Velocity analysis based on minimum entropy

VELOCITY ANALYSIS BASED ON

MINIMUM ENTROPY

V E L O C I T Y A N A L Y S I S B A S E D O N M I N I M U M E N T R O P Y n Ol f o 00 O 00 t o UI O «III t i t <t It M 111 = HI ) i IH ; ttî H1 IIHÜIi BIBLIOTHEEK TU Delft P 2092 5019 C 1 5309 865188

VELOCITY ANALYSIS BASED ON

MINIMUM ENTROPY

PROEFSCHRIFT ter verkrijging van

de graad van doctor in de

technische wetenschappen

aan de Technische Hogeschool Delft,

op gezag van de Rector Magnificus,

prof.ir. B.P.Th. Veltman,

in het openbaar te

verdedigen ten overstaan

van het College van Dekanen op

donderdag 7 juni 1984

te 16.00 uur door

DIEMER DE VRIES

natuurkundig ingenieur

geboren te Munnekeburen

Gebotekst Zoetermeer/1984

Dit proefschrift is goedgekeurd door de promotor

prof.dr.ir. A.J. Berkhout

Voor Nel, Tjalling en het nog ongeboren tweede kind

V I I

P R E F A C E

W i t h i n the group o f A c o u s t i c s o f the D e l f t U n i v e r s i t y o f T e c h n o l o g y , d e v e l o p -ment o f e c h o - a c o u s t i c a l p r o c e s s i n g t e c h n i q u e s has become the most i m p o r t a n t t o p i c on the r e s e a r c h program. I n e c h o - a c o u s t i c a l r e s e a r c h , v e l o c i t y a n a l y s i s r e q u i r e s ample a t t e n t i o n , s i n c e a p p r o p r i a t e p r o c e s s i n g of e c h o - a c o u s t i c a l d a t a i s p o s s i b l e o n l y i f the v e l o c i t y d i s t r i b u t i o n of the medium t h r o u g h w h i c h the a c o u s t i c waves have p r o p a g a t e d i s a c c u r a t e l y known. I n a d d i t i o n , v e l o c i t y may p r o v i d e v a l u a b l e i n f o r m a t i o n on the medium p r o p e r t i e s . O p e r a t i o n a l v e l o c i t y a n a l y s i s t e c h n i q u e s as d e s c r i b e d i n the l i t e r a t u r e a p p l y t o r e f l e c t i o n energy o n l y ; the v e l o c i t y i n f o r m a t i o n a v a i l a b l e i n d i f f r a c t i o n energy i s not u s e d . T h e r e f o r e , t h e r e e x i s t s a g r e a t need f o r v e l o c i t y a n a l y s i s t e c h n i q u e s w h i c h a r e a l s o a p p l i c a b l e t o d a t a w i t h an abundant amount o f d i f f r a c t i o n e n e r g y .

The i d e a t h a t l a t e r a l d i s p e r s i o n o f i n v e r t e d d a t a , as q u a n t i f i e d by minimum e n t r o p y norms, c o u l d be a s u i t a b l e c r i t e r i o n f o r v e l o c i t y a n a l y s i s o f d i f f r a c t i o n d a t a was b o r n i n D e l f t a few y e a r s ago. I n the p r e s e n t t h e s i s , the e v a l u -a t i o n of t h i s i d e -a i s d i s c u s s e d , r e s u l t i n g i n -an o p e r -a t i o n -a l t e c h n i q u e c -a l l e d MEVA: Minimum E n t r o p y V e l o c i t y A n a l y s i s .

I f e e l v e r y p r i v i l e g e d t h a t I had the o p p o r t u n i t y to do the r e s e a r c h f o r my t h e s i s i n a dynamic team l i k e the D e l f t group of A c o u s t i c s . I w i s h to e x p r e s s my s p e c i a l g r a t i t u d e t o the l e a d e r of t h a t g r o u p , my p r o m o t o r , p r o f e s s o r A . J . B e r k h o u t . I am a f r a i d t h a t , w i t h o u t h i s permanent c r i t i c a l , s t i m u l a t i n g and c o n s t r u c t i v e a t t e n t i o n - w h i c h o n l y seldom d r o v e me t o d e s p a i r - t h i s t h e s i s would never have been c o m p l e t e d .

I e x p r e s s my t h a n k s a l s o to my c o l l e a g u e s and some c o o p e r a t i n g s t u d e n t s , f o r b e i n g p r e p a r e d to d i s c u s s my i d e a s and p r o b l e m s , f o r p r o v i d i n g d a t a s e t s s u i t a b l e to t e s t my t h e o r i e s and t e c h n i q u e s , f o r g i v i n g me the o p p o r t u n i t y t o use t h e i r hardware and s o f t w a r e , and t o g e n e r a t e some i l l u s t r a t i v e f i g u r e s .

V I I I

A s p e c i a l word o f g r a t i t u d e d e s e r v e s our s e c r e t a r y Hanneke M u l d e r , who not o n l y gave me a l o t o f m e n t a l s u p p o r t , but a l s o typed out my p r e l i m i n a r y manus-c r i p t s , so t h a t I manus-c o u l d r e a d what I r e a l l y had w r i t t e n down.

I a l s o thank mr. De Knegt o f the d r a w i n g - o f f i c e and mr. S u i t e r s o f the photo-graphy s e r v i c e o f our d e p a r t m e n t , and mrs. Gerda Boone f o r t h e i r c o n t r i b u t i o n s t o the f i n a l d e s i g n o f t h i s book.

That MEVA w i l l f i n d u s e f u l a p p l i c a t i o n i n e c h o - a c o u s t i c a l p r a c t i c e i s my s i n c e r e w i s h .

Diemer de V r i e s J u n e 1984

IX

CONTENTS

I VELOCITY ANALYSIS - IMPORTANCE, POSSIBILITIES AND

LIMITATIONS 1

1.1 INTRODUCTION 1 1.2 IMPORTANCE OF VELOCITY ANALYSIS 2

1.3 SUMMARY OF APPROACHES TO VELOCITY ANALYSIS 6 1.4 SURVEY OF VELOCITY ANALYSIS TECHNIQUES 11

1.4.1 G e n e r a l remarks 11 1.4.2 V e l o c i t y a n a l y s i s t e c h n i q u e s c o n c e r n i n g r e f l e c t i o n energy . 11

1.4.3 V e l o c i t y a n a l y s i s t e c h n i q u e s c o n c e r n i n g r e f r a c t i o n energy . 15 1.4.4 V e l o c i t y a n a l y s i s t e c h n i q u e s c o n c e r n i n g d i f f r a c t i o n e n e r g y . 16

1.5 BASIC ELEMENTS OF MINIMUM ENTROPY VELOCITY ANALYSIS 17

1.6 OUTLINE OF THE FOLLOWING CHAPTERS 18

I I WAVE THEORETICAL APPROACH TO ACOUSTIC MODELING AND INVERSION 21

11.1 INTRODUCTION 21 11.2 PHYSICAL MODEL FOR DATA SIMULATION 22

11.2.1 G e n e r a l p h y s i c a l model 22 11.2.2 P h y s i c a l model f o r s i m u l a t i o n o f p r e s s u r e - t o - p r e s s u r e

p r i m a r i e s . 24 11.3 MODELING AND INVERSION I N ZERO-OFFSET TECHNIQUES 34

11.3.1 A d a p t i o n o f t h e p h y s i c a l p r i m a r y model t o t h e z e r o - o f f s e t

c o n f i g u r a t i o n 34 11.3.2 I n v e r s i o n scheme f o r z e r o - o f f s e t d a t a 37

11.4 EXAMPLES 43

I I I INFLUENCE OF VELOCITY ERRORS ON ACOUSTIC INVERSION 47

111.1 INTRODUCTION 47 111.2 INFLUENCE OF VELOCITY ERRORS ON THE SPATIAL WAVELET RESULTING

AFTER MATCHED FILTERING 47 111.3 INFLUENCE OF VELOCITY ERRORS ON THE IMAGED RESULT I N THE

SPACE-TIME DOMAIN 52

X

IV ENTROPY 63

I V . l INTRODUCTION 63 IV.2 SHANNON'S ENTROPY AND RELATED NORMS 64

IV.2.1 H i s t o r i c a l n o t e s 64 IV.2.2 Shannon's e n t r o p y : a measure o f a p r i o r i u n c e r t a i n t y . . . 65

IV.2.3 A measure o f a p r i o r i c e r t a i n t y 67 IV.2.4 E n t r o p y o f an a m p l i t u d e d i s t r i b u t i o n : a measure o f

r e s o l v i n g power 69 IV.2.5 E n t r o p y o f an a m p l i t u d e d i s t r i b u t i o n r e l a t e d t o u n c e r t a i n t y 72

IV.3 MINIMUM ENTROPY NORMS 72 IV.3.1 I n f l u e n c e o f a m p l i t u d e parameter a ^ 72

IV.3.2 G e n e r a l i z a t i o n o f t h e minimum e n t r o p y norm c o n c e p t s . . . . 75

IV.3.3 Minimum e n t r o p y norms i n l i t e r a t u r e 81 IV.4 SENSITIVITY OF MINIMUM ENTROPY NORMS TO VARIATIONS IN

ELEMENTARY DATA TRACE PROPERTIES 84 IV.5 MAXIMUM ENTROPY SPECTRAL ANALYSIS 91

I V . 6 CONCLUSIONS 91

V MINIMUM ENTROPY IN RELATION TO VELOCITY ANALYSIS 93

V. l INTRODUCTION 93 V.2 APPLICABILITY OF MINIMUM ENTROPY NORMS I N SPACE-FREQUENCY

DOMAIN 93 V.3 MINIMUM ENTROPY VELOCITY ANALYSIS 109

V.3.1 V e l o c i t y a n a l y s i s u s i n g minimum e n t r o p y norms i n the

space-f r e q u e n c y domain 109 V.3.2 V e l o c i t y a n a l y s i s u s i n g minimum e n t r o p y norms i n t h e

space-t i m e domain 115 V. 4 CONCLUSIONS 116

VI APPLICATIONS 117

V I . 1 INTRODUCTION 117 VI.2 SIMULATED ZERO-OFFSET DATA FROM DIFFRACTOR DISTRIBUTIONS . . 117

VI.2.1 D i s t r i b u t i o n s o f e l e m e n t a r y d i f f r a c t o r s 117 VI.2.2 D i f f r a c t i n g d i s c o n t i n u i t i e s on r e f l e c t o r s 124

VI.3 SEISMIC APPLICATIONS 127 VI.3.1 S i m u l a t e d CMP d a t a from one h o r i z o n t a l boundary 127

VI.3.2 S i m u l a t e d CMP d a t a from a model o f t h r e e d i p p i n g l a y e r s . . 129

XI

VI.4 MEDICAL APPLICATIONS 137 VI.4.1 Z e r o - o f f s e t d a t a measured on a phantom 137

VI.4.2 Z e r o - o f f s e t d a t a measured on r e a l t i s s u e 138 VI.5 PRACTICAL RECOMMENDATIONS FOR MEVA APPLICATION 138

APPENDIX A

A NOTE OF THE EFFECT OF VELOCITY ERRORS I N COMPUTERIZED

ACOUSTIC FOCUSING TECHNIQUES 141

APPENDIX B

MAXIMUM ENTROPY SPECTRAL ANALYSIS 145

REFERENCES 149 SUMMARY 157 SAMENVATTING 161 CURRICULUM VITAE 165

1

CHAPTER I:

VELOCITY ANALYSIS - IMPORTANCE,

POSSIBILITIES AND LIMITATIONS

I . l INTRODUCTION

The o b j e c t i v e o f t h i s t h e s i s i s t h e development o f a new v e l o c i t y a n a l y s i s t e c h n i q u e w h i c h makes use o f t h e s p a t i a l d i s p e r s i o n o f a c o u s t i c wave f i e l d s . The t e c h n i q u e c a n be employed i n a l l a r e a s where a c o u s t i c imaging i s a p p l i e d : e x p l o r a t i o n s e i s m o l o g y , ( s u b ) b o t t o m p r o f i l i n g , m e d i c a l d i a g n o s t i c s , non-d e s t r u c t i v e m a t e r i a l t e s t i n g . I n t h e above a p p l i c a t i o n s , t h e i m p o r t a n c e o f v e l o c i t y a n a l y s i s a p p e a r s i n s e v e r a l ways: ( 1 ) a c c u r a t e knowledge o f v e l o c i t y i n f o r m a t i o n i s r e q u i r e d t o o p t i m i z e wave f i e l d e x t r a p o l a t i o n r e s u l t s a s used i n s p a t i a l i n v e r s i o n t e c h n i q u e s , ( 2 ) knowledge o f p r o p a g a t i o n v e l o c i t y p r o v i d e s e x t r a i n f o r m a t i o n on c h a r a c t e r i s t i c p r o p e r t i e s o f a medium ( e . g . l i t h o l o g y , s t a t e o f human t i s s u e , c o m p o s i t i o n and c o n d i t i o n o f m a t e r i a l s ) . More s p e c i f i c a l l y i n s e i s m i c p r o c e s s i n g , v e l o c i t y i n f o r m a t i o n i s a l s o i n d i s -p e n s a b l e f o r t h e d a t a r e d u c t i o n -p r o c e d u r e known a s ' s t a c k i n g ' .

D u r i n g t h e p a s t d e c a d e s , many p a p e r s on v e l o c i t y a n a l y s i s have been p u b l i s h e d . I t a p p e a r s t h a t o p e r a t i o n a l v e l o c i t y a n a l y s i s t e c h n i q u e s a p p l y t o r e f l e c t i o n d a t a a s g e n e r a t e d by c o n t i n u o u s b o u n d a r i e s . I n t h i s c o n t e x t , d i f f r a c -t i o n energy a s g e n e r a -t e d by d i s c o n -t i n u i -t i e s a l o n g a boundary i s -t r e a -t e d a s d e t r i m e n t a l i n f o r m a t i o n w h i c h s h o u l d be s u p p r e s s e d . T h i s t h e more r e m a r k a b l e , s i n c e d i f f r a c t i o n d a t a c o n t a i n as much v e l o c i t y i n f o r m a t i o n a s r e f l e c t i o n

2

d a t a . I n t h i s t h e s i s , a v e l o c i t y a n a l y s i s t e c h n i q u e i s d e v e l o p e d w h i c h aims a t e x t r a c t i n g v e l o c i t y i n f o r m a t i o n from d i f f r a c t i o n d a t a , thus b e i n g a p p l i c a b l e t o c o n f i g u r a t i o n s as complex g e o l o g i c s t r u c t u r e s , human t i s s u e s , inhomogeneous m a t e r i a l s , e t c . On t h e o t h e r hand, t h e proposed t e c h n i q u e may be a p p l i e d t o r e f l e c t i o n d a t a as w e l l , t h u s f o r m i n g an i n t e r e s t i n g a l t e r n a t i v e t o e x i s t i n g methods. I n t h i s i n t r o d u c t o r y c h a p t e r , f i r s t l y t h e i m p o r t a n c e o f v e l o c i t y a n a l y s i s , as i n d i c a t e d above, w i l l be c o n s i d e r e d i n more d e t a i l . N e x t , d i f f e r e n t a p p r o a c h e s w i t h t h e i r a d v a n t a g e s and l i m i t a t i o n s a r e d i s c u s s e d , f o l l o w e d by a s u r v e y o f a v a i l a b l e t e c h n i q u e s . Then, t h e b a s i c i d e a s b e h i n d t h e v e l o c i t y a n a l y s i s t e c h n i q u e d e v e l o p e d i n t h i s t h e s i s a r e summarized. F i n a l l y , an o u t l i n e o f t h e o t h e r c h a p t e r s i s g i v e n .

1.2 IMPORTANCE OF VELOCITY ANALYSIS

The e s s e n t i a l r o l e o f p r o p a g a t i o n v e l o c i t y i n a c o u s t i c imaging i s most s i m p l y shown as f o l l o w s . Echo d a t a a r e r e c o r d e d as a f u n c t i o n o f t i m e . A f t e r s p a t i a l i n v e r s i o n , t h e r e s u l t r e p r e s e n t s t h e a c o u s t i c image o f t h e medium as a f u n c t i o n o f d e p t h . O b v i o u s l y , c o r r e c t 'mapping' o f time i n t o d e p t h i s d e t e r m i n e d by t h e v e l o c i t y d i s t r i b u t i o n o f t h e medium.

Hence, w i t h o u t a c c u r a t e knowledge o f t h i s v e l o c i t y d i s t r i b u t i o n a r e l i a b l e image c a n n e v e r be o b t a i n e d . When t h i s knowledge i s n o t a v a i l a b l e a p r i o r i , t h e c r u c i a l problem t o be s o l v e d i s how t o e x t r a c t v e l o c i t y i n f o r m a t i o n from t h e r e c o r d e d d a t a b e f o r e , o r i n c o r p o r a t e d i n , t h e i n v e r s i o n p r o c e s s . The e s s e n c e o f s p a t i a l i n v e r s i o n i s t h a t , f o r d i f f e r e n t p o s i t i o n s o f s o u r c e ( s ) and d e t e c t o r ( s ) , t h e r e c o r d e d r e f l e c t i o n and d i f f r a c t i o n d a t a a r e d o w n -w a r d e x t r a p o l a t e d t o t h e d i s c o n t i n u i t i e s i n t h e medium from w h i c h they o r i g i n a t e . I n c o m p u t e r i z e d i n v e r s i o n t e c h n i q u e s , downward e x t r a p o -l a t i o n i s performed u s i n g an i n v e r s i o n o p e r a t o r i n w h i c h p r o p a g a t i o n v e -l o c i t y p l a y s a v i t a l r o l e . A f t e r downward e x t r a p o l a t i o n t o a c e r t a i n d e p t h l e v e l z^, a n e s t i m a t i o n o f t h e r e f l e c t i v i t y a t d e p t h z i s o b t a i n e d by i m a g i n g , i . e . by s e l e c t i n g t h e d a t a i n a time window around t = 0. By r e p e a t i n g t h i s p r o c e d u r e f o r a l l d e p t h l e v e l s o f i n t e r e s t ( z = z. , z0, . . , z . , ) , an

m 1 2- M

3

(downward e x t r a p o l a t i o n ) and i m a g i n g i s i n s e i s m i c n o m e n c l a t u r e d e n o t e d as m i g r a t i o n .

I t s h o u l d be n o t e d t h a t i n r e a l - t i m e a c o u s t i c i m a g i n g - o f t e n a p p l i e d i n m e d i c a l d i a g n o s t i c s and then b e t t e r known as a c o u s t i c f o c u s s i n g -t h e above c o n c e p -t a l s o h o l d s : f o c u s s i n g i s an a p p r o x i m a -t i o n o f wave f i e l d i n v e r s i o n . F o r a w e l l - r e s o l v e d and c o r r e c t l y p o s i t i o n e d f o c u s p o i n t i t i s n e c e s s a r y t h a t a good e s t i m a t e of the medium v e l o c i t y o r v e l o c i t y d i s t r i b u -t i o n i s u s e d .

F i g . 1-1 shows the e f f e c t of a v e l o c i t y e r r o r on the image o f a f i n i t e , t i l t e d r e f l e c t o r w i t h a c e n t r a l o p e n i n g ( ' r e f l e c t i v i t y d r o p ' ) . I t i s seen i n F i g s . I - l c , e t h a t a v e l o c i t y e r r o r c a u s e s a p o s i t i o n i n g e r r o r as w e l l as d e t e r i o r a t i o n o f l a t e r a l r e s o l u t i o n : the edges o f the r e f l e c t o r image a r e not w e l l d e f i n e d , i . e . the image becomes more d i s p e r s e d .

B e s i d e s b e i n g a parameter o f h i g h i m p o r t a n c e i n the i n v e r s i o n p r o c e s s , p r o p a g a -t i o n v e l o c i -t y i s a c h a r a c -t e r i s -t i c m a -t e r i a l c o n s -t a n -t . Hence, a c c u r a -t e knowledge o f v e l o c i t i e s can be used f o r c h a r a c t e r i z a t i o n o f m a t e r i a l s . I n t a b e l 1/1, the p r o p a g a t i o n v e l o c L t i e s o f p r e s s u r e waves a r e g i v e n f o r s e v e r a l media as found i n the t h r e e a r e a s o f a p p l i c a t i o n mentioned i n the i n t r o d u c t i o n : s e d i m e n t s ( s e i s m o l o g y ) , t i s s u e s ( m e d i c a l d i a g n o s t i c s ) and c o n s t r u c t i o n m a t e r i a l s ( n o n - d e s t r u c t i v e t e s t i n g ) .

I t i s seen t h a t , f o r the human t i s s u e s c o n s i d e r e d ( e x c e p t f a t ) , t h e v e l o c i t y v a l u e s a r e found w i t h i n a s m a l l range around 1560 m/s. Hence, t i s s u e c h a r a c t e r i z a t i o n on v e l o c i t y c r i t e r i a i s o n l y p o s s i b l e i f the v e l o c i t y v a l u e s c a n be a c c u r a t e l y d e t e r m i n e d . F o r t i s s u e s i n v i v o , c h a r a c t e r i z a t i o n i s even more d i f f i c u l t , s i n c e p r o p a g a t i o n v e l o c i t y v a r i e s d u r i n g c o n t r a c t i o n ( M o l , 1 9 8 1 ) . F i n a l l y , we m e n t i o n the n e c e s s i t y o f v e l o c i t y a n a l y s i s f o r a s p e c i f i c s e i s m i c a p p l i c a t i o n . I n c o n v e n t i o n a l s e i s m i c p r o c e s s i n g , d a t a t r a c e s f o r w h i c h s o u r c e and d e t e c t o r p o s i t i o n s have the same m i d p o i n t (Common Mid P o i n t g a t h e r ) , a r e combined ( ' s t a c k e d ' ) t o one t r a c e , i . e . enhancement o f

s i g n a l t o n o i s e r a t i o i s o b t a i n e d by d a t a r e d u c t i o n . The p r i n -c i p l e i s i l l u s t r a t e d i n F i g . 1-2 f o r a h o r i z o n t a l r e f l e -c t o r . Note t h a t t h e CMP

F i g u r e 1-1; E f f e c t o f a v e l o c i t y e r r o r on t h e image o f a f i n i t e , t i l t e d r e f l e c t o r w i t h a c e n t r a l ' r e f l e c t i v i t y d r o p ' , a: model b: z e r o - o f f s e t r e s p o n s e c: image a f t e r e x t r a p o l a t i o n w i t h -10% r e l a t i v e v e l o c i t y e r r o r d: I d . w i t h c o r r e c t v e l o c i t y e: i d . w i t h +10% r e l a t i v e v e l o c i t y e r r o r

T a b l e 1/1

Some media and t h e i r p r o p a g a t i o n v e l o c i t i e s f o r p r e s s u r e waves.

medium v e l o c i t y [m/s] s e d i m e n t s w e a t h e r e d l a y e r c a . 1000 ( s e i s m o l o g y ) u n c o n s o l i d a t e d sands and c l a y s c a . 2000 c o n s o l i d a t e d sands and c l a y s c a . 3000 c a r b o n a t e r o c k s c a . 4000 t i s s u e s f a t 1445 + 35 ( m e d i c a l s p l e e n 1555 + 35 d i a g n o s t i c s ) l i v e r 1580 + 30 k i d n e y 1563 + 5 h e a r t c a . 1570 b l o o d c a . 1570 bone c a . 3000 m a t e r i a l s c o n c r e t e c a . 4500 ( n o n - d e s t r u c t i v e s t e e l c a . 5900 t e s t i n g ) p e r s p e x c a . 2700 g a t h e r forms a h y p e r b o l i c r e f l e c t i o n p a t t e r n ( F i g . I - 2 b ) . I t i s seen t h a t s t a c k i n g i s a two-step p r o c e s s : t h e CMP g a t h e r i s l i n e d up by 'nor-mal moveout c o r r e c t i o n ' ( F i g . I - 2 c ) and added t o a p p r o x i m a t e a z e r o - o f f s e t t r a c e a t t h e g a t h e r m i d p o i n t ( F i g . I - 2 d ) . F o r t h e s i m p l e case c o n s i d e r e d here a p p r o p r i a t e normal moveout c o r r e c t i o n i s f u l l y d e t e r m i n e d by t h e p r o p a g a t i o n v e l o c i t y between s u r f a c e and r e f l e c t o r .

6 *- X *• X *>x *-X I I

a

b

c

d

F i g u r e 1-2: S t a c k i n g o f a common m i d p o i n t g a t h e r . a: s o u r c e - d e t e c t o r c o n f i g u r a t i o n and r a y geometry f o r a h o r i z o n t a l r e f l e c t o r b: common m i d p o i n t g a t h e r

c: common m i d p o i n t g a t h e r a f t e r normal moveout c o r r e c t i o n d: p o s t - s t a c k t r a c e , a p p r o x i m a t i n g a z e r o - o f f s e t t r a c e a t t h e

From t h e c o n s i d e r a t i o n s i n t h i s s e c t i o n t h e c o n c l u s i o n c a n be drawn t h a t a c c u r a t e v e l o c i t y a n a l y s i s i s v e r y i m p o r t a n t i n a c o u s t i c ( e s p e c i a l l y s e i s m i c ) i m a g i n g .

I n o r d e r t o e x p l o r e t h e i n t e r i o r s t r u c t u r e o f a medium, t h e a r e a o f i n t e r e s t may be c o v e r e d w i t h a s u f f i c i e n t l y dense g r i d o f s o u r c e s and d e t e c t o r s , and an a l i a s i n g - f r e e c o m p l e t e d a t a s e t may be r e c o r d e d , i . e . f o r each s o u r c e t h e r e s p o n s e i s m o n i t o r e d by a f u l l range o f d e t e c t o r s , see F i g . I - 3 a . Then, c o m p l e t e a c o u s t i c i n f o r m a t i o n about t h e medium w i l l be o b t a i n e d .

m i d p o i n t

7 X O I X X X X X X X X X X ® X X X X ® X X X X X X X X X X X X X X X X X X X i X X X X X X X I X X X X X X X X X X X X X X X i x x x x x x x x a b F i g u r e 1-3: Schematic c o n f i g u r a t i o n s o f s o u r c e s ( o ) and d e t e c t o r s ( x ) t o o b t a i n a: a complete d a t a s e t b: a m u l t i - t r a c e d a t a s e t

N o t e t h a t each row r e p r e s e n t s t h e g e o m e t r i c c o n f i g u r a t i o n f o r one p h y s i c a l e x p e r i m e n t .

As w i l l be argued more e x t e n s i v e l y i n c h a p t e r I I , a complete d a t a s e t c a n e l e g a n t l y be r e p r e s e n t e d i n t h e f o r m o f a d a t a m a t r i x , see F i g . 1-4.

F i g u r e 1-4: R e p r e s e n t a t i o n o f a c o m p l e t e d a t a s e t as a d a t a m a t r i x .

I n t h i s m a t r i x , each column r e p r e s e n t s t h e d a t a g e n e r a t e d i n one p h y s i c a l e x p e r i m e n t , i . e . t h e r e s p o n s e s o f one s o u r c e a t t h e r e s p e c t i v e d e t e c t o r s . We d e n o t e such a column o f t h e d a t a m a t r i x a s a s o u r c e g a t h e r ^ . E a c h row

1) I n s e i s m i c n o m e n c l a t u r e , t h e d a t a from one p h y s i c a l e x p e r i m e n t i s c a l l e d a s e i s m i c r e c o r d .

8

r e p r e s e n t s the r e s p o n s e o f a l l s o u r c e s a t a g i v e n d e t e c t o r , i . e . a d e t e c t o r g a t h e r . I t i s e a s i l y seen t h a t common m i d p o i n t g a t h e r s as d i s c u s s e d i n t h e p r e -v i o u s s e c t i o n appear as a n t i d i a g o n a l s i n the d a t a m a t r i x . The z e r o - o f f s e t d a t a f o r m the main d i a g o n a l . The s u b d i a g o n a l s r e p r e s e n t common o f f s e t s e c t i o n s , g i -v i n g f o r each s o u r c e the r e s p o n s e a t a d e t e c t o r on a g i -v e n d i s t a n c e ( o f f s e t ) . A c o m p l e t e d a t a s e t may i n c l u d e r e f l e c t i o n as w e l l as r e f r a c t i o n and d i f f r a c -t i o n e n e r g y . Hence, s p a -t i a l i n v e r s i o n o f a c o m p l e -t e d a -t a s e -t w i -t h o u -t any d a -t a r e d u c t i o n ( ' p r e - s t a c k i n v e r s i o n ' ) y i e l d s f u l l a c o u s t i c i n f o r m a t i o n about t h e medium. I n s p a t i a l i n v e r s i o n o f a c o m p l e t e d a t a s e t , the d a t a o f each p h y s i c a l e x p e r i -ment, i . e . each column o f the d a t a m a t r i x , i s i n v e r t e d . The r e s u l t t h u s o b t a i n e d i s g a t h e r e d a c c o r d i n g t o t h e d e p t h p o i n t s o f o r i g i n , thus f o r m i n g Common TJepth P_oint g a t h e r s . A l l CDP g a t h e r s a r e s t a c k e d and combined t o form a t r u e CDP-stacked s e c t i o n , r e p r e s e n t i n g t h e a c o u s t i c image o f the medium

c o n s i d e r e d . P e r f o r m i n g t h i s m i g r a t i o n p r o c e s s , v e l o c i t y i n f o r m a t i o n c a n be e x t r a c t e d i n t h r e e d i f f e r e n t ways:

( 1 ) CDP g a t h e r s have maximum l a t e r a l c o h e r e n c y i f the c o r r e c t v e l o -c i t y d i s t r i b u t i o n o f the medium has been used i n t h e i n v e r s i o n p r o -c e s s , ( 2 ) CDP-stacked t r a c e s have maximum a m p l i t u d e ( o f the e n v e l o p e ) i f

o b t a i n e d w i t h c o r r e c t v e l o c i t y ,

( 3 ) the c o m p l e t e CDP-stacked s e c t i o n ( i . e . the p r e - s t a c k m i g r a t i o n r e s u l t ) has minimum l a t e r a l d i s p e r s i o n i f o b t a i n e d w i t h c o r r e c t v e l o c i t y . As shown i n t h i s t h e s i s , l a t e r a l d i s p e r s i o n can be p r o p e r l y q u a n t i -f i e d by norms based on the c r i t e r i o n o -f m i n i m u m e n t r o p y .

I n v e r s i o n o f c o m p l e t e d a t a s e t s r e q u i r e s a c q u i s i t i o n and p r o c e s s i n g o f huge numbers o f d a t a . Hence, f o r p r a c t i c a l and economic r e a s o n s , c o m p l e t e d a t a s e t s a r e a p p r o x i m a t e d by m u l t i - t r a c e d a t a s e t s o b t a i n e d w i t h a s p a r s e d s o u r c e - d e t e c t o r c o n f i g u r a t i o n , an example o f w h i c h i s g i v e n i n

F i g . I-3b. Note t h a t m i s s i n g d e t e c t o r s c o r r e s p o n d w i t h z e r o ' s i n the columns o f t h e d a t a m a t r i x .

T r a d i t i o n a l l y i n s e i s m i c a p p l i c a t i o n s , m u l t i - t r a c e d a t a s e t s a r e a r r a n g e d t o c o m m o n m i d p o i n t (CMP) g a t h e r s when used as i n p u t f o r v e l o c i t y a n a l y s i s p r o c e d u r e s . T h i s p r e f e r e n c e o f CMP g a t h e r s has two b a s i c r e a s o n s :

9

( 1 ) I n many p r a c t i c a l s i t u a t i o n s , r e f l e c t i o n energy w i t h i n one CMP g a t h e r i s g e n e r a t e d by a s m a l l p a r t o f each boundary: see F i g . I-2a f o r t h e s i m p l e s i t u a t i o n o f a h o r i z o n t a l r e f l e c t o r . Hence, i n many p r a c t i c a l s i t u a -t i o n s -t h e b o u n d a r i e s may be c o n s i d e r e d as f l a -t w i -t h i n a CMP g a -t h e r r a n g e .

( 2 ) I n many p r a c t i c a l s i t u a t i o n s , r e f l e c t i o n d a t a from a f l a t boundary w i t h i n one CMP g a t h e r form p a t t e r n s w h i c h may be c o n s i d e r e d as h y p e r -b o l i c , see F i g . I-2-b. The a s y m p t o t e s o f each h y p e r -b o l a a r e d e t e r m i n e d by t h e s t r u c t u r e and t h e v e l o c i t y d i s t r i b u t i o n o f t h e medium between s u r f a c e and r e f l e c t i v i t y boundary c o n s i d e r e d . Hence, v e l o c i t y i n f o r m a t i o n c a n be o b t a i n e d by p r o c e s s i n g s i m p l e h y p e r b o l i c r e s p o n s e s .

V e l o c i t y i n f o r m a t i o n c a n be e x t r a c t e d from CMP g a t h e r s i n s e v e r a l ways: ( 1 ) a f t e r n o r m a l moveout c o r r e c t i o n ( F i g . I - 2 c ) , a CMP g a t h e r has maximum

c o h e r e n c y i f t h e c o r r e c t i o n has been performed a c c o r d i n g t o t h e c o r r e c t h y p e r b o l a . A l s o , i n t h i s c a s e ,

( 2 ) t h e s t a c k e d t r a c e ( F i g . I-2d) has maximum r e f l e c t i o n a m p l i t u d e .

M o r e o v e r , I t c a n be shown t h a t an e s t i m a t i o n o f t h e s t a c k c a n be o b t a i n e d by m i g r a t i n g t h e CMP g a t h e r . Hence, v e l o c i t y i n f o r m a t i o n c a n be e x t r a c t e d from ( 3 ) t h e maximum r e f l e c t i o n a m p l i t u d e o f t h e m i g r a t e d CMP g a t h e r , o r , as p r o p o s e d i n t h i s t h e s i s , ( 4 ) t h e l a t e r a l e x t e n s i o n o f t h e m i g r a t e d CMP g a t h e r q u a n t i f i e d by m i n i -m u -m e n t r o p y nor-ms. I n p r a c t i c a l a p p l i c a t i o n s CMP v e l o c i t y a n a l y s i s i s c o n c e n t r a t e d on r e f l e c t i o n e n e r g y o n l y . To cope w i t h r e f r a c t i o n d a t a as w e l l , s p e c i a l t e c h n i q u e s a r e r e q u i r e d , e.g. xp mapping ( s e e s u b s e c t i o n 1.4.3). D i f f r a c t i o n energy i s a c t u -a l l y d e t r i m e n t -a l , due t o i n t e r f e r e n c e w i t h t h e h y p e r b o l i c r e f l e c t i o n p -a t t e r n s . I n c a s e o f h i g h l y d i s c o n t i n u o u s complex m e d i a , echo d a t a i s measured i n w h i c h d i f f r a c t i o n energy p r e d o m i n a t e s . Hence, f o r s u c h media CMP v e l o c i t y a n a l y s i s i s n o t a r e a l i s t i c p r o p o s i t i o n : o t h e r a p p r o a c h e s a r e t o be d e v e l o p e d .

As m e n t i o n e d i n the i n t r o d u c t i o n o f t h i s c h a p t e r , o p e r a t i o n a l v e l o c i t y a n a l y -s i -s t e c h n i q u e -s e x t r a c t i n g v e l o c i t y i n f o r m a t i o n -s p e c i f i c a l l y from d i f f r a c t i o n e n e r g y a r e s c a r c e l y i n d i c a t e d i n l i t e r a t u r e . I n t h i s t h e s i s s u c h a t e c h n i q u e i s p r o p o s e d . The b a s i c i d e a b e h i n d t h i s t e c h n i q u e i s t h a t t h e l a t e r a l

10 d i s p e r s i o n o f d i f f r a c t i o n energy i s m i n i m a l i f i n v e r t e d w i t h c o r r e c t v e l o c i t y , see a l s o F i g . 1-1. L a t e r a l d i s p e r s i o n appears a s a l a t e r a l e x t e n s i o n e f f e c t w h i c h p r o p e r l y c a n be q u a n t i f i e d by norms based on t h e c r i t e r i o n o f m i n i m u m e n t r o p y . I n s t e a d o f CMP g a t h e r s , c o m m o n o f f s e t ( i n c l u d i n g z e r o - o f f s e t ) d a t a s h o u l d be used f o r v e l o c i t y a n a l y s i s on d i f f r a c t i o n d a t a , i . e . t h e d i s t a n c e between s o u r c e s and d e t e c t o r s s h o u l d be c o n s t a n t ( i n c l u d i n g z e r o ) . Note t h a t , i n s t e a d o f z e r o -o f f s e t d a t a , s t a c k e d d a t a ( s e e F i g . I-2d) c a n be used a s w e l l .

The above c l a s s i f i c a t i o n o f v e l o c i t y a n a l y s i s approaches i s summarized i n t a b l e 1-2. I n t h e n e x t s e c t i o n a s u r v e y i s g i v e n how t h e s e p r i n c i p l e s a r e worked o u t i n v a r i o u s v e l o c i t y a n a l y s i s t e c h n i q u e s as proposed i n t h e l i t e r a t u r e . T a b l e 1/2 C l a s s i f i c a t i o n o f d i f f e r e n t approaches t o v e l o c i t y a n a l y s i s . d a t a arrangement d a t a t o be i n c l u d e d i n v e l o c i t y a n a l y s i s v e l o c i t y a n a l y s i s c r i t e r i a m u l t i - r e c o r d d a t a s e t r e f l e c t i o n r e f r a c t i o n d i f f r a c t i o n * c o h e r e n c y o f CDP g a t h e r *maximum a m p l i t u d e o f s t a c k e d CDP g a t h e r *minimum e n t r o p y o f CDP-stacked s e c t i o n CMP g a t h e r s r e f l e c t i o n * c o h e r e n c y o f normal moveout c o r r e c t e d CMP g a t h e r *maximum a m p l i t u d e o f s t a c k e d CMP g a t h e r *maximum a m p l i t u d e o r minimum e n t r o p y o f m i g r a t e d CMP g a t h e r common o f f s e t s e c t i o n s ( i n c l u -d i n g z e r o - o f f s e t and s t a c k e d d a t a ) d i f f r a c t i o n *minimum e n t r o p y o f m i g r a t e d common o f f s e t s e c t i o n

11

1.4 SURVEY OF VELOCITY ANALYSIS TECHNIQUES 1.4.1 General remarks

I n human t i s s u e s , the p r o p a g a t i o n v e l o c i t i e s o f p r e s s u r e waves a p p r o x i m a t e t h e v a l u e 1560 m/s w i t h i n a few p e r c e n t s , a s was seen i n T a b l e 1/1. Hence, i n i m a g i n g t e c h n i q u e s employed i n m e d i c a l d i a g n o s t i c s , t h i s v a l u e i s commonly used i n d e p e n d e n t l y o f the s p e c i f i c t i s s u e s c o n s i d e r e d . Due t o t h e random and r e l a t i v e l y s m a l l f l u c t u a t i o n s i n the a c t u a l v e l o c i t y d i s t r i b u t i o n , t h i s s i m p l i f i e d approach o f t e n y i e l d s imaged r e s u l t s o f a c c e p t a b l e q u a l i t y . I n n o n - d e s t r u c t i v e t e s t i n g o f m a t e r i a l s , t h e t e s t specimen u s u a l l y c o n s i s t s o f a s i n g l e b u l k medium ( e . g . c o n c r e t e , s t e e l ) , the v e l o c i t y o f w h i c h i s known by good a p p r o x i m a t i o n . I n s e i s m i c s , however, t h e s e d i m e n t s o f i n t e r e s t show l a r g e v e l o c i t y v a r i a t i o n s (500 m/s < c < 5000 m/s) i n b o t h v e r t i c a l and l a t e r a l d i r e c t i o n s . T h e r e f o r e , a c c u r a t e v e l o c i t y a n a l y s i s i s a t o p i c o f i n c r e a s i n g i n t e r e s t i n

t h i s f i e l d and most o p e r a t i o n a l v e l o c i t y a n a l y s i s t e c h n i q u e s have t h e i r o r i g i n i n g e o p h y s i c a l r e s e a r c h . I n t h i s p a r a g r a p h , we s h a l l g i v e a s u r v e y on v e l o c i t y a n a l y s i s t e c h n i q u e s o f c o n t e m p o r a r y i n t e r e s t , p a y i n g a t t e n t i o n t o u n d e r l y i n g p r i n c i p l e s and a p p l i c a -b i l i t y . F o r r e a s o n s o f s u r v e y a -b i l i t y , we a r r a n g e the v e l o c i t y a n a l y s i s t e c h n i q u e s i n t h r e e c a t e g o r i e s , v i z . : ( 1 ) t e c h n i q u e s u s i n g r e f l e c t i o n energy a s i n p u t d a t a , ( 2 ) t e c h n i q u e s u s i n g r e f r a c t i o n energy a s i n p u t d a t a , ( 3 ) t e c h n i q u e s u s i n g d i f f r a c t i o n energy as i n p u t d a t a .

1.4.2 V e l o c i t y a n a l y s i s techniques concerning r e f l e c t i o n energy

Most v e l o c i t y a n a l y s i s t e c h n i q u e s a r e based on p r o c e s s i n g o f CMP r e f l e c t i o n d a t a . As d i s c u s s e d i n s e c t i o n 1.3, such d a t a form a t l e a s t a p p r o x i -m a t e l y - h y p e r b o l i c p a t t e r n s i n the t w o - d i -m e n s i o n a l s p a c e - t i -m e ( x , t ) domain. The apex o f e a c h h y p e r b o l a i s g i v e n by the z e r o o f f s e t ( m i d -p o i n t ) t r a v e l time t o the boundary c o n s i d e r e d , whereas the a s y m -p t o t e s a r e d e s c r i b e d by

X — Vs t a c kt

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The v a l u e o f v e l o c i t y vstack ^s d e t e r m i n e d by d e p t h s , shapes and v e l o c i t i e s

o f a l l l a y e r s between s u r f a c e and boundary.vstack i s denoted a s s u c h , s i n c e

s t a c k i n g o f CMP d a t a y i e l d s o p t i m a l r e s u l t i f t h e n o r m a l moveout c o r r e c t i o n ( s e e F i g . I - 2 c ) i s performed a c c o r d i n g t o t h e h y p e r b o l a w i t h c o r r e c t apex and a s y m p t o t e s , i . e . c o r r e c t vstacjc* Thus, o p t i m i z a t i o n o f t h e m o v e o u t - c o r r e c t e d

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

The above v e l o c i t y a n a l y s i s t e c h n i q u e has some l i m i t a t i o n s . As mentioned b e f o r e , i t o n l y d e a l s w i t h r e f l e c t i o n d a t a as g e n e r a t e d by a p p r o x i m a t e l y f l a t b o u n d a r i e s . M o r e o v e r , t h e CMP r e s p o n s e i s p u r e l y h y p e r b o l i c o n l y f o r a homogeneous o v e r b u r d e n . F o r an inhomohomogeneous o v e r b u r d e n t h e r e s p o n s e s a r e a p p r o x i -m a t e l y h y p e r b o l i c f o r s -m a l l o f f s e t s o n l y . The a p p r o x i -m a t i o n d e t e r i o r a t e s w i t h i n c r e a s i n g d i p s and i n c r e a s i n g c u r v a t u r e s . Due t o t h e s e i m p e r f e c t i o n s , t h e a c c u r a c y o f t h e above v e l o c i t y a n a l y s i s t e c h n i q u e d e c r e a s e s w i t h d e p t h .

V e l o c i t y a n a l y s i s based on p r o c e s s i n g o f r e f l e c t i o n d a t a has been a p p l i e d l o n g s i n c e . DUrbaum (1954) p u b l i s h e d t h e t h e o r e t i c a l b a s i s f o r t h e h y p e r b o l i c

model. D i x (1955) i n d i c a t e s how, f o r a h o r i z o n t a l l y l a y e r e d system, t h e i n t e r v a l v e l o c i t i e s o f t h e r e s p e c t i v e l a y e r s c a n be r e c u r s i v e l y c a l c u l a t e d f r o m t h e e s t i m a t e d s t a c k i n g v e l o c i t i e svstack- Tne f o r m u l a d e s c r i b i n g t h i s

r e c u r s i v e r e l a t i o n has become a c l a s s i c under t h e name ' D i x f o r m u l a ' . Taner and K o e h l e r (1969) f o r m u l a t e c o h e r e n c y c r i t e r i a t o e s t i m a t e vstac]c ana they

a p p l y , w i t h l i m i t e d a c c u r a c y , t h e D i x f o r m u l a a l s o t o d i p p i n g b o u n d a r i e s . Brown (1969) c o n s i d e r s t h e i n f l u e n c e o f l o n g - o f f s e t d a t a . I n t h e e a r l y

seven-t i e s , much work has been done seven-t o e x seven-t e n d seven-t h e v a l i d i seven-t y o f seven-t h e D i x model seven-t o d i p p i n g b o u n d a r i e s : L e v i n ( 1 9 7 1 ) , L a r n e r and Rooney ( 1 9 7 3 ) , Shah ( 1 9 7 3 ) , E v e r e t t ( 1 9 7 4 ) . K r e y (1976) g e n e r a l i z e s , f o r s m a l l o f f s e t s , t h e t h e o r y o f DUrbaum t o a r b i t r a r y d i p p i n g o r c u r v e d b o u n d a r i e s i n t h r e e d i m e n s i o n s and H u b r a l (1976) p r o v e s t h e a p p l i c a b i l i t y o f t h i s t h e o r y f o r p r a c t i c a l s i t u a -t i o n s . May and S -t r a l e y (1979) r e f i n e -t h e m a -t h e m a -t i c a l d e s c r i p -t i o n o f -t h e d e e p e r boundary r e s p o n s e s , t a k i n g h i g h e r o r d e r terms i n t o a c c o u n t . B e r r y h i l l (1979) a p p l i e s w a v e f i e l d 'datuming' ( a k i n d o f v e l o c i t y r e p l a c e m e n t t e c h n i q u e ) t o o b t a i n b e t t e r CMP r e s u l t s f o r t h e u n d e r l y i n g s t r a t a . K h a t t r i e t a l . (1980) propose t h e F i b o n a c c i e s t i m a t i o n t e c h n i q u e as an a l t e r n a t i v e t o c o h e r e n c y t e c h n i q u e s . H a j n a l and Sereda (1981) g i v e an a n a l y s i s o f p o s s i b l e e r r o r s made i n a p p l y i n g t h e D i x f o r m u l a . R e c e n t l y , e x t e n s i o n o f r e f l e c t i o n d a t a v e l o c i t y

13

a n a l y s i s to media w i t h l a t e r a l v e l o c i t y v a r i a t i o n s has been d i s c u s s e d : H u b r a l ( 1 9 8 0 ) , L y n n and C l a e r b o u t ( 1 9 8 2 ) , L o i n g e r ( 1 9 8 3 ) .

S i n c e o n l y r e f l e c t i o n d a t a obey the s i m p l e h y p e r b o l i c CMP model i l l u s t r a t e d i n F i g . 1-2, the p r e s e n c e o f r e f r a c t i o n and p a r t i c u l a r l y

d i f f r a c t i o n energy d e t e r i o r a t e s the a p p l i c a b i l i t y o f CMP v e l o c i t y a n a l y s i s t e c h n i q u e s . T h i s was a l s o mentioned by Taner e t a l . (1970) and l a t e r by B l a c k b u r n ( 1 9 8 0 ) . A r g u i n g t h a t downward wave f i e l d e x t r a p o l a t i o n f o c u s s e s d i f f r a c t i o n energy and thus d i m i n i s h e s i t s s m e a r i n g e f f e c t , D o h e r t y and C l a e r b o u t (1976) p r o p o s e t o m i g r a t e the p r e - s t a c k m u l t i - t r a c e d a t a w i t h a r o u g h l y e s t i m a t e d v e l o c i t y b e f o r e p i c k i n g CMP g a t h e r s and e s t i m a t i n g

Vs t a c k kv a p p l lc a ti °n °f t n e u s u a l c o h e r e n c y t e c h n i q u e s . A more t h o r o u g h

a p p r o a c h i s t o m i g r a t e the p r e - s t a c k d a t a w i t h d i f f e r e n t v e l o c i t y v a l u e s , e s t i m a t i n g d i r e c t l y the m i g r a t i o n v e l o c i t y by a p p l y i n g c o h e r e n c y t e c h n i q u e s t o t h e m i g r a t e d r e s u l t . T h i s p r o c e d u r e was p r o p o s e d by Gardner e t a l . (1974) and S a t t l e g g e r ( 1 9 7 5 ) , and a p p l i e d i n p r a c t i c e by Dohr and S t i l l e r (1975) and S a t t l e g g e r e t a l . ( 1 9 7 6 ) . Owusu e t a l . (1983) e x t e n d the a p p l i c a t i o n o f t h i s method t o t h r e e - d i m e n s i o n a l d a t a u s i n g a f a s t m i g r a t i o n a l g o r i t h m .

As m e n t i o n e d i n s e c t i o n 1.3, m i g r a t i o n o f a CMP g a t h e r y i e l d s an e s t i m a t e o f t h e s t a c k t o be o b t a i n e d from t h a t g a t h e r . From p a r a x i a l ( i . e . s m a l l o f f s e t ) CMP d a t a an a c c u r a t e s t a c k e s t i m a t e can be found even a f t e r downward e x t r a -p o l a t i o n w i t h an e r r o n e o u s v e l o c i t y : t h e n , i m a g i n g must be -performed u s i n g a t i m e window not around t . =0 as u s u a l , but around a time t . ^0. The s t a c k

im lm i s w e l l - r e s o l v e d but p o s i t i o n e d a t an e r r o n e o u s d e p t h : v e l o c i t y e r r o r and

d e p t h e r r o r a r e exchanged. The a c t u a l v a l u e o f depends on the v e l o c i t y e r r o r , so t h a t t h i s v a l u e c o n t a i n s v e l o c i t y i n f o r m a t i o n . T h i s phenomenon was f i r s t d e s c r i b e d by D o h e r t y and C l a e r b o u t ( 1 9 7 4 ) . Y i l m a z and Chambers (1980) e v a l u a t e d the p o s s i b i l i t y o f v e l o c i t y e x t r a c t i o n , a p p l y i n g a s p e c i a l mapping p r o c e d u r e t o d e t e r m i n e t , and u s i n g maximum a m p l i t u d e as a c r i t e r i o n f o r o p t i m a l m i g r a t i o n . I n a paper on the e f f e c t s o f v e l o c i t y e r r o r s on m i g r a t i o n and f o c u s s i n g , De V r i e s and B e r k h o u t (1984) p l a c e the e x c h a n g e a b i l i t y o f v e l o -c i t y e r r o r s and d e p t h e r r o r s , as p r e s e n t i n r e -c u r s i v e m i g r a t i o n t e -c h n i q u e s , i n a wave t h e o r e t i c a l c o n t e x t . They a l s o i n d i c a t e the p o s s i b i l i t y t o e x t r a c t v e l o c i t y i n f o r m a t i o n from the time s h i f t d e n o t e d above as t , .

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F i g u r e 1-5: S c h e m a t i c i l l u s t r a t i o n o f T-p mapping ( s l a n t s t a c k i n g ) o f CMP r e f l e c t i o n and r e f r a c t i o n d a t a

a: CMP g a t h e r on a h o r i z o n t a l boundary between two media w i t h v e l o -c i t i e s -c^ and C 2 ( xc i s h a l f o f f s e t c o r r e s p o n d i n g w i t h c r i t i c a l a n g l e ) b: t o t h e d e f i n i t i o n o f p a r a m e t e r s p and T c: CMP g a t h e r a f t e r T-p mapping d: CMP g a t h e r on a h o r i z o n t a l 3-boundary c o n f i g u r a t i o n a f t e r T-p mapping

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1.4.3 V e l o c i t y a n a l y s i s techniques concerning r e f r a c t i o n energy

C o m p r e s s i o n a l p-waves i n c i d e n t on an a c o u s t i c boundary under c r i t i c a l a n g l e g e n e r a t e , b e s i d e s a r e f l e c t e d wave, a r e f r a c t e d wave. T h i s r e f r a c t e d p-wave ( i n g e o p h y s i c s a l s o c a l l e d 'head wave') t r a v e l s a l o n g the boundary w i t h t h e v e l o c i t y o f the l o w e r l a y e r . R e f r a c t i o n energy i s r a d i a t e d t o the s u r -f a c e , a p p e a r i n g i n the CMP r e s p o n s e a l o n g a s t r a i g h t l i n e , see F i g . I 5 a . By t r a n s f o r m i n g the x c o o r d i n a t e i n t o h o r i z o n t a l 'slowness' p a r a -meter p and the t - c o o r d i n a t e i n t o i n t e r c e p t t i m e T, see F i g . I-5b, the hyper-b o l i c p a r t o f the CMP r e s p o n s e i s t r a n s f o r m e d i n t o an e l l i p s e and the l i n e a r p a r t ( r e p r e s e n t i n g r e f r a c t i o n e n e r g y ) i n t o a ' f o c u s ' p o i n t a t p = l / c 2 , see F i g . I 5 c . F o r a c o n f i g u r a t i o n o f h o r i z o n t a l b o u n d a r i e s t h i s p r o -c e s s o f 'Tp mapping' r e s u l t s i n a p a t t e r n of e l l i p s e l i k e c u r v e s , the i n t e r -s e c t i o n -s o f w h i c h denote r e f r a c t i o n e n e r g y and a r e p o -s i t i o n e d a t p = l / c . ., ,

n+l see F i g . I-5d. Hence, the p o s i t i o n s o f the r e f r a c t i o n energy p o i n t s a f t e r T-p mapping y i e l d v e l o c i t y i n f o r m a t i o n .

As a l o g i c c o n t i n u a t i o n o f the work by D o h e r t y and C l a e r b o u t ( 1 9 7 6 ) , S c h u l t z and C l a e r b o u t (1978) i n t r o d u c e the p r i n c i p l e s o f the T-p mapping procedure m e n t i o n e d above, w h i c h they d e n o t e as s l a n t s t a c k i n g , s i n c e the t r a n s f o r m a t i o n i s c a r r i e d out by summing the d a t a a l o n g s l a n t i n g s t r a i g h t l i n e s ( s e e a l s o F i g . I - 5 b ) . They i n d i c a t e how v e l o c i t y a n a l y s i s t e c h n i q u e s u s i n g c o h e r e n c y c r i t e r i a can be a p p l i e d t o b o t h r e f l e c t i o n and r e f r a c t i o n d a t a a f t e r s l a n t s t a c k i n g , a l s o i n media w i t h l a t e r a l v e l o c i t y v a r i a t i o n s . D i e b o l d and S t o f f a (1981) f o l l o w the same p r i n c i p l e s , whereas S t o f f a , D i e b o l d and B u h l (1982) e x p l i c i t l y i n c l u d e w i d e a p e r t u r e d a t a . C l a y t o n and McMechan (1981) a p p l y v e l o c i t y a n a l y s i s to s l a n t s t a c k e d r e f r a c t i o n d a t a a f t e r m i g r a t i o n , l e a v i n g r e f l e c t i o n d a t a out o f c o n s i d e r a t i o n . I n a q u i t e d i f f e r e n t a p p r o a c h , W e g l e i n e t a l . (1981) use s l a n t s t a c k e d d a t a i n c l u d i n g r e f r a c t i o n s as i n p u t f o r an a l g o r i t h m i n v e r t i n g r e f l e c t i o n c o e f f i c i e n t s t o v e l o c i t y d a t a based on t h e Uppmann-Schwinger and S c h r o e d i n g e r e q u a t i o n s . S c h u l t z (1982) p r e s e n t s a method by w h i c h i n t e r v a l v e l o c i t i e s a r e d e t e r m i n e d a p p l y i n g c o h e r e n c y t e c h -n i q u e s t o r e c u r s i v e l y m i g r a t e d s l a -n t s t a c k e d d a t a . A k i a-nd R i c h a r d s (1980) i n d i c a t e how i n the (T,P) domain v e l o c i t y a n a l y s i s can be a p p l i e d to media w i t h v e l o c i t y g r a d i e n t s . R e s e a r c h on t h i s t o p i c i s nowadays done i n the group o f a c o u s t i c s o f the D e l f t U n i v e r s i t y o f T e c h n o l o g y .

As a f i n a l remark, i t s h o u l d be m e n t i o n e d t h a t a more e x t e n s i v e s u r v e y on v e l o c i t y a n a l y s i s t e c h n i q u e s as a p p l i e d i n s e i s m i c s i s g i v e n by H u b r a l and K r e y ( 1 9 8 0 ) .

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1.4.4 V e l o c i t y analysis techniques concerning d i f f r a c t i o n data

As mentioned i n t h e p r e v i o u s s u b s e c t i o n s , and more s y s t e m a t i c a l l y a n a l y z e d by Taner e t a l . (1970) and B l a c k b u r n ( 1 9 8 0 ) , CMP g a t h e r s l o o s e t h e i r s i m p l e geomet r i c p r o p e r geomet i e s when c o m p r i s i n g an abundangeomet amoungeomet o f s c a geomet geomet e r i n g d i f f r a c t i o n e n e r g y . Hence, f o r complex, l a t e r a l l y d i s c o n t i n u o u s s t r u c -t u r e s , v e l o c i -t y a n a l y s i s -t e c h n i q u e s based on -t h e h y p e r b o l i c CMP c o n c e p -t a r e n o t a p p l i c a b l e .

As i n d i c a t e d i n s e c t i o n 1.3, two a p p r o a c h e s may be adopted t o p e r f o r m v e l o c i t y a n a l y s i s i n t h e p r e s e n c e o f abundant d i f f r a c t i o n d a t a : ( 1 ) p r e - s t a c k i n v e r s i o n o f m u l t i - t r a c e d a t a s e t s . We saw ( s u b s e c t i o n 1.4.2) t h a t t h i s a p p r o a c h was d i s c u s s e d by G a r d n e r e t a l . ( 1 9 7 4 ) , S a t t l e g g e r ( 1 9 7 5 ) , D o h e r t y and C l a e r b o u t ( 1 9 7 6 ) , Owusu e t a l . ( 1 9 8 3 ) . I n t h e p h i l o s o -phy b e h i n d v e l o c i t y a n a l y s i s t e c h n i q u e s as c o n s i d e r e d by t h e above a u t h o r s d i f f r a c t i o n energy i s ' t o l e r a t e d ' a s an i n e v i t a b l e p a r t o f t h e i n p u t d a t a t o be coped w i t h , and c e r t a i n l y n o t as a s o u r c e o f v e l o c i t y i n f o r m a t i o n . ( 2 ) t r e a t m e n t o f d i f f r a c t i o n d a t a as a s p e c i f i c s o u r c e o f v e l o c i t y i n f o r m a t i o n . Then, r e c o r d e d d a t a s h o u l d be g a t h e r e d such t h a t d i f f r a c t i o n e n e r g y i s c l e a r l y p r o f i l e d . F o r t h i s p u r p o s e , c o m m o n o f f s e t d a t a ( i n c l u d i n g z e r o - o f f s e t d a t a and s t a c k e d d a t a ) a r e a p p r o p r i a t e ^ . I n common o f f s e t d a t a a c q u i s i t i o n , t h e r e s p o n s e o f a s i n g l e d i f f r a c t o r has a h y p e r b o l i c shape, as has t h e r e s p o n s e o f a r e f l e c t o r i n CMP d a t a a c q u i s i -t i o n .

S e v e r a l a u t h o r s have proposed i d e a s t o t a c k l e t h e problem o f v e l o c i t y a n a l y s i s i n complex media u s i n g t h e second a p p r o a c h o u t l i n e d above. S i n c e m e d i c a l a p p l i -c a t i o n s o f e -c h o — a -c o u s t i -c s a r e u s u a l l y performed w i t h z e r o - o f f s e t s -c a n n i n g d e v i c e s , i t c a n be u n d e r s t o o d t h a t r e s e a r c h e r s from t h i s f i e l d have l a r g e l y c o n t r i b u t e d t o t h e development o f d i f f r a c t i o n d a t a a n a l y s i s . 1) T h i s i s c o n f i r m e d by H u b r a l ( 1 9 7 5 ) , where he s t a t e s : " D i f f r a c t i o n c u r v e s w h i c h a r e o b s e r v e d on s t a c k e d s e c t i o n s ... p r o v i d e u s e f u l s u b s u r f a c e i n f o r -m a t i o n about a d i f f r a c t o r i n a s i -m i l a r way as CDP (= CMP, DdV) a r r i v a l t i m e s p r o v i d e i n f o r m a t i o n a b o u t a r e f l e c t o r . They c a n t h e r e f o r e be c o n s i -d e r e -d f o r t h e purpose o f i n t e r v a l v e l o c i t y c o m p u t a t i o n s an-d t i m e - t o - -d e p t h c o n v e r s i o n s as w e l l " .

17

S a n z g i r i (1977) d i s c u s s e s a z e r o - o f f s e t v e l o c i t y a n a l y s i s t e c h n i q u e w h i c h seems a p p l i c a b l e t o f l a t r e f l e c t o r s o n l y . Dameron (1979, 1980) d e s c r i b e s a z e r o - o f f s e t v e l o c i t y a n a l y s i s t e c h n i q u e a p p l i c a b l e t o media w i t h weak v e l o c i t y v a r i a t i o n s i n one d i m e n s i o n . Much work on v e l o c i t y a n a l y s i s has been done a t Denver U n i v e r s i t y , U.S.A., r e s u l t i n g i n a s e r i e s o f p a p e r s on z e r o - o f f s e t o r

common o f f s e t v e l o c i t y i n v e r s i o n a l g o r i t h m s a p p l i c a b l e t o media where t h e v e l o -c i t y v a r i a t i o n s a r e s m a l l p e r t u r b a t i o n s on a ( p r e f e r a b l y -c o n s t a n t ) r e f e r e n -c e v a l u e : Cohen and B l e i s t e i n ( 1 9 7 9 ) , Gray and B l e i s t e i n ( 1 9 8 0 ) , Gray e t a l .

( 1 9 8 0 ) , Gray (1981 a , b ) , B l e i s t e i n and Cohen ( 1 9 8 2 ) . Raz (1981 a,b) p r e s e n t s a g e n e r a l t h e o r y on v e l o c i t y p r o f i l e i n v e r s i o n , t h e p r a c t i c a l a p p l i c a b i l i t y o f w h i c h i s n o t y e t c l e a r . R o b i n s o n e t a l . (1982) use t h e c r o s s - c o r r e l a t i o n between two u l t r a s o u n d z e r o - o f f s e t s c a n s a c q u i r e d from t h e same t a r g e t t o d e t e r m i n e an 'image s h i f t ' from w h i c h v e l o c i t y i n f o r m a t i o n c a n be o b t a i n e d . W o r t h m e n t i o n i n g a r e a l s o t h e e f f o r t s t o e x t e n d v e l o c i t y a n a l y s i s t e c h n i q u e s a s used i n t r a n s m i s s i o n tomography t o r e f l e c t i o n and d i f f r a c t i o n d a t a : Wade e t a l . ( 1 9 7 8 ) , Kenue and G r e e n l e a f ( 1 9 8 2 ) , Kaveh e t a l . ( 1 9 8 2 ) , H i l l e r and E r m e r t

( 1 9 8 2 ) . U n t i l now, t h e r e s u l t s seem t o be n o t p r a c t i c a l l y a p p l i c a b l e .

The c o n c l u s i o n must be drawn t h a t a v e l o c i t y a n a l y s i s t e c h n i q u e w i t h g e n e r a l a p p l i c a b i l i t y t o common o f f s e t ( i n c l u d i n g z e r o - o f f s e t ) d i f f r a c t i o n d a t a i s n o t a v a i l a b l e . As a c o n t r i b u t i o n t o t h e f u l f i l l m e n t o f t h i s gap, t h e a u t h o r p r o -p o s e s t h e 'minimum e n t r o -p y ' t e c h n i q u e w h i c h i s t h e s u b j e c t o f t h i s t h e s i s . The b a s i c e l e m e n t s o f t h i s t e c h n i q u e a r e summarized i n t h e n e x t s e c t i o n .

1.5 BASIC ELEMENTS OF MINIMUM ENTROPY VELOCITY ANALYSIS

As m e n t i o n e d b e f o r e and seen i n F i g . I l b , d i f f r a c t i o n energy a p p e a r s i n z e r o -o f f s e t r e c -o r d s as h y p e r b -o l i c p a t t e r n s . M i g r a t i -o n -o f t h e d a t a has a f o c u s s i n g e f f e c t on t h e d i f f r a c t i o n e n e r g y : m i g r a t i o n w i t h a w e l l c h o s e n o p e r a t o r c o n t a i n i n g t h e r i g h t v e l o c i t y v a l u e makes t h e d i f f r a c -t i o n p a -t -t e r n s c o l l a p s e ( F i g . I - l c ) -t h u s a s s i g n i n g minimum l a -t e r a l d i s p e r s i o n t o t h e d a t a . Hence, t h e l a t e r a l e x t e n s i o n o f m i g r a t e d z e r o - o f f s e t (and, s i m i l a r l y , common o f f s e t ) d i f f r a c t i o n p a t t e r n s c a n be used a s a c r i t e r i o n f o r v e l o c i t y a n a l y s i s .

I t w i l l be shown t h a t i n t h e s p a c e - f r e q u e n c y domain t h e l a t e r a l e x t e n s i o n o f m i g r a t e d z e r o - o f f s e t ( o r common o f f s e t ) d i f f r a c t i o n p a t t e r n s - r e p r e s e n t i n g

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

A s u i t a b l e way o f q u a n t i f y i n g w a v e l e t d i s p e r s i o n i s t h e a p p l i c a t i o n o f norms known i n g e o p h y s i c a l l i t e r a t u r e a s 'minimum e n t r o p y ' norms. T h e r e f o r e , we s h a l l r e f e r t o o u r t e c h n i q u e as M i n i m u m E_ n t r o p y V e l o c i -t y A n a l y s i s , a b b r e v i a -t e d as MEVA. S i n c e t h e s p a t i a l w a v e l e t c o n c e p t d e s c r i b i n g i n v e r t e d z e r o - o f f s e t d i f f r a c t i o n d a t a i s a l s o v a l i d f o r i n v e r t e d CMP r e f l e c t i o n d a t a , MEVA a l s o a p p l i e s t o r e f l e c t i o n d a t a , f o r m i n g an a l t e r n a t i v e t o t h e t e c h n i q u e s mentioned i n sub-s e c t i o n 1.4.2.

The p r i n c i p l e s and p r e l i m i n a r y r e s u l t s o f MEVA were f i r s t p u b l i s h e d by

De V r i e s and Berkhout ( 1 9 8 2 ) . R e c e n t l y , H a n l a n e t a l . (1983) p r e s e n t e d a paper i n w h i c h a v e l o c i t y a n a l y s i s t e c h n i q u e i s proposed e x t r a c t i n g v e l o c i t y i n f o r -m a t i o n fro-m -m i g r a t e d d i f f r a c t i o n d a t a by a s t a t i s t i c a l w h i t e n i n g p r o c e s s .

1.6 OUTLINE OF THE FOLLOWING CHAPTERS

I n c h a p t e r I I , a wave t h e o r e t i c a l model f o r z e r o - o f f s e t d a t a i s p r e s e n t e d . I n v e r s i o n o f z e r o - o f f s e t d a t a i s d e s c r i b e d u s i n g a s p a t i a l matched f i l t e r p e r f o r m i n g phase c o r r e c t i o n as w e l l as a m p l i t u d e w e i g h t i n g . The i n v e r t e d r e s u l t i s p r e s e n t e d i n terms o f b a n d - l i m i t e d s p a t i a l w a v e l e t s d e t e r m i n e d by s p a t i a l a m p l i t u d e and phase s p e c t r a . In c h a p t e r I I I , t h e i n f l u e n c e o f v e l o c i t y e r r o r s on i n v e r s i o n i s d i s c u s s e d . P a r t i c u l a r l y , e f f e c t s on t h e phase s p e c t r u m and, hence, on t h e d i s p e r s i o n o f t h e s p a t i a l w a v e l e t s a r e c o n s i d e r e d .

I n c h a p t e r I V , t h e c o n c e p t o f ' e n t r o p y ' and e s p e c i a l l y 'minimum e n t r o p y ' i s c o n s i d e r e d . Minimum e n t r o p y norms a r e d e f i n e d and i n t e r p r e t e d as measures o f r e s o l v i n g power o f a d a t a s e t , s p e c i f i e d by d i s p e r s i o n and s p a r s i t y . B a s i c p r o p e r t i e s o f minimum e n t r o p y norms a r e e v a l u a t e d .

I n c h a p t e r V, t h e a p p l i c a b i l i t y o f minimum e n t r o p y norms t o s p a t i a l w a v e l e t s i s shown, making them a t o o l f o r v e l o c i t y a n a l y s i s (MEVA). E l e m e n t a r y a p p l i -c a t i o n s a r e p r e s e n t e d .

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I n c h a p t e r V I , a p p l i c a t i o n s o f MEVA a r e d i s c u s s e d . S i m u l a t e d as w e l l as measured d a t a s e t s a r e c o n s i d e r e d , i n c l u d i n g z e r o - o f f s e t as w e l l a s CMP d a t a f r o m v a r i o u s a p p l i c a t i o n f i e l d s . MEVA i s shown t o be a p o w e r f u l t e c h n i q u e w i t h a w i d e a p p l i c a b i l i t y .

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CHAPTER II:

WAVE THEORETICAL APPROACH TO ACOUSTIC

MODELING AND INVERSION

II.1. INTRODUCTION

I n any e c h o - a c o u s t i c a l t e c h n i q u e , two b a s i c p r o c e s s e s a r e t o be d i s c e r n e d , v i z . t h e f o r w a r d p r o c e s s and t h e i n v e r s e p r o c e s s .

I n t h e f o r w a r d p r o c e s s , a sound wave i s g e n e r a t e d a t t h e s u r f a c e o f t h e s y s t e m t o be i n v e s t i g a t e d . T h i s sound wave p r o p a g a t e s downward t h r o u g h t h e un-d e r l y i n g meun-dium, i s r e f l e c t e un-d anun-d un-d i f f r a c t e un-d by i n h o m o g e n e i t i e s anun-d l a y e r boun-d a r i e s , t h e n p r o p a g a t e s upwarboun-d, anboun-d f i n a l l y i s r e c o r boun-d e boun-d i n some a p e r t u r e a r e a a t t h e s u r f a c e . The f o r w a r d p r o c e s s i s a l s o c a l l e d m o d e l i n g p r o c e s s .

I n t h e i n v e r s e p r o c e s s , an a c o u s t i c image o f t h e medium below t h e a p e r t u r e a r e a i s formed by a p p r o p r i a t e p r o c e s s i n g o f t h e r e c o r d e d d a t a . T h i s p r o c e s s i n g i s c a l l e d image r e c o n s t r u c t i o n , s p a t i a l i n v e r s i o n , f o c u s s i n g o r , e s p e c i a l -l y i n s e i s m i c s , m i g r a t i o n . B a s i c a -l -l y , a -l -l i n v e r s e t e c h n i q u e s a i m a t e l i m i n a t i o n o f t h e p r o p a g a t i o n e f f e c t s i n t r o d u c e d by t h e f o r w a r d p r o c e s s , a s w e l l as t h e i n f l u e n c e s o f s o u r c e and d e t e c t o r c h a r a c t e r i s t i c s .

B o t h f o r w a r d and i n v e r s e p r o c e s s c a n be r e p r e s e n t e d by p h y s i c a l models based on a c o u s t i c wave t h e o r y . P r o p a g a t i o n i s d e s c r i b e d by t h e s c a l a r a c o u s t i c wave equa-t i o n , r e f l e c equa-t i o n and d i f f r a c equa-t i o n by equa-t h e boundary c o n d i equa-t i o n s imposed equa-t o equa-t h a equa-t e q u a t i o n . K i r c h h o f f used t h e wave e q u a t i o n , t o g e t h e r w i t h Huygens's p r i n c i p l e s t a t i n g t h a t any wave f r o n t may be c o n s i d e r e d a s a c o n f i g u r a t i o n o f s e c o n d a r y s o u r c e s , t o d e v e l o p h i s theorem w h i c h e n a b l e s f o r w a r d e x t r a p o l a t i o n o f a

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sound f i e l d on a c l o s e d s u r f a c e t o any p o i n t e n c l o s e d . R a y l e i g h m o d i f i e d t h i s theorem f o r d a t a on ( i n f i n i t e ) p l a n e s u r f a c e s . Based on t h i s , f o r w a r d propa-g a t i o n o f sound waves t h r o u propa-g h a medium can be d e s c r i b e d by e x t r a p o l a t i o n o p e r a t o r s w h i c h a r e o f t e n c a l l e d g e n e r a l i z e d R a y l e i g h - o p e r a t o r s . B e r k h o u t (1982) shows t h a t , i n g e n e r a l , t h e s e o p e r a t o r s can be e l e g a n t l y r e p r e s e n t e d i n m a t r i x n o t a t i o n , and i n s p e c i a l c a s e s as a s p a t i a l c o n v o l u t i o n . On the o t h e r hand, t h e i n v e r s e p r o c e s s where p r o p a g a t i o n e f f e c t s a r e e l i m i n a t e d can be d e s -c r i b e d by i n v e r s e e x t r a p o l a t i o n o p e r a t o r s , d e r i v e d from the f o r w a r d o p e r a t o r s by m a t r i x i n v e r s i o n o r d e c o n v o l u t i o n . I n l a t e r c h a p t e r s about v e l o c i t y a n a l y s i s ( c h p s . V, V I ) , m a i n l y z e r o - o f f s e t d a t a w i l l be c o n s i d e r e d , i . e . d a t a a c q u i r e d by p o s i t i o n i n g s o u r c e s and d e t e c t o r s a t the same p l a c e s i n the a p e r t u r e a r e a . T h e r e f o r e , i n t h i s c h a p t e r emphasis w i l l be l a i d on the d i s c u s -s i o n o f a model d e -s c r i b i n g f o r w a r d and i n v e r -s e p r o c e -s -s i n g o f z e r o - o f f -s e t d a t a . S i n c e many e l e m e n t s o f t h i s d e r i v a t i o n a r e c l e a r l y d e s c r i b e d by B e r k h o u t ( 1 9 8 2 ) , r e f e r e n c e s t o t h i s t e x t b o o k a r e made where p o s s i b l e .

II.2 PHYSICAL MODEL FOR DATA SIMULATION

I I . 2 . 1 General physical model

I n F i g . I I — 1 , the s o l i d l i n e s d e n o t e a t h r e e - d i m e n s i o n a l medium c o n s i s t i n g o f i r r e g u l a r l y shaped l a y e r s below a h o r i z o n t a l s u r f a c e p l a n e on w h i c h a c o n f i g u -r a t i o n o f sound s o u -r c e s (O) and d e t e c t o -r s ( x ) i s a -r -r a n g e d . I n o -r d e -r t o model t h e sound f i e l d a t the d e t e c t o r p o s i t i o n s , we c o v e r the medium w i t h a g r i d o f h o r i z o n t a l p l a n e s i n each p o i n t o f w h i c h the r e f l e c t i o n p r o p e r t i e s a r e g i v e n , d e p e n d i n g g e n e r a l l y on f r e q u e n c y and a n g l e o f i n c i d e n c e . F o r p e r f e c t m o d e l i n g ,

y

2 3

the v e r t i c a l d i s t a n c e between the h o r i z o n t a l p l a n e s s h o u l d be i n f i n i t e s i m a l l y s m a l l , but i n p r a c t i c a l s i t u a t i o n s a f i n i t e i n t e r v a l may be t a k e n d e p e n d i n g on t h e w a v e l e n g t h s o f the a c o u s t i c waves as w e l l as on the d e n s i t y and p r o p a g a t i o n v e l o c i t y g r a d i e n t s i n the medium c o n s i d e r e d . Between the h o r i z o n t a l p l a n e s , t h e p r o p a g a t i o n i s d e s c r i b e d by the a p p r o p r i a t e homogeneous e x t r a p o l a t i o n o p e r a t o r . T h e r e a r e t h r e e t y p e s o f o p e r a t o r s d e s c r i b i n g f o r w a r d e x t r a p o l a t i o n o f a sound f i e l d f r o m a p l a n e t h r o u g h a homogeneous medium t o a p o i n t , denoted as WI, WII and W i l l . T h e i r a p p l i c a b i l i t y depends on the f i e l d p a r a m e t e r s c o n s i d e r e d : o p e r a -t o r WI d e s c r i b e s e x -t r a p o l a -t i o n f r o m n o r m a l p a r -t i c l e v e l o c i -t y -t o p r e s s u r e , WII d e s c r i b e s e x t r a p o l a t i o n f r o m p r e s s u r e t o p r e s s u r e o r f r o m n o r m a l v e l o c i t y t o n o r m a l v e l o c i t y , W i l l d e s c r i b e s e x t r a p o l a t i o n from p r e s s u r e t o n o r m a l v e l o c i t y . The d e r i v a t i o n o f t h o s e o p e r a t o r s f r o m wave t h e o r y i s d e s c r i b e d by B e r k h o u t ( 1 9 8 2 , chp. 5 ) .

When the medium c o n s i s t s o f s o l i d m a t e r i a l s , a c o m p r e s s i o n a l sound wave, o f t e n d e n o t e d as p-wave, may be c o n v e r t e d i n t o o t h e r wave t y p e s , s u c h as s h e a r waves, b e n d i n g waves a.o.. S i n c e a - c e r t a i n l y i n t e r e s t i n g - s t u d y o f t h i s wave c o n v e r s i o n f a l l s beyond the scope o f t h i s t h e s i s , we c o n s i d e r i n o u r model o n l y t h e p - w a v e s t r e a t i n g the o t h e r wave t y p e s as n o i s e , as i s done i n most s t u d i e s on i n v e r s i o n . T h i s means t h a t , i n the h o r i z o n t a l p l a n e s c o n s i d e r e d , o n l y the r e f l e c t i o n p r o p e r t i e s f o r p-waves have t o be s p e c i f i e d . The form i n w h i c h t h o s e p r o p e r t i e s must be d e s c r i b e d depends on t h e s o u r c e and d e t e c t o r t y p e s . I f the s o u r c e has a monopole c h a r a c t e r and hence d e f i n e s n o r m a l p a r t i c l e v e l o c i t y , whereas the d e t e c t o r has a d i p o l e c h a r a c t e r i s t i c t h u s b e i n g p r e s s u r e - s e n s i t i v e , the r e f l e c t i o n p r o p e r t i e s s h o u l d be g i v e n i n terms o f l o c a l impedance Z. I f s o u r c e and d e t e c t o r a r e o f the same t y p e , the r e f l e c t i o n p r o p e r t i e s must be s p e c i f i e d i n terms o f r e f l e c t i v i t y R. F i n a l l y , i n c a s e o f a p r e s -s u r e -s o u r c e and a v e l o c i t y d e t e c t o r , the l o c a l a d m i t t a n c e Y i -s the -s u i t a b l e p a r a m e t e r to d e s c r i b e r e f l e c t i o n p r o p e r t i e s . Thus, the p r o p a g a t i o n between two p l a n e s can i n a l l c a s e s be f o r m u l a t e d by the homogeneous o p e r a t o r d e n o t e d above a s WII.

I n our m o d e l , we s h a l l assume p r e s s u r e s o u r c e s and p r e s s u r e d e t e c t o r s o n l y , so t h a t r e f l e c t i o n p r o p e r t i e s w i l l a l w a y s be d e s c r i b e d i n terms o f p r e s s u r e r e -f l e c t i v i t y . T h i s does n o t a -f -f e c t t h e g e n e r a l v a l i d i t y o -f o u r m o d e l , s i n c e the o t h e r c a s e s can s i m p l y be t r a n s f o r m e d t o the above one: a v e l o c i t y s o u r c e i s d e a l t w i t h by r e p l a c i n g i n the f i r s t s t e p (Zq •* z^) o p e r a t o r WII by

o p e r a t o r WI, a v e l o c i t y d e t e c t o r by r e p l a c i n g i n t h e l a s t s t e p (Z ^ ^ Z Q ) o p e r a t o r WII by o p e r a t o r W i l l . See B e r k h o u t (1982, pg. 162).

2 4

B e s i d e s p r i m a r y r e f l e c t i o n , between a p e r t u r e p l a n e and b o u n d a r i e s m u l t i p l e r e f l e c t i o n s w i l l o c c u r . I n our model, we s h a l l n e g l e c t such m u l t i p l e s f o r s i m p l i -c i t y r e a s o n s . B e r k h o u t (1982, s e -c t i o n s 6.9, 6.10, 7.6) i n d i -c a t e s how, i f d e s i r e d , m u l t i p l e s can be t a k e n i n t o a c c o u n t i n f o r w a r d as w e l l as i n i n v e r s e p r o c e s s i n g by r e l a t i v e l y s i m p l e a l g o r i t h m s .

A p p l y i n g the l i m i t a t i o n s mentioned above, the one-way wave e q u a t i o n may be used l e a d i n g t o a p r i m a r y s i m u l a t i o n model f o r pwaves w h i c h we s h a l l d i s -c u s s i n more d e t a i l i n the n e x t s u b s e -c t i o n .

II.2.2 Physical model f o r simulation of pressure-to-pressure primaries

I n F i g . I I - 2 the d a t a f l o w i n m o d e l i n g t w o - d i m e n s i o n a l m u l t i - r e c o r d p r e s s u r e p r i m a r i e s from one d e p t h l e v e l z i s shown. F i g . I I - 2 a shows the s c h e m a t i c

m °

geometry, whereas F i g . I I - 2 b r e p r e s e n t s the p h y s i c a l model i n t h e form o f a b l o c k d i a g r a m . S ( z0) D ( z0) a F i g u r e I I - 2 : P h y s i c a l t w o - d i m e n s i o n a l model f o r s i m u l a t i o n o f p r e s s u r e p r i m a r y d a t a from one d e p t h l e v e l . a . s c h e m a t i c geometry b. b l o c k d i a g r a m As B e r k h o u t (1982, s e c t i o n s 6.2, 6.6) e x p l a i n s , the e l e m e n t s o f the b l o c k scheme g e n e r a l l y c a n be r e p r e s e n t e d by m a t r i c e s , so t h a t the m o d e l i n g