalculata tha valua af j r n f

and t ualng aquatian ( 8 . 5 8 ) raapaativaly oarraapanding ta tha g ivan ^ • SftSpia faund frata£ within an accuraoy af ualng Nawtan'a appvaxlaatian ta find tha paint whara

( 3 . 6 7 )

Ualng aquatiana ( 2 . 5 9 ) and ( 2 . 6 0 ) wa knaw that

auat ba auah thatjo J ^ s t*7f < 3 - s 8 )

whara and aarraapand ta tha aalutiana

af tha abava aquatiana.t ia aalaulatad fram and £ and with

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission

F i g u r e 3 . 2 0 . The l o g i c o f s u b r o u t i n e TIMEV

The s u b r o u t i n e u s e s t h e f o l l o w i n g a l g o r i t h m t o f i n d t h e

v a r i a t i o n o f CC*/* a n d s/+* > w i t h t i m e r a f t e r s e t t i n g t h e

p a r a m e t e r s ( u s i n g e q u a t i o n C **”/>))

j - ( c Q u, j * '”/**?) <n * / "

DTU “ j f *'•*/*»} - /

DTL - 7 ” - J~(C®

( e q u a t i o n 3 . 6 5 )where

s'» a ) <6JF u r t h e r m o r e i s s o l v e d b y l i m i t i n g a s

i n e q u a t i o n ( 3 . 5 8 ) w h e r e m i n {s-tnjo ) i s g i v e n by t h e s o l u t i o n o f

T / - t w h e r e 7“'^ ~7~

THE ALGORITHM

NoEND

DTL = I ~ u CQL = cc&**4

t - T f DTUCQU

Ce>±J>C B (CQL x DTU + CQU x DTL) / ( DTU + DTL)

CALL PATH ( o r 'DPATH i f • T h i s s u b r o u t i n e c a l c u l a t e s

S t** f r o m c j - o a n d f r o m ^ ~ •________________f r o m


- 72 -

t h e t i m e C0 s p e c i f i e d . O n c e

/ £ ** £ « / “ ^ ( 3 . 5 9 )

t h e n t h e c a l c u l a t i o n i s c o m p l e t e .

E q u a t i o n ( 3 . 5 6 ) c a n b e r e - e x p r e s s e d a s

COS A = £*( 3 . 6 0 )

f t - — <£sv/y© -f* 3£-fccsA a ( 3 . 6 1 )

He nc e S m u s t b e s u f f i c i e n t l y s m a l l s o t h a : : SC c a n

be l e s s t h a n t?P$

H e n c * <?.’£ - <(<( S t ( 3 . 6 2 )

so t - h a t ^ *£ . £ cos A 42 Ce>sAj

a n d a s m a l l c h a n g e S c o s A ^ w i l l d i r e c t l y c a u s e a c h a n g e

i n SC

T h e r e f o r e «Ts/ * y p S/ ys>) ( 3 . 6 3 )

T h u s t h e s o l u t i o n o f e q u a t i o n ( 3 . 6 0 ) m u s t b e s u c h

t h a t t h e a m o u n t S j t h a t C[ d i f f e r s f r o m 0 . 0 m u s t b e g i v e n


- 73 -

b y

$ o •=■ SS/r> p > _ (3 .6 4 )

s# t h a t

S j «

3 . 7 4 2 S u b r o u t i n e s D I F F a n d RTCOEF

T h e s e t w e l e v e l 4 s u b r o u t i n e s m e r e l y c a l c u l a t e t h e

f u n c t i o n s A(-» ) a n d t h e r e f l e c t i o n c o e f f i c i e n t s d e s c r i b e d

i n C h a p t e r s Z a n d 3 .

3 . 8 M o d i f i c a t i o n t o A c c o u n t f o r D y n a m i c a n d K i n e m a t i c

E q u i v a l e n c e

C o n s i d e r t h e c a s e o f a l a y e r o v e r a h a l f s p a c e m o d e l .

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

a r r i v a l t i m e s o c c u r s w i t h p a r a l l e l l a y e r m o d e l s .

A l l r a y s w h i c h c o n t a i n a n e q u a l n u m b e r o f H a n d S

wa ve s e g m e n t s i n t h i s r e s t r i c t e d c a s e w e r e s e e n t o b e k i n e

m a t i c a l l y e q u i v a l e n t . H e n c e f o r a s u i t e o f o r d e r 4

t a k e s o n o n l y 5 d i f f e r e n t v a l u e s f o r a g i v e n t i m e c o r r : s p e n d i n g

t o £ ) . . . . . . j S w a v e s e g m e n t s . A l a r g e a m o u n t o f c o m -

p u t i n g t i m e i s t h u s s a v e d b y e n t e r i n g s u b r o u t i n e TIMEV K'h 1

t i m e s i n s t e a d o f Of* t i m e s f o r a s u i t e o f o r d e r / f •

We h a v e a l s o d e m o n s t r a t e d t h a t i f r a y s h a v e v a r i o u s


o t h e r p a r a m e t e r s e q u a l t h e y w o u l d b e d y n a m i c a l l y e q u i v a l e n t ,

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

L e v e l 3 t a k e s a d v a n t a g e o f t h i s b y s o r t i n g t h e r a y s o f a

s u i t e i n t o d y n a m i c a n d k i n e m a t i c e q u i v a l e n c e .

3 . 9 M o d i f i c a t i o n t o A c c o u n t f o r R e a l S o u r c e

T h i s p r o g r a m f o r t h e P a r a l l e l y l a y e r e d m o d e l s c a n

b e m o d i f i e d t o a c c o u n t f o r a r e a l s o u r c e . As w e l l , t h e

m o d i f i c a t i o n s d e s c r i b e d h e r e a p p l y e q u a l l y w e l l t o t h e d i p p i n g

a n d c u r v e d l a y e r e d c a s e s .

I f 4 (pCJ t ) r e p r e s e n t s a o n e r a y c o m p o n e n t

o f t h e r e s p o n s e t o a s p a t i a l } i n e a n d i m p u l s i v e t i m e s o u r c e ,‘i*

t h e n

5 =r f ( 3 . 6 6 ) *

Jo

r e p r e s e n t s t h e r e s p o n s e t o a s p a t i a l l i n e s o u r c e w i t h t i m e

r e s p o n s e fC * ) -

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

n u m e r i c a l t e c h n i q u e s a s J 'C ^ i s a f u n c t i o n e x p e r i m e n t a l l y

d e t e r m i n e d a s d i s c u s s e d i n C h a p t e r 5 .

T h e m a j o r p r o b l e m i s t h e s i n g u l a r i t i e s c o r r e s p o n d i n g

t o t h e d i r e c t a r r i v a l s i n -0C . N e a r d i r e c t a r r i v a l s i n g u

l a r i t i e s

c c - * . y . *

( 3 . 6 7 )


- 75 -

XjlT h u s a q u a t i o n ( 3 , 6 6 ) i s s p l i t i n t s t w o p a r t s

and A w h a r a

■U?* =■ C*A H (C- Q ( 3 . 6 8 )

a nd <X ^ ■=. - iX ,*41 t

( 3 . 6 9 )

T h a f u n a t i a n ^ t h u s a a r r a s p a n d s t a t h a d i r e c t

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

R a y l e i g h / S t a n e l e y w a v e s a n d h e a d w a v e c o m p o n e n t s .

T h u s we c a l c u l a t e

s f - -tl) £ * *Je, ( * c - t ey>-

( 3 ' 7 0 )•So

- + f C -“ a * • T) ^ ( S , ? i )

t & * f * j £ / l L ± ±J ( -c - c .y '*

w h e r e

T h a s a a a n d t e r m i s e a s i l y c a l c u l a t e d u s i n g G a u s s i a n

q u a d r a t u r e •

The a a n s t a n t C ** i 8 c o m p u t e d i n s u b r o u t i n e


- 76 -

TIMEV using squstisn (2.61) fsr each dynamically d ifferen t

set ef rays.The funetiens *nd ^ *nput i n i t i a l l y as

they are used fer every computation. These twe funetiens were found frem the experimental measurement ef a ref lected ray frem the lewer free surTaee ef -a plate as described in Chapter S. Mathematically i t w i l l be shewn that *nd

are eenneeted by an Abel and inverse Abel transferm-

atien in Chapter 6 .The m edifieatiens discussed abeve are implemented

in the lev e l 3 subreutines. (see foatnote following page)I t was e lse feund that eenvelving the singular

Rayleigh wave e f Lamb's problem presented numerical d i f f i c u l t i e s We assume that the denominator ef the expression

for Lamb's problem (see equation (3.1S))has the form near

the arrival time efo

£ ( t ) = + £ & ) } / c , (3.72)

where , £ ^

Therefore

£ ( t ) f ) - - z £ L - + I

(c-QL J(3.73)


Footnote

U n f o r t u n a t e l y t h i s n o d i f l o a t ! o n d o e s n o t o o n p l e t e l y

r i n o v t t h a s i n g u l a r i t y a a u a a d b y / 4 ^ d ^ * 0 i n a q u a t i o n ( 3 . 4 8 ) . I f

h e a d w a v e s o e o u r b e f o r e - t h e d i r e c t a r r i v a l t h e n w i l l beJn o n - z e r o b e f o r e t h e s i n g u l a r i t y a n d r a p i d l y i n e r e a s e immed

i a t e l y p r e c e d i n g t h e d i r e e t l y r e f l e c t e d a r r i v a l . T h i s same

e f f e c t i s n o t e d b y E w i n g , J a r d e t s k y a n d P r e s s ( 1 9 5 7 ) p a g e 1 0 4 .

I f t h e t h e o r e t i c a l s e l s m o g r a m i s s a m p l e d j u s t b e f o r e

a s i n g u l a r i t y , c o n v o l u t i o n may b r e a k down w i t h t h i s u n r e p r e s e n t a t i v e

v a l u e . T h i s e f f e c t wa s n o t s e r i o t i e f o r t h e p a r a l l e l l a y e r c a s e ,

b u t d i d c a u s e p r o b l e m s i n t h e d i p p i n g l a y e r c a s e .


As c a n b e s e e n f r o m F i g u r e ( 3 . 2 1 )

i s v e r y s n a i l f e r

T, 4 4 7S (»•«>w h e r e 7J* — 7 ^ ^ ^ f OO

T h u s t h e I n t e g r a t i o n i n e q u a t i o n ( 3 . 7 l ) i s d e n e b y

n u m e r i c a l t e c h n i q u e s e n * t h e i n t e g r a l

S f ' r 1 ■=■ t x / m - c ; « / g j c l t *c o = f ' f c * - e) “ t oJin,0 ( 3 . 7 5 )

+ ^ f / r - z O

J t, + C * I f c - i f j - u f r ' ) H (x -£ )< U *

« X , + ( 3 ' 76>*V9 3

N u m e r i c a l i n t e g r a t i o n o f a n d 2*c w e r e c a r r i e d

o u t a t i n t e r v a l s A t u s i n g G a u s s i a n q u a d r a t u r e . Tg i s

c a l c u l a t e d a s f o l l o w s , p u t t i n g

£ ( * ) = C * - C> V C , .( 3 . 7 7 )

J t ( t ) = £ ( &)

T h u s we h a v e n u m e r i c a l l y t o i n t e g r a t e t h e i n t e g r a l Xg


Figure 3

.21

.The behavior

of the

function E

(t) -

see equation

(3.7

2).

04ro

m

o

t o

zato


0.0

04

r

- 78 -

^ 8 = C \ Jr,

( 3 . 7 8 )

( 3 . 7 9 )

N.B. 7 ^ i t r a p l a t e d by T i f t

- c f Z ^ e) f o j f t - e j u t

= i + i*I n P » t £ ' » t - C a ( 3 . 8 0 )

i x . - U j M u r

/ e j / t ' / d ? ' ( 3 . 8 1 )

" T j ^ ^ ( 3 . 8 2 )

I n J 3 p u t C ' = £ /C â .~ ^ ° )T¥ p u t t / ( T , . Qa n d i n


- 79 -

= - c f ' [ £ & } f y M °ccu t T* - r* )

- - c f r - Z) , . J r ~ c [•* ■ # ) f j O l - t y *S i m i l a r l y

\ » c( T>-c:) ( '[ £ & } '° J £" o£r

. ' ■ ^ - c G - Q f c g C w 1' } * ' ° t c '

- c[* (V )i° jlT* -c4 - J'CT') ]* Cpift.) U j / Z -4.1 - uj / T'~ÎJ

N a t e t h a t t h a f i r a t a a n s t a n t i n ( 3 , 8 6 ) - ““ ^

• •■ h - - c a - Q l t e l . t r W ' " '

+ £ [ - * ( $ ,0j / ( T* - f y (T> - rJ IT h a a a i n t a g r a l a a r e f a u n d by G a u a a i a n q u a d r a t u

( 3 . 8 3 )

( 3 . 8 4 )

( 3 . 8 5 )

( 3 . 8 6 )

( 3 . 8 7 )

r e f a r


- 8 0 -

t h e kernel function ^ f e l l e w i n g B e r t h e d - Z a b e r e w s k i (1952)

f f 6 * ) **i f f c ) ( 3 - B 8 )

Varieus in t e r y a l s ^ ^ and value* af 7 and 7~f near war# utad and tha r e s u lt in g seismegrams eempared

ta anaura th a t tha in ta g ra t ia n was s t a b le .

3 -(lC Praoras Time EstimateTha time T required ta generate a synthetic seisms-

gram up ta the s u i t e i s faund fram the time i ttakas ta aa lau la ta - S jn *t a given time andfar a given ray in tha l e v e l 4 and 5 subreutines . The

upper l e v e l s a f tha pragraa are e s s e n t i a l l y supervisary as they sa t parameters - a preaess invalving very l i t t l e

eemputer t im e.Using tha r e s u l t a f a 30 minute eemputer run T

was faund t# be21 7 ~ — & & £ 0 * 0 ! seaands

H .n . . T - M * J l ( A + $A * o v

where N i s the number e f time paints and + i s thenumber e f rays which are net k inem atically equivalent in

a su ite a f erdar k.


- 81 -

3 1 ) Tht T h t i n t l t i l and Expt>l»tnttl Se law qramFigures (8 .2 8 ) and (3.2{|) d*a*nitratc the agreement

ebtelned between the twe seismegrams in the time in ter v a l preceding the a r r iv a l e f the aurfaee Rayleigh wave.

This p e r t ien e f the seismegram exh ib its a number ef in te r e s t in g fea tu res as the seuree receiver separatien increases . The nature e f the seismegram changes frem a wide-angle r e f l e e t i e n threugh refraetien type te a leaking nede type.

With in creas in g distance frem the seuree the head waves emerge frem behind the Rayleigh wave. They gradually increase in r e l a t iv e amplitude te ferm the leaking medes as the arr iv a l time e f PP and i t s multiples cenvergej (Figure (8.24))» the bettem experimental and synthetic seismegrams at a CCUrte receiver/flayer thickness ra t le (denete D /L ) e f 1 0 ex h ib its elements ef a l e a k i n g r i e d e

seismegram.The PL leaking medes which represent e a r l y a r r i v i n g

refracted events which have been multiply r e f l e c t e d b e t w e e n

the base e f the lnyer and the free surface (Laster e t al (1965)) are e v id e n t . This i s e e n f i r m e d by F i g u r e (3.2&.) which shews the sy n th e t ic seismegrams at i n c r e a s i n g D/L beeeming highly e s e i l l a t e r y with leaking medes.

The behavieur ef the d irect ly re f lec ted wave amplitudes with increasing distance i s shewn in Figure


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Figure 3

.25

. The

devplopempnt

of leaking

and sh

ear ‘

modes

- 8 2 -

( 3 . 2 6 ) . N o t e t h a t t h e a m p l i t u d e s e f a l l t h e w a v e s e x c e p t

p u r e P a n d p u r e S d r e p e f f r a p i d l y w i t h i n c r e a s i n g D / L . T h i s

i s t e b e e x p e c t e d a s t h e r e f l e c t i e n c e e f f i c i e n t s s h e w t h a t

a P S x e f l e o t i e n a t t h e i n t e r f a c e e r s u r f a c e a p p r e a e h e s z e r e

b e t w e e n t h e P a n d S w a v e s .

A t s m a l l e r D / L , t h e h e a d w a v e ( e . g . P i P g S ^ ) * n d

d i r e c t r e f l e a t i e n a r r i v a l ( e . g . P ^ S j ) o f t h e s a m e w a v e t y p e

i n t e r f e r e i n t h e s e - c a l l e d " i n t e r f e r e n c e z e n e 11 • T h e

s e i s m e g r a m w a s e a s i l y g e n e r a t e d i n t h i s z e n e a n d e x a m p l e s

e f w a v e - f e r m s i n t h e i n t e r f e r e n c e z e n e a r e i l l u s t r a t e d

F i g u r e ( 3 . 2 7 ) . I n c e n t r a s t t h e e r e t i c a l e v a l u a t i e n s i n

t h r e e d i m e n s i e n s u s i n g e t h e r t e c h n i q u e s i s e x t r e m e l y

t e d l e u s ( C e r v e n y ( 1 9 6 2 ^ ) ) .

s e i s m e g r a m s i s m a s k e d b y t h e R a y l e i g h a r r i v a l i n t h i s

m a d e 1 s e t h a t F i g u r e ( 3 . 2 7 ) w a s t a k e n f r e m s y n t h e t i c

s e i s m e g r a m s f e r a n i n d i v i d u a l s u i t e .

t h e e r y a n d e x p e r i m e n t b e y e n d t h e R a y l e i g h a r r i v a l ( F i g u r e s

3 . 2 8 ) a n d ( 3 . 2 9 ) ) . F r e m t h e s y n t h e t i c s e i s m e g r a m s f e r

e a c h s u i t e i n t h i s r e g i o n t h e w a y i n w h i c h t h e R a y l e i g h

w a v e b e c e m e s i n c r e a s i n g l y d i s p e r s e d c a n b e s e e n . I t

w a s f e u n d t h a t e a c h s u i t e e f r a y s c o n t r i b u t e s e n e h a l f

e s c i 1 l a t i e n t e t h e R a y l e i g h w a v e a s s c h e m a t i c a l l y s h e w n

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

U n f o r t u n a t e l y t h e i n t e r f e r e n c e z e n e i n t h e s e

G o o d a g r e e m e n t w a s o b t a i n e d f o r s m a l l D / L b e t w e e n


Reproduced with permission of the copyright owner Further reproduction prohibited without permission.


SOURCE - RECEIVER SEPARATION IN CM r o

iiiii>►f■

l V l N 3 W I H 3 d X 31VOII3HQ3H1

Reproduced with permission of the copyright owner. Further reproduction prohibited wfwithout permission.

• FIqure 3

.28

. A

comparision

of th

eoretical

and experim

ental seism

ograms

beyond the

Rayleigh

wave arriv

al.

Layer thickness

= 8.0cm

.

2 5 0TIME I N MI CROSECOi

~-1 0 a r f l 5 ‘ 2 Q - e x p e r i m e n t a l s e i s m o g r a m s f o r t h e p a r a l l e l l a y e r c< ** v e l o c i t y a n d d e n s i t y ct y d e t a i l s , h d e n o t e s l a y e r t h i c k n e s s

Reproduced w«p p e n s i o n oV,Pe copyrlgh, owner. pullher reproduct|on prohiblled w#tou, pem iss|on

.’RO SECONDS

L a y e r c a s e s h o w i n g R a y l e i g h w a v e d i s p e r s i o n . S e e f i g u r e ( 3 . s s

Reproduced with permission o tthe copyright owner Further reproduction prohibited w i t h o u lp e r m is s ^ T " .........

- 0 3 -

F i g u r e ( 3 . 3 0 ) . T h i s e f f e c t c a n b e <ir a<:r5. b e d f r o * t h e

t h e o r y b y c o n s i d e r i n g t h e s u i t e o f o r d e r 2 f o r e x a m p l e .

T h e t i m e - < £ > p a t h s i n t h e c o m p l e x p l a n e F e r t h e r a y s P P ,

PS o r S P , S S p a s s i n t h e v i c i n i t y o f t h e S t o n e l e y a n d

R a y l e i g h w a v e p o l e s ( s e e F i g u r e ( 2 , 7 ) a t d i f f e r e n t t i m e s

t o f o r m t h e h a l f o s c i l l a t i o n b y m u t u a l i n t e r f e r e n c e .

W i t h i n c r e a s i n g D / L t h e p a t h s o f a l l t h e s u i t e s a p p r o a c h

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

i n a g r e e m e n t w i t h t h e e x p e r i m e n t a l r e s u l t s ( F i g u r e ( 3 , 2 8 ) ) ,

A t l a r g e r d i s t a n c e s t h e p r o g r a m f o r g e n e r a t i n g*

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

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

n u m e r i c a l l y n e a r t h e p o l e ' .

F i n a l l y a n ' i m p o r t a n t f e a t u r e o f t h e t h e o r e t i c a l

s e i s m o g r a m i s i l l u s t r a t e d i n F i g u r e ( 3 . 3 l ) , E a c h s e i s m o

g r a m c a n b e b r o k e n i n t o t w o p a r t s c o r r e s p o n d i n g t o r e a l

a n d c o m p l e x a n g l e s o f p r o p a g a t i o n . I t i s t h e l a t t e r t y p e

t h a t g i v e t h e s m o o t h n o r m a l m o d e o s c i l l a t i o n s . T h e s h a r p

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

S h e r w o o d ( 1 9 7 0 - p e r s o n a l c o m m u n i c a t i o n ) h a s a l s o

g e n e r a t e d s y n t h e t i c s e i s m o g r a m s u s i n g t h e s a m e t e c h n i q u e — •

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

e e i s m o g r a m s a r e i l l u s t r a t e d i n F i g u r e s ( 3 , 3 2 ) a n d ( 3 . 3 3 ) ,


r c e

/

/

Suite of order 2 giver 1st* half o s c i l l a t i o n

\S u i ^ e o f o r d e r 4g i v e s 2 n d . h a l f o s c i l l a t i o n

//

R a y l e i g h wave

S u i t e o f o r d e r 6 g i v e s 3 r d . h a l f o s c i l l a t i o n

Suite of order 0 produces Rayleigh wave onset - equivalent to Lanb's problem

Figure 3 .30 , The d i s p e r s i o n of t h e R a y l e i g h w a v e i n a l a y e r over a h a l f s p a c e . See a l s o f i g u r e (3 .31).


3<+

<+u>oO3TJO303r+

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* D i s t a n c e

.3168

.6498

1.4531

3.9252

S 2.0000

2.7528

F i g u r e 3 . 3 2 . I m p u l s i v e r e s p o n s e 1f o r a t a n g e n t i a l l i n e f o r c e s o n r c ' e on t h e s u r f a c e o f an e l a s t i c p l a t e S e p a r a t i o n s i n d i c a t e d a r e n o r m a l i z e d t o l a y e r t h i c k n e s s ,

d e n s i t y d o e s n o t i n f l u e n c e s e i s m o g r a m * p e r s o n a l c o m m u n i c a t i o n )

( S h e r w o o d '


'-A

4

Z

0

f l l i u r * 3 . 3 3 . As f i g u r e ( 3 . 3 2 ) b u t w i t h a n o r i n ^ l f o r c ^ . ( S h e r w o o d p e r s o n a l

c o m m u n i c a t i o n )


- 8 4 -

4 . THE D I P P I N G AND CURVED LAYER CASES The f i f a t * case rep resen ts s g e n e r a l i z s t i e n ef the

th eere t ic s l technique used in Chapter 3 . The se lu t ien te this preblem i s merely a further ap p lica t ien e f the transmissien f a s t e r methed at i n t e r f a c e s .

'As the g en era l ized rays pregress threugh a medium we change the eeerd in a tes <2^) f*em which thedireetien e f the rays are defined t e new eeerdinates

( j X * w*'th a new d ir e e t i e n parameter .The 2 / * ) eeerd in ates are aligned with the

i**1 in te r fa c e . As a r e s u l t e f applying Snell* s Law the dependence e f S9 the e p t i c a l t r a v e l time ef a generalized

ray, en 2 ^ d isappears.

4.1 The S e lu t ien to the dipping layer caseCensider the 2-D wave with amplitude A er ig inating

at in the medal shewn in Figure (4 .1 ) where, asusual the p e s i t i v e d i r e e t i e n i s taken as dawn. As p rcv ieu s ly , the wave f i e l d in the f i r s t layer can be represented by

o (* - a - > o£ °°A o

. h .* . 1 . . . n . f u n . t i . n . f ^ /* '* ' c o s

ike wave may be represented as* C - Jfoc - Q -+-£&- Cos_&0? ?

c j? = 0/0 j o 4 <r ' * J


F i g u r e 4 . 1 . T h e g e o m e t r y o f t h e d i p p i n g l a y e r p r o b l e m .


- 8 5 -

where represents the eperatien

O f? = 5 > <g) ( 4 . 3 )

•New w h e n t h e w a v e r e a c h e s t h e i n t e r f a c e a t ( oct>iL*)

* 1 ** 4 . 4 )

1S u p p e s e t h a t t h e w a v e i s t r a n s m i t t e d t h r e u g h t h i s

i n t e r f a c e , t h e n_ . ~ ^ -t.Cfa-x?) SjJZ&t (& - 2?,)CO* fy Z

* ) = O /o fo ., <. t , J y r 1 ‘3

X f f A i n f a * #Tj ( 4 . S )

J R b e i n g t h e t r a n s m i s s i e n c e e f f i c i e n t w h i c h d e p e n d s e n

t h e a n g l e (&0 + t e t h e n e r m a l e f t h e i n t e r f a c e . N e t e

t h a t i s f e u n d f r e a

s m (& e + r f ) ~ S/*> ( @f + ? 0 ( 4 . 6 )

Co c *

a n d t h a t we h a v e p u t

- i f f a - = * 0 + (* ,-K )£ £ ± Q > ?a = c t - e ‘ c° c° J ( 4 . 7 )


- 86 -

E x p r e s s n * w c e e r d l n a t e s (p^/j z /* )

w h i c h a r c i l l u s t r a t e d i n t h e F i g u r e ( 4 . 2 ) .

x t =• X , ' c o s f i -

, . , j ( 4 * 8 )* ( * x t $ / » + zt C O S

P u t t h e s e u r e e s t t h e e r i g i n .

HT s i C -x s * *>(9, -f. M Sos (9, I

* * - * / * ' “ H r * ) * - ^ i ^ 4 . 9 )

- ' t f 'X l L n f l + * c o s &t? n ^ CL 4 6 c t c * r<

* Gxp I ~ [(*1 ? ^ +c<p&o *t* <$)- 0/ 0 ^ I

^ c o s pf - s / s iQ I

— &t ccsrf - s /^ ^ i s " > * )]J 1

a . * I c , * * R

x-ex/o f -c fx '(s/n(Q>+d) — s/r> (&, *■ rf))/ + Z ,' /c os (& t - J ,)~ ro s ^ g »/0 ) ? ( 4 , n )

H e n c e n e t i n g t h a t

&*’+ $ ') " ( 4 * 1 2 )

c /


F i g u r e 4 , 2 , R o t a t i o n o f r a y « o - o r d i n e t e * o f r e f e r e n c e .


- 87 -

then

f f c * ) ~ ®p\ ^ - if e .‘(c°s( $ .» r i) - Cos/'S,*d))f I (4.13)

where

O /o fie . J (4*14)

and that 3t/ - O represents the in ter fa ce . /*( i sa eenstant value equal te the perpendicular frem the erig in

te the in te r fa c e .This preeess nay be eentinued up te • • that

we end up with

S j

(see reaaen fer a lternate signs an the next page) where the $zT h j *nd C j are given and the ^ are

related te ^ by

s / / ? / < % 7 ^ ) - s /s ) ( f y - l O - ^

C° . C* (4.16)S /r>f& , *• <$J = S / s ( ' Q- f C ) ~ ^

cx


- 08 -

H e n c e t h i s i n t e g r a l c a n b e e v a l u a t e d b y t h e " S h e r w o o d -

C a g n i a r d " t e c h n i q u e b y t h e u s u a l r e q u i r e m e n t s t h a t

^ - ^ £ £ £ < 8 -f. F t /y/“>*(%-. cos (&,Z $ jjl ( 4 . 1 7 )

* J 3 Cj" = O ‘J ' JT h e a l t e r n a t e s i g n s b e f a r e A i . w r e f e r t e t h e f a c tJ> J

t h a t t h e w a v e may b e g o i n g i n a p o s i t i v e o r n e g a t i v e d i r e c t i o n s .

S u p p o s e a l l — &(/ . ♦ u

T h e n s t h i c k n e s s o f f i r s t l a y e r — n t

— /o COZ @3T -

j.= t ,

( 4 . 1 8 )C O S

— / - ( &¥ - £ f f

COS <9j_, ( 4 . 1 9 )

j - t S ' - 'w h i c h i s t h e s a m e r e s u l t g a i n e d i n t h e p a r a l l e l l a y e r c a s e .

T h e g e n e r a l s o l u t i o n f o r a r a y W i t h ^ s e g m e n t s i s

t h u s s e e n t o b e g i v e n b y


- 89 -

(P 0 CK->3 >

~ C Jfe. J «C<oj (4-20)

w h e r e T ( o $ = T [ R, ( ^w h e r e R t f r ) a r e t h e r e f l e e t i e n / r e f r a e t i e n c e a f f i c i e n t s

a t t h e i n t e r f a c e <*

S - x„,t im -Q 't •* h**t ggJ -^*2 tf/<

- s ,r i Q - ^ £ £ £ _ ^M ** * U . M )

J

w h e r e Ia/ ^ a n d x e s p e e t l v e l y l e f e r t # w h e t h e r t h e j j fave

I n t e a n d Wave O u t f r e m a n I n t e r f a c e 1« w a v i n g i n t h e p e e i t i v e

( W = + 7 ) e r n e g a t i v e - ** d i r e e t i e n .

A c l e s e d f e r n a e l u t i e n e f e q u a t i e n ( 2 0 ) i s f e u n d

i n p r e c i s e l y t h e s a n e way a a t h e p a r a l l e l l a y e r e a s e .

r # r e x a a p l e f r e n ( 4 » 2 0 )

■ u # = f s / n 0 ^ 1 ~ T fe .') ?x I CK A fa ) Jl>r - <\co

ind A fÔ " ^


4 . 1 1 B e h a v i o u r o f t h e S o l u t i o n

F r o m t h e d e r i v a t i o n we c e e t h a t i f we p l o t u p t h e

c o m p l e x p l a n e , t h e n t h e p o s i t i o n o f t h e b r a n c h p o i n t s

c o r r e s p o n d i n g t o co s ~ ® ( 4 . 2 2 )

d e p e n d on ^ , t h e a n g l e s o f t h e l a y e r s j t o t h e h o r i z o n t a l .

H e n c e h e a d w a v e a r r i v a l s d e p e n d ( a s e x p e c t e d ) o n t h e

“d i p s o f t h e - i n t e s r f a c e s .

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

t o t h e v a l u e o f *t> ~ w h e n

( 4 . 2 3 )

F i n a l l y t h e e x i s t e n c e o f R a y l e i g h a n d S t o n e l e y w a v e s

a r e d e t e r m i n e d b y t h e l o c a t i o n o f p o l e 3 i n t h e c o m p l e x

0 - p l a n e g e n e r a t e d b y t h e d e n o m i n a t o r s o f t h e

r e f l e c t i o n / r e f r a c t i o n c o e f f i c i e n t s a t e a c h i n t e r f a c e .

As o p p o s e d t o t h e p a r a l l e l l a y e r c a s e t h e d i p p i n g

l a y e r c a s e i s n o t s y m m e t r i c w i t h r e s p e c t t o s o u r c e a n d

r e c e i v e r . H e n c e t h e d e g e n e r a c y e f d y n a m i c s r .o rt -rr..-; .',r. I l y* 1

e q u i v a l e n c e o f v a r i o u s r a y s i s l o s t . C o n s e q u e n t l y f a r

m o r e c o m p u t i n g t i m e i s r e q u i r e d . t o b u i l d u p s e i s m o g r a m s t o r

a g i v e n n u m b e r o f r a y s .

N o t e t h a t a l t h o u g h t h i s s o l u t i o n r e p r e s e n t s a n e x a c t

c l o s e d e x p r e s s i o n , i t d o e s n o t a c c o u n t f o r a n y w a v e s s c a t t e r e d

b a c k f r o m p o i n t s w h e r e l a y e r s m e e t .

~ T h e e x p r e s s i o n f o r t h e d i p p i n g l a y e r e a s e i s e a s i l y


- 9 1 -

p r e g r a m m e d b y a s i m p l e e x t e n s i e n a f t h e p l a n e p a r a l l e l l a y e r

e a s e •

4 . 2 M e d l f i e a t l e n a f t h e p r c q r a n f a r t h e D i p p i n g L a v e r Cas*

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

s a a a . A d d i t i a n a l s u i t e p a t h p a r a m e t e r s } and

t h e d i p a n g l e 9 a r e r e q u i r e d t a r e l a t e t i m e ~C an dV

The v a r i a b l e ^ i n t h e p l a n e p a r a l l e l l a y e r e a s e b e s a m e s

a v a s t e r w i t h i t s e e m p e n e n t s c a r v e s p e n d i n g t a

= s " * ( &+ -*■ ^ A ) /< *( 4 . 2 4 )

a t a a a h i n t e r f a c e .

T h e e q u a t i a n s

f f a / j - - t

3 °

( 4 . 2 5 )

a r e m a r e a a m p l i e a t a d a s S n a l l , s Law new ê / j

by

cj J v

' ,

w h e r e a r e t h e a n g l e b e t w e e n t h e r a y i n t h e j s e g m e n tv/

and t h e v e r t i c a l a x i s


- 9 2 -

4 . 3 Theeretiaal and e x p er l i» n t> l t e l s i t o r m s f a r t h e d l p p l n g

layer i«»»

Plgurea ( 4 ,4 ) and (4 .5 ) i l l u s t r a t e the agreement between theery and p raat iae far the e n t ir e seismegrant wtth

the Rayleigh wave deminant far the medel e f f igure ( 4 . 3 ) .

As the la te r e s e i l l a t i e n s e f the Rayleigh Steneley waves h a v e

enly r e l a t i v e l y small amplitude the pregram eeuld hand..* t h e m

numerieally r e s u l t in g in the e x s e l le n t agreement shewn,Befere the Rayleigh #ave9 aenvelu tien areund d ire s t

arrivals far large separatien break down fer the reasens

given in the f e e tn e te e f s e e t ie n 3 , 9 . Even s e 9 a g r e e m e n t

i s s t i l l f a i r l y geed eut te a seurae re se iv er s e p a r a t i e n e f

3 1 . 0 sms., beyend whlah the large number e f d irest - r r i v a l s ,

head waves and in ter fex en ee zeftes beeeme tee m u s h f e r t h e

pregram te eepe with numerieally.

4 .4 The Curved Laver - CaseCensider the plane wave frent

f = £>) *

str ik ing a aurved in ter fa a e in F i g u r e ( 4 . 6 ) . T h e p l a a . v

represents the e igen fun atlen s used prwvieuely in t h e p l a n e

and dipping layered e a s e s .In erder te e a leu la te the farm e f th is wave a f t e r


R e c e i v e r m o v e s o v e r t h i s t eS o u r s e

8 . Ocm

0 . 1 9 6 r a d i a n s

F i g u r e 4 . 3 . G e o m e t r y o f t h e m o d e l u s e d i n t h e e x p e r i m e n t .


SOU

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FI n u v r . Tit r rt r r t i c. a ) r. r \ no g r amr, f o r

t h r d l p o l n y l ayer c n r- r t V/* I..? c I t l *'r and <Jr r»'-i 1 1 r r. ar. i n f i g u r e ( 3 , P.P.).

N,B . A c o p y o f t h i s d i a g r a m i s i n s i d e

b a c k c o v e r

150-0100-0'■>0.0Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

■ u >

0 0 0 100.0 IOUU 200.0TIME ID MICROSECONDS

F i g u r e d . d b . l - x p e r i m o n t a l s e i s m o g r . m s f o r t h e d i p p i n g l a y e r c a s e N*B. A c o p y o f t h i s d i a g r a m I s i n s i d e b a c k c o v e r


SOURCE - R E C E I VE R REPARATI ON I N CM;r\>

2 at• 300 a•

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Figure 4 . 6 . P l a n e waves s t r i k i n g a curv e d i n t e r f a c s

R e c e i v e r

F i g u r e 4 . 7 . E x a m p l e s o f d i r e c t l y r e f l e c t e d a r r i v a l s .


- 93 -

r e f l e c t i o n we h a v e t e e a l e u l a t e t h e t r a n s m i s s i o n f a c t o r

a p p r o p r i a t e t o t h e a n g l e o f i n a i d e n o e b e t w e e n w a v e a n d i n t e r

f a c e * Ac t h i s a n g l e i s n o t c o n s t a n t ) t h e p r o c e s s u s e d p r e

v i o u s l y i s n o t a p p l i c a b l e ) s o we r e s o r t t o a p p r o x i m e r j . s n «

F o r a r a y w h i c h p r o p a g a t e s a c c o r d i n g t o F e r m a t ' s

P r i n c i p l e - e . g . t h e d i r e c t l y r e f l e c t e d a r r i v a l s o f F i g u r e s

( 4 . 7 ) - t h e t r a n s m i s s i o n f a c t o r f o r t h e w a v e ( 4 . 2 7 ) i s e a s i l y

a p p r o x i m a t e d . N o t e t h a t t h e p r o p e r t y o f r a y s a c c o r d i n g t o

F e r m a t ' s P r i n c i p l e i s t h a t t h e y m o v e i n l i n e s n o r m a l t o t h e

w a v e f r o n t ( K l i n e & K a y 1 9 6 5 ) ) . T h u s we t a k e t h e p l a n e w a ’ e

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

e l e m e n t o f t h e w a v e f r o n t a l i g n e d a l o n g t h e n o r m a l f r o m t h e

s o u r c e .

a n d e v a l u a t e a s p r e v i o u s l y ) w h e r e S 9 t h e o p t i c a l t r « v . r> -

i s g i v e n b y

T h u s f o r t h e d i r e c t a r r i v a l

^ 4 . 2 8 )

' . 4 . 2 9 )

b y f i g u r e ( 4 . 8 ) .


C e n t r e o f c u r v a t u r e o f i n t e r f a c e

S o u r c e ^ X R e c e i v e r

F i g u r e 4 . 8 . The g e o m e t r y o f t h e c u r v e d l a y e r c a s e


f i g u r e 4 . 9 . The s i n g l e v a l u e d n e s s o f 3 f o r t h e h e a d w a v e s i n t h e

p l a n e p a r r a l l e l l a y e r e d c a s e


- 94 -

A t t i m e s e t h e r t h e n t h e d i r e e t l y r e f l e c t e d a r r i v a l

we h a v e t e m a k e a n u m b e r e f a d h e e a s s u m p t i o n s . F i r s t n o t e

t h a t S i n t h e p l a n e a n d d i p p i n g l a y e r e a s e s w a s s i n g l e v a l u e d

e v e n f e r t h e v a r i o u s h e a d w a v e s i l l u s t r a t e d i n F i g u r e ( 4 . 9 ) .

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

( F i g u r e ( 4 . 1 0 ) ) . F e r t h e h e a d vtrave f o r m e d a t t h e c u r v e d

i n t e r f a c e , f o r e x a m p l e , we c a n d e f i n e a f u n c t i o n f r o m

t h e o p t i c a l t r a v e l t i m e f u n c t i o n S a n d t h e p l a n e wa ve

r e f l e e t i o n / r e f r a c t i o n c o e f f i c i e n t s .

4 . 5 S t u d y o f a H e a d Wave

C o n s i d e r t h e r a y ( l o c u s o f a n e l e m e n t P o f t h e w a v e

f r o n t ) i n F i g u r e ( 4 . 1 l ) . I t h a s t h e g e n e r a l f o r m

f(P) - ( 4 . 3 0 )

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

t h e o r y s u c h a s K e l l e r ( 1 9 5 8 ) .

s f r ) t a k e s v a r i o u s s p e c i f i c f o r m s d e p e n d i n g u p o n

w h i c h s e g m e n t o f t h e r a y p a t h i s b e i n g c o n s i d e r e d .

A ( p ) = £ ( & )

$ C P ) = (?C - ^ O +■ (2 - S . ) C O S & o ' ( 4 . 3 1 ) .C + C o

P o** 5


S o u r c e B e c e l v e r

F i g u r e 4 . I O f h e m u l t i v a l u e d n e s s o f S f o r t h e c u r v e d l a y e r e a s e .

f i g u r e 4 . I I T h e p a t h o f t h e g e n e r a l i z e d r a y e o n a l d e r e d i n s e c t i o n 4 . 5 .


- 95 -

a ( p ) = £ ( 4 ) ^ < : 4 >

5 (p) = R tf + (rx,-x^SjrJ>t + fa -*')££*& (4'32)C n* 0

P o« BC

A ( p) = » * .* (* )N ( 4 . 3 3 )S s /n - <aa J c <?.s

A / ? <$ ~b 0r ) s*” Q + f r . ~ P o~

F u r t h e r m o r e a s a r e s u l t o f S n e l l ' s l a w

S P j r ^ ( 4 . 3 4 )

‘ *f. ' • ^

so t h a t ( f r o m e q u a t i o n ( 2 . # 4 ) a s w‘s>S Q ^ ^

w h i c h i s th u .s i d e n t i f i e d a s t h e r e f l e a t i e n c o e f f i c i e n t .

T h e a b e v e e x p r e s s i e n f a r S(fi) f a l l o w s t h e g e . m . t r i s

o p t i c s a p p r o x i m a t i o n .

S i m i l a r l y $ e o u l d b e d e f i n e d f a r t h e o t h e r two

h e a d w a v e s a f F i g u r e ( 4 . 1 1 ) and i n f a c t a l l g e n e r a l i z e d r a y s

whi Gh a r e p o s s i b l e i n b e t w e e n .

CO

\


- 96 -

4 . 6 The M u l t l v a l u c d n e s s > f t h e O p t i c a l T r a v e l T i m * F u n c t i o n

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

s i d e r t h a t S b e e e m e s a m u l t i - v a l u e d f u n c t i o n S f o r a g i v e n &

a n d m e a s u r e ^ • T h e s o l u t i o n t o t h e p r o b l e m i s t h u s g i v e n

by i n t e g r a t i n g o v e r a l l t h e p l a n e w a v e f u n c t i o n s % a s b e f o r e ,

= & f f T f r ) f y + J ( 4 . 3 6 )

♦

^ ^ f 7 ? ? ) £ ^ £ J Z ) } ( 4 . 3 7 )

w h e r e , a s p r e v i o u s l y f = 2 £ '

a n d I *

4 * ~ ( S» ) S 0 ( 4 . 3 8 ) ^

j & . ( % > } ' *T h u s

,u ( * > * ,* ) - fo

w h e r e ~ c L Mf y £ ) i s t h e m e a s u r e o f V a t a g i v e n t i m e o v e r

t h e p a t h d o m a i n D,

N o t e t h a t

( 4 . 3 9 )

By a s s u m i n g t h a t o n l y s i n g l e h e a d w a v e s e x i s t , t h a t


- 9 7 -

i s a generalized ray tan t r a v e l en ly an arb itrary distance aleng ene in t e r f a c e , then

e C / f f y 0 • Z(*J, O > j **, 2, 3 . - . (4.40)

y aerrespends te s i n g l e head wave pathsThat i s the measure e f a l l paths eerrespending te ndeublen head waves i s z e r e .

However, i t sheuld be p a ss ib le t h e e r e t i e a l ly te find Cfc/v/v, b y i n tegrating H4. ever a l l space c e v e r e d b y

a p a r t i c u l a r wave a t a l l in te r v a ls e f C . T h i s weu Id b e

e x t r e m e l y d i f f i c u l t .Nete th a t in the ease ef plane layers

(stn &) - f 0 ‘" (4.41)

a n d W ft* ) ■ V

where ^ denetes the varieus paths fer which 5^ ~ S g iv e n £

Henee

< £ > * & £ * ) ( 4 . 4 8 )

which i s a m ult i-va lued fu n ction .The problem of deciding which value o f j to take

i s decided h e u r i s t i c a l l y on the basis of each head wave


arrival. Thu* i f f y becomes f in i t e at C“ * £ t , f or y s X j £

for j s «2 e t c . then,

- (4*48>

Eeeh value of j defines a sub-suite . Thus a suite of order K has /< «* sub-suites eorresponding to the nunber of possible head wave paths.

Consider the su ite or order 2. At a d irest arrival a l l sub-suites for that ray are the sane. The calculation of Sv for the three sub-suites is simply a natter of geometry between in ter faces (see Figures (4.1?), (4.13) and (4 .14)) and Snell* s Law. The- real parts of ^ are related by geometry and the complex parts are seen to be equal between interfaces.

For complex ^ we simply took y •* 2 as the d i f f erence between the ^ are small for th is region. For real ^ ^ was ahosen according to the saheme of equation (4 .43).

F in a l ly , we approximated higher order su ites by

simply taking the form of the direct arrival.Thus wc suppose that

~ ^ (4.44)

w h e r e C , » f £ & ') - r & 0 ^ ) 2t - * C 2 A (a> Jy


F i g u r e s 4 . 1 2 a n d 4 . 1 3 . The 1 s t * s u b s u i t e o f a s u i t e o f o r d e r 2 .The 2 n d . s u b s u i t e i s o b t a i n e d b y i n t e r c h a n g i n g S o u r c e a n d R e c e i v e r


F i g u r e 4 . 1 4 . The 3 r d . s u b - s u i t e o f o r d e r 2 ,


- 99 -

Me n o t e t h a t I n a n a l p g y w i t h t h e p a r a l l a l a n d d i p p i n g l a y e r e d

e a s e * , t h e f o r m o f t h e c o m p l e x p a r t o f t h e Q e o n t o u r w i l l

d e p e n d o n l y w e a k l y o n t h e i n t h e s o l u t i o n o f

^ ( 4 . 4 4 )

H e n e e p u t V * i a e

f o r a l l l) ( 4 . 4 5 )

Re f i n d a l l b y a s s u m i n g t h a t

t h a t i t f o r Q f r o m a b o v e

t h e a r e f o u n d f r o a i

a t i n t e r f a a e s

Cb e t w e e n i n t e r f a a e s

e n d t l a g i v e n b y

( 4 . 4 6 )

( 4 . 4 7 )

1

«v U . 4 » >

( 4 . 5 0 )


- 100 -

4 . 7 T i i p o r r t l c a l a n d E x p e r i m e n t a l S e l s m o g r a m s f o r t h e C u r v e d L a y e r _Cas_e_

F i g u r e s ( 4 . 1 5 ) t o ’ ( 4 . 1 8 ) i l l u s t r a t e e x p e r i m e n t a l s e l s m o g r a m s

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

c h a r a c t e r i s t i c s a s t h e d i p p i n g a n d h o r i z o n t a l p l a n e l a y e r e d c a s e s .

The s e i s m o g r a m i n t h e t i m e i n t e r v a l s p r e c e d i n g t h e

R a y l e i g h wave i s m o r e c o m p l i c a t e d , a s e x p e c t e d , d u e t o t h e l a r g e

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

of head w a v e s . I n p a r t i c u l a r a c o m p a r i s i o n - b e t w e e n t h e c o n c a v e

up and c o n c a v e do wn c a s e s d e m o n s t r a t e t h a t t h e l e a d i n g wave

i s r e l a t i v e l y l a r g e i n t h e c o n c a v e down c a s e . T h i s c o u l d h a v e b e e n

a n t i c i p a t e d a s h e r e t h e w a v e i s a d i r e c t l y t r a n s m i t t e d a r r i v a l . I n t h e

c on cav e u p c a s e i t i s e i t h e r v e r y w e a k o r n o n - e x i s t e n t . The c o n c a v e

down e x p e r i m e n t a l s e i s m p g r a m s r e v e a l much more d e t a i l d u e t o o t h

waves l i k e " P ^ P , w h i c h - p a s s - t h r o u g h t h e - t o p o f t h e b o t t o m l a y e r .X X

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

f i g u r e ( 4 . 1 9 ) . Thete a p p e a r s t o b e g o o d a g r e e m e n t w i t h t h e e x p e r i m e n t a l

s e l s m o g r a m s o f f i g u r e ( 4 . 1 6 ) , a l l o w i n g f o r t h e s m o o t h i n g

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

v i e w o f t h e a p p r o x i m a t i o n s a n d ad. hoc . a s s u m p t i o n s r e q u i r e d

t h e o r y . C o m p a r i s i o n i s f a s c i l l i t a t e d b y i d e n t i f y i n g c o r r e s p o n d i n g

r a y s i n f i g u r e s ( 4 . 1 6 ) a n d ( 4 . 1 9 ) . F i g u r e ( 4 . 2 0 ) p l o t s t h e a m p l i t u d e s

o f t h e si pi'i P i S i » a n d S 1 S 1 r a y s * Xt Can b e SCen t h a tv a r i a t i o n o f t h e s e a m p l i t u d e s i s i n q u a l i t a t i v e a g r e e m e n t w i t h

e x p e r i m e n t . I n p a r t i c u l a r t h e P ^ w a ve a m p l i t u d e b e c o m e s n e g a t i v e

w i t h i n c r e a s i n g s o u r c e - r e c e i v e r s e p a r a t i o n a s d o e s t h i s d o m i n a n t


R e c e i v e r moves o v e r t h i s r e g i o n

1 6 . 0 em

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E lg u re 4 - T K . x h e g e o m e t r y o f t h e e u r v e d l a y e r e a s e c o n c a v e u p

®ee f i g u r e ( 3 . 2 2 ) f o r v e l o c i t i e s and d e n s i t i e s .


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F i g u r e 4 o I 7 T h e g e o m e t r y o f t h e c u r v e d l a y e r m o d e l c o n c a v e d o w n .

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i : i o u r o 4 . 2 0 Tho v a r i a t i o n o f t h r a m p l i t u d r * . o f t h r d l r r c t l y r c f l r c t r d

a r r i v a l * o f orcl*>r 2 I n t h r c o r v o d l a y e r c a ? c v e r s u s s o u r c e - r e c e i v e r

s e p a r a t i o n *


a r r i v a l i n e x p e r i m e n t * E v e n t h e h e a d w a v e s s e e m t o b e r e a s o n a b l y

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

do m a t c h e x p e r i m e n t a l v a r i a t i o n s a p a r t f r o m t h e d i r e c t l y r e f l e c t e d

a r r i v a l s •

A g r e e m e n t b e y o n d t h e R a y l e i g h wave a r r i v a l a p p e a r s g o o d ,

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

r e q u i r e d w i t h s o u r c e - r e c e i v e r s e p a r a t i o n *


- 10 2 -

5 . MODEL SEISMOLOGY

Two d i m e n s i o n a l s e i s m i s m o d e l l i n g i s a n e s t a b l i s h e d

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

p r o p a g a t i o n , ( N o r t h w o o d a n d A n d e r s o n , 1 9 5 3 ; O l i v e r e t a l ,

1 9 5 4 ; E v a n s a t a l , 1 9 5 4 ; R i z n i s h e n k o a n d S h a m i n a , 1 9 5 7 ;

H e a l y a n d P r e s s , 1 9 6 0 ; N a k a m u r a , 1 9 6 4 ; C o u r o n e a u , 1 9 6 5 ; e t a ) .

T he a l a s s i e p a p e r o f T o l s t o y a n d U s d i n ( 1 9 5 3 ) s h o w s t h a t

p r o p a g a t i o n i n a p l a t e a a n c l o s e l y a p p r o x i m a t e 2 - D p r o p a -Y

g a t i o n i n a n i n f i n i t e m e d i u m i f t h e m i n i m u m w a v e l e n g t h s

a r e n o t t o o s h o r t .

M o h a n t y ( 1 9 6 6 ) d e v e l o p e d a s e i s m i c m o d e l s y s t e m a t

t h e U n i v e r s i t y o f T o r o n t o w h i c h h a s g o o d r e s o l u t i o n a n d

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

b e r a t i o n s , y e t w h i c h i s s i m p l e a n d e e e n e m i c a l t o c o n s t r u c t .

V a r i o u s i m p r o v e m e n t s s u c h a s a t w i n - T n o t c h f i l t e r t o

e l i m i n a t e r e v e r b e r a t i o n , t h e i n s e r t i o n o f a n o p e r a t i o n a l

a m p l i f i e r f o r d i f f e r e n t i a t i o n h a v e b e e n m a d e t o M o h a n t y 1s

s y s t e m .

5 . 1 T h e S e i s m i c M o d e l S y s t e mi

T h e s y s t e m u s e s f r e e p r o b e s w h i c h n e e d o n l y b e

p l a c e d o n t h e t o p e d g e o f t h e m o d e l t o o b t a i n c o n s i s t e n t

s e l s m o g r a m s . C o n t a c t b e t w e e n t h e p r o b e a n d t h e m o d e l i s

m a i n t a i n e d b y a l l o w i n g t h e p r o b e s t o s l i d e f r e e l y u n d e r

t h e i n f l u e n c e o f t h e i r own w e i g h t , ( p l u s a n a d d e d w e i g h t )


103 -

o n t o t h e m o d e l s u r f a c e . S t r a i g h t f o r w a r d e a l e u l a t i e n s a r e

s h e w n t a a c c o u n t f a r t h e c h a r a c t e r i s t i c s o f t h e p r o b e s w h a n

c o m p a r i n g t h e o r e t i c a l a n d e x p e r i m e n t a l s e l s m o g r a m s .

T h e s y s t e m i s u s e d w i t h m o d e l s l / l 6 t h i n c h t h i c k

w h i c h g i v e s a

j L . ’ » . . >

m a r k i n g t h e u p p e r l i m i t o f t h e a l l o w a b l e f r e q u e n c y r a n g e f o r

a b o u t 1% d e v i a t i o n a s c a l c u l a t e d f r o m T o l s t o y a n d U s d i n ( 1 9 5 3 ) .

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

m a t e r i a l s w h i c h e x h i b i t e d o n l y m o d e s t a t t e n u a t i o n f o r t h e

s o u r c e - r e c e i v e r s e p a r a t i o n s u s e d .

T h e s y s t e m , s h o w n i n m o d u l a r f o r m i n F i g u r e ( 5 . 1 )

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

s e i s m i c m o d e l .

B a s i c a l l y t h e s y s t e m f i r e s t h e T h y r a t r o n P u l s e

G e n e r a t o r ( l ) , c a u s i n g a v o l t a g e s t e p t o b e a p p l i e d t o t h e

t r a n s m i t t e r p r e b e w h i c h c o n s e q u e n t l y u n d e r g o e s a r a p i d

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

t h e s e i s m i c m o d e l ( 3 ) . T h e e l a s t i c w a v e s a r e t h e n d e t e c t e d

b y t h e r e c e i v e r c i r c u i t b e g i n n i n g a t t h e r e c e i v e r p r o b e ( 4 ) .

Th e p r o b e c o n v e r t s t h e e l a s t i c wa ve e s c i l l a t i e n s t o v o l t a g e

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

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© v e r y 0 . I s e c o n d s b y t h e t i m e m a r k g e n e r a t o r ( 9 ) t e m a i n t a i n

a s t a t i o n a r y t r a c e c n t h e c s c i l l c s c c p e w i t h a n a c c o m p a n y i n g

t i m e c o m b .

As t h o e n t i r e s y s t e m i s l i n e a r , i n t h e o r y a n d i n

p r a c t i c e , wo may c o n c e p t u a l l y i n t e r c h a n g e t h e o r d e r o f t h e

c o m p o n e n t s o f F i g u r e ( 5 . 1 ) t o t h e t h r e e b a s i c m o d u l e s ( F i g u r e

5 . 2 ) .

( 1 ) The p u l s e g e n e r a t o r a n d t h e e n t i r e e l e c t r o n i c

c o m p o n e n t s — t h e e l e c t r o n i c m o d u l e ;

( 2 ) The s o u r c e a n d r e c e i v e r p r o b e s m a k i n g u p t h e

e l e c t r o m e c h a n i c a l m o d u l e a nd

( 3 ) t h e s e i s m i c m o d e l i t s e l f .

H e nc e t h e s t e p - f u n c t i e n p u l s e g e n e r a t o r i n p u t a n d d i f f e r e n t i

a t o r a r e c o n s i d e r e d t o g e t h e r a s a ( f r e c j h e n c y l i m i t e d ) d e l t a -

f u n c t i e n i n p u t . The a d v a n t a g e s o f a h i g h p o w e r i n p u t

p o s s i b l e w i t h t h e T h y r a t r o n P u l s e G e n e r a t o r a n d t h e s i m u l a t i o n

o f a d e I t a - f u n c t i o n i n p u t a r e t h u s b o t h i n c o r p o r a t e d . We

a l s o r e g a r d t h e e l e c t r o n i c a n d e l e c t r o m e c h a n i c a l m o d u l e s a s

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

Th o h e a r t o f t h e s y s t e m i s t h O p r o b e s . By u s i n g

d i s c s h a p e d p i e z o - e l e c t r i c t r a n s d u c e r s i d i t H a c a r b o n s t e e l

b a c k i n g r o d , w h i c h r e d u c e s r e s o n a n t r e v e r b e r a t i o n s , h i g h

q u a l i t y s o i s m o g r a m s a r e o b t a i n e d . To f u r t h e r r e d u c e t h e s e

v i b r a t i o n s t h e t w i n - T n o t c h f i l t e r ( 7 ) i s i n t r o d u c e d i n t o

t h e r e c e i v e r c i r c u i t . H i g h f r e q u e n c y c u t - o f f w a s i m p l i c i t i y


T h e e l e c t r o n i c m o d u l e c o n s i s t i n g o f t h e

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

a m p l i f i e r , d i f f e r e n t i a t o r , f i l t e r a n d

o s c i 1 l o s c o p e

T h e e l e c t r o m e c h a n i c a l m o d u l e - t h e s o u r c e a n d

r e c e i v e r p r o b e s

The t w o - d i m e n s i o n a l s e i s m i c mouel

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- 1 0 5 -

p r e s e n t i n t h e a m p l i f i e r , d i f f e r e n t l a t e r a n d n a t c h f i l t e r

t e s a t i s f y t h e h i g h f r e q u e n c y c u t e f f r e q u i r e m e n t s , ( E q u a t i e n

5 . 1 ) .

5 . 1 , 1 P r e b e A s s e m b l y

M e h a n t y ( 1 9 6 6 ) d e v e l e p e d t h e f e l l e w i n g t e c h n i q u e

f a r p r e b e s e n s t r u c t i e n • C e r a m i c P i e z e - e l e e t r i c t r a n s d u c e r s

s f t h e B a r i u m - T i t a n a t e a n L e a d - Z i r c e n a t e - T i t a n a t e t y p e s a r e

u s e d . T h e s e t r a n s d u c e r s h a v e l i n e a r s t r a i n r e s p e n s e w i t h

c h a r g e w h i c h s u g g e s t s t h a t a l e w i m p e d a n c e a c t i v a t i n g

c i r c u i t s h e u l d b e u s e d s e a s t e e n s u r e t h e c h a r g e a n t h e '

t r a n s d u c e r s i s p r e p e r t i e n a l t e t h e v e l t a g e . T y p i c a l

c h a r a c t e r i s t i c s e f t h r e e t r a n s d u c e r s c f t h i s t y p e u s e d a r e

l i s t e d i n F i g u r e 5 . 3 .

I n c e n s t r u c t i n g t h e p r e b e s p a r t i c u l a r c a r e w a r

d e v e t e d t e e n s u r i n g t h a t t h e t a p a n d b e t t e m s u r f a c e s e f t h e

t r a n s d u c e r s a n d t h e b e t t e m s u r f a c e e f t h e b a c k i n g r e d w e r e

p r e c i s e l y p e r p e n d i c u l a r t e t h e a x i s e f t h e r e d . F a i l u r e

t e e n s u r e t h i s p r e d u c e s a n u m b e r e f r e s e n a n t f r e q u e n c i e s

w i t h i n , t h e a d j u s t m e n t r a n g e e f t h e n e t s h f i l t e r a n d t h e

f i l t e r i s t h u s i n c a p a b l e e f e l i m i n a t i n g a l l r e s e n a n c e s .

T h e t r a n s d u c e r a s s e m b l y i s s h e w n i n F i g u r e 5 v t f .

T h e c e r a m i c t r a n s d u c e r s w e r e s e n n e c t e d i n p a i r s ( e l e c t r i c a l l y

i n p a r a l l e l , m e c h a n i c a l l y i n s e r i e s ) . T h i s h a l v e d t h e

r e s e n a n c e f r e q u e n c y t e a v a l u e t h a t i s s t i l l * a b e v s'


F i g u r e 5 . 3 . T h e c h a r a c t e r i s t i c s o f a t y p i c a l p i e z o - e l e c t r i c

c e r a m i c t r a n s d u c e r .

M a t e r i a l s G l e n n i t e L e a d Z i r c o n a t e T i t a n a t e

D e n s i t y s 7 . 6 g r m . / c c .

c o u p l i n g c o e f f i c i e n t s 0 . 6 0( m e a s u r e s t h e a b i l i t y o f t h e c r y s t a l t o c o n v e r t e n e r g y f r o m

e l e c t r i c a l t o m e c h a n i c a l a n d v i c e - v e r s a )

d g 3 2 8 0 x 1 0 “ 1 2 m e t e r s / v o l t

( s t r a i n d e v e l o p e d f o r a g i v e n e l e c t r i c a l f i e l d )O

g 3 3 2 3 x 1 0 “ v o l t m e t e r s / n e w t o n

( o p e n c i r c u i t e l e c t r i c a l f i e l d d e v e l o p e d f o r a n a p p l i e d s t r e s s ) '

• Y o u n g * s m o d u l u s ^ 3 3 6 . 7 x 1 0 ^ ^ n e w t o n s / m

L o s s t a n g e n t 0 . 0 0 6%

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- 1 0 6 -

i n e q u a t i e n ( 5 . 1 ) . T h i s a r r a n g e m e n t y i e l d s g r e a t e r m e c h a n i c a l

e u t p u t f e r t h e s a m e I m p r e s s e d v e l t a g e . I t a l s e s i m p l i f i e s

t h e s h i e l d i n g p r e b l e m a s b e t h t h e b a c k i n g r e d a n d f r e n t e n d

a r e g r e u n d e d w i t h t h e h e t e l e c t r e d e p l a c e d b e t w e e n t h e d i s c s .

The d i s c s w e r e b e n d e d t e e a c h e t h e r a n d t e t h e b a c k i n g r e d

b y c e n d u c t i n g e p e x y c e m e n t ( H y s e l — K 8 - 4 2 3 8 w i t h H 2 - 2 4 7 5

h a r d e n e r ) . F i n e a l u m i n u m f e i l l i n k e d t h e g r e u n d t e r m i n a l s

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

b a c k i n g r e d w e r e c e n n e c t e d t e a c e a x i a l c a b l e . T he d e t a i l e d

c e n s t r u c t i e n e f t h e t r a n s d u c e r i s e v i d e n t f r e m F i g u r e 5 . 4 .

S i l i c e n d i e l e c t r i c g r e a s e a n d e e r e n a d e p e w e r e u s e d i n s i d e

t h e p e l y v i n y l s h e e t f e r i n s u l a t i e n a n d s e a l a n t .

T h e p r e b e s w e r e s h i e l d e d t e r e d u c e h i g h f r e q u e n c y

e l e c t r e m a g n e t i c c e u p l i n g b e t w e e n t h e t r a n s m i t t e r a n d r e c e i v e r

c i r c u i t s j w h i c h w a s t h e m a i n d i s t u r b i n g i n f l u e n c e . I n

a d d i t i e n , M e h a n t y ' s t e c h n i q u e w a s m e d i f i e d s e t h a t t h eI

p r e b e s w e r e s e t u p s e t h a t t h e y w e r e f r e e t e s l i d e d e w n w a r d s

u n d e r t h e i n f l u e n c e e f a f i x e d w e i g h t a n t e t h e m e d e l . T h i s

g a v e a c e n s i s t e n t c e n t a c t b e t w e e n p r e b e a n d m e d e l a n d a c e n -

s i s t e n t b i a s i n g s t r e s s e n t h e t r a n s d u c e r s . The p r e b e s w e r e

v e r t i c a l l y a l i g n e d u s i n g p l u m b - b e b s a n d c e u p l i n g w a s i m p r c v e i $

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

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- 1 0 7 -

5 , 2 S y s t e m R c i p t n x

I n a n a l y z i n g t h e s y s t e m a p l e x i g l a s s p l a t e wa s u s e d

a s a m e d e l a n d t h e f e l l e w i n g i n i t i a l a s s u m p t i e n s w e r e made

a b s u t i t :

a ) T h e p r e b e m e d e l c e n t a e t i s e e n s i s t e n t a n d d e e s

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

e e n d i t i e n e x c e p t a t t h e s e u r e e p r e b e c o n t a c t .

b ) T h e t r a n s m i t t e r i s e s s e n t i a l l y a t r a n s i e n t

s e u r e e e f v e r t i c a l f e r e e d i s t r i b u t e d e v e n l y e v e r i t s w i d t h .

e ) T h e r e c e i v e r e b s e r v e s a v e r a g e v e r t i c a l d i s

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

d ) A t t e n u a t i e n a n d d i s p e r s i e n e f f e c t s a r e n e g l i g i b l e

A s s u m p t i o n ( a ) i s p a r t i a l l y j u s t i f i e d f r o m F i g u r e

5 . 5 w h i c h p l e t s t h e e x p e r i m e n t a l r a t i o o f m in im um o v e r m a x i

mum d e f l e c t i o n f e r v a r i o u s v a l u e s o f — t h e s o u r c e - r e c e i v e r

s e p a r a t i o n . The s m o o t h e x p e r i m e n t a l c u r v e f o u n d f e r t h e

d e f l e c t i o n r a t i o a r e a c h e c k o n t h e p r o b e m e d e l c o n t a c t a s

a n o n - v e r t i c a l c o n t a c t w o u l d e r r a t i c a l l y c h a n g e t h i s r a t i o

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

o r n e g a t i v e . A s s u m p t i e n s ( b ) a n d ( c ) c a n b e m o d i f i e d a f t e r

t e s t i n g . F i n a l l y a s s u m p t i o n ( d ) i s an e s s e n t i a l I n i t i a l

a s s u m p t i o n .

The r e s p o n s e s o f t h e e l e c t r o n i c a n d e l e c t r o

m e c h a n i c a l m o d u l e s a r e i n i t i a l l y u n k n o w n . To e l u c i d a t e t h e

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- 1 0 8 -

1 ) d e t e r m i n e t h e i n p u t w a v e l e t i n t o t h e m o d e l * t h e n

2 ) c a l c u l a t e t h e e x p e c t e d o u t p u t s i g n a l ( t h e

s y n t h e t i c s e i s m o g n a m ) b y t i m e c o n v o l v i n g w i t h t h e t h e o r e t i c a l

r e s p o n s e e f t h e m o d e l ( w h i c h d e p e n d s on t h e i n i t i a l a s s u m p t i o n s

( a ) , ( b ) a n d ( c ) ; a n d f i n a l l y

3 ) c o m p a r e t h e s y n t h e t i c a n d e x p e r i m e n t a l s i g n a l s

( s e l s m o g r a m s ) .

P i e z o - e l e c t r i c t r a n s d u c e r s e x h i b i t a c o m p l i c a t e d

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

o u t p u t e l e c t r i c a l a n d m e c h a n i c a l i m p e d a n c e s ( e . g . b i a s i n g

s t r e s s ) i n t h e f r e q u e n c y d o m a i n . I t i s t h e o r e t i c a l l y f e a s i b l e

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

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

c o m p o n e n t s . H o w e v e r , a s d e r i v e d b e l o w , I t i s e a s i e r t o

c a l c u l a t e t h e i n p u t w a v e l e t f r o m a n e x p e r i m e n t a l d e t e r m i n a t i o n

o f . a r e f l e c t e d s e i s m i c w a v e f r o m t h e f r e e s u r f a c e o f t h e m o d e l

( u s i n g t h e i n i t i a l a s s u m p t i o n ) . " T h i s a p p r o a c h t h e r e f o r e o n l y

c a l i b r a t e s t h e s y s t e m o n a r e l a t i v e r a t h e r t h a n a b s o l u t e

b a s i s . G i v e n t h e f o r m o f t h e r e f l e c t e d wa ve we w i l l b e a b l e —

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

p r o v i d i n g t h e t h e o r e t i c a l s e i s m i c r e s p o n s e i s k n o w n .

T h e i n p u t w a v e l e t i s c a l c u l a t e d f r o m a r e f l e c t e d P

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

t h e b o t t o m o f t h e m o d e l , o s t h a t t h e p r o b e s a r e p o s i t i o n e d

a s n o r m a l . U n l i k e 3 - D p r o p a g a t i o n , 2 - D w a v e s a r e d i s p e r s e d


- 10 9 -

i s t h e y n t v t t h x t u g h s p a t e . At sh ew n i n C h a p ter B, e q u a t i e n

( 2. . 6 1 ) t h e f e r a e f t h e d i r e e t r e f l e e t e d wave f r e a a. f r e e

a u r f e e e e> I n t e r f a c e due t e a d e l t a f u n a t i e n t im e a e u r a e i s

g i v e n by

( S . 2 )

= o c < e 0

a t t i a e £ w h i c h i s w e l l s e p a r a t e d f r e a i n t e r f e r i n g head and

R a y l e i g h / S » e n e l e y w a v e s . C+ i s t h e f i r s t a r r i v a l t i a e e f

t h e r e f l e c t e d . , wave and C i s a s e n s t e n t .

The. r e f l e c t e d wave due t e t h e i n p u t w a v e l e t

i e t h e r e f e r e b y e e n v e l u t i e n

j - ( b > # )

J ( J t.fr-* ? -* :)* ■P u t t i n g ~ Co and T " = T - C0 and t h e n

d r e p p i n g p r i m e s

o A i r - X O = C f S ' / f e ( 5 - 4 )

- c ' j / ( i % l — p - j t e ? 4 - - } <‘ -sl


- n o -

then*

where C, = C /'/sc .

P x e v i d i n g £9 7 " where 7 “ i . su eh t h a t

f (* * * T ) ^ * <5 , 6 )

- C, f £C + Q c /t (5,7)

Jo - £ ) y*

Pax s i m p l i c i t y we s h i f t th e t im e e x i e e f r e f e r e n c e

fa r £ and J ee t h a t

/ f * * $ ) — * f C * ) ; j C t * C ) - ♦ j ( T) <6 ’ 8 )

f (’*') = / “ ■fC^dt'----- (6.9)

^ y o ( T - t ) *

T h is l a s t e q u a t i s n i s th e fsr is a f A b e l ' s e q u a t i s n

s f s r d e r a t/ltC

(5.10)

= z * i t ( , - u )


- I l l -

E q u a t i o n s ( 5 . 9 ) a n d ( 5 . 1 0 ) i n v o l v e t h e i n t e g r a t i o n

T h u s i f ) * 8 f°und bY e x p e r i m e n t , i t - i s p o s s i b l e

t o c a l c u l a t e t h e f o r m o f t h e i n i t i a l w a v e l e t ^ ^ V ^

I n o r d e r t o e n s u r e t h a t j - f e ' ) i s s t a b l y c a l c u l a t e d

h i g h d e n s i t y d i g i t i z i n g a n d s u b s e q u e n t s m o o t h i n g # f )

i s n e c e s s a r y . As w e l l , t h e n u m e r i c a l i n t e g r a t i o n £

m u s t b e a c c u r a t e l y c a r r i e d o u t .

F o l l o w i n g A b r a r a o w i t z ( 1 9 6 5 , p . 8 8 9 )

A/

I(jCO) - = 2 t,/3Yl 0~ %)).t*2

•whe-ne i s t h e i* '*1 . p o s i t - i v e z e r o o f ancl ^

a r e t h e G a u s s i a n w e i g h t s o f o r d e r A/%The p o i n t s a r e f o u n d f r o m t h e

d i g i t i z e d g r i d o f i n t e r v a l u s i n g L a g r a n g e 4 - p o i n t i n t e r

p o l a t i o n . F i n a l l y t i m e d i f f e r e n t i a t i o n i s d o n e by a s i m p l e

d i f f e r e n c i n g a l g o r i t h m .

As a c h e c k o n t h e s t a b i l i t y o f t h e n u m e r i c a l

t e c h n i q u e s u s e d , J £& ) was r e c a l c u l a t e d u s i n g e q u a t i o n ( 5 . 9 )

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

m ax i m u m v a l u e a t a l l p o i n t s .

F i g u r e ( 5 . 6 ) i l l u s t r a t e s t h e e x p e r i m e n t a l l y m e a s u r e d

r e f l e c t e d p u l s e J £ ? ) , t h e i n p u t w a v e l e t c a l c u l a t e d

f r o m i t a n d t h e c h e c k r e f l e c t e d p u l s e J t (?)> o a l c u l a t e d f r o m


t.o

g( t )0 8

UJ

g 0 6_jCL2 0.4<

0.2

200IOO 3 0 0 50.04 0 0TIME IN MICROSECONDS

0.20

f ( t )Q

OIO

5 0 03 0 020 010.0

0 8

LUO 0 6

a! 0 4

0.2

5 0 03 0 020010.0TIME IN MICROSECONDS

Hi ou r p 5 . G . The o x p p r i m o n t a l l y ' I n t o r n i n*»d r e f l e c t e d w a v e l e t o ( t ) j t h e c a l c u l a t e d f u n d a m e n t a l w a v e l e t f ( t ) a n d t h e c h e c k r e f l e c t e d w a v e O l Q t J

Reproduced with permission of the copyright owner. Further reproduction p ro h ib ite ^T ith o u ê m !ss io rr," fc™ ™ ^ ^ ^ " —" " ,™ " " —

- 1 1 2 -

^ • T h e i n p u t w a v e l e t w a s a l s o d e t e r m i n e d d i r e c t l y b y

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

c i r c u l a r r o d a s i l l u s t r a t e d ( F i g u r e 5 . 7 ) . P r o v i d i n g t h e

r o d i s t h i n * a c c o r d i n g t o t h e P o c k h h a m m e r - C l e e e q u a t i o n s

t h e w a v e l e t w i l l n o t b e d i s p e r s e d . F i g u r e ( 5 . 7 ) g i v e s t h i s

w a v e l e t s h o w i n g s o m e a g r e e m e n t w i t h F i g u r e ( 5 . 6 ) a s t h e r o d

o n l y m a r g i n a l l y f u l f i l l e d t h e c o n d i t i o n s f o r n o n - d i 6 p e r s i v e

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

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

F i g u r e ( 5 . 8 ) i l l u s t r a t e s t h e f o r m o f t h e c h a r g e

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

t h e n m o d i f i e d b y th*e e l e c t r o m e c h a n i c a l m o d u l e t o g i v e t h e

i n p u t w a v e l e t o f " F i g u r e 5 . 6 . F o r r e f e r e n c e t h e f r e q u e n c y r e s p o n s e

o f t h e e l e c t r o n i c c o m p o n e n t s a r e g i v e n i n f i g u r e ( 5 . 9 ) .

5 . 3 C h e c k i n g t h e S y s t e m u s i n g L a m b * s P r o b l e m . .

W i t h t h e t i m e f o r m o f t h e i n p u t w a v e l e t a n d t h e

s p a t i a l d i s t r i b u t i o n o f a p p l i c a t i o n kn ow n i t i s p o s s i b l e t o

c a l c u l a t e L a m b ’ s p r o b l e m s e i s r a o g r a m i t p r o d u c e s b y c o n v o l v i n g

i n t i m e a n d s p a c e . L a m b ’ s p r o b l e m i s i d e a l l y s u i t e d f o r a

c h e c k c a l i b r a t i o n a s t h e R a y l e i g h wa ve i s a- s e n s i t i v e

i n d i c a t o r o f w a v e f o r m i n t h a t i t r e p r e s e n t s i n t e r f e r e n c e

b e t w e e n P . a n d S w a v e s . T h u s i t i s p o s s i b l e t o c h e c k t h e

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

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


hiQDhDQ.

TIME I N MICROSECONDS J____________I____________ I

20.0 30 .040.0 50 .0

<t - 0 . 2 -

F i g u r e 5 . 7 . The f o r m o f t h e wave t r a n s m i t t e d t h r o u g h a t h i n r o d

160Ui0 120+mJhZj 80 CL

1 40

2 ~

ELECTRONIC TIME RESPONSE

MICROSECONDS

100

F i g u r e 5 . 8 . The f o r m o f t h e c h a r g e p u l s e p r o d u c e d by t h e t h y r a t r o n

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


AMPLIFIER

FREQUENCY (kHz)

1 0 1 0 0

DIFFERENTIATOR(NORMALISED)

100

NOTCH FILTER

2 0 5 kHz

I 10 IOOF i n u r p 5 . 0 . The f r e q u e n c y r e s p o n s e o f t h p e l e c t r o n i c c o m p o n e n t s o f t h e model s y s t e m , K o t o t h a t 205 kHz. c o r r e s p o m i s t o t h e t r a n s d u c e r r e s o n a n c e f r e q u e n c y . Th« d i f f e r e n t i a t o r i s n o r r s a l i z e d t o t h e t h e o r e t i c a l r e s j p o n s e .


- 1 1 3 -

th eore t ica l response of L a a b ' s problem with the o b s e r v e d

experimental response. The comparison w il l allow further

c r i t i c a l , an a lys is of the system and possible modification

of our assumptions and t h e f o r m o f t h e f u n d a m e n t a l wavelot.Equations (3 .18) to (3 .23) give the response of

an Lafi.ai.to half apace to a d e lta time spatia l l ine source. A plot of t h i s impulse response i s given in Figure (5 .1 0 ) .

In order .to compare the theoretica l seismograms

of-.figure (5 .10) with an experimental seismegram i t i s ne-eseaary to convolve the impulse respense as described in Chapter 3*

The r e su lt ef time cenvolutien i s given in Figure (5 .1 1 ) , As the p os it ive and negative d e f lec t io n s about the Rayleigh wave arrival are not exactly a n t isymmetric the e f f e e t e f the t in e respense cenvolutien i s te. increase the negative def lect ion ever the p os it ive d ef le c t io n increas ing ly with small R as was observed

e x p e r i m e n t a l l y i n F i g u r e ( 5 . 5 ) .

As well for greater accuracy space cenvolutien ^scalculated as follows* from equations (3.18) to (3 .23) note that apart from the factor l / R the displacement

depends only on the variableTe ca lcu la te the displacement at a point P due te

a probe e f f i n i t e width we assume that the probe i s equivalent to a ser ie s ef point sources spread ever the face


Figure 5,10.

Lamb*s problem

delta

function resR

onse on

the surface

of plexiglass

See figure

(S.l) for

velocities*

- 0 . 5 0I

DISPLACEMENT- 0 . 2 L 0 . 2 5 0

“ To00

00o

00

00 Mo zl-loVO»- o

. O CO ro m • ooo o o zCO

oo

00o


Figure 5*11.

Figure (5.10)

corrected for

real tim

e response.

D I S P L A C E M E N T 0 . 1 30 . 6 3 3 6 0 . 1 2

roCD

roOD

oo

ooo

oro

oo


- 1 1 4 -

ef the prebe. Hence i f the probe diameter Is A R

s ( cs R) = £ I ^ C r, (* -£ * « /" )^ - “/* (5.13)

-- H « ( * * ; R )

• ■ R and <6t varies slowly with R

A s t ( R ~ < A * / " ) / * (5.14)

Henee the e f f e c t ef a f in i te prebe width is

equivalent te smeething <44 with respeet te time. The receiverpaste a l«e has f i n i t e width se i t i s necessary te again smooththe soi smogram. I t i s equally valid te apply these processes after the time cenvolutien rather than before. The f ina l resu lt e f space and time cenvolutien i s shewn in Figure (5.12).

The exce l len t agreement between theoretical and

experimental ceismegrams are shewn in Figure (5.13) for a

variety ef prebe separations,and in Figure (5 .5 ) .The ex ce l len t agreement obtained calibrates the

system by Justify ing the use ef the f u n s t i e n ^ ^ Figure (5.6) as the fundamental wavelet ef the system. Furthermore i t shews that the Influence ef the attenuation in p lex ig lass

i s n eg lig ib le for these separations.In order t e use the function ** the fundamental

wavelet ef ether models censtructed ef d ifferen t materials


Figure 5.12.

Figure (5•11)

corrected for

real source

and real

receiver*

D I S P L A C F . M E H T0 . 6 5 0 . 1 5 0.10 0 . 3 5

roooo

oo. woo o o w o

co no o

oCO

ooo

ooo

CO

C Do


DIS

PL

AC

EM

EN

T

TIME IN MICROSECONDS 84.00 108.00100.0092.004 100 75.0052.00 68.00

o ' *

experimental /

theo re t ica lo--

Figure 5.13a. Theoretical and experimental selsmograms for R s 9.3 cm. on the surface

COCO

CDQ _

of p lex ig la ss . See figure (3.1) for ve loc i t ie s

Rep

rodu

ced

with

perm

issi

on

of the

co

pyrig

ht o

wne

r. Fu

rther

rep

rodu

ctio

n pr

ohib

ited

with

out

Figure 5,13b,

Theoretical

and experim

ental seism

ograms

D I S P LACE ME NTt f . 143 96 4

tn

to

oH9 0

tl

CDO3

tnmO

co

ro


Figure 5*130.

Theoretical

and experim

ental seisraogram

s for

R'=

15*3

DI SP LA CE ME NT0 . 6 4 0 . 3 9 0 . 1 4 0.11 0 . 3 6

o

i—i

cn »—« O Z

wO5 0oCOmooo oco

co

TOO


Figure 5»13d.

Theoretical

and experim

ental seism

ograms

for R

- 17*0 cm

DI SP LACE ME NT0 . 3 9 0 . 1 4 0.11 0 . 3 6

CD

O

H

CDo

HoVOoCD m m

CD ooto

ro


- 1 1 5 -

( s u c h a s e p e x y ) we h i v e t e s a k e t h e a s s u n p t i e n : T h e

I n f l u e n i e e f t h e p r o p e r t i e s e f t h e m o d e l l i n g m a t e r i a l s e n

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

t h e f u n d a m e n t a l w a v e l e t a t f r e q u e n c i e s b e l e w c u t - o f f . S i n c e

t h e t h i e k n e s c e f t h e p l a s t i c s h e e t i s m u c h l e s s t h a n t h e

d i a m e t e r e f t h e t r a n s d u c e r a n d t h e e l a s t i c m e d u l i e f t h e

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

c e r a m i c 9 t h e f e r e g e i n g s e e m s a r e a s e n a b l e a s s u n p t i e n .

5 * 4 T h e C e n s t r u c t l e n e f L a y e r e d M o d e l s

S c h w a b ( 1 9 6 7 ) a n d S c h w a b a n d B u r r i d g e ( 1 9 6 8 ) h a v e

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

i t z u c t i n g 2 - 0 l a y e r e d m e d e l s . T he f u n d a m e n t a l d i f f i c u l t y

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

a b r u p t c h a n g e e f v e l e c i t y . The i n t e r f a c e m u s t b e a c e n t i n u e u s

c h e m i c a l l y w e l d e d c e n t a c t b e t w e e n m a t e r i a l s e f d i f f e r e n t

v e l e e i t i e s • I n t h e c a s e e f 2 - D m e d e l s t h e c e n t a c t i s a l e n g

a n e d g e e f t h e s h e e t .

I f t h e c e n t a c t i s n e t c e n t i n u e u s t h e n t h e i n t e r f a c e

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

t h e b e u n d a r y c e n d i t i e n s c h a n g e . I f t h e v e l e c i t y c h a n g e i s

n e t a b r u p t n e r m a l t e t h e i n t e r f a c e t h e n t h e m a g n i t u d e e f

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

a p p l y .

I n e r d e r t e e v e r e e m e t h i s d i f f i c u l t y a n u m b e r e f


- 116 -

w o r k e r s h a v e t a k e n a s h e e t e f a m a t e r i a l a n d t h e n a t t e m p t e d

t e p h y s i c a l l y a l t e r p a r t e f t h e s h e e t .

A t e s h n i q u e p e p u l a r w i t h R u s s i a n a u t h e r s ( i v a k i n

( 1 9 5 6 ) a n d ( i 9 6 0 ) ) , G i l ' b r s h t e i n a n d G u r v i c h ( i 9 6 0 ) i s d r i l l i n g

h e l e s s m a l l c e m p a r e d t e w a v e l e n g t h t h r e u g h a s h e e t t e l e e a l l y

a l t e r i t s a v e r a g e d e n s i t y a n d e l a s t i e i t y . W h i l e t h i s t e e h n l q u e

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

e x p e r i m e n t a s a n a p p r e x i m a t i e n t e c e n t i n u e u s m e d i a i s f e r m i d a b l e .

P e l y m e r s , i n t h e i r g r e a t d i v e r s i t y , r e p r e s e n t a

f e r t i l e f i e l d e f i n v e s t i g a t i e n t e f i n d a s u i t a b l e t e c h n i q u e

t a - ~ a c n a l x u e t a. l a y e r e d m e d e l . A l a r g e n u m b e r e f e f f e c t s a r e

m e A t i e n e d i n t h e l i t e r a t u r e w h i c h c e u l d l e e a l l y a l t e r t h e

p h y s i c a l p r e p e r t i e s e f a p e l y m e r s h e e t . I n p a r t i c u l a r , c r e s s -

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

The e f f e c t e f e r a s s - l i n k i n g i s t e a l t e r t h e e l a s t i c p a r a m e t e r s ,

e f t h e m a t e r i a l a n d h e n c e i t s v e l e c i t y .

An e x p e r i m e n t w a s e e n d u c t e d i n w h i c h p e l y e t h y l e n e

wa s i r r a d i a t e d b y u p t e 1 0 8 r a d s e f b r e n s t r a h l u n g p r b d t i c e d

b y t h e U n i v e r s i t y e f T e r e n t e L i n a c a c e e l e r a t e r o The r e a s e n

f a r c h e e s i n g p e l y e t h y l e n e wa s t h a t i t r e a c t s r e l a t i v e l y

s t r e n g l y t e i r r a d i a t i e n a l t h e u g h i t s Q v a l u e m a k e s i t m a r g i n a l l y

s u i t a b l e f a r m e d e l w e r k . The e x p e r i m e n t f a i l e d h e w e v e r ,

p r e b a b l y b e c a u s e e x y g e n d e g r a ^ a t i e n p r e v e n t e d t h e m a t e r i a l

f r e e c r e s s - l i n k i n g e n e u g h t e i n c r e a s e t h e v e l e c i t y b y m e r e

t h a n 5 %•


- 1 1 7 -

A s e c o n d t y p e o f t e c h n i q u e f o r m o d e l c o n s t r u c t i o n

i s b o n d i n g t w o d i f f e r e n t m a t e r i a l s t o g e t h e r , , M o s t o f t h e s e

t e c h n i q u e s i n v o l v e t h e a p p l i c a t i o n o f a v e r y t h i n l a y e r o f

a c h e m i c a l t o t h e e d g e o f o n e m a t e r i a l t o " g l u e * 1 i t t o t h e

o t h e r . As G i l * b r s h t e i n a n d G u r v i c h ( 1 9 6 6 ) h a v e s h o w n e v e n

b o n d i n g s u b s t a n c e t h i c k n e s ^ w i l l make m o d e l l i n g

r e s u l t s u n r e l i a b l e . He~ad w a v e a m p l i t u d e s w i l l be u n d e s i r a b l y

a f f e c t e d b y e v e n s m a l l i n h o r a o g e n e i t i e s a n d t h i c k n e s s o f t h e

b o n d i n g s u b s t a n c e a t i n t e r f a c e s ( F a i z u l t a n ( 1 9 6 6 ) ) .$N e v e r t h e l e s s , d e s p i t e t h e s e d r a w b a c k s w h o s e

c o n s e q u e n c e s h a v e b e e n d e t a i l l e d e x p e r i m e n t a l l y b y S c h w a b

a n d F a s z u l t a n , a l a r g e n u m b e r o f a u t h o r s h a v e u s e d s u c h

m o d e l l i n g t e c h n i q u e s . T h e a b o v e b o n d i n g t e c h n i q u e s w e r e u s e d

S h i m a r a u r a a n d S a t o ( 1 9 6 5 ) a n d M o h a n t y ( 1 9 6 6 ) m a i n l y b e c a u s e

o f t h e l a c k o f o t h e r s u i t a b l e b o n d i n g t e c h n i q u e s . I n f a c t ,

M o h a n t y ( 1 9 6 6 ) a c h i e v e d a d m i r a b l e q u a l i t a t i v e a g r e e m e n ta

b e t w e e n t h e o r y a n d p r a c t i c e

T h e r e c e n t p r o d u c t i o n ©f a n u m b e r o f e p o x y r e b i n s

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

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

h a v e b e e n u s e d b y s u c h a u t h o r s a s G i l 5b r s h t e i n ( 1 9 6 6 ) , a

n u m b e r o f J a p a n e s e a u t h o r s , a n d H e a l y a n d P r e s s ( i 9 6 0 ) . A

m a j o r p r o b l e m w i t h s o m e e p o x i e s h a s b e e n t h a t c u r i n g t i m e

i s o f t h e o r d e r o f d a y s , o r s h r i n k i n g o c c u r s r u i n i n g i n t e r

f a c e c o n t a c t . H o w e v e r , H e a l y a n d P r e s s o b t a i n e d e x c e l l e n t


- 1 1 8 -

m o d e l s e i s r a o g r a m s u s i n g a l a m i n a t e d a l u m i n u m / e p o x y m o d e l

c o n s t r u c t i o n . T h e i r m a j o r p r o b l e m wa s m a i n t a i n i n g u n i f o r m i t y

• f t h i c k n e s s . T h e i r c o n s t r u c t i o n t e c h n i q u e i n v o l v e d t h e u s e

o f a h o r i z o n t a l m i l l i n g m a c h i n e t o c u t u n i f o r m c o n t o u r s i n

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

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

A n o t h e r r e e e n t d i s c o v e r y - c o u p l i n g a g e n t s - l e a d s

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

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

t r a t e d i n F i g u r e ( 5 . 1 5 ) . B r i e f l y , t h e p o l y m e r i s m e l d e d

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

c o n t r o l l e d b y a s h e e t - s h a p e d m o u l d . T he p o l y m e r g l a s s b o n d

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

p a i n t e d o n t o t h e g l a s s e d g e i n a n e a r m o n o —mo 1 o c u l a r l a y e r .

I n t h e f o l l o w i n g t w o s e c t i o n s t h e c h e m i s t r y o f

p o l y m e r s a n d c o u p l i n g a g e n t s i s d e s c r i b e d . An o v e r - r i d i n g

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

b o n d l i n k i n g . I n F i g u r e ( 5 . 1 4 ) d u e t o P l u e d e m a n n ( 1 9 6 8 )

t h e c o v a l e n t b o n d I s s e e n t o b e e x t r e m e l y p o w e r f u l compared

t o o t h e r t y p e s o f b o n d i n g b e t w e e n m o d e l s .

5 . 4 1 The P o l y m e r i z a t i o n o f Epoxy R e s i n s

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

w a s e p o x y r e s i n . T he r e a s o n f o r t h i s c h o i c e was t h a t e p o x i e s

i n g e n e r a l h a v e t h e f o l l o w i n g f a v o u r a b l e p r o p e r t i e s ( F l y n n


F I m i r r» 5 . I S AMWI'WICI IQ.’iAM IY 0 ! Ml. AM: COWLING A U II IS

R “ Si(— OH*)| T y jiic .il S il.n * C i'n p lin c A gen t

R = pi cn u o fim c t iona l j;ioup , w h ich i c . u h w ith th e n u t i ix i c i i n .

Ofl* ~ liy ilio ly /.iM c jy c u{*. w h ich I ty d io ly /e s to y ie ld on H — S i ( — O H ),, w h ich then cond e n s e s w ith th e —Si — O lt £.iol;>s on the su b s t i .i lc to y ie ld on — Si — O — Si —b ond .

EX AM PLES O F A S ll.A N C C O U PL IN G A G EN T’S REACTIONS:

S y s tem s = g lo s s ic in fo rc c d ep o x y , p h e n o lic , and m rlan iinc i c s in s

Si la n e = c a in in v am m optopyH f ie th o x y sil.m e ' *

R E A C T ION V.1T H GLA SS:

H ty C H ,) ,S i - (O C H jC H ,), + — (OH), + 3Clf*CH2OH

OH OH

N H jI C H ^ S i— OH + H O —-Si - g la s s - t -K H ^ C H ^ S i O — Si — c l a s s

X 0 H OH

R EA C TIO N WITH RESIN S:

E poxy

(R O L— S K C H ^ N H , + C — C — R -^ (R O ]^ — S K C H jT jI . 'C -C —-R\ ✓ Io o

H

4 0

3 0

VAN D ER V/AALS BOND20

+ 10

-E(kcol)

V /HYDROGEN BOND- 1 0

20o

re 3 0

40

.50 COVALENT BONO

GO

7 0 *

F W i u r o s . i a .-•..I *-»»«•! ?:y n i r v r s

o f inl«-i.tl**mi»: Inn*’'


- 1 1 9 -

and Luker, 1968; t h i s a r t i c l e a lse d escr ibes the ch ea lstry • f epexy r e s i n s ) :

1) e ea p le te r e a c t iv i t y with n# v c l a t l l e s prcducedj*during the cure ( p e ly a e r iz a t ia n ) , hence very l i t t l e shrinkage

2) engineered prepert ies far w idely d iverse per- fera a n ee , as a wide range ef curing agents i s a v a i la b le .

3) gced f lew c h a r a c te r i s t ic s se th a t centact pressure between the epexy and ether a a t e r i a l s i s s u f f i c ie n t far adhesien

4) grea t strength which can be s u f f i c i e n t l y high*

t e w ithstand pressures ef5 ) epexy res in s adhere te n a c ie u s ly te a large

v a r ie t y e f a a t e r i a l s - a preperty which i s iapreved further

by us ing ceup ling agents .Epexy r e s in i s a pelyaer (see S t i l l s ( i960) far the

eh e a ls tx y e f p e ly a e r s ) . Pelyaers can change th e ir physical ^ p r e p e r t ie s d r a a a t ic a l ly by in te r —react ien between t h e ir leng

l in e a r a s l e c u la r chains te fera cress l i n k s . This preeess i s knewn as curing and can change a pelyaer frea a l iqu id

v ise e u s substance te a s e l i d .In the case c f the epexy we used, S ty ca st 2651

4

(a. preduct e f Eaaersen & Cuaaing iaf^ curing caused t h i s l iq u id t e s e l i d change and was e f fe c te d by a c a t a ly s t (er c r e s s - l i n k i n g ) agent . The agents are eeupled d i r e c t l y in te


- 1 2 0 -

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

T h e e p o x y w a s f o u n d t o h a v e a n i d e a l v i s c o s i t y f o r

- p o u r i n g i n t o t h e c a s t u s e d . C u r i n g w a s a c c o m p l i s h e d a t r o o m

t e m p e r a t u r e a n d w a s c o m p l e t e i n 2 4 h o u r s . H e n c e n o c o m p l i c a t e d

h e a t i n g e q u i p m e n t w a s r e q u i r e d . As w e l l , o n h a r d e n i n g t h e

e p o x y w a s f o u n d t o b e u n i f o r m , h a v e a h i g h a n d y e t b e

r a a c h i n e a b l e b y c o n v e n t i o n a l m i l l i n g e q u i p m e n t .

5 . 4 2 C o u p l i n g A g e n t s

" S i l a n e c o u p l i n g a g e n t s a r e m o n o m e r i c s i l i c o n

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

t h e i r u n i q u e a b i l i t y t o c h e m i c a l l y b o n d o r g a n i c p o l y m e r s

( e p o x i e s ) t o i n o r g a n i c m a t e r i a l s — g l a s s , m i n e r a l f i l l e r s ,

m e t a l s a n d m e t a l l i c o x i d e s ( s e e F i g u r e ( 5 . 1 5 )*' ( S t e r r a a n

a n d M a x s d e n ( 1 9 6 6 ) ) . As t h e f i g u r e s h o w s c o u p l i n g a g e n t s

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

o f t h e i r m o l e c u l a r c h a i n a s s i l a n e s i n a c o n d e n s a t i o n r e a c t i o n

w i t h g l a s s ( s e e a l s o i l e r ( l 9 5 5 ) ) o r m e t a l a n d a t t h e o t h e r

e n d - a s h a r d e n e r s o r o a t a l y s t s w i t h p o l y m e r s . T h e r e a e t i o n s

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

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

Falconmx e t a l ( 1 9 6 4 ) h a d s h o w n t h a t h i g h h u m i d i t y e n v i r o n

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

b e t w e e n t h e r m o s e t t i n g r e s i n s ( e . g . e p o x y ) a n d v a r i o u s

s u b s t r a t e s s u c h a s g l a s s f i b e r . T h i s d e g r a d a t i o n i s t h e

Reproduced with permission of the copyright owner Fnrth*owner. Further reproduction prohibited without permission.

- 1 2 1 -

f u n d a m e n t a l j u s t i f i c a t i o n f a r u s i n g c o u p l i n g a g e n t s <>

E v e n v a n d e r W a a l f o r c e s , w h i c h s h o u l d e x i s t a l o n g

a n e p o x y / g l a s s i n t e r f a c e , a r e s t r o n g e n o u g h s o t h a t f a i l u r e

a l o n g p e r f e c t c o n t a c t b e t w e e n p o l y m e r s a n d s u b s t r a t e s s h o u l d

p r a c t i c a l l y a l w a y s b e i n o n e o f t h e m a t e r i a l s ( c o h e s i v e )

r a t h e r t h a n i n t h e t w o m a t e r i a l s ( a d h e s i v e ) ( P l u e d d e m a n n ( 1 9 6 8 )

B i k e r a a n ( 1 9 6 7 ) p r o p o s e s t h a t o b s e r v e d f a i l u r e s i n a d h e s i o n

a r e u s u a l l y a c o h e s i v e b r e a k o f a w e a k b o u n d a r y l a y e r a t t h e

i n t e r f a c e , . T h i s w e a k b o u n d a r y l a y e r may b e a i r , w a t e r , o i l s ,

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

t y p e o f f a i l u r e i s p r o b a b l y r e s p o n s i b l e f o r t h e b o n d i n g

f a i l u r e s o b s e r v e d b y M o h a n t y ( 1 9 6 6 ) i n h i s l a y e r e d m o d e l s .

F o r p o l y m e r / s o l i d a d h e s i o n t h e p o l y m e r m u s t c o m p e t e

w i t h m a n y p o t e n t i a l w e a k b o u n d a r y l a y e r s f o r a d h e s i o n t o t h e

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

t o e x c l u d e . To o v e r c o m e t h e p r o b l e m s i l a n e c o u p l i n g a g e n t s

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

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

b e c o m e p a r t o f t h e p o l y m e r .■

We c h o s e t h e c o u p l i n g a g e n t P i - 1 e r t i a r y - b u t y 1 p e r o y y -

d l m o t h v l s i l a n e . a d e r i v a t i v e o f LUCIDOL ( t r a d e n a m e ) , o n

t h e a d v i c e o f W e e d h a m ( 1 9 6 8 ) ( p e r s o n a l c o m m u n i c a t i o n ) , w h i c h

r e a c t s w i t h i t s - ^ R r a d i c a l ( R = m e t h y l ) w i t h g l a s s i n a

c o n d e n s a t i o n r e a c t i o n .y

T h e o t h e r e n d o f t h e b i f u n c t i o n a l m o l e c u l e a c t s a s


• c a t a l y s t w i t h t h e e p e x y r e s i n ( S t y s a r t 2651 -MM) u s e d .

T h u a t h a u s e e f t h a a a u p l i n g a g a n t a a k e s t h e e p a x y

l i t e r a l l y p e l y a e r i z e i n t a t h a g l a s s , a d h e r i n g t h r e u g h a

s a v a i a n t b a n d . H a n a e we a r e a t l e a s t t h e e r e t i e a l l y a s s u r e d

t h a t t h a b a n d i s a s h e a i s a l l y w e l d e d s a n t i n u a u s a a n t a s t and

t h a t t h e r e f e r e t h a b a u n d a r y s a n d i t i a n s a r e a b e y a d .

I n a r d a r t a i n e r e a s e t h a n u a b e r a f a s t i v e s i t e s

a l a n g t h a e d g e e f t h a g l a s s ( l i t * ( 1 9 5 5 ) ) t h a a r e a a f t h e

s u x f a s e e d g e a a n b a i n s r e a s e d . Tha g l a s s w a s s u t e v e n when

n a t n e s e s s a r y s a t h a t t h a e d g e s u r f a s a wa s r a u g h a n a

a i c r e s s e p i s s c a l e a n d t h e r e f e r e h a d a g r e a t e r a r e a .

S e a e t e s t s w a r e c a r r i e d c u t w h i c h d e a e n s t r a t e d

t h a t a b a n d e d e d g e t a e d g e g l a s s - e p a x y s h e e t d i d n a t p r e f e r

e n t i a l l y b r e a k a l a n g t h a s e n t a s t .

5 . 4 3 M e c h a n i c a l C c n s t r u a t l e n

F i g u r e ( 5 . 1 6 ) s h e w s t h e e p e r a t i e n a f t h a a a l d u s e d

t a s u t t h e l a y e r e d a e d e l s .

I n i t i a l l y t h a l i q u i d e p e x y w i t h h a r d e n e r a d d e d was

p e u r e d i n t a t h a a a d a l i n t h a h e r i z e n t a l p e s i t i e n . Tha u p p e r

p a r t a f t h a a a l d w a s t h a n b r a u g h t d a w n e n s u r i n g t h a t a i r

bubbles .ware excluded frea tha systea.F i n a l l y t h a e n t i r e s y s t e a was r a t a t e d t a t h a v e r

t i c a l , p a e i b l a n . Tha u p p e r c a n t c i n i n g s t e a l w a s t h a n z e a e v e d

f r e a t h a t a p t a a l l a w a n y r e a a i n i n g a i r b u b b l e s t a e s c a p e


T e f l o n c o a t e d f i b r e g l a s s f a b r i c

e p o x y p o u r e d i n t o mo%d

removable s i d e o f mo 1 d

g l a s s

Upper p l a t e o f mold i s lowered and alamped, t h e e n t i r e mold

i s t h e n r o t a t e d t o t h e v e r t i c a l and t h e t o p removed.

epoxy wi th a i r b u b b l e s e s c a p i n g

F i g u r e S . 1 6 . T h e o p e r a t i o n o f t h e

mold u s e d t o a o n s t r u a t m o d e l s .

&b o l t e d t o e n s u r e mold p l a t e s remain p a r r a l l e l

.g la s s


frea the epexy as It cured.T h e m e l d i t s e l f w a s e e n s t r u e t e d f r e a t h i e k

• t a e l p i s t e s . Te p r e v e n t t h e s e p i s t e s w a r p i n g u n d e r t h e

c h a n g i n g s t r e s s e s e f a e v i n g t h e s y s t e a f r e a h e r i z e n t a l t e

v e r t i c a l a d d i t i e n s t r u t s w e r e a d d e d .

T h e B u s i e r p r e b l e a e f s e n s t r u s t i e n w a s t h e a a i n -

t e n a n e e e f u n i f e r a t h i c k n e s s e f t h e a e l d . Te d e t e r a i n e

t h i c k n e s s u n i f e x a i t y a n e p e x y s h e e t w a s c u t a n d i t s t h i c k n e s s

a e a s u r e d b y a i s r e a e t e r . F r e a t h i s we e e n e l u d e d t h a t t h i c k

n e s s w a s u n i f e r a t e 5 $ . H e n c e v e l o c i t y i n t h e e p e x y w a s

u n i f e r a t e 2J6. T h i s v a r i a t i e n c a n b e d i s r e g a r d e d e s p e c i a l l y

a s i t w a s v e r y s a a l l c e a p a r e d t e t h e 5 0% c c n t r a s t b e t w e e n

t h e e p e x y a n d g l a s s v e l e e i t i e s ( s e e F i g u r e ( 3 J L 2 ) ) .

I n e x d e x t e p r e v e n t a d h e s i c n c f t h e e p e x y t e t h e

a e l d , t h e a e l d w a e s e v e r e d w i t h t c f l c n - c e a t e d g l a s s f a b r i c

w h i c h i s w c l l - k n e w n f e r i t s e h e n i e a l i n e r t n e s s a n d r e s i s t a n c e

t e a d h e s i e n . As a f u r t h e r p r e v e n t i e n a a e l d r e l e a s e c e a p e u n d

( E a a e r s e n & C u a w i n g p r e d u e t N e . 1 2 2 - S ) w a s s p r a y e d e n t c t h e

T e f l e n .

An e u t s t a n d i n g a d v a n t a g e c f t h i s s y s t e a w a s t h a t

t h e e p e x y —g l a s s i n t e r f a c e c e u l d t a k e a n y s h a p e d e s i r e d .

H e n c e t h e s e n s t r u s t i e n e f c u r v e d i n t e r f a c e s t e e e a p a r e w i t h

t h e t h e c r y e f C h a p t e r 4 w a s e a s i l y a c h i e v e d .

F i n a l l y t h e e p e x y w a s a l l i e d t e - f e r n t h e u p p e r

f r e e s u r f a c e e f t h e a e d e l . T h e a a c h i n a b i l i t y e f t h e e p e x y


- 1 2 4 -

w a s a i 1 - i m p o r t a n t f a r t h i s p r a s e s * • T h e me d e l w a s l a i d

h e x i z e n t a l l y a n d e l a a p e d t # a t r a v e l l i n g t a b l e w h i e h a e v e d

u n i f e r a l y p a s t a v e r t i e a l l y s u t t i n g m i l l . T h e r e s u l t w a s a

s a e e t h f r e e s u r f a a e t r u e t e l / l 9 0 0 0 t h e f a n i n e h *


APPENDIX A

The Sh»TW»d^C»qnltrd-I«»hniqut

In 1939l-Cagniard dtt1«td~ln a highly rigereus nannar t h e - s a lu t ia n ta tha a la a t ia wav a aquatian in a medium cen- s U i i n g • f . t w a a«Al-5vnfinita ha lf spaces af d i f feren t

u i a z i a l . N a l d a d tags the it, ana a f which. oantainad t paint seurae* Ha damanatxatad. that h is s a lu t ia n was cample* and

unique* Cagniard. (1932), (1962) la id ta rest arguments eancatning tha e x i s t e n s e - e f haad waves and surface waves.

Russian authar Sabalav (1932, 1933, 1934) alsa davalapad a technique a la se ly re la ted ta Cagniard's sa lu t ian .

A s a lu t ia n ta tha e l a s t i c wave aquatian

ta a b s fsuad by sa lv in g tha wave aquatians far the d i l a t i a n a l and r a ta t ia n a l cempenents af f f as darived in tha ia t r s d u c t ie n (£quatiens (1 .2 ) and ( 1 .3 ) ) . Tha sa lu t ian

af tha d i l a t i a n a l camp an ant wave aquatian

( A.2 )

in sp hax iaa l ( f i j $ aaardinatas i s given by (Cagniard,

(L962) aquatian ( 2 . 3 ) ) ,


'"f- £ f f p ' - o + f <*-*>}w h e r e ^ a n d J a r e a r b i t r a r y c o n t i n u o u s f u n c t i o n s h a v i n g

c o n t i n u o u s d e r i v a t i v e s u p t o a t l e a s t t h e s e c o n d o r d e r .

S u p p o s e t h e i n i t i a l c o n d i t i o n s i m p l y t h a t t h e

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

t h e n

J * 0 ( A . 4 )

T h e i n i t i a l c o n d i t i o n s a l s o i m p l y t h a t ^ O

Y e t t h i s . c o n d i t i o n i s i n c o n f l i c t w i t h t h e c o n c e p t t h a t a

w a v e - f r o n t c o n s i s t s o f a k i n e m a t i c d i s c o n t i n u i t y ( H a d a m a r d

( 1 9 0 0 ) ) a n d t h a t t h e r e f o r e ^ * T h e s e a r e

a l s o t h e „ i n i t i a l c o n d i t i o n s u s e d b y S h e r w o o d ( 1 9 5 8 ) a n d -

i n t h i s t h e s i s . As w e l l , we w i l l u s e ' t h e i d e a , e q u a t i o n

( 2 , 6 ) , t h a t a ~ . k i n e m a t i c o r v e l o c i t y d i s c o n t i n u i t y e x i s t s

a t t i m e z e r o b y p o s t u l a t i n g a n i m p u l s i v e p o i n t £>rce , t h a t

i s a n i m p u l s i v e a c c e l e r a t i o n , - e q u i v a l e n t t o a s t e p f u n c t i o n

v e l o c i t y s o u r c e . T h u s t h e m a j o r m a t h e m a t i c a l d i f f i c u l t y

c o n f r o n t i n g t h e . s o l u t i o n o f ( A . 3 ) i s t h a t a n d ( t h e

r o t a t i o n a l c o m p o n e n t o f <^ ) a n d t h e i r d e r i v a t i v e s a r e

n o t c o n t i n u o u s d u e t o t h e f o r m a t i o n o f w a v e f r o n t s * H o w e v e r

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

a n d o r t h e i r d e r i v a t i v e s o c c u r i n a n a r b i t r a r i l y s m a l l

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

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

t i m e . C a g n i a r d w r o t e h i s t h e s i s , h e m a d e h i s s o l u t i o n


- A 3 -

rigorous by employing the Laplace trensferm er Careen in teg ra l

fu n ction s and a l low s a rigereua s e lu t ie n te be developed,We can cen s id ex th e s e generalized funatiens (represented by Cagniard in Laplace transferaed ferns) te represent s e lu t ie n

in a g e n e r a l iz e d s e n s e .I t l a I n t e r e s t in g t e nets that Cagniard had the

eeneept e f g e n e r a l iz e d funetiena in mind when he wrete

(1962) page 7 ) t"But a sharp bend — a small radius e f curvature - in the curve r e p r e s e n t i n g i e physical ly eempletely equiv a len t te the mathematical idea ef a d isco n t in u ity ef v e l o c i t y . Such a way ef leaking at a kinematic d isco n t i n u i t y thus dees net c o n f l i c t with eur accepting the fa c t th a t (equation (A .3 ) ) always expresses the s e l u t i e n . ”

Cagniard considered the s e lu t ie n ef (A,2) through

the .fun ction

e f This g iv e s a representation e f generalized

e f equation (A l) by supposing that and are continuous

n t l n g ( p . 2 1 C a g n i a r d )

where.where. A may have d i s c o n t in u i t i e s ef various orders and

at-.a~ denu me rah 1 o .. number..of-values ~af TTrelated by


< f = £ [ £ " & ) } < * • »

This p r e a e s s h e w e v e r •11winstar d l e a a A t l f t u l t l e * . ef A (see U u i t i»d Ftihbtik (1953)) a* 2 A extent at dlaten* ti&ttitiaa*

Cagniard than thaws that

[' -L £ , ] £ ( A . 8 ,a t 3

= owhere £.

f = ( Q C * ' * )Jo

.l w- a sa ln t l an af tha wava aquatian whLeh la unique* Cagniard

thus taJcea tha funatlan G (t- t) * (aee Cagniard1aqifc*tinn„Xa^l&a4>.jLt..-thla..ataga te. ranava d ia een t ln u it ie s •

— Xhia—praaaaa..g i .vesus a sannaatian between geenetrle

aptlaa>.whiah aa.au war tha. presents af wavafrantt defined by a-Jdaejaatle. d iscantinu lty (Kara! It Kallar (195b), Yanavakaya (JL9H4)Jl. and th a . Cagniard appraaah as Cagniard nates that

tha~funetiena

t m 7 f a y * ’) ( A *9 )

in his selutien where T f representv i j U m*

Cagniard f in a l ly pravas that wa way aannaat tha

Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission

ftalutlaiL »f tad. la diifaxtnt Btdii9 by supposing..thatM *

tbaix. Caaitn in ta g a a l . XApsaitAtailtB (UpXaaa. txansfaxa) ataâalatad and in t i s z a la tn d by txansa iss ian a a a f f i s ie n t s . Ha than pxavaa th a t tha n a l ia e t ia n af wavaa ahawn aahaaat-

i a n l ly by. th a ix wavafxanta in Figuza ( A . l ) a l l a x i s t , and axa . uniqua and aaaplata aalutiana fax a la s t ia wave pxa-

pagatinn... GanaxaXixad- funatiana haya naw baan. xlgaxaualy

daxi-vad. and tha ix . pxapaxtlea a.tudlad^ Fallawing L ighth il l

(X9&0X aa. ganaxalizad funstiana a an naw ba pxapdxly xapxas- antad by th a ix Fauxiax. txanafaxaa, wa w i l l uae Fauxiar s y n th a s is , whiah givaa tha aiganfunatian axpansian af in tha anna.- f a a i l l a x faxa a f plana wavaa pxapagating with

a xaal^fxaqmaaay • CagnXaxd* a..aantx ihutian ta aaianalagy (apaxt fxaa

h> j p p A g f** ta th* s a lu t ia n ) ia that ha shawad that I f ** ia,.£uxthax. txanafaxnad with xaapeat ta J t and *

than I t i n pan a i b i s ta "play ana intagral againat anathax" (D Ix*lSB 4) taxaduaa tha 3-D tx tp le intagxal sa lutian ta

a a in g la in ta g x a l . In twa dinanaiana, Shaxwaad plays tha

daubla in ta g x a l dapn ta a alasad faxn sa lut ian .


Tlic head waves at an interface where K | > f i t \ f i t < f i t *

$

X>* X> Medium I

Medium 2

(incident IMedium t

S

Medium 2

» The head wave*at an interface where

' ai < 0*.

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

a n i n c i d e n t w a v e a s p r o o v e d b y C a g n i a r d , ( s e e t e x t ) *

Reproduced w » p e r .s s io o * the c „p yrigm ow ller. Furthef ^ ^

APPENDIX B

I h e L o c u s e f T i m e I n t h e C o m p l e x (P P l a n e

N o t e t h a t i f £ i s a s e l u t i e n t # t h e e q u a t i o n s

( 2 . 5 4 ) a n d ( 2 . 5 5 ) t h e n w i l l a l s o b e a s o l u t i o n . As we

a r e d e a l i n g w i t h t h e f o u r t h q u a d r a n t e f t h e <9 p l a n e ,

a n d ^ m u s t b o t h b e p o s i t i v e ( s e e e q u a t i o n ( 2 . 5 3 ) ) . S n e l l * s

Law g i v e s t h e r e l a t i o n s h i p b e t w e e n j5)j a n d

~ * g t ) = s , n ( f > - ± £ ) ------ ( J . 1 )

CJ

E q u a t i n g r e a l a n d i m a g i n a r y p a r t s

N o t i n g t h a t

A 32 + c o s * f > s ' " t 'Z2

= C osA *f, “ C » S a/>

- Si»Ay = * * Cs*y)

( B . 2 )S /n ^ cos A r st*/> cosAp

c o s f y SrinAfj = co sp S ' " * / ( b . 3 )

w h . , . * f r C - / C < / C B . 4 )

J J /

( B . 5 )


- B2 -

we h a v e

j 2COSh 0. - c o s > J s '”j ( c °sh *% - COSzf ) (B . 6 )

J + s '” > j - *>y

F r o m e q u a t i o n s ( 8 . 2 ) t o ( B . 5 )

3 , , 7 ^ ( / - ~ ”j S m Y COS**% ( B . 7 )

2,tfo J>4 V - CosA^-t - *>f s,ny cos Ay

P u tZ

(B . 8 )

s tn * & c o s A * ? - ^

E q u a t i o n s r e p r e s e n t q u a d r a t i c s i n a nd C O S A ^

T h a i * s o l u t i o n s a r e g i v e n by

- - i f O * ' * ! * 1) - F * ] ( b . 9 )

C 6 f / > y [(* + * £ * ) * r * j

T h e r e f o r e

~ * / 3 * t { 0 - f <*•■»>

5 " * ^ * j / V ' " 7 1 " ? '

/ r * ( i +

- ( ^ ~ ° * )

w h e r e


- B3 -

where <%Z - - C O S * ^> ( b - 1 2 )

I f ' £ t h e n . e q u e t i e n ( B . 1 0 ) s h e w s t h s t a l l & j ? 0T

H e n e e t h e c o n t e u r p a t h c a n b e d i v i d e d i n t o two

p a r t s d e p e n d i n g o n w h e t h e r i s z e r o o r n o t . The p a t h

s t a r t s “i t t h e o r i g i n , t h e n f o l l o w s t h e r e a l a x i s o f & t o

s e m e p o i n t <4 w h e r e i t d e p a r t s a l o n g t h e c o m p l e x p a r t

e f t h e p a t h i n (P •

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

p a t h a s ^ ^ ^ a n d t h e n a s ^ and

t h e v a l u e e f t a l o n g t h e p a t h .

F r o m t h e e q u a t i o n s ( B . 2 ) and ( B . 3 ) t h e f i r s t s t e p

i s t e f i n d

** / T f* " * /? =

2 . / t m r COS p *% -*>o L J

3 ' *A

4 ‘ J ' T l ( cos/>J cosA% l *A

F r o m e q u a t i o n s ( B . 2 ) t e (B:. 5 )

^ ( B . 1 7 )

L ,(B .1 3 )

( B . 1 4 )

v , ( B . 1 5 )

£( B . 1 6 )

Str> s t


B4 -

c°sf j cosA?j m£ l O +

T h e r e f o r e f i n d

h r *i F ,/x - F * 9, <»*#/ ft** F ^ - Fup - + o %

. *. ^2. - i ,h * J ~( ~ c e s ^ \^ -&0 C

f r o n e q u a t i o n ( B . l l )a

= h ~ n f t + 4 ” ? S '» A * 9 ****/>)P ' + o ' ' J ^ (1 - •y * y x

yC-vi _ / / _ o t; * « * ) / / + 2 o*? S‘» A v f g « > #•. .*

1 J ' O

To f i n d t h e u p p e r l i m i t F^

F = / -

+*?(*"'*/* + 2s"'2/ 3iir'*>y * *'*h *2 )+ 4 ‘t l j ’ C O S^f SJ*

- s , ~ A y + 2 coSy _ ( ,

- 4 c o s y ( c o s y - > ■ + r f* " ,a/ a)

( B . 18 )

( B*19)

( 0 . 2 0 )

( B . B I )

( e . e e )

( B . s a )

( * 2 4 )


- B 5 -

• - £ yff ' CL* v 2(esxf> - t jx f t — 2 c o s zio ( tos*f> - 1 +■ '»?***>y ° ) ? ’/z

( ^ j aZf + 2 ces* p - fy** ^

( 8 . 2 5 )

Const J***

L =• h^n f s**> p . 5/*9 A * 2% - + * € J ' j j

r -jL . / m f i t .2*0 L J

( B . 2 6 )

( B . 2 7 )

'2 ° 9 - ^ Ot ! m ^ 2 ( t['^— / y > y-^2 6 2~ a ^ j

W'_ - i - / / / * » f t v f ( ‘2 6 :i*a '1)- i-2 'n fs f* A Z2 eesyo ?'A (B.ea) " i / 2 f , - + o I J y > ~ -> y-a ^ 2

{-- > f e 5“/>7 ^l /

■f s / s i A 2

? 7 - Vc * s *s> 7 ,/z

L. =- 5 /f»A & COSp

( { - - 7 ^ s

( B.29)


B6 -

L2 / / / ? - ?

//~? J . f t + Q F'h + f -■*?»« ?/z %-*o 2 c J J

- - ^ o f l 3 (' ^ / ( • ' - ^ ' ) ■> S * ) / '

= h

( B . 3 0 )

m -L. [ 2 - 2 “m3'bz -f Z'nf s/* cos> 6 f z l J J( , ' / J

fAJ

j

t - z

d - l n * ( i , ^ '3J° ~ c o s ,f> - o f s ' ^ y o )

~2 J (,J

( B* 31 )

( B, 32 )

(Jf r / / / ^/ ^

_ L / * r>?(2<>*-«*)}if~2 9 -*><=*

•A ( B . 3 3 )

from e q u a t i o n ( B . 2 5 )

- J -~ / / /■ * * »_»oo

PwJ-ci*'-f~ / — 2 s**a'f> ~ t -f- ty

= / / W "

/ V- n z(2b'l- a i)Z</z( ta34) J


= /7 5/r} *s> s /s t/t a f t — 0 - * * / )J ' ? I „ * * , „ V - J

( B . 3 5 )

(B. 36)

S i m i l a r l y ^ - / / * * ^ C o s C o sA j^ .J

( B . 3 7 )

T h u s c a n s i d e r t h e l i m i t e f a s f r e m

e q u a t i o n ( 2 . 5 5 ) . T h i s g i v e s a l i m i t i n g p a s i t i a n a l a n g t h e

r e a l a x i s a f t h a o a n t a u r a f i n t e g r a t i a n .

T h u s / / * v i ^ — > C?

- CoSJ^ z '* /9oc c o s p st * - yC CtJ . ' -J as,

- f \ .

j* > O - v

A d 2 f_ 2 !tL & .W S Z / r * ' A >

As ^ ®»c a n t a u r l i n e a c c u r s a t

t h e l i m i t i n g p o s i t i a n a f t h e

M.

OC CO$f> = A Aj S /V °( B . 3 9 )


- B8 -

VI

Figure 2 .7 gives the general fern ef the path in the eeaplex 9 plane•

The peint eerrespends te the d irest arrival singula r i ty ef a given ray as i t eerrespends aatheaatleally te the i d e n t i t i e s (see equatien (2 .50 )) .

A O (B.40)

P r e e f

F i0M aquatian. (2*50) and (2*51)

A

A f t * ) — ^ t y s fSi (B.41)1/

a s 2 f e e s £>•) — S / / » & / c o s Q (B.42)P v 1 y / </ ' y

. \ ^ 0 * * ) - ^ ^ ^ Cst*7f> COsMp-b CCOSjK? S /» A p ?j - i sttoAjpj-J

- hj hj ^ c o s f y c e t * g j

j * % + cosf* s,M/ j s

+ {st> » focosA p z

- S m & &


- -

Now **

X — > s " j e / c n j j .

corresponding t o t h e c o n d i t i o n th a t ^ ^ £>

^ - ► *

= t 2 6 'Ju S tsS L -s* * ? 7

-h COS J* ( / - '*1??'” *;&)) St* A?

- < y 1 c ° £ s > ______

S tn Y ') '* ’

Hence ^ a '" ?

The var ia t ion of ^ with ^ _A s ^ ^ o o we s e e f r o m e q u a t i o n ( 2 . 5 5 ) t h a t

t*C ^ o r Sir*j/o -A Co*.f^ S)CoS 7

yf v/

Thus time increases preportienally with CosAj, along the complex path. I f a l l ^ ■= / t h e n

2T — C C O S A ^ / a * ct / / ^

( B . 4 4 )

( B. 45 )

( B. 4 6 )

( B . 4 7 )

( B . 4 8 )


- BIO -

The v a r i a t i o n of with as î s found from equat ion (B.47) squaring both s ide s

C O S A Z Q ■= I *■ S / n A 7 £ ( B . 4 9 )

♦ a - t * * c * ' " * * /

= ( £7- c' r / c°

( B . 5 0 )

( B . 5 2 )


A b r a m o w l t z , M . , a n d S t e g u n , I . A . , 1 9 6 5 . H a n d b o o k o f M a t h -.

e r a a t i c a l F u n c t i o n s ; D o v e r P u b l i c a t i o n s , I n c . ,

New Y o r k .

A l t e r r a a n , Z . a n d K a r a l , F . C . , 1 9 6 8 . P r o p a g a t i o n o f e l a s t i c

w a v e s I n l a y e r e d m e d i a by f i n i t e d i f f e r e n c e m e t h o d s :

B u l l . S e i s . S o s . Ant. . 5 8 . 3 6 7 - 3 9 8 .

B a k e r , B . B . a n d C o p s o n , E . T . , 1 9 3 2 . The M a t h e m a t i c a l T h e o r y

o f H u q g e n ' s P r i n c i p l e : O x f o r d U n i v e r s i t y P r e s s .

B e n - M e n a h e m , A . , 1 9 6 4 . M o d e - R a y D u a l i t y : B u l l . S e i s . S o o . Ant. .

5 4 , 1 3 1 5 - 1 3 2 1 .

B e r r y , M. J . a n d W e . s t , G. F . , 1 9 6 6 . R e f l e c t e d a n d h e a d wave

a m p l i t u d e s i n a med ium o f s e v e r a l l a y e r s : G e o p h y

s i c a l Mena g r a p h No. 1 0 . The e a r t h b e n e a t h t h e c o n

t i n e n t s , A m e r i c a n G e o p h y s i c a l U n i o n .

B e r t h o d - Z a b o r o w s k i , 1 9 5 2 . Le c a l c u l d e s i n t e g r a l e s d e l a

f o r m e : f - f t x ) l o g x dx i n H. M i n e u r , T e c h n i q u e sJo ^

d e c a l c u l n u m e r i q u e . 5 5 5 - 5 5 6 ( L i b r a i r e P o l y t e c h n i q u e

C h . B e r a n g e r , P a r J . 3 , F r a n c e ) .

B i k e r m a n , J . J . , 1 9 6 7 . I n d . E n g . Che,m._» 5 9 , 4 1 .

B r e k h o v s k i k h , L . M . , 1 9 6 0 . Waves i n L a y e r e d . M e d i a , ( E n g l i s h

t r a n s l a t i o n ) : Ac ad e m ic P r e s s .

B u l l e n , K. E . , 1 9 6 2 . I n t r o d u c t i o n t o t h e T h e o r y o f S e i s , mo-

l o q y : C a m b r i d g e U n i v e r s i t y P r e s s ./ N

C a g n i a r d , L . , 1 9 3 2 . S u r l a p r o p a g a t i o n d ' u n s e i s m e a✓

1* i n t e ' r i e u r d ' u n c o l i d e h o n o g e n e i s o t r o p e , e l a s t i q u e ,

s e m i - i n d e f i n i , l i m i t e ^ p a r u n e s u r f a a e p l a n e : C o m t .

R e n d . A c a d . S c l . P a r i s * 1 9 4 , 8 9 9 .

permission * * . oopyrigM o w n ,. Furtn , re„ n p rohM ed withoul

C a g n i a r d , L ( y 1 9 6 2 . Re f 1 e c t i o n a n d R e f r a c t i o n o f P r o g r e s s i v e

S e i s m i c W a v e s . ( T r a n s l a t e d a n d r e v i s e d b y F l i n n , E .

A . , a n d D i x f C . H. ) M e G r a w - H i l l B o o k C o m p a n y I n * *

C e r v e n y , V . , 1 9 6 2 * On r e f l e c t e d a n d h e a d w a v e s a r * u n d t h e

f i r s t a n d s e c o n d c r i t i c a l p o i n t s s G e o f y s l k a l n l

S b c r n i k . 1 9 6 2 , 4 3 - 9 4 .

C e r v e n y , V . a n d R a v i n d r a , R . , 1 9 7 0 . T h e o r y o f S e i s m i c Head

W a v e s 8 U n i v e r s i t y e f T o r o n t o P r e s s ( i n p r e p a r a t i o n ) •

C o u r o n e a u , J « , 1 9 6 5 . E t u d e ^ d u " P o i n t b r i l l a n t " s u r m o d e l e s

s i s m i q u e s s G e o p h y s i c a l P r o s p e c t i n g , 1 3 . 4 0 5 - 4 3 2 .

D i x , C . H . s 1 9 5 4 . The m e t h o d o f C a g n i a r d i n s e i s m i c p u l s e

p r o b l e m s s O e » p h y s 1 c s . 2 9 7 2 2 - 7 3 8 .

D u f f , G . F . D . a n d N a y l o r , D . , 1 9 6 6 . P i f f e r_*n t i a 1 E qu a t l o n s

- o f A p p l i e d M a t h e m a t i c s s J o h n - W i l e y & S o n s , I n c .

E v a n s , J . P . , H a d l e y . C . F . , E i s l e r , J® D. a n d S i l v e r m a n , D . ,

1 9 5 4 . A t h r e e d i m e n s i o n a l s e i s m i c m o d e l w i t h b o t h

e l e c t r i c a l a n d v i s u a l o b s e r v a t i o n s s Geophysio_s_+ .19,,

2 2 0 - 2 3 6 •

E w i n g , W. M . , J a r d e t z k y , W . S . , a n d P r e s s , F®, 1 9 5 7 . H1 a s _t l_o

W a v e s i n L a y e r e d M e d i a 8 M c G r a w - H i l l B o o k C o m p a n y ,

I n * •

F a i z u l l i n , I . S ® , 1 9 6 6 . I n c r e a s e s i n r a d i a t i o n I n t e n s i t y i n

s e i s m i c m o d e l l i n g s E a r t h P h y s i c s , N o . 6 . , 3 9 1 - 3 9 4 ,

( E n g l i s h T r a n s l a t i o n ) .

Reproduced w i,h permlssion o f the copyrigh, o w n ff_ ^

permission.

F a l c o n e r , D . J . e t a l , 1 9 6 4 . T h a e f f e c t e f h i g h h u m i d i t y

e n v i r o n m e n t s e n t h e s t r e n g t h e f a d h e s i v e j o i n t s *

C h e m i s t r y a n d I n d u s t r y . J u l y . 1 9 6 4 .

F l y n n , C . R . , a n d L u k e r , H. C . , E p e x y r e s i n s s 1 9 6 8 M o d e r n

P l a s t i e s E n s v s l c p s d l a . 1 5 3 - 1 6 0 .

G a r v i n 9 W. W . , 1 9 5 6 . E x a e t t r a n s i e n t s e l u t i e n e f t h e b u r i e d

l i n e s e u r e e p r o b l e m s P r e s . R e v . S e e . ( L o n d o n ) . A,

2 3 4 . 5 2 8 - 5 4 1 .

G i l ' b e r s h t e i n , P . G . , a n d G u r v i s h , I . I . , I 9 6 0 . On t h e u s e

• f p e r f o r a t e d m a t e r i a l s f o r t w o d i m e n s i o n a l s e i s m i e

m o d e l l i n g s B u l l e t i n f f t h e I n s t i t u t e e f H i g h e s t

E d u c a t i o n s G e o l o g y a n d P r o s p e c t i n g . 1 9 6 0 . N e . 1 .

G i l b e r t , F . a n d L a s t e r , S . J " . , 1 9 6 2 . E x a l t a t i o n a n d p r o p a

g a t i o n e f p u l s e s e n a n i n t e r f a c e s B u l l . S e i s m . S e e .

A m . . 5 2 . 2 9 9 - 3 1 9 .

G r a n t , F . S . a n d W e f t , G . F . , 1 9 6 5 . I n t e r p r e t a t i o n T h e o r y

i n A p p l i e d G o s p h v s l o s g M c G r a w - H i l l B o o k C o m p a n y .

H a d a m a r d , J . , 1 9 0 0 . L e m o n s s u r l a p r o p a g a t i o n d e s e n d o s a t .

I P s e q u a t i o n s d e 1 < f c u d r o d v n a m l g u e s H e r m a n n e t C l e . ^ l

P e r i s .

H a n n o n , W. J . , 1 9 6 4 . A p p l i c a t i o n e f t h e Has k a l i - T h e m e e n

m a t r i x m e t h o d t e t h e s y n t h e s i s e f t h e s u r f a c e

m o t i o n d u e t e d i l a t i e n a l w a v e s * B u l l . B e i s . S e e .

A m . , 5 4 , 2 0 0 7 - 2 0 7 9 .

with permission of the copyright owner. Further reproduction prohibited without permission.

H a s k e l l , N. A * } 1 9 5 3 . T h e d i s p e r s i o n o f s u r f a c e w a v e s i n

m u l t i l a y e r e d m e d i a * B u l l . S e i s . S o c . Am. . 4 3 .

1 7 - 3 4 .

H e a l y f J . H. a n d P r e s s , F . , 1 9 6 0 . t w o - d i m e n s i o n a l s e i s m i c

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

a n d d e n s i t y f u n c t i o n s * G e o p h y s i c s . 2 5 , 9 8 7 - 9 9 7 .

H o l m e s , A . , 1 9 6 6 . P r i n c i p l e s o f P h y s i c a l Geo loci v s T h o m a s

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d e H o o p , A . T . , I 9 6 0 . A m o d i f i c a t i o n o f C a g p i a r d ' s m e t h o d

f o r s o l v i n g s e i s m i c p u l s e p r o b l e m s * Ap p I . S c l . R e s .

3 , 8 , 3 4 9 - 3 5 6 .

f t i y g e n , C . , 1 6 9 0 . T r a i t e d e l a L u m i e r e . An E n g l i s h t r a n s

l a t i o n I s a v a i l a b l e f r o m t h e U n i v e r s i t y o f C h i s a g o

P r e s s , C h i c a g o , 1 9 4 5 , o r D o v e r P u b l i c a t i o n s , I n c . ,

New Y o r k , 1 9 6 2 .

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S i l i c a t e s * C o r n e l l U n i v e r s i t y P r e s s .

I v a k i n , B . N . , 1 9 5 6 . S i m i l a r i t y o* f e l a i t i c wa ve p h e n o m e n a *

B u l l . A c a d . S c l . USSR s e r i e s * G e o p h y s . 1 9 5 6

N o s . 11 a n d 1 2 .


I v a k i n , B . N . , I 9 6 0 . M e t h o d s f e r e e n t x e l l i n g t h e d e n s i t y en d

e l a s t i c i t y e f a a e c f l u a d u r i n g t h e t w e - d l a e n s i e n a l

a e d e l l i n g e f s e i s a i e w a v e s i B u l l , ( i z v e s t l a ) A e a d .

S e i . US SR . G e c p h y s l e a S e r i e s . 1 9 6 0 . N e . 8 .

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G e n e r a l i z e d F u n a t l a n a i U a a b r i d g e U n i v e r s i t y P r e s s .

M e h a n t y , B . B . , 1 9 6 5 . M a d e l s e i s a e l e g y s M. A. T h e s i s . D e p t .

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P h y a l a a 8 M e G r a w - H i l l , New Y a r k .

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A a . . 5 4 , 1 - 8 .

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with permission of the copyright owner. Further reproduction prohibited without permission.

O l i v e r , ^ • y P r e s s , F . , a n d E w i n g , M . , 1 9 5 4 . Two d i m e n s i o n a l

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m u l t i p l e s a n d t * a n s a i s s i e n • • • * f i « l « n t s * G e e p h y s i_ s s

2 5 . 1 0 6 - 1 2 9 .

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g e a l a g i a a l a a d e l s s U . S . D e p t , a f t h e I n t e r l a s ,

G a a l a q l a a l S u r v e y . O p e n - f i l e r e p a r t . ,

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p r e p a g a t i e n a f s e i s m i c w a v e s . 7 , 6 1 - 7 6 .


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