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
G e o p h y s i c s L a b o r a t o r i e s
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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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 ) .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
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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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page*
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
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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 $
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. 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 *
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
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- 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 )
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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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 *
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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
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- 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 ) ‘ ^i « , « > * ( 2 . 16)
$ ^ £ 2 r U l E
4 r r f r ) ,/s*££
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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?
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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 ) .
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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
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- 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 #
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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
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- 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
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- 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
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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..
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- 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 .
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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
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- 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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 ) )
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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
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- 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*
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- 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 )
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- 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
_ 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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 ,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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 seismo- grams 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 )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 ) )
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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^* ^
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 )
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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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 ) .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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 )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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 )
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- 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 * .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 ) .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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 •
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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 ,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 ) <
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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 ^An 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 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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 £
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
»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)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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 )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
* //<& // *■*
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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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= - 1 5 0 . 0 .
permission.
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 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
• •
- 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 . *
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 -
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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 )
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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
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- 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.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
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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
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- 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
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- 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 >
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- 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
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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
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- 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
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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
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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 •
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- 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
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,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
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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
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- 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
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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
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- 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 ) ) .
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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
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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
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- 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)
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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
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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
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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
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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
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- 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
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- 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
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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 )
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- 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
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- 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)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission
Figure 3
.21
.The behavior
of the
function E
(t) -
see equation
(3.7
2).
04ro
m
o
t o
zato
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 ^a .~ ^ ° )T¥ p u t t / ( T , . Qa n d i n
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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'~^IJ
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
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- 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.
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- 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
m l c r o r, r>c
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SOU
RCE
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ION
IN
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Flfiur*- T h r t h r o r o t l c n l s ^ i rmogram*. f o r a l a y f ' r o v e r
a h a l f s p a c r f o r a J a y o r t h i c k n e s s o f 0 . 0 cm.
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 .
15-0
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SOUR
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IVER
SE
PARA
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IN
eras
.
Fiqurp d . £ 3 a . The experimental sei smogr ains
thickness
v e l o c i t i e s and d e n s i t i e s .
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s p a c e ) f o r a l a y e r t h i c k n e s s o f 0 « 0 cmr>«
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
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005
W9 NI NOIlVHHdHS U3AI30HH ~ 30H n0S
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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
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Reproduced with permission of the copyright owner Further reproduction prohibited without permission.
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 ) ,
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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).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3<+
<+u>oO3TJO303r+
*ao>Hrt-w
•»joa•jco*
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A)<A T> * H* (A3OIDH0»3
001 3
orT»
trHo? c■»3 .Q .O£3
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cnQ .H*
(A
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Suit** of
ordrr 2
5N o r m a l i z e d
* 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 '
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'-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 )
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- 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
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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 .
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- 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 )
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- 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 /
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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 .
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- 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
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- 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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 f^O " ^
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
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- 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 ^e / 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
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- 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
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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 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
SOU
RCE
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IN
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M
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
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SOURCE - R E C E I VE R REPARATI ON I N CM;r\>
2 at• 300 a•
a>a
0 >!>0 H*
TJ*< r t
f'*0 ■o
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SOUHCE - RKCHI VUU S E P A R A T I O N I I I CM
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 .
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- 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 ) .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
R e c e i v e rS o u r c 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
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- 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 .
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- 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
\
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- 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
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- 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
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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
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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
R e c e i v e rS o u r c e
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 ,
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- 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 )
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- 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
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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 *
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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 *
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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 )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
( 6 ) , f i l t e r e d ( 7 ) , a n d d i s p l a y e d ( 8 ) . T he s y s t e m i s p u l s e d
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Filter
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Differen-tiator Time Mark
Generator< r
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Time Base Generator
DecodeAmplifier
Trigger
Time Comb
TransmitterReceiver
/ / / 7 / / / / / / / / / / / M o d e l / / / / / / / / / / / / / / / / / / /
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- 104 -
© 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
Figure 5 . 2 , The t h r e e b a s i c modules of t h e s e i s m i c model sys tem*
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission
- 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'
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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%
C u r i e T e m p e r a t u r e g r e a t e r t h a n 3 3 0 ° F
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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
a n d m e d e l .
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DETAIL OF T R A N S D U C E R H E A D — 4 x
✓Transducer head (See Detail)
7
I 'd
^Electrode connection
cm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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
e n t i r e s y s t e m r e s p o n s e i n c l u d i n g t h e m o d e l we
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MAX
IMUM
/MIN
IMUM
DE
FLEC
TIO
N0.70
X e x p e r i m e n t a l
^ t h e o r e t i c a l
0-65
0.60
0.55
25.020.015.010.0
RF i q u r e r». r> Tho e x p e r i m e n t a l a n d t h e o r e t i c a l c u r v e s f o r t h o R a y l o i g h r~ T ~ / r a t i o v e r s u s s o u r c n - r e c e i v e r s e p a r a t i o nwave m i n i m u m / maximum r a t i o
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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 )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
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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 ^e 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
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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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
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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
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- 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
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- 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 %•
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- 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
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- 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
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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
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F W i u r o s . i a .-•..I *-»»«•! ?:y n i r v r s
o f inl«-i.tl**mi»: Inn*’'
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- 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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
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• 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
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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
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- 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 *
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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 ) ) ,
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'"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
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- 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
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< 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) ata^aalatad 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 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
B4 -
c°sf j cosA?j m£ l O +
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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 « > #•. .*
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F = / -
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- B 5 -
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L =• h^n f s**> p . 5/*9 A * 2% - + * € J ' j j
r -jL . / m f i t .2*0 L J
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( 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
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B6 -
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= /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
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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 .
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M.
OC CO$f> = A Aj S /V °( B . 3 9 )
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- 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)
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- hj hj ^ c o s f y c e t * g j
j * % + cosf* s,M/ j s
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- -
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
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Hence ^ a '" ?
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t*C ^ o r Sir*j/o -A Co*.f^ S)CoS 7
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2T — C C O S A ^ / a * ct / / ^
( B . 4 4 )
( B. 45 )
( B. 4 6 )
( B . 4 7 )
( B . 4 8 )
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission
- BIO -
The v a r i a t i o n of with as ^i 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 )
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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 .
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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
N e l s o n a n d S o n s L t d .
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 .
H e r , R . K . , 1 9 5 5 . T h e C o l l o i d C h f c i i l s t r y o f S i l i c a a n d
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 .
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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 .
K e l l e r , J . B . , 1 9 5 8 . A g e e a e t r i e t h e e x y e f d i f f x a e t i e n s
P x e e e e d i n g s e f S v a p s a i a i n A p p l i e d M a t h e a a t l c s . 8 ,
2 7 - 5 2 . MeGxaw-Hi 1 1 , New Y e r k .
K e l l e x , J . B . , 1 9 5 8 . D i f f r a e t i e n b y a e e n v e x s e e y l l n d e x t
E l e a t x c a a g n e t l s Wave T h e e x y S y a p e s l u a . 3 1 2 - 3 2 1 .
K l i n e , M. a n d K a y , I . W. , 1 9 6 5 . E l e e t r e a a q n e t l e T h e e x y a n d
G e e a e t r i e O p t i c s * I n t e x s e i e n e e P u b l i s h e r s .
K n e p e f f , L . , G i l b e r t , F . a n d P l i a n t , W. L , I 9 6 0 . Wave p r e -
p a g a t i e n i n a a e d l u a w i t h a s i n g l e l s y e r i J . G e e p h y s
R e s e a r e h . 6 5 , 2 6 5 - 2 7 8 .
K n e t t , C . G . , 1 8 9 9 . On t h e r e f l e e t l e n a n d x e f r a e t l e n e f
e l a a t i e w a v e s , w i t h s e l s a e l e g i e a l a p p l l e a t l e n s t
P h i l . M a g . . 5 t h s e x . , 4 8 , 6 4 - 9 7 .
L a a f o , H . , 1 9 0 4 . On t h e p x e p a g - a t l e n e f t r e a e x s e v e r t h e
s u r f a c e e f an e l a s t i c s e l i d s P h i l . T r a n s . Re_y*
S e e . ( L e n d e n ) . A, 2 0 3 . 1 - 4 2 .
L a p w e e d , E . R . , 1 9 4 9 . The d i s t u r b a n c e d u e t e a l i n e s e u x c e
i n a s e a l - i n f i n i t e n e d i u a . P h i l . T r a n s . R e v . S e e . .
( L e n d e n ) . A, 841. . 6 3 - 1 0 0 .
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L a t t e r , S . J , , F e x e a a n , J . G. a n d L l n v l l l e , A. F . , 1 9 6 S .
T h e e r e t l e a l i n v e s t i g a t i e n a f a e d e l s e i s a e g r a a s f a s
a l a y e r e v e r a h a l f - s p a e e s G e e p h v s l e s . 3 0 , 5 7 1 - 5 9 6 .
L i g h t h i l l , M. J . , I 9 6 0 . I a t r e d u s t l e n t a F e u r i e r A n a l y s i s an d
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 .
a f P h y a i s a . U n i v e r s i t y a f T a r e n t e .
M a r a a , P . M. a n d F e s h b a a h , H . , 1 9 5 3 . M e t h a d s a f T h a a r e t l s a l
P h y a l a a 8 M e G r a w - H i l l , New Y a r k .
M a s g r a v e , A. W . , 1 9 5 2 . W a v e f r e n t e h a r t s and r a y p a t h p l a t t e r s *
Q u a r t e r l y a f t h a C a l a r a d a S a h e a l a f M i n e s . 4 7 . n a . 4
N a k a a u r a , Y . , 1 9 6 4 . M a d a l e x p e r i a e n t s a n r e f r a a t l e n a r r i v a l s
f r a a a l i n e a r t r a n s l t i a n l a y e r s B u l l . S e i s . S e e .
A a . . 5 4 , 1 - 8 .
N e w l a n d a * M . , 1 9 5 3 . The d l s t u r b a n e e d u e t a a l i n e s e u r e e i n
e e a i - l n f i n i t a e l a s t i a a e d i u a w i t h a s i n g l e s u r f a e e
l a y e r s P h i l . T r a n s . R e v . S e a . L a n d . S e r . A . , 2 4 5 ,
2 1 3 - 3 0 8 .
N a r t h w a a d , T . D. a n d A n d e r s e n , D, V . , 1 9 5 3 . M e d a l s e i s a e l e g y s
B u l l . S e i s . S e a . A s . . 4 3 . 2 3 9 - 2 4 6 .
O l d h a a , R . D . , 1 9 0 0 . On t h e p r e p a g a t i e n e f e a r t h q u a k e a e t i e n
t a g r e a t d i a t a n a e s . P h i l . T r a n s . R a y . S e a . (Le .nda j i )
A, 1 3 5 - 1 7 4 .
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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
m o d e l s e i s m o l o g y : G e o p h y s i c s . 1 9 . 2 0 2 - 2 1 9 .
P e k e r i s , C . L . , 1 9 4 1 . T he p r e p a g a t i e n e f a n SH p u l s e i n a
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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 . ,
Y a n a v s k a y a , T . B . , 1 9 6 4 i n P r e b l e a s a f t h e d y n a m i c , t h e c ry , .a f
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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