Pre-stack migration by single shot record inversion

PRE-STACK MIGRATION

BY SINGLE SHOT RECORD INVERSION

BY SINGLE SHOT RECORD INVERSION

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PRE-STACK MIGRATION

BY SINGLE SHOT RECORD INVERSION

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

dinsdag 20 november 1984 te 16.00 uur door MATTHIJS PIETER DE G R A A F F wiskundig ingenieur geboren te Vlissingen

| 0 ö 0 2 2 9 , j

Gebotekst Zoetermeer/1984

PREFACE

The o r i g i n a l s u b j e c t o f the r e s e a r c h w h i c h l e d t o t h i s t h e s i s was o r i e n t e d on a p p l i c a t i o n s i n m e d i c a l d i a g n o s t i c s . However, the emphasis s h i f t e d by and by toward s e i s m i c a p p l i c a t i o n s , because o f t h e r a p i d l y i n c r e a s i n g s i g n i f i c a n c e o f p r e - s t a c k t e c h n i q u e s i n the s e i s m i c i n d u s t r y . Never-t h e l e s s , Never-t h e r e r e m a i n v e r y i m p o r Never-t a n Never-t a p p l i c a Never-t i o n s w i Never-t h i n m e d i c a l d i a g n o s t i c s and n o n - d e s t r u c t i v e t e s t i n g . The r e s u l t s w h i c h have been o b t a i n e d w i t h the p r e - s t a c k m i g r a t i o n t e c h n i q u e p r e s e n t e d i n t h i s t h e s i s a r e v e r y p r o m i s i n g . Y e t , t h e r e i s s t i l l much work t o be done i n r e f i n i n g and o p t i m i z i n g the method, i n d e s i g n i n g a s o f t w a r e c o n c e p t c o r r e s p o n d i n g w i t h new r e q u i r e m e n t s o b t a i n e d from p r a c t i c e , e t c . I t i s my s i n c e r e d e s i r e t h a t the work o f t h i s t h e s i s may c o n t r i b u t e t o t h e f a s t development o f p r e - s t a c k t e c h n i q u e s i n a l l d i s c i p l i n e s f o r e m e n t i o n e d .

I w i s h to thank g r a t e f u l l y my promotor p r o f e s s o r B e r k h o u t f o r h i s h e l p and ' d r i v i n g f o r c e ' . Due t o h i s e v e r l a s t i n g o p t i m i s m w i t h r e s p e c t t o t h e i m p l e m e n t a t i o n o f the p r o b l e m I was a b l e t o c o m p l e t e the s o f t w a r e development f o r t h i s t h e s i s .

I am a l s o g r e a t l y o b l i g e d t o my c o l l e a g u e s from TNO/DGV i n D e l f t f o r t h e i r p r o f e s s i o n a l a s s i s t a n c e . E s p e c i a l l y , I would l i k e t o thank Ad van d e r S c h o o t and P a u l de B e u k e l a a r who were o f g r e a t h e l p t o me.

I thank Renée Breeuwer and U i l k e S t e l w a g e n from TNO/TPD i n D e l f t f o r t h e i r e f f o r t i n d o i n g w a t e r t a n k measurements. Much t o my r e g r e t the r e s u l t s c o u l d n o t be i n c l u d e d i n t h i s t h e s i s .

I thank J a n R i d d e r f o r d o i n g the measurements w i t h the SAS system, w h i c h e v e n t u a l l y c o u l d be i n c l u d e d i n t h i s t h e s i s by a v e r y narrow m a r g i n . I thank my c o l l e a g u e Kees Wapenaar f o r many f r u i t f u l d i s c u s s i o n s , mr. De Knegt o f the d r a w i n g o f f i c e and my f r i e n d K l a a s Laansma f o r t h e i r

a s s i s t a n c e i n making the f i g u r e s and mr. S u i t e r s o f the p h o t o g r a p h i c s e r v i c e f o r p r o d u c i n g the huge amount o f photo p r i n t s f o r t h i s t h e s i s . I thank mrs. Gerda Boone f o r t y p i n g the m a n u s c r i p t and l a s t but n o t l e a s t I w i s h t o thank my dear f r i e n d Hanneke M u l d e r f o r her m o r a l s u p p o r t . The r e s e a r c h f o r t h i s t h e s i s was f i n a n c i a l l y s u p p o r t e d by the D u t c h f o u n d a t i o n f o r Fundamental R e s e a r c h o f M a t t e r (FOM).

CONTENTS

CHAPTER 1

MIGRATION OF SEISMIC DATA 1 1.1 INTRODUCTION 1 1.2 MIGRATION METHODS 4 1.3 CONSIDERATIONS ABOUT PROPAGATION VELOCITY DETERMINATION. . . 7

1.4 MODEL FAILURES 8 1.5 THE OUTLINE OF THIS THESIS 9

CHAPTER 2

BASIC CONSIDERATIONS ON THE FORWARD AND INVERSE PROBLEM 11 2.1 THE LIPPMANN-SCHWINGER INTEGRAL EQUATION 11 2.2 A GENERALIZED VERSION OF THE RAYLEIGH INTEGRALS 15

2.3 THE DISCRETIZED INTEGRAL REPRESENTATIONS OF THE BORN

APPROXIMATION 18 2.4 THE INVERSE PROBLEM 21

CHAPTER 3

INVERSION PROCEDURES 23 3.1 INVERSION BY DOWNWARD CONTINUATION 23

3.2 INVERSION OF SINGLE SHOT RECORDS 25

CHAPTER 4

TRUE COMMON DEPTH POINT STACKING BY SINGLE SHOT RECORD MIGRATION . .29

4.1 INTRODUCTION 29 4.2 THE COMMON MIDPOINT STACKING PROCEDURE 30

GENERATION OF MULTI-FOLD ZERO-OFFSET SECTIONS BY MEANS OF WAVE FIELD

EXTRAPOLATION 37 5.1 THE SCHEME FOR GENERATION OF ZERO-OFFSET SECTIONS 37

5.2 CONSIDERATIONS CONCERNING THE STACK LEVEL 39

5.3 SYNTHETIC EXAMPLES 41

CHAPTER 6

NUMERICAL EXPERIMENTS 45 6.1 INTRODUCTION 45 6.2 SHORT COMMENTS ON THE DEVELOPED SOFTWARE 47

6.3 PULSE SHAPE DISTORTION 49 6.4 SYNTHETIC NOISE CHARACTERISTICS 50

6.5 SYNTHETIC EXPERIMENTS ON RESOLVING POWER 51 6.6 SYNTHETIC EXPERIMENTS ON SIGNAL-TO-NOISE ENHANCEMENT . . . .63

CHAPTER 7

PRACTICAL APPLICATIONS 75 7.1 MEASUREMENTS ON THE RMI-PHANTOM 75

7.2 MIGRATION OF MARINE DATA 81

REFERENCES 89

SUMMARY 93

SAMENVATTING 95

MIGRATION OF SEISMIC DATA

1.1 INTRODUCTION E x p l o r a t i o n f o r o i l and gas c a n be r o u g h l y d i v i d e d i n t o s i x f i e l d s ( J a i n and De F i g u e i r i d o ) . One o f t h o s e i s e x p l o r a t i o n g e o p h y s i c s . E x p l o r a t i o n g e o p h y s i c s i n c l u d e s measurement o f e a r t h ' s m a g n e t i c , g r a v i t a t i o n a l and e l e c t r i c a l f i e l d s , a s w e l l a s t h e r e c o r d i n g o f a r t i f i c i a l l y g e n e r a t e d e l a s t o - a c o u s t i c a l wave f i e l d s . E x p l o r a t i o n P e t r o p h y s i c s E x p l o r a t i o n Well Stimulation & C o m p l e t i o n F i g u r e 1.1: E x p l o r a t i o n hexagon ( a f t e r Thomasson).

s t a r t e d i n the 1 9 t h c e n t u r y ; d i r e c t mapping o f s t r u c t u r e s i n the e a r t h c r u s t u s i n g those f i e l d s began i n the mid 1920's. A l t h o u g h y o u n g e r , the use o f s e i s m i c d a t a forms t h e main p o i n t i n the s e a r c h f o r h y d r o - c a r b o n s i n modern g e o p h y s i c a l s e i s m o l o g y . T i l l the 1960's the l i m i t a t i o n s o f a n a l o g d e v i c e s p r o h i b i t e d e x t e n s i v e d a t a p r o c e s s i n g . The i n t r o d u c t i o n o f d i g i t a l t e c h n o l o g y b r o u g h t a b o u t a t r u e r e v o l u t i o n i n s e i s m i c d a t a

p r o c e s s i n g . P r e s e n t l y , even w o r l d s f a s t e s t computer i s not y e t c a p a b l e t o p r o c e s s a n o r m a l 3D s e i s m i c s u r v e y i n an a c c e p t a b l e t i m e , w i t h o u t a s i g n i f i c a n t d a t a r e d u c t i o n on f o r e h a n d (CMP methods). However, development i s g o i n g on v e r y r a p i d l y . A new g e n e r a t i o n computers i s t o be b o r n , w h i c h may cope w i t h p r o p e r CDP p r o c e s s i n g t e c h n i q u e s and 3D d a t a - s e t s a t the same t i m e . The m a t e r i a l i n t h i s t h e s i s f u l l y d e v i a t e s from the c u r r e n t CMP methods and a p r o p e r 2D CDP t e c h n i q u e i s proposed w h i c h a l g o r i t h m can be e a s i l y extended t o 3 d i m e n s i o n s . S e i s m i c d a t a c o n t a i n i n f o r m a t i o n about the a c o u s t i c p r o p e r t i e s of e a r t h l a y e r s . However, t h i s i n f o r m a t i o n i s h i d d e n b e h i n d many d i s t o r t i o n s o r i g i -n a t i -n g from d a t a a c q u i s i t i o -n i m p e r f e c t i o -n s , p r o p a g a t i o -n e f f e c t s , wave c o n v e r s i o n s and n o i s e s o u r c e s . By a p p l y i n g advanced i n v e r s i o n t e c h n i q u e s i t i s p o s s i b l e t o e x t r a c t r e l e v a n t g e o l o g i c i n f o r m a t i o n from the d a t a . A w e l l - k n o w n t e c h n i q u e f o r s i g n a l - t o - n o i s e enhancement and d a t a r e d u c t i o n i s t h e method o f s t a c k i n g common-midpoint t r a c e s a f t e r a c o r r e c t i o n f o r n o r m a l move-out (CMP t e c h n i q u e ) . The m u l t i p l i c i t y g a i n e d i n t h i s way

s u p p r e s s e s u n d e s i r e d e v e n t s . However, s t a c k i n g of d a t a n o r m a l l y c a u s e s a l o s s o f a p a r t of the i n f o r m a t i o n as w e l l , as i s t r u e f o r most d a t a r e d u c t i o n t e c h n i q u e s . CMP s t a c k i n g i s o f t e n succeeded by a s o - c a l l e d ' p o s t - s t a c k ' m i g r a t i o n t e c h n i q u e . S e i s m i c m i g r a t i o n d e f i n e s an i n v e r s i o n method w h i c h c o r r e c t s f o r t h e d i s t o r t i o n i n p o s i t i o n , shape and a m p l i t u d e o f r e f l e c t o r r e s p o n s e s and f o c u s s e s the r e s p o n s e s from d i f f r a c t o r s . T h i s

t h e s i s d e a l s w i t h ' p r e - s t a c k ' m i g r a t i o n , w h i c h o b v i o u s l y t a k e s p l a c e b e f o r e s t a c k . I n t h i s way the CMP p r i n c i p l e i s r e p l a c e d by the CDP p r i n c i p l e . A l t h o u g h s i g n i f i c a n t l y more d a t a have to be p r o c e s s e d , t h e r e s u l t s a r e e x p e c t e d to be g r e a t l y s u p e r i o r t o p o s t - s t a c k m i g r a t e d d a t a w i t h r e s p e c t t o b o t h s i g n a l - t o - n o i s e r a t i o and r e s o l v i n g power.

A p a r t from p r e - and p o s t - s t a c k m i g r a t i o n t h e r e e x i s t i n t e r m e d i a t e ways o f p r o c e s s i n g , c o n s i s t i n g o f a p r e - s t a c k p a r t and a p o s t - s t a c k p a r t ( e . g . mapping o f common-offset d a t a t o z e r o - o f f s e t d a t a and s t a c k i n g o f r e l a t e d z e r o - o f f s e t t r a c e s ) .

F i g u r e 1.2; P r e - s t a c k p r o c e s s i n g sequence.

The u n i q u e n e s s o f t h e method p r e s e n t e d i n t h i s t h e s i s i s t h e s e p a r a t e t r e a t m e n t o f p h y s i c a l e x p e r i m e n t s . S m a l l amounts o f d a t a o u t o f a whole s e i s m i c f i l e c a n be p r o c e s s e d w i t h o u t any need f o r an e x t e n s i v e r e a r r a n g e -ment o f t h e d a t a . As f a r as we know, up t o now no such method i s used i n s e i s m i c p r o c e s s i n g . The e l e g a n t a p p r o a c h o f p r o c e s s i n g s i n g l e s e i s m i c r e c o r d s r e m a i n s p r a c t i c a l l y a p p l i c a b l e , p a r t i c u l a r l y when h a n d l i n g 3D-data. T h i s t h e s i s a l s o i n d i c a t e s t h e r e l a t i o n s h i p w i t h c o n v e n t i o n a l common m i d p o i n t s t a c k i n g t e c h n i q u e s (CMP). I t p r o v e s t h a t t h e method p r e s e n t e d i s i n f a c t a ' t r u e ' common d e p t h p o i n t s t a c k (CDP). Some i n t e r e s t i n g consequences w h i c h f o l l o w from t h i s i m p o r t a n t p r o p e r t y a r e d i s c u s s e d . 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 o show t h a t , by l o o k i n g f o r a r i g o r o u s a t t a c k o f t h e problem c o n c e r n i n g r e s o l u t i o n and s i g n a l - t o - n o i s e enhancement f o r a c o m p l i c a t e d g e o l o g y , t h e approach o f i n v e r s i o n by p r o c e s s i n g s i n g l e s e i s m i c r e c o r d s a p p e a r s t o be t h e o n l y p r o p e r way. A p a r t from t h e s e i s m i c f i e l d , i m p o r t a n t a p p l i c a t i o n s c a n be found w i t h i n t h e realms o f n o n - d e s t r u c t i v e t e s t i n g as w e l l as i n m e d i c a l d i a g n o s t i c s .

S e v e r a l methods o f m i g r a t i n g s e i s m i c d a t a have been d e v e l o p e d . We w i l l g i v e a r o u g h l y summary o f t h e s e methods c l a s s i f y i n g them c o n f o r m the b a s i c s t e p s i n w h i c h t h e y can be d i v i d e d .

The c o n v e n t i o n a l way o f p e r f o r m i n g s e i s m i c m i g r a t i o n i s a s o - c a l l e d ' p o s t - s t a c k ' scheme w h i c h c o n s i s t s o f two s t e p s :

1) S t a c k i n g CMP-data a l o n g h y p e r b o l i c nmo-curves. T h i s p r o c e d u r e maps a s e i s m i c f i l e i n t o a z e r o - o f f s e t ( Z O - ) s e c t i o n .

2 ) Z O - m i g r a t i o n o f the s t a c k e d d a t a .

The a s s u m p t i o n i n s t a c k i n g i s t h a t the nmo-curves a r e h y p e r b o l i c . I t i s a l s o assumed t h a t i n f o r m a t i o n d e f i n e d by one nmo-curve i s g e n e r a t e d by a s m a l l r e f l e c t i o n a r e a a t a r e f l e c t o r . I t s p e r f o r m a n c e depends l a r g e l y on t h e v a l i d i t y o f t h e s e a s s u m p t i o n s . F u r t h e r m o r e the method i s d i p

-s e l e c t i v e . Hence r e f l e c t i o n d a t a and d i f f r a c t i o n d a t a can n e v e r be t r e a t e d p r o p e r l y a t the same t i m e .

P o s t - s t a c k m i g r a t i o n i s based on the z e r o - o f f s e t a s s u m p t i o n and the e x p l o d i n g r e f l e c t o r model. The l a t t e r a s s u m p t i o n i n t r o d u c e s a m p l i t u d e e r r o r s .

A p a r t i a l ' p r e - s t a c k ' scheme has been p r o p o s e d by D e r e g o w s k i and Rocca ( 1 9 8 1 ) . T h e i r p r o c e d u r e c o n s i s t s of t h r e e s t e p s :

1) D i p move-out c o r r e c t i o n a p p l i e d t o CO-data. T h i s p r o c e d u r e maps a C O - s e c t i o n i n t o a Z O - o f f s e t s e c t i o n .

2 ) S t a c k i n g of the i n d i v i d u a l Z O - s e c t i o n s . 3) Z O - m i g r a t i o n of the s t a c k e d d a t a .

The method i s based on r a y t h e o r y . I t assumes a o n e - d i m e n s i o n a l

sub-s u r f a c e . The main advantage o v e r C M P - sub-s t a c k i n g i sub-s t h a t c o n f l i c t i n g d i p sub-s can be h a n d l e d p r o p e r l y . The t h e o r y does not p r o v i d e e x a c t s o l u t i o n s i n c l o s e d f o r m .

O t t o l i n i and C l a e r b o u t ( 1 9 8 4 ) . T h e i r p r o c e d u r e c o n s i s t s o f t h r e e s t e p s :

1 ) T,p-mapping ( s l a n t s t a c k i n g ) a p p l i e d on CMP-data.

2 ) M i g r a t i o n i n the T,x-domain f o r e a c h r a y parameter p, where x r e p r e s e n t s t h e m i d p o i n t c o o r d i n a t e .

3 ) S t a c k i n g t h e m i g r a t e d d a t a a l o n g c o n s t a n t p c u r v e s .

The a s s u m p t i o n o f t h i s method i s t h a t f o r each common m i d p o i n t t h e

v e l o c i t y d i s t r i b u t i o n i s o n e - d i m e n s i o n a l . The method i s a c c u r a t e f o r s t e e p d i p s and wide o f f s e t s . Z O - m i g r a t i o n c o n v e n t i o n a l napping from C O - d a t a to Z O - d a t a s t a c k i n g of C M P - t r a c e a s t a c k i n g of Z O - t r a c e s s t a c k i n g along c o n s t a n t p Z O - m i g r a t i o n P e r e o o w s k i a R o c c a T-p mapping of C M P - d a t a migration for e a c h p Ottolini a C l e a r b o u t F i g u r e 1.3: E x i s t i n g m i g r a t i o n methods. The p r o c e d u r e f o r p r e - s t a c k m i g r a t i o n p r e s e n t e d i n t h i s t h e s i s i s based on t h e s c a l a r wave e q u a t i o n f o r inhomogeneous f l u i d s . A d o u b l e i n v e r s i o n p r o c e d u r e i s d e r i v e d on i n d i v i d u a l s h o t r e c o r d s , w h i c h d e f i n e s t h e f u l l p r o c e s s o f p r e - s t a c k m i g r a t i o n i n a d i r e c t o r r e c u r s i v e way ( f i g . 1 . 4 ) . A m a t r i x f o r m u l a t i o n o f t h i s p r o c e s s was g i v e n by B e r k h o u t ( 1 9 8 2 ) .

I t a d m i t s b o t h a x i a l and l a t e r a l v e l o c i t y changes. The p r e - s t a c k m i g r a t i o n scheme by s i n g l e shot r e c o r d i n v e r s i o n i s f o r p r a c t i c a l r e a s o n s more

inverse extrapolation C S G reorder: C S G — C R G inverse extrapolation CRG reorder: C R G — C S G next frequency »

add all frequency components next depth level

«< imaged result F i g u r e 1.4: R e c u r s i v e p r e - s t a c k , m u l t i - r e c o r d m i g r a t i o n scheme. a t t r a c t i v e a s o n l y one s h o t r e c o r d i s t r e a t e d a t a t i m e . I t i n v o l v e s t h e f o l l o w i n g s t e p s ( s t a r t i n g a t t h e s u r f a c e ) : 1) F o r w a r d e x t r a p o l a t i o n o f t h e s o u r c e wave f i e l d t o t h e n e x t d e p t h l e v e l . 2) I n v e r s e e x t r a p o l a t i o n o f t h e r e f l e c t e d wave f i e l d t o t h e n e x t d e p t h l e v e l .

3) C o r r e l a t i o n between t h e e x t r a p o l a t e d s o u r c e and r e f l e c t e d wave f i e l d . 4 ) S e l e c t i o n o f t h e z e r o t r a v e l t i m e i n f o r m a t i o n from t h e c o r r e l a t i o n

r e s u l t ( i m a g i n g ) .

s e i s m i c r e c o r d s . i f o r w a r d e x t r a p o l a t i o n o f s o u r c e w a v e f i e l d s . i . i i n v e r s e e x t r a p o l a t i o n o f r e f l e c t e d w a v e f i e l d s . i n e x t f r e q u e n c y . i i m a g e f o r c u r r e n t d e p t h n e x t d e p t h l e v e l n e x t s h o t r e c o r d c o m b i n e a l l r e c o r d s i m a g e d r e s u l t F i g u r e 1.5: Scheme f o r p r e - s t a c k m i g r a t i o n by i n v e r s i o n o f s i n g l e s e i s m i c s h o t r e c o r d s .

1.3 CONSIDERATIONS ABOUT PROPAGATION VELOCITY DETERMINATION

The r o l e o f t h e p r o p a g a t i o n v e l o c i t y i s o f c r u c i a l i m p o r t a n c e i n any m i g r a t i o n scheme. I t i s t h e p h y s i c a l parameter w h i c h g o v e r n s t h e phase o f

t h e wave f i e l d s i n t h e medium i n v e s t i g a t e d . A l l i m p o r t a n t p r o p a g a t i o n e f f e c t s ( s p a t i a l d i s p e r s i o n ) a r e d i r e c t l y r e l a t e d t o i t . C o n s e q u e n t l y , t h e r e m o v a l o f s p a t i a l d i s p e r s i o n w i l l be e f f e c t i v e o n l y when a good

a s s u m p t i o n has been made f o r t h e v e l o c i t y d i s t r i b u t i o n i n t h e medium. An e r r o n e o u s l y chosen v e l o c i t y p r o f i l e may cause phase e r r o r s , w h i c h r e s u l t s i n a bad s i g n a l - t o - n o i s e r a t i o , a l o w r e s o l u t i o n and i n c o r r e c t p o s i t i o n i n g o f r e f l e c t o r s .

These symptoms a r e on t h e i r t u r n used i n p r o c e d u r e s t o f i n d t h e o p t i m a l v e l o c i t i e s , w h i c h may be p a r t o f an i t e r a t i v e m i g r a t i o n scheme. V e l o c i t y

a n a l y s i s may be based on c o h e r e n c y c r i t e r i a f o r CMP o r CDP d a t a ( T a n e r and K o e h l e r , 1969; S c h u l t z and C l a e r b o u t , 1978), a n a l y s i s o f r e f r a c t i o n d a t a w i t h T-p t e c h n i q u e s ( D i e b o l d and S t o f f a , 1981), minimum e n t r o p y c r i t e r i a (De V r i e s , 1984) o r parameter e s t i m a t i o n (Van R i e l e t a l . , 1984). F i n a l l y , v e l o c i t y i n f o r m a t i o n may be r e t r i e v e d f r o m w e l l - l o g d a t a .

I n t h i s t h e s i s we w i l l not pay a t t e n t i o n t o the p r o b l e m of f i n d i n g 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 . F o r our p r e - s t a c k m i g r a t i o n method i t w o u l d i n v o l v e i n v e s t i g a t i o n o f the t r u e CDP-gather p r i o r to s t a c k i n g .

A p a r t from the above, a second p r o b l e m r i s e s when d e a l i n g w i t h media w h i c h a d m i t the e x i s t a n c e of s h e a r w a v e s . Shear waves a r e t r a v e l l i n g a t a c o n s i d e r a b l y lower speed t h a n p r e s s u r e waves. A l t h o u g h h a v i n g a d i f f e r e n t p h y s i c a l c h a r a c t e r , s h e a r waves may be a p p r o x i m a t e l y i n c l u d e d i n o u r m i g r a t i o n scheme f o r p r e s s u r e waves by u s i n g e.g. P-wave v e l o c i t i e s i n t h e f o r w a r d e x t r a p o l a t i o n p r o c e s s and s h e a r wave v e l o c i t i e s i n t h e i n v e r s e e x t r a p o l a t i o n p r o c e s s (P-S c o n v e r s i o n ) .

1.4 MODEL FAILURES

The model o f the m i g r a t i o n t e c h n i q u e p r e s e n t e d i n t h i s t h e s i s assumes the one-way s c a l a r wave e q u a t i o n t o be a p p l i c a b l e i n the d e s c r i p t i o n of wave f i e l d p r o p a g a t i o n . C o n s e q u e n t l y , the o c c u r r e n c e of m u l t i p l e r e f l e c t i o n s , c r i t i c a l a n g l e e f f e c t s as w e l l as the above mentioned wave c o n v e r s i o n s a r e n o t i n c l u d e d . However, w i t h r e s p e c t t o m u l t i p l e r e f l e c t i o n s , t h e r e can be added a m u l t i p l e e l i m i n a t i o n p r o c e s s t o the p r e - s t a c k m i g r a t i o n scheme ( B e r k h o u t , 1 9 8 2 ) . The o t h e r model f a i l u r e s can not be overcome p r o p e r l y and the model would have t o be based on the f u l l ( e l a s t i c ) wave e q u a t i o n . Use o f the s o - c a l l e d p h a s e - s h i f t o p e r a t o r i n wave f i e l d e x t r a p o l a t i o n r e q u i r e s the v e l o c i t y d i s t r i b u t i o n t o be c o n s t a n t w i t h i n the o p e r a t o r l e n g t h . Hence, o n l y weak l a t e r a l v e l o c i t y changes w i t h i n one s e i s m i c r e c o r d can be h a n d l e d p r o p e r l y (De V r i e s and B e r k h o u t , 1984). T u r n i n g o v e r t o f i n i t e d i f f e r e n c e o p e r a t o r s w i l l s o l v e t h i s p r o b l e m a t the c o s t of an i n c r e a s e i n c o m p u t a t i o n t i m e . The s o f t w a r e has been d e s i g n e d i n s u c h way, t h a t i n f u t u r e more c o m p l i c a t e d e x t r a p o l a t i o n o p e r a t o r s can be e a s i l y i n c l u d e d i n the package.

I n p r a c t i c a l a p p l i c a t i o n s i t m i g h t be o f t e n s u f f i c i e n t t o s t a y w i t h t h e e c o n o m i c a l l y a t t r a c t i v e p h a s e - s h i f t o p e r a t o r and t o c o r r e c t f o r 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 w i t h i n one s h o t r e c o r d by a p p l y i n g r e s i d u a l

n m o - c o r r e c t i o n s i n t h e CDP-gather j u s t p r i o r t o s t a c k i n g .

1.5 THE OUTLINE OF THIS THESIS

I n c h a p t e r 2 t h e b a s i c i n t e g r a l e q u a t i o n s a r e d e r i v e d , w h i c h d e s c r i b e t h e p r o p a g a t i o n o f a c o u s t i c waves i n an inhomogeneous f l u i d . A r e f e r e n c e medium i s used w i t h smooth t r a n s i t i o n zones and c r i t i c a l a n g l e e f f e c t s a r e n o t c o n s i d e r e d . Under t h e s e c o n d i t i o n s a B o r n a p p r o x i m a t i o n c a n be found f o r t h e i n t e g r a l e q u a t i o n d e s c r i b i n g t h e f o r w a r d problem. By d i s c r e t i -s a t i o n o f t h e r e -s u l t i n g e q u a t i o n a m a t r i x f o r m u l a t i o n f o l l o w -s , t h a t a l r e a d y has been g i v e n by B e r k h o u t ( 1 9 8 0 ) . The i n v e r s e p r o b l e m i s s h o r t l y i n d i c a t e d . I n c h a p t e r 3 t h e method i s p r e s e n t e d t o c a r r y o u t p r e - s t a c k m i g r a t i o n by i n v e r s i o n o f s i n g l e s e i s m i c s h o t r e c o r d s . I t i n v o l v e s a t r u e common d e p t h p o i n t s t a c k . C h a p t e r 4 d i s c u s s e s t h e c o n c e p t 'common d e p t h p o i n t s t a c k ' . The d i f f e -r e n c e s w i t h -r e s p e c t t o common m i d p o i n t s t a c k a -r e e x p l a i n e d . E x a m i n a t i o n o f common d e p t h p o i n t g a t h e r s i s p r o p o s e d as a t o o l f o r v e l o c i t y a n a l y s i s . I n c h a p t e r 5 a method i s p r e s e n t e d t o g e n e r a t e a m u l t i - f o l d z e r o - o f f s e t s e c t i o n o u t o f s e i s m i c s h o t r e c o r d s . I n c h a p t e r 6 s y n t h e t i c e x p e r i m e n t s a r e p r e s e n t e d w h i c h i l l u s t r a t e and s u p p o r t t h e t h e o r y o f t h e p r e v i o u s c h a p t e r s . The r e s o l v i n g power and s i g n a l - t o - n o i s e enhancement o f t h e p r e - s t a c k m i g r a t i o n method i s e v a l u a t e d .

C h a p t e r 7 c o n c l u d e s t h i s t h e s i s by a p p l i c a t i o n o f t h e p r e - s t a c k m i g r a t i o n method on u l t r a s o n i c and s e i s m i c measurements.

CHAPTER 2

BASIC CONSIDERATIONS ON THE FORWARD

AND INVERSE PROBLEM

I n t h i s c h a p t e r we w i l l d e r i v e t h e b a s i c e q u a t i o n s , w h i c h d e s c r i b e t h e f o r -ward p r o b l e m o f wave f i e l d p r o p a g a t i o n and r e f l e c t i o n i n an inhomogeneous f l u i d . The r e l a t i o n s h i p w i t h t h e l i n e a r i z e d i n v e r s e s c a t t e r i n g t h e o r y w i l l be i n d i c a t e d . F i n a l l y , t h e b a s i c p r e - s t a c k i n v e r s i o n p r o c e s s i s p r e s e n t e d and t h e most i m p o r t a n t c o m p u t a t i o n a l consequences a r e m e n t i o n e d .

To a d d r e s s a p o i n t i n space we w i l l u s e an o r t h o g o n a l C a r t e s i a n c o o r d i n a t e s y s t e m as frame o f r e f e r e n c e . The c o o r d i n a t e s a r e d e n o t e d by x, y and z. F o l l o w i n g t h e s e i s m i c l i t e r a t u r e t h e z - c o o r d i n a t e i s t a k e n t o be p o s i t i v e i n t h e downward d i r e c t i o n .

2.1 THE LIPPMANN-SCHWTNGER INTEGRAL EQUATION

Z

r

t

F i g u r e 2.1: O r t h o g o n a l s p a t i a l c o o r d i n a t e s y s t e m and time c o o r d i n a t e .

The v e c t o r r i s d e f i n e d as r = x i + y i + z i . The t i m e c o o r d i n a t e

x J y z

w i l l be denoted by t . The p a r t i a l d e r i v a t i v e i n the d i r e c t i o n normal t o a s u r f a c e w i l l be denoted by 3 . The p a r t i a l d e r i v a t i v e s toward x, y and z w i l l be denoted by 9^, 9^ and 9z r e s p e c t i v e l y .

L e t us c o n s i d e r a volume bounded by a s u r f a c e A t e a c h p o i n t on t h e s u r f a c e we d e n o t e the v e c t o r normal t o and p o i n t i n g outward by n.

F i g u r e 2.2: Volume bounded by s u r f a c e sr.

We assume t h e space i n s i d e SP t o be f i l l e d w i t h an inhomogeneous f l u i d . The space o u t s i d e ^ i s c o n s i d e r e d t o be homogeneous. The F o u r i e r t r a n s f o r m e d s c a l a r f i e l d i n s i d e SPdue t o s o u r c e s o u t s i d e

SPwill be d e n o t e d by P(r,0J). I t obeys the wave e q u a t i o n

v \ ( - i - VP) + k2P = 0 (2.1.1a)

P. o

o r , making use o f a r e f e r e n c e medium,

p v". ( — vT?) + k P = -yk P - Vlnß.VP, (2.1.1b) P0

where

p ( r ) = the d e n s i t y o f some r e f e r e n c e medium a t r e s t ; p ( r ) = the d e n s i t y o f the a c t u a l medium a t r e s t ; S ( r ) = p0/ p0;

c ( r ) = t h e phase v e l o c i t y o f l o n g i t u d i n a l waves i n the r e f e r e n c e medium; c ( r ) = t h e phase v e l o c i t y o f l o n g i t u d i n a l waves i n the a c t u a l medium;

Y( r ) = ( c / c )2 - 1;

The s c a l a r f i e l d i n s i d e SP due t o a monopole s o u r c e l o c a t e d i n p o i n t r i n s i d e SP w i l l be denoted by G ( r , r , to). F o r a r e f e r e n c e medium c h a r a c t e r -i z e d by d e n s -i t y p and phase v e l o c -i t y c the f -i e l d G w -i l l be denoted by G. I t i s t h e s o l u t i o n o f t h e s c a l a r wave e q u a t i o n p V . ( — V G ) + k2G = - 4 I T 6 ( 1 ? - r I ) , (2.1.2) o - m o where <$(.) s t a n d s f o r t h e D i r a c p u l s e . A c c o r d i n g t o t h e p r i n c i p l e o f r e c i p r o c i t y G ( r , r ,u) = G ( r ,r,o)). S u b s t i t u t i o n o f 1 / p [PVG - GVP] i n m m o

t h e Gauss theorem y i e l d s t h e f o l l o w i n g r e p r e s e n t a t i o n o f Greens s e c o n d theorem: [ [ PV\ (-|- VG) - G^.(-5- V " P ) ] d ^ = ' p p yp^ o o I-- i I-- [P3„G I-- G3^P]dOT (2.1.3)

SP

0 S u b s t i t u t i o n o f (2.1.1) and (2.1.2) i n (2.1.3) y i e l d s t h e s o - c a l l e d Lippmann-Schwinger i n t e g r a l e q u a t i o n P ( ?m, t o ) = P(?m.a>) +~ ƒ 5 ( ?m, r " ) G ( ?m, ? , u ) Q ( ? , a ) ) d T ^ (2.1.4)

y

where P ( ? ,w) = ^ I a ( r , r ) [ G ( r " ,r,u))3 P(r,«) -m 4TT I m m n 3 G ( r , ?)o j ) P ( r , u ) ] d ^ (2.1.5) a ( r m ) r ) = ^ L , (2.1.6a) m' P ( r ) o and Q(r,u)) = [Y( r ) k2 + V l n g ( r ) . V ] P ( r , o ) ) . (2.1.6b)

I n t h e f o r w a r d p r o b l e m P ( rQ, U ) ) i s t o be computed f o r a g i v e n

r e f e r e n c e medium and g i v e n f u n c t i o n s g and v . I n t h e i n v e r s e p r o b l e m P ( r oi) i s known, and f o r a g i v e n r e f e r e n c e medium g and y have t o be computed. I n most p r a c t i c a l s i t u a t i o n s f u l l s o l u t i o n s o f b o t h p r o b l e m s a r e d i f f i c u l t t o f i n d as t h e i n t e g r a l i n (2.1.4) c o n t a i n s a l s o t h e unknown t o t a l wave f i e l d P ( r, u ) w i t h i n t h e volume y

E q u a t i o n (2.1.4) c a n be r e w r i t t e n as

P ( rm, a ) ) = P ( rm, o j ) + A P ( rm, c o ) , (2.1.7)

w i t h

A P ( rm, t o ) = ^ L ƒ S(?m, ? ) G ( ?m, r " , a ) ) Q ( r * , oJ) d/K ' (2.1.8)

The ' d e v i a t i o n p r e s s u r e ' AP i s due t o t h e d i f f e r e n c e between t h e a c t u a l medium and t h e r e f e r e n c e medium. I t c o n s i s t s o f two s e p a r a t e p a r t s p e r t a i n i n g t o d e v i a t i o n s i n v e l o c i t y and d e n s i t y r e s p e c t i v e l y . E x p r e s s i o n (2.1.8) f o r m u l a t e s an i m p l i c i t e q u a t i o n f o r AP. T h i s c a n be s e e n by s u b s t i t u t i n g (2.1.7) i n (2.1.8) and an i t e r a t i v e s o l u t i o n c a n be w r i t t e n as A P( i ) - _ L f a G Q d T + 7L f a 5 A Q( 1-1 )d/r , (2.1.9a) 4TT J 4TT J

y y

where A Q( 1 _ 1 ) - [ Y k2 + V l n 6 . V ] A P( l 1 ). (2.1.9b)

The i n i t i a l s t e p i n (2.1.9) might be chosen as

A P( o ) = 0 (2.1.10)

I f t h e r e f e r e n c e medium i s chosen c l o s e t o the a c t u a l medium, t h e n we may w r i t e

-2 •*• •+

T h i s means t h a t t h e c o n t r i b u t i o n s o f y k Ap and Vlng.VAP may be n e g l e c t e d and AP^ ^ i s a good e s t i m a t e of AP:

Y

= J_f

a G [ y k + V l n g . V ] P d Y (2.1.12)

T h i s a p p r o x i m a t i o n i s u s u a l l y c a l l e d t h e 'Born a p p r o x i m a t i o n ' o f AP. I n p h y s i c a l terms i t i n v o l v e s two a s s u m p t i o n s :

a ) I n h o m o g e n e i t i e s as d e f i n e d by y and B have a minor i n f l u e n c e on t h e p r o p a g a t i o n o f the i n c i d e n t wave f i e l d .

b ) M u l t i p l e s c a t t e r i n g can be n e g l e c t e d .

2 . 2 A GENERALIZED VERSION OF THE RAYLEIGH INTEGRALS

I n many a p p l i c a t i o n s wave f i e l d s a r e g e n e r a t e d by c a u s a l s o u r c e s l o c a t e d on a p l a n e s u r f a c e ( a c q u i s i t i o n p l a n e ) . As t o the volume Yintroduced

above, l e t us c o n s i d e r t h e volume bounded by t h e p l a n e SP f o r z = z

J r o o

and a hemisphere i n t h e h a l f s p a c e z > z^ w i t h a r a d i u s i n c r e a s i n g w i t h o u t any l i m i t .

F i g u r e 2.3: Geometry f o r R a y l e i g h i n t e g r a l s .

Assume t h e p o i n t r t o be l o c a t e d on the s u r f a c e and t h e p o i n t r

o o r

somewhere i n the i n t e r i o r o f Y. The h a l f s p a c e z < z i s c o n s i d e r e d t o be

= o

c o n f i g u r a t i o n e x p r e s s i o n (2.1.5) c a n be r e f o r m u l a t e d as

i:

P ( r , u ) =T T a ( r , r ) G ( r , r l ( l))3 P ( t ,to)d SP, (2.2.1a) ¿11 J O O z o o o r P ( r , u ) = i a ( r , r )3 G ( r , r ,U) P ( ?

,u)d&*

(2.2.1b) 2TT J o z o o

SP

0 These e x p r e s s i o n s may be v i e w e d as g e n e r a l i z e d v e r s i o n s o f t h e R a y l e i g h i n t e g r a l s . They d e f i n e t h e p r e s s u r e i n t h e r e f e r e n c e medium z > z due t o

o a monopole ( 2 . 2 . 1 a ) o r a d i p o l e (2.2.1b) d i s t r i b u t i o n on t h e boundary z = zQ. When d e a l i n g w i t h a h o m o g e n e o u s r e f e r e n c e medium, e x p r e s s i o n s (2.2.1) w i l l r e d u c e t o t h e c l a s s i c a l R a y l e i g h i n t e g r a l s , w i t h : G ( r , r ,w) o""' A r rft t ^ - - j k A r ( 1 + j k A r ) , ,„ . „. 3 G ( r , r ,o)) - e J - — * — c o s 4>, (2.2.2) o ( A r )

where A r = |r - rQ| , and <}> d e n o t e s t h e a n g l e between r - r Q and t h e

n o r m a l t o SPQ. I n a d d i t i o n , i f Born's a p p r o x i m a t i o n (2.1.11) a p p l i e s as w e l l , t h e n i t i s a l l o w e d t o i d e n t i f y A? i n (2.1.12) w i t h t h e u p g o i n g r e f l e c t e d w a v e f i e l d P . M o r e o v e r , P e q u a l s t h e d o w n e o -up _ ^ 6 i n g i l l u m i n a t i n g s o u r c e f i e l d S ( r ) and (2.1.12) c a n be r e f o r m u l a t e d as Pu p " 2 i ƒ 4 r ^[ ^ + ^ln^]S" * Y ( 2-2-3 > T h i s e x p r e s s i o n f o r m u l a t e s t h e b a s i c f o r w a r d model w h i c h j u s t i f i e s c u r r e n t s e i s m i c m i g r a t i o n t e c h n i q u e s . The f i r s t term t a k e s i n t o a c c o u n t inhomoge-n e i t i e s due t o v e l o c i t y chainhomoge-nges, whereas t h e secoinhomoge-nd term t a k e s i inhomoge-n t o a c c o u n t i n h o m o g e n e i t i e s due t o d e n s i t y changes.

H e r e a f t e r we assume the r e f e r e n c e medium t o be chosen as f o l l o w s :

a ) The t o t a l f i e l d i n t h e r e f e r e n c e medium c a n be s p l i t i n t o two p a r t s

P ( r ) = S ( r ) + P ( r ) , up s u c h t h a t | Pu p< r ) I « S ( r ) . T h i s a s s u m p t i o n means t h a t we may t a k e i n (2.1.12) P e q u a l t o S. b ) The r e f l e c t e d f i e l d s i n t h e r e f e r e n c e medium a r e s m a l l : N ( r ) r e p r e s e n t i n g t h e n o i s e c o n t r i b u t i o n . T h i s a s s u m p t i o n means t h a t we may t a k e i n (2.1.12) AP e q u a l t o P •

I f t h e f i r s t c o n d i t i o n i s f u l f i l l e d , then t h e components S and P obey up t h e one-way wave e q u a t i o n s 9zS + j k H ^ S = 0 3 P - jkH.P = 0, (2.2.4) z up J 1 up ' where and H^ a r e l a t e r a l o p e r a t o r s ( B e r k h o u t , 1984).

I f t h e above c o n d i t i o n s a r e f u l f i l l e d , the r e f e r e n c e medium c o n s i s t s o f smooth t r a n s i t i o n zones and c r i t i c a l a n g l e e f f e c t s a r e n o t c o n s i d e r e d . F o r t h i s c h o i c e o f t h e r e f e r e n c e medium t h e B o r n a p p r o x i m a t i o n c a n be r e f o r m u -l a t e d as

4 i

ƒ

5

w i t h

S(r,u>) f a ( r , rQ) 3z G ( ? , ?o, a ) ) S ( ?o, u ) d ^ . (2.2.6)

k

2 . 3 THE DISCRETIZED INTEGRAL REPRESENTATIONS OF THE BORN

APPROXIMATION A s s u m i n g b a n d l i m i t e d d a t a , i n t e g r a l r e p r e s e n t a t i o n (2.2.5) c a n be d i s c r e t -i z e d a l o n g t h e z — d -i r e c t -i o n . M r puP " £ I J [ S ë W * 2 + ^ ^ W m A z d ^ > (2.3.1) where Az < \\ . , X . b e i n g t h e s m a l l e s t w a v e l e n g t h o f i n t e r e s t and mm mm ° MAz b e i n g t h e maximum d e p t h o f i n t e r e s t . The f u n c t i o n G i s chosen t o be g i v e n by ( 2 . 2 . 2 ) . E x p r e s s i o n (2.3.1) f o r m u l a t e s t h a t P c o n s i s t s o f M

up

c o n t r i b u t i o n s from d e p t h l e v e l s A z , 2Az,..., MAz. Next we d i s c r e t i z e a l o n g t h e x- and y - d i r e c t i o n as w e l l A A A K L M p _ ûxflyflz V V V — «P 4TT L L L l a G ]k A x , l A y , m A z K=—K 1=—L m= i [ y k2 + V l n B . V ]kAx>1Ay)mAzSkAX)1Ay>mAz (2.3.2)

A more handsome f o r m i s found by u s i n g t h e m a t r i x n o t a t i o n . L e t us i n t r o -duce m a t r i c e s , M, Y and G as w e l l as v e c t o r s P and S:

P.c

M ( zm) r e p r e s e n t s t h e m a t r i x s c a l i n g f o r t h e d e n s i t y a t d e p t h l e v e l

z = zm- I t has a d i a g o n a l form. I t s d i a g o n a l c o n t a i n s t h e

d i s c r e t i z e d v e r s i o n o f p (z ). 'o m

Yc( zm) r e p r e s e n t s t h e m a t r i x a c c o u n t i n g f o r v e l o c i t y c o n t r a s t s due

t o d e v i a t i o n s between t h e r e f e r e n c e medium and t h e a c t u a l medium. I t s d i a g o n a l c o n t a i n s t h e d i s c r e t i z e d v e r s i o n o f

Y ( z ) r e p r e s e n t s t h e m a t r i x a c c o u n t i n g f o r d e n s i t y c o n t r a s t s due t o

p m' r J

d e v i a t i o n s between t h e r e f e r e n c e medium and t h e a c t u a l medium. I t s e l e m e n t s r e p r e s e n t t h e d i s c r e t i z e d v e r s i o n o f o p e r a t o r ^z( V l n B . V )m A z. G ( z » zm) r e p r e s e n t s t h e upward p r o p a g a t i o n m a t r i x between d e p t h l e v e l z and z . I t s columns r e p r e s e n t d i s c r e t i z e d v e r s i o n s o f AxAy -» . AxAy -»• . " I F " ( ro, rm) = ft ( rm, ro ' ' a monoP °le r e s p o n s e . S ( zm) r e p r e s e n t s t h e d i s c r e t i z e d v e r s i o n o f t h e downgoing s o u r c e wave f i e l d i n t h e r e f e r e n c e medium a t d e p t h l e v e l z = z . m P ( zQ) r e p r e s e n t s t h e d i s c r e t i z e d v e r s i o n o f t h e u p g o i n g r e f l e c t e d wave f i e l d a t t h e s u r f a c e z = z . o U s i n g t h e s e d e f i n i t i o n s (2.3.2) c a n be t r a n s f o r m e d i n t o i t s m a t r i x r e p r e -s e n t a t i o n ?

<

z0

) - I

M ( zo) G ( zo, zm) M -1( zm) Y ( zm) S>( zm) , (2.3.3) m=l

where we combined t h e m a t r i c e s Y and Y i n t o one m a t r i x Y = Y +

c p c Y p. L e t us a l s o combine a l l i n f o r m a t i o n c o n c e r n i n g p r o p a g a t i o n e f f e c t s i n t o one m a t r i x :

V(z

o

,z

m

) M M ( z

o

) G (

Z o ( Z m

) M

_ 1

( z

m

)

V ( zm )zo)

U M-

1

(z

m

)G(z

m

,

Z o

)M(z

o

),

(2.3.4) where V ( zm, zo) = VT( zo, zm) . The f i r s t o p e r a t o r r e p r e s e n t s t h e u p w a r d p r o p a g a t i o n m a t r i x between d e p t h l e v e l z and z . whereas t h e second one r e p r e s e n t s t h e d o w n

-m o r

w a r d p r o p a g a t i o n m a t r i x between d e p t h l e v e l z and z . o m

Hence, we may w r i t e a c c o r d i n g t o (2.2.6) S ( zm) = - V ( zm, zo) 3zS ( zo) . (2.3.5) S u b s t i t u t i o n o f (2.3.5) and (2.3.4) i n t o (2.3.3) y i e l d s M P ( z ) = - Y V ( z ,z ) Y ( z ) V(z ,z ) 3 S ( z ). (2.3.6) o t—• o' m m m' o z o m= 1

As we assumed t h e one-way wave e q u a t i o n t o be v a l i d , t h e d e r i v a t i v e toward z c a n be e x p r e s s e d as a d i s c r e t i z e d v e r s i o n o f a b a n d l i m i t e d , l a t e r a l o p e r a t o r j H ^ : 3ZS(z,w) = -H^S(Z,ÜU) (2.3.7) 3 P (z,w) = H"p ( z . u ) . z up ' 1 up '

M a k i n g use o f (2.3.6) and (2.3.7) we may w r i t e

?<z 0) " I

[ V (

Z o

,

Z m

) Y (

Z i i i

) V ( z

m

, z

o

) ] H + (

Z o

) r ( z

o

)

(2.3.8) m= 1

and

3z? ( zo > "

ZH7(*

0

)[VU

0

.«

m

)Y(«

11

)V<«

11

.«o)]Ht<"o>s<«o)-

( 2

-

3

'

9 ) m=l E x p r e s s i o n s (2.3.8) and (2.3.9) f o r m u l a t e v i z . a 3 p •*• p t r a n s i t i o n and a p * 3zp t r a n s i t i o n . S i m i l a r e x p r e s s i o n s c a n be d e r i v e d f o r p •* p and 3 p •+ 3 p t r a n s i t i o n s ( B e r k h o u t , 1982). H e r e a f t e r we w i l l u s e t h e z zr e x p r e s s i o n + M + P ( z ) = Y W ( z ,z ) R ( z ) W ( z ,z )S(z ), (2.3.10) v o t—' o' m m m' o o m= 1 where

= [H^(

Z m

)]-

1

Y(z

m

)

> (2.3.11)

W ( z0, zm) = V ( zo, zm) H - ( zm) ,

(2.3.11)

W(z .z ) = V ( zm, Zn) H1" ( z ).

m o m o 1 o

M a t r i x R contains r e f l e c t i v i t y information due to d e v i a t i o n s i n v e l o c i t y and d e n s i t y between the r e f e r e n c e medium and t h e a c t u a l medium. Combining many e x p e r i m e n t s i n t o one m a t r i x and t a k i n g i n t o a c c o u n t d e t e c t o r

p a t t e r n s , one f i n a l l y a r r i v e s a t

P ( zQ) = D ( zo) [ X W ( zo, zM) R ( zm) W ( zm, zo) ] S ( zo) . (2.3.12)

m

T h i s i s the f o r w a r d model used by B e r k h o u t i n h i s d e s c r i p t i o n of p r e - s t a c k d e p t h m i g r a t i o n ( B e r k h o u t , 1982).

2.4 THE INVERSE PROBLEM

T h u s f a r we i n v e s t i g a t e d the f o r w a r d problem. The f u l l i n v e r s e p r o b l e m ( w i t h i n the v a l i d i t y o f the one-way a c o u s t i c a l model) aims a t e s t i m a t i n g t h e d e v i a t i o n p a r a m e t e r s 3 and y. I n t h i s t h e s i s we w i l l l i m i t o u r s e l v e s t o e s t i m a t i n g z e r o - o f f s e t r e f l e c t i v i t y ( t h e d i a g o n a l e l e m e n t s of R ) .

The i n v e r s i o n p r o c e s s i n v o l v e s i n f a c t two major s t a g e s . F i r s t l y , we have t o compensate f o r p r o p a g a t i o n e f f e c t s on the upward and downward t r a v e l -l i n g wave f i e -l d . S e c o n d -l y , we have t o r e t r i e v e r e f -l e c t i v i t y i n f o r m a t i o n f r o m the i n v e r t e d d a t a . I n v e r s i o n f o r t h e p r o p a g a t i o n e f f e c t s i n v o l v e s t h e c o n s t r u c t i o n of o p e r a t o r F w i t h the d e s i r e d p r o p e r t y FW = I (2.4.1) T h i s i s a s p a t i a l c o n v o l u t i o n a l type of o p e r a t i o n as a l r e a d y d i s c u s s e d by B e r k h o u t ( 1 9 7 9 ) . However, f u l l i n v e r s i o n i s an u n s t a b l e p r o c e s s due t o the e v a n e s c e n t p a r t of the wave f i e l d . C o n s e q u e n t l y , the i n v e r s i o n s h o u l d be c a r r i e d out i n a s p a t i a l l y b a n d l i m i t e d way. The band l i m i t a t i o n of the d a t a imposes a f u n d a m e n t a l bound on the r e s o l v i n g power o f the i n v e r s i o n method ( B e r k h o u t , 1984).

B e f o r e d i s c u s s i n g t h e i n v e r s i o n p r o c e s s i n t o d e t a i l we remark, t h a t f o r a l l examples i n t h i s t h e s i s we d i d choose t h e r e f e r e n c e medium t o be o n e - d i m e n s i o n a l : c ( r ) = c ( z ) p ( r ) = p ( z ) (2.4.2) Note t h a t we e x c l u d e d c r i t i c a l a n g l e e f f e c t s , w h i c h r e q u i r e s t h e o p e r a t o r a n g l e t o be r e s t r i c t e d ($ < 90°). Under t h e s e c o n d i t i o n s a r e c u r s i v e phase-s h i f t o p e r a t o r can be u phase-s e d .

CHAPTER 3

INVERSION PROCEDURE

I n t h i s c h a p t e r we w i l l c a r r y o u t i n v e r s i o n by downward c o n t i n u a t i o n o f s o u r c e s and d e t e c t o r s . I t w i l l be shown t h a t i n v e r s i o n o f s i n g l e s h o t r e c o r d s i s most s u i t a b l e when r e c o v e r i n g z e r o - o f f s e t r e f l e c t i v i t y o n l y . By c o m b i n i n g t h e i n v e r t e d r e s u l t s f o r d i f f e r e n t r e c o r d s a t one d e p t h p o i n t , a common d e p t h p o i n t s t a c k i s o b t a i n e d .

3.1 INVERSION BY DOWNWARD CONTINUATION

I n t h i s s e c t i o n we d i s c u s s t h e g e n e r a l method o f i n v e r t i n g a m u l t i - r e c o r d d a t a s e t and t h e e x t r a c t i o n o f r e f l e c t i v i t y p a r a m e t e r s . Our s t a r t p o i n t w i l l be eq. ( 2 . 3 . 1 2 ) . Assume f o r t h e moment t h a t t h e m a t r i x D ( Zq) , w h i c h d e s

-c r i b e s t h e d e t e -c t o r p a t t e r n s , e q u a l s t h e u n i t y m a t r i x . T h i s means t h a t a l l g r o u p s c o n s i s t o f one geophone o n l y . The f i r s t s t e p o f t h e i n v e r s i o n i s c o m p e n s a t i o n f o r p r o p a g a t i o n e f f e c t s f o r each s e p a r a t e d e p t h l e v e l z . m M u l t i p l y i n g (2.3.12) by i n v e r s i o n o p e r a t o r F ( zN >ZQ) , where i n a b a n d l i m i t e d sense

F (z ,z )W(z ,z )

v o' n n' o I

F (z ,z )W(z ,z )

n' o o' n I, we y i e l d :

F(

Z n

, z

0

) P ( z

0

) =

X

F(z

n

, z

0

)W(z

0

, z

m

) R ( z

M

)S(z

m

)

(3.1.2) m

P(z

J = X F ( Z . Z

)W(z

. z ) R ( z

M

)S(z),

(3.1.3) n m n o o m m m'

P(z ) =

F(z ,z ) P(z )

n n o o S(z ) = W(z ,z )S(z ) (3.1.4) m m o o

r e p r e s e n t the downward c o n t i n u e d s o u r c e and r e f l e c t e d wave f i e l d s a t d e p t h l e v e l z = z and z = z r e s p e c t i v e l y . From (3.1.3) we y i e l d

n m

P(z

n

) = R ( z

N

)S(z

n

)

+ £ [ . . . ] (3.1.5) m^n

A c t u a l l y the summation i n d e x i n (3.1.4) can be t a k e n t o s t a r t a t n + 1, as one may s u b s t r a c t a t each d e p t h l e v e l the c o n t r i b u t i o n of the e s t i m a t e d r e f l e c t i v i t y a t h i g h e r d e p t h l e v e l s . N o t e , t h a t o n l y the i n s t a n t a n e o u s p a r t o f the r e f l e c t i v i t y i s a l l o w e d t o be s u b t r a c t e d , o t h e r w i s e r e l e v a n t r e f l e c t i v i t y i n f o r m a t i o n a t o t h e r d e p t h l e v e l s i s a f f e c t e d . T h i s r e q u i r e s t h e d a t a t o be p r o c e s s e d f o r a l l f r e q u e n c y components, b e f o r e p e r f o r m i n g t h e n e x t downward e x t r a p o l a t i o n s t e p . o r where

O n l y the f i r s t term on the r i g h t h a n d s i d e of (3.1.5) i s p r o p e r l y c o r r e c t e d f o r p r o p a g a t i o n e f f e c t s . The second s t e p of the i n v e r s i o n c o n s i s t s of c o m p e n s a t i n g f o r

S(z

n

),

the downward c o n t i n u e d s o u r c e wave f i e l d :

P(z

n

)

F

s

( z

n

) = R(z

n

) + I

[...] (3.1.6) m^n

and i n v e r s i o n o p e r a t o r FG i s d e f i n e d i n a b a n d l i m i t e d sense by

S(zn >Fs <zn > = 1 ( 3 -1-7)

As a l r e a d y m e n t i o n e d , c o m p e n s a t i o n f o r p r o p a g a t i o n e f f e c t s between the s u r f a c e and d e p t h l e v e l z = z^ has been done, so r e l e v a n t r e f l e c t i v i t y i n f o r m a t i o n f o r the a c t u a l d e p t h l e v e l w i l l be found a t z e r o t r a v e l t i m e . N o t e the p r e s e n c e o f p r o p a g a t i o n e f f e c t s under the summation s i g n . The

c o n t r i b u t i o n s from t h o s e terms a r e m e r e l y found a t n o n - z e r o t r a v e l t i m e s . Thus a s e p a r a t i o n has been made between m i g r a t e d and u n m i g r a t e d d a t a .

3 . 2 INVERSION OF SINGLE SHOT RECORDS

The f u l l p r o c e s s o f m i g r a t i o n as d e s c r i b e d above i n v o l v e s a p r o c e s s o f c o m p e n s a t i o n f o r p r o p a g a t i o n e f f e c t s on t h e r e g i s t e r e d d a t a , s e q u e n t i a l l y r e o r d e r i n g t h e d a t a f r o m common s h o t g a t h e r s (CSG) t o common r e c e i v e r g a t h e r s (CRG) and v i c e v e r s a ( B e r k h o u t , 1982). T h i s p r o c e s s r e c o v e r s t h e t o t a l r e f l e c t i v i t y m a t r i x R. I f o n l y z e r o - o f f s e t r e f l e c t i v i t y i s t o be r e c o v e r e d , t h e n t h e main d i a g o n a l o f R i s t o be computed o n l y . I n t h a t c a s e a much s i m p l e r i n v e r s i o n p r o c e s s c a n be u s e d . H e r e a f t e r we w i l l c o n f i n e o u r s e l v e s t o r e c o v e r y o f z e r o - o f f s e t r e f l e c t i v i t y . L e t us c o n s i d e r a medium w i t h a c o n t r a s t a t d e p t h z = z . m A s o u r c e i s p o s i t i o n e d a t t h e s u r f a c e and a common s h o t g a t h e r i s a c q u i r e d . These d a t a w i l l occupy t h e e l e m e n t s o f m a t r i c e s S ( Zq) and

P ( Zq) i n (2.3.12) as i n d i c a t e d i n f i g u r e 3.2a. A f t e r c o m p l e t i n g t h e p r o c e s s o f downward c o n t i n u a t i o n toward d e p t h l e v e l z = z ^ ( f i g . 3 . 1 ) , t h e energy w i l l be d i s t r i b u t e d o v e r t h e m a t r i x e l e m e n t s as shown i n f i g u r e 3.2b. o zQ x x X

\ I i \

--=o—zm x x X F i g u r e 3.1: Downward c o n t i n u a t i o n o f s o u r c e and d e t e c t o r a r r a y s f o r one p h y s i c a l e x p e r i m e n t .

\

S < Z J — \ _ P ( 2 j

\

(a)

forward I extrapolation inversel extrapolation

•J

S

(zm)

\

t

1

( b )

F i g u r e 3.2: Nonzero e l e m e n t s i n d a t a m a t r i c e s S and P f o r one s e i s m i c e x p e r i m e n t . a ) a t s u r f a c e b) a f t e r downward c o n t i n u a t i o n U s i n g a matched f i l t e r p r o c e d u r e f o r i n v e r s i o n i n ( 3 . 1 . 6 ) , t h e elements o f R ( z ) a r e found as t h e z e r o t r a v e l time v a l u e s i n _m

P ( z

n

) S *

T

( z

m

) - [ W *

T

( z

o

, z

m

) W ( z

o

, z

m

) ] R (

Z i n

) [ S ( z

m

) S *

T

( z

m

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(3.2.1) n^m

The z e r o t r a v e l t i m e i n f o r m a t i o n i s found by summing (3.2.1) f o r a l l f r e q u e n c i e s :

For one s e i s m i c r e c o r d t h e r e s u l t a n t a p p r o x i m a t i o n o f R ( zm) i s s c h e m a t i c a l l y drawn i n f i g u r e 3.3. \

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< zm) F i g u r e 3.3: Nonzero e l e m e n t s i n d a t a m a t r i c e s f o r one s e i s m i c e x p e r i m e n t and r e s u l t a n t a p p r o x i m a t i o n o f m a t r i x R.(z ). m When r e c o v e r i n g z e r o - o f f s e t r e f l e c t i v i t y o n l y t h e m a t r i x m u l t i p l i c a t i o n c a n be r e d u c e d t o c o m p u t a t i o n o f t h e d i a g o n a l e l e m e n t s . T h i s means a 2 r e d u c t i o n f r o m n toward n d a t a s a m p l e s , where n d e n o t e s t h e d i m e n s i o n s o f t h e m a t r i c e s . Combining a l l e x p e r i m e n t s r e s u l t s i n a d d i t i o n o f t h e s e p a r a t e l y computed d i a g o n a l s . As we w i l l d i s c u s s i n c h a p t e r 4 t h i s r e p r e s e n t s a common d e p t h p o i n t s t a c k . F i n a l l y , we summarize t h e c o m p u t a t i o n scheme f o r i n v e r s i o n o f s i n g l e s h o t r e c o r d s : 1) F o r w a r d e x t r a p o l a t i o n o f t h e s o u r c e wave f i e l d t o t h e n e x t d e p t h l e v e l . 2) I n v e r s e e x t r a p o l a t i o n o f t h e r e f l e c t e d wave f i e l d t o t h e n e x t d e p t h l e v e l . 3) C o r r e l a t i o n o f t h e e x t r a p o l a t e d s o u r c e wave f i e l d w i t h t h e e x t r a p o l a t e d r e f l e c t e d wave f i e l d . 4 ) A p p l i c a t i o n o f t h e i m a g i n g p r i n c i p l e . 5) Combining a l l m i g r a t e d s h o t r e c o r d s ( C D P - s t a c k ) .

CHAPTER 4

TRUE COMMON DEPTH POINT STACKING BY

SINGLE SHOT RECORD MIGRATION

I n t h i s c h a p t e r we w i l l b r i e f l y d i s c u s s t h e d i f f e r e n c e s between common m i d p o i n t (CMP) s t a c k and common d e p t h p o i n t (CDP) s t a c k . We w i l l show t h a t t h e i n v e r s i o n method w h i c h t r e a t s s i n g l e s e i s m i c r e c o r d s as d i s c u s s e d i n c h a p t e r 3 r e p r e s e n t s a ' t r u e common d e p t h p o i n t s t a c k ' .

4.1 INTRODUCTION

The a i m o f any s t a c k i n g p r o c e d u r e i s s i g n a l - t o - n o i s e r a t i o enhancement by some s o r t o f s y n t h e t i c f o c u s i n g p r o c e s s ( i n - p h a s e a d d i t i o n ) . I t g e n e r a l l y i n v o l v e s a s i g n i f i c a n t d a t a r e d u c t i o n . A l l t y p e s o f s t a c k i n g p r o c e d u r e s i n c l u d e t r a v e l t i m e c o r r e c t i o n s o f r e l a t e d t r a c e s f o r a s u i t a b l y chosen r e f e r e n c e p o i n t . Here t h e d e s i r e d c o h e r e n t s i g n a l s ( p r i m a r y r e f l e c t i o n s and d i f f r a c t i o n s ) i n t h e t r a c e s s h o u l d add o p t i m a l l y d u r i n g t h e s t a c k . B e f o r e s t a r t i n g o u r d i s c u s s i o n on s t a c k i n g p r o c e d u r e s we s h o r t l y want t o draw a t t e n t i o n t o t h e c o n f u s i o n i n s e i s m i c l i t e r a t u r e on common m i d p o i n t and common d e p t h p o i n t s t a c k i n g . H i s t o r i c a l l y t h e s e c o n c e p t s a r e based on r a y t h e o r e t i c a l models f o r s i m p l e g e o l o g i c a l s i t u a t i o n s , where

C M P - s t a c k i n g ( a l m o s t ) e q u a l s C D P - s t a c k i n g . When t u r n i n g o v e r t o more c o m p l i c a t e d m o d e l s , C M P - s t a c k i n g i s f u n d a m e n t a l l y d i f f e r e n t from C D P - s t a c k i n g . U n f o r t u n a t e l y , CMP i s o f t e n r e f e r r e d t o as CDP.

J 1 i I I I I I I a I I I I I [ I I I I I ;, I I I I I I ;' I 1 I I I . I I I

F i g u r e 4.1; Comparison o f a CMP g a t h e r ( a ) and a CUP g a t h e r ( b ) f o r a s i m p l e g e o l o g i c a l s i t u a t i o n .

We i l l u s t r a t e t h e d i f f e r e n c e by p r e s e n t i n g i n f i g u r e 4.1 a CMP g a t h e r as w e l l as a CDP g a t h e r f o r a s i m p l e g e o l o g i c a l s i t u a t i o n . The CMP g a t h e r shows a s m e a r i n g a t t h e d e p t h p o i n t w i t h a maximum o f f s e t o f a p p r o x i m a t e l y 500 m due t o t h e d i p o f t h e o v e r l a y i n g i n t e r f a c e .

The v e l o c i t y i n t h e f i r s t l a y e r amounts t o 2000 m/s; i n t h e second l a y e r i t amounts t o 3000 m/s. The d i p a n g l e o f t h e f i r s t r e f l e c t o r amounts a p p r o x i m a t e l y t o 6°.

4.2 THE COMMON MIDPOINT STACKING PROCEDURE

The common m i d p o i n t s t a c k i s p e r f o r m e d on a s e t o f r e l a t e d t r a c e s (common m i d p o i n t g a t h e r o r CMP g a t h e r ) , f o r w h i c h s h o t and r e c e i v e r p o s i t i o n s a r e l o c a t e d s y m m e t r i c a l l y around a p r e f i x e d s u r f a c e p o i n t (common m i d p o i n t ) . A s s u m i n g t h e same d i r e c t i v i t y p r o p e r t i e s f o r a l l s o u r c e s and d e t e c t o r s , one c a n p r o v e t h e r e g i s t r a t i o n s t o be s y m m e t r i c a l w i t h r e s p e c t t o t h e common m i d p o i n t . G e n e r a l l y , o n l y t h e ' r i g h t h a l f o f a C M P - s e c t i o n i s d i s p l a y e d . H i s t o r i c a l l y t h e d e s c r i p t i o n o f t h e CMP-stack s t a r t s w i t h a s i m p l e g e o l o g i c a l s i t u a t i o n as d e p i c t e d i n f i g u r e 4.2.

The s u b s u r f a c e i s homogeneous a p a r t from a h o r i z o n t a l c o n t r a s t a t a c e r t a i n d e p t h l e v e l . F o l l o w i n g r a y p a t h a n a l y s i s a l l r a y s w i l l r e f l e c t i n a

F i g u r e 4.2: Geometry o f a common m i d p o i n t g a t h e r o f a h o r i z o n t a l r e f l e c t o r w i t h a homogeneous o v e r b u r d e n .

common r e f l e c t i o n p o i n t on t h e r e f l e c t o r below t h e common m i d p o i n t . W i t h r e s p e c t t o t h e t r a v e l t i m e s , t h e r e s u l t i n g time s e c t i o n w i l be h y p e r b o l i c ( f i g . 4 . 3 ) .

N o t e t h a t a r e f l e c t i o n p o i n t i s n e v e r a r e a l p o i n t b u t a l w a y s an a r e a ( B e r k h o u t , 1984). T h i s r e f i n e m e n t w i l l be i g n o r e d h e r e .

x—-F i g u r e 4.3: Raypaths and t i m e s e c t i o n o f a CMP g a t h e r f o r one h o r i z o n t a l r e f l e c t o r . The f o r m u l a , w h i c h d e s c r i b e s t h e two-way t r a v e l t i m e s as a f u n c t i o n o f t h e o f f s e t between s o u r c e and r e c e i v e r , i s g i v e n by t h e h y p e r b o l i c r e l a t i o n : T2( x ) = T2( o ) + ^_ • (4.1.1) where x = t h e o f f s e t between s o u r c e and r e c e i v e r ; T ( o ) = t h e two-way, ZO t r a v e l t i m e ; c = t h e p r o p a g a t i o n v e l o c i t y .

F o r a s i n g l e c l i p p i n g l a y e r w i t h d i p a n g l e 8 t h i s e x p r e s s i o n becomes: 2 T2( x ) = T2( o ) + — 2 =- , (4.1.2)

I—? '

\cos6I w h i c h i s s t i l l h y p e r b o l i c . However, as shown i n f i g u r e 4.4 t h e r e f l e c t i o n p o i n t has been smeared. When d e a l i n g w i t h more complex s i t u a t i o n s the r e l a t i o n s h i p becomes n o n - h y p e r b o l i c . H i g h e r o r d e r a p p r o x i m a t i o n s have been g i v e n by a.o. May and S t r a l e y ( 1 9 7 9 ) . B e f o r e p e r f o r m i n g any s t a c k i n g one commonly a p p l i e s a normal moveout (n.m.o.) c o r r e c t i o n on the d a t a based on t h e one o f the above mentioned r e l a t i o n s . The v e l o c i t y g i v i n g the b e s t ( h y p e r b o l i c ) f i t i n the n m o - c o r r e c t i o n i s u s u a l l y r e f e r r e d t o as ' s t a c k i n g v e l o c i t y ' (Taner and K o e h l e r , 1979). X •« •>

CMP

F i g u r e 4.4: F o r a t i l t e d l a y e r t h e nmo-curve remains h y p e r b o l i c . The l a t e r a l smearing AL o f the d e p t h p o i n t i s i n d i c a t e d .

F i g u r e 4.5: Raypaths and time s e c t i o n i n a CMP g a t h e r f o r one r e f l e c t o r i n a more complex s u b s u r f a c e .

I n the s i m p l e case o f one h o r i z o n t a l r e f l e c t o r w i t h a homogeneous o v e r -burden t h e s t a c k i n g v e l o c i t y w i l l e q u a l t h e p r o p a g a t i o n v e l o c i t y . I f t h e r e f l e c t o r i s t i l t e d , o n l y a c o r r e c t i o n w i t h c o s S i s needed: c =

Vs t a c k c o s® ' ^n more complex s i t u a t i o n s more c o m p l i c a t e d c o r r e c t i o n s

have t o be done ( H u b r a l and K r e y , 1980). F i n a l l y , i n v e r y complex s i t u a t i o n s t h e s t a c k i n g v e l o c i t y does n o t e x i s t anymore.

4.3 THE COMMON DEPTH POINT STACKING PROCEDURE

The common d e p t h p o i n t s t a c k i n g t e c h n i q u e ( C D P - s t a c k ) i s based on the r e m o v a l o f wave p r o p a g a t i o n e f f e c t s between the r e g i s t r a t i o n p l a n e and any d e p t h p o i n t o f i n t e r e s t . E s s e n t i a l l y , the w a v e f i e l d s p e r t a i n i n g t o s o u r c e s and r e c e i v e r s a r e l o w e r e d i n t o the s u b s u r f a c e (downward c o n t i n u a t i o n ) .

F i g u r e 4.6: Downward c o n t i n u a t i o n i s an e s s e n t i a l p r o c e s s i n the c r e a t i o n o f a common d e p t h p o i n t g a t h e r .

Then the s t r u c t u r e o f the o v e r b u r d e n w i l l n o t any more i n f l u e n c e ( a p a r t f r o m m u l t i p l e s ) the d a t a a t the c h o s e n d e p t h l e v e l . The problem o f c o r r e c t i n g the n m o_c u r v e , as o c c u r r i n g i n t h e CMP-stack, has been changed

i n t o the p r o b l e m o f p e r f o r m i n g a downward c o n t i n u a t i o n o f the s o u r c e wave f i e l d and the r e f l e c t e d wave f i e l d s . The CMP g a t h e r i s r e p l a c e d by a CDP g a t h e r . Note t h a t i n t h e downward c o n t i n u a t i o n p r o c e s s n m o - c o r r e c t i o n i s accompanied w i t h a m p l i t u d e c o r r e c t i o n s as w e l l .

A CDP g a t h e r i s composed from a s e t o f m i g r a t e d s h o t r e c o r d s by r e o r d e r i n g o f the d a t a ( f i g . 4.7). I t may be c o n s i d e r e d as a CMP g a t h e r where a l l e v e n t s have a l r e a d y been p r o p e r l y m i g r a t e d . C o n s e q u e n t l y , i n a CDP g a t h e r a l l e v e n t s a r e p o s i t i o n e d a t t h e i r d e p t h l e v e l o r c o r r e s p o n d i n g v e r t i c a l t i m e and t h e r e f o r e t h e y w i l l be a l i g n e d w i t h each o t h e r . They w i l l add

m i g r a t e d s h o t g a t h e r s a s a function of v e r t i c a l t i m e T = 2 / c -1( z ) d z c o m m o n d e p t h point g a t h e r s for lateral p o s i t i o n x shot 2 shot N F i g u r e 4.7: A CDP g a t h e r i s composed f r o m a s e t m i g r a t e d s h o t r e c o r d s . ' i n - p h a s e ' when c o m b i n i n g a l l t r a c e s i n t h e CDP g a t h e r .

F i g u r e 4.8 shows t h e r e s u l t i n g common d e p t h p o i n t g a t h e r from a s e t m i g r a t e d , s y n t h e t i c s h o t r e c o r d s o b t a i n e d from a s i n g l e p o i n t d i f f r a c t o r m o d e l . The CDP g a t h e r t r a c e s have t h e l a t e r a l p o s i t i o n o f t h e p o i n t d i f f r a c t o r l o c a t i o n .

F i g u r e 4.8: A CDP g a t h e r f o r a s i n g l e p o i n t d i f f r a c t o r model.

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

R e t u r n i n g t o t h e f u l l p r o c e s s o f m i g r a t i o n , we remark t h a t CDP d a t a a r e g a t h e r e d and s t a c k e d f o r a l l d e p t h p o i n t s o f i n t e r e s t . N o t e , t h a t by a p p l y i n g downward c o n t i n u a t i o n i n p r i n c i p l e a l l a m p l i t u d e s and phases a r e t r e a t e d c o r r e c t l y . Hence, t h i s p r o c e d u r e r e p r e s e n t s a ' t r u e ' common d e p t h p o i n t s t a c k . S i m i l a r t o CMP d a t a , t h e i n s p e c t i o n o f CDP g a t h e r s i s a q u i t e i n t e r e s t i n g manner o f d e t e c t i n g d e v i a t i o n s from t h e c o r r e c t m i g r a t i o n 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 h o u l d be used as a new t o o l f o r v e l o c i t y e s t i m a t i o n . F i g u r e 4.9 shows a few CDP g a t h e r s r e s u l t i n g f r o m a s e t m i g r a t e d s h o t r e c o r d s o u t o f a marine l i n e ( t h i s example w i l l be d i s c u s s e d l a t e r on more e x t e n s i v e l y ) . Note t h e n o n - h o r i z o n t a l e v e n t s i n t h e deeper o f t h e d a t a . O b v i o u s l y an i n c o r r e c t m i g r a t i o n v e l o c i t y p r o f i l e has been u s e d , w h i c h i n f a c t a p p e a r e d t o be t o o low f o r t i m e s g r e a t e r t h a n 2 s e c o n d s . A l s o d e p t h l e v e l s c a n be i n d i c a t e d where t h e m i g r a t i o n v e l o c i t y was i n c i d e n t a l l y t o o h i g h .

F i g u r e 4.9: CDP g a t h e r s from a marine l i n e showing m i g r a t i o n v e l o c i t y e r r o r s .

N o t e t h a t an i n c o r r e c t l y d e f i n e d m i g r a t i o n v e l o c i t y p r o f i l e might be a d j u s t e d by a p p l y i n g r e s i d u a l n m o - c o r r e c t i o n on t h e CDP g a t h e r s .

F i n a l l y , we c o n c l u d e from t h e p r e v i o u s :

- The r a y geometry o f CMP d a t a d i f f e r s from CDP d a t a f o r l a t e r a l l y v a r i a n t m e d i a .

- The nmo-curve o f b o t h d a t a s e t s i n complex g e o l o g i c a l s i t u a t i o n s i s n o n - h y p e r b o l i c (more pronounced a t h i g h e r o f f s e t s ) . - Downward c o n t i n u a t i o n r e p l a c e s n m o - c o r r e c t i o n ( p o s s i b l y n o n - h y p e r b o l i c ) . M o r e o v e r , a m p l i t u d e s a r e t r e a t e d c o r r e c t l y . - The f u l l p r o c e s s o f m i g r a t i o n as d e s c r i b e d i n c h a p t e r 3 r e p r e s e n t s a t r u e common d e p t h p o i n t s t a c k . - I n s p e c t i o n o f CDP g a t h e r s w i l l be a v a l u a b l e t o o l i n t h e d e t e c t i o n o f m i g r a t i o n v e l o c i t y e r r o s . - R e s i d u a l n m o - c o r r e c t i o n s c a n be c o n s i d e r e d i n c a s e o f a p p a r e n t m i g r a t i o n v e l o c i t y e r r o r s .