UBC Theses and Dissertations

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UBC Theses and Dissertations

Inversion of reflection seismograms Levy, Shlomo 1985

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INVERSION OF REFLECTION SEISMOGRAMS by SHLOMO LEVY M.Sc. The U n i v e r s i t y Of B r i t i s h Columbia B.Sc. U n i v e r s i t y of C a l i f o r n i a a t Los A n g e l e s A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Department of G e o p h y s i c s And Astronomy We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e a u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA MAY 1985 © SHLOMO LEVY, 1985 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of (yEdfdy^rcS A^X) Asrrt***Qs« The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 ( i i ) A b s t r a c t A method f o r the e s t i m a t i o n of impedance or p s e u d o - v e l o c i t y s e c t i o n s from the i n f o r m a t i o n c o n t a i n e d i n CMP s t a c k e d s e c t i o n s , the c o r r e s p o n d i n g s t a c k i n g v e l o c i t i e s and s o n i c and d e n s i t y l o g s (when a v a i l a b l e ) i s p r e s e n t e d . The method r e l i e s on a l i n e a r programming approach f o r the r e c o n s t r u c t i o n of f u l l - b a n d r e f l e c t i v i t i e s , and u t i l i z e s l i n e a r i z e d r e l a t i o n s between the m u l t i p l e f r e e r e f l e c t i v i t y f u n c t i o n s and a v e r a g e or p o i n t - w i s e impedance or v e l o c i t y v a l u e s . The r e c o n s t r u c t i o n p r o c e d u r e r e q u i r e s the s o l u t i o n of an u n d e r d e t e r m i n e d s e t of e q u a t i o n s and hence a minimum s t r u c t u r e c o n d i t i o n i s imposed on the d e s i r e d s o l u t i o n . T h i s c o n d i t i o n g u a r a n t i e s the u n i q u e n e s s of the o b t a i n e d s o l u t i o n i n the sense t h a t i t i s the s o l u t i o n t h a t f e a t u r e s the l e a s t amount of impedance v a r i a t i o n s as a f u n c t i o n of t r a v e l - t i m e (or d e p t h ) . S i n c e the p r e s e n t e d i n v e r s i o n y i e l d s minimum s t r u c t u r e s o l u t i o n s , i t i s argued t h a t f e a t u r e s which appear on the o b t a i n e d r e s u l t a r e s t r i c t l y demanded by the d a t a and a r e not a r t i f a c t s of the i n v e r s i o n scheme. A number of p h y s i c a l a ssumptions a r e r e q u i r e d by the p r e s e n t e d i n v e r s i o n . These are summarized below i n p o i n t form: ( 1 ) The e a r t h r e f l e c t i v i t y f u n c t i o n i s non-white and can be r e a s o n a b l y r e p r e s e n t e d by a s p a r s e s p i k e t r a i n . ( i i i ) (2) The o b s e r v e d CMP s t a c k e d s e c t i o n i s a r e a s o n a b l e r e p r e s e n t a t i o n of the m u l t i p l e - f r e e n o r m a l - r a y s e c t i o n w i t h r e a s o n a b l y c o r r e c t r e l a t i v e a m p l i t u d e r e l a t i o n s . (3) The r e s i d u a l w a v e l e t on the s t a c k e d s e c t i o n i s t o a good a p p r o x i m a t i o n a zero-phase w a v e l e t w i t h a r e l a t i v e l y f l a t s p ectrum. (4) The 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 s can be i n v e r t e d t o y i e l d an a c c e p t a b l e r e p r e s e n t a t i o n of the a v e r a g e s of the t r u e e a r t h v e l o c i t y model. S i n c e i n a r e a l i s t i c e nvironment some of the above a s s u m p t i o n s may be v i o l a t e d , a l l the c o r r e s p o n d i n g r e l a t i o n s i n the p r e s e n t e d i n v e r s i o n scheme i n c l u d e a p p r o p r i a t e u n c e r t a i n t y terms. That i s , a l l the i n f o r m a t i o n components c o n s i d e r e d i n the i n v e r s i o n a r e s a t i s f i e d o n l y t o w i t h i n some p r e s p e c i f i e d e r r o r bounds. A number of p o s s i b i l i t i e s f o r s p e e d i n g up the i n v e r s i o n scheme a r e d e s c r i b e d . I t i s shown t h a t u t i l i z i n g t h e e x p e c t e d t r a c e - t o - t r a c e c o h e r e n c y of s e i s m i c r e f l e c t i o n d a t a y i e l d s c o n s i d e r a b l e r e d u c t i o n i n c o m p u t a t i o n a l e f f o r t s . F i n a l l y , a number of s t e p s r e q u i r e d f o r a s u c c e s s f u l c o m p l e t i o n of the i n v e r s i o n a r e d e s c r i b e d . In p a r t i c u l a r , the problems of p r e i n v e r s i o n d a t a s c a l i n g and the c o r r e c t i o n of the r e s i d u a l w a v e l e t ' s phase a r e d i s c u s s e d i n some d e t a i l . ( i v ) TABLE OF CONTENTS A b s t r a c t ( i i ) L i s t of T a b l e s (x) L i s t of F i g u r e s ( x i ) Acknowledgments ( x v i ) C h a p t e r I : I n t r o d u c t i o n 1 1 . 1 Stage I : E a r l y Developments 2 1 . 2 Stage I I : E x t e r n a l C o n s t r a i n t s i n the R e c o n s t r u c t i o n of F u l l - B a n d R e f l e c t i v i t i e s 4 1 . 3 Stage I I I : P r a c t i c a l I m p l e m e n t a t i o n of the I n v e r s i o n . 6 Chapter I I : R e c o n s t r u c t i o n of a S p a rse S p i k e T r a i n From a P o r t i o n of i t s Spectrum and A p p l i c a t i o n t o H i g h R e s o l u t i o n D e c o n v o l u t i o n 8 2 . 1 I n t r o d u c t i o n 8 2 . 2 R e c o n s t r u c t i o n of a Sparse S p i k e T r a i n from a P o r t i o n of i t s Spectrum 10 (v) 2.3 Examples w i t h N o i s e - F r e e Data 13 2.3.1 Example 1 13 2.3.2 Example 2 14 2.4 S e l e c t i o n of Data i n the Frequency Domain i n the P r e s e n c e of N o i s e 16 2.4 .1 E m p i r i c a l C u t o f f 19 2.4.2 S t a t i s t i c a l C u t o f f 19 2.5 F o r m u l a t i o n of the I n v e r s e Problem i n the P r e s e n c e of N o i s e . 22 2.5.1 I n e q u a l i t y C o n s t r a i n t s 22 2.5.2 E q u a l i t y C o n s t r a i n t s 24 2.6 Examples w i t h N o i s y Data 26 2.6.1 Example 3 . 27 2.6.2 Example 4 29 2.7 P r a c t i c a l Recommendations 31 2.8 Comparison w i t h L e a s t - S q u a r e s T e c h n i q u e s 34 2.8.1 Example 5 35 2.8.2 Example 6 36 2.9 C o n c l u s i o n s 39 Chapter I I I : A c o u s t i c Impedance I n v e r s i o n and W e l l - L o g C o n s t r a i n t s 41 3.1 I n t r o d u c t i o n 41 3.2 E x t e r n a l C o n s t r a i n t s i n Impedance I n v e r s i o n 43 3.3 P o i n t V e l o c i t y C o n s t r a i n t s - W e l l Log I n f o r m a t i o n .... 44 3.4 D i s c u s s i o n 46 ( v i ) C h apter IV: RMS V e l o c i t i e s and Recovery of the A c o u s t i c Impedance 48 4.1 I n t r o d u c t i o n 48 4.2 I n v e r s i o n of RMS V e l o c i t i e s : Model C o n s t r u c t i o n 54 4.3 L i n e a r A p p r a i s a l 67 4.4 D e r i v a t i o n of the Impedance C o n s t r a i n t s 73 4.5 Summary 88 Chapter V: M u l t i - T r a c e Simplex A l g o r i t h m i n S e i s m i c Data A n a l y s i s 92 5.1 I n t r o d u c t i o n 92 5.2 Simple x A l g o r i t h m (background) 93 5.3 B a s i c F o r m u l a t i o n of the A l g o r i t h m 98 5.4 S o l v i n g M u l t i p l e R e l a t e d Problems 1 0 1 5.5 A p p l i c a t i o n t o S e i s m i c Data 103 5.6 Example .....105 5 . 7 Summary 115 Chapter V I : P r e - I n v e r s i o n S c a l i n g of R e f l e c t i o n Seismograms 116 6.1 I n t r o d u c t i o n 116 6.2 Method 1 1 8 6.2.1 G e n e r a l 118 6.2.2 The C o n s t r u c t i o n of S p i k y R e f l e c t i v i t y F u n c t i o n -R e l a t i v e I n f o r m a t i o n 119 ( v i i ) 6.2.3 W e l l - L o g s or S t a c k i n g V e l o c i t i e s - A b s o l u t e I n f o r m a t i o n 120 6.2.4 Comparison of the A b s o l u t e and the R e l a t i v e I n f o r m a t i o n 121 6.2.5 C o n s i d e r a t i o n s P e r t a i n i n g t o the Use of S t a c k i n g V e l o c i t i e s f o r S c a l i n g 123 6.2.6 C o n s i d e r a t i o n s P e r t a i n i n g t o L o c a l i z e d S p i k i n g D e c o n v o l u t i o n F a i l u r e s .124 6.3 S y n t h e t i c Examples .....125 6.3.1 L a y e r e d E a r t h Model w i t h W e l l D e f i n e d L a y e r B o u n d a r i e s 125 6.3.2 L a y e r e d E a r t h Model w i t h S l o w l y V a r y i n g Impedance Zones 131 6.4 R e a l Data Examples ...135 6.4.1 S c a l i n g Data U s i n g W e l l - L o g I n f o r m a t i o n 1 3 5 6.4.2 S c a l i n g Data U s i n g S t a c k i n g V e l o c i t y I n f o r m a t i o n 143 6.5 C o n c l u s i o n 147 Chapter V I I : P r e - I n v e r s i o n C o r r e c t i o n of Wavelet R e s i d u a l Phase 150 7.1 I n t r o d u c t i o n 150 7.2 Method and E x p l a n a t o r y Examples .....155 7.3 R e a l Data Examples 172 7.4 C o n c l u d i n g Comments 179 ( v i i i ) C h a pter V I I I : Summary 182 R e f e r e n c e s 201 Appendix 2-A: Mean N o i s e Power and the V a r i a n c e of a Random N o i s e P r o c e s s 206 Appendix 2-B: The Choice of S t a t i s t i c a l C u t o f f i n LP D e c o n v o l u t i o n 209 Appendix 2-C: N o i s e C o n s i d e r a t i o n i n the E q u a l i t y C o n s t r a i n t s F o r m u l a t i o n of LP D e c o n v o l u t i o n 211 Appendix 6-A: AR S p i k i n g D e c o n v o l u t i o n of B a n d - L i m i t e d Data .213 Appendix 7-A: The D e c o n v o l u t i o n of P h a s e - S h i f t e d W a v e l e t s ...217 7-A.1 I n t r o d u c t i o n 217 7-A.2 Theory 220 7-A.3 C o n c l u s i o n s 246 Appendix 7-B: A p p l i c a t i o n s of A n a l y t i c Common S i g n a l A n a l y s i s i n E x p l o r a t i o n G e o p h y s i c s 248 7-B.1 I n t r o d u c t i o n 248 7-B.2 M a t h e m a t i c a l Background 249 7-B.3 M o d e l l i n g D i s p e r s i o n by a C o n s t a n t P h a s e - S h i f t 255 ( i x ) 7-B.4 D e n s i t y and V e l o c i t y E s t i m a t i o n from S u p e r - C r i t i c a l R e f l e c t i o n s 260 7-B.5 C o n c l u s i o n s 265 (x) LIST OF TABLES 4 . 1 Seventeen a c c u r a t e and i n a c c u r a t e RMS v e l o c i t i e s 6 2 5 . 1 CPU run t i m e s f o r the m o d i f i e d S i m p l e x a l g o r i t h m . . ' . . . 1 1 4 6 . 1 S c a l e f a c t o r e v a l u a t i o n f o r a s i m p l e model 127 6 . 2 S c a l e f a c t o r e v a l u a t i o n f o r impedance model w i t h ramp-l i k e components 1 3 4 7 . 1 The varimax s t a n d o u t f o r a b a n d l i m i t e d s i n e f u n c t i o n 1 6 0 7 . 2 Summary of the a p p l i c a t i o n of the a u t o m a t i c phase c o r r e c t i o n a l g o r i t h m t o w e l l - l o g s y n t h e t i c r e f l e c t i v i t i e s 1 6 5 7 . 3 Summary of the a p p l i c a t i o n of the a u t o m a t i c phase c o r r e c t i o n a l g o r i t h m t o w e l l - l o g s y n t h e t i c r e f l e c t i v i t i e s a f t e r AR s p e c t r a l e x t e n s i o n 1 6 9 7 - A . l Phase and a m p l i t u d e e s t i m a t e s as a f u n c t i o n of r e f l e c t o r s time s e p a r a t i o n 2 3 9 7 - B . 1 O f f s e t , a n g l e of i n c i d e n c e and phase a n g l e f o r r e f l e c t i o n s a s s o c i a t e d w i t h a f i r s t - o r d e r m u l t i p l e . . . 2 6 4 7-B.2 D e n s i t y and sound speed r a t i o s o b t a i n e d v i a the i n v e r s i o n of phase s h i f t i n f o r m a t i o n . 2 6 6 ( x i ) LIST OF FIGURES 2-1 Recovery of a s p a r s e s p i k e s e r i e s - s i m p l e w a v e l e t . 15 2-2 Recovery of a s p a r s e s p i k e s e r i e s - complex w a v e l e t 17 2-3 Recovery of a s p a r s e s p i k e s e r i e s from n o i s y d a t a s i m p l e w a v e l e t 30 2-4 Recovery of a s p a r s e s p i k e s e r i e s from n o i s y d a t a complex w a v e l e t 32 2-5 Recovery of a r e f l e c t i v i t y s e r i e s u s i n g f r e q u e n c y and t i m e domain l e a s t squares d e c o n v o l u t i o n t e c h n i q u e s s i m p l e w a v e l e t 37 2-6 Recovery of a r e f l e c t i v i t y s e r i e s u s i n g f r e q u e n c y and time domain l e a s t squares d e c o n v o l u t i o n t e c h n i q u e s complex w a v e l e t . . . 38 4-1 V e l o c i t y p r o f i l e and the c o r r e s p o n d i n g RMS v e l o c i t y and r e f l e c t i v i t y f u n c t i o n s 58 4-2 C o n s t r u c t e d i n t e r v a l v e l o c i t y models a c c u r a t e d a t a 61 4-3 C o n s t r u c t e d i n t e r v a l v e l o c i t y models - i n a c c u r a t e d a t a 65 4-4 C o n s t r u c t e d i n t e r v a l v e l o c i t y models, d a t a i n c l u d e b o t h RMS and p o i n t v e l o c i t i e s 68 4-5 T r a d e - o f f diagram f o r RMS v e l o c i t y i n v e r s i o n 71 ( x i i ) 4-6 Sample t r a d e - o f f c u r v e s f o r t o=0.55 seconds and t 0=1.25 seconds 72 4-7 Impedance i n v e r s i o n w i t h RMS v e l o c i t y c o n s t r a i n t s a c c u r a t e d a t a 80 4-8 Impedance i n v e r s i o n w i t h RMS v e l o c i t y c o n s t r a i n t s i n a c c u r a t e d a t a . 84 4- 9 Impedance i n v e r s i o n w i t h RMS and p o i n t v e l o c i t y c o n s t r a i n t s - i n a c c u r a t e d a t a 87 5- 1 G e o m e t r i c a l i n t e r p r e t a t i o n of a l i n e a r programming problem 94 5-2 G e o m e t r i c a l and a l g o r i t h m i c c l o s e n e s s i n the c o n t e x t of l i n e a r programming 97 5-3 S t a c k e d s e c t i o n f e a t u r i n g a r e a s o n a b l e t r a c e t o t r a c e c o r r e l a t i o n 106 5-4 The d a t a of F i g u r e 5-3 a f t e r s p e c t r a l w h i t e n i n g , phase c o r r e c t i o n and t h r e e - t r a c e p r i n c p a l component SNR enhacement 107 5-5 The d a t a of F i g u r e 5-4 a f t e r l i n e a r programming d e c o n v o l u t i o n w i t h the m u l t i t r a c e S i m p l e x mode .......109 5-6 The d a t a of F i g u r e 5-4 a f t e r l i n e a r prograrnming d e c o n v o l u t i o n w i t h o b j e c t i v e f u n c t i o n w e i g h t s and p o l a r i t y c o n s t r a i n t s .......110 5-7 The pseudo-impedance s e c t i o n o b t a i n e d v i a the i n t e g r a t i o n of the d a t a i n F i g u r e 5-5 112 5-8 Pseudo-impedance s e c t i o n o v e r l a i d on the f u l l band LP r e f l e c t i v i t y 113 ( x i i i ) 6 - 1 Data f o r a s i m p l e s c a l i n g example 1 2 8 6 - 2 True pseudo-impedance f u n c t i o n and the c o r r e s p o n d i n g pseudo-impedances c a l c u l a t e d from the LP s o l u t i o n and i n v e r t e d i n t e r v a l v e l o c i t i e s . . . 1 2 9 6 - 3 The r e c o n s t r u c t e d a c o u s t i c impedance f u n c t i o n o b t a i n e d v i a c o n s t r a i n e d i n v e r s i o n of s c a l e d d a t a 1 3 0 6 - 4 Input d a t a w i t h a r a m p - l i k e component. Second s y n t h e t i c example f o r s c a l i n g 1 3 2 6 - 5 True pseudo-impedance f u n c t i o n and the c o r r e s p o n d i n g pseudo-impedances c a l c u l a t e d from the LP s o l u t i o n and i n v e r t e d i n t e r v a l v e l o c i t i e s (second example) 1 3 6 6 - 6 The v e l o c i t y f u n c t i o n o b t a i n e d from the c o n s t r a i n e d i n v e r s i o n of the second s e t of s y n t h e t i c d a t a 1 3 7 6 - 7 Input seismograms a f t e r p o s t s t a c k s p e c t r a l w h i t e n i n g and phase c o r r e c t i o n 1 3 8 6 - 8 S o n i c l o g from a w e l l s i t e a t the v i c i n i t y of t r a c e 7 0 1 3 9 6 - 9 The d a t a of F i g u r e 6 - 7 a f t e r a p p l i c a t i o n of l i n e a r programming d e c o n v o l u t i o n 141 6 - 1 0 The pseudo-impedance s e c t i o n o b t a i n e d v i a the i n t e g r a t i o n of the d a t a i n F i g u r e 6 - 9 1 4 2 6 - 1 1 The r e s u l t of a c o n s t r a i n e d i n v e r s i o n 1 4 4 6 - 1 2 Input s t a c k e d s e c t i o n 1 4 5 6 - 1 3 I n t e r v a l v e l o c i t y s e c t i o n o b t a i n e d by the i n v e r s i o n of a s e t of s t a c k i n g v e l o c i t y p r o f i l e s 1 4 6 (xiv) 6- 14 The r e s u l t of a c o n s t r a i n e d i n v e r s i o n ( w i t h s t a c k i n g v e l o c i t i e s ) 1 4 8 7- 1 Phase r o t a t i o n s of a R l a u d e r w a v e l e t 157 7-2 Phase c o r r e c t i o n of mixed d e l a y w a v e l e t s 162 7-3 Phase c o r r e c t i o n and d i p o l e r e f l e c t i v i t y 164 7-4 E i g h t v e l o c i t y l o g s and the c o r r e s p o n d i n g r e f l e c t i v i t y f u n c t i o n s 1 66 7-5 Example of an a l t e r n a t i v e d e c o n v o l u t i o n a p p r o a c h 173 7-6a CMP seismograms r e p r e s e n t i n g t h e o u t p u t of a s t a n d a r d p r o c e s s i n g sequence 176 7-6b The d a t a of F i g . 7-6a a f t e r z e r o - p h a s e d e c o n v o l u t i o n .177 7-6c Phase c o r r e c t e d d a t a 178 7-7 Comparison of u n c o n s t r a i n e d impedance i n v e r s i o n b e f o r e and a f t e r the a p p l i c a t i o n of phase c o r r e c t i o n 180 7-8 The r e s u l t of u n c o n s t r a i n e d i n v e r s i o n a p p l i e d t o phase c o r r e c t e d d a t a 1 8 1 7-A.1 The b a s i c q u a n t i t i e s i n v o l v e d i n the d e c o n v o l u t i o n of phase s h i f t e d w a v e l e t s 229 7-A.2 Simple example of the d e c o n v o l u t i o n of phase s h i f t e d d a t a 24 1 7-A.3 C o n v e n t i o n a l d e c o n v o l u t i o n of the d a t a shown i n F i g u r e 7-A.2 244 7-A.4 Wavelet s h i f t e d by v a r y i n g amounts of p h a s e - s h i f t a n g l e s 245 (xv) 7-B.1 D i s p e r s e d R i c k e r w a v e l e t s and the c o r r e s p o n d i n g p r i n c i p a l components 257 7-B.2 D i s p e r s e d and a t t e n u a t e d R i c k e r w a v e l e t s and the c o r r e s p o n d i n g p r i n c i p a l components 259 7-B.3a Reduced time p l o t of the d a t a f o r phase i n v e r s i o n example 262 7-B.3b,c Phase e s t i m a t i o n w i t h the K-L t r a n s f o r m 263 ( x v i ) ACKNOWLEDGEMENTS I am proud t o be l o n g t o the group w h i c h a t a r e l a t i v e l y low c o s t d e v e l o p e d a working s o l u t i o n t o an i m p o r t a n t i n d u s t r i a l p roblem. S i n c e t h e r e weren't many road s i g n s on the r o u t e t h a t we have chosen, the m a j o r i t y of the components p r e s e n t e d here are o r i g i n a l and have not been p r e v i o u s l y t r e a t e d i n the p u b l i s h e d l i t e r a t u r e . My deepest t h a n k s a r e g i v e n t o t h o s e r e s e a r c h e r s (Dr. D.W. Oldenburg, Dr. P.K. F u l l a g a r , K. S t i n s o n , T. Scheuer, Dr. T.J. U l r y c h , C. W a l k e r , B.T. P r a g e r and I . F . Jones) who took the whole or s e l e c t e d p o r t i o n s of t h i s r e s e a r c h road and c o n t r i b u t e d g r e a t l y t o the s o l u t i o n of the pr o b l e m p r e s e n t e d i n t h i s work. I am a l s o g r a t e f u l t o my f r i e n d s J . J . C a b r e r a and S. C l e g g f o r t h e i r h e l p i n the p r e p a r a t i o n of t h i s t h e s i s . 1 CHAPTER I : INTRODUCTION The f o l l o w i n g work i s d e d i c a t e d t o the probl e m of seismogram i n v e r s i o n ( t h a t i s , the c o n s t r u c t i o n of impedance or v e l o c i t y s e c t i o n s from the i n f o r m a t i o n c o n t a i n e d i n r e f l e c t i o n s eismograms). The r e s u l t s of t h i s i n v e r s i o n s h o u l d enhance the i n t e r p r e t a b i l i t y of the d a t a and s u p p l y a d d i t i o n a l i n f o r m a t i o n which may be s u b s e q u e n t l y used f o r the i n f e r e n c e of p e t r o -p h y s i c a l p a r a m e t e r s . The problem, which i s complex and e x h i b i t s a h i g h degree of non-uniqueness, has been t a c k l e d i n t h r e e s t a g e s each of which r e f l e c t s the l e v e l of n a i v e t y p r e v a i l i n g a t the c o r r e s p o n d i n g time of development. A f t e r a l o n g p e r i o d of s t r u g g l e a com p l e t e s e t of a l g o r i t h m s which p r e s e n t s a w o r k i n g s o l u t i o n t o the impedance i n v e r s i o n p r o b l e m has been d e v e l o p e d . An o u t l i n e of the s o l u t i o n p r e s e n t e d i n t h i s work i s g i v e n below. 2 1.1 STAGE I : EARLY DEVELOPMENTS At the e a r l y s t a g e s of t h i s work, the problem of the a c o u s t i c impedance i n v e r s i o n of r e f l e c t i o n seismograms was not t r e a t e d d i r e c t l y . R a t h e r , I have c o n c e n t r a t e d my e f f o r t s on the c o n s t r u c t i o n of f u l l - b a n d r e f l e c t i v i t y f u n c t i o n s from the a v a i l a b l e b a n d - l i m i t e d i n f o r m a t i o n . I s t a r t e d by assuming a model i n which the r e f l e c t i o n seismograms c o n s i s t of the c o n v o l u t i o n of a b a n d - l i m i t e d zero-phase ( f l a t - s p e c t r u m ) w a v e l e t w i t h a s p a r s e s p i k e e a r t h response f u n c t i o n . To the p r o d u c t of t h i s c o n v o l u t i o n we have a l l o w e d the a d d i t i o n of a c e r t a i n l e v e l of random n o i s e . Then the problem under c o n s i d e r a t i o n was the r e c o v e r a b i l i t y of the f u l l - b a n d r e f l e c t i v i t y s e r i e s from the a v a i l a b l e b a n d - l i m i t e d s i g n a l . Chapter II shows t h a t v i a the use of t h e L i n e a r Programming (LP) scheme, and the m i n i m i z a t i o n of the L1 norm of the d e s i r e d r e f l e c t i v i t y model, a good r e p r e s e n t a t i o n of a s p a r s e s p i k e s e r i e s can be r e c o v e r e d from a r e l a t i v e l y s m a l l p o r t i o n of i t s spectrum. A l a t t e r work by Scheuer (1981) and Oldenburg e t a l . (1983) has p o i n t e d out t h a t f o r data i n which the m u l t i p l e s were removed from t h e measured seismograms p r i o r t o LP r e c o n s t r u c t i o n of the f u l l - b a n d r e f l e c t i v i t y f u n c t i o n , d i r e c t i n t e g r a t i o n of the r e c o v e r e d r e f l e c t i v i t y w i l l y i e l d a r e a s o n a b l e 3 r e p r e s e n t a t i o n of the t r u e e a r t h r e l a t i v e impedance f u n c t i o n . A l t h o u g h s t a c k e d s e c t i o n s ( t h e i n p u t t o the f o l l o w i n g i n v e r s i o n scheme) do not g e n e r a l l y r e p r e s e n t the i d e a l p r i m a r y -o n l y s e c t i o n r e q u i r e d h e r e , p r o c e s s i n g c h a i n s w h i c h c o n t a i n o p e r a t i o n s l i k e s t a c k w i t h p r i m a r y s t a c k i n g v e l o c i t i e s , gap-d e c o n v o l u t i o n and frequency-wave number f i l t e r s w i l l l a r g e l y s u p p r e s s m u l t i p l e e f f e c t s . Energy c o n t a i n e d i n the r e m a i n i n g m u l t i p l e s can then be t r e a t e d as n o i s e and be f u r t h e r s u p p r e s s e d by the i n t e r n a l mechanism of the a l g o r i t h m as was d e s c r i b e d i n Oldenburg e t a l . 1983. The r e a d e r who i s s t i l l s k e p t i c a l as t o the a p p l i c a b i l i t y of the p r i m a r i e s - o n l y model t o r e a l s e i s m i c d a t a i s reminded t h a t the g e n e r a t i o n of a s t a c k e d s e c t i o n i n which p r i m a r y e v e n t s and a l l t h e i r c o r r e s p o n d i n g m u l t i p l e s are p r e s e n t i s e q u a l l y as h a r d as the g e n e r a t i o n of a m u l t i p l e - f r e e s e c t i o n . C o n s e q u e n t l y , the s t a r t i n g p o i n t of the i n v e r s i o n p r o c e d u r e t o be d e s c r i b e d i n t h i s work i s no worse than t h a t u n d e r l y i n g wave-equation approaches t o the impedance i n v e r s i o n p roblem. Scheuer (1981), and Oldenburg et a l . (1983) have a l s o p r e s e n t e d a d i f f e r e n t approach t o the r e c o n s t r u c t i o n of the f u l l - b a n d r e f l e c t i v i t y f u n c t i o n . T h i s approach uses the Auto-R e g r e s s i v e model and e x t r a p o l a t e s the a v a i l a b l e f r e q u e n c y i n f o r m a t i o n toward the m i s s i n g low and h i g h f r e q u e n c i e s by means of a p r e d i c t i o n o p e r a t o r . Most i m p o r t a n t l y , they have i n c o r p o r a t e d v e l o c i t y c o n s t r a i n t s i n t o the d e c o n v o l u t i o n problem 4 and t h e r e b y c r e a t e d the b a s i s of what w i l l be l a t e r r e f e r r e d t o as g e o l o g i c a l d e c o n v o l u t i o n . In Chapter I I I I d e s c r i b e the e s s e n t i a l s of t h i s l a t e r a d d i t i o n . I s h o u l d add t h a t my c o n t r i b u t i o n t o t h i s p o r t i o n of the work was r a t h e r m i n o r . 1.2 STAGE I I : EXTERNAL CONSTRAINTS IN THE RECONSTRUCTION OF FULL-BAND REFLECTIVITIES At t he e a r l y s t a g e s of t h i s work, i t was b e l i e v e d t h a t g i v e n the i d e a l d a t a , an i n v e r s i o n p r o c e d u r e i n v o l v i n g the i n t e g r a t i o n and e x p o n e n t i a t i o n of the f u l l - b a n d r e f l e c t i v i t y f u n c t i o n r e c o n s t r u c t e d from the i n f o r m a t i o n c o n t a i n e d i n the seismogram a l o n e s h o u l d y i e l d a r e a s o n a b l e e s t i m a t e of the c o r r e s p o n d i n g impedance f u n c t i o n . A l t h o u g h t h i s o p e r a t i o n seemed to be s u c c e s s f u l i n a f a i r l y l a r g e number of c a s e s ( p a r t i c u l a r l y on s h o r t time windows f e a t u r i n g a r e l a t i v e l y s m a l l number of s h a r p impedance c o n t r a s t s ) , t h e r e a r e a c o n s i d e r a b l e number of g e o l o g i c a l - p h y s i c a l models i n which s a t i s f a c t o r y impedance r e c o n s t u c t i o n r e q u i r e s a d d i t i o n a l i n f o r m a t i o n . For example, the i n c o r p o r a t i o n of i n f o r m a t i o n c o n t a i n e d i n the v e l o c i t y and d e n s i t y l o g s a c q u i r e d a t a w e l l s i t e i n the v i c i n i t y of the l i n e of i n t e r e s t i s l i k e l y t o y i e l d a s i g n i f i c a n t r e d u c t i o n i n the a l l o w e d s o l u t i o n space. Hence, the s o l u t i o n o b t a i n e d from the combined i n f o r m a t i o n s e t i s c o n s i d e r a b l y more r e l i a b l e . 5 S i n c e w e l l - l o g s a r e not always a v a i l a b l e and s i n c e i t i s not a l w a y s p o s s i b l e t o e s t a b l i s h a c l e a r t i e between the w e l l -l o g i n f o r m a t i o n and the o bserved s t a c k e d s e c t i o n , we had t o l o o k f o r a d d i t i o n a l i n f o r m a t i o n s o u r c e s which w i l l r o u t i n e l y be a t our d i s p o s a l . The most n a t u r a l s o u r c e of i n f o r m a t i o n c o n c e r n i n g g r o s s v e l o c i t y s t r u c t u r e i s s t o r e d i n the s e t of v e l o c i t i e s used i n the s t a c k i n g p r o c e s s . In Chapter IV we p r e s e n t a scheme t h r o u g h which a l i n e a r i z e d r e l a t i o n between the s t a c k i n g v e l o c i t i e s and the c o n s t r u c t e d f u l l - b a n d r e f l e c t i v i t i e s i s e s t a b l i s h e d . U s i n g t h i s r e l a t i o n , the d e c o n v o l u t i o n / i m p e d a n c e i n v e r s i o n p r o c e s s i s b e t t e r c o n s t r a i n e d ; t h a t i s , we have f u r t h e r l i m i t e d the s o l u t i o n space and t h e r e b y d e c r e a s e d the non-uniqueness a s s o c i a t e d w i t h the problem. I t i s i m p o r t a n t t o note t h a t u s i n g the s t a c k i n g v e l o c i t i e s and the CMP s t a c k e d seismograms s i m u l t a n e o u s l y , the i n v e r s i o n scheme proposed here i s u t i l i z i n g b o t h the dynamic and the k i n e m a t i c p r o p e r t i e s of the r e c o r d e d d a t a . The a d d i t i o n a l i n f o r m a t i o n s u p p l i e d from the w e l l - l o g s i s used t o f u r t h e r d e c r e a s e the a l l o w e d s o l u t i o n space and t h e r e b y i n c r e a s e the s o l u t i o n ' s r e l i a b i l i t y . 6 1.3 STAGE I I I : PRACTICAL IMPLEMENTATION OF THE INVERSION A c a d e m i c a l l y s p e a k i n g the combined approach p r e s e n t e d i n C h a p t e r s I I , I I I and IV, c o n s t i t u t e s a r e a s o n a b l e s o l u t i o n t o the problem we s e t f o r t h t o s o l v e . However, a number of p r a c t i c a l c o n s i d e r a t i o n s had t o be a d d r e s s e d i n o r d e r t o t r a n s f o r m t h i s work i n t o an e c o n o m i c a l l y v i a b l e and g e o p h y s i c a l l y r e l i a b l e p r o d u c t i o n a l g o r i t h m . The e f f o r t s i n t h i s s t a g e a r e d i r e c t e d toward the f o l l o w i n g a s p e c t s : A. Speeding up the l i n e a r - p r o g r a m m i n g a l g o r i t h m . B. S c a l i n g the observed s t a c k e d s e c t i o n so t h a t the i n c o r p o r a t i o n of the t h r e e i n f o r m a t i o n s e t s i n t o the i n v e r s i o n w i l l y i e l d p h y s i c a l l y m e a n i n g f u l r e s u l t s , and C. C o r r e c t i n g the phase of the r e s i d u a l ( i n t e r p r e t e r ' s ) w a v e l e t so t h a t the i n p u t seismograms w i l l c o n s t i t u t e a r e a s o n a b l e a p p r o x i m a t i o n t o the e a r t h r e f l e c t i v i t y f u n c t i o n c o n v o l v e d w i t h a zero-phase w a v e l e t . C hapter V d e s c r i b e s p r o c e d u r e s f o r i m p r o v i n g the c o m p u t a t i o n a l e f f i c i e n c y of the l i n e a r programming a l g o r i t h m , C hapter VI d i s c u s s e s the p r e - i n v e r s i o n s c a l i n g of s t a c k e d s e c t i o n s and C h a p t e r V I I d e s c r i b e s a p a r t i a l s o l u t i o n t o the problem of phase c o r r e c t i o n . In the a p p e n d i c e s of C h a p t e r V I I , 7 we take the r e a d e r t h r o u g h the concept of the c o n s t a n t phase s h i f t model and p r e s e n t a number of methods f o r the e s t i m a t i o n of the d e s i r e d phase s h i f t a n g l e . However, the e s s e n t i a l c o n t r i b u t i o n of Chapter V I I i s the a u t o m a t i c phase c o r r e c t i o n i n which we use a s i m p l e scheme t o e f f e c t the d e s i r e d c o r r e c t i o n . An i n t e r e s t i n g secondary outcome of the phase c o r r e c t i o n s tudy i s p r e s e n t e d i n Appendix V I I - B . There, we d e s c r i b e the i n v e r s i o n of phase s h i f t s a s s o c i a t e d w i t h p o s t - c r i t i c a l r e f l e c t i o n s t o o b t a i n both v e l o c i t y and d e n s i t y i n f o r m a t i o n . T h i s method was t r i e d f o r the d e t e r m i n a t i o n of the d e n s i t y and v e l o c i t y of the uppermost l a y e r of sediments i n a deep water environment and was found t o y i e l d r e a s o n a b l e r e s u l t s . F i n a l l y t he c o n c l u s i o n of t h i s work c o n t a i n s a d e s c r i p t i o n of the s t e p s r e q u i r e d f o r the c o m p l e t i o n of a c o u s t i c impedance i n v e r s i o n , t o g e t h e r w i t h some recommendations w h i c h may ease some of the i n i t i a l p a i n s t h a t w i l l a f f l i c t the r e a d e r who i s i n t e r e s t e d i n t r y i n g h i s hand i n the s u b j e c t . 8 CHAPTER 2: RECONSTRUCTION OF A SPARSE SPIKE TRAIN FROM A PORTION OF ITS SPECTRUM AND APPLICATION TO HIGH RESOLUTION DECONVOLUTION 2.1 INTRODUCTION In s e i s m o l o g y and o t h e r branches of a p p l i e d s c i e n c e , i t i s o f t e n n e c e s s a r y t o e s t i m a t e a f u n c t i o n from a p o r t i o n of i t s F o u r i e r t r a n s f o r m , e i t h e r because i t s complete t r a n s f o r m cannot be measured or because s e c t i o n s of the t r a n s f o r m a r e u n r e l i a b l e . F o u r i e r t r a n s f o r m a t i o n of an i n c o m p l e t e spectrum i s an u n d e r d e t e r m i n e d l i n e a r i n v e r s e problem. T h e r e f o r e i t a d m i t s an i n f i n i t e number of s o l u t i o n s (Backus and G i l b e r t , 1967, p. 251). I f , however, the form of the unknown f u n c t i o n can be r e s t r i c t e d on p h y s i c a l grounds, i t may be p o s s i b l e t o d e t e r m i n e the most p h y s i c a l l y a c c e p t a b l e s o l u t i o n by m i n i m i z i n g an a p p r o p r i a t e norm. I f a s o l u t i o n c o n s i s t i n g of i s o l a t e d s p i k e s i s r e q u i r e d , m i n i m i z a t i o n of the L,-norm i s most a p p r o p r i a t e . A c c o r d i n g l y , a l i n e a r programming method has been d e v e l o p e d t o r e c o n s t r u c t a " s p a r s e s p i k e t r a i n " from a p o r t i o n of i t s spectrum by m i n i m i z i n g the L,-norm. W i t h t he assumption t h a t the e a r t h r e s p o n s e i s a s e t of d e l t a - f u n c t i o n s of unknown a m p l i t u d e and d e l a y , one immediate f i e l d of a p p l i c a t i o n of the method i s h i g h - r e s o l u t i o n s e i s m i c d e c o n v o l u t i o n . I t i s commonly o b s e r v e d t h a t the s i g n a l - t o - n o i s e (S/N) r a t i o i s v e r y poor i n c e r t a i n p o r t i o n s of a s e i s m i c s pectrum, and i n c l u s i o n of these h i g h l y c o n t a m i n a t e d s p e c t r a l 9 d a t a i n the c a l c u l a t i o n of the r e s p o n s e f u n c t i o n can produce g r o s s e r r o r s . I f the u n r e l i a b l e data a r e r e j e c t e d , the i n v e r s e F o u r i e r t r a n s f o r m a t i o n i s u n d e r d e t e r m i n e d and t h e r e f o r e has no unique s o l u t i o n . Indeed, i f the r e l i a b l e ( a l b e i t n o i s y ) s p e c t r a l d a t a c o n s t i t u t e the t o t a l i t y of the a v a i l a b l e i n f o r m a t i o n , a h i g h l y r e s o l v e d e s t i m a t e of the response f u n c t i o n i s n e c e s s a r i l y a s s o c i a t e d w i t h l a r g e u n c e r t a i n t i e s ( O l d e n b u r g , 1976, 1981). However, by e n f o r c i n g the p h y s i c a l r e q u i r e m e n t t h a t the e a r t h response s h o u l d c o n s i s t of i s o l a t e d s p i k e s , i . e . , by i n t r o d u c i n g a d d i t i o n a l i n f o r m a t i o n , the l i n e a r programming method d e s c r i b e d here i s a b l e t o c o n s t r u c t a response f u n c t i o n e s t i m a t e which i s both a c c e p t a b l y a c c u r a t e and w e l l r e s o l v e d . The e f f i c a c y of the method i s d e m o n s t r a t e d i n s i n g l e - t r a c e examples w i t h 10 p e r c e n t random n o i s e , and the q u a l i t y of our r e s u l t s s u r p a s s e s t h a t a t t a i n a b l e u s i n g c o n v e n t i o n a l l e a s t - s q u a r e s t e c h n i q u e s . L i n e a r programming has been a p p l i e d p r e v i o u s l y t o s e i s m i c d e c o n v o l u t i o n ( C l a e r b o u t and M u i r , 1973; T a y l o r e t a l . 1979), but i t s p o t e n t i a l was not f u l l y e x p l o i t e d i n the time-domain f o r m u l a t i o n a d o pted. The most s i g n i f i c a n t d i f f e r e n c e s i n the p r e s e n t a p p r o a c h r e l a t e t o the t r e a t m e n t of e r r o r s . F i r s t , by f o r m u l a t i n g the problem i n the f r e q u e n c y domain, i t i s p o s s i b l e t o i d e n t i f y (and then t o d i s c a r d ) the s p e c t r a l d a t a which ar e most s e r i o u s l y c o n t a m i n a t e d ; no such "winnowing" of d a t a i s p o s s i b l e i n the time domain. Second, t h e r e i s no a t t e m p t t o m i n i m i z e the m i s f i t e r r o r s ; the response f u n c t i o n w i t h minimum L,-norm i s a l w a y s sought i n our f o r m u l a t i o n because m i n i m i z a t i o n of t h i s norm f a v o r s s o l u t i o n s w i t h as few nonzero v a l u e s as 1 0 p o s s i b l e , i . e . , w i t h i s o l a t e d s p i k e s . The d a t a a r e s a t i s f i e d t o an a c c u r a c y c o m p a t i b l e w i t h the p e r c e i v e d n o i s e l e v e l . What degree of a c c u r a c y i s " c o m p a t i b l e " i s governed by the p r o p a g a t i o n of random n o i s e : t h i s i s c o n s i d e r e d i n some d e t a i l i n a p p e n d i c e s 2-A, 2-B and 2-C. 2.2 RECONSTRUCTION OF A SPARSE SPIKE TRAIN FROM A PORTION OF ITS SPECTRUM The unknown time s e r i e s a ( t ) may be v i s u a l i z e d as a g e n e r a l i z e d D i r a c comb, i . e . , a ( t ) = I a ^ 6 ( t - O , ( 2 . 1 ) where t^, = nAt, f o r some time increment A t . The c o e f f i c i e n t s {a :n = 0, N - 1} are the s p i k e a m p l i t u d e s ; t h u s a ^ = 0 when t h e r e i s no s p i k e a t time t^,. D e n o t i n g the F o u r i e r t r a n s f o r m of the time s e r i e s a ( t ) by A(w), i t f o l l o w s from e q u a t i o n ( 2 . 1 ) t h a t A(w) = 1/N I a ^ e x p t - i w t ^ ) . Hence, Re{A j } = M - l 1/N I a c o s ( w j t ), (2.2) N-1 Im{Aj} = - 1/N I a s i n ( w j t ^ ) , where wj = 2J7i/(NAt) w i t h j = 0, 1, 2,..., N/2. The d e f i n i n g r e l a t i o n s (2.2) f o r the s p e c t r u m of a ( t ) c o n s t i t u t e a s e t of l i n e a r c o n s t r a i n t s on t h e unknowns {a :n = 0, N - 1}. I f the sp e c t r u m were known e x a c t l y f o r a l l (N/2 + 1) d i s t i n c t f r e q u e n c i e s , the time s e r i e s c o u l d be r e c o v e r e d p e r f e c t l y by s t r a i g h t f o r w a r d i n v e r s e F o u r i e r t r a n s f o r m a t i o n . When t h e r e a r e gaps i n the s p e c t r u m , however, the i n v e r s e problem i s u n d e r d e t e r m i n e d and a d m i t s an i n f i n i t e number of s o l u t i o n s , c o r r e s p o n d i n g t o time s e r i e s of g r e a t l y d i f f e r e n t appearance. I f the time s e r i e s i s assumed t o be a sparse s p i k e t r a i n , by no means a r e a l l of t h e s e p o s s i b l e m a t h e m a t i c a l s o l u t i o n s p h y s i c a l l y a c c e p t a b l e , and i t i s d e s i r a b l e t o i n t r o d u c e a c r i t e r i o n which d i s c r i m i n a t e s a g a i n s t those s o l u t i o n s of i n a p p r o p r i a t e form. The c r i t e r i o n most s u i t a b l e i n t h i s c o n n e c t i o n i s the m i n i m i z a t i o n of the L,-norm | j a i l , of the time s e r i e s a ( t ) where a = Z|a (2.3) L i n e a r programming may now be in v o k e d t o f i n d the 12 c o e f f i c i e n t s {a} which s a t i s f y the r e l i a b l e s p e c t r a l data and f o r which • | |a| | , i s minimum. The m i n i m i z a t i o n of | |a| |, w i l l f a v o r s o l u t i o n s which are zero except at a sma l l number of i s o l a t e d times, and i t w i l l t h e r e f o r e tend to produce sparse s p i k e t r a i n s , as d e s i r e d . Furthermore, the number of nonzero v a l u e s i n the l i n e a r programming output never exceeds the number of c o n s t r a i n t s ; i n other words, the number of c o n s t r a i n t s i s an upper l i m i t on the number of s p i k e s i n the s o l u t i o n . The Simplex a l g o r i t h m i s w r i t t e n f o r nonnegative unknowns only, whereas both p o s i t i v e and n e g a t i v e s p i k e s are u s u a l l y e q u a l l y a c c e p t a b l e p h y s i c a l l y . T h i s d i f f i c u l t y i s overcome by e x p r e s s i n g each s p i k e amplitude as the d i f f e r e n c e between two p o s i t i v e q u a n t i t i e s , i . e . , • *\v " ^ > n = 0 /N " 2. (2.4) where b > 0 and c > 0 f o r a l l n = 0, N - 1. The q u a n t i t y a c t u a l l y minimized i s the sum N" 1 . s = Z [b + c ] -YV-C The minimum i s a t t a i n e d when at l e a s t one of b and c i s zero at each va l u e of n; hence la I = lb -c I = | b + c I , n = 0, N - 1 n,1 ' T y 1\. 1 I T V T V I 13 .2.3 EXAMPLES WITH NOISE-FREE DATA The recovery of s p i k e s from an incomplete spectrum may be v i s u a l i z e d as d e c o n v o l u t i o n of a n o i s e - f r e e seismogram when the wavelet i s known. Without l o s s of g e n e r a l i t y , t h e r e f o r e , the e f f i c a c y of the technique f o r ( s p e c i a l i z e d ) i n v e r s e F o u r i e r t r a n s f o r m a t i o n w i l l be e s t a b l i s h e d by means of s e i s m i c examples. The assumption t h a t a seismogram x°(t) i s a c o n v o l u t i o n of a source wavelet w(t) with an e a r t h response f u n c t i o n a ( t ) i s widely used i n a p p l i e d seismology. In the frequency domain, t h i s c o n v o l u t i o n may be expressed as a product, whence A(u) = Z°(w)/W(«), (2,5) where upper case symbols denote F o u r i e r - t r a n s f o r m e d q u a n t i t i e s . 2.3.1 EXAMPLE 1 The input time s e r i e s i n F i g u r e 2-1C r e p r e s e n t s the c o n v o l u t i o n of the R i c k e r wavelet i n F i g u r e 2-1a with the s p i k e t r a i n i n F i g u r e 2 - l b . Our aim here i s to show t h a t the sp i k e s e r i e s can be r e c o n s t r u c t e d from a s m a l l p o r t i o n of i t s spectrum 1 4 without s i g n i f i c a n t l o s s of r e s o l u t i o n . In implementing the technique d e s c r i b e d i n S e c t i o n 2.2, only those deconvolved s p e c t r a l data k(u) c o r r e s p o n d i n g to f r e q u e n c i e s between the arrows i n F i g u r e 2-1d were i n c l u d e d i n the computations which produced the times s e r i e s shown i n F i g u r e 2 - l e . Thus 20 percent of the spectrum s u f f i c e d f o r a p e r f e c t r e c o n s t r u c t i o n of the g e n e r a t i n g s p i k e sequence. 2.3.-2 EXAMPLE 2 The seismogram in F i g u r e 2-2c i s the r e s u l t of c o n v o l u t i o n of the wavelet i n F i g u r e 2-2a with the response f u n c t i o n in F i g u r e 2-2b. As i n example 1, only the deconvolved data a s s o c i a t e d with f r e q u e n c i e s between the arrows i n F i g u r e 2-2d e n t e r e d i n t o the c a l c u l a t i o n s ; both h i g h and low f r e q u e n c i e s have been d i s c a r d e d i n t h i s case. The output i n F i g u r e 2-2e e x h i b i t s a c l o s e resemblance to the o r i g i n a l s p i k e t r a i n i n F i g u r e 2-2b, a l t h o u g h some of the s p i k e s are s p l i t i n two. T h i s phenomenon i s a m a n i f e s t a t i o n of nonuniqueness a r i s i n g from the n e g l e c t of some of the f r e q u e n c i e s . The degree of s p l i t t i n g p r o v i d e s a q u a l i t a t i v e i n d i c a t i o n of the r e s o l v i n g power of the method; a s p l i t s p ike i n d i c a t e s t h a t r e s o l u t i o n i s p l u s or minus one time u n i t at best.. F i g u r e 2-1 (a) A t a p e r e d R i c k e r wavelet, (b) G e n e r a t i n g s p i k e s e r i e s , (c) Normalized i n p u t t r a c e , o b t a i n e d by c o n v o l v i n g the wavelet in (a) with the s e r i e s i n ( b ) . (d) Norma l i z e d amplitude spectrum of the seismogram' in ( c ) . (e) Norma l i z e d response f u n c t i o n r e c o v e r e d u s i n g o n l y the data c o r r e s p o n d i n g to the f r e q u e n c i e s between the arrows i n ( d ) . 1 6 2.4 SELECTION OF DATA IN THE FREQUENCY DOMAIN IN THE PRESENCE OF NOISE Let e ( t ) denote the n o i s e i n an observed seismogram x ( t ) , i . e . , x ( t ) = x°(t) •+ e ( t ) , where x°(t) r e p r e s e n t s the n o i s e - f r e e seismogram as before.. I f the wavelet i s known, the n o i s y seismogram may be deconvolved i n the frequency domain by d i v i d i n g by the wavelet spectrum - W ( C J ) . The r e s u l t i n g e stimate A'(a;) of the spectrum A(CJ) i s given by A'(CJ) = X°(u)/W(u) + E(w)/W(w), (.2.6) where upper case l e t t e r s denote F o u r i e r t ransformed q u a n t i t i e s as b e f o r e . I t may be observed t h a t a c o n v e n t i o n a l i n v e r s e F o u r i e r t r a n s f o r m of A' (CJ) w i l l not produce an a c c e p t a b l e e s t i m a t e of a ( t ) i f at some f r e q u e n c i e s |E/W| i s a p p r e c i a b l e , i n comparison w i t h |X°/W|. To understand the i n f l u e n c e of e r r o r s b e t t e r , i t i s i n s t r u c t i v e to review the g e n e r a l c h a r a c t e r i s t i c s of the component s p e c t r a which shape the spectrum of a seismogram. The amplitude spectrum of a t y p i c a l wavelet f a l l s o f f r a p i d l y at 1 7 F i g u r e 2-2 (a) A complex wavelet, d e v i s e d by Wiggins (1978). (b) N o r m a l i z e d response f u n c t i o n , (c) The n o r m a l i z e d i n p u t t r a c e , the r e s u l t of c o n v o l v i n g the wavelet i n (a) w i t h the s p i k e t r a i n i n ( b ) . (d) N o r m a l i z e d amplitude spectrum of the seismogram i n ( c ) . (e) R e c o n s t r u c t i o n of the response f u n c t i o n a ( t ) computed from s p e c t r a l data a s s o c i a t e d w i t h f r e q u e n c i e s w i t h i n the band d e l i n e a t e d by the arrows i n ( d ) . 18 h i g h frequency, whereas the spectrum of a s p i k e s e r i e s does not decay with frequency. The spectrum of a n o i s e - f r e e seismogram i s the product of the wavelet and response s p e c t r a [per e q u a t i o n (2.5)] and i t t h e r e f o r e decays with frequency l i k e the wavelet spectrum. Consequently, i f a seismogram i s contaminated wi t h pseudo-white n o i s e , the S/N r a t i o i s very poor at h i g h f r e q u e n c i e s because the n o i s e power i s almost independent of frequency. For the same reason, the r e l i a b l e data are a s s o c i a t e d with f r e q u e n c i e s at which the wavelet c a r r i e s s i g n i f i c a n t power. If a l t ) i s est i m a t e d u s i n g only those deconvolved data c o r r e s p o n d i n g to f r e q u e n c i e s from a c e r t a i n band, i t w i l l not be r e c o n s t r u c t e d p e r f e c t l y u n l e s s some independent i n f o r m a t i o n i s i n t r o d u c e d . Some r e s o l v i n g power must be s a c r i f i c e d i f , f o r example, the u n r e l i a b l e data are r e p l a c e d by zero s or i f a p e n a l t y f u n c t i o n i s i n t r o d u c e d to d i s c r i m i n a t e a g a i n s t the n o i s y d a t a . What d i s t i n g u i s h e s our approach i s t h a t we make no a p r i o r i assumptions about the u n r e l i a b l e s p e c t r a l d a t a . I n s t e a d we use only the r e l i a b l e data, i n combination w i t h the a d d i t i o n a l knowledge t h a t the d e s i r e d output c o n s i s t s of i s o l a t e d s p i k e s . Whether or not the s i g n a l power at a c e r t a i n frequency i s s i g n i f i c a n t must be as s e s s e d i n r e l a t i o n to the n o i s e power at th a t frequency. I f the no i s e i s pseudo-white, i t i s r e a s o n a b l e to i n t r o d u c e a c u t o f f and to regard a datum as " r e l i a b l e " i f i t exceeds the c u t o f f , and " u n r e l i a b l e " o t h e r w i s e . The c h o i c e of c u t o f f i s governed by the p e r c e i v e d n o i s e l e v e l ; however, t h e r e 19 are no u n e q u i v o c a l q u a n t i t a t i v e r u l e s , and two d i f f e r e n t procedures have been d e v i s e d . 2.4.1 EMPIRICAL CUTOFF An a p p r o p r i a t e c h o i c e of c u t o f f can u s u a l l y be made a f t e r i n s p e c t i o n of the amplitude spectrum of the seismogram or on the b a s i s of p r i o r knowledge of the l e v e l of n o i s e to be expected. I t i s c o n v e n i e n t to express the c u t o f f K as a f r a c t i o n of max {|X(u)|}; o n l y the deconvolved data at those f r e q u e n c i e s f o r which |X(w)|> K f i g u r e i n the subsequent computations. 2.4.2 STATISTICAL CUTOFF I t i s p o s s i b l e to develop a c u t o f f c r i t e r i o n based on the s t a t i s t i c a l p r o p e r t i e s of random n o i s e . F o l l o w i n g J e n k i n s and Watts ( 1 9 6 9 , p. 231), i t i s assumed that the n o i s e i n the time domain i s a r e a l i z a t i o n of a random process c h a r a c t e r i z e d by the random v a r i a b l e s n = 0, N - 1 ] , each having mean zero and v a r i a n c e a2. I f i t i s p o s s i b l e to i s o l a t e a p o r t i o n of the spectrum of the seismogram (e.g., the h i g h - f r e q u e n c y s e c t i o n ) which i s due to n o i s e a l o n e , then (as shown i n Appendix 2-A) the mean observed power at these f r e q u e n c i e s c o n s t i t u t e s an e s t i m a t e 20 a2 of a2. C o n s i d e r e d as a random v a r i a b l e , a2 i s d i s t r i b u t e d as a c h i - s q u a r e d v a r i a b l e , so c o n f i d e n c e l i m i t s can be a s s i g n e d u s i n g , f o r example, F i g u r e 2-3-10 of J e n k i n s and Watts (1969). A l t h o u g h t h e n o i s e power i s e s t i m a t e d from the spectrum of the o b s e r v e d seismogram X(u) i n the manner j u s t d e s c r i b e d , i t i s the d e c o n v o l v e d spectrum X/W or A' which i s used i n the s e l e c t i o n of f r e q u e n c i e s because the r e l i a b i l i t y of the r e a l and i m a g i n a r y p a r t s of A', not X, governs the a c c u r a c y of the d a t a e q u a t i o n s ( 2 . 6 ) . The r e a l and i m a g i n a r y p a r t s of A' a r e s c r u t i n i z e d i n d e p e n d e n t l y , s i n c e t h e r e i s no guarantee t h a t b o t h a r e r e l i a b l e a t a p a r t i c u l a r f r e q u e n c y , i . e . ]Re{X/W}| may be l a r g e when |lm{X/W}| .is s m a l l , or v i c e v e r s a . The same c u t o f f i s a p p l i e d t o b o t h r e a l and i m a g i n a r y p a r t s , so i t i s w i t h o u t l o s s of g e n e r a l i t y t h a t t he r e a l p a r t i s c o n s i d e r e d i n the development below. From e q u a t i o n (2.6) i t f o l l o w s t h a t R e 2 {A' } = Re 2{X°/W} •+ 2Re{X°/W}Re{E/W} + Re 2{E/W}. (2.7) u n d e r s t a n d i n g t h a t t h i s e q u a t i o n a p p l i e s a t a l l f r e q u e n c i e s . \ I f Re{E/W} i s v i s u a l i z e d as the r e a l i z a t i o n of a random v a r i a b l e , say Q, then e[Q] i s z e r o , and by e q u a t i o n ( 2 - B - 2 ) , e [ Q j 2 = a 2 / ( 2 | W | 2 ) , (2.8) where e(x) i s the e x p e c t e d v a l u e of x. 21 Combining equations (2.7) and (2.8), e[Re 2{A'}] = Re 2{X°/W} + cr2/{ 2 j W [ 2 ) . Thus the r e l a t i v e magnitudes of Re 2{A'} and e[Q 2] p r o v i d e a measure of the r e l i a b i l i t y of Re{A'}. I n t r o d u c i n g the f r e e parameter a, the c u t o f f K j , at frequency C J J i s d e f i n e d by Kj = 2a 2e[Q 2.] = a 2 a 2 / | W j | 2 (2.9) and the r e a l p a r t i s c o n s i d e r e d r e l i a b l e at t h a t frequency p r o v i d e d Re 2{Aj} > K j . The c h o i c e of a i s guided by the p r o b a b i l i t y d i s t r i b u t i o n of Q 2; s p e c i f i c a l l y , from Appendix 2-B, Pr{Q 2>Kj} = e r f c ( a ) . (2.10) In view of the u n c e r t a i n t y i n the e s t i m a t e a2, Kj cannot be c a l c u l a t e d p r e c i s e l y u s i n g equation ( 2 . 9 ) . Consequently, the p r o b a b i l i t y that pure n o i s e w i l l exceed the c u t o f f at a p a r t i c u l a r value of a can be determined only approximately from e q u a t i o n (2.10). 22 2.5 FORMULATION OF THE INVERSE PROBLEM IN THE PRESENCE OF NOISE Although the d i s t u r b i n g i n f l u e n c e of the n o i s e w i l l be g r e a t l y reduced by winnowing the most erroneous s p e c t r a l data, as d e s c r i b e d above, the remaining ( r e l i a b l e ) data are not n o i s e -f r e e . I t i s t h e r e f o r e i n a p p r o p r i a t e to s o l v e the data equations e x a c t l y . Two ways to take the e r r o r s i n t o account are d i s c u s s e d here, u s i n g i n e q u a l i t y and e q u a l i t y c o n s t r a i n t s , r e s p e c t i v e l y . 2.5.1 INEQUALITY CONSTRAINTS In the i n e q u a l i t y f o r m u l a t i o n , the . l i n e a r programming a l g o r i t h m i s r e q u i r e d t o f i t the n o i s y data only to w i t h i n c e r t a i n l i m i t s and, r e c a l l i n g e q u a t i o n s (2.2) and (2.4), the c o n s t r a i n t s are w r i t t e n i n the f o l l o w i n g form: +Re{A',} + ej> + + Im{A|-} + ej > + where ej i s the a s s i g n e d u n c e r t a i n t y a t frequency uj. These u n c e r t a i n t i e s do not appear i n the o b j e c t i v e f u n c t i o n ; the L,-norm of the response f u n c t i o n alone i s to be minimized. 1/N £ o ( b ^ - cjcos(^t^) , (2 . 1 1 ) 1/N V ( b ^ - c , J s i n ( a ; j t , J , 23 I f the u n c e r t a i n t i e s are i n c r e a s e d , t h e r e i s a g r e a t e r l i k e l i h o o d t h a t the t r u e s p e c t r a l v a l u e l i e s w i t h i n the p r e s c r i b e d l i m i t s . The g r e a t e r the freedom allowed, the g r e a t e r the p r o b a b i l i t y t h a t the a l g o r i t h m w i l l f i n d a f e a s i b l e s o l u t i o n with fewer s p i k e s ( i . e . w i t h s m a l l e r L^norm) than the t r u e response f u n c t i o n . T h i s f e a t u r e can be e x p l o i t e d to determine the most prominent s p i k e s . An i n c r e a s e i n the u n c e r t a i n t i e s a l s o reduces the c o m p u t a t i o n a l l a b o r and hence, the c o s t . C o n v e r s e l y , the c o s t i n c r e a s e s as the u n c e r t a i n t i e s are decreased, and th e r e i s a tendency f o r more and more s p i k e s to appear. There i s no f a i l - s a f e way t o d i s t i n g u i s h t r u e s p i k e s from s p u r i o u s s p i k e s , but p h y s i c a l l y untenable response f u n c t i o n s can r e s u l t when the u n c e r t a i n t i e s are too s m a l l . T h i s w i l l be d i s c u s s e d f u r t h e r . l a t e r . J u s t as t h e r e i s more than one way to choose a c u t o f f , so too i s there some f l e x i b i l i t y i n the c h o i c e of u n c e r t a i n t i e s ; two p o s s i b i l i t i e s are d e s c r i b e d below. (1) E m p i r i c a l u n c e r t a i n t i e s . - The u n c e r t a i n t y may be chosen to be some s m a l l f r a c t i o n .of the c u t o f f K ( d e f i n e d under " E m p i r i c a l c u t o f f " i n the p r e v i o u s s e c t i o n ) . The most s t r a i g h t f o r w a r d approach i s then t o a s s i g n the same u n c e r t a i n t y to a l l the d a t a . (2) S t a t i s t i c a l u n c e r t a i n t i e s . - The c h o i c e of u n c e r t a i n t i e s may be guided by c o n s i d e r a t i o n of the s t a t i s t i c a l c h a r a c t e r of the n o i s e . S p e c i f i c a l l y , i t i s a p p r o p r i a t e i n view of e q u a t i o n (2.6) to d e f i n e the u n c e r t a i n t i e s i n terms of the standard d e v i a t i o n 24 of Q, where Q i s a random v a r i a b l e r e p r e s e n t i n g e i t h e r Re{E/w] or Im{E/W} as b e f o r e . Using e q u a t i o n (2-B-2), the d e f i n i n g r e l a t i o n i s e.j = I | V a r [ Q j ] = 0.7071 j3o/|Wj|, where J3 i s a f r e e parameter. Note tha t the u n c e r t a i n t i e s are frequency dependent i n t h i s i n s t a n c e . I f the n o i s e i s normal or near-normal, the f a m i l i a r Gaussian c o n f i d e n c e l i m i t s serve to guide the c h o i c e of J3. 2.5.2 EQUALITY CONSTRAINTS In the p r o c e s s of s o l v i n g f o r the response f u n c t i o n with minimum ,L,—norm, the l i n e a r programming a l g o r i t h m a l s o i n d i r e c t l y determines an e s t i m a t e of the n o i s e . In the i n e q u a l i t y f o r m u l a t i o n t h i s mathematical n o i s e i s bounded at each frequency, but i t s o v e r - a l l s t a t i s t i c a l c h a r a c t e r i s very weakly c o n s t r a i n e d . For example, there i s no guarantee that the mean of the n o i s e removed by the a l g o r i t h m i s a c c e p t a b l y c l o s e to i t s ^ e x p e c t e d value of zero, nor t h a t i t s power i s compatible with the p e r c e i v e d noise power. In order to c o n t r o l the removal of the n o i s e more c l o s e l y , the e r r o r can be i n t r o d u c e d e x p l i c i t l y i n t o each equation as an a d d i t i o n a l unknown, say r j j . R e c a l l i n g e q u a t i o n s (2.6) and (2.2) the r e s u l t i n g e q u a l i t y c o n s t r a i n t s can be w r i t t e n i n the form: 25 R e { A ' . - } = R e d i j / w j } + 1/N K b - c ^ ) c o s ( < j j t ) , and (2.12) I m { A ' } = I m { r , j / W j } - lIl/Nlfb^ - c j s t n l w j t ^ ) . In s o l v i n g these e q u a t i o n s , the a l g o r i t h m i s r e q u i r e d to minimize the L,-norm of the response f u n c t i o n as b e f o r e , and i n so doing to determine that s t a t i s t i c a l l y c o n s i s t e n t set most f a v o r a b l e to the m i n i m i z a t i o n . The most obvious s t a t i s t i c a l r e s t r i c t i o n on the n o i s e i s t h a t i t have mean z e r o . I n a d d i t i o n , i t i s p o s s i b l e to d e v i s e a c o n s t r a i n t on i t s L,-norm. S p e c i f i c a l l y , the expected v a l u e of the q u a n t i t y S i s p r e s c r i b e d , with S d e f i n e d by S = (1/M) I (i> l + l c j ) ' (2.13: where M = Re{Ej(m)} and v = Im{Ej(m)}, with j(m) the index of the mth a c c e p t a b l e frequency, t h e r e being M such f r e q u e n c i e s i n a l l . A formula f o r e[S] i s d e r i v e d i n Appendix 2-C: e [ S ] = 2a/fV. S i n c e S i s an average of M q u a n t i t i e s , i t s v a r i a n c e d e c r e a s e s as M i n c r e a s e s , i . e . , as the number of r e l i a b l e data i n c r e a s e s . Thus the lower the c u t o f f , the more l i k e l y i s S to 26 assume i t s expected v a l u e . In t h i s sense, the n o i s e compensation u s i n g e q u a l i t y c o n s t r a i n t s improve as more and more data are i n c l u d e d i n the computations; t h i s i s i n c o n t r a s t to the i n e q u a l i t y f o r m u l a t i o n where the n o i s e compensation i s u n a f f e c t e d by the number of a c c e p t a b l e d a t a . The use of e q u a l i t y c o n s t r a i n t s does i n t r o d u c e c o n s i d e r a b l y more v a r i a b l e s i n t o the l i n e a r programming problem, an e x t r a four at each a c c e p t a b l e frequency i n f a c t . However, the number of c o n s t r a i n t s i s reduced by a f a c t o r of ap p r o x i m a t e l y two with r e s p e c t to the c o r r e s p o n d i n g problem with i n e q u a l i t i e s , and i n g e n e r a l the computer c o s t s are l i t t l e more than those i n c u r r e d u s i n g i n e q u a l i t i e s . 2.6 EXAMPLES WITH NOISY DATA The examples s t u d i e d e a r l i e r w i l l now be reworked i n the presence of Gaussian random n o i s e . Three v a r i a n t s of our l i n e a r programming approach are t e s t e d below. Two of the v a r i a n t s , denoted by l e and I s , employ i n e q u a l i t y c o n s t r a i n t s but d i f f e r i n t h e i r c h o i c e s of c u t o f f and u n c e r t a i n t i e s : V a r i a n t l e adopts the e m p i r i c a l c r i t e r i a and the e m p i r i c a l u n c e r t a i n t i e s , while v a r i a n t Is makes use of the s t a t i s t i c a l c r i t e r i a and u n c e r t a i n t i e s . The t h i r d v a r i a n t , denoted Es, employs s t a t i s t i c a l c u t o f f and e q u a l i t y c o n s t r a i n t s . .27 Whereas both l o c a t i o n and r e l a t i v e amplitude of the s p i k e s are u s u a l l y f a i t h f u l l y reproduced when the data are p e r f e c t l y a c c u r a t e , the presence of e r r o r s i n the data o f t e n renders i t i m p o s s i b l e to d i s c r i m i n a t e between the e f f e c t of a s i n g l e s p i k e at time t u with amplitude c, say, and two s p i k e s at times t and •t with amplitudes d . and c - d, r e s p e c t i v e l y . The t r a c e o b t a i n e d by c o n v o l v i n g the wavelet with the double s p i k e w i l l d i f f e r from t h a t o b t a i n e d u s i n g the s i n g l e s p i k e by an amount which i s everywhere n e g l i g i b l e i n comparison to the n o i s e i n the seismogram. As noted e a r l i e r with the n o i s e - f r e e data, the i n s u f f i c i e n c y of r e l i a b l e data i s a l s o a c o n t r i b u t i n g f a c t o r to t h i s " s p l i t t i n g " . I f s p l i t t i n g i s taken i n t o account, the amplitude r e c o v e r y i s o f t e n b e t t e r than i t appears at f i r s t g l a n c e because the amplitude of a s p l i t s p i k e should be regarded as the sum of the amp l i t u d e s of i t s components. At r e l a t i v e l y h i g h n o i s e l e v e l s the e n t i r e amplitude of a s p i k e may be a s s i g n e d t o an ad j a c e n t time, i . e . , the s p i k e i n the output i s s h i f t e d by one time i n t e r v a l with r e s p e c t to the l o c a t i o n of the c o r r e s p o n d i n g s p i k e i n the t r u e response f u n c t i o n . 2.6.1 EXAMPLE 3 The time s e r i e s i n F i g u r e 2-3a was o b t a i n e d by the a d d i t i o n of 10 percent random n o i s e to the s y n t h e t i c seismogram i n F i g u r e 2-1c. As i n the p r e v i o u s example, the s i m i l a r i t y of the 28 amplitude s p e c t r a i n F i g u r e s 2-3b and 2 - l d t e s t i f y to the r e l i a b i l i t y of the data i n the frequency band of the wavelet. V a r i a n t l e ( i n e q u a l i t y c o n s t r a i n t s with e m p i r i c a l c u t o f f and u n c e r t a i n t i e s ) : S e t t i n g c u t o f f K to 0.2 {|X(w)|} max and u n c e r t a i n t y e to 0.015 { |X(o>) | } max, the s p i k e t r a i n d e p i c t e d i n F i g u r e 2-3c was r e t u r n e d . T h i s s o l u t i o n e x h i b i t s a h i g h l y s a t i s f a c t o r y degree of agreement with F i g u r e .2-lb. The most s i g n i f i c a n t e f f e c t of the e r r o r s has been to a l t e r the s e p a r a t i o n of the t h i r d and f o u r t h s p i k e s . V a r i a n t Is ( i n e q u a l i t y c o n s t r a i n t s with s t a t i s t i c a l c u t o f f and u n c e r t a i n t i e s ) : The s p i k e s s e r i e s d e p i c t e d i n F i g u r e 2-3d was computed usi n g the parameter v a l u e s a = 3 and £ - 3. There i s good correspondence with the g e n e r a t i n g s p i k e s e r i e s ( F i g u r e 2-1b), a l t h o u g h a small s p u r i o u s s p i k e has appeared at time 7 and the s p i k e at time 25 has been s h i f t e d by one time u n i t . Some of the s p i k e s have been s p l i t , e s p e c i a l l y t h a t a t time 42. V a r i a n t Es ( e q u a l i t y c o n s t r a i n t s with s t a t i s t i c a l c u t o f f ) : The e s t i m a t e of a ( t ) o b t a i n e d u s i n g the e q u a l i t y c o n s t r a i n t f o r m u l a t i o n with a = 0.2 i s shown i n F i g u r e 2-3e; there i s very good agreement with F i g u r e 2-1b. A s m a l l a [and hence, by e q u a t i o n (2.9), a low c u t o f f ] was adopted in order to demonstrate the s u p e r i o r n o i s e t o l e r a n c e of t h i s f o r m u l a t i o n ; n o t w i t h s t a n d i n g the c o n s i d e r a b l y lower c u t o f f , the output i n F i g u r e 2-3e i s f r e e of s p u r i o u s s p i k e s , i n c o n t r a s t to F i g u r e 2-29 3d. 2.6.2 EXAMPLE 4 The c o m p l i c a t e d wavelet ( F i g u r e 2-2a) employed i n t h i s second example i s that used p r e v i o u s l y by Wiggins (1978). The s y n t h e t i c e a r t h response i n F i g u r e 2-2b i n c l u d e s e i g h t s p i k e s i n an i n t e r v a l of approximately f o u r - f i f t h s of the l e n g t h of the wavelet; i n the presence of n o i s e , t h e r e f o r e , t h i s example taxes the r e s o l v i n g power of any d e c o n v o l u t i o n method. The seismogram i n F i g u r e 2-4a was c o n s t r u c t e d by the a d d i t i o n of 10 percent random no i s e to the t r a c e i n :F.igure 2-2c. The s i m i l a r i t y between the amplitude s p e c t r a of the n o i s y and n o i s e - f r e e seisinograms at low f r e q u e n c i e s ( F i g u r e s 2-4b and .2-2d) j u s t i f i e s our r e l i a n c e on the data at those f r e q u e n c i e s . The dashed h o r i z o n t a l l i n e i n F i g u r e 2-4b r e p r e s e n t s a p o s s i b l e c u t o f f l e v e l . Note tha t the r e l i a b l e f r e q u e n c i e s i n t h i s case do not l i e i n a s i n g l e continuous band, but are d e r i v e d from a number of narrow bands; t h i s i l l u s t r a t e s the f a c t t h a t the d i s t r i b u t i o n of r e l i a b l e f r e q u e n c i e s i s unimportant. V a r i a n t l e : The response f u n c t i o n o b t a i n e d w i t h R = 0.2 max {|X(w)|} and e =0.01 max {|X(u)|} i s shown i n F i g u r e 2-4c. Except f o r the small s p i k e at time 10, a l l the s p i k e s i n F i g u r e 30 TIME - 1 1 TIME (At = 1) F i g u r e 2-3 (a) N o i s y i n p u t t r a c e r e s u l t i n g from the a d d i t i o n of 10 p e r c e n t random n o i s e to the t r a c e i n F i g u r e 2-1c. (b) Normalized amplitude spectrum of the t r a c e shown i n ( a ) . At low f r e q u e n c i e s t h i s spectrum bears a s t r o n g resemblance to t h a t f o r the n o i s e -f r e e seismogram ( F i g u r e 2 - l d ) . (c) N o r m a l i z e d response f u n c t i o n r e c o v e r e d from the r e l i a b l e data u s i n g v a r i a n t l e . (d) R e c o n s t r u c t e d response f u n c t i o n o b t a i n e d by a p p l i c a t i o n of v a r i a n t Is wit h a = 3 and J3 = 3. (e) No r m a l i z e d e s t i m a t e of the response f u n c t i o n determined w i t h v a r i a n t Es when a = 0.2. 31 2-2b have been l o c a t e d to w i t h i n one time i n t e r v a l . There i s some i n d i c a t i o n of the sp i k e at time 10, but t h i s i s much s m a l l e r than the s p u r i o u s s p i k e at time 44. Since the f o u r t h s p i k e i s s p l i t , i t s e f f e c t i v e amplitude i s i n f a c t very s i m i l a r to the amplitude of the t h i r d s p i k e . V a r i a n t I s : The sp i k e s e r i e s shown in F i g u r e 2-4d was ob t a i n e d w i t h a = .2.5 and j3 = 1 . S i x of the s p i k e s i n F i g u r e 2-2b have been r e c o v e r e d f a i t h f u l l y . However, the s p i k e s at 17 and 21 have been r e p l a c e d by a s i n g l e l a r g e s p i k e spread over times 18 and 19, and a s m a l l s p u r i o u s s p i k e has been i n t r o d u c e d at time -47. V a r i a n t Es: When a - 0.2, the e q u a l i t y c o n s t r a i n t f o r m u l a t i o n r e t u r n s the response f u n c t i o n shown i n F i g u r e 2-4e. Six of the s p i k e s i n F i g u r e .2-2b have been re c o v e r e d , and no sp u r i o u s s p i k e s have been i n t r o d u c e d . The net i n f l u e n c e of the e r r o r s has been to mask the presence of two s m a l l s p i k e s at times 10 and 40, as w e l l as c a u s i n g some s p l i t t i n g . 2.7 PRACTICAL RECOMMENDATIONS When u s i n g the l i n e a r programming technique to deconvolve n o i s y seismograms, the f o l l o w i n g p o i n t s deserve s p e c i a l c o n s i d e r a t i o n . 32 T I M E (At=1) F i g u r e 2-4 (a) N o i s y i n p u t t r a c e o b t a i n e d by a d d i n g 10 p e r c e n t random n o i s e to the t r a c e i n F i g u r e 2-2c. (b) Amplitude spectrum of the t r a c e d e p i c t e d i n F i g u r e 2-4a. The h igh-energy p o r t i o n s of t h i s spectrum bear a c l o s e resemblance to the c o r r e s p o n d i n g p o r t i o n s of the n o i s e - f r e e spectrum ( F i g u r e 2-2d) . (c) N o r m a l i z e d r e c o n s t r u c t e d response f u n c t i o n f u r n i s h e d by the e m p i r i c a l v a r i a n t l e . (d) Response f u n c t i o n e s t i m a t e computed u s i n g v a r i a n t Is w i t h a = 2.5 and £ = 1. As f o r ( c ) , t h e r e i s good agreement w i t h the g e n e r a t i n g s p i k e s e r i e s ( F i g u r e 2-2b). (e) N o r m a l i z e d response f u n c t i o n r e c o v e r e d u s i n g the e q u a l i t y c o n s t r a i n t s v a r i a n t Es, with a =0.2. 33 (1) The computing c o s t s i n c r e a s e i f the a l g o r i t h m i s r e q u i r e d t o s a t i s f y the data more p r e c i s e l y . The accuracy of s o l u t i o n i s governed by the magnitude of the u n c e r t a i n t i e s when us i n g i n e q u a l i t i e s , or by the p r e s c r i b e d n o i s e l e v e l f o r e q u a l i t y c o n s t r a i n t s . I f the accurac y demanded i s i n c o n s i s t e n t with the a c t u a l l e v e l of n o i s e i n the da t a , the Simplex a l g o r i t h m w i l l perform many i t e r a t i o n s before i t can l o c a t e the s o l u t i o n w i t h minimum L,-norm. If an u n j u s t i f a b l y c l o s e f i t i s demanded, the s o l u t i o n o b t a i n e d w i l l be c h a r a c t e r i z e d by an unreasonably l a r g e number of s p i k e s . At the other extreme, l a r g e " u n c e r t a i n t i e s f a c i l i t a t e s o l u t i o n i n a s m a l l number of i t e r a t i o n s and at a c o r r e s p o n d i n g l y lower c o s t . However, only the most prominent s p i k e s w i l l be r e c o v e r e d , and t h e i r amplitudes may be underestimated. In o r d e r to a v o i d n e e d l e s s computation, i t i s recommended that i f p o s s i b l e a statement be i n t r o d u c e d i n t o the l i n e a r programming r o u t i n e t o t e r m i n a t e e x e c u t i o n when the o b j e c t i v e f u n c t i o n exceeds a c e r t a i n l i m i t . I f the maximum amplitude of the seismogram i s nor m a l i z e d t o u n i t y , the upper l i m i t on the 1,-norm s h o u l d c o n s t i t u t e a c o n s e r v a t i v e e s t i m a t e of the number of s p i k e s e x p e c t e d . (2) I d e n t i f i c a t i o n of contaminated f r e q u e n c i e s i s a v a l u a b l e a i d i n the d e t e r m i n a t i o n of the s p i k y s e r i e s , and i t i s recommended t h a t a r e c o r d of the background n o i s e i n the study .34 area be a c q u i r e d . In c o n j u n c t i o n w i t h some knowledge of the frequency band of the wavelet, the spectrum of the n o i s e i n d i c a t e s which f r e q u e n c i e s should be omi t t e d . Furthermore, a knowledge of the n o i s e guides the c h o i c e of u n c e r t a i n t i e s t o be a s s i g n e d to the r e l i a b l e s p e c t r a l d a t a . 2.8 COMPARISON WITH LEAST-SQUARES TECHNIQUES In order to compare the l i n e a r programming method with c o n v e n t i o n a l t e c h n i q u e s , the examples have been re p e a t e d u s i n g (1) a l e a s t - s q u a r e s frequency domain f i l t e r (Berkhout 1977) and (2) a l e a s t - s q u a r e s time domain f i l t e r (Wood e t . a l . , 197.8). In terms of the n o t a t i o n adopted here, the frequency domain f i l t e r F ( w ) i s d e f i n e d by -W*(w) F(a>) = _ T _ r _ _ ^ ^ _ _ _ ^ _ _ (2.14) W(u)W*(a>) + c2 where W* denotes the complex conjugate of W. The " w a t e r - l e v e l " parameter c i s a d j u s t e d a c c o r d i n g to the l e v e l of n o i s e ; c i s zero f o r p e r f e c t data and i s i n c r e a s e d as the l e v e l of n o i s e r i s e s . The time domain f i l t e r f ( t ) i s d e f i n e d by 35 f ( t ) = w ( - t ) * w - 1 ( t ) * w - 1 ( - t ) , (2.15) where w-1 i s the z e r o - d e l a y Wiener i n v e r s e of w, and where the argument - t i s used to denote time r e v e r s a l . Noise s u p p r e s s i o n i s a c h i e v e d i n t h i s case by the a d d i t i o n of a constant c' to the main d i a g o n a l of the a u t o c o r r e l a t i o n matrix which f e a t u r e s i n the d e t e r m i n a t i o n of w-1 (Berkhout, 1 977 ) .. The higher the l e v e l of n o i s e , the l a r g e r the val u e of c' adopted. 2.8.1 EXAMPLE 5 To p r o v i d e a p o i n t of r e f e r e n c e , the frequency domain f i l t e r was f i r s t a p p l i e d to the n o i s e - f r e e t r a c e i n F i g u r e 2-1c, us i n g the wavelet i n F i g u r e 2-1a as b e f o r e . The output ( F i g u r e 2-5a) i s an exact r e p l i c a of the g e n e r a t i n g spike sequence ( F i g u r e 2-1b, p-15), as expected. When the f i l t e r i n e q u a t i o n (2.14) was a p p l i e d to the n o i s y seismogram i n F i g u r e 2-3a, the response f u n c t i o n shown i n F i g u r e 2-5b was o b t a i n e d . In t h i s case the most s u i t a b l e v a l u e of c z was found to be 0.0.1 max {|W(*j)| 2}, c o n s i s t e n t with the known 10 percent n o i s e l e v e l . The output i n F i g u r e 2-5c, on the other hand, r e s u l t e d when the time domain f i l t e r was a p p l i e d to the same example with c' = 0.1R 0, where R 0 denotes the zero l a g a u t o c o r r e l a t i o n . S u b s t a n t i a l 36 d e l a y s have been caused by the s u c c e s s i v e c o n v o l u t i o n s performed d u r i n g f i l t e r c o n s t r u c t i o n . Comparing F i g u r e s 2-3c, 2-3d, and 2-3e with F i g u r e s 2-5b and 2-5c, i t i s c l e a r t h a t t h e l i n e a r programming method p r o v i d e s a much more h i g h l y r e s o l v e d output than the c o n v e n t i o n a l l e a s t - s q u a r e s t e c h n i q u e s . 2.8.2 EXAMPLE 6 As f o r the p r e v i o u s example, the frequency domain f i l t e r f o r the wavelet i n F i g u r e 2-2a was f i r s t a p p l i e d to the exact seismogram i n F i g u r e 2-2c, and the o r i g i n a l s pike s e r i e s ( F i g u r e 2-2b) was reproduced p e r f e c t l y ( F i g u r e 2-6a). Subsequently, the same f i l t e r was a p p l i e d to the n o i s y seismogram i n F i g u r e 2-4a; the most a p p r o p r i a t e v a l u e of c 2 was found to be 0.02 max {|W(o))|2}, and the r e s u l t i n g output i s presented i n F i g u r e 2-6b. When the time domain f i l t e r was a p p l i e d to the n o i s y t r a c e with c' = 0.05R o, the response f u n c t i o n i n F i g u r e 2-6c r e s u l t e d . Comparison of F i g u r e s 2-4c, 2-4d, and 2-4e with F i g u r e s 2-6b and 2-6c r e v e a l s t h a t the l i n e a r programming output i s much more h i g h l y r e s o l v e d than the c o r r e s p o n d i n g output o b t a i n e d u s i n g the c o n v e n t i o n a l f i l t e r s [ e quations (2.14) and (2 . 1 5 ) ] . 37 CL Z < 0-z <r o -1- r — .25 TIME r 50 1-1 a: z < OH cr o -1 25 TIME .50 1-i o_ Z < 0 H z cr o -1 — i 1 1 25 .50 75 TIME ( A t - 1 ) 100 125 F i g u r e 2-5 (a) N o r m a l i z e d r e c o n s t r u c t e d response f u n c t i o n o b t a i n e d by a p p l y i n g the frequ e n c y domain f i l t e r (2.14) to the n o i s e - f r e e seismogram i n F i g u r e 2-1C. (b) No r m a l i z e d response f u n c t i o n r e c o v e r e d from the n o i s y seismogram i n F i g u r e 2-3a u s i n g the freque n c y domain f i l t e r with c 2 = 0.01 max {|w|:}. (c) R e c o n s t r u c t i o n of response f u n c t i o n r e s u l t i n g from a p p l i c a t i o n of the time domain f i l t e r (2 . 1 5 ) to the t r a c e i n F i g u r e 2-3a, wit h c' = 0 . 1R 0. 38 1n o Q_ Z <-o-> cr o z -1 2 5 T I M E 50 -i 1 r 0 2 5 50 7 5 100 T I M E ( A t = 1 ) 1 2 5 F i g u r e 2-6 (a) N o r m a l i z e d r e c o n s t r u c t e d response f u n c t i o n o b t a i n e d by a p p l y i n g the frequency domain f i l t e r (2.14) t o the n o i s e - f r e e seismogram i n F i g u r e 2-2c. (b) N o r m a l i z e d response f u n c t i o n r e c o v e r e d from the n o i s y seismogram i n F i g u r e 4a u s i n g the frequency domain f i l t e r w i t h c 2 = 0.02 max {|W|2}. (c) R e c o n s t r u c t i o n of response f u n c t i o n r e s u l t i n g from a p p l i c a t i o n of the time domain f i l t e r (2.15) t o the t r a c e i n F i g u r e 2-4a, with c' = 0.05R o. 39 2.9 CONCLUSIONS Using a l i n e a r programming f o r m u l a t i o n , we have shown t h a t i t i s p o s s i b l e to r e c o n s t r u c t a s p i k y s i g n a l from a sma l l p o r t i o n of i t s F o u r i e r spectrum. The recovered s i g n a l bears a c l o s e resemblance to the o r i g i n a l s p i k e sequence. The minor d i f f e r e n c e s which can sometimes be observed are m a n i f e s t a t i o n s of the non-uniqueness i n h e r e n t i n any underdetermined l i n e a r i n v e r s e problem. To t e s t i t s e f f e c t i v e n e s s i n the presence of n o i s e , the r e c o n s t r u c t i o n method has been a p p l i e d to the s e i s m i c d e c o n v o l u t i o n problem when the wavelet i s known. The i n f l u e n c e of n o i s e has been suppressed by d i s c a r d i n g those s p e c t r a l data which are most s e r i o u s l y contaminated and by i n c l u d i n g only the remaining r e l i a b l e data i n the computations. Furthermore, when s o l v i n g f o r the s p i k e amplitudes, the r e l i a b l e data are not s a t i s f i e d e x a c t l y but r a t h e r to an accurac y c o n s i s t e n t with the p e r c e i v e d l e v e l of n o i s e . In s y n t h e t i c examples with 10 p e r c e n t random n o i s e , t h i s method has produced h i g h l y r e s o l v e d deconvolved t r a c e s . In each case the output bears a c l o s e s i m i l a r i t y to the g e n e r a t i n g spike sequence. Although the e f f e c t of random e r r o r s i n d e c o n v o l u t i o n has been c o n s i d e r e d i n some d e t a i l , the r e c o n s t r u c t i o n method i s r e a d i l y adaptable to other types of n o i s e . Large amplitude s i n u s o i d a l n o i s e , f o r example, would produce a very obvious 40 signature in the frequency domain, thus f a c i l i t a t i n g recognition (and subsequent rejection) of u n r e l i a b l e frequencies. The deconvolution method proposed here is simple to implement, given the wide a v a i l a b i l i t y of linear programming routines. Although we have not conducted a serious study of comparative costs, i t is to be expected that the lin e a r programming method w i l l be somewhat more expensive than most existing deconvolution techniques. However, any a d d i t i o n a l expense is o f f s e t by the high q u a l i t y of the linear programming output, consisting as i t does of spikes separated by zeros, in contrast to the broadened peaks and side lobes which t y p i f y conventional least-squares deconvolution. 41 CHAPTER I I I : ACOUSTIC IMPEDANCE INVERSION AND WELL-LOG CONSTRAINTS 3.1 INTRODUCTION The f o l l o w i n g i s a s h o r t summary of some of the e s s e n t i a l p o i n t s brought up by Scheuer (1981) and Oldenburg et a l . (1983). For complete d e t a i l s the reader i s r e f e r r e d to the o r i g i n a l papers. G e o l o g i c a l environments which are c h a r a c t e r i z e d by a r e l a t i v e l y s m a l l number of sharp impedance d i s c o n t i n u i t i e s are l i k e l y t o produce r e f l e c t i v i t y f u n c t i o n s which are w e l l approximated by sparse s p i k e t r a i n s . In such environments, the f u l l - b a n d r e f l e c t i v i t i e s produced by l i n e a r programming (LP) d e c o n v o l u t i o n c o n s t i t u t e a good r e p r e s e n t a t i o n of the true e a r t h r e f l e c t i v i t y f u n c t i o n and can be subsequently used f o r the c a l c u l a t i o n of the a c o u s t i c impedance f u n c t i o n . As shown by Peterson et a l . (1955) and Waters (1978, p. 219), the r e l a t i o n s h i p between the r e f l e c t i v i t y f u n c t i o n r ( t ) and the a c o u s t i c impedance £(t) i s a p p r o x i m a t e l y : 42 r ( t ) = l / 2 d [ l n £(t)]/dt (3.1) T h i s a p p roximation i s a c c e p t a b l e f o r r e f l e c t i o n c o e f f i c i e n t s s m a l l e r than 0.3, i n which case we may w r i t e : | ( t ) = |(0) exp [2 J r(u) du] (3.2) o or a f t e r t a k i n g the l o g a r i t h m of (3.2): T?(t)= In [£(t)/{(0)] = 2 J r ( u ) du (3.3) From e q u a t i o n (3,3) we see that seismogram i n v e r s i o n can be r e a d i l y a c h i e v e d i f f u l l - b a n d , m u l t i p l e - f r e e seismograms are a v a i l a b l e . The remainder of t h i s work i s d e d i c a t e d to the development of t o o l s which w i l l a l l o w r e l i a b l e r e c o n s t r u c t i o n of f u l l - b a n d r e f l e c t i v i t y s e c t i o n s from the observed b a n d - l i m i t e d s t a c k e d s e c t i o n s . We s t a r t by s t a t i n g t h a t the s t r a i g h t forward technique p r e s e n t e d i n chapter II i s not s u f f i c i e n t to ensure s u c c e s s f u l i n v e r s i o n . Although a f a i r l y . l a r g e number of g e o l o g i c a l environments conform with the sparse r e f l e c t i v i t y assumption, an even l a r g e r number w i l l c o n t a i n at l e a s t a number of time windows i n which t h i s assumption i s v i o l a t e d . C o n s i d e r f o r example a g e o l o g i c a l environment c o n t a i n i n g a number of t r a n s i t i o n zones c h a r a c t e r i z e d by a sl o w l y changing impedance f u n c t i o n . Such t r a n s i t i o n zones g i v e r i s e to a den s e l y populated 4 3 r e f l e c t i v i t y s e r i e s which cannot be d e s c r i b e d as sparse. Consequently, the r e f l e c t i v i t i e s o b t a i n e d by the LP d e c o n v o l u t i o n are not expected to c o n s t i t u t e a good r e p r e s e n t a t i o n of the t r u e e a r t h r e f l e c t i v i t y and hence t h e i r c o n v e r s i o n to l o g r e l a t i v e impedance (Equation 3.2) w i l l not y i e l d meaningful r e s u l t s . To overcome t h i s problem and i n c r e a s e the r e l i a b i l i t y of the ob t a i n e d impedance s e c t i o n i t i s necessary to i n t r o d u c e a d d i t i o n a l i n f o r m a t i o n i n t o the i n v e r s i o n p r o c e s s . The t a r g e t of t h i s c hapter and Chapter I V i s the g e n e r a t i o n of a complete set of l i n e a r r e l a t i o n s combining i n f o r m a t i o n from the stacked s e c t i o n , the s o n i c w e l l - l o g and the s t a c k i n g v e l o c i t i e s . T h i s set i s subsequently s o l v e d to y i e l d a f u l l - b a n d r e f l e c t i v i t y f u n c t i o n which i s c o n s i s t e n t with a l l the a v a i l a b l e i n f o r m a t i o n . 3.2 EXTERNAL CONSTRAINTS IN IMPEDANCE INVERSION E x t e r n a l i n f o r m a t i o n about the r e f l e c t i v i t y can be i n t r o d u c e d i n t o the i n v e r s i o n i f the c o r r e s p o n d i n g knowledge can be expressed i n the l i n e a r form: a- = Z A•• r- (3.4) 44 For each such c o n s t r a i n t , a and a l l the A 's must be known. The p o s s i b i l i t y t h a t a i s not known e x a c t l y can be h'andled by s p e c i f y i n g only i t s bounds, to w i t h i n the d e s i r e d u n c e r t a i n t y . That i s , a. - 5. < Z A - • r; < a. + 6- ( 3 . 5 ) - j . , J J The f u l l set of equ a t i o n s c o n t a i n s a number of F o u r i e r t r a n s f o r m i n e q u a l i t i e s (as per e q a t i o n . 2 . 1 1 ) p l u s e x t e r n a l c o n s t r a i n t s of the form s p e c i f i e d i n equation ( 3 . 5 ) above. 3 . 3 POINT VELOCITY CONSTRAINTS-WELL LOG INFORMATION When w e l l - l o g i n f o r m a t i o n i s a v a i l a b l e , i t s c a u t i o u s i n c o r p o r a t i o n i n t o the i n v e r s i o n i s h i g h l y d e s i r a b l e . In t h i s s e c t i o n we develop the set of r e l a t i o n s r e q u i r e d f o r t h i s o p e r a t i o n . For the purpose of the f o l l o w i n g development we model the d i g i t i z e d r e f l e c t i v i t y f u n c t i o n as a D i r a c comb, we w r i t e : r ( t ) = z ' r S(t-nA) ( 3 . 6 ) ••vv where, 6 symbolizes the D i r a c d e l t a f u n c t i o n , and A the sampling i n t e r v a l . S u b s t i t u t i n g equation ( 3 . 6 ) i n t o ( 3 . 3 ) , exchanging the order of 4 5 summation and i n t e g r a t i o n , and i n t e g r a t i n g the r e s u l t term by term, we get: with H d e n o t i n g the H e a v i s i d e step f u n c t i o n . E quation (3.7) s t a t e s that the l o g a r i t h m of the r e l a t i v e impedance at time ' t ' i s equal to twice the sum of the r e f l e c t i o n c o e f f i c i e n t s i n the time window z e r o to ' t ' seconds. T h i s e q u a t i o n a l l o w s c o n s t r a i n t s to be a p p l i e d t o the c o n s t r u c t e d r e f l e c t i v i t y i f the d e n s i t y and v e l o c i t y are known f o r s p e c i f i c two-way t r a v e l times. In g e n e r a l , we assume that both r ( t ) and 7 7(t) are not known a c c u r a t e l y ; hence, a c o n s t r a i n t at a g i v e n time ' t ' i s of the form: r?(t)= 2 *Z r H(t-nA) ( 3 . 7 ) N - l 7?(t)-6T?(t) < 2 Z r H(t-nA) ( . 3 . 8 ) and with 8rj(t) being the e s t i m a t e d e r r o r . One can a l s o assume a d e n s i t y power law, such as: p(t)= c o n s t a n t [ v ( t ) J ( 3 . 9 ) with v ( t ) being the v e l o c i t y versus two-way t r a v e l time p r o f i l e . 46 Invoking t h i s assumption equation (3.7) becomes: rj(t)= In [ v ( t ) / v ( 0 ) ] = [ 2 / ( 1+a) ] z ' r H(t-nA) (3.10) 3.4 DISCUSSION The importance of e x t e r n a l c o n s t r a i n t s in the r e c o n s t r u c t i o n of f u l l - b a n d r e f l e c t i v i t y f u n c t i o n s cannot be overemphasized. In a normal s e i s m i c experiment the a c q u i r e d data i s c h a r a c t e r i z e d by a r e l a t i v e l y narrow frequency band and hence the c o n s t r u c t i o n of f u l l - b a n d r e f l e c t i v i t y f u n c t i o n s i s a s t r o n g l y underdetermined i n v e r s e problem. Consequently, the technique p r e s e n t e d i n chapter II w i l l a t t a i n an a c c e p t a b l e s o l u t i o n only i f the t r u e e a r t h r e f l e c t i v i t y f u n c t i o n i s w e l l approximated by a sparse s p i k e s e r i e s . The i n c o r p o r a t i o n of the v e l o c i t y c o n s t r a i n t s i n t o the i n v e r s i o n reduces the allowed s o l u t i o n space and r e l a x e s the s t r i c t n e s s of the c o n d i t i o n s p r e v i o u s l y imposed on the sparseness of the c o n s t r u c t e d r e f l e c t i v i t y f u n c t i o n . For example, the problem of a r e f l e c t i v i t y f u n c t i o n which i n c l u d e s a number of t r a n s i t i o n zones i n an otherwise sparse r e f l e c t i v i t y background can now be t a c k l e d with very encouraging r e s u l t s . Furthermore, the i n v e r s i o n of p r o p e r l y s c a l e d stacked seismograms supported by the c o r r e s p o n d i n g w e l l - l o g i n f o r m a t i o n can be t r a n s l a t e d d i r e c t l y i n t o p h y s i c a l l y meaningful impedance or p s e u d o - v e l o c i t y f u n c t i o n s , thereby s u p p l y i n g v a l u e a b l e i n f o r m a t i o n f o r quantitative interpretation of the observed data. 48 CHAPTER IV: RMS VELOCITIES AND RECOVERY OF THE ACOUSTIC IMPEDANCE 4.1 INTRODUCTION The paper by Oldenburg, Scheuer, and Levy (1983) ( h e r e a f t e r r e f e r r e d t c as OSL) i n v e s t i g a t e d the problem of r e c o v e r i n g the a c o u s t i c impedance from normal i n c i d e n c e r e f l e c t i o n seismograms. That work began by u s i n g the simple c o n v o l u t i o n a l model f o r a r e f l e c t i o n seismogram where x ( t ) i s the seismogram, r ( t ) i s the r e f l e c t i v i t y f u n c t i o n , w(t) i s the source wavelet, and the symbol * denotes the c o n v o l u t i o n o p e r a t i o n . I f v ( t ) i s an i n v e r s e f i l t e r which shapes w(t) "as w e l l as p o s s i b l e " i n t o a D i r a c d e l t a f u n c t i o n , then the best e s t i m a t e s f o r r ( t ) are the unique averages where a ( t ) = w ( t ) * v ( t ) i s c a l l e d the a v e r a g i n g f u n c t i o n . Note t h a t a ( t ) i s b a n d l i m i t e d because w(t) i s , and hence <r(t)> i s , at best, a b a n d - l i m i t e d r e p r e s e n t a t i o n of the r e f l e c t i v i t y f u n c t i o n . The consequences of i n t e g r a t i n g <r(t)>, or u s i n g i t s d i s c r e t i z e d form i n the standard a c o u s t i c impedance r e c u r s i o n x(.t)«r(t)*w(t) (4. 1 ) < r ( t ) > = r ( t ) * a ( t ) (4.2) 49 formula, was shown to y i e l d a b a n d - l i m i t e d r e p r e s e n t a t i o n of the t r u e impedance. More p r e c i s e l y , i t was shown that only averages <rj(t)> - - 7 j ( t ) * a ( t ) ' (4.3) c o u l d be r e c o v e r e d . In e q u a t i o n (4.3), T?( t) = l n ( £ ( t ) / | ( 0 )) where £(t) i s the a c o u s t i c impedance at time t , and a ( t ) i s the same b a n d - l i m i t e d a v e r a g i n g f u n c t i o n t h a t was o b t a i n e d i n d e c o n v o l v i n g the i n i t i a l seismogram. The l o s s of the low f r e q u e n c i e s makes a g e o l o g i c a l i n t e r p r e t a t i o n of <rj(t)> very d i f f i c u l t . The s o l u t i o n o f f e r e d by OSL was to abandon the a p p r a i s a l t e chnique which y i e l d s unique averages of the a c o u s t i c impedance and i n s t e a d , attempt to c o n s t r u c t a broadband r e f l e c t i v i t y f u n c t i o n which reproduced the s e i s m i c d a t a . The .basic d i f f i c u l t y w ith such an approach i s the i n h e r e n t nonuniqueness i n the s o l u t i o n . There e x i s t s an i n f i n i t y of a c c e p t a b l e models and these can d i f f e r g r e a t l y from one another. T h i s nonuniqueness can be conquered only i f model space i s r e s t r i c t e d s u f f i c i e n t l y so t h a t the o n l y a c c e p t a b l e models are l i k e the t r u e e a r t h and d i f f e r from each other i n at most s m a l l s c a l e f e a t u r e s . In OSL, model space was r e s t r i c t e d by f i r s t i n t r o d u c i n g the l a y e r e d e a r t h so t h a t the r e f l e c t i v i t y f u n c t i o n was r e p r e s e n t e d as r ( t ) = Z r 6 ( t - r w ) (4.4) In e q u a t i o n (4.4) r k i s the r e f l e c t i o n c o e f f i c i e n t at the base 50 of the k'th l a y e r , and r K i s the two-way t r a v e l time to that l a y e r . C l e a r l y r ( t ) as r e p r e s e n t e d by equation (4.4) has the p o t e n t i a l f o r being broadband s i n c e a d e l t a f u n c t i o n has energy : at a l l f r e q u e n c i e s . Moreover, an a c o u s t i c impedance having a minimum of s t r u c t u r a l d e t a i l can be found by c o n s t r u c t i n g an r ( t ) i n the form of e q u a t i o n (4.4) which has the fewest number of l a y e r s . T h i s l e d to the development of two s o l u t i o n s . The f i r s t was a l i n e a r programming (LP) a l g o r i t h m which minimized . , <t> = / |d7?(t)/dt | dt = 21 | r | • k (4.5) The LP computations were c a r r i e d out i n the frequency domain usin g the method of Levy and F u l l a g a r (1981). That a l g o r i t h m c o n s t r u c t s a broadband r e f l e c t i v i t y whose spectrum agrees with the measured spectrum w i t h i n the energy band of the wavelet, and has a spectrum c o n s i s t e n t with th a t of equation (4.4) o u t s i d e the band. The second s o l u t i o n modelled the F o u r i e r t r a n s f o r m of the r e f l e c t i v i t y f u n c t i o n as an a u t o r e g r e s s i v e (AR) p r o c e s s and p r e d i c t e d unmeasured p o r t i o n s of the r e f l e c t i v i t y spectrum from the s p e c t r a l v a l u e s w i t h i n the band of the wavelet. I t was found that the nonuniqueness c o u l d be f u r t h e r c o n t r o l l e d by i n t r o d u c i n g a d d i t i o n a l c o n s t r a i n t s i n t o the c o n s t r u c t i o n a l g o r i t h m s . The LP s o l u t i o n was p a r t i c u l a r l y 51 amenable to such m o d i f i c a t i o n . OSL were a b l e to use the unique averages of the r e f l e c t i v i t y f u n c t i o n to c o n s t r a i n the p o l a r i t y of the r e f l e c t i o n c o e f f i c i e n t s and a l s o to weight the o b j e c t i v e f u n c t i o n so t h a t s i z e a b l e r e f l e c t i o n c o e f f i c i e n t s would not appear at those times where the unique averages i n d i c a t e d they should not. L a s t l y , and i m p o r t a n t l y , they a l s o showed how impedance c o n s t r a i n t s c o u l d be i n c o r p o r a t e d d i r e c t l y i n t o the s o l u t i o n . In extending^the a u t o r e g r e s s i v e approach, Walker and D l r y c h (1983) showed how the o r i g i n a l AR f o r m u l a t i o n c o u l d be m o d i f i e d so t h a t i t a l s o c o u l d i n c l u d e impedance c o n s t r a i n t s . Thus, impedance i n f o r m a t i o n from a nearby w e l l , an e s t i m a t e of a basement impedance, or a r e c o g n i z e d marker zone with known impedance c o u l d be input d i r e c t l y i n t o the LP or AR a l g o r i t h m s . I t would seem that such c o n s t r a i n t s would so r e s t r i c t model space t h a t the only nonuniqueness would be r e l e g a t e d to ambiguity of f i n e s c a l e s t r u c t u r e . OSL showed t h a t t h i s o f t e n appears to be the case when the e a r t h i s adequately modelled by a set of homogeneous l a y e r s . However, t h e r e are numerous i n s t a n c e s when a l a y e r e d e a r t h i s not j u s t i f i e d because the impedance changes s l o w l y and c o n t i n u o u s l y with depth. Such t r a n s i t i o n zones or 'ramps' can cause problems with the c o n s t r u c t i o n procedures because r ( t ) •« 1/2 d/dt [ - l n { { ( t ) / { ( 0 ) } ] can be a r b i t r a r i l y s m a l l i f d£(t)/dt i s s u f f i c i e n t l y s m a l l . Slow changes i n . the impedance t h e r e f o r e produce very s m a l l 52 r e f l e c t i o n s and hence such changes are e s s e n t i a l l y a n n i h i l a t o r s f o r the r e f l e c t i o n s e i s m i c problem; t h a t i s , they are f u n c t i o n s which produce no o b s e r v a b l e d a t a . Large components of such a n n i h i l a t o r s can be added to any a c c e p t a b l e r e f l e c t i v i t y f u n c t i o n without d e g r a d i n g s u b s t a n t i a l l y the m i s f i t to the d a t a . C l e a r l y , more i n f o r m a t i o n i s r e q u i r e d to c o n t r o l t h i s form of nonuniqueness. A major purpose of t h i s chapter i s to show how e x t r a i n f o r m a t i o n d e r i v e d from RMS ( s t a c k i n g ) v e l o c i t i e s can be i n c o r p o r a t e d d i r e c t l y i n t o the LP and AR a l g o r i t h m s . There are two ways i n which RMS v e l o c i t i e s can be i n c l u d e d i n the c o n s t r u c t i o n a l g o r i t h m s . The f i r s t , and s i m p l e s t , would be to compute a minimum s t r u c t u r e v e l o c i t y f u n c t i o n from the RMS v e l o c i t i e s and then i n c o r p o r a t e approximate impedance c o n s t r a i n t s d i r e c t l y by s e t t i n g the d e n s i t y equal to u n i t y and p r e s c r i b i n g | ( t ) = v ( t ) as p o i n t c o n s t r a i n t s i n the c o n s t r u c t i o n a l g o r i t h m s . T h i s method has the p o t e n t i a l f o r producing good r e s u l t s but i t b r i n g s to l i g h t a problem of fundamental importance. The RMS v e l o c i t i e s are averages of the i n t e r v a l v e l o c i t y v ( t ) , but knowledge of o n l y a few i n a c c u r a t e RMS v e l o c i t i e s does not permit one to p l a c e p o i n t w i s e bounds on v ( t ) . M a t h e m a t i c a l l y , v ( t ) can have any value at a p a r t i c u l a r time, but p h y s i c a l l y , i t i s c o n s t r a i n e d w i t h i n the l i m i t s imposed by the a v a i l a b l e g e o l o g i c a l l i t h o l o g i e s . In p r i n c i p l e t h e r e f o r e , i t i s not p o s s i b l e to e s t i m a t e e r r o r s f o r v ( t ) and hence i t i s g e n e r a l l y poor p r a c t i c e to i n c o r p o r a t e impedance c o n s t r a i n t s i n f e r r e d d i r e c t l y from a c o n s t r u c t e d model. An a l t e r n a t i v e method i s p o s s i b l e . I t s b a s i s i s the 53 r e a l i z a t i o n t h a t the RMS v e l o c i t i e s are averages of the i n t e r v a l v e l o c i t y and t h e r e f o r e the only unique i n f o r m a t i o n a v a i l a b l e i s encompassed i n the averages <v(t)>. Our goal w i l l be to i n v e s t i g a t e these averages and show how i n f o r m a t i o n c o n t a i n e d in them can be used to produce c o n s t r a i n t s f o r the LP and AR c o n s t r u c t i o n a l g o r i t h m s . The development of the s o l u t i o n to our o b j e c t i v e i s d i v i d e d i n t o t h r e e s e c t i o n s . We begin by i n v e r t i n g RMS v e l o c i t i e s to recover i n f o r m a t i o n about the i n t e r v a l v e l o c i t y . A l g o r i t h m s f o r computing f l a t t e s t L, ( a b s o l u t e v a l u e ) and L 2 ( l e a s t squares) norm models w i l l be p r e s e n t e d . Both of these a l g o r i t h m s are s t a b l e when a p p l i e d to n o i s y data and both can i n c o r p o r a t e known c o n s t r a i n t s on the v e l o c i t y . The next stage i n v o l v e s .use of the l i n e a r a p p r a i s a l methods of Backus and G i l b e r t (1970) to summarize our i n f o r m a t i o n about the i n t e r v a l v e l o c i t y . . T h i s a n a l y s i s q u a n t i f i e s the l o s s of r e s o l v i n g power at i n c r e a s e d time al o n g the r e c o r d and a l s o i l l u s t r a t e s the importance of data e r r o r s . The f i n a l s e c t i o n shows how to i n c l u d e the i n f o r m a t i o n c o n t a i n e d i n the RMS v e l o c i t i e s i n t o the a c o u s t i c impedance c o n s t r u c t i o n a l g o r i t h m s . T h i s i s done by f i r s t c o n s t r u c t i n g an a c c e p t a b l e i n t e r v a l v e l o c i t y model and then usi n g l i n e a r a p p r a i s a l to o b t a i n unique averages of v 2 ( t ) or v ( t ) . The d e s i r e d c o n s t r a i n t s , i n the form of l i n e a r combinations of the r e f l e c t i o n c o e f f i c i e n t s , can be computed d i r e c t l y with t h a t i n f o r m a t i o n . 54 •4.2 : INVERSION OF RMS VELOCITIES: MODEL CONSTRUCTION In t h i s s e c t i o n we s h a l l show how t o c o n s t r u c t an i n t e r v a l v e l o c i t y .model from a g i v e n s e t of RMS v e l o c i t i e s . T h i s problem has been a d d r e s s e d i n a l a r g e number of p u b l i c a t i o n s over the l a s t t h r e e decades [ c . f , Dix (1955) r. Taner and K o e h l e r ( 1 9 6 9 ) , Taner e t a l . (1970), L e v i n (1971), S c h n e i d e r (1971, E v e r e t t (1974), Rrey (1976), and H u b r a l and Krey (198 0 ) ] , but our approach w i l l be somewhat d i f f e r e n t . In p a r t i c u l a r we s h a l l show how o b s e r v a t i o n a l e r r o r s and p o i n t v e l o c i t y c o n s t r a i n t s can be i n c o r p o r a t e d d i r e c t l y i n t o the c o n s t r u c t i o n of an i n t e r v a l -v e l o c i t y . A l s o , we p r e s e n t two c o n s t r u c t i o n a l g o r i t h m s : one produces a c o n t i n u o u s v e l o c i t y s t r u c t u r e , and the ot h e r g e n e r a t e s a .layered e a r t h w i t h t r a n s i t i o n zones. The f l e x i b i l i t y a f f o r d e d by these a l g o r i t h m s a l l o w s one t o gene r a t e many d i f f e r e n t t y p e s of v e l o c i t y s t r u c t u r e s which f i t the o b s e r v a t i o n s . The i n t e r p r e t e r can t h e r e f o r e choose t h a t a l g o r i t h m (and a d j u s t a b l e parameters) which d e v e l o p s an i n t e r v a l v e l o c i t y . s t r u c t u r e c o i n c i d i n g w i t h the d e s i r e d type of g e o l o g i c a l s t r a t i f i c a t i o n . Our methods i n c o r p o r a t e e r r o r s from the o u t s e t and c o n s e q u e n t l y i t i s important t o u n d e r s t a n d t h e i r s o urces and magnitudes. An e x c e l l e n t d e s c r i p t i o n of the p o s s i b l e s o u r c e s of e r r o r a s s o c i a t e d w i t h the e s t i m a t i o n of s t a c k i n g v e l o c i t y i s g i v e n i n H u b r a l and Krey (1980). There a r e two p r i n c i p a l types of e r r o r s : those which r e s u l t i n b i a s e d s t a c k i n g v e l o c i t y e s t i m a t e s , f o r example, s i g n a l bandwidth (Stone, 1974) and 55 s p r e a d - l e n g t h b i a s ( A l - C h a l a b i , 1974); and those which r e s u l t i n random, e r r o r s (such as s t a t i c s , event i n t e r f e r e n c e , e t c ) . In most of the f o l l o w i n g work we s h a l l assume t h a t the t o t a l o b s e r v a t i o n a l i n a c c u r a c y of each datum can be r e p r e s e n t e d by a G a u s s i a n random v a r i a b l e w i t h z e r o mean. However, i n the l i n e a r programming c o n s t r u c t i o n , b i a s e d e r r o r s a r e i n c l u d e d i n a n a t u r a l way. We b e g i n by c a s t i n g the problem i n an a p p r o p r i a t e framework, t h a t of l i n e a r i n v e r s e t h e o r y . The b a s i c e q u a t i o n r e l a t i n g the RMS v e l o c i t y V ( t ) t o the i n t e r v a l v e l o c i t y v ( t ) i s •t V 2 ( t ) = 1/t / v 2 ( u ) d u (4.6) o When the RMS v e l o c i t i e s a r e known on l y a t d i s c r e t e times t j J=1,N, the a p p p r o p r i a t e e q u a t i o n s a r e V* = V 2 ( t ; ) = 1 / t ; / v 2 ( t ) d t j=1,N o (4.7) The o b j e c t i v e of a c o n s t r u c t i o n a l g o r i t h m i s t o f i n d a model v ( t ) which s a t i s f i e s the N e q u a t i o n s (4.7) to a degree c o n s i s t e n t w i t h the assumed s t a t i s t i c a l e r r o r s on V j . T h i s i s e a s i l y a c c o m p l i s h e d by m i n i m i z i n g some norm of the model w h i l e u s i n g e q u a t i o n s (4.7) as c o n s t r a i n t s . A v a r i e t y of u s e a b l e norms are a v a i l a b l e but our s e l e c t i o n i s m o t i v a t e d by a d e s i r e t o c o n s t r u c t a model t h a t has a minimum of s t r u c t u r a l d e t a i l . The two norms used here are 56 *, = .; w(t) |m(t) |dt (4.8) 0 and <t>2 •- / w ( t ) m 2 ( t ) d t (4.9) o where m ( t ) = d v 2 ( t ) / d t (4.10) In these e q u a t i o n s t ^ i s the maximum r e c o r d time, and w(t) i s a p o s i t i v e w e i g h t i n g f u n c t i o n which we s h a l l l a t e r a d j u s t to s u i t our needs. Both and <t>2 i n v o l v e the g r a d i e n t of v 2 ( t ) and hence m i n i m i z i n g e i t h e r of t h e s e w i l l produce a c o n s t r u c t e d v e l o c i t y model w i t h few o s c i l l a t i o n s . To b e g i n the c o n s t r u c t i o n we f i r s t i n t e g r a t e e q u a t i o n (4.7) by p a r t s and use (4.10) to o b t a i n V? - v 2 ( 0 ) = J l/ty(t^-t)H(t£-t)m(t)dt (4.11) where H( t ) i s the H e a v i s i d e s t e p f u n c t i o n . Another datum e q u a t i o n can be found by i n t e g r a t i n g (4.10) t o get v 2 ( t f c ) - v 2 ( 0 ) = / m ( t ) H ( t f e - t ) d t (4.12) E q u a t i o n (4.12) i s important i f some a p r i o r i knowledge about the v e l o c i t y a t time t ^ i s a v a i l a b l e . . I t w i l l p e r mit us to c o n s t r u c t i n t e r v a l v e l o c i t i e s s u b j e c t to p o i n t c o n s t r a i n t s . A c o n s t r u c t e d v e l o c i t y model can be o b t a i n e d by m i n i m i z i n g e i t h e r or <f>2 s u b j e c t t o u s i n g e q u a t i o n s (4.11) and/or (4.12) as c o n s t r a i n t s . The v e l o c i t y o b t a i n e d by m i n i m i z i n g tf>, w i l l be r e f e r r e d t o as the f l a t t e s t L, norm model, w h i l e the model o b t a i n e d by m i n i m i z i n g <t>2 w i l l be c a l l e d the f l a t t e s t model. A v e l o c i t y s t r u c t u r e which w i l l be used as a common example throughout t h i s i n v e s t i g a t i o n i s shown i n F i g . 4-1a. The c o r r e s p o n d i n g RMS v e l o c i t i e s , computed from e q u a t i o n (4.6), and the r e f l e c t i o n seismogram, b a n d l i m i t e d to 10-35 Hz, a r e shown i n f i g . 4 - l b and 4 - l c , r e s p e c t i v e l y . Seventeen RMS data were s e l e c t e d at times c o i n c i d i n g w i t h l a r g e r e f l e c t i o n e v e n t s ; the times and v e l o c i t i e s a r e g i v e n i n T a b l e 4-1 and i n d i c a t e d by arrows i n F i g . 4-1c. To b e g i n the c o n s t r u c t i o n s we s h a l l compute the f l a t t e s t model. T h i s i s a c c o m p l i s h e d by u s i n g a s t a n d a r d s p e c t r a l e xpansion t e c h n i q u e . S i n c e the method i s w e l l known, i t w i l l not be o u t l i n e d h e r e . However, a r e a d e r who i s u n f a m i l i a r with the approach i s r e f e r r e d t o Parker (1977) f o r d e t a i l s and t o Oldenburg and Samson (1979) or OSL (1981) f o r t y p i c a l a p p l i c a t i o n s . The f l a t t e s t model which reproduces the a c c u r a t e RMS d a t a i s shown i n F i g . 4-2a. The t r u e v e l o c i t y i s superposed upon the c o n s t r u c t e d model and i t i s apparent t h a t the f l a t t e s t model i s 5 8 F i g u r e 4-1 The t r u e v e l o c i t y s t r u c t u r e i s shown i n ( a ) . The RMS v e l o c i t i e s c o r r e s p o n d i n g to the v e l o c i t y i n (a) are shown i n ( b ) . The arrows i n d i c a t e those RMS v e l o c i t i e s t h a t w i l l be used i n the i n v e r s i o n . For both (a) and (b) the s c a l e s on the r i g h t are i n k f t / s e c . The r e f l e c t i o n seismogram, b a n d l i m i t e d to 10-35 Hz. i s shown i n (c ) . 59 a good approximation to the t r u e model. The f l a t t e s t model norm e x p r e s s i o n f o r <j>2 ( e q u a t i o n 4.9) i n c l u d e s a w e i g h t i n g f u n c t i o n w ( t ) . That such a f u n c t i o n might be u s e f u l a r i s e s from the f o l l o w i n g o b s e r v a t i o n . The f l a t t e s t model i s t h a t one whose g r a d i e n t (squared) i s as s m a l l as p o s s i b l e . Yet s t a c k i n g v e l o c i t i e s can be o b t a i n e d only when there i s a s i g n i f i c a n t r e f l e c t i o n , and t h a t .is at times c o r r e s p o n d i n g - t o l a r g e g r a d i e n t s i n impedance. T h i s p h y s i c a l i n f o r m a t i o n can be i n c o r p o r a t e d by d e s i g n i n g a weighting f u n c t i o n which i s s m a l l ( s i g n i f i c a n t l y l e s s than u n i t y ) near times t c o r r e s p o n d i n g to s t a c k i n g v e l o c i t y d a t a , and u n i t y at other times. The weig h t i n g f u n c t i o n used here i s a sum of Gaussians. We have chosen w(t) = max{l - I c e x p [ - 7 ( t - t )2'] ,w .} (4.13) where the c o e f f i c i e n t s c c o n t r o l the amplitude of the wei g h t i n g , 7 c o n t r o l s i t s width, and w i s a parameter which pre v e n t s the weig h t i n g from g e t t i n g too s m a l l . In p r a c t i c a l a p p l i c a t i o n s reasonable e s t i m a t e s of c and 7 may be i n f e r r e d d i r e c t l y from the v e l o c i t y a n a l y s i s diagram. For the c u r r e n t example, the e f f e c t s of a d m i t t i n g a Gaussian w e i g h t i n g with c = 0.9 and 7 = 1000 are shown i n F i g . 4-2c. The r e s u l t a n t v e l o c i t y s t r u c t u r e i s more 'blocky' than t h a t given i n F i g . 4-2a, and i s very s i m i l a r to the Dix r e s u l t . T h i s i s expected. As c approaches 1.0 and f o r s u f f i c i e n t l y s m a l l w and 60 s u f f i c i e n t l y l a r g e 7j , l a r g e g r a d i e n t s w i l l be p e r m i t t e d o n l y at the RMS times t j . These a r e the same times at which the v e l o c i t i e s can change w i t h the Dix f o r m u l a t i o n . L a s t l y , the e f f e c t s of i n c o r p o r a t i n g s i x v e l o c i t y c o n s t r a i n t s (see T a b l e 4-1 f o r magnitudes and times) a r e shown i n F i g . 4-2d. No Gaussian w e i g h t i n g was a p p l i e d i n t h i s example. The p r e v i o u s example shows t h a t the f l a t t e s t model c o n s t r u c t i o n can produce good r e s u l t s when the d a t a a r e a c c u r a t e . R e a l data however, ar e always i n a c c u r a t e , and i f an a l g o r i t h m i s t o be u s e f u l i t must be s t a b l e i n the p r e s e n c e of n o i s e . The s t a n d a r d Dix formula f a i l s i n t h i s r e g a r d when a p p l i e d d i r e c t l y to n o i s y d a t a . The a n a l y t i c i n v e r s e t o e q u a t i o n (4.6) i s v ( t ) = V ( t ) {1 + 2 t V ' ( t ) / V ( t ) } (4.14) and the Dix formula i s merely a d i s c r e t i z e d v e r s i o n of (4.14). I n s t a b i l i t y a r i s e s because e r r o r s i n V ( t ) a r e g r e a t l y a m p l i f i e d when e s t i m a t i n g V ( t ) . Moreover, the i n s t a b i l i t e s worsen w i t h i n c r e a s e d r e c o r d time because of the l i n e a r dependence on t i n the r i g h t hand s i d e of (4.14). To i l l u s t r a t e the e f f e c t s of e r r o r s we s h a l l i n v e r t RMS data o b t a i n e d by adding G a u s s i a n random n o i s e to the t r u e data i n T a b l e 4-1. The i n a c c u r a t e d a t a and t h e i r s t a n d a r d d e v i a t i o n s 61 0.0 0.4 0.8 1.2 1.6 t(sec) F i g u r e 4-2 The f l a t t e s t model o b t a i n e d from i n v e r t i n g seventeen a c c u r a t e RMS v e l o c i t i e s i s shown i n ( a ) . That model, as w e l l as a l l o t h e r s i n t h i s f i g u r e , a r e superposed on the t r u e v e l o c i t y s t r u c t u r e . The model gen e r a t e d from the Dix formula i s shown i n (b ) . The v e l o c i t y i n (c) was o b t a i n e d by u s i n g the f l a t t e s t model f o r m u l a t i o n but weighting, the .norm so t h a t l a r g e g r a d i e n t s c o u l d appear at those times c o r r e s p o n d i n g to RMS v e l o c i t y d a t a . The f l a t t e s t model c o n s t r u c t e d from the RMS da t a and s i x a d d i t i o n a l p o i n t v e l o c i t y c o n s t r a i n t s i s shown i n ( d ) . V e l o c i t y Standard deviation Accurate 5.104 rms 5.556 velocities 6.067 6.540 — 6.798 — 7.405 8.043 8.742 — 9.034 9.397 9.736 9.977 — 10.711 — 11.544 — 12.306 12.385 — 12.568 — Inaccurate 5.126 .08 rms 5.521 .08 velocities 6.061 .10 6.547 .10 7.012 .10 7.483 .10 7.879 .13 8.655 .13 9.096 .15 9.675 .15 9.621 .15 9.915 .15 10.791 .17 11.449 .18 12.324 .20 12.583 .20 11429 .20 Point-wise 6.647 .01 velocity 12.828 .o: constraints 11.236 .01 20.278 .01 13.029 .01 16.815 .01 TABLE 4-1 The f i r s t p o r t i o n of the t a b l e c o n t a i n s the seventeen a c c u r a t e RMS v e l o c i t i e s and t h e i r times ( d e p i c t e d by arrows i n F i g 4-1 b) c o r r e s p o n d i n g to the example used i n t h i s c h a p t e r . The second p o r t i o n of the t a b l e c o n t a i n s the seventeen i n a c c u r a t e RMS v e l o c i t i e s and t h e i r standard d e v i a t i o n s . The l a s t p o r t i o n c o n t a i n s the va l u e s , standard d e v i a t i o n s , and times of the s i x po i n t w i s e v e l o c i t y c o n s t r a i n t s used in the i n v e r s i o n s . 63 are a l s o g i v e n i n T a b l e -4-1. The r e s u l t s from i n v e r t i n g w i t h the Dix formula a r e shown i n F i g . 4-3a. C l e a r l y , r e l a t i v e l y s m a l l data e r r o r s have caused s i g n i f i c a n t d e t e r i o r a t i o n i n the c o n s t r u c t e d v e l o c i t y . In p a r t i c u l a r , the Dix model d i s p l a y s s t r u c t u r a l d e t a i l near t=i.O seconds which i s p u r e l y an a r t i f a c t of the a d d i t i v e n o i s e . I f an a l g o r i t h m i s to be r o b u s t , i t must not be r e q u i r e d to f i t i n a c c u r a t e data e x a c t l y . The a l g o r i t h m s h o u l d produce p r e d i c t e d responses Vj , which a r e a c c e p t a b l y c l o s e , but not e x a c t l y e q u a l t o , the o b s e r v a t i o n s Vj 0. We s h a l l assume t h a t the e r r o r s on the RMS v e l o c i t i e s a r e independent and G a u s s i a n w i t h z e r o mean, and t h a t the s t a n d a r d d e v i a t i o n of the j t h datum i s cj . Then X 2 = .'Z {(vP-vf) / c } 2 (4.15) denotes the c h i - s q u a r e d m i s f i t between the o b s e r v a t i o n s and the p r e d i c t e d RMS v e l o c i t i e s . In c o n s t r u c t i o n , we s h o u l d attempt t o g e n e r a t e a model which has a x 2 v a l u e t h a t i s n e i t h e r too b i g nor too s m a l l . The e x p e c t e d v a l u e of x 2 i s a p p r o x i m a t e l y N i f N i s g r e a t e r than about 5. A model t h a t generates a x 2 much l e s s than N reproduces the o b s e r v a t i o n s too w e l l and i t w i l l d i s p l a y f e a t u r e s t h a t a r e merely a r t i f a c t s of the n o i s e . A l t e r n a t i v e l y , a model t h a t g e n e r a t e s a x 2 much g r e a t e r than N has f i t the data too p o o r l y and hence i n f o r m a t i o n about the i n t e r v a l v e l o c i t y which i s c o n t a i n e d i n the data w i l l have been d i s c a r d e d . F i g s . 4-3b and 4-3e i l l u s t r a t e t h i s e f f e c t i v e l y . The c o n s t r u c t e d 64 models have x2 v a l u e s of 36, 17, 11 and 1. S i n c e N=17, the p r e f e r r e d model based upon a x 2 c r i t e r i o n i s t h a t shown i n F i g . 4-3c. That v e l o c i t y i s a smooth r e p r e s e n t a t i o n of the t r u e v e l o c i t y and a comparison w i t h the c o n s t r u c t e d model i n F i g . 4-2b g i v e s some i n s i g h t about the l o s s of i n f o r m a t i o n due to d a t a e r r o r s . We s h a l l now t u r n to the c o n s t r u c t i o n of the L^-norm f l a t t e s t model.. Our g o a l i s to minimize the norm g i v e n i n e q u a t i o n (4.8) s u b j e c t to d a t a c o n s t r a i n t s i n e q u a t i o n (4,11) and (4.12). We do t h i s by f i r s t i n t r o d u c i n g a p a r t i t i o n { 0 = t , , t 2 , . . . t ^ = t m } and p a r a m e t e r i z i n g m(t) such t h a t i t i s a c o n s t a n t m(- on the i'.th p a r t i t i o n element. E q u a t i o n s (4.11) and (4.1.2) can then be w r i t t e n as V 2 ( f ) - v 2 ( 0 ) = X a - m- (4,16) where a i ' j ' " 1 / ^ ./ ( t j - t ) H ( t j - t ) dt and v 2 ( t ; ) - v 2 (0) = I b-. m • (4.17) where b-j = _/ H ( t j - t ) - dt The d i s c r e t i z e d form of the o b j e c t i v e f u n c t i o n i s = I w- |m£ | (4.18) where w^_ i s a weight f o r the i ' t h p a r t i t i o n element. As i n the 65 0.0 0.4 0.8 1.2 1.6 t(sec) F i g u r e 4-3 Seventeen i n a c c u r a t e RMS data a r e i n v e r t e d . The r e s u l t s from the Dix formula are shown i n ( a ) . P r o f i l e s i n (b) - (e) r e p r e s e n t f l a t t e s t model v e l o c i t y c o n s t r u c t i o n s whose x 2 m i s f i t v a l u e s are r e s p e c t i v e l y 36, 17, 11, and 1. The t r u e v e l o c i t y s t r u c t u r e i s superposed upon a l l c u r v e s . 66 f l a t t e s t model c o n s t r u c t i o n , these weights can be a l t e r e d to produce d i f f e r e n t models. Here, however, we s h a l l s e t W.-» ( t : - t . ).. • -v I I-l T h i s c h o i c e n o r m a l i z e s the changes i n d ( v 2 ( t ) ) / d t c o r r e s p o n d i n g to each time i n t e r v a l . The m i n i m i z a t i o n of e q u a t i o n (4.18) s u b j e c t to the c o n s t r a i n t s i n (4.16) and (4.17) i s e a s i l y c a r r i e d out u s i n g l i n e a r programming t e c h n i q u e s . E s t i m a t e d e r r o r s i n the data a r e r e a d i l y i n c o r p o r a t e d by w r i t i n g the c o n s t r a i n t s as i n e q u a l i t i e s . I f t h e r e a r e N d a t a c o n s t r a i n t s , the l i n e a r programming s o l u t i o n w i l l r e t u r n at most N nonzero v a l u e s of m , meaning t h a t the c o n s t r u c t e d v e l o c i t y model w i l l have a t most - N p a r t i t i o n s w i t h nonzero g r a d i e n t s . A l s o by p a r a m e t e r i z i n g more f i n e l y about the RMS t i m e s , t j , we ensure t h a t the model c o n s t r u c t i o n can put i n l a r g e g r a d i e n t s of v 2 ( t ) a t those t i m e s . I t i s e x p e c t e d t h e r e f o r e , t h a t the L, norm f l a t t e s t model w i l l produce good e s t i m a t e s of the t r u e v e l o c i t y s t r u c t u r e when the e a r t h i s l a y e r e d . The L, norm f l a t t e s t model o b t a i n e d by i n v e r t i n g the a c c u r a t e RMS d a t a i s shown i n F i g . 4-4a. I t i s noted t h a t t h i s model i s s i m i l a r t o i t s L 2 c o u n t e r p a r t i n b e i n g g e n e r a l l y a smooth v e r s i o n of the t r u e v e l o c i t y . However, i t has a l s o c o r r e c t l y a c h i e v e d the l a r g e g r a d i e n t s e x p e c t e d near the RMS data t i m e s . The c o n s t r u c t e d model o b t a i n e d by i n c o r p o r a t i n g the a c c u r a t e p o i n t v e l o c i t y c o n s t r a i n t s i s shown i n F i g . 4-4b. 67 The s t a b i l i t y of t h i s a l g o r i t h m i n the presence of n o i s e i s i l l u s t r a t e d i n F i g s . 4-4c and 4-4d. In F i g . 4-4c we have shown the model c o n s t r u c t e d from the i n a c c u r a t e RMS d a t a of T a b l e 4-1. T h i s r e s u l t i s c l e a r l y s u p e r i o r to t h a t of the Dix c o n s t r u c t i o n on the same d a t a , shown i n F i g . 4-3a. Our r e s u l t s a r e f u r t h e r improved by a d d i n g i n a c c u r a t e p o i n t v e l o c i t y c o n s t r a i n t s ; see F i g . 4-4d. 4.'3 : LINEAR APPRAISAL The c o n s t r u c t i o n methods p r o v i d e ways t o o b t a i n a minimum s t r u c t u r e v e l o c i t y which w i l l reproduce the o b s e r v a t i o n s . Yet t h e r e are i n f i n i t e l y many a c c e p t a b l e models and i t i s important to determine what unique i n f o r m a t i o n about the i n t e r v a l v e l o c i t y can be found d i r e c t l y from the d a t a . Such an a p p r a i s a l a n a l y s i s i s c a r r i e d out by u s i n g the methods of Backus and G i l b e r t (1970). Aga i n , t h e i r work i s w e l l known and o n l y the b a r e s t d e t a i l s w i l l be p r e s e n t e d h e r e . The r e a d e r i s r e f e r r e d to Backus and G i l b e r t (1970) f o r a t h e o r e t i c a l b a s i s , and t o Parker (1977) or Oldenburg and Sampson (1.979) f o r a d d i t i o n a l e x p o s i t i o n and examples. Given e q u a t i o n s of the type e- = / v 2 ( t ) - G; ( t ) d t (see e q u a t i o n ( 4 . 7 ) ) , then unique averages of the form 68 Figure 4-4 The Li-norm f l a t t e s t model obtained by inv e r t i n g seventeen accurate RMS v e l o c i t i e s i s shown in (a). The addition-of six accurate point v e l o c i t y constraints i s shown in (b). Analogous re s u l t s with inaccurate data are shown in (c) and (d) res p e c t i v e l y . 69 < v 2 ( t 0 ) > •- / v 2 ( t ) A ( t , t 0 ) d t 0 ' . (4.19) can be g e n e r a t e d by f i n d i n g a s e t of N c o e f f i c i e n t s {aj} such t h a t the k e r n e l f u n c t i o n s , G j ( t ) , can be shaped i n t o an a v e r a g i n g f u n c t i o n c e n t e r e d on t 0 - That i s , we want t o f i n d {aj} so t h a t the a v e r a g i n g f u n c t i o n A ( t , t 0 ) •- Z a : ( t 0 ) G ; ( t ) (4.20) i i s , i n some sense, l i k e a D i r a c d e l t a f u n c t i o n c e n t e r e d on t 0 The v a r i a n c e of < v 2 ( t 0 ) > i s e 2 ( t 0 ) = X af of (4.21) where oj i s the s t a n d a r d d e v i a t i o n of each datum ej . The q u a d r a t i c form t o be m i n i m i z e d to o b t a i n the o p t i m a l {aj} i s <t>(t0) - cos0 - r ( A ( t , t 0 - 5 ( t - t 0 ) 2 dt+sine-2 .a 2a? + X(1- / * A ( t , t 0 ) d t ) (4.22) In e q u a t i o n (4.22), 6 (0 £ (9 £ ir/2) i s the t r a d e o f f parameter used to s a c r i f i c e r e s o l u t i o n i n < v 2 ( t 0 ) > f o r g a i n i n s t a t i s t i c a l a c c u r a c y , and X i s a Lagrange m u l t i p l i e r f o r the unimodular c o n s t r a i n t . E s t i m a t e s of < v 2 ( t 0 ) > and i t s s t a n d a r d e r r o r c ( t 0 ) are a v a i l a b l e f o r any v a l u e of 6. In p r a c t i c a l a p p l i c a t i o n s 70 where the data are i n a c c u r a t e , the v a l u e of 6 s h o u l d always be g r e a t e r than z e r o ; t h a t i s , some r e s o l u t i o n s h o u l d always be s a c r i f i c e d . We s h a l l r e s t r i c t our a t t e n t i o n - to the i n a c c u r a t e d a t a g i v e n i n T a b l e 4-1. The t r a d e o f f diagram f o r those data a r e shown i n F i g . 4-5. The l o s s of r e s o l v i n g power w i t h i n c r e a s i n g r e c o r d time i s c l e a r l y d i s p l a y e d . In f a c t , f o r t<0.5 averages h a v i n g s t a n d a r d d e v i a t i o n l e s s than 0.5 km/sec and h a v i n g r e s o l u t i o n widths e q u a l t o the d i f f e r e n c e between two a d j a c e n t RMS times a r e a v a i l a b l e . T h i s c o r r e s p o n d s t o the g r e a t e s t r e s o l v i n g power p o s s i b l e . For t £ 1.5 and e £ 0.5, the b e s t r e s o l u t i o n i s 0.45 seconds even though a d j a c e n t RMS t i m e s ^ a r e .12 seconds. In F i g . 4-6a we have p l o t t e d t r a d e o f f c u r v e s f o r to-0.55 seconds and t 0 = l . 2 5 seconds. S e l e c t e d a v e r a g i n g f u n c t i o n s are shown i n F i g s . 4-6b and 6c. Those i n F i g . 4-6b c o r r e s p o n d t o (9=0 and those i n F i g . 4-6c each have a width of 0.45 seconds. T h i s s e r v e s t o i l l u s t r a t e how the a v e r a g i n g f u n c t i o n s s p r e a d out as r e s o l u t i o n i s s a c r i f i c e d i n o r d e r t o g a i n s t a t i s t i c a l r e l i a b i l i t y . 7 1 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 t (sec) Figure 4-5 Trade-off diagram for RMS v e l o c i t y inversion. The standard deviation of the average value i s p l o t t e d as the ordinate. Contour values are in seconds and they represent a width of the averaging function evaluated by computing 1 . 0 / A ( t o , t 0 ) • No widths e x i s t for t 0 > 1.78 seconds since t h i s exceeds the maximum time for an RMS datum. Consequently, a l l of the kernel functions are zero beyond t=l.78 and so i s the averaging function. 7.2 2.0 F i g u r e 4-6 T r a d e - o f f c u r v e s f o r t o=0.55 seconds and t 0 = l . 2 5 seconds are shown i n ( a ) . Maximum r e s o l u t i o n a v e r a g i n g f u n c t i o n s f o r these two times are shown i n (b) . A v e r a g i n g f u n c t i o n s which have a width of 0.45 seconds are shown i n ( c ) . 73 4.4 : DERIVATION OF THE IMPEDANCE CONSTRAINTS The LP or AR broadband r e f l e c t i v i t y c o n s t r u c t i o n a l g o r i t h m s can i n c o r p o r a t e c o n s t r a i n t s of the form J l l ^ i = 7k± 5-yh (4.23) where r^ are the unknown r e f l e c t i v i t y c o e f f i c i e n t s i n a time window ( t , , t 2 ) , i.lk a r e a s e t of c o n s t a n t s f o r the k'th c o n s t r a i n t , 7^ i s the c o n s t r a i n t v a l u e , and 67^ i s an e s t i m a t e d e r r o r . Our g o a l i s t o use the RMS v e l o c i t i e s V ( t j ) , j = i , . . . N and t h e i r e s t i m a t e d e r r o r s t o f i n d c o n s t r a i n t s i n the form of e q u a t i o n (4.23). We s h a l l a c c o m p l i s h t h i s by f i r s t c h o o s i n g a time t 0 i n the i n t e r v a l ( t , , t 2 ) and then computing the Backus-G i l b e r t l o c a l i z e d averages < v 2 ( t 0 ) > •• J v 2 ( t ) A ( t , t 0 ) d t "(4.24) o E q u a t i o n (4.24) w i l l - t h e n be m a n i p u l a t e d so t h a t i t has the form of e q u a t i o n (4.23). The d e s i r e d s e t of c o n s t r a i n t s to be i n p u t i n t o the c o n s t r u c t i o n a l g o r i t h m s i s o b t a i n e d by c a r r y i n g t h i s p r o c e d u r e out f o r a number of times t 0 w i t h i n the i n t e r v a l ( t , , t 2 ) . We begin w i t h the l i n e a r i z e d form of the a c o u s t i c impedance, 74 i •|(t) = $(0) exp[2-J r ( u ) d u ] o (4.25) and d e f i n e t v(t) = l n { | ( t ) / $ ( 0 ) } . - 2-S r ( u ) du 0 (4.26) In o r d e r t o proceed f u r t h e r , we need t o determine the r e l a t i o n s h i p between v e l o c i t y and impedance. T h i s s p e c i f i c a t i o n depends upon the assumptions made c o n c e r n i n g the d e n s i t y and u s u a l l y one of two' c h o i c e s i s s e l e c t e d . : E i t h e r the d e n s i t y v a r i a t i o n s a r e assumed to be unimportant, i n which case £(t) i s se t e q u a l t o v(.t) and hence v ( t ) = v(0)-exp[i?(t)'] (4.27) or the d e n s i t y i s assumed to depend upon a power of the v e l o c i t y . That i s , p ( t ) - C - v * ( t ) . (4.28) where C i s a c o n s t a n t . The v a l u e of a i s o f t e n chosen t o be near 0.25. I f e q u a t i o n (4.28) i s assumed then v ( t ) = v(0) exp[pi?(t) ] 7 5 (4.29) where p=l/0+a). C l e a r l y (4.27) i s a s p e c i a l form of (4.29) and so we s h a l l keep the g e n e r a l i t y a f f o r d e d by the power law r e p r e s e n t a t i o n and l e t p ^ l when d e n s i t y v a r i a t i o n s a r e t o be n e g l e c t e d . By s q u a r i n g e q u a t i o n (4.29), and s u b s t i t u t i n g i n t o (4.24) we get < v 2 ( t o ) > = ./ v 2 ( 0 ) exp[2pT?(t)] A(t,t 0)4i ± 6 < > (4.30) where 6 < i > i s the e r r o r i n < v 2 ( t 0 ) > a r i s i n g from o b s e r v a t i o n a l u n c e r t a i n t i e s . A f t e r expanding the e x p o n e n t i a l and u s i n g the u n i m o d u l a r i t y of the a v e r a g i n g -function t h i s may be w r i t t e n as J"\?(t)-A(t,.t 0). dt = l/2p{ < v 2 ( t o ) > / v 2 ( 0 ) - 1 -I (.2p)*/k! f~T,k{t) A ( t , t 0 ) dt} ± 6 o / 2 p v 2 ( 0 ) (4.31 ) The i n t e r g r a l on the l e f t can be w r i t t e n as ; Tf(t) A ( t , t 0 ) d t - I , + / T j ( t ) - A ( t , t 0 ) . dt + I 2 6 t, (4.32) where 76 .1 i - J i?(t)- A ( t , t 0 ) dt and I i = J i?(t)- A ( t , t 0 ) - dt By s u b s t i t u t i n g e q u a t i o n (4.26) i n t o (4.32) and i n t e r c h a n g i n g the o r d e r of i n t e g r a t i o n we get J r ? ( t ) - A ( t , t 0 ) dt - I, + l 2 + I 3 + 2-S r ( u ) £ ( u , t 0 ) du (4.33) where <2 I j " v(t,). J A ( t , t 0 ) - d t and £ ( u , t 0 ) = / A ( t , t 0 ) - d t E q u a t i o n (4.33) can now be w r i t t e n as ±2 S r(u)-£(u,t 0) du = -1/2 {I, + I 2 + I 3} + 1/4p { < v 2 ( t o ) > / v ' ( 0 ) - 1 - I (2p)*.L^} ± 6 o / 4 p v a ( 0 ) (4,34) where 77 L = 1/k! / IJ ( t ) A ( t , t 0 ) dt ° (4.35) A d i s c r e t i z e d v e r s i o n of the i n t e g r a l on the l e f t hand s i d e of e q u a t i o n (4.34) i s of the form r e q u i r e d f o r the c o n s t r a i n t . The only d i f f i c u l t y t h a t remains i s to e v a l u a t e the r i g h t hand s i d e of that e q u a t i o n . There are some s p e c i a l c i r c u m s t a n c e s i n which t h i s e v a l u a t i o n i s e a s i l y c a r r i e d out. I f the a v e r a g i n g f u n c t i o n i s c o n f i n e d t o the r e g i o n ( t , , t 2 ) , and i f r,(t) i s s m a l l enough so t h a t terms L K can be n e g l e c t e d f o r k>2, then the r i g h t hand s i d e of equation (4.34) reduces t o - 0 / 2 p ) nit,) + 1/4p { < v 2 ( t , ) > / v 2 ( 0 ) - 1 } ± 6^/4p v 2 ( 0 ) (4.36) T h i s q u a n t i t y i s determined i f the impedance a t t , i s known. U n f o r t u n a t e l y , these c o n d i t i o n s r a r e l y occur i n p r a c t i c e . U s u a l l y , the a v e r a g i n g f u n c t i o n cannot be co m p l e t e l y c o n f i n e d t o the r e g i o n ( t , , t 2 ) . A l s o , the impedance i s o f t e n observed t o vary by a f a c t o r of 2 or more i n r e a l i s t i c data s e c t i o n s . T h i s means that TJ i s g r e a t e r than In 2, and hence the power s e r i e s r e p r e s e n t a t i o n r e q u i r e s more than j u s t the l i n e a r term; tha t i s , the i n t e g r a l s L k cannot be n e g l e c t e d . The b a s i c d i f f i c u l t y then i s t h a t an e v a l u a t i o n of the r i g h t hand s i d e of e q u a t i o n (4.34) r e q u i r e s knowledge of 7?(t), or e q u i v a l e n t l y the model f o r which we are attempting to s o l v e . Our c h o i c e s are to r e f o r m u l a t e the problem and s o l v e the f u l l y 78 n o n l i n e a r problem or to attempt to f i n d a good e s t i m a t e of the r i g h t hand s i d e through some other means. We s h a l l adopt the l a t t e r c h o i c e and a c c o m p l i s h our o b j e c t i v e by u s i n g the RMS v e l o c i t i e s and i n t e r v a l v e l o c i t y c o n s t r a i n t s to c o n s t r u c t a best e s t i m a t e , v c ( t ) , of the t r u e e a r t h v e l o c i t y v ( t ) . The p r e v i o u s s e c t i o n of t h i s paper showed t h a t the c o n s t r u c t e d models can r e a s o n a b l y be expected t o emulate the broad s c a l e f e a t u r e s of v(t). but w i l l e x h i b i t few, i f any, of the r a p i d changes of v e l o c i t y t h a t a r e so important t o the f i n a l g e o l o g i c a l i n t e r p r e t a t i o n . N e v e r t h e l e s s , such a v c ( t ) w i l l s u f f i c e here s i n c e the c o n s t r a i n t s i n (4.34) i n v o l v e o n l y i n t e g r a l s of r ? ( t ) . In the f o l l o w i n g examples we have e l e c t e d t o use the f l a t t e s t model w i t h a w e i g h t i n g f u n c t i o n of u n i t y . As a type example, we s h a l l i n v e r t the bandlimi.ted seismogram i n F i g . 4-1 and attempt t o r e c o v e r the v e l o c i t y s t r u c t u r e i n the t i m e window (0.4,1.4) seconds. The r e s u l t s of a p p l y i n g an u n c o n s t r a i n e d LP a l g o r i t h m t o the seismogram w i t h i n the window of i n t e r e s t are shown i n F i g 4-7a. The f a i l u r e of the c o n s t r u c t i o n i s d r a m a t i c a l l y i l l u s t r a t e d by s u p e r p o s i n g the t r u e v e l o c i t y on the LP s o l u t i o n . I t s h o u l d be p o i n t e d out t h a t t h i s example was c o n t r i v e d t o make the LP s o l u t i o n f a i l ; n e v e r t h e l e s s , the chosen v e l o c i t y s t r u c t u r e does not seem u n r e a l i s t i c and we can a p p r e c i a t e t h a t s i m i l a r f a i l u r e s of the LP or AR a l g o r i t h m s may occur i n p r a c t i c e . I n f o r m a t i o n from the seventeen a c c u r a t e RMS v e l o c i t i e s i n T a b l e 4-1 was added i n the f o l l o w i n g manner. F i r s t we g e n e r a t e d h i g h r e s o l u t i o n a v e r a g i n g f u n c t i o n s (0=0) and o b t a i n e d the 79 c o r r e s p o n d i n g v a l u e s of < v 2 ( t 0 ) > at twelve e q u a l l y spaced v a l u e s of t 0 i n the i n v e r v a l (0.4, 1.4). The f l a t t e s t model ( F i g . 4-2a) was c o n s t r u c t e d , and the c o n s t r a i n t s of e q u a t i o n (4.34) were c a l c u l a t e d . The LP s o l u t i o n was computed and the v e l o c i t y was r e c o v e r e d by u s i n g the s t a n d a r d r e c u r s i o n f o r m u l a . That v e l o c i t y i s shown i n F i g 4-7b. The e x c e l l e n t agreement w i t h the t r u e v e l o c i t y i l l u s t r a t e s the importance of i n c l u d i n g i n f o r m a t i o n from RMS v e l o c i t i e s . The c o r r e s p o n d e n c e between the i n t e r v a l v e l o c i t y r e c o v e r e d from the LP c o n s t r u c t i o n and the f l a t t e s t model from which c o n s t r a i n t s were g e n e r a t e d i s shown i n F i g . 4-7c. We have shown how an a p p r a i s a l a n a l y s i s which ge n e r a t e s l i n e a r averages of < v 2 ( t ) > can be used to p r o v i d e c o n s t r a i n t s f o r the c o n s t r u c t i o n a l g o r i t h m s . T h i s average has the form of e q u a t i o n ( 4 . 1 9 ) and i t i s unique i n t h a t any model v 2 ( t ) which reproduces the RMS d a t a w i l l have the same average when i t s i n n e r p r o d u c t i s taken w i t h the a v e r a g i n g f u n c t i o n A ( t ) . There i s , however, another p o s s i b l i t y f o r computing an average. E q u a t i o n (4.10) can be w r i t t e n as ej= J R j ( t ) - v ( t ) d t J=1,...N (4.37) where R j ( t ) = G j ( t ) - v c ( t ) a r e m o d i f i e d k e r n e l f u n c t i o n s and v c ( t ) i s a c o n s t r u c t e d v e l o c i t y f u n c t i o n . U s i n g e q u a t i o n (4.37), i t i s p o s s i b l e to o b t a i n B a c k u s - G i l b e r t averages of the v e l o c i t y r a t h e r than v e l o c i t y s quared. That i s , 80 F i g u r e 4-7 The t r u e v e l o c i t y s t r u c t u r e , and the r e c o v e r e d i n t e r v a l v e l o c i t y when no RMS " c o n s t r a i n t s were used, are shown i n ( a ) . The i n v e r s i o n was c a r r i e d out o n l y on the time window (0.4-1.4) seconds. Shown i n (b) i s the r e c o v e r e d i n t e r v a l v e l o c i t y when RMS c o n s t r a i n t s u s i n g averages < v 2 ( t 0 ) > were i n c o r p o r a t e d i n t o the i n v e r s i o n . The r e c o v e r e d v e l o c i t y i n the time window (0.4-1.4) seconds i s i n s e r t e d i n t o the f l a t t e s t model o b t a i n e d by i n v e r t i n g the RMS data a l o n e . The e n t i r e p r o f i l e (0.0-2.0) seconds i s then superposed upon the t r u e v e l o c i t y s t r u c t u r e . The co r r e s p o n d e n c e between the f l a t t e s t model and the LP r e c o n s t r u c t i o n i s shown i n (c) where these two f u n c t i o n s have been superposed. The r e s u l t s i n (d) and (e) are analagous to those i n (b) and (c) except t h a t averages < v ( t 0 ) > were used to generate c o n s t r a i n t s f o r the l i n e a r programming a l g o r i t h m . 81 < v ( t 0 ) > = / v ( t ) - A ( t f . t 0 ) ' d t o (4.38) where the a v e r a g i n g f u n c t i o n A ( t ) i s now a l i n e a r c ombination of the k e r n e l s R j ( t ) . The i n t e r p r e t a t i o n of the uniqueness of < v ( t 0 ) > i s t h a t a l l models which a r e l i n e a r l y c l o s e t o v t ( t ) w i l l have the same average < v ( t 0 ) > . I f v c ( t ) i s a good e s t i m a t e f o r the g r o s s e a r t h v e l o c i t y , t h i s a d d i t i o n a l r e s t r i c t i o n s h o u l d prove b e n e f i c i a l . M a n i p u l a t i n g (4.38) i n the same manner as used t o generate e q u a t i o n (4.34) produces / r ( u ) -jHu/to) -du •- -1/2 {I, -+ 1 2 + i 3 } + i' <*> k l/2p {<v(t o)>/v(0) - 1 •+ I p -LK} ± 6 o / 2 p v 2 ( 0 ) . (4.39) where 8 0 i s the e r r o r of < v ( t 0 ) > . A g a i n , the d i s c r e t i z e d form, of the l e f t hand s i d e y i e l d s the d e s i r e d c o n s t r a i n t . A comparison of e q u a t i o n (4.39) w i t h (4.34) i l l u s t r a t e s another advantage of the n o n l i n e a r form of the B a c k u s - G i l b e r t a v e r a g e s . The i n t e g r a l s i n the i n f i n i t e summation i n (4.39) a r e no l o n g e r p r e m u l t i p l i e d by 2 as they were i n e q u a t i o n (4.34). Thus fewer terms need be taken, and computations are t h e r e b y reduced. The r e s u l t s of u s i n g c o n s t r a i n t s based upon averages <v(t)> are shown i n F i g . 4-7d. The r e c o n s t r u c t e d v e l o c i t y has been superposed upon the t r u e v e l o c i t y and agreement between the two 82 f u n c t i o n s i s good. We a l s o note t h a t f o r t h i s example t h e r e appears t o be l i t t l e d i f f e r e n c e i n the r e s u l t s from u s i n g <v(t)> r a t h e r than < v 2 ( t ) > . We do not know whether t h i s i s a g e n e r a l statement but i t appears t o be so i n the examples t h a t we have done. Because of the advantages a l r e a d y s t a t e d , the remainder of the paper w i l l use o n l y averages <v(t)> t o generate the c o n s t r a i n t s . The example i n F i g . 4-7 shows t h a t h i g h q u a l i t y c o n s t r a i n t s can be g e n e r a t e d from RMS v e l o c i t i e s which ar e a c c u r a t e . Yet such p r e c i s e d a t a are never a v a i l a b l e i n p r a c t i c e and i t i s important t o i n v e s t i g a t e the e f f e c t s of d a t a i n a c c u r a c i e s . R e a l i s t i c c o n s t r a i n t s must be w r i t t e n as 7. - 67- £ I l r r L S 7: •+ 67 • (4.40) where 67J i s an e r r o r e s t i m a t e f o r 7 j . An e x a m i n a t i o n of (4.39) shows t h a t o b s e r v a t i o n a l e r r o r s i n c r e a s e 57j i n two ways. F i r s t , the averages < v ( t 0 ) > (or < v 2 ( t 0 ) > ) ar e i n a c c u r a t e because of s t a t i s t i c a l e r r o r s i n the d a t a . The s t a n d a r d d e v i a t i o n of < v ( t 0 ) > i s g i v e n by e q u a t i o n (4.21) and t h i s must be i n c o r p o r a t e d as an e r r o r i n the r i g h t hand s i d e of (4.39). S e c o n d l y , the e v a l u a t i o n of the i n t e g r a l s . I , , I 2 or L ^ r e q u i r e s t h a t a good r e p r e s e n t a t i o n of the low f r e q u e n c y components of v ( t ) be p r e s e n t i n the c o n s t r u c t e d v e l o c i t y v c ( t ) . Yet from S e c t i o n 4.2, i t i s expected t h a t s i m i l a r i t i e s between v c ( t ) and v ( t ) w i l l be degraded as the e r r o r s i n c r e a s e . The e r r o r s i n e v a l u a t i n g the i n t e g r a l s , i n c u r r e d by u s i n g v c ( t ) r a t h e r than 83 the t r u e v ( t ) , s h o u l d a l s o be i n c l u d e d i n the § 7 j i n e q u a t i o n (4.40). For the work done here we s h a l l assume t h a t t h i s c o n t r i b u t i o n t o the e r r o r can' be i g n o r e d and we w i l l c o n c e n t r a t e upon the s t a t i s t i c a l e r r o r s a r i s i n g from the B a c k u s - G i l b e r t a p p r a i s a l . At each time t 0 , many averages < v ( t 0 ) > , t h e i r s t a n d a r d d e v i a t i o n s , e ( t 0 ) # and t h e i r a s s o c i a t e d a v e r a g i n g f u n c t i o n s , A ( t , t 0 ) , are a v a i l a b l e . The q u e s t i o n a r i s e s as t o which average p r o v i d e s ~the most u s e f u l c o n s t r a i n t . Choosing an average w i t h a l a r g e s t a n d a r d d e v i a t i o n can make 6 7 - so l a r g e t h a t the c o n s t r a i n t becomes impotent. A l t e r n a t i v e l y forming averages which a r e v e r y a c c u r a t e may r e q u i r e the s a c r i f i c e of so much r e s o l u t i o n t h a t i n f o r m a t i o n c o n t a i n e d i n the RMS d a t a has been n e e d l e s s l y l o s t . I t i s d o u b t f u l t h a t a s t r a t e g y e x i s t s which w i l l work o p t i m a l l y i n a l l c a s e s , but the f o l l o w i n g a t t a c k i s r e a s o n a b l e . Of p r i m a r y c o n c e r n i s the e r r o r on the c o n s t r a i n t . To keep t h i s s u f f i c i e n t l y s m a l l we r e q u i r e t h a t < v ( t 0 ) > have a r e l a t i v e e r r o r no g r e a t e r than a p e r c e n t . I f no average e x i s t s w i t h t h i s e r r o r , then the average h a v i n g two times the minimum s t a n d a r d d e v i a t i o n (0 = ir/2) i s s e l e c t e d . The r e s u l t s i n F i g s . 4-8b and 8c show the r e c o v e r e d i n t e r v a l v e l o c i t i e s when a = 25% and 5%. They are not g r e a t l y d i f f e r e n t .and hence the v a l u e of a does not appear to be too c r i t i c a l . The r e c o v e r e d v e l o c i t i e s i n F i g s . 4-8b and 8c are g r e a t l y improved over those i n F i g . 4-7a where no RMS c o n s t r a i n t s were used, but as expected, the agreement between the t r u e and 84 F i g u r e 4-8 The f l a t t e s t i n t e r v a l v e l o c i t y model o b t a i n e d by i n v e r t i n g the i n a c c u r a t e RMS c o n s t r a i n t s i n T a b l e 4-1 i s shown i n ( a ) . I t i s superposed upon the t r u e v e l o c i t y . F i g u r e s (b) - (c) r e s p e c t i v e l y show the i n t e r v a l v e l o c i t y r e c o v e r e d from the LP s o l u t i o n when averages < v ( t 0 ) > have a r e l a t i v e e r r o r of a = 25% and 5%. F i g u r e (d) shows the f l a t t e s t model when the RMS e r r o r s i n T a b l e 4-1 are t r e b l e d . The LP c o n s t r u c t i o n , u s i n g these erroneous d a t a and a=25%, i s shown i n ( 8 e ) . 85 r e c o v e r e d v e l o c i t i e s i s not as good as when the data were a c c u r a t e . ' To see how the d e g r a d a t i o n i n c r e a s e s w i t h i n c r e a s e d e r r o r s on the RMS d a t a we have redone the c a l c u l a t i o n s a f t e r t r e b l i n g the e r r o r s i n T a b l e 4-1. The r e s u l t s a r e shown i n F i g s . 4-8d and 8e. They ar e somewhat d i s a p p o i n t i n g ; major d i s c r e p a n c i e s between the t r u e and r e c o n s t r u c t e d v e l o c i t i e s are a p p a rent, p a r t i c u l a r l y i n the i n t e r v a l 1.0 - 1.4 seconds. I t appears t h a t the s t r a t e g y o u t l i n e d so f a r can work w e l l when the d a t a . a r e r e a s o n a b l y a c c u r a t e . However, the u s e f u l n e s s of the RMS d a t a d i m i n i s h e s r a p i d l y i f the e r r o r s become too l a r g e . When l a r g e e r r o r s are t o be contended w i t h , i t i s b e s t t o a l t e r s l i g h t l y the above approach. We propose the f o l l o w i n g s t e p s . We f i r s t c o n s t r u c t a best e s t i m a t e f o r the i n t e r v a l , v e l o c i t y by i n v e r t i n g RMS v e l o c i t i e s a l o n g w i t h p o i n t v e l o c i t y c o n s t r a i n t s i f they are a v a i l a b l e . Next we form a h i g h r e s o l u t i o n a v e r a g i n g f u n c t i o n f o r an a r b i t r a r y time - t 0 i n the same manner as b e f o r e . However, i n s t e a d of u s i n g < v ( t 0 ) > c a l c u l a t e d from e q u a t i o n (4.38), we compute <v ( t 0 ) > - S v c ( t ) A ( t , t 0 ) dt o where v c ( t ) i s the c o n s t r u c t e d i n t e r v a l v e l o c i t y model. In a d d i t i o n , we a s s i g n a r e l a t i v e e r r o r of v p e r c e n t t o < v c ( t ) > where v depends upon how c l o s e l y we wish the f i n a l c o n s t r u c t e d model t o l o o k l i k e v c ( t ) . In e f f e c t , t h i s approach f o r c e s the f i n a l v e l o c i t y o b t a i n e d from the LP s o l u t i o n to look l i k e the 8 6 c o n s t r u c t e d model. I t must be understood t h a t the r e s u l t s o b t a i n e d from t h i s method a r e l e s s c e r t a i n than those o b t a i n e d from the i n i t i a l s t r a t e g y . There w i l l be o t h e r models which s a t i s f y the RMS v e l o c i t i e s and the r e f l e c t i o n seismogram c o n s t r a i n t s , but we are r e s t r i c t i n g our best guess t o be t h a t one which l o o k s most l i k e v c ( t ) . T h i s means t h a t i f v £ ( t ) i s a good a p p r o x i m a t i o n t o v ( t ) , our f i n a l v e l o c i t y s t r u c t u r e s h o u l d be c l o s e t o the t r u e e a r t h v e l o c i t y , but i f v c ( t ) i s g r e a t l y wrong, then our r e s u l t s w i l l be i n e r r o r . In F i g . 4-9 we show the r e s u l t s of u s i n g averages < v £ ( t ) > to d e v e l o p the LP c o n s t r a i n t s . The LP v e l o c i t y r e c o v e r e d a f t e r a s s i g n i n g an e r r o r of v - 5% i s shown i n F i g . 4-9b. The r e s u l t i s c o n s i d e r a b l y b e t t e r than t h a t shown i n F i g . 4-8e which began with the same erroneous RMS d a t a -I t i s o b v i o u s t h a t the c u r r e n t s t r a t e g y w i l l - w o r k b e t t e r as v c ( t ) becomes c l o s e r t o the t r u e v ( t ) . Improved r e s u l t s s h o u l d t h e r e f o r e be e xpected i f p o i n t v e l o c i t y c o n s t r a i n t s a r e i n c l u d e d i n the i n v e r s i o n . The degree of improvement i s shown i n F i g s . 4-9c and 9d where s i x i n t e r v a l v e l o c i t y c o n s t r a i n t s (see T a b l e 4-1 f o r l o c a t i o n s and v a l u e s ) were i n c l u d e d i n the c o n s t r u c t i o n of v _ ( t ) . The agreement between the t r u e and r e c o v e r e d i n t e r v a l v e l o c i t i e s i n F i g . 4-9d i s e x c e l l e n t . I t i s perhaps too much t o expect t h a t s i x v e l o c i t y c o n s t r a i n t s might be known. More commonly, o n l y the v e l o c i t y at a l a r g e r e f l e c t o r (perhaps c o i n c i d i n g w i t h a major u n c o n f o r m i t y ) might be a v a i l a b l e . With t h i s i n mind the a n a l y s i s was redone .87 I i 1 1 1 1 1 1 f -0.0 0.4 0.8 1.2 1.6 t (sec) F i g u r e 4-9 The f l a t t e s t model o b t a i n e d by i n v e r t i n g the i n a c c u r a t e RMS data i n T a b l e 1 i s shown i n ( a ) . That v e l o c i t y i s used to form averages <v(t)> which are a r b i t r a r i l y a s s i g n e d an e r r o r of v=5h and then used as c o n s t r a i n t s i n the LP s o l u t i o n . The r e s u l t s are shown i n ( b ) . The e f f e c t s of adding s i x i n a c c u r a t e p o i n t v e l o c i t y c o n s t r a i n t s i n t o the c o n s t r u c t i o n of the i n t e r v a l v e l o c i t y i s shown i n (c) and ( d ) . The f l a t t e s t model i n t e r v a l v e l o c i t y has been reproduced i n (c) and the v e l o c i t y r e c o v e r e d from the LP c o n s t r u c t i o n i s shown i n ( d ) . F i g u r e s (e) and ( f ) are the same as (c) and (d) except t h a t only one v e l o c i t y c o n s t r a i n t (at t=1.4 seconds) has been used i n the i n t e r v a l v e l o c i t y c o n s t r u c t i o n . 8 8 a f t e r u s i n g o n l y the s i n g l e v e l o c i t y c o n s t r a i n t a t 1.4 seconds. The c o n s t r u c t e d v e l o c i t y v c ( t ) i s shown i n F i g . 4-9e and the i n t e r v a l v e l o c i t y o b t a i n e d from the LP a l g o r i t h m i s shown i n F i g . 4-9f. The agreement between the two v e l o c i t i e s i s v e r y good. •4.5 SUMMARY The p r i m a r y g o a l of t h i s c h a p t e r has been t o i n c o r p o r a t e i n f o r m a t i o n from s t a c k i n g v e l o c i t i e s d i r e c t l y i n t o the LP or AR c o n s t r u c t i o n a l g o r i t h m s . To a c c o m p l i s h t h i s we c o n s i d e r e d two s e p a r a t e i n v e r s e problems. In the f i r s t problem, RMS v e l o c i t i e s as w e l l as p o i n t v e l o c i t y c o n s t r a i n t s were i n v e r t e d to f i n d a low fre q u e n c y i n t e r v a l v e l o c i t y . Our p h i l o s o p h y has been t o c o n s t r u c t o n l y v e l o c i t i e s h a v i n g a minimum of s t r u c t u r e and we have a c h i e v e d t h i s g o a l by m i n i m i z i n g the g r a d i e n t of the squared v e l o c i t y i n a 1-norm or 2-norm sense. Some f l e x i b i l i t y has been i n t r o d u c e d by i n c l u d i n g a w e i g h t i n g f a c t o r i n the norm to be m i n i m i z e d . The r a t i o n a l e f o r the w e i g h t i n g was t h a t s t a c k i n g v e l o c i t i e s a r e o b t a i n e d o n l y a t those times c o r r e s p o n d i n g t o s i g n i f i c a n t r e f l e c t i o n s and hence the c o n s t r u c t e d model s h o u l d be a l l o w e d t o have l a r g e g r a d i e n t s a t those t i m e s . Both the L^-norm f l a t t e s t model, and the L 2-norm f l a t t e s t model w i t h a G a u s s i a n w e i g h t i n g , w i l l generate models of t h i s t y pe. I t i s d i f f i c u l t to say which norm and which 89 w e i g h t i n g w i l l produce the best r e s u l t s . Most l i k e l y t h e r e are data s e t s and p a r t i c u l a r g e o l o g i c environments i n which e i t h e r a l g o r i t h m w i l l be s u p e r i o r to the o t h e r . For the work c a r r i e d out here we have used o n l y the L 2-norm f l a t t e s t model without any w e i g h t i n g and have found i t t o p e r f o r m s a t i s f a c t o r i l y . The important improvement of our c o n s t r u c t i o n a l g o r i t h m s over the Dix formula i s one of s t a b i l i t y . The Dix formula i s a d i s c r e t i z e d v e r s i o n of the a n a l y t i c i n v e r s e , and i s i n h e r e n t l y u n s t a b l e . O s c i l l a t i o n s i n the r e c o v e r e d v e l o c i t y w i l l always appear when the data are i n a c c u r a t e because of the need t o f i n d the time d e r i v a t i v e of the RMS v e l o c i t y . A l t e r n a t i v e l y , the a l g o r i t h m s p r e s e n t e d here f i t the o b s e r v a t i o n s o n l y t o w i t h i n a degree j u s t i f i e d ' by t h e i r e r r o r . C o n sequently, s m a l l s c a l e s t r u c t u r a l d e t a i l t h a t may be an a r t i f a c t of the n o i s e i s g e n e r a l l y not i n c l u d e d i n our models. A l s o our methods admit the d i r e c t i n c o r p o r a t i o n of p o i n t v e l o c i t y c o n s t r a i n t s i n t o the i n v e r s i o n . T h i s has no c o u n t e r p a r t i n the Dix f o r m u l a . In the second i n v e r s e problem, we use the RMS v e l o c i t i e s t o e x t r a c t unique i n f o r m a t i o n about the average v a l u e of the v e l o c i t y or i t s square. The e x p r e s s i o n f o r these averages i s m a n i p u l a t e d i n t o a form t h a t can be h andled by the LP and AR a l g o r i t h m s . The methods developed i n the paper seem t o work w e l l when the RMS v e l o c i t i e s are r e a s o n a b l y a c c u r a t e . However, i f the o b s e r v a t i o n a l u n c e r t a i n t i e s become too l a r g e , the e r r o r i n the c o n s t r a i n t v a l u e may i n c r e a s e to the p o i n t where the c o n s t r a i n t i s i n e f f e c t i v e . Attempts to reduce the e r r o r s by i n c r e a s i n g the 9 0 width of the a v e r a g i n g f u n c t i o n may not be p r o d u c t i v e i f too much r e s o l u t i o n must be l o s t to a c h i e v e the d e s i r e d a c c u r a c y . In such c i r c u m s t a n c e s an a l t e r n a t e s t r a t e g y must be developed. We have proposed t h a t the c o n s t r a i n t s s u p p l i e d to the .LP or AR a l g o r i t h m s be g e n e r a t e d u s i n g h i g h r e s o l u t i o n a v e r a g i n g f u n c t i o n s . However, the l a r g e c a l c u l a t e d u n c e r t a i n t i e s i n the a s s o c i a t e d c o n s t r a i n t v a l u e s s h o u l d be i g n o r e d , and new a r t i f i c i a l l y .small e r r o r s a s s i g n e d . By d o i n g t h i s we a r e presuming t h a t the c a l c u l a t e d smooth v e r s i o n of the i n t e r v a l v e l o c i t y i s ' c l o s e r ' t o the t r u e v e l o c i t y than the e r r o r s and the nonuniqueness would t e c h n i c a l l y i n d i c a t e . In a d d i t i o n t o the RMS d a t a , any a v a i l a b l e bounds on v e l o c i t i e s a t s p e c i f i c t imes s h o u l d be i n c o r p o r a t e d i n t o t h e i n v e r s i o n . . The norm t o be min i m i z e d and t h e w e i g h t i n g f u n c t i o n s h o u l d be chosen by t h e i r a b i l i t y to produce the r i g h t 'type' of v e l o c i t y f u n c t i o n f o r the p a r t i c u l a r g e o l o g i c a l environment. When an e n t i r e s e c t i o n i s t o be i n v e r t e d i t may be a d v i s a b l e to i n v e r t s t a c k i n g v e l o c i t i e s from many t r a c e s a c r o s s the s e c t i o n , smooth the o u t p u t , and use these r e s u l t s as a best guess f o r the low f r e q u e n c y v e l o c i t y s t r u c t u r e . The r e c o v e r y of the a c o u s t i c impedance may then c o n t i n u e a f t e r forming averages of the i n t e r v a l v e l o c i t y over r e a s o n a b l e r e s o l u t i o n w i d t h s . These averages can be a s s i g n e d e r r o r s i n accordance w i t h the c o n f i d e n c e t h a t the i n t e r p r e t e r has i n h i s c o n s t r u c t e d low frequency v e l o c i t y . The computations to o b t a i n the n u m e r i c a l v a l u e of the c o n s t r a i n t and the a p p r o p r i a t e l i n e a r combination of the r e f l e c t i o n c o e f f i c i e n t s can proceed e x a c t l y as b e f o r e . 91 The LP or AR a l g o r i t h m s can then be used to c o n s t r u c t a best e s t i m a t e of the f u l l band r e f l e c t i v i t y and s u b s t i t u t i o n of those r e s u l t s i n t o the s t a n d a r d r e c u r s i o n formula produces a b e s t estimate- of the impedance. I t i s now seen t h a t the r e c o v e r y of the a c o u s t i c impedance from the band l i m i t e d seismograms i n v o l v e s f o u r l i n e a r i n v e r s e problems. The' f i r s t i s a p p r a i s a l d e c o n v o l u t i o n t o determine unique averages of the r e f l e c t i v i t y f u n c t i o n . The i n c l u s i o n of i n f o r m a t i o n from s t a c k i n g v e l o c i t i e s r e q u i r e s the c o n s t r u c t i o n of a low frequency i n t e r v a l v e l o c i t y which i s c o n s i s t e n t w i t h e s t i m a t e d RMS v e l o c i t i e s and p o i n t v e l o c i t y c o n s t r a i n t s , and i t a l s o r e q u i r e s a B a c k u s - G i l b e r t a p p r a i s a l to o b t a i n unique averages of the v e l o c i t y . The F o u r i e r t r a n s f o r m of the averages of the r e f l e c t i v i t y f u n c t i o n , the c o n s t r a i n t s - from the RMS v e l o c i t i e s , and any a d d i t i o n a l impedance c o n s t r a i n t s a r e then used i n a f i n a l c o n s t r u c t i o n a l g o r i t h m t o produce an a c o u s t i c impedance which i s c o n s i s t e n t w i t h a l l a v a i l a b l e g e o l o g i c a l and g e o p h y s i c a l i n f o r m a t i o n . 92 CHAPTER V: MULTI-TRACE SIMPLEX ALGORITHM IN SEISMIC DATA ANALYSIS 5.1 INTRODUCTION Advantages a s s o c i a t e d with the use of l i n e a r programming i n the s o l u t i o n of c e r t a i n g e o p h y s i c a l problems have been p o i n t e d out i n a number of recent p u b l i c a t i o n s . A p p l i c a t i o n s of the method to s i g n a l a n a l y s i s , wavelet d e c o n v o l u t i o n , amplitude spectrum e s t i m a t i o n , and a c o u s t i c impedance i n v e r s i o n have been d e s c r i b e d i n C l a e r b o u t and Muir (1973), T a y l o r et a l . ( 1979) , Levy and F u l l a g a r (1981),' Deeming and T a y l o r ( 1 9 8 1 ) , Oldenburg et a l . ( 1 9 8 3 ) , Levy et - a l . (1982), and Oldenburg et a l . (1984). However, d e s p i t e i t s u t i l i t y and robust performance when a p p l i e d to these problems, the method i s s t i l l not w i d e l y used. To a l a r g e degree t h i s i s p r o b a b l y a t t r i b u t a b l e to the l a r g e c o mputational e f f o r t u s u a l l y r e q u i r e d in the s o l u t i o n of l i n e a r programming problems. Attempts t o i n c r e a s e the e f f i c i e n c y of l i n e a r programming are d e s c r i b e d i n Deeming and T a y l o r (1981) and Oldenburg et a l . (1983). The former e l e c t e d to use the conjugate s u b - g r a d i e n t s method, which i s shown to have computational advantages over the standard Simplex method when the problem s o l v e d exceeds a c e r t a i n s i z e . The l a t t e r i n t r o d u c e d ' p o l a r i t y c o n s t r a i n t s ' and o b j e c t i v e f u n c t i o n w eighting to both reduce the s i z e of the problem, and to f o r c e a s h o r t e r s o l u t i o n s e a r c h . Although the 93 c o m p u t a t i o n a l e f f o r t was d e c r e a s e d s i g n i f i c a n t l y , the preimposed c o n d i t i o n s on the s o l u t i o n c o u l d r e s u l t i n d i s t o r t i o n or b i a s i n c e r t a i n c a s e s , such as f o r s m a l l a m p l i t u d e events i n near p r o x i m i t y t o l a r g e amplitude r e f l e c t i o n s . We p r e s e n t here an a l t e r n a t i v e approach based on c e r t a i n m a n i p u l a t i o n s of the Simplex method, which when a p p l i e d t o s e i s m i c s e c t i o n s e x h i b i t i n g a r e a s o n a b l e t r a c e - t o - t r a c e c o r r e l a t i o n , w i l l r e s u l t i n s i g n i f i c a n t l y s h o r t e r p r o c e s s i n g time. 5.2 SIMPLEX ALGORITHM (BACKGROUND) In the b e g i n i n g of t h i s s e c t i o n we g i v e an i n t u i t i v e d e s c r i p t i o n of the Simplex a l g o r i t h m . A s h o r t mathematical p r e s e n t a t i o n w i l l f o l l o w . As the r e a d e r w i l l n o t e, our d e s c r i p t i o n i n c l u d e s o n l y the i n f o r m a t i o n we f e e l i s n e c e s s a r y f o r the u n d e r s t a n d i n g of the proposed m a n i p u l a t i o n s ; f o r more d e t a i l s the reader i s r e f e r r e d t o Gass (1964), and C l a e r b o u t and Muir (1973). A g e n e r a l l i n e a r programming problem i s of the form: minimize (or maximize) c T - x s u b j e c t t o : A x £ b ( or A x £ b ) and I•x > 0 where c i s an (Nxl) v e c t o r of o b j e c t i v e f u n c t i o n c o e f f i c i e n t s , 94 x i s an (Nxl) v e c t o r of unknowns, b i s the (Mxl) v e c t o r c o n t a i n i n g the RKS, A i s the (MxN) mat r i x of c o n s t r a i n t s , I i s the (NxN) i d e n t i t y m a t r i x , 0 i s an (Nx1) v e c t o r whose elements a r e z e r o s , and M and N, are the number of c o n s t r a i n t s and v a r i a b l e s , r e s p e c t i v e l y . We w i l l use a simple two v a r i a b l e example t o d e f i n e and d e s c r i b e c e r t a i n n e c e s s a r y terms by g e o m e t r i c a l means. C o n s i d e r the problem; maximize c,x, + c 2 x 2 s u b j e c t t o : a,,x, + a , 2 x 2 £ b, a 2 i x , + a 2 2 x 2 £ b 2 x, £ 0 x 2 2 0 G e o m e t r i c a l l y t h i s problem i s ex p r e s s e d a s : FIGURE 5.1 95 TERMINOLOGY (a) The hatched a r e a i s c a l l e d the s o l u t i o n s o l i d ; a l l p o s s i b l e s o l u t i o n s t o the problem a r e c o n t a i n e d i n t h i s space, (note, i n 2-space t h i s i s an a r e a , i n 3-space t h i s i s a volume, e t c . ) (b) The s o l u t i o n s o l i d i s bounded by N+M c o n s t r a i n t e q u a t i o n s ; A x = b, and I x - 0. (note, i n 2-space these are l i n e s , i n 3-space they a r e p l a n e s , e t c . ) The p o i n t s where N of these e q u a t i o n s i n t e r s e c t a r e c a l l e d 'extreme f e a s i b l e p o i n t s ' . I t i s easy t o see that" each of these i n t e r s e c t i o n s d e f i n e s a v e c t o r w i t h a t most M non-zero elements. In F i g u r e 5-1, the extreme f e a s i b l e p o i n t s a r e marked by the numbers 1, 2, 3, and 4.. (c) The o b j e c t i v e f u n c t i o n i s r e p r e s e n t e d by the f a m i l y of l i n e s c,x, + c 2 x 2 == c o n s t a n t . T h i s f u n c t i o n . s p e c i f i e s a unique s o l u t i o n to the problem p r o v i d e d i t i s not ' p a r a l l e l ' t o any of the c o n s t r a i n t s . A b a s i c theorem of l i n e a r programming s t a t e s t h a t the o p t i m a l - s o l u t i o n t o a l i n e a r programming problem i s an extreme f e a s i b l e p o i n t ( Gass, 1964, pg.46-53) ( i . e . i n F i g u r e 5-1 the s o l u t i o n w i l l be one of the p o i n t s 1, 2, 3, or 4 ) . The Simplex a l g o r i t h m i s based on t h i s theorem. The a l g o r i t h m s t a r t s by f i n d i n g an extreme f e a s i b l e p o i n t ( u s u a l l y X J = 0 , j = i . . N , f o r example, p o i n t 1 i n F i g u r e 5-1 above), and then hops from one extreme f e a s i b l e p o i n t to another i n a way t h a t w i l l ensure a c o n t i n u o u s improvement i n the v a l u e of the o b j e c t i v e f u n c t i o n . 96 When no f u r t h e r improvement i s p o s s i b l e , the c u r r e n t extreme f e a s i b l e p o i n t i s the d e s i r e d o p t i m a l s o l u t i o n . S i n c e the number of extreme f e a s i b l e p o i n t s s e a r c h e d i n the p r o c e s s of f i n d i n g the s o l u t i o n d etermines the run time and e x e c u t i o n c o s t , we wish t o minimize t h i s number. To a c h i e v e t h i s g o a l one may take any of the f o l l o w i n g r o u t e s : (a) u s i n g a g r a d i e n t method ( C l a e r b o u t and M u ir, 1973), one may seek a t each s t e p the d i r e c t i o n of search- which w i l l y i e l d the best p o s s i b l e improvement i n the o b j e c t i v e f u n c t i o n . Thus the number of s t e p s n e c e s s a r y to a c h i e v e the d e s i r e d extremum i s h o p e f u l l y d e c r e a s e d . U n f o r t u n a t e l y , t h e r e i s no g u a r a n t e e t h a t the t o t a l number of s t e p s w i l l be d e c r e a s e d , and i n any event the amount of c a l c u l a t i o n r e q u i r e d a t each s t e p of t h e g r a d i e n t method i s l a r g e enough t o add doubt t o the p r o f i t a b i l i t y of t h i s o p e r a t i o n . (b) Assume t h a t some 'a p r i o r i ' knowledge of t h e sought s o l u t i o n i s g i v e n . I t i s then p o s s i b l e to reduce the number of Simplex s t e p s r e q u i r e d i n the s o l u t i o n of a l i n e a r programming problem u s i n g the f o l l o w i n g o p t i o n s : (1) m o d i f y i n g the d e s i r e d o b j e c t i v e f u n c t i o n i n such a way t h a t the Simplex a l g o r i t h m w i l l choose a ' s h o r t ' p a t h toward the s o l u t i o n . In the d e c o n v o l u t i o n problem t h i s m o d i f i c a t i o n i n v o l v e s the assumption t h a t the p r o c e s s e d s e i s m i c t r a c e c o n t a i n s i n f o r m a t i o n c o n c e r n i n g r e f l e c t o r p o s i t i o n s and r e l a t i v e a m p l i t u d e s which i s a p p r o x i m a t e l y t r u e f o r a t l e a s t the major r e f l e c t i o n s . T h i s type of m o d i f i c a t i o n i s d e s c r i b e d i n d e t a i l i n 97 Oldenburg e t a l . (1983), and Scheuer (1981). (2) s t a r t i n g the Simplex s e a r c h at an extreme f e a s i b l e p o i n t which i s ' c l o s e ' to the sought s o l u t i o n . For example, c o n s i d e r the problem o u t l i n e d i n F i g u r e 5-1. Assume t h a t t h e s o l u t i o n we seek i s extreme f e a s i b l e p o i n t ( 3 ) , A Simplex s e a r c h s t a r t i n g e i t h e r a t p o i n t ( 2 ) , or ( 4 ) , w i l l r e a c h the s o l u t i o n i n a s m a l l e r number of s t e p s than a s e a r c h i n i t i a l i z e d a t ( 1 ) . I n the p r o c e s s i n g of s t a c k e d s e i s m i c s e c t i o n s we g e n e r a l l y do not expect the r e f l e c t i v i t i e s t o change d r a s t i c a l l y a c r o s s a d j a c e n t t r a c e s ; t h a t i s , we expect the s o l u t i o n of the i t h t r a c e t o d e f i n e an extreme f e a s i b l e p o i n t which i s ' c l o s e ' t o the s o l u t i o n of the ith+1 t r a c e . By s t a r t i n g the Simplex s e a r c h a t t h i s p o i n t we w i l l a v o i d the c o s t a s s o c i a t e d w i t h the o p e r a t i o n s n e c e s s a r y t o move from an a r b i t r a r y s t a r t i n g p o i n t to the ' v i c i n i t y ' of the s o l u t i o n . The meaning of the terms ' c l o s e ' and ' v i c i n i t y ' i n the c o n t e x t of t h i s work can be u n d e r s t o o d through the example shown i n F i g u r e 2 below: FIGURE 5.2 9 8 In t h i s f i g u r e , 'extreme f e a s i b l e p o i n t ' (7) i s a l g o r i t h m i c a l l y c l o s e r t o 'extreme f e a s i b l e p o i n t (5) than i s p o i n t ( 2 ) , al t h o u g h p o i n t (2) i s c l o s e r to (5) i n a E u c l i d i a n sense. Keeping t h i s i n mind, we use the term ' c l o s e r ' i n the r e s t of t h i s work as meaning E u c l i d i a n c l o s e r , but w i t h the hope t h a t i t a l s o t r a n s l a t e s i n t o meaning a l g o r i t h m i c a l l y c l o s e r . 5.3 BASIC FORMULATION OF THE ALGORITHM C o n s i d e r the problem; !' minimize H(x) = c T- x, s u b j e c t t o x ^ O , A x * b , ( i = 1 N) where A i s an (MxN) m a t r i x . We can c o n v e r t the i n e q u a l i t i e s i n t o e q u a l i t i e s by i n t r o d u c i n g s l a c k v a r i a b l e s y - ( y ^ £ 0 , i=1...M). 1 V x' = b r i k 11 where V i s an (M x N+M) m a t r i x . Denoting the j t h column of V by vj we have Now, assume t h a t we a l r e a d y know an extreme f e a s i b l e p o i n t , f o r example, x'° : X.' " = X; i =1...M, t I = 0 i>M ( R e c a l l t h a t an extreme f e a s i b l e p o i n t i s c h a r a c t e r i z e d by at most M non-zero v a r i a b l e s ) . We then w r i t e : 99 .Ix' ; 0 • v. = b (5.1) The c o r r e s p o n d i n g v a l u e of the o b j e c t i v e f u n c t i o n i s : H, = Z x ! 0 . C; (5.2) Assuming t h a t the v e c t o r s v^  , i = 1..M a r e l i n e a r l y independent, they form a b a s i s of the M space. I f the MxM m a t r i x B i s -c o n s t r u c t e d u s i n g v^ , i = l . . . M as i t s columns then we may w r i t e : M V-. = .Zs-; • v: f o r j=1 N+M. (5.3) where S •• B " 1 • V D e f i n e We now wish t o e s t a b l i s h whether r e p l a c i n g any of the v a r i a b l e s c o r r e s p o n d i n g t o the c u r r e n t b a s i s w i t h some ot h e r v a r i a b l e , f o r example, x^, w i l l y i e l d f a r t h e r r e d u c t i o n i n the o b j e c t i v e f u n c t i o n v a l u e . S u b t r a c t i n g 'p' times e q u a t i o n (5.3) from e q u a t i o n ( 5 . 1 ) , w i t h j=k, we o b t a i n : ZU[° " p sik) \ + p.vk = b -(5.5) where p i s an a r b i t r a r y non-negative c o n s t a n t t o be determined, and M < k S N+M. Our new s o l u t i o n i s then x' 1 ( w i t h p o t e n t i a l l y M+1 non-zero values).: x? 1 = x/ 0 - p. s ^ , i = 1....M x - ' - p The r e s u l t i n g v a l u e of the o b j e c t i v e f u n c t i o n would be: H 2 = I i x | 0 - p - S i i j J - c ^ + p - c ^ = H, - p - h t (5.6) By c h o o s i n g p = x^° / s ^ (1£1£M), we s e t the v a l u e of the 1th element of x' 1 t o z e r o , thereby i n t r o d u c i n g x £ 1 i n t o the set of a c t i v e v a r i a b l e s and e x c l u d i n g x^ 1 from t h i s s e t . The hew v a l u e s of the b a s i s v a r i a b l e s w i l l .be: 100 X'-1 = X | ° - Xl° slk /SU i ¥ 1 (:5.7) * { 1 = x j , 0 /sLk i - . 1 To ensure the n o n - n e g a t i v i t y of the c u r r e n t s o l u t i o n , the v a l u e of p must be: p - min (x'.° / s - j j ) ; f o r a l l i w i t h s ^ s 0 (5.8) S i n c e we wish H 2 to be s m a l l e r than H,, the v a r i a b l e i n t r o d u c e d i n t o the b a s i s must have i t s c o r r e s p o n d i n g v a l u e of h ^ > 0, as seen from e q u a t i o n ( 5 . 6 ) . In g e n e r a l , t h i s v a r i a b l e i s a s s o c i a t e d w i t h the l a r g e s t h j , a l t h o u g h t h i s does not n e c e s s a r i l y ensure the b e s t improvement i n H, s i n c e the c o r r e s p o n d i n g p may be s m a l l . Once the d e c i s i o n i s made as t o which v a r i a b l e l e a v e s the b a s i s , and which ( x L 1 ) e n t e r s , the new se t of v a r i a b l e s i s c a l c u l a t e d v i a e q u a t i o n (.5.7), and t h e hj v a l u e s n e c e s s a r y f o r the next d e c i s i o n u s i n g e q u a t i o n ( 5 . 4 ) . T h i s p r o c e s s can then be c o n t i n u e d u n t i l the o p t i m a l s o l u t i o n i s r e a c h e d . I t i s important t o u n d e r s t a n d t h a t i n d e t e r m i n i n g x'° and x 1 1 , we have a c t u a l l y s o l v e d M e q u a t i o n s w i t h M unknowns of the form ; B°-x^°=b, and B 1-x^ ^ b , r e s p e c t i v e l y . The v e c t o r s x^ 0 and x ^ 1 a r e j u s t t r u n c a t e d v e r s i o n s of the complete s o l u t i o n s x'° and x' 1 from which the z e r o elements have been e x c l u d e d . The columns of B° and B 1 are the column v e c t o r s of V a p p r o p r i a t e f o r the non-zero elements It* 0 and xt' \ r e s p e c t i v e l y . At each e n s u i n g i t e r a t i o n , the c h o i c e of the new b a s i s determines a new B 1 , and the v a l u e s of x^ 1 (and thus x' 1 ) can then be computed. We c a l l the a t t e n t i o n of the reader to the f a c t t h a t the v e c t o r s x'* c o n s t i t u t e extreme f e a s i b l e p o i n t s . Thus the 101 correspondence between the g e o m e t r i c a l and a l g o r i t h m i c d e s c r i p t i o n s i s c l e a r . A l t h o u g h each Simplex i t e r a t i o n n e c e s s i t a t e s a s o l u t i o n of the system B-xt = B, t h i s s o l u t i o n does not take the f u l l number of s t e p s u s u a l l y r e q u i r e d . S t a n d a r d Simplex r o u t i n e s f i n d i t c o m p u t a t i o n a l l y more e x p e d i e n t to s t o r e B* 1-A, and B _ 1 - b , and update the s e a t each i t e r a t i o n (Gass, 1964, pg.96-113). Because A c o n t a i n s B, B"^ A w i l l always c o n t a i n the u n i t m a t r i x I . As t h i s needs not be s t o r e d i n m a t r i x form, i t i s p o s s i b l e t o s t o r e B" 1 i n i t s p l a c e . We w i l l use t h i s i n f o r m a t i o n f o r the d e f i n i t i o n of a s t a r t i n g extreme f e a s i b l e p o i n t i n the next s e c t i o n . 5.4 SOLVING MULTIPLE RELATED PROBLEMS C o n s i d e r a s e t of problems of the form; minimize c.T- x^ s u b j e c t t o A;- 1 ^ = b^ , x^ £ 0 f o r a l l i and j ; w i t h the c o n d i t i o n t h a t xt-+j i s ' c l o s e ' to x; . A t e c h n i q u e f o r an e f f i c i e n t s o l u t i o n t o t h i s type of a problem was proposed i n Gass,1964, pg.143-144. The g i s t of i t was t h a t the s t a r t i n g extreme f e a s i b l e p o i n t not be chosen a r b i t r a r i l y , but r a t h e r i s chosen u s i n g some a p r i o r i i n f o r m a t i o n about the s o l u t i o n form. In the f o l l o w i n g , we a p p l y t h i s t e c h n i q u e ( m u l t i - t r a c e Simplex) wit h some m o d i f i c a t i o n s . We d i s t i n g u i s h the f o l l o w i n g c a s e s : (a) A; and c: are i n v a r i a n t f o r a l l i . 102 (b) A{ and b; a r e i n v a r i a n t f o r a l l i . (c) h-L i s i n v a r i a n t f o r a l l i . (d) A l l of A;, b^ , and c- change w i t h i , but x;_, i s expected to be s i m i l a r t o x,-.. Case (a) Suppose t h a t we have a l r e a d y s o l v e d the f i r s t problem; min c " i T - x 1 f s u b j e c t t o A r x , = b,, ; x, j £ 0, f o r a l l j , by the Simplex method. At t h i s p o i n t we a l s o p o s s e s s the i n v e r s e b a s i s B," 1. The v e c t o r x 2 t • B 1 " 1 - b 2 , when p r o p e r l y o r d e r e d i n x 2 , d e f i n e s an extreme f e a s i b l e point.. We then use t h i s p o i n t which h o p e f u l l y i s ' c l o s e ' t o the d e s i r e d s o l u t i o n x 2 , to s t a r t the new Simplex s e a r c h . When x 2 i s a t t a i n e d , we a l s o have the new i n v e r s e b a s i s B 2 " 1 which t o g e t h e r w i t h b 3 d e f i n e s the s t a r t i n g extreme f e a s i b l e p o i n t x 3 t o be used i n the s o l u t i o n of the t h i r d problem. We c o n t i n u e i n t h i s f a s h i o n u n t i l the d e s i r e d number of problems i s s o l v e d . Case (b) In t h i s case the s o l u t i o n x, i s a l r e a d y an extreme f e a s i b l e p o i n t f o r the next problem. U t i l i z i n g e q u a t i o n (5.4) the v a l u e s of hj can be updated u s i n g x., and c 2 , and the Simplex s e a r c h i s r e s t a r t e d . The p r o c e s s c o n t i n u e s s i m i l a r l y f o r x 2 , e t c . An a l t e r n a t i v e approach to the one o u t l i n e d above i s t o work on the d u a l problem, (Gass 1964, pg.83-95) u s i n g the sequence of o p e r a t i o n s d e s c r i b e d i n case ( a ) . Case (c) Given the s o l u t i o n x,, B,, and B,"1, we proceed as i n case (a) 103 to f i n d x 2 (an extreme f e a s i b l e p o i n t p e r t a i n i n g t o the second problem). U s i n g x 2 , and c 2 , we u t i l i z e e q u a t i o n (5.4) to update h\ , and p r o c e e d w i t h the Simplex s e a r c h s t a r t i n g at x 2 . The p r o c e s s i s r e p e a t e d u n t i l the d e s i r e d number of problems i s solved.. Case (d) In t h i s case we use a c o m b i n a t i o n of the c a s e s o u t l i n e d above. However, the c r u x of the a p p r o a c h i s heavy o b j e c t i v e f u n c t i o n w e i g h t i n g used i n a f i r s t pass' on each problem. Given the s o l u t i o n x,, we weight the o b j e c t i v e f u n c t i o n c 2 i n s uch a way t h a t the non-zero elements i n x 2 w i l l be i n n e a r l y the same l o c a t i o n s as t h o s e of x,. We then u t i l i z e e q u a t i o n (5.4) t o update hj w i t h the d e s i r e d o b j e c t i v e f u n c t i o n c 2 (as -o u t l i n e d i n c a s e ( b ) ) , and r e s t a r t the a l g o r i t h m a t extreme f e a s i b l e p o i n t x*2. Once x 2 i s o b t a i n e d we use i t t o weight the i n i t i a l o b j e c t i v e f u n c t i o n f o r JT3 , then p r o c e e d i n t h i s manner u n t i l the d e s i r e d number of problems i s s o l v e d . 5.5 APPLICATION TO SEISMIC DATA Given a s t a c k e d s e c t i o n , our o b j e c t i v e i s t o r e c o v e r a s p i k y r e f l e c t i v i t y s e c t i o n c o r r e s p o n d i n g to the g i v e n d a t a . The d e t a i l s of how a f u l l - b a n d r e f l e c t i v i t y s e c t i o n i s determined from the b a n d - l i m i t e d i n p u t s e c t i o n u s i n g l i n e a r programming 1 04 (LP) i s d e s c r i b e d i n c h a p t e r s 2, .3 and 4 and i n Levy and F u l l a g a r (1981), Oldenburg e t a l . (1983), Scheuer ( 1 9 8 1 ) , and Oldenburg e t a l (1984). Consequently, we w i l l g i v e o n l y the b a r e s t d e t a i l s . The problem t o be s o l v e d i s of t h e form: minimize c T- x, s u b j e c t t o , A x £ h, -and x.Lt 0 f o r a l l i . The m a t r i x A may c o n s i s t of t h r e e d i s t i n c t b l o c k s : A = I FT WLC RMS C The b l o c k IFT .represents the c o n s t r a i n t s a s s o c i a t e d w i t h the i n v e r s e F o u r i e r t r a n s f o r m , and i t i s assumed t o be i n v a r i a n t f o r a g i v e n s et of d a t a . The b l o c k s WLC and RMS C a r e a s s o c i a t e d w i t h w e l l - l o g c o n s t r a i n t s and RMS v e l o c i t y c o n s t r a i n t s , r e s p e c t i v e l y . These b l o c k s may or may not be p r e s e n t i n the problem. Furthermore, when p r e s e n t , they may be changing a c r o s s the s e c t i o n , ( i . e . A may then be changing a c r o s s the s e c t i o n ; case (d) of the p r e v i o u s s e c t i o n ) . We s t a r t by a s s e r t i n g t h a t f o r r e a s o n a b l e q u a l i t y s e i s m i c data the assumption t h a t the ith+1 t r a c e i s s i m i l a r to the i t h 1 05 t r a c e h o l d s t r u e except f o r s e c t i o n s c o r r e s p o n d i n g t o v e r y complex geology. Furthermore, s i n c e the same energy source i s used i n the experiment, i t can be expected t h a t a f t e r the a p p r o p r i a t e b a l a n c i n g ( p r o c e s s i n g ) i s used the s i g n a l energy i s c o n c e n t r a t e d i n a g i v e n f r e q u e n c y band which i s independent of t r a c e index. I f we a v o i d the use of o b j e c t i v e f u n c t i o n w e i g h t i n g and p o l a r i t y c o n s t r a i n t s , we encounter a problem c o r r e s p o n d i n g to the one o u t l i n e d i n case ( a ) . I f we d e s i r e o b j e c t i v e f u n c t i o n w e i g h t i n g , the problem we -confront i s the one o u t l i n e d i n case ( c ) . Imposing p o l a r i t y c o n s t r a i n t s i n v o l v e s problems s i m i l a r t o the types d e s c r i b e d i n c a s e s (b) and ( d ) . Other c o m b i n a t i o n s of problem c h a r a c t e r i s t i c s a r e p o s s i b l e , but f o r p r a c t i c a l purposes we r e s t r i c t our a t t e n t i o n to those o u t l i n e d above. 5.6 EXAMPLE For the purpose of a s s e s s i n g the performance of the a l g o r i t h m we have used the s e t of r e a l d a t a shown i n F i g u r e 5-3. Be f o r e a p p l y i n g the a l g o r i t h m we have p h a s e - c o r r e c t e d and deconvolved the da t a u s i n g a minimum en t r o p y approach. A d d i t i o n a l l y , t o improve the s i g n a l t o n o i s e r a t i o we have used 3 - t r a c e p r i n c i p a l component a n a l y s i s (Oldenburg e t a l (1983)). The r e s u l t s are shown i n F i g u r e 5-4. We then a p p l i e d the LP a l g o r i t h m t o the data of F i g u r e 5-4 u s i n g the f o l l o w i n g modes: a) o b j e c t i v e f u n c t i o n w e i g h t i n g , no p o l a r i t y c o n s t r a i n t s . 1 06 F i g u r e 5-3 c o r r e c t i o n 6 " 1 0 " f e a t U r i n 5 a r e a s o n a b l e t r a c e to t r a c e 1 0 7 F i g u r e 5-4 The data of F i g u r e 5-3 a f t e r s p e c t r a l whitening, phase c o r r e c t i o n and the a p p l i c a t i o n of t h r e e - t r a c e p r i n c i p a l component SNR enhancement. 108 b) p o l a r i t y c o n s t r a i n t s , no o b j e c t i v e f u n c t i o n w e i g h t i n g . c) no p o l a r i t y c o n s t r a i n t s , no o b j e c t i v e f u n c t i o n weighting d) p o l a r i t y c o n s t r a i n t s and o b j e c t i v e f u n c t i o n w e i g h t i n g Modes (a) to (d) were executed twice,once with the m u l t i -t r a c e Simplex approach, and again with the t r a c e - b y - t r a c e Simplex approach. For reasons of run e f f i c i e n c y and memory requirements, the data were pr o c e s s e d i n short windows of 64 samples each. The CPU run times (Vax 11/780) are g i v e n i n Table 5.1. In most cases the m u l t i - t r a c e Simplex approach proves to be more e f f i c i e n t than the t r a c e - b y - t r a c e approach, with the e x c e p t i o n being mode ( c ) . For t h i s mode numerical checks showed th a t although the l a s t s o l u t i o n seems to be ' c l o s e ' to the c u r r e n t one, the Simplex search may l e a d to i t i n a round-about path. T h i s i s not unexpected, s i n c e i n t h i s case both the s t a r t i n g s o l u t i o n and the o b j e c t i v e f u n c t i o n are updated from the p r e v i o u s , so that our s u p p o s i t i o n t h a t geometric ' c l o s e n e s s ' e q u a l s a l g o r i t h m i c ' c l o s e n e s s ' breaks down. I t sho u l d be noted however, that f o r a l i n e a r programming run executed i n mode ( c ) , the CPU run time i s reduced ( f o r t h i s set of data) by a f a c t o r of f o u r . F i n a l l y , i n order to compare p r o c e s s i n g r e s u l t s , we present i n F i g u r e 5-5 the r e f l e c t i v i t y o b t a i n e d v i a the m u l t i - t r a c e Simplex (mode ( c ) ) , and t h a t a s s o c i a t e d with the t r a c e - b y - t r a c e Simplex (mode ( d ) ) , shown i n F i g u r e 5-6. These r e s u l t s g e n e r a l l y agree q u i t e w e l l , except i n some s m a l l amplitude e v e n t s . T h i s s h o u l d be expected, s i n c e the o b j e c t i v e f u n c t i o n w e i g h t i n g tends to d i s c r i m i n a t e a g a i n s t those events. 200 180 160 i40 120 ! 1 X - X . 1 i •f \ . 6 -§- i . 9 4-2.0 J-2.1 — 2.2 Figure 5-5 The data of figu r e 5-4 a f t e r l i n e a r programming deconvolut with the multitrace simplex mode. 1 1 0 200 180 160 m0 120 Figure 5-6 The data of figure 5-4 a f t e r l i n e a r programming deconvolution with objective function weights and p o l a r i t y c o n s t r a i n t s . 111 For completeness we a t t a c h the pseudo-impedance s e c t i o n c o r r e s p o n d i n g t o the r e f l e c t i v i t y of F i g u r e 5-5. T h i s s e c t i o n i s shown i n F i g u r e 5-7 i n l i n e p l o t , and i n F i g u r e .5-8 i n grey shades p l o t . The impedance produced t i e s w e l l w i t h a v e l o c i t y l o g o b t a i n e d i n a nearby w e l l (not shown h e r e ) . The improvement of the s e c t i o n i n F i g u r e 5-8 over t h a t of F i g u r e 5-3 i s q u i t e c l e a r and does not r e q u i r e e l a b o r a t i o n . 1 1 2 F i g u r e 5-7 The pseudo-impedance data i n f i g u r e 5-5. s e c t i o n o b t a i n e d v i a the i n t e g r a t i o n of the 1 1 3 Figure 5 - 8 Pseudo-impedance section o v e r l a i d on the f u l l band LP r e f l e c t i v i t y . This type of dis p l a y seems to enhance the i n t e r p r e t a b i l i t y of the data. 114 T a b l e 5.1 MW MP NFAST CPU (sec.) 1 1 0 789.7 1 1 1 1232.7 0 0 0 2976.4 0 0 1 713.4 1 0 0 1 398.-4 1 0 1 1358.1 0 1 0 1305.8 0 1 1 1 175.0 Note: MP 0 no p o l a r i t y c o n s t r a i n t s used ... ...1 p o l a r i t y c o n s t r a i n t s used MW 0 no o b j e c t i v e f u n c t i o n w e i g h t i n g ......1 o b j e c t i v e f u n c t i o n w e i g h t i n g used NFAST..0 t r a c e by t r a c e Simplex used ^ ..1 m u l t i - t r a c e Simplex used Data c o n t a i n s : 100 t r a c e s , one second long each, sampling i n t e r v a l 0.004 sec, frequency band 12-40 Hz. 115 5.7 SUMMARY A number of o p t i o n s which y i e l d an i n c r e a s e i n the e f f i c i e n c y of the l i n e a r programming method were d i s c u s s e d . These methods do not r e q u i r e t h a t c e r t a i n f e a t u r e s of the model be predetermined or s e l e c t i v e l y weighted, as i s the case with p r e v i o u s t e c h n i q u e s (Oldenburg et a l . 1983, and Scheuer, 1 9 8 1 ) . The r e s u l t i s t h a t now the l i n e a r programming method can be used to p r o c e s s r e f l e c t i o n s e i s m i c data i n a v a r i e t y of d i f f e r e n t modes, a l l i n reasonable e x e c u t i o n time. Comparison of the r e s u l t s of the p r o c e s s i n g ( F i g u r e s 5-8) to the input data ( F i g u r e 5-3) demonstrates the e f f i c a c y of the method. 116 CHAPTER V I : PRE-INVERSION SCALING OF REFLECTION SEISMOGRAMS 6.1 INTRODUCTION As d e s c r i b e d i n the p r e v i o u s c h a p t e r s , approaches to the e s t i m a t i o n of a c o u s t i c impedance from r e f l e c t i o n seismograms i n v o l v e c o n s t r u c t i o n of a f u l l - b a n d r e f l e c t i v i t y s e c t i o n from the i n p u t b a n d l i m i t e d s e c t i o n . Some of these approaches a l s o advocate the d i r e c t i n c o r p o r a t i o n of w e l l - l o g and s t a c k i n g v e l o c i t y i n f o r m a t i o n i n t o the e s t i m a t i o n p r o c e s s . An impediment t o these methods f o r both the c o n v e r s i o n of t h e f u l l - b a n d r e f l e c t i v i t y t o impedance, as w e l l as the i n c o r p o r a t i o n of the v e l o c i t y i n f o r m a t i o n , i s the l a c k of t r u e a m p l i t u d e i n f o r m a t i o n i n the s t a c k e d s e c t i o n . Due t o d i f f i c u l t i e s i n da t a a c q u i s i t i o n and subsequent p r o c e s s i n g , the . f i n a l s t a c k e d s e c t i o n w i l l g e n e r a l l y r e q u i r e s c a l i n g p r i o r t o the impedance i n v e r s i o n p r o c e s s . To u n derstand the importance of c o r r e c t s c a l i n g , l e t us assume t h a t the e s t i m a t e d r e f l e c t i o n s e r i e s ?j , j = l , . . . N , i s a s c a l e d v e r s i o n of the t r u e r e f l e c t i o n s e r i e s rj , J=1,...N; t h a t i s : r^ = a - r j f o r a l l j (6.1) wi t h a b e i n g a r e a l c o n s t a n t , and rj being the j t h r e f l e c t i o n c o e f f i c i e n t . The impedance, $j = Pj • v j >' i s c a l c u l a t e d from the e s t i m a t e d r e f l e c t i o n s e r i e s u s i n g the l i n e a r i z e d a p p r o x i m a t i o n (see Oldenburg e t a l , 1984): S j - So exp [2-.l' x- .] (6.2) = So exp [2'a- JZ T{ ] ( S I I t i s c l e a r t h a t i n c o r r e c t v a l u e s of the a c o u s t i c impedance are o b t a i n e d i f o i s much l a r g e r or much s m a l l e r than 1. Furthermore, i n t h i s c a s e , the s e t of e x t e r n a l c o n s t r a i n t s a p p l i e d t o t h e r e f l e c t i v i t y c o n s t r u c t i o n w i l l be i n c a p a b l e of s u p p l y i n g t r u e p h y s i c a l meaning. Hence, the i n c o n s i s t e n c y between the i n f o r m a t i o n c o n t a i n e d i n the seismograms and t h a t s u p p l i e d by the e x t e r n a l c o n s t r a i n t s w i l l a n n i h i l a t e the advantage to be g a i n e d by i n v e r s i o n of the combined data s e t . To a v o i d t h i s i n c o n s i s t e n c y , we propose a method f o r the e s t i m a t i o n of t h e s c a l e f a c t o r a u s i n g the observed seismograms, and t h e i r a s s o c i a t e d s t a c k i n g v e l o c i t i e s and w e l l - l o g impedances when a v a i l a b l e . 118 6.2 METHOD 6.2.1 GENERAL For the purpose of s c a l i n g we r e c o g n i z e two types of i n f o r m a t i o n r o u t i n e l y a v a i l a b l e t o p r o c e s s o r s of s e i s m i c d a t a . The f i r s t type can be o b t a i n e d d i r e c t l y from the b a n d l i m i t e d s t a c k e d s e c t i o n and we c o n s i d e r i t as the r e l a t i v e i n f o r m a t i o n . The second s e t of i n f o r m a t i o n i s a v a i l a b l e from w e l l - l o g s or from s t a c k i n g v e l o c i t i e s and i t i s termed here as the a b s o l u t e i n f o r m a t i o n . The proposed s c a l i n g a l g o r i t h m makes use of a l i n e a r r e l a t i o n s h i p ( t r u e f o r r e f l e c t i o n c o e f f i c i e n t s s m a l l e r than 0.3), which a l l o w s d i r e c t comparison between those two i n f o r m a t i o n s e t s and hence the e s t i m a t i o n of the s c a l i n g f a c t o r a. A l t h o u g h we w i l l be c o n s i d e r i n g o n l y the case of a time independent s c a l i n g f a c t o r (as r e f l e c t e d by e q u a t i o n ( 6 . 1 ) ) , our method can be a p p l i e d t o cases of a time dependent s c a l i n g f u n c t i o n by a p p l y i n g the a n a l y s i s t o a number of s u c c e s s i v e time windows and then i n t e r p o l a t i n g the r e s u l t s . 1 19 6.2.2 THE CONSTRUCTION OF SPIKY REFLECTIVITY FUNCTION - RELATIVE INFORMATION The proposed method of s c a l e f a c t o r e v a l u a t i o n i s based on r e c e n t l y i n t r o d u c e d a l g o r i t h m s which use the i n f o r m a t i o n c o n t a i n e d i n b a n d - l i m i t e d seismograms t o p r e d i c t the m i s s i n g p o r t i o n s of the frequency band. These a l g o r i t h m s i n c l u d e a l i n e a r programming approach ( T a y l o r et a l . 1979; Levy a'nd F u l l a g a r 1981; Scheuer 1981; and Oldenburg et a l . 1983), an a u t o r e g r e s s i v e technique (Scheuer,1981; Oldenburg et a l . 1983; and Walker and U l r y c h 1983), and a n o n - l i n e a r r e f l e c t i v i t y e s t i m a t i o n ( B i l g e r i and • C a r l i n i , 1 981 ). For the b e n e f i t of the i n t e r e s t e d reader, we have i n c l u d e d i n Appendix 6-A a s h o r t d e s c r i p t i o n of the a u t o r e g r e s s i v e d e c o n v o l u t i o n scheme. The f i n a l r e s u l t i n a l l these schemes i s a ' s p i k y ' , f u l l band r e f l e c t i v i t y s e r i e s , * r j , j=1,...N. In the f o l l o w i n g , we assume t h a t the p r e d i c t e d low f r e q u e n c i e s o b t a i n e d v i a the a p p l i c a t i o n of these d e c o n v o l u t i o n a l g o r i t h m s , f o l l o w e d by i n t e g r a t i o n and e x p o n e n t i a t i o n of the r e s u l t a n t r e f l e c t i v i t i e s (as i n eq u a t i o n ( 6 . 2 ) ) , w i l l y i e l d r e a s o n a b l e e s t i m a t e s of the t r e n d of the impedance f u n c t i o n i n at l e a s t some time windows. Since the r e f l e c t i v i t y f u n c t i o n s r j to be i n t e g r a t e d are s c a l e d by some a r b i t r a r y r e a l f a c t o r yet to be determined, these f u n c t i o n s c o n s t i t u t e our s c a l e d , or r e l a t i v e i n f o r m a t i o n . 120 6.2.3 WELL-LOGS OR STACKING VELOCITIES - ABSOLUTE INFORMATION Our a b s o l u t e i n f o r m a t i o n i s o b t a i n e d from e i t h e r the i n v e r s i o n of s t a c k i n g v e l o c i t i e s to g i v e i n t e r v a l v e l o c i t i e s , or d i r e c t l y from w e l l - l o g d a t a . 1. A b s o l u t e I n f o r m a t i o n from Well-Log Data When both v e l o c i t y and d e n s i t y l o g s are a v a i l a b l e , we can d i r e c t l y c a l c u l a t e the a c o u s t i c impedance as a f u n c t i o n of time. I f the d e n s i t y - l o g i s not a v a i l a b l e we assume a d e n s i t y - v e l o c i t y power law ( t h a t i s , p = A - v ^ ( t ) ) , and e s t i m a t e the a c o u s t i c impedance u s i n g the r e l a t i o n s h i p : $ ( t ) = A . v * p ( t ) . (6.3) Note tha t s u b s t i t u t i n g A=1, and p=0 i n the above d e n s i t y power law amounts to the c o n s t a n t d e n s i t y a p p r o x i m a t i o n , while A=0.23 and p=0.25 i s the u s u a l power law as suggested i n Anstey, 1'977 (p. 88).. 2. A b s o l u t e i n f o r m a t i o n from S t a c k i n g - V e l o c i t i e s When on l y s t a c k i n g v e l o c i t i e s are a v a i l a b l e , the i n v e r s i o n a l g o r i t h m s d e s c r i b e d i n Oldenburg et a l . (1984) g i v e a c c e p t a b l e r e p r e s e n t a t i o n s (vV, j=1,,..N) of the t r u e i n t e r v a l v e l o c i t y (v^» j = i , . . . N ) . Again, u s i n g the d e n s i t y power law given above, we c o n v e r t v L ( t ) to e s t i m a t e d impedance. A d e t a i l e d d e s c r i p t i o n 121 of the s t a c k i n g t o i n t e r v a l v e l o c i t y i n v e r s i o n schemes used i n t h i s work was g i v e n i n Chapter IV. 6.2.4 COMPARISON OF THE ABSOLUTE AND THE RELATIVE INFORMATION.. The e s t i m a t i o n of the s c a l i n g f a c t o r a i s e f f e c t e d by comparison of the a b s o l u t e and r e l a t i v e i n f o r m a t i o n a t s e l e c t e d time windows. To a l l o w t h i s comparison, we r e w r i t e the approximate r e l a t i o n between' r e f l e c t i v i t y and impedance g i v e n i n e q u a t i o n ( 6 . 2 ) , t h e r e b y i n t r o d u c i n g the pseudo-impedance, 7 7 : l n ( S ; / $ o ) = r?;= 2-X r { j > 1 = 0 j = 1 (6.4) Assuming a time independent s c a l i n g f u n c t i o n , we s u b s t i t u t e e q u a t i o n (6.1) i n t o (6.4) to get our r e l a t i v e pseudo-impedance e s t i m a t e s from the c o n s t r u c t e d f u l l - b a n d r e f l e c t i v i t i e s : j > 1 j = 1 (6.5) Our e s t i m a t e of the a b s o l u t e pseudo-impedance from w e l l - l o g data w i l l be: V- = 2a-I r t = 0 122 (a) d e n s i t y l o g a v a i l a b l e i?-- •- ln(*j / S o ) . or (using, e q u a t i o n 6.3): (6.6) (b) d e n s i t y l o g u n a v a i l a b l e -q- « ( i + p ) • l n ( v | / v 0 ) S i m i l a r l y , from i n t e r v a l v e l o c i t i e s i n v e r t e d from s t a c k i n g v e l o c i t i e s , our e s t i m a t e of the a b s o l u t e pseudo-impedance i s : •tfj = (1+p)- l n ( v j /vo) (6.7) S i n c e the pseudo-impedance rjj i s l i n e a r w i t h r e s p e c t t o the s c a l e f a c t o r a, comparison of T J J and rj j , (j=l,.,.,N) w i l l y i e l d the d e s i r e d s c a l i n g f a c t o r a . 123 6.2.5 CONSIDERATIONS PERTAINING TO THE USE OF STACKING VELOCITIES FOR SCALING. As the r e a d e r may have a l r e a d y r e a l i z e d , the 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 d e s c r i b e d i n Chapter IV a r e s t r i c t l y a p p l i c a b l e t o a one d i m e n s i o n a l e a r t h . Consequently, when s t e e p l y d i p p i n g e v e n t s are e v i d e n t on the s t a c k e d s e c t i o n , the proposed a l g o r i t h m i s not l i k e l y to y i e l d r e a s o n a b l e r e s u l t s u n l e s s some c o r r e c t i o n i s a p p l i e d t o the s t a c k i n g v e l o c i t i e s p r i o r t o i n v e r s i o n (see f o r example, Kesmarky 1 9 7 7 ) . Furthermore, s i n c e the i n t e r v a l v e l o c i t y p r o f i l e o b t a i n e d from the i n v e r s i o n of the s t a c k i n g v e l o c i t i e s i s at best .an average of the t r u e v e l o c i t y c u r v e , a p o i n t by p o i n t comparison of rf and T? i s u n d e s i r a b l e . I n s t e a d , box-car averages over a number (N/M) of windows (each M samples l o n g ) , a r e used t o get a .series of e s t i m a t e s of the s c a l e f a c t o r : o.; i-1,N/M (6.8) Z The f i n a l e s t i m a t e of the s c a l e f a c t o r f o r the seismogram i s o b t a i n e d from the median of the s e r i e s { a-}, so as to a v o i d i n c o r p o r a t i o n of s t a t i s t i c a l o u t l i e r s i n t h i s e s t i m a t e . 1 24 6.2.6 CONSIDERATIONS PERTAINING TO LOCALIZED SPIKING DECONVOLUTION FAILURES. The approach summarized i n e q u a t i o n s (6.5), (6.6), (6.7) and (6.8) s u f f e r s from the dependence of v a l u e s of rj at any given time on the c o r r e c t n e s s of p r e c e d i n g v a l u e s at s m a l l e r times, t h a t i s : j - i 7j. = .2 X r = 77. •+ 2r- . Since slow changes i n e a r t h impedance as a f u n c t i o n of depth w i l l not produce a c l e a r m a n i f e s t a t i o n i n the r e c o r d e d r e f l e c t i o n d a t a , they may not appear :in the pseudo-impedances 77 o b t a i n e d v i a summation of the r e f l e c t i v i t i e s from e i t h e r l i n e a r programming or a u t o - r e g r e s s i v e s p i k i n g d e c o n v o l u t i o n . However, the shape of the pseudo-impedance curve i n some g i v e n time windows may s t i l l be c o r r e c t . Thus, to reduce the e f f e c t s of the dependency of 7? on p r e v i o u s v a l u e s , o n l y the r e l a t i v e changes i n pseudo-impedance in each window are c o n s i d e r e d . That i s : 125 a (6.9) Furthermore, due t o the n a t u r e of both the LP and the AR d e c o n v o l u t i o n a l g o r i t h m s (both s e a r c h f o r minimum s t r u c t u r e s o l u t i o n s ) , i t i s expected t h a t b e t t e r shape-wise e s t i m a t e s of rj w i l l be o b t a i n e d at r e g i o n s of r a p i d impedance changes. T h e r e f o r e , a v a l u e s from windows c o r r e s p o n d i n g t o t h i s type of impedance zone may be weighted more h e a v i l y i n the f i n a l median c a l c u l a t i o n . The d e t e c t i o n of t h i s type of impedance zone and the assignment of the a p p r o p r i a t e weights i s based on the degree of c o r r e l a t i o n e x h i b i t e d by rj and r? i n the g i v e n time window. T h i s c o r r e l a t i o n f a c t o r i s e s t i m a t e d from the dot p r o d u c t of these two f u n c t i o n s . 6.3 SYNTHETIC EXAMPLES 6.3.1 LAYERED EARTH MODEL WITH WELL DEFINED LAYER BOUNDARIES. The r e f l e c t i v i t y f u n c t i o n ( F i g u r e 6-ia) c o r r e s p o n d i n g to the t r u e v e l o c i t y model of F i g u r e 6-1d.has been c a l c u l a t e d under the assumption of c o n s t a n t d e n s i t y . A f t e r b a n d - l i m i t i n g t h i s 0 126 r e f l e c t i v i t y f u n c t i o n (lO-50Hz), adding 5% random n o i s e , and s c a l i n g the r e s u l t by a f a c t o r of 100, we o b t a i n e d the s y n t h e t i c d a t a shown i n F i g u r e 6-1b. Usi n g t h i s s y n t h e t i c as an in p u t to the l i n e a r programmming (LP) d e c o n v o l u t i o n r o u t i n e , the r e s u l t a n t f u l l band r e f l e c t i v i t y r i s c o n s t r u c t e d and p l o t t e d i n F i g u r e 6 - 1C . 7 c o n s t i t u t e s our r e l a t i v e i n f o r m a t i o n . U s i n g the t r u e v e l o c i t y model of F i g u r e 6-1d, we have e s t i m a t e d the RMS v e l o c i t y c u r v e shown i n F i g u r e 6 - l e . F i v e s e l e c t e d p o i n t s of the l a t t e r (marked on F i g u r e 6 - l e ) , t o g e t h e r w i t h a p o i n t c o n s t r a i n t of 12 K f e e t / s e c a t 1.8 sec (assumed to be a v a i l a b l e from w e l l - l o g i n f o r m a t i o n ) were used i n the Backus-G i l b e r t f l a t t e s t - m o d e l i n v e r s i o n (Oldenburg e t a l . 1983), to y i e l d the e s t i m a t e d i n t e r v a l v e l o c i t y , our a b s o l u t e i n f o r m a t i o n ( F i g u r e 6-1f).. U s i n g e q u a t i o n s ( 6 . 5 ) , and (6 . 7 ) , we o b t a i n pseudo-impedances 77 and 77, shown i n F i g u r e 6-2b and 2c, r e s p e c t i v e l y . We then a p p l i e d e q u a t i o n (6.8) w i t h N=512 and M=128, t o get a s e r i e s of f o u r e s t i m a t e s of the s c a l e f a c t o r o. These a r e l i s t e d i n T a b l e 6-1 f o r the c o r r e s p o n d i n g time windows. As seen from F i g u r e 6-2, both rf and rj resemble the t r u e pseudo-impedance ( F i g u r e 6-2a) q u i t e w e l l . Hence, as expected, the f i n a l e s t i m a t e d v a l u e of 122 f o r the s c a l e f a c t o r i s i n good agreement w i t h t h e t r u e v a l u e of 100. 1 2 7 •T 0 T a b l e 6--1 : S c a l e f a c t o r e v a l u a t i o n f o r example i n F i g . 2 and 3. N=512, M=128 True s c a l e factor=!00 i a 1 71 2 148 .3 122 4 105 • Median - 1:22. Mean = 112. 0 o To complete the p r o c e s s , we have r e s c a l e d the i n p u t seismogram ( F i g u r e 6-lb) by a=1/122, and performed the a c o u s t i c impedance i n v e r s i o n (LP), u s i n g the s y n t h e t i c seismogram and a s e t of s t a c k i n g v e l o c i t y c o n s t r a i n t s (Oldenburg e t a l . 1984). The f i n a l r e s u l t o v e r l a y e d on the t r u e impedance c u r v e i s shown i n F i g u r e 6-3. 1 2 8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 ^ 1 1 1 H 1 r F i g u r e 6-1 (a) The r e f l e c t i v i t y s e r i e s c o r r e s p o n d i n g to the v e l o c i t y model of F i g u r e 6 - l d . (b) The s e r i e s i n (a) a f t e r the a p p l i c a t i o n of a 10-50 hz. bandpass f i l t e r and the a d d i t i o n of random n o i s e , (c) The s e r i e s i n (b) a f t e r LP d e c o n v o l u t i o n . (d) The t r u e v e l o c i t y model. (e) RMS v e l o c i t y c o r r e s p o n d i n g to the v e l o c i t y model i n ( d ) . ( f ) The e s t i m a t e d i n t e r v a l v e l o c i t y o b t a i n e d from the f i v e RMS p i c k s marked on (e) v i a the B a c k u s - G i l b e r t i n v e r s i o n . 129 F i g u r e 6-2 (a) The t r u e pseudo-impedance f u n c t i o n , (b) The pseudo-impedance f u n c t i o n c a l c u l a t e d from the c o n s t r u c t e d LP r e f l e c t i v i t y , (c) The pseudo-impedance f u n c t i o n c a l c u l a t e d from the c o n s t r u c t e d i n t e r v a l v e l o c i t y model. 130 F i g u r e 6-3 The r e c o n s t r u c t e d a c o u s t i c impedance f u n c t i o n o b t a i n e d v i a c o n s t r a i n e d i n v e r s i o n . The seismogram i n F i g u r e 6-1b was s c a l e d by a f a c t o r of 1/122 and the r e s u l t t o g e t h e r w i t h the s e t of s t a c k i n g v e l o c i t i e s (marked on F i g u r e 6-1e) was used i n the i n v e r s i o n . The t r u e v e l o c i t y model i s o v e r l a i d on the f i g u r e . 131 6.3.2 LAYERED EARTH MODEL WITH SLOWLY VARYING IMPEDANCE ZONES. As was d i s c u s s e d p r e v i o u s l y , both the LP and AR d e c o n v o l u t i o n a l g o r i t h m s minimize an o b j e c t i v e f u n c t i o n which i m p l i e s minimum e a r t h s t r u c t u r e . Consequently, i t can be e x pected t h a t the r e l a t i v e l y s m a l l r e f l e c t i o n c o e f f i c i e n t s ( r e l a t i v e a m p l i t u d e i s d e f i n e d by frequency band-width and e r r o r a l l o w a n c e used "in the d e c o n v o l u t i o n ) w i l l not be m a n i f e s t e d i n the e s t i m a t e d r e f l e c t i v i t y f u n c t i o n "r. Due to t h e s e o m i s s i o n s , c o n v e r s i o n of r i n t o pseudo-impedance 7? may y i e l d r e s u l t s which are s u b s t a n t i a l l y d i f f e r e n t ( i n a g l o b a l sense) from those g i v e n by the t r u e e a r t h model. However, i n time windows which do not i n c l u d e l a r g e r a m p - l i k e impedance components, the r e p r e s e n t a t i o n g i v e n by Ij may s t i l l be shape-wise a c c e p t a b l e . When t h i s i s the c a s e , the s c a l i n g a l g o r i t h m summarized by e q u a t i o n s ( 6 . 5 ) , ( 6 . 6 ) , (6.7) and (6.9) i s s t i l l e x p ected t o y i e l d r e a s o n a b l e r e s u l t s . To demonstrate the v i a b i l i t y of the proposed s c a l i n g scheme in an environment which c o n t a i n s r a m p - l i k e impedance components, we have c o n s t r u c t e d the t r u e v e l o c i t y p r o f i l e shown i n F i g u r e 6-4d. The a s s o c i a t e d t r u e r e f l e c t i v i t y , and RMS v e l o c i t y c u r v e are shown i n f i g u r e s 6-4a and 4e, r e s p e c t i v e l y . We band-pass (10-50Hz) the r e f l e c t i v i t y of F i g u r e 6-4a, add 5% random n o i s e , and s c a l e i t by a f a c t o r of 100 to o b t a i n the s y n t h e t i c seismogram ( F i g u r e 6-4b) t o be used i n t h i s example. 1 32 F i g u r e 6-4 Same as F i g u r e 6-1. T h i s time, a ra m p - l i k e component i s added to the v e l o c i t y model between 0.5 and 0.7 seconds. (a) The r e f l e c t i v i t y s e r i e s c o r r e s p o n d i n g to the v e l o c i t y model of F i g u r e 6-4d. (b) The s e r i e s i n (a) a f t e r the a p p l i c a t i o n of a 10-50 hz. bandpass f i l t e r and the a d d i t i o n of random n o i s e , (c) The s e r i e s i n (b) a f t e r LP d e c o n v o l u t i o n . (d) The t r u e v e l o c i t y model. (e) RMS v e l o c i t y c o r r e s p o n d i n g to the v e l o c i t y model in ( d ) . ( f ) The e s t i m a t e d i n t e r v a l v e l o c i t y o b t a i n e d from the f i v e RMS p i c k s marked on (e) v i a the B a c k u s - G i l b e r t i n v e r s i o n . 133 A p p l i c a t i o n of the l i n e a r programming d e c o n v o l u t i o n t o t h e da t a of F i g u r e 6-4b, y i e l d s the f u l l - b a n d r e f l e c t i v i t y e s t i m a t e 7 ( F i g u r e 6-4c), w h i l e the i n v e r s i o n of the RMS v e l o c i t i e s ( p i c k p o s i t i o n s f o r i n v e r s i o n purposes a r e marked i n F i g u r e 6-4e) g i v e s the e s t i m a t e d i n t e r v a l v e l o c i t y p r o f i l e of F i g u r e 6-4f. U s i n g e q u a t i o n s (6.5) and (6.7) we c o n v e r t e d the d a t a i n f i g u r e s 6-4d, 4c, and 4f i n t o pseudo-impedance and the r e s u l t s a r e shown i n F i g u r e 6-5a , 5b, and 5c, r e s p e c t i v e l y . I n s p e c t i o n of F i g u r e 6-5 r e v e a l s good shape-wise agreement between the t r u e pseudo-impedance and the one e s t i m a t e d from r f o r windows 1, 3, and 4. However, the c o r r e l a t i o n e x h i b i t e d i n the second window ( c o r r e s p o n d i n g t o the r a m p - l i k e impedance s t r u c t u r e ) i s q u i t e poor. U s i n g e q u a t i o n s (6.8) and (.6.9) on the data of F i g u r e s 6-5b and 5c w i t h N=512 and M=128, we o b t a i n the s e r i e s of e s t i m a t e d s c a l e f a c t o r s a which i s summarized i n T a b l e 6-2. The median of the s e r i e s c a l c u l a t e d by e q u a t i o n (6.8) i s 61, w h i l e t h a t c o r r e s p o n d i n g t o e q u a t i o n (6.9) i s 97. O b v i o u s l y , the l a t t e r e s t i m a t e i s good and s u g g e s t s t h a t a s c a l i n g a l g o r i t h m based on e q u a t i o n s (6.5), (6.6) (or ( 6 . 7 ) ) , and (6.9) can produce a c c e p t a b l e r e s u l t s even when the LP d e c o n v o l u t i o n scheme f a i l s a t some i s o l a t e d time windows. I t i s t h a t set of e q u a t i o n s which w i l l . b e used throughout the remainder of t h i s work. T a b l e 6-2: S c a l e f a c t o r e v a l u a t i o n f o r example i n F i g . 6-5. N=512, M=1.28 True s c a l e f a c t o r = = 100 I a ( f r o m eq.6 .6) a( f r o m eq.6.7) 1 72. 7.2. 2 23. -4. .3 44. 129. 4 61 . 97. Median (from eq. 6.8) = 61.. Mean (from eq. 6.8) = 50. Median (from eq. 6.9) = 97. Mean (from eq. 6.9) = 73. 1 35 Again we complete the p r o c e s s by r u n n i n g the LP a c o u s t i c impedance i n v e r s i o n a l g o r i t h m on the d a t a of F i g u r e 6-4b s c a l e d by 1/97 w i t h a s e t of c o n s t r a i n t s o b t a i n e d from the i n v e r t e d i n t e r v a l v e l o c i t y (shown i n F i g u r e 6 - 4 f ) . The r e s u l t , o v e r l a y e d on the t r u e v e l o c i t y f u n c t i o n i s shown i n F i g u r e 6-6. 6.4 REAL DATA EXAMPLES In the f o l l o w i n g examples we demonstrate the a p p l i c a t i o n of the above s c a l i n g method t o c a s e s where: (a) o n l y w e l l - l o g i n f o r m a t i o n i s a v a i l a b l e and (b) o n l y s t a c k i n g v e l o c i t i e s are a v a i l a b l e . Our i n p u t s t a c k e d s e c t i o n s (sampled a t 4Ms), have been s u b j e c t e d t o a normal ( p r e - s t a c k ) p r o c e s s i n g sequence which i n c l u d e d p r e d i c t i v e d e c o n v o l u t i o n f o r s h o r t p e r i o d m u l t i p l e s u p p r e s s i o n . Subsequently ( p o s t - s t a c k ) , we have whitened the s p e c t r a l band (10-40Hz) and c o r r e c t e d the phase of the i n t e r p r e t e r ' s wavelet .so t h a t the r e s i d u a l wavelet i s presumed zero-phase. These f i n a l s e c t i o n s c o n s t i t u t e the i n p u t t o our s c a l i n g a l g o r i t h m . 6.4.1 SCALING DATA USING WELL-LOG INFORMATION Given the s t a c k e d s e c t i o n shown i n F i g u r e 6-7 and the v e l o c i t y l o g c o r r e s p o n d i n g t o a w e l l i n the v i c i n i t y of t r a c e 70 ( F i g u r e 6-8), we wish t o p r e s c a l e our data f o r subsequent .1 36 Figure 6-5 (a) The true pseudo-impedance function, (b) The pseudo-impedance function c a l c u l a t e d from the reconstructed LP r e f l e c t i v i t y . (c) The pseudo-impedance function c a l c u l a t e d from the constructed i n t e r v a l v e l o c i t y model. 137 \ Figure 6-6 The v e l o c i t y function obtained from the constrained inversion superimposed on the true v e l o c i t y model. Input consists of the scaled seismogram (Figure 6-4b) and f i v e stacking v e l o c i t y estimates (marked on Figure 6-4e). 138 Figure 6-7 Input seismograms a f t e r post stack c o r r e c t i o n . s p e c t r a l whitening and phase 139 Figure 6-8 Sonic log from a well at the v i c i n i t y of trace 70. 1 40 impedance i n v e r s i o n p r o c e s s i n g . The f i r s t s t e p r e q u i r e d i n our s c a l i n g scheme i s the s p i k i n g d e c o n v o l u t i o n of the i n p u t data s e t . F i g u r e 6-9 r e p r e s e n t s t h e output of the l i n e a r programming d e c o n v o l u t i o n a p p l i e d t o our i n p u t . T h i s wide-band r e f l e c t i v i t y i s assumed t o be a r e a s o n a b l e r e p r e s e n t a t i o n of the l o c a l p r i m a r y r e f l e c t i v i t y f u n c t i o n , and upon i n t e g r a t i o n i t y i e l d s the c a l c u l a t e d pseudo-impedance s e c t i o n shown i n F i g u r e 6-10.. I n s p e c t i o n of t h e l a t t e r r e s u l t shows t h a t the low f r e q u e n c y t r e n d e x h i b i t e d by the v e l o c i t y l o g has been repr o d u c e d f a i t h f u l l y . C o n s e q u e n t l y , d i r e c t comparison of the c a l c u l a t e d pseudo-impedance t o the w e l l - l o g pseudo-impedance i s l i k e l y t o produce the d e s i r e d s c a l i n g f a c t o r . We have compared thes e q u a n t i t i e s f o r the near-w e l l t r a c e s (#*s -60 t o 80) and the r e s u l t a n t s c a l i n g f a c t o r was found t o be 1.3. As a l a s t s t e p i n our p r o c e s s we have r e s c a l e d the o r i g i n a l d a t a , i n t e r p o l a t e d a number of v e l o c i t i e s w i t h t h e i r a s s o c i a t e d c o n f i d e n c e bounds a l o n g a number of c o n s t r a i n t h o r i z o n s (marked on F i g u r e 6-10), and executed the a c o u s t i c impedance i n v e r s i o n p r o c e d u r e u s i n g the s c a l e d data and the a f o r e mentioned c o n s t r a i n t s . F i g u r e 6-11 r e p r e s e n t s the f i n a l output of t h e — impedance i n v e r s i o n scheme. In t h i s f i g u r e we have o v e r l a y e d the p s e u d o - v e l o c i t i e s (grey shades) over the wide band c o n s t r a i n e d r e f l e c t i v i t y s e c t i o n ( t h i s r e f l e c t i v i t y s e c t i o n i s c o n s i s t e n t with the observed w e l l - l o g i n f o r m a t i o n ) . Our f i n a l r e s u l t s a r e i n a form s u i t a b l e f o r normal 141 Figure 6-9 The data of Figure 6-7 a f t e r a p p l i c a t i o n of l i n e a r programming deconvolution. ^ y 1 42 F i g u r e 6- 1 0 The pseudo-impedance s e c t i o n o b t a i n e d v i a the i n t e g r a t i o n of the data i n F i g u r e 6-9. These data were used in c o n j u n c t i o n with the s o n i c l o g f o r the e s t i m a t i o n of a s c a l i n g f a c t o r . 1 43 i n t e r p r e t a t i o n work. Furthermore, with a p p r o p r i a t e c a u t i o n (as i n d i c a t e d by the c o n s t r a i n t ' s c o n f i d e n c e bounds s u p p l i e d t o the i n v e r s i o n ) , they may a l s o be used f o r i n f e r e n c e of some p e t r o -p h y s i c a l parameters l i k e p o r o s i t y , f l u i d s a t u r a t i o n and o v e r p r e s s u r e ( A n g e l e r i and C a r p i , 1982, B i l g e r i and Ademeno, 1982).. These parameters are measurable at the w e l l s i t e , and may be e x t r a p o l a t e d away from t h i s s i t e by comparing p s e u d o - v e l o c i t y v a l u e s along a given formation to those observed on the near w e l l t r a c e s . 6.4.2 SCALING DATA USING STACKING VELOCITY INFORMATION When w e l l - l o g i n f o r m a t i o n i s not a v a i l a b l e , and the observed l o c a l geology i s a p p r o p r i a t e ( t h a t i s , no s t e e p l y d i p p i n g s t r u c t u r e s are p r e s e n t ) , i t i s p o s s i b l e t o use the s t a c k i n g v e l o c i t i e s f o r s c a l i n g purposes. Given the data of F i g u r e 6-12, and the c o r r e s p o n d i n g set of s t a c k i n g v e l o c i t i e s we s t a r t our s c a l i n g scheme by i n v e r t i n g the s t a c k i n g v e l o c i t y set v i a the B a c k u s - G i l b e r t methodology. The r e s u l t of t h i s i n v e r s i o n , t h a t i s the smooth r e p r e s e n t a t i o n of the i n t e r v a l v e l o c i t i e s ( F i g u r e 6-13), i s c o n v e r t e d t o a pseudo-impedance s e c t i o n . Next, we have a p p l i e d the l i n e a r programming d e c o n v o l u t i o n to the data of F i g u r e 6-12 and c o n v e r t e d the output wide-band r e f l e c t i v i t i e s to a pseudo-impedance s e c t i o n . D i r e c t comparison of these two i n f o r m a t i o n s e t s y i e l d s a s c a l i n g f a c t o r of .008. 1 44 F i g u r e 6-11 The r e s u l t of a c o n s t r a i n e d i n v e r s i o n . F u l l band r e f l e c t i v i t y s e c t i o n i s o v e r l a i d on top of the p s e u d a - v e l o c i t y s e c t i o n . Grey shades r e p r e s e n t v e l o c i t i e s i n K f e e t / s e c . 1 45 Figure 6-12 Input stacked section. 146 256 192 128 64 |- 0.00 -I 0.25 0.50 „ •f-0.75 ~1 •r 1.00 J- 1.25 1.50 -1-1.75 •f- 2.50 2. 75 -I- 3.00 ZD Q f-2.00 ~ m -|-:2.25 CD O F i g u r e 6-13 I n t e r v a l v e l o c i t y s e c t i o n o b t a i n e d by the i n v e r s i o n of a set of s t a c k i n g v e l o c i t y p r o f i l e s c o r r e s p o n d i n g to the data i n F i g u r e 6-12. These i n t e r v a l v e l o c i t i e s were used i n c o n j u n c t i o n w i t h the u n c o n s t r a i n e d p s e u d o - v e l o c i t i e s i n the c a l c u l a t i o n of the r e q u i r e d s c a l i n g f a c t o r . 147 L a s t l y , we have r e s c a l e d the o r i g i n a l d a t a , and e x e c u t e d the a c o u s t i c impedance i n v e r s i o n procedure u s i n g s t a c k i n g v e l o c i t y c o n s t r a i n t s (Oldenburg et a l 1984). The o b t a i n e d p s e u d o - v e l o c i t i e s o v e r l a y e d on the c o n s t r a i n e d wide-band r e f l e c t i v i t i e s a r e shown i n F i g u r e 6-14. A g a i n , the f i n a l output of our i n v e r s i o n appears t o produce h i g h q u a l i t y r e s u l t s i n which the o b t a i n e d r e f l e c t i v i t i e s a r e c o n s i s t e n t w i t h the o b s e r v e d s t a c k i n g v e l o c i t i e s . I n t e r p r e t a t i o n of t h e s e r e s u l t s i s i n our o p i n i o n e a s i e r than t h a t of the o r i g i n a l s e c t i o n , w h i l e a h i g h e r degree of c o n f i d e n c e can be p l a c e d on t h e s e r e s u l t s s i n c e a l a r g e r i n f o r m a t i o n s e t has been used i n i t s c o n s t r u c t i o n . 6.5 CONCLUSION We have p r e s e n t e d a s c a l i n g method d e s i g n e d as a p r e p a r a t o r y s t e p to the a c o u s t i c impedance i n v e r s i o n of r e f l e c t i o n seismograms. T h i s method, which i s a p p l i c a b l e t o c a s e s where e i t h e r w e l l - l o g or s t a c k i n g v e l o c i t i e s are a v a i l a b l e , i s shown t o y i e l d good r e s u l t s on d a t a c o l l e c t e d i n a number of e x p l o r a t i o n a r e a s . N e v e r t h e l e s s , i n a r e a s where the l o c a l impedance — f u n c t i o n s do not c o n t a i n a number of sharp impedance d i s c o n t i n u i t i e s the proposed scheme may f a i l t o y i e l d a c c e p t a b l e r e s u l t s . The d e s i g n of our a l g o r i t h m i s l i m i t e d t o t h o se i n v e r s i o n t e c h n i q u e s which are c a p a b l e of y i e l d i n g a f i r s t o r der a p p r o x i m a t i o n to the low f r e q u e n c i e s g e n e r a l l y m i s s i n g i n normal 148 Figure 6-14 The r e s u l t of a constrained inversion (with stacking v e l o c i t i e s ) as applied to the data of Figure 6-12. Full-band r e f l e c t i v i t i e s are o v e r l a i d on the pseudo-velocity s e c t i o n . Grey shades represent v e l o c i t i e s in Kfeet/sec. 1 49 s e i s m i c work. The use of these a l g o r i t h m s , supplemented by w e l l -l o g i n f o r m a t i o n and s t a c k i n g v e l o c i t i e s , opens the door t o po w e r f u l data p r o c e s s i n g t e c h n i q u e s which u t i l i z e a l a r g e r i n f o r m a t i o n base t o o b t a i n both more r e l i a b l e r e f l e c t i v i t i e s , as w e l l as the a s s o c i a t e d p s e u d o - v e l o c i t y s e c t i o n . 150 CHAPTER V I I : PRE-INVERSION CORRECTION OF WAVELET RESIDUAL PHASE 7.1 INTRODUCTION The g o a l of most s t a n d a r d p r o c e s s i n g sequences i s t o produce a CDP s t a c k e d s e c t i o n i n which each t r a c e can be c o n s i d e r e d t o be the e a r t h ' s r e f l e c t i v i t y f u n c t i o n c o n v o l v e d w i t h a z e r o phase wavelet. D u r i n g the p r o c e s s i n g sequence e v e r y e f f o r t i s made t o e l i m i n a t e the e f f e c t s of the sou r c e s i g n a t u r e and m u l t i p l e s , and t o c o r r e c t f o r instr u m e n t r e c o r d i n g response, t r a n s m i s s i o n l o s s e s , and a t t e n u a t i o n . The d e c o n v o l u t i o n p r o c e d u r e s n o r m a l l y used to c o r r e c t f o r these e f f e c t s make a number of assumptions c o n c e r n i n g the phase p r o p e r t i e s of the e f f e c t s t h e m s e l v e s . When th e s e assumptions a r e v i o l a t e d , as they o f t e n a r e , the d e c o n v o l v e d output may not e x h i b i t the d e s i r e d zero-phase i n t e r p r e t e r ' s wavelet and a f u r t h e r phase c o r r e c t i o n w i l l be r e q u i r e d . Phase d i s t o r t i o n s a r i s e f o r a v a r i e t y of r e a s o n s . We w i l l g i v e some of the causes here and i n d o i n g so, we s h a l l d i s t i n g u i s h between e f f e c t s which o p e r a t e d i r e c t l y on the s o u r c e s i g n a t u r e ( f o r example, d i s p e r s i o n or p r o c e s s i n g f i l t e r s ) , and those which a r i s e due t o our l i m i t e d r e s o l u t i o n and the r e b y cause 'apparent' phase s h i f t s (e.g. i n t e r f e r i n g e v e n t s l i k e ghosts or s h o r t p e g - l e g m u l t i p l e s ) . The e f f e c t s of d i s p e r s i o n , a t t e n u a t i o n , and s u p e r c r i t i c a l 151 r e f l e c t i o n on a given source s i g n a t u r e have been e x t e n s i v e l y d i s c u s s e d i n the s e i s m o l o g i c a l l i t e r a t u r e over the past 50 y e a r s . Examples of the phenomenae f o r simple models are given i n Robinson (1979, 1980a and 1980b) where the m o d e l l i n g and removal of a t t e n u a t i o n r e l a t e d d i s p e r s i o n v i a the F o u r i e r s c a l i n g theorem i s d e s c r i b e d . The e f f e c t s of s u p e r c r i t i c a l r e f l e c t i o n s have been s t u d i e d by Arons and Yennie (1954). They i s o l a t e d a set of water bottom r e f l e c t i o n s and showed how the phase of the wavelet i s a l t e r e d f o r p o s t - c r i t i c a l r e f l e c t i o n s . The phenomenon of s u p e r c r i t i c a l r e f l e c t i o n i s d i s c u s s e d i n d e t a i l by Aki and R i c h a r d s (1980) . Phase d i s t o r t i o n s are a l s o i n c u r r e d through the a p p l i c a t i o n of d e c o n v o l u t i o n f i l t e r s . Most readers are f a m i l i a r with these e f f e c t s and hence we w i l l o n l y touch upon the matter l i g h t l y . The c a l c u l a t i o n of d e c o n v o l u t i o n f i l t e r s i s based upon a number of assumptions which are, at times, i n c o n s i s t e n t with the conducted f i e l d experiment. In the case of an e x p l o s i v e source, i t i s g e n e r a l l y assumed tha t both the c o r r e s p o n d i n g source s i g n a t u r e and the p r o p a g a t i o n e f f e c t s are minimum phase (Futterman, 1962; Knopoff, 1964; Sherwood and T r o r e y , 1965; and Wuenschel, 1965). Due to r e c o r d i n g and p r o c e s s i n g e f f e c t s , the above assumption i s only p a r t l y met, and hence the a p p l i c a t i o n of minimum phase d e c o n v o l u t i o n to f i e l d data may i n t r o d u c e an u n d e s i r e d phase d i s t o r t i o n . A n u m e r i c a l problem of great importance a l s o a r i s e s because of the b a n d - l i m i t e d nature of the recorded seismogram. Even i f the e x c i t a t i o n source i s t r u l y minimum phase, the c a l c u l a t i o n of the 152 phase spectrum from the observed amplitude spectrum r e q u i r e s i n f o r m a t i o n from p o r t i o n s of the spectrum which are not a v a i l a b l e to the p r o c e s s o r . In c o n v e n t i o n a l minimum phase d e c o n v o l u t i o n a l g o r i t h m s , the above problem m a n i f e s t s i t s e l f i n the form of an i l l c o n d i t i o n e d a u t o c o r r e l a t i o n matrix whose c o n d i t i o n number can be improved by the a d d i t i o n of a " w h i t e - n o i s e " c o n s t a n t to i t s d i a g o n a l terms. As expected, the magnitude of the "white n o i s e " parameter i s an important f a c t o r i n the c a l c u l a t i o n of the d e s i r e d i n v e r s e o p e r a t o r , but i t ' s value can a l s o s i g n i f i c a n t l y a l t e r the phase of the r e s i d u a l wavelet. When a V i b r o s e i s source i s c o n s i d e r e d , t h e r e are c o m p l e x i t i e s i n v o l v i n g the i n t e r a c t i o n of the v i b r a t o r h y d r a u l i c s , the b a s e p l a t e , and the e a r t h ' s s u r f a c e . The t o t a l e f f e c t of t h i s i n t e r a c t i o n i s to in t r o d u c e an unknown phase d i s t o r t i o n onto the observed Klauder wavelet ( F r a z e r , 1 9 8 3 ) . Furthermore, t h e c o n v e n t i o n a l p r o c e s s i n g route which employs minimum phase d e c o n v o l u t i o n f o l l o w e d by phase c o r r e c t i o n (Ristow and J u r c z y k , 1975), s u f f e r s from the same numerical problem d i s c u s s e d p r e v i o u s l y (see f o r example Gibson and Larn e r , 1983). F i n a l l y , g i v e n the frequency band l i m i t a t i o n s imposed by f i e l d and p r o c e s s i n g procedures, we may a l s o encounter 'apparent' phase d i s t o r t i o n s which are caused by event i n t e r f e r e n c e . T h i s type of d i s t o r t i o n i s extremely troublesome s i n c e i t may g i v e r i s e to r a p i d phase v a r i a t i o n s a c r o s s the observed d a t a . For example, l e t us c o n s i d e r the case of a stack of t h i n beds with l a t e r a l l y changing t h i c k n e s s e s . I f the two-way t r a v e l time of the wave through any of 1 53 these t h i n beds i s s h o r t e r than the time d u r a t i o n of the source s i g n a t u r e , a r r i v a l s from h o r i z o n s which are deeper than t h i s stack of t h i n beds w i l l f e a t u r e a c h a r a c t e r i s t i c s i g n a t u r e which may d i f f e r s u b s t a n t i a l l y from that a s s o c i a t e d with the s h a l l o w e r e v e n t s . Furthermore, s i n c e the t h i c k n e s s e s of the beds are a l s o changing l a t e r a l l y , the a s s o c i a t e d 'apparent' phase d i s t o r t i o n s may c r e a t e severe i n t e r p r e t a t i o n d i f i c u l t i e s . The c o m p l i c a t i o n s i n h e r e n t i n e s t i m a t i n g the phase d i s t o r t i o n i n t r o d u c e d by the many e f f e c t s mentioned above mean t h a t the phase of the f i n a l i n t e r p r e t e r ' s wavelet i s not w e l l determined even when the most c a r e f u l p r o c e s s i n g has been c a r r i e d out. T h i s p r e s e n t s major d i f f i c u l t i e s f o r f u r t h e r i n t e r p r e t a t i o n and p o s t - s t a c k p r o c e s s i n g t e c h n i q u e s . For example, l i n e a r programming s p i k i n g d e c o n v o l u t i o n (Levy and F u l l a g a r , 1981) and a c o u s t i c impedance i n v e r s i o n (Oldenburg ;et a l . , 1983, and Oldenburg et a l . , 1984) r e q u i r e t h a t the i n t e r p r e t e r ' s wavelet i s zero phase. The g o a l of t h i s work i s to show how the r e s i d u a l phase of a s e i s m i c wavelet may be estimated d i r e c t l y from the data and how the data may then be phase a d j u s t e d . We assume at the o u t s e t t h a t a l l of the phase d i s t o r t i o n s mentioned p r e v i o u s l y can be modelled by a wavelet whose phase i s a l t e r e d by a frequency independent c o n s t a n t ( f o r f u r t h e r d e t a i l s see Appendix 7-A). T h i s i s c e r t a i n l y the c o r r e c t assumption f o r m o d e l l i n g s u p e r c r i t i c a l r e f l e c t i o n s but i t i s only approximately true f o r other e f f e c t s l i k e d i s p e r s i o n which a r i s e s from constant Q a t t e n u a t i o n (Futterman, 1965; Robinson, 1980(a)). N e v e r t h e l e s s , Levy et a l . (1983) have used the c o n s t a n t 1 54 p h a s e - s h i f t c o n v o l u t i o n a l model and the Karhunen-Loeve t r a n s f o r m a t i o n i n order to e f f e c t phase c o r r e c t i o n of i s o l a t e d events e x h i b i t i n g frequency dependent phase d i s t o r t i o n s (see Appendix 7-B). The r e s u l t s of t h e i r approach were very encouraging so long as the a n a l y s e d frequency band-width of the s i g n a l s d i d not exceed about two and a h a l f o c t a v e s . T h i s g i v e s us added c o n f i d e n c e that we can apply the c o n s t a n t phase s h i f t model to the work pr e s e n t e d here. In t h i s work we w i l l a l s o adopt the c o n v o l u t i o n a l model f o r the s e i s m i c s i g n a l s . The r e s i d u a l phase of the s e i s m i c wavelet w i l l be e s t i m a t e d by f i r s t r o t a t i n g each s e i s m i c s i g n a l u s i n g a frequency independent phase s h i f t . T h i s r o t a t i o n w i l l be c a r r i e d out f o r a number of phases between - i r / 2 and i r / 2 . The varimax norm of the phase r o t a t e d s i g n a l s i s measured, and the t r a c e which e x h i b i t s the h i g h e s t norm value i s chosen as the d e s i r e d phase c o r r e c t e d r e p r e s e n t a t i o n . The varimax norm thus l i e s at the heart of our computational procedure and the j u s t i f i c a t i o n f o r u s i n g t h i s norm i s p r e s e n t e d i n the next s e c t i o n . 1 55 7.2 M E T H O D A N D E X P L A N A T O R Y E X A M P L E S The goal of t h i s s e c t i o n i s to develop a method whereby the phase of the r e s i d u a l wavelet can be determined d i r e c t l y from the data. The a l g o r i t h m to be developed here i s based on a model i n which the seismogram s ( t ) i s r e p r e s e n t e d by the c o n v o l u t i o n of a r e f l e c t i v i t y f u n c t i o n r ( t ) with a p o s s i b l y phase s h i f t e d zero-phase wavelet w(e,t), t h a t i s , s-( t) = r (t) * w ( e , t) (7.1) Th i s i s the same model used by Levy and Oldenburg (1981> i n t h e i r approach f o r c a r r y i n g out a d e c o n v o l u t i o n i n the presence of phase s h i f t e d s i g n a l s (see Appendix 7-A). In equation (7.1), w(e,t) i s r e l a t e d to the zero-phase wavelet w(t) by w(e,t) = cose w(t) + s i n e X f w ( t ) ] (7.2) where}-^[ ] denotes the H i l b e r t t r a n s f o r m . The form of equations (7.1) and (7.2) i s , as we have a l r e a d y d i s c u s s e d i n the i n t r o d u c t i o n , a reasonable mathematical model f o r many of the e f f e c t s observed on r e a l s i g n a l s . L e t us c o n s i d e r the set of phase s h i f t e d r e p l i c a t i o n s w(e,t) shown i n F i g u r e 7-1. Our f i r s t g o a l i s to f i n d a norm, which when i t acc e p t s w(e,t) as i t s argument, has an extremum when e = 0 . That i s , the norm i s extremized 1 56 when i t operates on a zero phase wavelet. The Varimax norm has p r e c i s e l y the p r o p e r t i e s t h a t we look f o r . The varimax of the time s e r i e s {s^} (i=l,N) i s d e f i n e d by I s." V(s) = (7.3) (Z s 2 ) 2 T h i s norm was used by Wiggins (1978) to i n d i c a t e ' s p i k i n e s s ' and i t serve d as the core of h i s minimum entropy d e c o n v o l u t i o n a l g o r i t h m . The maximum v a l u e of the varimax f o r any time s e r i e s i s u n i t y and t h i s o c c u r s when ( s - J i s a s i n g l e s p i k e at some a r b i t r a r y time sample. On the other hand, the varimax a t t a i n s a minimum when the s e r i e s c o n s i s t s of equal amplitude elements at each time sample. S i n c e a zero-phase wavelet c o n t a i n s the m a j o r i t y of i t s energy i n a r e l a t i v e l y narrow time d u r a t i o n ( i . e . i t s major l o b e ) , i t i s expected th a t V(w(0,t)) i s l a r g e r than V(w(e,t)) f o r e*0. Indeed, a p l o t of the varimax as a f u n c t i o n of e f o r a Klauder wavelet ( F i g . 7-1b) v e r i f i e s our e x p e c t a t i o n s . We note t h a t V(e) i s maximized when e=0, as d e s i r e d , and that the varimax c u r v e l o o k s very much l i k e a s i n u s o i d . We a l s o note t h a t the p e r i o d of t h i s s i n u s o i d i s 180° and that V(e) i s a l s o maximized when e=7r. T h i s curve i n d i c a t e s a fundamental ambiguity which i s e v i d e n t d i r e c t l y from e q u a t i o n (7.3); the varimax of a time s e r i e s and of i t s p o l a r i t y r e v e r s e d c o u n t e r p a r t are e q u a l . T h i s ambiguity i n p o l a r i t y i s something tha t may have to be c o r r e c t e d at a l a t e r s t a g e , a f t e r the phase c o r r e c t i o n has been completed. 1 57 1.80 I • \ • I • 1 • 1 • ' ' 1 -60 0 60 120 180 240' 300 Phase (Degrees) F i g u r e 7-1 In p a n e l (a) a K l a u d e r w a v e l e t has been phase r o t a t e d i n s t e p s of 4 5°. The v a r i m a x of t h e r o t a t e d w a v e l e t s , p l o t t e d as a f u n c t i o n of phase s h i f t a r e g i v e n i n ( b ) . A p p l i c a t i o n of t h e a u t o m a t i c phase c o r r e c t i o n a l g o r i t h m t o each of the t r a c e s i n (a) i s shown i n ( c ) . 1 5 8 T h e g o a l o f a n a u t o m a t i c p h a s e c o r r e c t i o n ( A P C ) a l g o r i t h m i s f i r s t t o e s t i m a t e t h e r e s i d u a l p h a s e o f e a c h t r a c e a n d t h e n t o p e r f o r m t h e d e s i r e d r o t a t i o n t o z e r o - p h a s e . T o e f f e c t t h i s c o r r e c t i o n , o u r a l g o r i t h m o p e r a t e s i n t h e f o l l o w i n g s e r i e s o f s t e p s : I . E q u a t i o n s ( 7 . 2 ) a n d ( 7 . 3 ) a r e u s e d t o c o m p u t e t h e v a r i m a x V ( e ) o f t h e a n a l y s e d s i g n a l f o r r o t a t i o n a n g l e s e r a n g i n g f r o m + 9 0 ° t o - 9 0 ° . I I . V ( e ) i s i n s p e c t e d t o i d e n t i f y t h e a n g l e w h i c h y i e l d s t h e h i g h e s t v a l u e o f t h e v a r i m a x . I I I . T h e o b s e r v e d s i g n a l i s r o t a t e d b y t o o b t a i n t h e p h a s e c o r r e c t e d s i g n a l . I V . F o r m u l t i t r a c e d a t a t h e r e s u l t s a r e i n s p e c t e d a n d p o s s i b l e c o r r e c t i o n s f o r p o l a r i t i e s a r e m a d e . T h e r e s u l t s o f t h e a p p l i c a t i o n o f o u r a l g o r i t h m t o t h e s e t o f w a v e l e t s s h o w n i n F i g u r e 7 - i a , a r e g i v e n i n F i g u r e ' 7 - 1 c . . T o d e m o n s t r a t e t h e e f f e c t s o f t h e b a n d l i m i t a t i o n u p o n t h e v a r i m a x n o r m m e a s u r e m e n t w e d e f i n e a q u a n t i t y S w h i c h m e a s u r e s t h e s t a n d o u t o f t h e v a r i m a x p e a k a b o v e t h e v a r i m a x l o w f o r t h e s i g n a l b e i n g a n a l y s e d : 159 S = x 100 U s i n g t h i s d e f i n i t i o n we c a l c u l a t e S f o r a s e t of s i n e f u n c t i o n s b a n d l i m i t e d by a t r a p e z o i d a l f i l t e r w i t h c o r n e r f r e q u e n c i e s f , , f 2 , f 3 , and f„. T y p i c a l v a r i a t i o n s of S are seen i n T a b l e 7-1. We n o t i c e t h a t the standout i s p a r t i c u l a r l y a f f e c t e d by the low fr e q u e n c y c u t o f f . T h i s i s an important o b s e r v a t i o n and i t has meant t h a t i n d o i n g the varimax c a l c u l a t i o n s on c o m p l i c a t e d s e i s m i c s i g n a l s we have made c o n s i d e r a b l e e f f o r t t o keep ( f , , f 2 ) as low as p o s s i b l e . In those c a s e s where the low f r e q u e n c y c u t o f f of the s e i s m i c s i g n a l has been too h i g h , we have f i r s t extended the s e i s m i c spectrum toward lower f r e q u e n c i e s by u s i n g the a u t o r e g r e s s i v e approach of Oldenburg e t a l . , (1983) and U l r y c h and Walker, (1983) ; see Appendix 6-A. We s h a l l d i s c u s s t h i s i n f u r t h e r d e t a i l l a t e r when we c o n s i d e r more c o m p l i c a t e d s i g n a l s which have a s m a l l varimax s t a n d o u t . To g a i n some i d e a about the a p p l i c a b i l i t y of our c o n s t a n t phase model t o a more r e a l i s t i c s i t u a t i o n , we w i l l c o n s i d e r the wavelet r e p r e s e n t e d by the 2 t r a n s f o r m (1 .1-Z) 2 x (1.. 7-Z) 1 3 . F i f t e e n d i f f e r e n t w a v e l e t s , r a n g i n g s u c c e s s i v e l y from minimum t o maximum phase ( d e l a y ) , have been g e n e r a t e d by t r a n s f e r r i n g one z e r o at a time from the o u t s i d e t o the i n s i d e of the u n i t c i r c l e . These wavelets and the c o r r e s p o n d i n g varimax cu r v e a r e shown i n f i g u r e s 7-2a and 7-2b, r e s p e c t i v e l y . We see t h a t the varimax i s maximized f o r the second wavelet and t h a t t h i s wavelet i s a r e a s o n a b l e a p p r o x i m a t i o n to a symmetric, t h a t i s z e r o phase w a v e l e t . The e s t i m a t e d phase a n g l e 1 60 " ^ \ £ f 3 ,fOHz (fl ,f 2 )Hz 30-35 40-45 50-55 60-65 1-5 44 53 60 65 5-10 6 15 23 31 10-15 .004 .8 4 10 TABLE 7-1 The varimax standout for the bandlimited sine function i s pl o t t e d as a function of ( f , , f 2 ) , the low frequency taper and (f 3,f«), the high frequency taper. The numbers shown in the table are the varimax peak standouts defined by the equation given in the text. 161 adjustments r e q u i r e d t o r o t a t e each t r a c e t o z e r o phase ar e shown i n F i g . 7-2c and the r o t a t e d t r a c e s a r e d i s p l a y e d i n F i g . 7-2d. I t seems, a t l e a s t f o r the simple case p r e s e n t e d above, t h a t the c o n s t a n t p h a s e - s h i f t a p p r o x i m a t i o n d i d a r e a s o n a b l e j o b i n c o n v e r t i n g v a r i o u s d e l a y w a v e l e t s i n t o an approximate zero-phase r e p r e s e n t a t i o n . The p r e v i o u s two examples show t h a t the varimax norm can be used to s e l e c t the z e r o phase wavelet when the r e f l e c t i v i t y f u n c t i o n c o n s i s t s of a s i n g l e s p i k e . The next q u e s t i o n t o be i n v e s t i g a t e d i s whether the approach w i l l g i v e good r e s u l t s when the r e f l e c t i v i t y f u n c t i o n i s more c o m p l i c a t e d . We know a t the o u t s e t t h a t f o r any g i v e n s o u r c e s i g n a l , a r e f l e c t i v i t y f u n c t i o n can be d e v i s e d t o make the method f a i l . C o n s i d e r the example o f f e r e d i n F i g . 7-3a. There a z e r o phase wavelet has been c o n v o l v e d w i t h a d i p o l e r e f l e c t i v i t y f u n c t i o n t o produce a seismogram which l o o k s v e r y much l i k e a 90° p h a s e - s h i f t e d wavelet. A p p l i c a t i o n of the varimax norm i n d i c a t e s t h a t t h i s t r a c e s h o u l d be s h i f t e d by 90° t o make i t zero phase and t h i s phase r o t a t i o n produces the phase c o r r e c t e d t r a c e shown i n F i g u r e 7-3b. The method can t h e r e f o r e f a i l and i t i s important t o remember t h i s . T h i s s i n g l e c o u n t e r example a l s o shows t h a t no mathematics can be d e v e l o p e d t o prove t h a t our method of u s i n g the varimax norm w i l l always produce the c o r r e c t phase s h i f t . N e v e r t h e l e s s , a s i n g l e d i p o l e i s not a g e o l o g i c a l l y r e a l i s t i c r e f l e c t i v i t y f u n c t i o n . G e o l o g i c r e f l e c t i v i t i e s a r e much more complex and w i l l o f t e n c o n s i s t of a number of w e l l s e p a r a t e d major r e f l e c t o r s i n a d d i t i o n to a l a r g e number of d i p o l e r e f l e c t o r s h a v i n g v a r i a b l e d i s t a n c e between the up and down s p i k e s . The c u m u l a t i v e 1 6 2 0.24 Wavelet * Time Figure 7-2 Fif t e e n wavelets having d i f f e r e n t delay are shown .in (a). The varimax of the wavelets in (a) are shown in (b). The phase s h i f t s required to rotate each of the wavelets in (a) so that the varimax is maximized i s shown in (c). Panel (d) shows the r e s u l t s of applying the phase s h i f t s given in ( c ) . 1 6 3 e f f e c t of the d i f f e r e n t d i p o l e s would be t o c o n t r o l the l e v e l of the varimax norm and one would hope t h a t the i s o l a t e d r e f l e c t o r s would have enough i n f l u e n c e on the varimax t h a t they determine the proper phase s h i f t . To t e s t the r e l i a b i l i t y of our phase e s t i m a t i o n method i n a r e a l i s t i c environment we a p p l y the t e c h n i q u e to r e f l e c t i v i t y f u n c t i o n s o b t a i n e d from w e l l l o g s . E i g h t v e l o c i t y l o g s used f o r t h i s t e s t a r e shown i n F i g . 7-4a. The r e f l e c t i v i t i e s c o r r e s p o n d i n g t o t h e s e l o g s a r e shown i n F i g . 7-4b. Each r e f l e c t i v i t y i s then bandpassed and used as i n p u t f o r the APC a l g o r i t h m . The phase c o r r e c t i o n r e t u r n e d by the APC a l g o r i t h m i s t a b u l a t e d i n T a b l e 7-2 as a f u n c t i o n of the bandpass p a r a m e t e r s . S i n c e the r e f l e c t i v i t y f u n c t i o n i n p u t i n t o the APC program i s the t r u e r e f l e c t i v i t y c o n v o l v e d w i t h a z e r o phase wavelet, i t would be hoped t h a t the a l g o r i t h m would have r e t u r n e d a z e r o v a l u e f o r the phase r o t a t i o n f o r a l l t e s t s . However, r o t a t i o n s of up t o 30° do not g r e a t l y a l t e r the shape of a wavelet and t h e r e f o r e we s h a l l c o n s i d e r t h a t the a l g o r i t h m has produced a s u c c e s s f u l r e s u l t i f -30° < e < 30°. Having adopted t h i s as a c r i t e r i o n of s u c c e s s we can examine T a b l e 7-2 more c l o s e l y . We n o t i c e s p e c i f i c a l l y t h a t f o r f i x e d ( f 3 , f 4 ) , the r e s u l t s i n T a b l e 7-2a are g e n e r a l l y s u p e r i o r t o those i n T a b l e 7-2b. T h i s s u g g ests t h a t our phase c o r r e c t i o n a l g o r i t h m i s h e a v i l y dependent on the r a t i o of the main energy lobe t o the s i d e l o b e s e x h i b i t e d by the r e s i d u a l w a v e l e t . Thus every e f f o r t s h o u l d be made to maximize t h i s r a t i o by m a i n t a i n i n g the lower frequency components and a p p r o p r i a t e l y shaping the wavelet's a m p l i t u d e spectrum to a v o i d e x c e s s i v e s i d e l o b e s r i n g i n g . t i m e Figure 7-3 Trace (a) i s a bandpassed dipole r e f l e c t i v i t y . It resembles a 90° wavelet. A p p l i c a t i o n of the automatic phase c o r r e c t i o n algorithm produces the rotated trace (b). 1 6 5 (f , , f : ) - (5,10)Hz (a) well ) 30-35 40-45 50-55 60-65 a 27 0 -27 18 b 54 -63 72 -72 c 45 -27 27 -9 d -27 18 0 -18 e 63 18 27 9 f -36 -63 36 18 g 9 9 -27 -36 h -18 -9 0 -9 ( f i . f a ) - (10,15)H* (b) \ ( f i , f « ) w e l l t f ^ ^ .30-35 40-45 50-55 60-65 .a 90 -54 -45 36 b 81 72 81 54 c -81 45 72 -36 d -90 45 -18 -18 -e 81 9 9 f -81 63 -72 -9 g .27 9 -36 9 h 90 0 27 54 TABLE 7-2 The automatic phase c o r r e c t i o n a l g o r i t h m i s a p l i e d to each of the r e f l e c t i v i t i e s shown i n F i g . 7-4. The r e f l e c t i v i t i . e s are f i r s t bandpassed with a l i n e a r taper extending ( f , , f 2 ) on the low frequency s i d e and ( f 3 f f » ) on the high frequency s i d e . The numbers shown i n the t a b l e are the phase r o t a t i o n s i n degrees r e q u i r e d to maximize the varimax a f t e r the bandpassing. 1 66 0 . 4 0 . 6 0 . S time (sec) 4>~ (b) J i * r 1 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1.0 time (sec) Figure 7-4 Eight v e l o c i t y logs used to test the automatic phase correction algorithm are shown on the l e f t . The f u l l band r e f l e c t i v i t i e s corresponding to these v e l o c i t i e s are shown on the r i g h t . 167 Now l e t us c o n s i d e r a s p e c i f i c column (f 3,f„) = (40,45) Hz i n Ta b l e 7-2a. We n o t i c e t h a t the APC a l g o r i t h m has ' p r o v i d e d a s u c c e s s f u l r e s u l t f o r 6 of the 8 l o g s ; o n l y r e f l e c t i v i t i e s from l o g s (b) and ( f ) would have been a d j u s t e d i n c o r r e c t l y . We f e e l t h a t t h i s i s a good s u c c e s s r a t i o f o r the APC program. There are some d i f f e r e n c e s i n the a d j a c e n t columns but the d i f f e r e n c e s a r e not extreme, except when ( f 3 , f s ) i s reduced to (30,35), a v a l u e which i s too low and produces too many f a i l u r e s . Such d i f f e r e n c e s w i t h the change i n the h i g h f r e q u e n c y f i l t e r parameters a r e expected because of the g e o l o g i c t u n i n g of the s i g n a l . That i s , whether a r e f l e c t i v i t y d i p o l e l o o k s l i k e a s i n g l e 90° wavelet or two z e r o phase wa v e l e t s depends upon the s e p a r a t i o n between the r e f l e c t o r s and the bandpass parameters. We a r e v e r y encouraged by these r e s u l t s . The a l g o r i t h m has proven r e l i a b l e i n the m a j o r i t y of cases p r o v i d i n g t h a t low f r e q u e n c i e s have been used i n the p r o c e s s . We a l s o r e a l i z e t h a t the suc c e s s r a t i o can be i n c r e a s e d by u s i n g the APC a l g o r i t h m o n l y i n c e r t a i n t y p e s of g e o l o g i c s t r u c t u r e s . We know, f o r i n s t a n c e , t h a t the APC a l g o r i t h m w i l l f a i l ( g e n e r a l l y ) i f the r e f l e c t i v i t y f u n c t i o n i s G a u s s i a n . In such cases the a l g o r i t h m has an e q u a l p r o b a b i l i t y of r e t u r n i n g a phase s h i f t between -90° and 90°. R e t u r n i n g to F i g . 7-4 we see t h a t l o g b has a r e f l e c t i v i t y f u n c t i o n which resembles a Gau s s i a n d i s t r i b u t i o n ; t h i s may account f o r the f a i l u r e of the APC a l g o r i t h m t o o p e r a t e upon t h a t r e f l e c t i v i t y . We would expect the APC a l g o r i t h m t o work w e l l when the r e f l e c t i v i t y i s non-Gaussian. T h i s i s c o n f i r m e d by the r e s u l t s i n T a b l e 7-2a. 1 68 The importance of the low frequency i-nf ormat i o n i n a p p l y i n g the varimax c r i t e r i o n has been c l e a r l y i n d i c a t e d i n the p r e c e d i n g examples. I t o f t e n happens, however, that' the ' f i e l d d a t a are m i s s i n g i n f o r m a t i o n below 10 or 15 Hz. In such cases b e t t e r r e s u l t s can be expected by f i r s t e x t e n d i n g the s e i s m i c spectrum i n t o the low frequency r e g i o n by u s i n g a u t o r e g r e s s i v e (AR) t e c h n i q u e s and then a p p l y i n g the APC a l g o r i t h m . To use the AR approach we f o l l o w the work i n i t i a l l y proposed by Oldenburg e t a l . (1983) and m o d i f i e d by Walker and U l r y c h (1983). The low and h i g h frequency c u t o f f s of the s e i s m i c spectrum a r e f i r s t e s t i m a t e d from the d a t a and the spectrum i s m o d e l l e d as an AR p r o c e s s of o r d e r p. The Yule-Walker a l g o r i t h m i s used t o e v a l u a t e the AR f i l t e r c o e f f i c i e n t s and the c o n s t r u c t e d f i l t e r i s then used to p r e d i c t the m i s s i n g low f r e q u e n c y v a l u e s . The . b e n e f i t s of u s i n g the AR f r e q u e n c y e x t e n s i o n a r e shown i n the next example. We have taken the r e f l e c t i v i t i e s i n F i g . 7-4 and bandpassed them w i t h the t r a p e z o i d f i l t e r . The low f r e q u e n c y parameters f o r the t r a p e z o i d were always ( f , , f 2 ) = ( 1 0 , 1 5 ) Hz but t h e h i g h f r e q u e n c y v a l u e s ( f 3 , f A ) were v a r i a b l e . A f t e r each bandpassing we used the frequency band ( f 2 , f 3 ) t o generate the s p e c t r a l v a l u e s a t f r e q u e n c i e s s m a l l e r than f 2 . The r e s u l t s from the APC a l g o r i t h m when i n f o r m a t i o n 5-10 Hz was used and when i n f o r m a t i o n 1-5 'Hz was used a r e shown i n T a b l e 7-3. We see t h a t the s u c c e s s r a t i o i s v e r y good, e x p e c i a l l y f o r T a b l e 7-3b. A l t h o u g h our a l g o r i t h m seems to y i e l d r e a s o n a b l y r e l i a b l e r e s u l t s , i t i s recommended t h a t , whenever p o s s i b l e , a t e s t t r i a l on a w e l l - l o g s y n t h e t i c r e f l e c t i v i t y s h o u l d be made p r i o r to the 169 (f i.f :) - (5,10)Hi 30-35 40-4 5 50-55 60-65 a -27 -9 -18 -18 b 81 72 -18 -18 c 18 -81 18 -9 d 45 9 -18 -27 e -63 -36 45 -9 f -9 9 -36 9 g 0 9 72 54 h 54 9 9 ( f l . f j ) - (l,5)Hz •30-35 40-4 5 50-55 60-65 a -18 -18 -54 9 b 63 -18 -9 -9 c 18 -90 9 -9 36 9 -9 -18 •e 0 -27 27 -9 f -9 0 -27 9 8 1 0 9 63 36 h i 1 36 -9 0 0 TABLE 7-3 The automatic phase c o r r e c t i o n a l g o r i t h m i s a p l i e d to each of the r e f l e c t i v i t i e s shown i n F i g . 7-4. The r e f l e c t i v i t i e s ' a r e f i r s t bandpassed w i t h a l i n e a r t a p e r e x t e n d i n g (10,15)Hz -pn the low frequency s i d e and (f 3,f„) on the h i g h f r e q u e n c y : s i d e . U s i n g the AR a l g o r i t h m we have extended the freq u e n c y i n f o r m a t i o n band ( l 5 , f 3 ) H z i n t o the low frequency p o r t i o n of the spectrum and then bandpassed the r e s u l t as i n d i c a t e d by the c o r n e r f r e q u e n c i e s ( f , , f 2 ) and ( f . 3 , f -.».) . The'numbers shown i n the t a b l e are the phase r o t a t i o n s - r e q u i r e d to maximize the varimax a f t e r the bandpassing. 170 a p p l i c a t i o n of the a l g o r i t h m to a g i v e n set of d a t a . T h i s t e s t s h o u l d e s t a b l i s h the r e l i a b i l i t y of the phase c o r r e c t i o n o p e r a t i o n f o r the g i v e n zone of i n t e r e s t . A d d i t i o n a l l y , i f the c o r r e c t i o n a n g l e e s t i m a t e d from the w e l l - l o g s y n t h e t i c d i f f e r s l a r g e l y from 0°, i t may be prudent to f o l l o w the APC a l g o r i t h m run by the a p p l i c a t i o n of a s i n g l e phase c o r r e c t i o n t o the whole s e c t i o n so t h a t the f i n a l r e s u l t w i l l conform w i t h the w e l l - l o g i n f o r m a t i o n . B e f o r e we p r o c e e d t o some p r a c t i c a l c o n s i d e r a t i o n s and r e a l d a t a c a s e s , we would l i k e t o p r e s e n t one more i d e a . I f we c o n s i d e r the example summarized i n F i g u r e 7-2 as r e p r e s e n t a t i v e of a l a r g e number of c a s e s , i t seems t h a t our a l g o r i t h m opens the door f o r an a d d i t i o n a l approach t o s i g n a t u r e d e c o n v o l u t i o n . The t a r g e t of s i g n a t u r e d e c o n v o l u t i o n schemes i s t o r e c o v e r a b a n d l i m i t e d z e r o phase r e p r e s e n t a t i o n of the r e f l e c t i v i t y f u n c t i o n . L e t the seismogram be s ( t ) = r ( t ) * w ( t ) (7.4) where r ( t ) i s the t r u e r e f l e c t i v i t y and w(t) i s a s e i s m i c w a v e l e t . T a k i n g the F o u r i e r t r a n s f o r m of (7.4) l e a d s to S(u) = R ( L > ) W(CJ) In most c a s e s the a m p l i t u d e spectrum |W(u)| of the wavelet can be r e l i a b l y d e termined but i t s phase spectrum <j>{u) cannot. The work i n F i g . 7-2 however shows t h a t the e f f e c t s of </>(to) can a p p r o x i m a t e l y be compensated f o r by a p p l y i n g a f r e q u e n c y independent c o n s t a n t phase 171 adjustment. That i s , f o r any of the mixed d e l a y wavelet, t h e r e e x i s t s a phase s h i f t e such t h a t ^ J - - 1 { W(u) exp[i<*>(w)] exp[±ie] } (7.5) i s a p r o x i m a t e l y a zero phase s i g n a l . 1 denotes the i n v e r s e F o u r i e r t r a n s f o r m ) . In e q u a t i o n (7.5) the + s i g n i s taken f o r p o s i t i v e f r e q u e n c i e s and the - s i g n f o r n e g a t i v e f r e q u e n c i e s . We now suggest t h a t s i g n a t u r e d e c o n v o l u t i o n can be c a r r i e d out u s i n g two s u c c e s s i v e independent o p e r a t i o n s . In the f i r s t s t e p we whiten the r e f l e c t i v i t y spectrum by d i v i d i n g the spectrum of the seismogram by the amplitude spectrum of the e s t i m a t e d wavelet. We then attempt to n u l l i f y the e f f e c t s of the wavelet's phase spectrum u s i n g the o p e r a t o r e x p [ i e ] . Our f i n a l r e s u l t w i l l be S(u) exp[±ie] _ R ( w ) exp[ i U ( u ) + e) ]•» R (>o) |W(u)| The i n v e r s e F o u r i e r t r a n s f o r m of t h i s f u n c t i o n w i l l y i e l d a r e f l e c t i v i t y s e r i e s r e p r e s e n t i n g (approximately) the c o n v o l u t i o n of the t r u e r e f l e c t i v i t y s e r i e s with a zero-phase r e s i d u a l wavelet. To demonstrate t h i s d e c o n v o l u t i o n approach we use the f o l l o w i n g example. The mixed delay wavelet of F i g u r e 7-5b i s convolved with the r e f l e c t i v i t y f u n c t i o n of F i g u r e 7-5a to y i e l d the input seismogram shown i n F i g u r e 7-5d. D i v i d i n g the spectrum of the input seismogram by the amplitude spectrum of the wavelet (weighted by a 5% water l e v e l parameter) and i n v e r s e F o u r i e r t r a n s f o r m i n g the 1 72 r e s u l t , y i e l d s the t r a c e of F i g u r e 7-5e. A p p l i c a t i o n of the p h a s e - c o r r e c t i o n a l g o r i t h m to the l a t t e r gave the f i n a l d e c o n v o l v e d output which i s shown i n F i g u r e 7-5f. Comparison of t h i s r e s u l t w i t h the b a n d - l i m i t e d r e f l e c t i v i t y f u n c t i o n ( F i g u r e 7-5c) shows t h a t t h i s d e c o n v o l u t i o n r o u t e can y i e l d the d e s i r e d s o l u t i o n . C o n s i d e r i n g the d i f f i c u l t i e s so o f t e n a s s o c i a t e d w i t h the e s t i m a t i o n of the wavelet's phase spectrum i t seems t h a t the proposed d e c o n v o l u t i o n r o u t e o f f e r s an a t t r a c t i v e a l t e r n a t i v e to c o n v e n t i o n a l methods. 7.3 REAL DATA EXAMPLES In a p p l y i n g the phase c o r r e c t i o n p r o c e d u r e to r e a l data s e t s we want to enhance the a b i l i t y f o r the a l g o r i t h m t o produce c o r r e c t r e s u l t s . We o u t l i n e below a number of s t e p s which are d e s i g n e d t o i n c r e a s e the s u c c e s s r a t i o of the proposed a l g o r i t h m . I . I f low freq u e n c y i n f o r m a t i o n i s m i s s i n g , the spectrum of the data s h o u l d be extended i n t o the range (5-10 Hz) u s i n g the AR p r e d i c t i o n scheme. I I . To a v o i d end e f f e c t s , the da t a s h o u l d be t a p e r e d p r i o r to the a p p l i c a t i o n of the APC a l g o r i t h m . I I I . To a v o i d i n c o n s i s t e n t phase r o t a t i o n s which may a r i s e due to l a r g e l a t e r a l v a r i a b i l i t y of the p r o c e s s e d d a t a , the a l g o r i t h m s h o u l d determine the a p p r o p r i a t e phase c o r r e c t i o n v i a the a n a l y s i s of s u c c e s s i v e groups of seismograms. The number of t r a c e s i n each 1 73 +— h 0.0 0.2 0.4 time (sec) F i g u r e 7-5 The work summarized by F i g u r e 7-2 suggests t h a t zero phase d e c o n v o l u t i o n f o l l o w e d by a phase c o r r e c t i o n may p r o v i d e an a l t e r n a t e way of c a r r y i n g out s i g n a t u r e d e c o n v o l u t i o n . The t r u e r e f l e c t i v i t y f u n c t i o n i s shown i n (a) and a mixed d e l a y wavelet i s gi v e n i n ( b ) . The bandpassed r e f l e c t i v i t y f u n c t i o n i s shown i n ( c ) . The seismogram i n (d) has undergone zero phase d e c o n v o l u t i o n by d i v i d i n g i t s spectrum by the spectrum of the wavelet i n ( b ) . The r e s u l t of t h i s d e c o n v o l u t i o n i s shown i n ( e ) . A p p l i c a t i o n of the phase c o r r e c t i o n a l g o r i t h m y i e l d s the f i n a l r e s u l t i n ( f ) ; i t may be compared d i r e c t l y w i t h the bandpassed r e f l e c t i v i t y i n ( c ) . 1 74 a n a l y s e d group depends on the v a r i a b i l i t y e x h i b i t e d by the d a t a . As a r u l e , the number of t r a c e s i n an a n a l y s e d group s h o u l d not be s m a l l e r than t e n or twenty. I f i t i s f e l t t h a t the l a t e r a l v a r i a b i l i t y of the e s t i m a t e d phases i s too l a r g e , then the APC a l g o r i t h m s h o u l d be r e r u n u s i n g a l a r g e r number of s e i sinograms i n each group. IV. The e s t i m a t e d phase c o r r e c t i o n s s h o u l d be smoothly i n t e r p o l a t e d so t h a t each t r a c e can be a s s i g n e d a phase s h i f t . In a d d i t i o n , i t may be n e c e s s a r y t o c o r r e c t f o r the p o l a r i t y t o con s e r v e t r a c e - t o - t r a c e t r u e p o l a r i t y r e l a t i o n s . V. To a v o i d smoothing e f f e c t s due to source s i g n a t u r e time dependency, the a n a l y s e d time window f o r the phase c o r r e c t i o n a l g o r i t h m s h o u l d be c e n t e r e d around the zone of i n t e r e s t . I t i s important t o remember t h a t the phase c o r r e c t i o n which i s s u i t a b l e f o r a g i v e n time window may perf o r m p o o r l y o u t s i d e t h i s window. Our e x p e r i e n c e suggests t h a t the a n a l y s e d time window s h o u l d not be s h o r t e r than about 500 m i l l i s e c o n d s nor l o n g e r than about 1.0 second. V I . When w e l l - l o g i n f o r m a t i o n i s a v a i l a b l e , the a l g o r i t h m s h o u l d be a p p l i e d to the b a n d - l i m i t e d s y n t h e t i c r e f l e c t i v i t y f u n c t i o n to determine the phase a n g l e e ^ . I f e , ^ d i f f e r s l a r g e l y from 0 ° , the output from the phase c o r r e c t i o n a l g o r i t h m s h o u l d be f u r t h e r r o t a t e d by -e^i degrees. V I I . The a l g o r i t h m s h o u l d not be a p p l i e d to time s e c t i o n s i n which the g e o l o g i c r e f l e c t i v i t y i s a p p r o x i m a t e l y G a u s s i a n . 1 75 Having s p e c i f i e d these p r a c t i c a l c o n s i d e r a t i o n s we s h a l l now t u r n to the r e a l data examples. As with many other p r o c e s s i n g t e c h n i q u e s , the g o a l of our newly de v e l o p e d a l g o r i t h m i s to improve the i n t e r p r e t a b i l i t y of the g i v e n s e i s m i c s e c t i o n . To show t h a t t h i s g o a l can be a c h i e v e d we w i l l r e c o v e r the l o g a r i t h m of the r e l a t i v e impedance f u n c t i o n (Oldenburg et a l . 1983) of a CMP s t a c k e d s e c t i o n both b e f o r e and a f t e r phase c o r r e c t i o n . The s u c c e s s of the APC a l g o r i t h m w i l l be shown by comparing the r e c o v e r e d r e l a t i v e impedance f u n c t i o n s t o a v e l o c i t y l o g measured a t a w e l l s i t e s l i g h t l y o f f the p r o c e s s e d l i n e . The d a t a of F i g u r e 7-6a c o n s t i t u t e the output of a c o n v e n t i o n a l p r o c e s s i n g sequence. We have f u r t h e r a p p l i e d a p o s t - s t a c k zero-phase d e c o n v o l u t i o n t o y i e l d the r e s u l t s i n F i g u r e 7-6b. Those data were i n p u t i n t o the APC a l g o r i t h m to produce the r e s u l t s shown i n F i g u r e 7-6c. The u n c o n s t r a i n e d (no e x t e r n a l w e l l - l o g or s t a c k i n g v e l o c i t y c o n s t r a i n t s - a p p l i e d ) l i n e a r programming a c o u s t i c impedance i n v e r s i o n a l g o r i t h m was a p p l i e d to the d a t a i n f i g u r e s 7-6b and 7-6c and the r e s u l t s f o r twenty c o n s e c u t i v e t r a c e s near the w e l l are shown i n F i g u r e 7-7. The impedance t r a c e s t o the r i g h t of the w e l l - l o g c o r r e s p o n d t o the phase c o r r e c t e d data (e=*80°) w h i l e those those to the l e f t c o r r e s p o n d to the u n c o r r e c t e d d a t a . O b v i o u s l y , the a p p l i c a t i o n of the phase c o r r e c t i o n a l g o r i t h m t o t h i s data r e s u l t e d i n a f a i r l y good match between the w e l l - l o g and the r e c o v e r e d r e l a t i v e impedance f u n c t i o n s . Furthermore, comparison of the f a u l t e d zone ( t r a c e s 50 to 70, between 1.6 t o 2.0 seconds) i n f i g u r e s 7-6a, b, and c, shows a b e t t e r f a u l t d e f i n i t i o n on the phase c o r r e c t e d o u t p u t . To complete t h i s example we show i n F i g u r e 7-8 the r e s u l t of 1 76 Figure 7-6a CMP seismograms representing the output of a standard sequence. 1 77 Figure 7-6b The data of F i g . 7-6a a f t e r zero-phase deconvolution. 178 Figure 7-6c Applica t i o n of the.APC algorithm to the data, shown in F i g . 7-6b produced the phase-corrected data shown in t h i s f i g u r e . Successive groups of 10 traces were analysed for the whole data set and a l l phase co r r e c t i o n s were close to 90°. 179 the u n c o n s t r a i n e d l i n e a r programming a l g o r i t h m a p p l i e d to the complete data set of F i g u r e 7-6c. We have o v e r l a y e d the rec o v e r e d r e f l e c t i v i t y on the grey shaded r e l a t i v e impedance f u n c t i o n s . The c o n s i s t e n t behaviour of these f u n c t i o n s a f t e r the a p p l i c a t i o n of the phase c o r r e c t i o n i s very n o t i c e a b l e . 7.4 CONCLUDING COMMENTS In t h i s chapter we have put f o r t h a method f o r e s t i m a t i n g the r e s i d u a l phase of the s e i s m i c wavelet. The combined use of the varimax norm and the concept of a c o n s t a n t phase r o t a t i o n p ermits us to e s t i m a t e t h i s phase d i r e c t l y from the d a t a . The method i s not guaranteed to work i n every i n s t a n c e but work done with a c q u i r e d v e l o c i t y l o g s shows t h a t the method should work i n the m a j o r i t y of c a s e s . Moreover, m o d i f i c a t i o n s such as extending the frequency band of the spectrum u s i n g a u t o r e g r e s s i v e t e c h n i q u e s , s e l e c t i n g p a r t i c u l a r g e o l o g i c environments i n which to.use the a l g o r i t h m , and a p p l y i n g the a l g o r i t h m to groups of s e i s m i c t r a c e s , w i l l i n c r e a s e the chances f o r success. Our work with t h i s a l g o r i t h m to date has been very p r o m i s i n g , and we now a p p l y the APC a l g o r i t h m on a r o u t i n e b a s i s to c o n v e n t i o n a l l y p r o c e s s e d d a t a . In a great may cases, we f i n d t h a t the i n t e r p r e t a b i l i t y of the s e c t i o n i s improved and we a l s o f i n d b e t t e r agreement w i t h w e l l logs when the i n v e r s i o n i s c a r r i e d out on phase c o r r e c t e d d a t a . 180 80 70 80 70 Figure 7-7 The r e l a t i v e acoustic impedance shown on the l e f t has been obtained by i n v e r t i n g twenty seismograms near trace #70 in Figure 7-6a. These r e s u l t s do not compare favourably with the v e l o c i t y log plo t t e d in the center of the diagram. The r e l a t i v e acoustic impedance obtained by i n v e r t i n g the phase corrected data in Figure 7-6c i s shown on the right side of t h i s f i g u r e . 181 Figure 7-8 The l i n e a r programming r e f l e c t i v i t i e s obtained from the phase corrected data in Figure 7 -6c are shown in t h i s f i g u r e . They are o v e r l a i d with the grey shade of the r e l a t i v e acoustic impedance. 182 CHAPTER V I I I : SUMMARY It i s the i r o n y of the s u b j e c t of t h i s work that at i t s c o n c l u s i o n I s t i l l have to say t h a t a l l the p r e c e d i n g m a t e r i a l may not be s u f f i c i e n t f o r p r a c t i c a l implementation of the i n v e r s i o n scheme. Indeed, a number of st e p s are r e q u i r e d i f b a n d l i m i t e d s e i s m i c data are to be s u c c e s s f u l l y i n v e r t e d to y i e l d a f u l l band a c o u s t i c impedance. Each i s important and f a i l u r e of any s t e p can r e s u l t i n a degraded end pr o d u c t . In the f o l l o w i n g I w i l l o u t l i n e i n s t e p wise form some c o n s i d e r a t i o n s and o p e r a t i o n s which when f o l l o w e d w i l l enhance the l i k e l i h o o d of success f o r the complete i n v e r s i o n . STEP 1: P r e i n v e r s i o n P r o c e s s i n g The p r o c e s s i n g sequence a p p l i e d to s e i s m i c data p r i o r to impedance i n v e r s i o n w i l l have important e f f e c t s on the outcome of the i n v e r s i o n . Of p a r t i c u l a r importance are those p r o c e s s e s which a f f e c t the r e l a t i v e amplitudes of the r e f l e c t o r s , the e f f e c t i v e bandwidth of the seismogram, and the phase of the r e s i d u a l source s i g n a l . I t i s g e n e r a l l y i m p o s s i b l e to o b t a i n t r u e amplitude s e c t i o n s without 'a p r i o r i ' knowledge of the su b s u r f a c e geology. Sin c e t h i s i n f o r m a t i o n i s not u s u a l l y a v a i l a b l e , one should do h i s best u s i n g 183 programmed gain c o n t r o l with the hope t h a t subsequent i n c o r p o r a t i o n of w e l l - l o g i n f o r m a t i o n and s t a c k i n g v e l o c i t y i n f o r m a t i o n i n t o the i n v e r s i o n w i l l overcome any amplitude d i s c r e p a n c i e s present i n the data. I t i s recommended that AGC with s h o r t window l e n g t h s not be a p p l i e d to the d a t a . In many cases, l a t e r a l energy b a l a n c i n g of the data i s r e q u i r e d . A module which n o r m a l i z e s t r a c e amplitudes s h o u l d be a p p l i e d to produce a balanced s e c t i o n , t h a t i s , one i n which the energy or the sum of the a b s o l u t e v a l u e s of each of the seismograms i s n o r m a l i z e d to a given c o n s t a n t . I f t h i s i s not done and the s e c t i o n i s unbalanced then i t i s l i k e l y t h a t the c o n s t r u c t e d impedance s e c t i o n w i l l d i s p l a y an apparent decrease i n l a t e r a l c o n t i n u i t y . The input to the a c o u s t i c impedance i n v e r s i o n a l g o r i t h m s i s assumed to be a b a n d l i m i t e d r e p r e s e n t a t i o n of the t r u e r e f l e c t i v i t y . T h e r e f o r e r o u t i n e p r o c e s s e s l i k e s p i k i n g d e c o n v o l u t i o n and g a p - d e c o n v o l u t i o n which are a p p l i e d t o the data i n order to -whiten the t r a c e spectrum and e l i m i n a t e s h o r t to medium p e r i o d m u l t i p l e s are r e q u i r e d p r i o r to impedance i n v e r s i o n runs. Some c a u t i o n i s recommended when gap-deconvolution i s a p p l i e d to the data because t h i s p r o c e s s may i n t r o d u c e a phase s h i f t i n t o the seismogram. In Chapter VII we showed how t h i s phase s h i f t may be r e c o v e r e d and how to a pply the c o r r e s p o n d i n g c o r r e c t i o n to the s e i s m i c s e c t i o n . I t i s important to remember t h a t the g o a l of p r e i n v e r s i o n p r o c e s s i n g i s a stacked s e c t i o n i n which: (a) m u l t i p l e s are l a r g e l y a t t e n u a t e d , (b) the e f f e c t s of the wavelet w i t h i n the frequency 1 84 i n f o r m a t i o n band have been removed to our best a b i l i t y , and (c) r e l a t i v e amplitude r e l a t i o n s are approximately c o r r e c t . That i s , the observed seismograms are a reasonable b a n d - l i m i t e d r e p r e s e n t a t i o n of the primary r e f l e c t i v i t y f u n c t i o n . STEP 2: S e c t i o n P o l a r i t y C o r r e c t i o n In some i n s t a n c e s t h e r e i s doubt about the t r u e p o l a r i t y of the s e i s m i c s e c t i o n . Such ambiguity may be the r e s u l t of f i e l d p rocedures or due to subsequent p r o c e s s i n g ( f o r example minimum phase d e c o n v o l u t i o n f o l l o w e d by phase c o r r e c t i o n a p p l i e d to Vibro.seis d a t a ) . I f t h i s i s the case, d e t e r m i n a t i o n of the c o r r e c t p o l a r i t y i s an e s s e n t i a l s t e p which should precede the impedance i n v e r s i o n . P o l a r i t y d e t e r m i n a t i o n can be accomplished to a c e r t a i n extent u s i n g the r e s u l t s of an u n c o n s t r a i n e d impedance i n v e r s i o n run. We d i s t i n g u i s h the f o l l o w i n g cases: A. W e l l l o g i n f o r m a t i o n i s a v a i l a b l e ( i ) When there i s c o r r e l a t i o n between some r e f l e c t o r s on the stacked s e c t i o n and c o r r e s p o n d i n g events on the w e l l - l o g s y n t h e t i c r e f l e c t i v i t y , then d i r e c t comparison of these events may be used t o determine the c o r r e c t p o l a r i t y . ( i i ) I f d i r e c t c o r r e l a t i o n i s not a v a i l a b l e ( t h a t i s , the s y n t h e t i c w e l l - l o g r e f l e c t i v i t y does not match the ne a r - w e l l stacked seismograms), an u n c o n s t r a i n e d impedance i n v e r s i o n should be executed. The p o l a r i t y of the s e c t i o n 185 may be est i m a t e d from the g e n e r a l behavior of the pseudo-impedance l o g ( i . e . g e n e r a l r i s e w i t h i n some g i v e n time windows which match ( c o r r e c t p o l a r i t y ) or m i r r o r image ( r e v e r s e p o l a r i t y ) the w e l l - l o g impedance). Due t o the nature of the i n v e r s i o n a l g o r i t h m s (the s o l u t i o n s are minimum s t r u c t u r e r e p r e s e n t a t i o n s of s u b s u r f a c e g e o l o g y ) , AREAS OF sharp changes i n impedance on the w e l l - l o g are more l i k e l y to be c o r r e c t l y r e c o n s t r u c t e d . These areas are p r e f e r a b l e f o r the purpose of p o l a r i t y d e t e r m i n a t i o n . B. Well l o g i n f o r m a t i o n i s not a v a i l a b l e In t h i s case the i n f o r m a t i o n c o n t a i n e d i n the s t a c k i n g v e l o c i t i e s should be used. S t a c k i n g v e l o c i t i e s are p i c k e d and i n v e r t e d u s i n g the Dix formula or one of the 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 o u t l i n e d i n Chapter IV. The c o n s t r u c t e d v e l o c i t y p r o f i l e i s used i n c o n j u n c t i o n with the pseudo-impedance l o g e s t i m a t e d by an u n c o n s t r a i n e d impedance i n v e r s i o n of the d a t a . The p o l a r i t y i s e s t i m a t e d by comparing the g e n e r a l behavior of the two p r o f i l e s w i t h i n chosen time windows. The user i s c a u t i o n e d that an i n t e r p r e t e r ' s wavelet which c o n t a i n s a l a r g e r e s i d u a l phase may s e v e r e l y i n h i b i t the succ e s s of the methods d i s c u s s e d here. 186 STEP 3 ; S e c t i o n Phase C o r r e c t i o n The i n v e r s i o n a l g o r i t h m s proposed i n t h i s work, assume t h a t the i n t e r p r e t e r ' s wavelet, (or r e s i d u a l wavelet) i s zero-phase. In many cases t h i s assumption i s i n v a l i d and a c o r r e c t i o n to the wavelet phase i s r e q u i r e d . For t h i s c o r r e c t i o n , we adopt the c o n s t a n t phase s h i f t model i n which we approximate the r e s i d u a l phase of the wavelet by a frequency independent phase s h i f t . T h i s c o r r e c t i o n c o n s i s t s of c o n v e r t i n g the observed seismogram x ( t ) i n t o an a n a l y t i c a l s i g n a l x ( t ) , and then m u l t i p l y i n g x ( t ) by the f a c t o r e x p [ i e ] w i t h e being the c o r r e c t i o n phase a n g l e . The phase c o r r e c t e d t r a c e i s r e c o v e r e d by t a k i n g the r e a l p a r t of t h i s f i n a l q u a n t i t y (See C h a p t e r _ V T I ) . Our e x p e r i e n c e shows t h a t (with a p p r o p r i a t e c h o i c e of e) t h i s o p e r a t i o n s a t i s f a c t o r i l y c o r r e c t s the phase and t h a t the subsequent impedance i n v e r s i o n y i e l d s markedly b e t t e r r e s u l t s . The d e t e r m i n a t i o n of the ' c o r r e c t ' s h i f t angle s h o u l d be based on the r e s u l t s of the u n c o n s t r a i n e d impedance i n v e r s i o n and i n f o r m a t i o n c o n t a i n e d i n e i t h e r the w e l l - l o g or i n the s t a c k i n g v e l o c i t i e s , in the manner d e s c r i b e d below: 187 A. W e l l l o g i n f o r m a t i o n i s a v a i l a b l e ( I ) . When w e l l - l o g impedance or v e l o c i t y i s a v a i l a b l e , e f f o r t s s h o u l d be c o n c e n t r a t e d on a window which c o n t a i n s a l a r g e impedance d i s c o n t i n u i t y (both the AR and LP a l g o r i t h m s a r e b e s t s u i t e d f o r the d e t e c t i o n of t h i s type of impedance s t r u c t u r e ) . S u c c e s s i v e r o t a t i o n of the n e a r - w e l l s t a c k e d t r a c e s by a number of s h i f t a n g l e s (say = -9 0 ° , - 6 0 ° , - 3 0 ° , +30°, and +60°) f o l l o w e d by impedance i n v e r s i o n w i l l y i e l d a s e t of pseudo-impedance c u r v e s . These a r e then compared to the w e l l - l o g impedance, and the t r a c e which produces the best match i s used t o determine the c o r r e c t i o n a n g l e . I t i s important t o r e a l i z e t h a t p o l a r i t y u n c e r t a i n t y may s t i l l e x i s t a f t e r the a p p l i c a t i o n of t h i s a n a l y s i s , but t h i s u n c e r t a i n t y can be r e s o l v e d by i n s p e c t i n g t h e s e t of pseudo-impedance p r o f i l e s produced by an u n c o n s t r a i n e d i n v e r s i o n run. ( I I ) . I f no d i s t i n c t impedance d i s c o n t i n u i t y i s p r e s e n t on the g i v e n w e l l - l o g , the o p e r a t i o n d e s c r i b e d i n ( i ) above i s performed on some a r b i t r a r y data window. T h i s time, however, the c h o i c e of the c o r r e c t i o n i s based on the comparison of the g e n e r a l b e h a v i o r of the pseudo-impedance p r o f i l e s and the w e l l - l o g impedance i n the chosen window. E x p e r i e n c e has shown t h a t the r e s u l t s of the a n a l y s i s f o r t h i s type of i n f o r m a t i o n a r e l e s s r e l i a b l e and must be a p p l i e d w i t h c a u t i o n . 188 B. No w e l l - l o g i n f o r m a t i o n i s a v a i l a b l e In t h i s case the r e s u l t s of the u n c o n s t r a i n e d impedance i n v e r s i o n a l g o r i t h m s a p p l i e d to the phase c o r r e c t e d t r a c e s a r e compared t o the v e l o c i t y c u r v e o b t a i n e d from the Dix formula or from the i n v e r s i o n of s t a c k i n g v e l o c i t i e s u s i n g a l g o r i t h m s o u t l i n e d i n Chapter IV. Again however, thes e r e s u l t s a r e not c o m p l e t e l y r e l i a b l e and the t e c h n i q u e must be a p p l i e d w i t h c a u t i o n . C. Automatic Phase C o r r e c t i o n The p r o c e d u r e o u t l i n e d i n Chapter VII has p roved s u c c e s s f u l i n a l a r g e number of examples. However, i t i s q u i t e c o n c e i v a b l e t h a t t h i s scheme w i l l f a i l i n some g e o l o g i c a l environments. Consequently, i t i s always recommended to compare the r e s u l t s of an u n c o n s t r a i n e d i n v e r s i o n run on the phase c o r r e c t e d s e c t i o n t o the a v a i l a b l e w e l l - l o g and s t a c k i n g v e l o c i t y i n f o r m a t i o n . I f any doubt as t o c o r r e c t n e s s of the a p p l i e d phase remains, i t i s recommended t o use the u n c o r r e c t e d d a t a i n the e n s u i n g i n v e r s i o n . 189 STEP 4 : Improving S i g n a l to Noise R a t i o When the q u a l i t y of the s t a c k e d s e c t i o n i s low, impedance i n v e r s i o n a l g o r i t h m s which op e r a t e d i r e c t l y on these data are not l i k e l y t o produce r e l i a b l e r e s u l t s . In such c a s e s poor q u a l i t y output might be expected even when the i n v e r s i o n i s done by i n c o r p o r a t i n g e x t e r n a l i n f o r m a t i o n from w e l l - l o g s and s t a c k i n g v e l o c i t i e s . An improvement i n the s i g n a l t o n o i s e r a t i o i n the o bserved s t a c k e d s e c t i o n can add s i g n i f i c a n t l y t o the f i n a l q u a l i t y of r e s u l t s . T r a c e mixing i s the s i m p l e s t scheme f o r s i g n a l t o n o i s e r a t i o improvement. However, m i x i n g r e s u l t s i n a l o s s of l a t e r a l r e s o l u t i o n which may obscure some f e a t u r e s of i n t e r e s t . An a l t e r n a t e approach i s based on p r i n c i p a l component a n a l y s i s as o u t l i n e d i n Appendix 7-B. Here we s e a r c h f o r t h e most c o r r e l a t a b l e ( s i m i l a r ) component i n a s e t of 'n' s u c c e s s i v e t r a c e s ( g e n e r a l l y n •« 3 i s s u f f i c i e n t ) . T h i s method d e t e r i o r a t e s t o a s t r a i g h t s t a c k i f the s i g n a l s are u n c o r r e l a t e d but i t w i l l a c h i e v e a somewhat b e t t e r s i g n a l t o n o i s e r a t i o , accompanied by a somewhat s m a l l e r l o s s of l a t e r a l r e s o l u t i o n , o t h e r w i s e . 190 STEP 5 : D e t e r m i n i n g the ' R e l i a b l e Frequency Bandwidth' The b a s i c assumption made i n the i n v e r s i o n a l g o r i t h m s d e s c r i b e d here i s t h a t the s e c t i o n to be i n v e r t e d c o n s i s t s of a b a n d l i m i t e d r e p r e s e n t a t i o n of the r e f l e c t i v i t y f u n c t i o n . The LP a l g o r i t h m attempts to c o n s t r u c t a f u l l band r e f l e c t i v i t y whose spectrum i s c o n s i s t e n t w i t h the. data w i t h i n the i n f o r m a t i o n band. I t i s t h e r e f o r e important t h a t the width of t h i s f r e q u e n c y band be r e l i a b l y d e termined. In d e t e r m i n i n g the r e l i a b l e f r e q u e n c y band one sho u l d use the f o l l o w i n g g u i d e - l i n e s : (a) The r e s u l t s o f " t h e impedance i n v e r s i o n a l g o r i t h m s a r e more c e r t a i n and e x h i b i t h i g h e r r e s o l u t i o n when the a l g o r i t h m s use a wider f r e q u e n c y bandwidth. (b) E a r t h a t t e n u a t i o n causes a severe l o s s of energy a t the h i g h e r f r e q u e n c i e s w i t h the r e s u l t t h a t the amount of energy t h a t r e t u r n s t o the r e c e i v e r s a t these f r e q u e n c i e s f o r ' l a r g e ' a r r i v a l times i s s m a l l . I f t h i s energy i s comparable i n magnitude to t h a t of the background n o i s e , the i n f o r m a t i o n c o n t a i n e d i n these f r e q u e n c i e s i s not r e l i a b l e and s h o u l d not be used i n the i n v e r s i o n . The a c t u a l d e t e r m i n a t i o n of the e n d p o i n t s of the r e l i a b l e frequency band (FLO, FHI) i s based upon i n s p e c t i o n of a s e t of amplitude s p e c t r a c o r r e s p o n d i n g t o a r b i t r a r i l y s e l e c t e d t r a c e s from a c r o s s the s e c t i o n . The p r o c e s s o r s h o u l d note the band of 191 f r e q u e n c i e s a t which r e l a t i v e l y h i g h energy i s c o n c e n t r a t e d and the l e v e l of n o i s e a t those f r e q u e n c i e s which ar e o u t s i d e the frequency band of the s o u r c e . The upper l i m i t of the r e l i a b l e f r e q u e n c y band i s the h i g h e s t frequency of t h i s band a t which energy i s about twice the n o i s e l e v e l . The lower l i m i t can be computed l i k e w i s e , but one i s c a u t i o n e d not t o s e t the lower l i m i t l e s s than about 8 Hz because the r e l i a b i l i t y of the phase below t h i s f r e q u e n c y i s r a t h e r poor. STEP 6; U n c o n s t r a i n e d I n v e r s i o n In the absence of any a d d i t i o n a l i n f o r m a t i o n , the i n v e r s i o n a l g o r i t h m can o p e r a t e on the i n f o r m a t i o n i n c l u d e d i n the s t a c k e d s e c t i o n a l o n e . In t h i s c a s e , the r e f l e c t i v i t y o b t a i n e d from the a l g o r i t h m s r e p r e s e n t s a good d e c o n v o l u t i o n of the s t a c k e d s e c t i o n , and can be s a f e l y used as such. However, the pseudo-impedance s e c t i o n o b t a i n e d from t h i s i n v e r s i o n c a r r i e s a t best r e l a t i v e i n f o r m a t i o n (as i t r e p r e s e n t s o n l y the l o g a r i t h m of the r a t i o of the impedance a t time Vt' t o t h a t a t the t o p of the s e c t i o n ) . T h i s i n f o r m a t i o n may be c o r r e c t i f the geology of the area does not c o n t a i n many r e g i o n s where the impedance i s a s l o w l y changing f u n c t i o n of depth. G e n e r a l l y , however, t h i s impedance e s t i m a t e has a lower r e l i a b i l i t y than t h a t produced from a c o n s t r a i n e d impedance i n v e r s i o n r u n . Moreover, because the i n f o r m a t i o n a v a i l a b l e i s not s u f f i c i e n t f o r proper s c a l i n g of the s t a c k e d s e c t i o n , subsequent c o n v e r s i o n of the pseudo-impedance i n t o impedance or p s e u d o - v e l o c i t y i s not l i k e l y t o produce p h y s i c a l l y m e a n i n g f u l i n f o r m a t i o n c o n c e r n i n g f o r m a t i o n c h a r a c t e r i s t i c s . 192 STEP 7: S c a l i n g The Data When c o n s t r a i n t s are used, i t i s important t h a t the data are p r o p e r l y s c a l e d . The primary reason i s t h a t both p o i n t v e l o c i t y c o n s t r a i n t s and s t a c k i n g v e l o c i t y c o n s t r a i n t s r e q u i r e that the sum of the r e f l e c t i o n c o e f f i c i e n t s o b t a i n e d i n the i n v e r s i o n w i l l a t t a i n some d e s i r e d value to w i t h i n a determined e r r o r . In order f o r the sum to be p h y s i c a l l y meaningful i t i s necessary th a t t h e data are p r o p e r l y s c a l e d . The magnitude of the r e f l e c t i o n c o e f f i c i e n t s o b t a i n e d from the u n c o n s t r a i n e d i n v e r s i o n run p r o v i d e an immediate i n d i c a t i o n about the n e c e s s i t y t o s c a l e the d a t a . For i n s t a n c e , r e f l e c t i o n c o e f f i c i e n t s l a r g e r than u n i t y are not p h y s i c a l l y r e a s o n a b l e . S i m i l a r l y , r e f l e c t i o n c o e f f i c i e n t s of magnitude 0.5 i n an area where the r e f l e c t i o n c o e f f i c i e n t should be i n the order of 0.05 i n d i c a t e t h a t the data s h o u l d be r e s c a l e d . To a v o i d such i n s t a n c e s we recommend that c o n s i d e r a b l e e f f o r t be spent i n s c a l i n g the data p r i o r to an i n v e r s i o n run. The procedures suggested here are q u i t e s u b j e c t i v e and are l a r g e l y dependent on the g e o l o g i c a l i n t u i t i o n possessed by the p r o c e s s o r . We d i s t i n g u i s h two c a s e s : A. W e l l - l o g i n f o r m a t i o n i s a v a i l a b l e The s i m p l e s t and most s t r a i g h t forward method i s to compare the s i z e of the r e f l e c t i o n c o e f f i c i e n t s o b t a i n e d from the u n c o n s t r a i n e d i n v e r s i o n t o those observed i n the w e l l - l o g r e f l e c t i v i t y . The s c a l i n g f a c t o r o b t a i n e d from t h i s comparison i s t h a t number which w i l l s c a l e the c o n s t r u c t e d r e f l e c t i v i t y so t h a t i t s r e f l e c t i o n 1 93 c o e f f i c i e n t s are comparable to those of the w e l l - l o g r e f l e c t i v i t y . The user i s reminded to use the same frequency i n f o r m a t i o n band i n the u n c o n s t r a i n e d run as he w i l l when he runs the c o n s t r a i n e d v e r s i o n . Once an approximate s c a l i n g f a c t o r has been determined, i t i s recommended t h a t the c o n s t r a i n e d i n v e r s i o n a l g o r i t h m be run on small s e t s of s c a l e d data. One sho u l d now i n s p e c t the computed impedance or v e l o c i t y e s t i m a t e s to v e r i f y t h a t these e s t i m a t e s are g e o l o g i c a l l y r e a s o n a b l e . I f they are not, f u r t h e r adjustment of the s c a l e s h o u l d be made on the b a s i s of t h i s i n s p e c t i o n . I f the pr o c e s s o r i s s a t i s f i e d , the whole s e c t i o n may be s c a l e d and the c o n s t r a i n e d run can be submitted.-B. Automatic s c a l i n g The t e c h n i q u e o u t l i n e d i n Chapter VI has proved s u c c e s s f u l i n a number of c a s e s . However, r e s i d u a l phase and severe r e l a t i v e amplitude d i s c r e p a n c i e s may cause i t s f a i l u r e . I t i s then recommended t h a t t h i s scheme be used o n l y when the l o g r e l a t i v e impedance from an u n c o n s t r a i n e d run e x h i b i t s a reas o n a b l e match with the w e l l - l o g or the v e l o c i t y p r o f i l e o b t a i n e d from the i n v e r s i o n of the s t a c k i n g v e l o c i t i e s . s 1 94 Step 8; W e l l - l o g C o n s t r a i n t s P o i n t v e l o c i t y c o n s t r a i n t s can be used i n the i n v e r s i o n when w e l l - l o g i n f o r m a t i o n i s a v a i l a b l e and a reaso n a b l e t i e i s e s t a b l i s h e d between t h i s i n f o r m a t i o n and e i t h e r the observed stack e d s e c t i o n or the r e f l e c t i v i t y output from an u n c o n s t r a i n e d i n v e r s i o n . The format of the c o n s t r a i n t s i n c l u d e s the time of the c o n s t r a i n t , the e s t i m a t e d v e l o c i t y and a c o r r e s p o n d i n g e s t i m a t e of the u n c e r t a i n t y i n the v e l o c i t y . Step 9: S t a c k i n g V e l o c i t y C o n s t r a i n t s S t a c k i n g v e l o c i t i e s are always a v a i l a b l e f o r use i n c o n s t r a i n i n g the impedance i n v e r s i o n , but t h e i r a p p l i c a b i l i t y to t h i s i n v e r s i o n may decrease c o n s i d e r a b l y i n r e g i o n s of complex geology. I f i t i s p o s s i b l e to u t i l i z e them, t h e i r i n f o r m a t i o n s u p p l i e s a d d i t i o n a l c o n t r o l s on the low-frequency t r e n d s i n the rec o v e r e d impedance. To d e r i v e i n f o r m a t i o n from the s t a c k i n g v e l o c i t i e s we assume that to good approximation they are i n f a c t RMS v e l o c i t i e s . Thus, from the s t a c k i n g v e l o c i t i e s we can get averages of the v e l o c i t y over s e l e c t e d time windows. V e l o c i t y a n a l y s i s , c o n t i n u o u s v e l o c i t y a n a l y s i s maps, or constan t v e l o c i t y s t a c k s can a l l be used f o r s e l e c t i n g s t a c k i n g v e l o c i t i e s . However, the p i c k i n g of s t a c k i n g v e l o c i t i e s w i t h the c o r r e s p o n d i n g times and the a s s o c i a t e d u n c e r t a i n t y v a l u e s i s most e a s i l y performed on a contour v e l o c i t y a n a l y s i s p l o t . One should 195 s e l e c t as many reasonable s t a c k i n g v e l o c i t i e s as p o s s i b l e , p a r t i c u l a r l y i n those time windows where the stacked s e c t i o n i n d i c a t e s major s t r u c t u r a l changes. V e l o c i t y c o n s t r a i n t s ( i f a v a i l a b l e ) should be used along with the s t a c k i n g v e l o c i t i e s i n the v e l o c i t y i n v e r s i o n . The f i n a l c o n s t r u c t e d model then i n c o r p o r a t e s a l l of the a v a i l a b l e v e l o c i t y i n f o r m a t i o n . The 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 ( d e s c r i b e d i n Chapter IV) o f f e r important advantages i n both s t a b i l i t y and approach over other one di m e n s i o n a l schemes. Out of a l l p o s s i b l e v e l o c i t y models which f i t the data, these a l g o r i t h m s seek t h a t s o l u t i o n which e x h i b i t s the s m a l l e s t amount of change as a f u n c t i o n of time (d e p t h ) . T h i s r e s u l t s g e n e r a l l y i n a smooth v e r s i o n of the t r u e s t r u c t u r e . The c o r r e c t n e s s of the low frequency f e a t u r e s i n t h i s model i s f u r t h e r ensured by the a d d i t i o n of p o i n t v e l o c i t y c o n s t r a i n t s (from w e l l - l o g i n f o r m a t i o n ) when a v a i l a b l e . S t a b i l i t y i s a c h i e v e d by not r e q u i r i n g t h a t the s o l u t i o n f i t the data e x a c t l y , but r a t h e r a l l o w i n g i t to m i s f i t i n accordance with the s u p p l i e d e r r o r s . The RMS i n v e r s i o n f o l l o w e d by an a p p r o p r i a t e i n t e r p o l a t i o n generates an i n t e r v a l v e l o c i t y p r o f i l e a t each CDP l o c a t i o n which i s a smooth es t i m a t e of the t r u e v e l o c i t y p r o f i l e . In u s i n g t h i s v e l o c i t y model to c o n s t r a i n the impedance i n v e r s i o n , we must remember t h a t i t i s s i m i l a r to the t r u e v e l o c i t y only- i n terms of averages over c e r t a i n time windows. T h i s p r e s c r i b e s the form of the 196 c o n s t r a i n t u t i l i z e d by the impedance a l g o r i t h m . The average value of the impedance change a c r o s s a s p e c i f i e d time window i s c o n s t r a i n e d to be tha t c a l c u l a t e d from the input i n t e r v a l v e l o c i t y , to w i t h i n a given m i s f i t (see Chapter I V ) . One i s c a u t i o n e d again t h a t the r e l i a b i l i t y of the c o n s t r u c t e d i n t e r v a l v e l o c i t y curves depends on the q u a l i t y of the v e l o c i t y a n a l y s i s maps and the d e v i a t i o n of the l o c a l geology from the h o r i z o n t a l l y l a y e r e d e a r t h model (which i s assumed i n the c o n v e r s i o n from s t a c k i n g to i n t e r v a l v e l o c i t y p r o f i l e s ) . In ar e a s where s t e e p l y d i p p i n g events are p r e s e n t , the e s t i m a t e d v e l o c i t y c u r v e s may e x h i b i t p r o h i b i t i v e l y l a r g e e r r o r s . Step 10; P r i n c i p a l Component A n a l y s i s and Low  Fr e q u e n c i e s M i x i n g . In order to i n c r e a s e the r e l i a b i l i t y and enhance the p r e s e n t a t i o n of the r e s u l t s , we recommend that p r i n c i p a l component a n a l y s i s w i l l be a p p l i e d to the f i n a l r e f l e c t i v i t y s e c t i o n produced by the i n v e r s i o n . T h i s o p e r a t i o n (which has been d e s c r i b e d p r e v i o u s l y ) should be a p p l i e d t o the r e f l e c t i v i t y s e c t i o n p r i o r to p l o t t i n g . I t s a p p l i c a t i o n w i l l i n c r e a s e c o n s i d e r a b l y the s i g n a l to n o i s e r a t i o of the r e f l e c t i v i t y s e c t i o n and r e s u l t i n o n l y a small l o s s of l a t e r a l r e s o l u t i o n . Due to the inherent non-uniqueness i n the a c o u s t i c impedance 1 97 i n v e r s i o n problem, i t i s necessary to mix the p r e d i c t e d low f r e q u e n c i e s , t h a t i s , to average those f r e q u e n c i e s generated by the i n v e r s i o n a l g o r i t h m which are s m a l l e r than 'FLO' (the lower l i m i t of the i n f o r m a t i o n band). T h i s mixing operates on the premise that the very low f r e q u e n c i e s (say 0-1.5 Hz) s h o u l d e x h i b i t o n l y a very s m a l l change a c r o s s the s e c t i o n and .hence the i n f o r m a t i o n about the r e f l e c t i v i t y c o n t a i n e d i n those f r e q u e n c i e s i s averaged over a l a r g e number of t r a c e s . C o n v e r s e l y , f o r f r e q u e n c i e s c l o s e to 'FLO' only the i n f o r m a t i o n c o n t a i n e d i n a s m a l l number of s u c c e s s i v e t r a c e s i s mixed. T h i s p e r m i t s l a t e r a l v a r i a b i l i t y to o c c u r . Between these two extremes, the number of t r a c e s used i n the mixing d e c r e a s e s r a p i d l y by a r u l e which behaves ap p r o x i m a t e l y as the i n v e r s e frequency ( 1 / f ) . STEP 1 1 : Conversion of F u l l - b a n d R e f l e c t i v i t y t o P s e u d o - V e l o c i t y or  Pseudo-Impedance The output of the a c o u s t i c impedance i n v e r s i o n a l g o r i t h m s i s a set of ' f u l l - b a n d ' r e f l e c t i v i t y f u n c t i o n s which s a t i s f y a l l c o n s t r a i n t s imposed d u r i n g the i n v e r s i o n , to w i t h i n the p r e s p e c i f i e d e r r o r s . That i s , the r e f l e c t i v i t y f u n c t i o n has a spectrum which matches the spectrum of the input seismograms w i t h i n the i n f o r m a t i o n band, and i t a l s o s a t i s f i e s w e l l - l o g v e l o c i t y c o n s t r a i n t s and s t a c k i n g v e l o c i t y c o n s t r a i n t s i f they were imposed. Co n v e r s i o n of the f u l l band r e f l e c t i v i t i e s i n t o v e l o c i t y or impedance i s c a r r i e d out by e x p o n e n t i a t i n g the p a r t i a l sums of the r e f l e c t i v i t i e s and then m u l t i p l y i n g them by the s t a r t i n g v e l o c i t y at each CDP l o c a t i o n . 198 STEP 12: D i s p l a y i n g Impedance S e c t i o n s P r e s e n t a t i o n of impedance s e c t i o n s depends l a r g e l y on the type of i n f o r m a t i o n d e s i r e d . When the goal of the i n t e r p r e t e r i s to d e l i n e a t e g e o l o g i c a l formations with the a s s o c i a t e d r e l a t i v e impedance i n f o r m a t i o n , a simple l i n e p l o t seems to o f f e r a u s e f u l p r e s e n t a t i o n . T h i s type of p r e s e n t a t i o n i s p a r t i c u l a r l y advantageous s i n c e the i n t e r p r e t e r ' s eye w i l l tend to f o l l o w impedance d i s c o n t i n u i t i e s and not be a f f e c t e d a d v e r s e l y by s m a l l t r a c e to t r a c e impedance v a r i a t i o n s . I f the i n t e r p r e t e r wishes to a s s i g n a v e l o c i t y or impedance v a l u e t o the formation of i n t e r e s t , c o l o r or grey shades c o n s t i t u t e a b e t t e r form of p r e s e n t a t i o n . When such a p r e s e n t a t i o n i s d e s i r e d , the c h o i c e of the number of c o l o r . l e v e l s and t h e i r a c t u a l v a l u e s i s of paramount importance. A poor c h o i c e of these parameters may cause a s i g n i f i c a n t decrease i n the observed r e s o l u t i o n of the p r e s e n t e d impedance. I t i s recommended t h a t the number of c o l o r l e v e l s be chosen such t h a t . t h e i n t e r p r e t e r can r e c o g n i z e e a s i l y any t r a n s i t i o n from one v e l o c i t y / i m p e d a n c e l e v e l to another. Choosing too many c o l o r l e v e l s a l l o w s s u b t l e t r a n s i t i o n s from one c o l o r l e v e l to another, thereby o b s c u r i n g s m a l l t a r g e t s of economic importance. Furthermore, although i t may seem d e s i r a b l e to a s s i g n an a r b i t r a r y v e l o c i t y / i m p e d a n c e v a l u e to each c o l o r l e v e l so that the t o t a l number of c o l o r s w i l l cover the range of the expected v e l o c i t y v a l u e s i n a l i n e a r (equi-spaced) f a s h i o n , t h i s c o l o r i n g scheme i s s t r o n g l y not recommended. T h i s scheme w i l l cause a severe l o s s i n r e s o l u t i o n ( r e d u c t i o n i n the number of a c t i v e c o l o r l e v e l s ) i n any 199 g e o l o g i c a l environment which i n c l u d e s a sma l l number of very low and very high impedance formations embedded i n a set of fo r m a t i o n s c h a r a c t e r i z e d by medium impedance v a l u e s . A b e t t e r c h o i c e of l e v e l s i s one which a s s i g n s v a l u e s to c o l o r l e v e l s so that each c o l o r l e v e l r e p r e s e n t s about the same number of p i c t u r e c e l l s (a p i c t u r e c e l l i s a p o r t i o n of the impedance p i c t u r e d e f i n e d by the time and the space sampling i n t e r v a l s ) . T h i s approach ensures t h a t a l l the a v a i l a b l e c o l o r s w i l l be a c t i v e i n the c o l o r i n g process and thereby y i e l d a b e t t e r apparent r e s l u t i o n than th a t o f f e r e d by the l i n e a r c o l o r i n g scheme. F i n a l l y , the r e s u l t s of an i n v e r s i o n run executed with the f u l l s e t of steps d i s c u s s e d p r e v i o u s l y i s pr e s e n t e d i n the f o l d e d data set marked as L i n e C. These data have been a c q u i r e d i n A l b e r t a and f e a t u r e a number of e x p l o r a t i o n t a r g e t s i n the time window 0.5 to 0.7 seconds. The processed r e s u l t p r e s e n t e d here d e p i c t s the f u l l - b a n d r e f l e c t i v i t y f u n c t i o n s o v e r l a y e d by the p s e u d o - v e l o c i t i e s c a l c u l a t e d from these r e f l e c t i v i t i e s u s i n g the assumption of co n s t a n t d e n s i t y . D e t a i l s which are not e a s i l y d e c i p h e r e d from the st a c k e d s e c t i o n are c l e a r l y observed on the f i n a l i n v e r s i o n r e s u l t . In p a r t i c u l a r , l a t e r a l v a r i a t i o n s i n formation v e l o c i t y are c l e a r l y v i s i b l e even though only nine grey shade l e v e l s r e p r e s e n t i n g v e l o c i t y i n k i l o m e t e r s / s e c o n d s are d i s p l a y e d . Comparison of the c a l c u l a t e d p s e u d o - v e l o c i t i e s with s o n i c l o g i n f o r m a t i o n (not a v a i l a b l e d u r i n g the i n v e r s i o n p r o c e s s ) r e v e a l e d e x c e l l e n t 200 agreement. Past e x p e r i e n c e with the pro c e s s d e s c r i b e d i n t h i s work has shown tha t c a r e f u l u t i l i z a t i o n of the p r e s e n t e d a l g o r i t h m s g e n e r a l l y y i e l d s h i g h l y r e l i a b l e r e s u l t s . O R I G I N A L D A T A o C J LU >-CC <c _ l o 0_ CC o Q LU O _ l Q-CJ Z UJ o LU CO CD CO CO LU o o CC Q_ LD LO I CM O z CD CC O LD CO z z O C J ' »-•<;<: I— I — CC ID CO I — I CO CD O Z > z O z: <: o o U LD > LU O I— Q I '—1 CJ U J o •—• I I LD 111 111 O ^ > _ l O I LU LD CM I — CO LU CJ <c CC ro ro <: i — CO CD CO > 2Z CC i n CO LU LU >-LU CC LD 2= O CC CO _J LU > LU _ J CC ZD o _ J o CJ CD >-CO Q C O C O O O C C D CJ CO z o <: CJ D_ D_ <: >-CC o LU x LU CO CC C O N S T R A I N E D L P R E F L E C T I V I T Y / P S E U D O V E L O C I T Y o CO CM I i lo C M o o C M 4 » " o C D aa o C M I mm I'. \ o CO I O m -liiili 11111111111111111111111111 < 111111111111111111111111111111111 [ 11111111111111111111111111111M11 111111111111111 J 1111 CO • o D o C M • o C O • o • o L O • o CD t"» • o o o CO C O a o o a C M o rffifhtml HBI CO • C M C D • C M C O • C M O a CO 00 • C D • i n C O • C D 201 REFERENCES A k i , K., and R i c h a r d s , P. G., 1980, Q u a n t i t a t i v e Seismology, Theory and Methods, v . l : W. H. Freeman and Co. 557 p. A l - C h a l a b i , M., 1974, An a n a l y s i s of s t a c k i n g , RMS, average, and i n t e r v a l v e l o c i t i e s over a h o r i z o n t a l l y l a y e r e d ground: G e o p h y s i c a l P r o s p e c t i n g . , v.22, p. 458-475. A n g e l e r i , G.P., and C a r p i , R., 1982, P o r o s i t y p r e d i c t i o n from s e i s m i c d a t a : G e o p h y s i c a l P r o s p e c t i n g , v.30, p. 580-607. Anstey, N.A., 1977, Seismic I n t e r p r e t a t i o n - The P h y s i c a l A s p e c t s : I n t e r n a t i o n a l Human Resources Development C o r p o r a t i o n . Arons, A. B., and Yennie, D. R., 1950, Phase d i s t o r t i o n of a c o u s t i c p u l s e s o b l i q u e l y r e f l e c t e d from a medium o f h i g h e r sound v e l o c i t y : J . Ac. Soc. Am. v. 22, p. 231-237. Backus, G.E., and G i l b e r t , F., 1970, Uniqueness i n the i n v e r s i o n of i n a c c u r a t e gross e a r t h d a t a : P h i l . Trans. R. Soc. London Ser. v.A-266, p. 123-192, Backus, G. E., and G i l b e r t , J . F., 1967, Numerical a p p l i c a t i o n s of a formalism f o r g e o p h y s i c a l i n v e r s e problems: Geophys. J . Roy. A s t r . S o c , v. 13, p. .247-276. Berkhout, A. J . , 1977, Least squares i n v e r s e f i l t e r i n g and wavelet d e c o n v o l u t i o n : Geophysics, v. 42, p. 1369-1383. B i l g e r i , D. , and Ademeno, E.B., 1982, P r e d i c t i n g abnormally p r e s s u r e d sedimentary r o c k s : G e o p h y s i c a l P r o s p e c t i n g , v.30, p. 608-621. B i l g e r i , D., and C a r l i n i , A., 1981, N o n - l i n e a r e s t i m a t i o n of r e f l e c t i o n c o e f f i c i e n t s from s e i s m i c d a t a : G e o p h y s i c a l P r o s p e c t i n g , v.29, p. 672-686. B r a c e w e l l , R. , N., 1978, The F o u r i e r Transform and i t s A p p l i c a t i o n s : New York, McGraw-Hill Book Co., Inc., 2nd. ed. 444 p. Burg., J . P., 1975, Maximum Entropy S p e c t r a l A n a l y s i s : Ph.D. T h e s i s , S t a n f o r d Univ. Choy, G., L., and R i c h a r d s , P., G., 1975, Pulse d i s t o r t i o n and H i l b e r t t r a n s f o r m a t i o n by m u l t i p l y r e f l e c t e d and r e f r a c t e d body waves: SSA B u l l . , v. 65, p. 55-70. 202 C l a e r b o u t , J.. F., 1 976, Fundamentals of G e o p h y s i c a l Data P r o c e s s i n g : McGraw-Hill Inc., 274 p. C l a e r b o u t , J . F., and Muir, F., 1973, Robust m o d e l l i n g with e r r a t i c Data: Geophysics, v. 38, p. 826-844. Deeming, T. J . and T a y l o r , H. L., 1981, Use of an improved 1, norm m i n i m i z a t i o n f o r wavelet p r o c e s s i n g of s e i s m i c s e c t i o n s : P r e s e n t e d at 51st. Annual I n t e r n a t i o n a l SEG Meeting, Oct.11-15, 1981, i n Los Angeles. Dix, C. H., 1955, Seismic v e l o c i t i e s from s u r f a c e measurements: Geophysics., v.20, p. 68-86. E v e r e t t , J . E., 1974, O b t a i n i n g i n t e r v a l v e l o c i t i e s .from s t a c k i n g v e l o c i t i e s when d i p p i n g h o r i z o n s are i n c l u d e d : G e o p h y s i c a l P r o s p e c t i n g , v.22, p. 122-142. F r a z e r , J . , 1983, V i b r o s e i s down the h o l e : Proc. A u s t r a l i a n Petroleum E x p l o r a t i o n A s s o c i a t i o n 23rd Annual Meeting, p. 203-210, Melbourne. Futterman, W. I., 1962, D i s p e r s i v e body waves: J . of G e o p h y s i c a l Research, v.67, p. 5279-5291. Gass, S., 1964, L i n e a r Programming- Methods and A p p l i c a t i o n s : McGraw-Hill Book Co. Gibson, B., and K. L a r n e r , 1984, P r e d i c t i v e d e c o n v o l u t i o n and the zero-phase source: Geophysics, v.49, p. 379-397. Hemon, C , and Mace, D. , 1 978, The use of the Karhunen-Loeve t r a n s f o r m a t i o n i n s e i s m i c data p r o c e s s i n g : G e o p h y s i c a l P r o s p e c t i n g , v. 26, p. 600-626. Huang, T.S., and Narenda, P.M., 1975, Image r e s t o r a t i o n by s i n g u l a r v a l u e decomposition: A p p l . Opt., v.14, p.2213-2216. Hu b r a l , P., and Krey, T., 1980, I n t e r v a l V e l o c i t i e s from Seismic R e f l e c t i o n Time Measurements: T u l s a , SEG, 203 p. J e n k i n s , G., M., and Watts, D. G., 1969, S p e c t r a l A n a l y s i s and i t s A p p l i c a t i o n s : San F r a n c i s c o , Holden-Day, 525 p. Karhunen, K., 1947, Uber L i n e a r e Methoden i n der W a h r s c h e i n l i c h k e i t s r e c h n u n g : Ann. Acad. S c i . Fenn. , (Suomalainen T i e d e a k a t e m i a ) , v.37, p.1-79. ( T r a n s l a t i o n by I . S e l i n , 1960: "On L i n e a r Methods i n P r o b a b i l i t y Theory." t-131 RAND Corp., Santa Monica, C a l i f o r n i a ) . Kesmarky, I., 1977, E s t i m a t i o n of r e f l e c t o r parameters by the v i r t u a l image te c h n i q u e : G e o p h y s i c a l P r o s p e c t i n g , v.25, p. 621-635. 203 Knopoff, L., 1964, Q: Reviews of Geophysics, v.2, p. 625-660. Kramer, H. P., and Mathews, M. V,, 1968, A l i n e a r coding f o r t r a n s m i t t i n g a set of c o r r e l a t e d s i g n a l s : Presented at the 38th Annual I n t e r n a t i o n a l SEG meeting, Denver, Co. Krey, T., 1976, Computation of i n t e r v a l v e l o c i t i e s from common r e f l e c t i o n p o i n t movemout times f o r N l a y e r s with a r b i t r a r y d i p s and c u r v a t u r e s i n thre e dimensions when assuming s m a l l shot-geophone d i s t a n c e s : G e o p h y s i c a l P r o s p e c t i n g , v.24, p. 91-111. L e v i n , F. K., 1971 , Apparent v e l o c i t y from d i p p i n g i n t e r f a c e r e f l e c t i o n s : Geophysics, v.36, p. 510-516. Levy, S., and F u l l a g a r , P. K., 1981, The r e c o n s t r u c t i o n of a sparse spike t r a i n from a p o r t i o n of i t s spectrum and a p p l i c a t i o n to h i g h r e s o l u t i o n d e c o n v o l u t i o n : Geophysics, v.46, p. 1235-1243. Levy, S., Walker, C , U l r y c h , T. J . , and F u l l a g a r , P. K. , 1982, A l i n e a r programming approach to the e s t i m a t i o n of the power s p e c t r a of harmonic p r o c e s s e s : I.E.E.E. A c o u s t i c S e c t i o n , v. 30 (4), p. 675-679. Levy, S., and Oldenburg, D. W., 1982, The d e c o n v o l u t i o n of phase s h i f t e d wavelets: Geophysics, v.47, p. 1285-1294. Levy, S., U l r y c h , T. J . , Jones, I. F., and Oldenburg, D. W., 1983, A p p l i c a t i o n s of complex common s i g n a l a n a l y s i s i n e x p l o r a t i o n seismology: Presented at the 53rd Annual SEG Meeting Las Vegas. Loeve, M., 1955, F u n c t i o n s A l t e a t o r i e s de Second Odre: Chapter 8, p.299-352, Hermann, P a r i s . O ' b r i e n , P. N. S., 1967, Q u a n t i t a t i v e d i s c u s s i o n on s e i s m i c amplitudes produced by e x p l o s i o n s i n Lake S u p e r i o r : J . of Geoph. Res., v.72, p. 2569-2575. Oldenburg, D. W., 1976, C a l c u l a t i o n of F o u r i e r t r ansforms by the B a c k u s - G i l b e r t method: Geophys. J . Roy, A s t r . Soc. v.44, p. 413-431. -N Oldenburg, D. W. , and Samson, J . C , 1979, I n v e r s i o n of i n t e r f e r o m e t r i c data from c y l i n d r i c a l l y symmetric r e f r a c t i o n l e s s plasmas: J . Opt. Soc. Am., v.69, p. 927-942. Oldenburg, D. W., 1981, A comprehensive s o l u t i o n to the l i n e a r d e c o n v o l u t i o n problem: Geophys. J . Roy. A s t r . S o c , v. 65, p. 331-357. Oldenburg, D. W., Levy, S., W h i t t a l l , K. P., 1981, Wavelet e s t i m a t i o n and d e c o n v o l u t i o n : Geophysics, v. 46, p. 204 1528-1542. Oldenburg, D. W., Scheuer, T. E., and Levy, S., 1983, Recovery of the a c o u s t i c impedance from r e f l e c t i o n seismograms: Geophysics, v.48, p. 1318-1337. Oldenburg, D. W., Levy, S., and S t i n s o n , K. , 1984, RMS v e l o c i t i e s and r e c o v e r y of the a c o u s t i c impedance: Submitted to Geophysics. Parker, R. L., 1977, Understanding i n v e r s e t h e o r y : Ann. Rev. E a r t h P l a n . S c i . , v.5, p. 35-64. P e l a t , D., 1974, Karhunen-Loeve S e r i e s Expansion: A new approach f o r s t u d y i n g a s t r o p h y s i c a l d a t a : A s t r o n . and A s t r o p h y s . , v.33, p.321-329. P e t e r s o n , R.A., F i l l i p o n e , W.R., and Coker, F.B., 1955. The s y n t h e s i s of seismograms from w e l l l o g d a t a : Geophysics, v.20, p.516-538. Ready, R.J., and Wintz, P.A., 1973, I n f o r m a t i o n E x t r a c t i o n , SNR improvement and data compression i n m u l t i s p e c t r a l imagery: IEEE T r a n s . Commun., v.COM-21, p.1123-1130. Ristow, D., and J u r c z y k , D., 1975, V i b r o s e i s d e c o n v o l u t i o n , G e o p h y s i c a l P r o s p e c t i n g , v..23, p. 363-377. Robinson, J . C , 1979, A t e c h n i q u e f o r the continuous r e p r e s e n t a t i o n of d i s p e r s i o n i n s e i s m i c d a t a : Geophysics, v.44, p. 1345-1351. 1980a, Reply to d i s c u s s i o n of Geophysics, v.44, p. 1345-1351: Geophysics, v.45, p. 1316-1317. 1980b, Reply to d i s c u s s i o n of Geophysics, v.44, p. 1345-1351: Geophysics, v.45, p. 1881 - 1 8 8 3 . Scheuer, T. E., 1981, The Recovery of Subsurface R e f l e c t i v i t y and Impedance S t r u c t u r e from R e f l e c t i o n Seismograms: M.Sc. T h e s i s , U n i v e r s i t y of B r i t i s h Columbia. Sherwood, J . W. C , and T r o r e y , A. W. , 1965, Minimum phase and r e l a t e d p r o p e r t i e s of the response of a h o r i z o n t a l l y s t r a t i f i e d a b s o r p t i v e e a r t h to plane a c o u s t i c waves: Geophysics, v.30, p. 191-197. Sc h n e i d e r , W. A., 1971, Developments i n s e i s m i c data p r o c e s s i n g and a n a l y s i s (1968-1970): Geophysics, v.36, p. 1043-1073. Stone, D. G., 1974, V e l o c i t y and bandwidth: Paper p r e s e n t e d at 44th annual i n t e r n a t i o n a l SEG meeting, D a l l a s Taner, M.T., K o e h l e r , F., and S h e r i f f , R.E.,1979, Complex 205 s e i s m i c t r a c e a n a l y s i s : Geophysics, v.44, p.1041-1057. Taner, M. T., Cook, E. E., and N e i d e l l , N. S., 1970, L i m i t a t i o n s of the r e f l e c t i o n s e i s m i c method. Lessons from computer s i m u l a t i o n s : Geophysics, v.35, p. 551-573. T a y l o r , H. L. , Banks S. C , and McCoy, J . F., 1979, D e c o n v o l u t i o n w i t h the L1 Norm: Geophysics, v.44, p. 39-52, U l r y c h , T. J . , and Walker, C , 1982, A n a l y t i c minimum entropy d e c o n v o l u t i o n : Geophysics, v 47, p. 1295-1302. U l r y c h , T. J . , and Bishop, "T. N., 1975, Maximum entropy s p e c t r a l a n a l y s i s and a u t o r e g r e s s i v e d e c o m p o s i t i o n : Rev. Geophys. and Space P h y s i c s , v.13, p. 183-200. U l r y c h , T. J . , and C l a y t o n , R. W., 1976, Time s e r i e s m o d e l l i n g and maximum en t r o p y : Phys. E a r t h P l a n . I n t . , v.12, p. 188-200. U l r y c h , T. J . , Levy, S., Oldenburg, D. W., and Jones, I. F., 1983, A p p l i c a t i o n s of the Karhunen-Loeve t r a n s f o r m a t i o n i n r e f l e c t i o n s e ismology: p r e s e n t e d at the 53rd Annual SEG meeting i n Las Vegas. Watanabe, S., 1965, Karhunen-Loeve expansion and f a c t o r a n a l y s i s . T h e o r e t i c a l remarks and a p p l i c a t i o n . R e p r i n t e d i n : P a t t e r n R e c o g n i t i o n , J . Sklansky, ed., Stroudsburg, P e n n s y l v a n i a , 1973, (p.146-171). Walker C., and U l r y c h T. J . , 1983, A u t o r e g r e s s i v e recovery of the a c o u s t i c impedance: Geophysics, v.48, p. 1338-1350. Waters, K.H., 1978, R e f l e c t i o n seismology, a t o o l f o r energy r e s o u r c e e x p l o r a t i o n : New York, John Wiley and Sons. Wiggins, R. A., 1978, Minimum entropy d e c o n v o l u t i o n : G e o e x p l o r a t i o n , v.16, p. 21-35. Wood., L. C , H e i s e r , R.C., T r e i t e l , S., and R i l e y , P. L. , 1978, The d e b u b b l i n g of marine source s i g n a t u r e : Geophysics, v. 43, p. 715-7.29. Wuenschel, P. C , 1965, D i s p e r s i v e body waves - an e x p e r i m e n t a l study: Geophysics, v.30, p. 539-551. 206 APPENDIX 2-A: MEAN NOISE POWER AND THE VARIANCE OF A RANDOM NOISE T h i s appendix j u s t i f i e s the use of the mean n o i s e power as an e s t i m a t e of the v a r i a n c e of the random n o i s e p r o c e s s i n the time domain, u s i n g the treatment of J e n k i n s and Watts ( 1 9 6 9 ) , p. 230-232) as a s t a r t i n g " p o i n t . L e t the obse r v e d n o i s e power a t fr e q u e n c y u j be reg a r d e d as a r e a l i z a t i o n of the random v a r i a b l e Cj d e f i n e d by where M J and r j a r e random v a r i a b l e s r e p r e s e n t i n g the r e a l and imaginary p a r t s of the n o i s e i n the fr e q u e n c y domain, i . e . , PROCESS Cj (2.A - 1 ) and (2.A-2) 207 The random n o i s e p r o c e s s Z i s assumed to be normal w i t h mean zero and v a r i a n c e er2; hence, V a r [ y ] - a 2/2 = Var.[*»] = V, f o r example. (2.A-3) Suppose the h i g h - f r e q u e n c y h a l f of the spectrum of the seismogram X(u) can be re g a r d e d as pure n o i s e , and we d e f i n e the random v a r i a b l e C by C * (2/N*) Z C j , where N* = N/2 and N" ^ N/4 + 1. The sample mean a2 d e f i n e d i n the t e x t " s t a t i s t i c a l c u t o f f " s e c t i o n .under " S e l e c t i o n of data.....'" i s a r e a l i z a t i o n of C: 5 2 = (2/N*) Z | X j | 2 , >«" Prom e q u a t i o n s (2.A-1) and (2.A-3) i t may be observed t h a t Cj i s the sum of squares of two normal random v a r i a b l e s w i t h v a r i a n c e V, so C j / V i s d i s t r i b u t e d as x 2 2 - Hence, r e c a l l i n g e q u a t i o n (2.A-3), c [ C j ] = e t c ] = 2V = a 2 . (2.A-4) T h e r e f o r e the sample mean of the n o i s e power i s an unbiased 208 e s t i m a t o r of the v a r i a n c e of the random n o i s e p r o c e s s , i . e . , a 2 i s a v a l i d e s t i m a t e of a 2.. C may be regarded as near normal on the s t r e n g t h of the c e n t r a l l i m i t theorem and Gaussian c o n f i d e n c e l i m i t s a s s i g n e d with r e s p e c t to Var[C] = 8V 2/N' = 2o*/N'. (2.A-5) In the computer program, the (upper) Gaussian 95 perc e n t c o n f i d e n c e l i m i t i s adopted as the e s t i m a t e of the v a r i a n c e . T h i s c o n s e r v a t i v e e s t i m a t e can cause overcompensation f o r n o i s e , p a r t i c u l a r l y i n the e q u a l i t y c o n s t r a i n t f o r m u l a t i o n , w i t h the r e s u l t t h a t only the most prominent s p i k e s are r e c o v e r e d . 209 APPENDIX 2-B: THE CHOICE OF STATISTICAL CUTOFF IN LP DECONVOLUTION The- theory u n d e r l y i n g t h e c h o i c e of the s t a t i s t i c a l c u t o f f i n the " s t a t i s t i c a l c u t o f f " s e c t i o n under " S e l e c t i o n of data " i n the t e x t i s p r e s e n t e d i n t h i s appendix. The r e a l p a r t of the e r r o r i n the deconvolved data i s regarded as a r e a l i z a t i o n of the random v a r i a b l e Q g i v e n by Q = Re{E/W} = [ jiRe{W} + v Im{W} ]/|W| 2 , (2.B-1) where t h i s e q u a t i o n i s understood to apply at each f r e q u e n c y . The treatment r e q u i r e s only t r i v i a l m o d i f i c a t i o n i f Q i s regarded as Im{E/W} i n s t e a d . The r e l i a b i l i t y of data i s a s s e s s e d by comparing Re2{X/W} with a c u t o f f K d e f i n e d i n equa t i o n (2.9) i n terms of e[Q 2] and a parameter a. Hence, the d i s t r i b u t i o n of Q 2 i s of primary i n t e r e s t here. S i n c e u and v i n equa t i o n (2.A-2) are normal random v a r i a b l e s with mean z e r o , so too i s Q; t h e r e f o r e , Q 2/Var[Q] i s d i s t r i b u t e d as X 21. Hence, u s i n g equations (2.B-1) and (2.A-3), e[Q2] = Var[Q] = a 2/(2|W| 2), (2.B-2) 210 which c o r r e s p o n d s to e q u a t i o n (.2.8). As a i n c r e a s e s , the p r o b a b i l i t y t h a t pure n o i s e w i l l exceed the c u t o f f d e c r e a s e s . More p r e c i s e l y , Pr{Q 2>K} = (1/^2*) r e x p ( - x / 2 ) d x / l T x (2.B-3) = e r f c ( a ) . Thus, i f a 2 were known e x a c t l y , s e t t i n g a = 2 would reduce the p r o b a b i l i t y of the i n c l u s i o n of a pure n o i s e v a l u e t o l e s s than 0.005. .21 1 APPENDIX 2-C: NOISE CONSIDERATION IN THE EQUALITY CONSTRAINTS FORMULATION OF LP DECONVOLUTION E s s e n t i a l t o the e q u a l i t y c o n s t r a i n t s f o r m u l a t i o n i s a c o n d i t i o n t o ensure the s t a t i s t i c a l c o m p a t i b i l i t y of the n o i s e removed i n the c o u r s e of the computations; such a c o n d i t i o n i s d e r i v e d below. The r e a l p a r t of the n o i s e n w i l l be c o n s i d e r e d , without l o s s of g e n e r a l i t y . S i n c e u i s a normal random v a r i a b l e , u2/V i s d i s t r i b u t e d as x 2 , , w i t h V d e f i n e d i n e q u a t i o n (.2.A-3). W r i t i n g i t f o l l o w s from J e n k i n s and Watts (1969, p. 71) t h a t -e [ | M | ] = ^ V/2*/exp(-y/2)dy = a/ f~i„ Hence, e[S] = 2a/ f ? (2.C-1) f o r the random v a r i a b l e S d e f i n e d i n e q u a t i o n (2.13). 21.2 In the l i n e a r programming problem, M and v a r e exp r e s s e d as d i f f e r e n c e s between p o s i t i v e q u a n t i t i e s , i . e . , urn = urn - vm and vm = xm - ym, m = 1, ...... M, where M i s the number of a c c e p t a b l e f r e q u e n c i e s as b e f o r e . The a c t u a l c o n s t r a i n t imposed on the n o i s e i s rA 0/M)I. (urn + vm + xm + ym) » e[S] = 2a/'/ff. (2.C-2). T h i s c o n d i t i o n i s c o n s i s t e n t w i t h e q u a t i o n (2.13), p r o v i d e d |,ym| = urn + vm and | PXR | = xm + ym. (2.C-3) The v a l i d i t y of e q u a t i o n s (2.C-3) d e r i v e s from t h e f a c t t h a t m i n i m i z a t i o n of the Li-norm of a ( t ) i s f a v o r e d by r e l a t i v e l y l a r g e n o i s e v a l u e s ; w i t h r e f e r e n c e t o e q u a t i o n (.2.12), i t i s apparent t h a t the t r i v i a l s o l u t i o n i s f e a s i b l e f o r s u f f i c i e n t l y s e r i o u s n o i s e p o l l u t i o n . T h e r e f o r e , g i v e n the c o n s t r a i n t e q u a t i o n (2.C-2), t h e n o i s e i s most e f f e c t i v e i n r e d u c i n g | | <x. | | Y when i t s a t i s f i e s e q u a t i o n s (2.C-3). 2 1 3 APPENDIX 6-A: AR SPIKING DECONVOLUTION OF BAND-LIMITED DATA The problem of o b t a i n i n g a s p a r s e s p i k e r e p r e s e n t a t i o n of b a n d - l i m i t e d r e f l e c t i o n seismograms has been t r e a t e d r e c e n t l y by a number of a u t h o r s i n the g e o p h y s i c a l l i t e r a t u r e . The b a s i c assumptions u n d e r l y i n g t h i s o p e r a t i o n a r e : (a) An e a r t h model which c o n s i s t s of a s e t of p h y s i c a l l y d i s t i n c t l a y e r s . (b) A s p a r s e r e f l e c t i v i t y s e r i e s , t h a t i s , r e f l e c t i o n s a r e g e n e r a l l y s e p a r a t e d by a number of z e r o s , and (c) The r e c o r d e d seismogram i s a r e a s o n a b l e b a n d - l i m i t e d r e p r e s e n t a t i o n of the e a r t h response f u n c t i o n . That i s , t h e s o u r c e ' s phase and a m p l i t u d e d i s t o r t i o n s a r e l a r g e l y removed. We e x p r e s s the second assumption by the e q u a t i o n : A H r ( t ) - I r •5(t-nA) (6.A-1) where N i s the number of samples i n the i n p u t seismogram, r n i s the r e f l e c t i o n c o e f f i c i e n t a t the nth sample, 214 A i s the sampling i n t e r v a l , and S(t-nA) r e p r e s e n t s a D i r a c d e l t a f u n c t i o n at time nA. The d i g i t a l F o u r i e r t r a n s f o r m of a s p a r s e r e f l e c t i v i t y s e r i e s r ( t ) c o n s i s t s of the sum of a number of s i n u s o i d s and i s w r i t t e n a s : R(wj)=Rj= X r n-exp(-i'2TJn/N) (6.A-2) w i t h j b e i n g the index of w the a n g u l a r f r e q u e n c y , and n the time sampling index. I t i s obvious t h a t each f r e q u e n c y i n the s e r i e s R( W J ) c o n t a i n s c o n t r i b u t i o n s from each of the s p i k e s p r e s e n t i n the r e f l e c t i v i t y s e r i e s . Hence, i t i s expected t h a t under the assumptions s p e c i f i e d p r e v i o u s l y , a f u l l - b a n d e s t i m a t e of the r e f l e c t i v i t y f u n c t i o n r ( t ) can be o b t a i n e d from an inc o m p l e t e s e t of R(w (')'s v i a the use of a p p r o p r i a t e n u m e r i c a l t e c h n i q u e s . In t h i s work, we have used the L i n e a r Programming and the A u t o - R e g r e s s i v e t e c h n i q u e which w i l l be d e s c r i b e d next. A u t o - R e g r e s s i v e D e c o n v o l u t i o n As was shown i n e q u a t i o n (6.A-2), the f r e q u e n c y r e p r e s e n t a t i o n of a s p a r s e - s p i k e s e r i e s c o n s i s t s of a sum of a number of complex s i n u s o i d s each of which c o r r e s p o n d s t o a s p e c i f i c non-zero r e f l e c t i o n c o e f f i c i e n t . The problem of s p e c t r a l e x t e n s i o n can then be viewed as an A u t o - R e g r e s s i v e p r o c e s s where a complex p r e d i c t i o n f i l t e r {£k} i s c a l c u l a t e d and the a v a i l a b l e i n f o r m a t i o n i s e x t r a p o l a t e d by a c o n v o l u t i o n w i t h t h i s f i l t e r . The p r o c e s s i s 215 simply summarized by the f o l l o w i n g s e t of e q u a t i o n s . L e t the forward AR p r e d i c t i o n be g i v e n by: and the r e v e r s e AR p r e d i c t i o n by.: f R*= Z flLR*- . w i t h p b e i n g the o r d e r ( l e n g t h ) of the p r e d i c t i o n o p e r a t o r {j3k} , and * d e n o t i n g the complex c o n j u g a t e . We seek a p r e d i c t i o n f i l t e r j3 such t h a t the sum of the f o r w a r d and r e v e r s e p r e d i c t i o n e r r o r s i s m i n i m i z e d i n a l e a s t - s q u a r e s sense. That i s , the f i l t e r c o e f f i c i e n t s are found through the m i n i m i z a t i o n o f : e* = ^ ( N - p h Z j R j - l l ^ l 2 D e t a i l s of some of the approaches to the s o l u t i o n of t h i s m i n i m i z a t i o n problem are g i v e n i n Burg (1975), U l r y c h and C l a y t o n (1976), U l r y c h and B i s h o p (19-7-5-)- and C l a e r b o u t (1976). The r e a d e r who i s f a m i l i a r w i t h the AR p r o c e s s p r o b a b l y r e a l i z e s t h a t the o r d e r of the p r e d i c t i o n f i l t e r i s q u i t e important t o a s u c c e s s f u l s p e c t r a l e x t e n s i o n p r o c e s s . S i n c e one p r e d i c t i o n c o e f f i c i e n t i s r e q u i r e d f o r the e x t r a p o l a t i o n of each s i n u s o i d , the i d e a l o r d e r of the AR p r o c e s s s h o u l d be equal to or l a r g e r than the 216 number of l a y e r s NL i n our model. However, f o r b a n d - l i m i t e d r e a l d a t a , where we a r e g i v e n o n l y M<(N+i)/2 freq u e n c y o b s e r v a t i o n s the requirement p>NL cannot be met. Hence, i n t h i s c a s e , we hope t h a t the p r o c e s s w i l l d e t e c t o n l y the l a r g e r r e f l e c t o r s i n our model, th e r e b y , r e c o n s t r u c t i n g o n l y the i n f o r m a t i o n c o n c e r n i n g the major f e a t u r e s of the sought r e f l e c t i v i t y model. Our e x p e r i e n c e t o date w i t h a l a r g e number of s y n t h e t i c as w e l l as r e a l d a t a examples have proven the p r o c e s s t o be q u i t e s u c c e s s f u l . However, i n c a s e s where the l o c a l geology cannot be approximated by a r e l a t i v e l y s m a l l number of r e f l e c t o r s the AR p r o c e s s , and a l s o t h e LP w i l l have a somewhat lower l i k e l i h o o d of s u c c e s s . 217 APPENDIX 7-A: THE DECONVOLUTION OF PHASE-SHIFTED WAVELETS 7-A.1 INTRODUCTION In both r e f l e c t i o n and r e f r a c t i o n s e i s m i c s t u d i e s , i t i s o f t e n assumed t h a t the e a r t h i s composed of a number of homogeneous l a y e r s . A t y p i c a l - seismogram from such an e a r t h model can be approximated by c o n v o l v i n g a source wavelet w i t h a set of r e f l e c t i v i t y c o e f f i c i e n t s . The s o u r c e energy i s assumed to be p r o p a g a t i n g as a p l a n e wave, and the r e f l e c t i v i t y sequence i s i d e a l l y a s e t of D i r a c d e l t a f u n c t i o n s l o c a t e d at times c o r r e s p o n d i n g t o p r i m a r y and m u l t i p l e r e f l e c t i o n s from the l a y e r e d e a r t h model. Given t h a t t h i s " c o n v o l u t i o n a l model" f o r a seismogram has some v a l i d i t y , a s t a n d a r d p r o c e d u r e i s f i r s t t o e s t i m a t e a s o u r c e wavelet and then t o remove i t s e f f e c t by d e c o n v o l u t i o n . The r e s u l t a n t e s t i m a t e of the r e f l e c t i v i t y sequence i s used to i n t e r p r e t the parameters of the l a y e r e d e a r t h s t r u c t u r e . A c o m p l i c a t i n g f a c t o r t o t h i s s i m p l i s t i c approach i s i n t r o d u c e d when wide-angle r e f l e c t i o n s a r e c o n s i d e r e d or when the wave-front or r e f l e c t i v e boundary i s c u r v e d . Under t h e s e c o n d i t i o n s , the a n g l e of i n c i d e n c e f o r the incoming ray can be s u p e r c r i t i c a l and inhomogeneous waves w i l l be c r e a t e d on the boundary (Aki and R i c h a r d s , 1980, p. 155). The r e f l e c t i v i t y c o e f f i c i e n t f o r the l a y e r becomes complex; c o n s e q u e n t l y , the shape of the emerging wave packet d i f f e r s from the i m p i n g i n g one. The amount of d i s t o r t i o n depends upon the phase e of the complex r e f l e c t i o n c o e f f i c i e n t . F o l l o w i n g 218 Arons and Yennies (1950) [or A k i and R i c h a r d s (1980)], the a l t e r e d waveform w(e;t) can be w r i t t e n as w(e;t) = c o s ( e ) - w ( t ) + s i n ( e ) • H [ w ( t ) ] (7-A.1) where H[ ] denotes the H i l b e r t t r a n s f o r m . B a s i c a l l y , upon i n t e r a c t i n g w i t h the boundary, the p o s i t i v e f r e q u e n c i e s i n w(t) are advanced by e and the n e g a t i v e f r e q u e n c i e s a r e r e t a r d e d by e; i t i s t h i s s i g n e f f e c t which i s r e s p o n s i b l e f o r the H i l b e r t t r a n s f o r m i n e q u a t i o n (7-A.1). The importance of an a l t e r a t i o n i n the waveform can be a p p r e c i a t e d when one attempts t o e s t i m a t e an : a r r . i v a l time f o r a wavelet t h a t has been s i g n i f i c a n t l y phase s h i f t e d . F o r example, i f e = 7r/2, then w(e;t) * H [ w ( t ) ] , and the a l t e r e d waveform w i l l d i f f e r g r e a t l y from w ( t ) . In p a r t i c u l a r , i f w(t) i s d e l t a - l i k e , w(e;t) w i l l have an emergent p o r t i o n of s i g n i f i c a n t a m p l i t u d e a t times p r e c e d i n g those e x p e c t e d from g e o m e t r i c a l ray computations. E s t i m a t i n g the a r r i v a l time t o be near the onset of the emergent energy can produce a p p r e c i a b l e e r r o r . Choy and R i c h a r d s (1975) p r e s e n t e d whole e a r t h seismograms f o r SH waves i n which v a r i o u s S a r r i v a l s are c l e a r l y phase s h i f t e d by a p p r o x i m a t e l y w/2; they suggested t h a t e r r o r s i n e s t i m a t i n g the a r r i v a l time of the p h a s e - s h i f t e d w a v e l e t s c o u l d be as l a r g e as 3 to 5 s e c . Another a r e a i n which waveform a l t e r a t i o n can have s i g n i f i c a n t consequences i s i n e x p l o r a t i o n s e i smology where wide-angle 219 r e f l e c t i o n s , nonplanar waves, and d i p p i n g l a y e r s produce c o n d i t i o n s of s u p e r c r i t i c a l r e f l e c t i o n . Attempting to f o l l o w an event h o r i z o n by c o r r e l a t i n g peak (or troughs) on s u c c e s s i v e t r a c e s i n a r e c o r d s e c t i o n can l e a d to erroneous r e s u l t s i f the c h a r a c t e r of the wavelet changes from t r a c e t o t r a c e . The purpose of t h i s paper i s to overcome these d i f f i c u l t i e s by i n t r o d u c i n g a complex r e f l e c t i v i t y f u n c t i o n from the o u t s e t . The c o n v o l u t i o n of t h i s r e f l e c t i v i t y f u n c t i o n with an a n a l y t i c source wavelet produces an a n a l y t i c seismogram. Once r e c a s t as a c o n v o l u t i o n problem, l i n e a r i n v e r s e theory i s used to deconvolve the a n a l y t i c seismogram to o b t a i n unique averages of the r e a l and imaginary p a r t s of a complex r e f l e c t i v i t y f u n c t i o n . I n f o r m a t i o n about the amplitude and phase of the r e f l e c t i v i t y f u n c t i o n i s r e c o v e r a b l e through t e c h n i q u e s of model c o n s t r u c t i o n or by u s i n g a s i m p l i f i e d i n t e r p r e t a t i o n .based upon the modulus of the deconvolved output. T h i s l a t t e r method i s not exact, but i t works w e l l i f the s e i s m i c a r r i v a l s are not too c l o s e t o g e t h e r . Here we apply the a n a l y t i c d e c o n v o l u t i o n technique only t o s y n t h e t i c d a t a , but U l r y c h and Walker (1982) use our a n a l y t i c f o r m u l a t i o n and a complex v e r s i o n of minimum entropy d e c o n v o l u t i o n (MED) to deconvolve a s e t of normal i n c i d e n c e seismograms. T h e i r work p r e s e n t s an important and p r a c t i c a l e x t e n s i o n of the a n a l y t i c f o r m u l a t i o n , s i n c e the MED a l g o r i t h m does not r e q u i r e knowledge of the wavelet; i n f a c t , an e s t i m a t e of the source wavelet can be o b t a i n e d by i n v e r t i n g the MED f i l t e r . In those cases where the source f u n c t i o n i s not known, such an e s t i m a t e d wavelet i s a c r u c i a l f i r s t s t e p to c a r r y i n g out the d e c o n v o l u t i o n by the methods 220 p r e s e n t e d here. 7-A.2 THEORY In many p h y s i c a l problems, a data set x ( t ) i s generated by the s u p e r p o s i t i o n of t i m e - d i s p l a c e d r e p l i c a t i o n s of a source wavelet w(t) t h a t i s , x ( t ) = r ( t ) * w ( t ) , (7-A.2) where * denotes the c o n v o l u t i o n o p e r a t o r . In s e i s m o l o g i c a l problems, r ( t ) i s c a l l e d the r e f l e c t i v i t y f u n c t i o n . For normal i n c i d e n c e seismograms over a l a y e r e d e a r t h i t can be w r i t t e n as r ( t ) = X r- 6 (t - T ), (7-A.3) so t h a t M x ( t ) = Z r-w(t - T • ) , (7-A.4) where r are r e f l e c t i o n c o e f f i c i e n t s and 7N are the delay times r e l a t e d t o l a y e r t h i c k n e s s e s and m a t e r i a l v e l o c i t i e s . When the data are adequately r e p r e s e n t e d by equa t i o n (7-A.2), the rec o v e r y of r ( t ) , when w(t) i s known, i s a well-posed l i n e a r i n v e r s e problem f o r which a v a r i e t y of s o l u t i o n s have been given i n the l i t e r a t u r e . But th e r e e x i s t cases of p r a c t i c a l importance where n e i t h e r equation (7-A.2) nor equation (7-A.4) i s adequate because w(t) has been phase 221 s h i f t e d upon r e f l e c t i o n from a boundary. C o n s e q u e n t l y , the data c o n s i s t of a s u p e r p o s i t i o n of w a v e l e t s , each phase s h i f t e d by an a r b i t r a r y and unknown amount. Any attempt to deconvolve without a l l o w i n g f o r t h e s e phase s h i f t s w i l l produce a poor e s t i m a t e or r ( t ) . I f w(t) i s used t o denote the H i l b e r t t r a n s f o r m H [ w ( t ) ] , . t h e n a wavelet, phase s h i f t e d by an amount e, can be w r i t t e n as w(e;t) » cos e-w(t) + s i n e.'w(t) = Re{w(t )• exp[ i e ]}, where Re denotes the r e a l p a r t and w(t) i s the a n a l y t i c wavelet w(t) = w(t) - iw(.t) ( B r a c e w e l l , 1978, p.268). A l l o w i n g f o r p h a s e - s h i f t e d w a v e l e t s , e q u a t i o n (7-A.4) can be r e w r i t t e n as x ( t ) = .1 r: w( e: , t - r: ) j s l J 1 i = Re { I r- e x p f i e ] - w ( t - r; )}, (7-A.5) By i n t r o d u c i n g the c o n t i n u o u s f u n c t i o n r ( t ) . e x p t i e ( t ) ] = Z r exp[ i e-, ] • 6 (t - r: ) , 222 e q u a t i o n (7-A.S) may be r e w r i t t e n as x ( t ) •= R e { r ( t ) - e x p [ i e ( t ) ] * w ( t ) } . (7-A.6) From e q u a t i o n (7-A.6) i t can be seen t h a t the seismogram x ( t ) i s the r e a l p a r t of the c o n v o l u t i o n p r o d u c t of a complex r e f l e c t i v i t y f u n c t i o n and the a n a l y t i c r e p r e s e n t a t i o n of the wav e l e t . A l t h o u g h e q u a t i o n (7-A.6) i s n e a r l y i n the form of e q u a t i o n (7-A.2), the r e a l p a r t on the r i g h t - h a n d s i d e p r e v e n t s us from a p p l y i n g known d e c o n v o l u t i o n t e c h n i q u e s to r e c o v e r r(.t) and e ( t ) . I t i s c o n v e n i e n t , t h e r e f o r e , to i n t r o d u c e the a n a l y t i c s i g n a l x ( t ) = x ( t ) - i x * ( t ) . Then, s i n c e the b r a c k e t e d term i n e q u a t i o n (7-A.6) i s an a n a l y t i c s i g n a l , i t f o l l o w s t h a t x ( t ) = r ( t ) • e x p [ i e ( t ) ] *w(t).. (7-A.7) T h i s i s the d e s i r e d c o n v o l u t i o n f o r m u l a , and e v i d e n t l y s t a n d a r d p r o c e d u r e s can be used to r e c o v e r r ( t ) and e ( t ) i f w(t) i s known. Here we s h a l l t r e a t the d e c o n v o l u t i o n as a l i n e a r i n v e r s e problem and use the f o r m a l i s m d e v e l o p e d i n Oldenburg ( 1981) . L e t us f i r s t d e f i n e a complex model m(t) as m(t) = r ( t ) e x p [ i e ( t ) ] . (7-A.8) 223 Then e q u a t i o n (7-A.7) can be r e w r i t t e n as x ( t ) = m(t)*w(t) + n ( t ) , ' (7-A.9) where t h e . a n a l y t i c n o i s e n ( t ) has been i n c l u d e d t o account f o r a d d i t i v e n o i s e n ( t ) i n the measured s i g n a l . Sometimes, the power spectrum of n ( t ) may be known; or, i n the absence of t h a t i n f o r m a t i o n , one can assume t h a t n ( t ) i s u n c o r r e l a t e d G a u s s i a n n o i s e w i t h an e s t i m a t e d s t a n d a r d d e v i a t i o n , o 0 . The n o i s e s t a t i s t i c s of n ( t ) a r e e a s i l y computed from n ( t ) , but the r e s u l t w i l l depend upon the a l g o r i t h m used t o compute the H i l b e r t t r a n s f o r m . I f the d i g i t a l F o u r i e r t r a n f o r m i s used, the power spectrum of n ( t ) w i l l be the same as t h a t of n ( t ) when n ( t ) i s u n b i a s e d . The d e c o n v o l u t i o n of e q u a t i o n (7-A.9) w i l l be c a r r i e d out i n the f r e q u e n c y domain,. .It i s assumed here t h a t a l l time f u n c t i o n s a r e p e r i o d i c w i t h p e r i o d T. The F o u r i e r t r a n s f o r m of any such f u n c t i o n g ( t ) i s g i v e n by G ( f - ) = G L = 1/T / g ( t ) - e x p [ - i 2 i r k t / T ] d t , k = 0 , +1 , + 2, ..., and the i n v e r s e F o u r i e r t r a n s f o r m i s g ( t ) = 1 G,.exp[ i 2 i r k t / T ] 224 Unique i n f o r m a t i o n about m(t) w i l l be found by c o n s t r u c t i n g an a n a l y t i c f i l t e r v ( t ) such t h a t an a n a l y t i c a v e r a g i n g f u n c t i o n a ( t ) = w ( t ) * v ( t ) (7-A.10) i s "as c l o s e as p o s s i b l e " t o some d e s i r e d f u n c t i o n ' l i ( t ) . I d e a l l y we want v ( t ) t o c o n t r a c t w(t) i n t o a r e a l d e l t a - l i k e f u n c t i o n . However, A t h i s i s i m p o s s i b l e i n the a n l y t i c f o r m u l a t i o n because = 0 f o r k<0; hence the n e g a t i v e f r e q u e n c i e s a r e l o s t from the s o l u t i o n . T h i s i s the p r i c e p a i d so t h a t p h a s e - s h i f t e d wavelets can be t r e a t e d i n a c o n v o l u t i o n format. The unique averages of the model a r e found by c o n v o l v i n g v ( t ) wi t h the a n a l y t i c s i g n a l . That i s , <m(t)> •» x ( t ) * v ( t ) = m ( t ) * a ( t ) , (7-A.11) The s t a t i s t i c a l v a r i a n c e of the a v e r a g e s i s g i v e n by Var[<m(t)>] = V a r [ n ( t ) * v ( t ) ] . (7-A.12) The concept of unique a v e r a g e s i s sometimes m i s u n d e r s t o o d , so an e l a b o r a t i o n i s u s e f u l . L e t us f i r s t suppose t h a t the d a t a are a c c u r a t e . The f i r s t e q u a l i t y i n e q u a t i o n (7-A.11) shows t h a t <m(t)> i s o b t a i n e d from l i n e a r c o m b i n a t i o n s of the d a t a , and the second e q u a l i t y shows t h a t any model which i d e n t i c a l l y r e p r o d u c e s the data w i l l have these same a v e r a g e s . I t i s i n t h i s sense t h a t the averages are unique; c o n s e q u e n t l y , <m(t)> and a ( t ) summarize our knowledge 2 2 5 about a l l p o s s i b l e models which might reproduce the d a t a . However, i f the data a r e i n a c c u r a t e , the averages must a l s o be i n a c c u r a t e . T h i s s t a t i s i c a l i n a c c u r a c y i s q u a n t i f i e d by the v a r i a n c e i n e q u a t i o n (7-A.12). The a n a l y t i c f i l t e r can be found by m i n i m i z i n g "*7j . A <t> = cos 6-J | a ( t ) - h ( t ) 1 2 d t - T / i + sin0-Var[<m(t)>], O<0<jr/2-. (7-A.13) The u s u a l t r a d e - o f f parameter 6 has been i n c l u d e d i n e q u a t i o n (7-A.13), and i t c o n t r o l s the t r a d e - o f f between r e s o l u t i o n and a c c u r a c y of the a v e r a g e s . F o l l o w i n g Oldenburg (1981), we use P a r s e v a l ' s theorem and the c o n v o l u t i o n theorem t o r e w r i t e e q u a t i o n (7-A.13) a s \fr = cos 6-Z ' T l T O ^ - H j J 2 + s i n 6-T2Z \\\\2- (7-A. 14) The lower l i m i t on the summation i s k = 0 i n s t e a d of the u s u a l k = - <x, because the s p e c t r a l components of a n a l y t i c f u n c t i o n s a r e ze r o f o r n e g a t i v e f r e q u e n c i e s . The F o u r i e r c o e f f i c i e n t s of the d e s i r e d f i l t e r , found by d i f f e r e n t i a t i n g e q u a t i o n (7-A.14) w i t h A r e s p e c t to V* and s e t t i n g a ^ / sV* = 0, are 2 2 6 A cos 0-%w* v, = — _ k > o, * T| wj 2 c o s f l •+ |Nj 2 s i n e , V = 0, k,< 0, (7-A.15! where the * denotes the complex c o n j u g a t e . In the s p e c i a l case h ( t ) = S ( t ) , a D i r a c d e l t a f u n c t i o n c e n t e r e d at the o r i g i n , cos 9> W* Vi = ; k > 0 T 2 | W j J 2 . c o s 6 + T | N f e | 2 s i n 8, V = 0, k<0, (7-A.16) E q u a t i o n (7-A. 16) reduces t o = l/(T 2Wj^) when 6 = 0; t h i s r e s u l t i s i d e n t i c a l to t h a t o b t a i n e d when s o l v i n g 5 ( t ) = w ( t ) * v ( t ) by u s i n g the c o n v o l u t i o n theorem and d i v i d i n g i n the frequency domain. With v ( t ) computed, the a v e r a g i n g f u n c t i o n a ( t ) i s e a s i l y found from e q u a t i o n (7-A.10). The averages [from e q u a t i o n (7-A.11)] have the form <m(t)> = x ( t ) * v ( t ) = m ( t ) * a ( t ) - im(.t) *a ( t ) .. (7-A.17) I t i s seen t h a t <m(t)> i s the sum of two averages of the model. One of these averages i s the i n n e r produce to m(t) w i t h a d e l t a - l i k e f u n c t i o n a ( t ) , and the o t h e r average i s the i n n e r product of m(t) 227 w i t h a ( t ) . Only i n the s p e c i a l case where m(t) i s p u r e l y r e a l or p u r e l y imaginary can t h e i n t e r p r e t a t i o n proceed w i t h the same ease t h a t i t does i n o r d i n a r y d e c o n v o l u t i o n a l g o r i t h m s . In g e n e r a l , the model m(t) = r ( t ) - c o s e ( t ) + i r ( t ) - s i n e ( t ) (7-A.18) w i l l have both r e a l and imaginary components; c o n s e q u e n t l y , the r e a l and imaginary p a r t s of <m(,t)> have the form Re<m(t)> = Re{x*v} = r ( t ) - c o s «(t)*a(.t) + r ( t ) - s i n e ( t ) * a ( t ) , Im<m(t)> = Im{x*v} = r ( t ) - s i n c ( t ) * a ( t ) -r(t)cos«(t)*a(-t)'. (7-A.19) The averages i n e q u a t i o n (7-A.19), t h e i r s t a t i s t i c a l e r r o r , and the a v e r a g i n g f u n c t i o n s c o n t a i n the d e s i r e d i n f o r m a t i o n about r ( t ) and e ( t ) . The g e n e r a l problem of r e c o v e r i n g i n f o r m a t i o n about r ( t ) and e ( t ) from e q u a t i o n (7-A.19) w i l l be c o n s i d e r e d , but f i r s t a si m p l e , i n s i g h t f u l example w i l l be p r e s e n t e d . We c o n s i d e r a s i n g l e complex r e f l e c t i v e c o e f f i c e n t l o c a t e d a t t = T, so t h a t the model i s 228 m(t) = r , e x p f i e , ] 6 ( t - T , ) . From e q u a t i o n (7-A. 1.9) the r e a l and imaginary p a r t s of the averages become Re<m(t)> = r v c o s e r a ( t - T , ) + r ^ s i n - e ^ l ' t - r , ) . Im<m(t)> • r r s i n e , a ( t - T , ) - r.,cos e 1 a ( t - T 1 ) . C o nsequently, Re<m(t)> w i l l look l i k e a r e p r o d u c t i o n of a ( t ) , and the imaginary p a r t w i l l have the form of 'a(t) i f c, i s c l o s e t o z e r o or ir. A l t e r n a t i v e l y , i f e, i s c l o s e t o v/2 (or 3ir/2), the Re<m(t)> w i l l resemble a ( t ) and the imaginary p a r t w i l l look l i k e a ( t ) . For i n t e r m e d i a t e v a l u e s of e,, t h e d e l t a f u n c t i o n s p i k e i n the model w i l l be seen as a l i n e a r c o m b i n a t i o n of a d e l t a - l i k e a v e r a g i n g f u n c t i o n and i t s H i l b e r t t r a n s f o r m . For the c u r r e n t example, we s h a l l choose e, = TT/4. The wavelet and data a r e shown i n F i g u r e 7-A.1, and the r e s u l t s of the d e c o n v o l u t i o n o b t a i n e d by u s i n g e q u a t i o n (7-A. 16) w i t h 0 = 0 a r e shown i n F i g u r e s 7 - A . l c - l . f . C l e a r l y , both the r e a l and ima g i n a r y p a r t s of the averages are l i n e a r c o m b i n a t i o n s of a ( t ) a n d a ( t ) . I f the da t a were i n a c c u r a t e the d e c o n v o l u t i o n c o u l d not be c a r r i e d — o u t at 6 = 0; some s a c r i f i c e i n r e s o l u t i o n would have t o be made i n o r d e r t o a c h i e v e s t a t i s t i c a l r e l i a b i l i t y i n the a v e r a g e s . T h i s c o u l d be e f f e c t e d by u s i n g e q u a t i o n (7-A.16) w i t h a v a l u e of d g r e a t e r than z e r o . T y p i c a l r e s u l t s a r e shown i n F i g u r e s 7 - A . l g - l j . Our u l t i m a t e g o a l i s to determine the v a l u e s of the parameters 2 2 9 F i g u r e 7-A.1 Shown i n s u c c e s s i v e p a n e l s a r e (a) the da t a o b t a i n e d by p h a s e - s h i f t i n g the wavelet i n ( b ) ; (c) the r e a l p a r t of the averages o b t a i n e d by d e c o n v o l v i n g w i t h 8 = 0; (d) the a v e r a g i n g f u n c t i o n a ( t ) ; (e) the imaginary p a r t of the averages o b t a i n e d by d e c o n v o l v i n g w i t h 6 = 0; ( f ) the H i l b e r t t r a n s f o r m of a ( t ) ; (g)-(j) are the same as the r e s u l t s i n (c) - ( f ) except t h a t the d e c o n v o l u t i o n was c a r r i e d out w i t h 6 > 0 ; (k) the modulus of the averages i n (g) and ( i ) ; (1) the modulus A ( t ) of the a v e r a g i n g f u n c t i o n [ e q u a t i o n (7-A.20)]; (m) the r e c o v e r e d r e f l e c t i v i t y f u n c t i o n ; (n) the r e c o v e r e d phase f u n c t i o n . 230 (:,,r,,£,) from the averages i n F i g u r e s 7-A.1g-1i (or 1c and l e ) . There are a number of ways to a c c o m p l i s h t h i s . The f i r s t method uses the' modulus of the averages. For the example of a s i n g l e r e f l e c t o r , the modulus of <m(t)> i s e a s i l y shown t o be independent of the phase e, and e q u a l to |<m(t)>| = | x ( t ) * v ( t ) | = r , f a ' ( t - T/) 2 •+ a ( t - T , ) 2 = r , A ( t - T,). where (7-A.20) A ( t ) » f a ( t ) 2 + a ( t ) 2 . Thus |<m(t)>| i s s c a l e d and t r a n s l a t e d r e p l i c a t i o n of A ( t ) , and the best v a l u e s f o r the s c a l i n g f a c t o r r , and the time d e l a y T, are those which minimize the d i f f e r e n c e between the l e f t - and r i g h t - h a n d s i d e s of e q u a t i o n (7-A.20). The modulus of the averages and A ( t ) a r e shown i n F i g u r e s 7-A.1k - 11. In an a l t e r n a t i v e method, r , and r , can be e s t i m a t e d by r e c o g n i z i n g t h a t A ( t ) i s a d e l t a - l i k e f u n c t i o n whose maximum i s at t = 0; the time d i f f e r e n c e between z e r o and the l o c a t i o n of the maximum i n |<m(t)>| w i l l thus p r o v i d e an e s t i m a t e f o r r , . Moreover, *a(t = 0) = 0, so r , can be r e c o v e r e d d i r e c t l y from r , = 1/A(0)-max|<m(t)>|, (7-A.21) where max i n d i c a t e s the maximum v a l u e . The phase e, can be e s t i m a t e d 231 from the f u n c t i o n 0(t) = t a n - 1 [Im<m(t)> /Re<m(t)>] = t a n - 1 { [ s i n « ( t ) a ( t - T , ) - c o s e ( t ) a ( t - T , ) ] / [cos e ( t ) a ( t - T , j + s i n e ( t ) a ( t - r , )]} (7-A. .22) which, a t time t = r , , reduces t o *(.t •• r,) = t a n - ' [ s i n - e ( t ) / c o s -e(t)] = e ( r , ) . (7-A.23) Thus the phase s h i f t e, can be found by e v a l u a t i n g <t>(t) a t p e r c i s e l y the a r r i v a l time of the w a v e l e t . The r e c o v e r e d a m p l i t u d e and phase are shown i n F i g u r e s 7-A.lm and i n . I t i s c l e a r t h a t i f the data were a c c u r a t e and c o n s i s t e d of a •single p h a s e - s h i f t e d wavelet t h e r e would be no problem i n computing the a r r i v a l time of the p u l s e , the magnitude of the r e f l e c t i v e c o e f f i c i e n t , and the phase s h i f t . "In s p i t e of i t s s u c c e s s , we emphasize t h a t e q u a t i o n (7-A.23) s h o u l d be used w i t h some c a u t i o n because a ( t ) i s l a r g e i f t i s not p r e c i s e l y z e r o . A s m a l l e r r o r i n e s t i m a t i n g r , can l e a d t o a l a r g e e r r o r i n e,. The i n t e r p r e t a t i o n of the d e c o n v o l u t i o n r e s u l t s i s l e s s s t r a i g h t - forward when the d a t a are i n a c c u r a t e or when m u l t i p l e r e f l e c t o r s are s u f f i c i e n t l y c l o s e so t h a t a v e r a g i n g f u n c t i o n s from 232 d i f f e r e n t r e f l e c t i v e events o v e r l a p . These c o m p l e x i t i e s a f f e c t the averages d i f f e r e n t l y so they are t r e a t e d s e p a r a t e l y . The e f f e c t s of o b s e r v a t i o n a l e r r o r s are c o n s i d e r e d f i r s t . L e t n ( t ) r e p r e s e n t the a d d i t i v e n o i s e on the seismogram, and n ( t ) and n ( t ) , r e s p e c t i v e l y , r e p r e s e n t the H i l b e r t t r a n s f o r m of the n o i s e and the a n a l y t i c n o i s e s i g n a l . . The averages of the model can be w r i t t e n as where v ( t ) i s the a n a l y t i c f i l t e r [ e q u a t i o n (7-A.16)] and 5<m(t)> r e p r e s e n t s the d i s c r e p a n c y between the t r u e average and t h a t (7-A.24) computed from the erroneous d a t a . S i n c e ,x(t) = x ( t ) + n ( t ) , i t f o l l o w s t h a t 6<m(t)> n ( t ) * v ( t ) . (7-A.25) and thus 6 Re<m(t)> 2 n ( t ) * v ( t ) , 5Im<m(t)> 2 H [ n ( t ) * v ( t ) ] . (7-A.26) 233 By u s i n g P a r s e v a l ' s e q u a t i o n , the c o n v o l u t i o n theorem f o r F o u r i e r t r a n s f o r m s , and the assumption t h a t n ( t ) i s s t a t i o n a r y , i t f o l l o w s from e q u a t i o n (7-A.26) t h a t the v a r i a n c e of the averages i s e v e n l y d i v i d e d between the r e a l and imaginary p a r t s , t h a t i s , Var[Re<m(t)>] = Var[Im<m(t)>] = T 2 / 2 I |N^V^ j 2 . (7-A..27) [The sum i n e q u a t i o n (7-A..27) be g i n s .at k ••- 1 i n s t e a d of k = 0 because th e n o i s e i s assumed to be u n b i a s e d , and hence N 0 - 0.] In a d d i t i o n , approximate formulas f o r the v a r i a n c e s of |<m(t)>| and 0 ( t ) can be computed from the T a y l o r e x p a n s i o n t e c h n i q u e p r e s e n t e d by J e n k i n s and Watts (1969, p. 76). We have Var[ | <m) (t)> | ] « Var [Re<m(t )>.], (7-A.28) and Var[«(t)] » Var[Re<m(t)>]/|m(t)| 2 (7-A.29) In d e r i v i n g e q u a t i o n s (7-A.27), (7-A.28), and (7-A.29), we have used the f a c t t h a t f o r s t a t i o n a r y n o i s e C o v [ n ( t ) , n ( t ) ] = 0. F u r t h e r , i n e q u a t i o n (7-A.29), the denominator s h o u l d r e a l l y be 234 f E[Re<m(t)>] } 2 + { E [Im<m(t)>]} 2 where E denotes the expected v a l u e ; t h e s e q u a n t i t i e s , however, are unknown and have been r e p l a c e d with t h e i r measured v a l u e s . A second i n t e r p r e t i v e c o m p l i c a t i o n a r i s e s when the r e f l e c t i v e h o r i z o n s a r e so c l o s e t h a t t h e r e i s s i g n i f i c a n t o v e r l a p of the a v e r a g i n g f u n c t i o n s i n the d e c o n v o l v e d o u t p u t . An attempt t o i n f e r d i r e c t l y from <m(t)> the c o r r e c t v a l u e s of the phase s h i f t s , the a r r i v a l times of the w a v e l e t s , and the magnitude of the r e f l e c t i v e c o e f f . i c e n t s may produce poor r e s u l t s i f e q u a t i o n s (7-A.21) and (7-A.23) are used a l o n e . In f a c t , the problem encountered i s s i m i l a r to t h a t f a c e d i n i n t e r p r e t i n g the r e s u l t s from s t a n d a r d d e c o n v o l u t i o n t e c h n i q u e s . There, each r e f l e c t i v e event appears as an a v e r a g i n g f u n c t i o n m u l t i p l i e d by some amplitude f a c t o r . I f r e f l e c t i v e events a r e so c l o s e t h a t t h e a v e r a g i n g f u n c t i o n s o v e r l a p s i g n i f i c a n t l y , then the l o c a t i o n of the maxium of <m(t)> may no l o n g e r c o i n c i d e w i t h the t r u e a r r i v a l time of a wavelet. In the extreme, two r e f l e c t i v e e v e n t s might appear as a s i n g l e peaked f u n c t i o n i n the averages w i t h the time a t which the peak o c c u r s l y i n g between the two t r u e a r r i v a l t i m e s . Our problem i n d e c o n v o l v i n g p h a s e - s h i f t e d s i g n a l s i s somewhat worse than i n the s t a n d a r d d e c o n v o l u t i o n problem • because the averages <m(t)> are complex c o m b i n a t i o n s of d e l t a - l i k e a v e r a g i n g f u n c t i o n s and t h e i r H i l b e r t t r a n s f o r m s . Here we p r e s e n t two methods f o r r e c o v e r i n g r ( t ) and e ( t ) from the averages i n e q u a t i o n (7-A.17). The f i r s t method i s one of many p o s s i b l e model c o n s t r u c t i o n t e c h n i q u e s ; i t w i l l y i e l d good r e s u l t s even when the r e f l e c t i v e 235 events are c l o s e t o g e t h e r . The second method i s based upon the approximate formulas i n (7-A.21) and (7-A.23); i t w i l l work w e l l i f the r e f l e c t i v e e v e n t s are s u f f i c i e n t l y s e p a r a t e d i n time. In f a c t , the approximate f o r m u l a s appear to be q u i t e good and s h o u l d p r o v i d e a c c e p t a b l e r e s u l t s f o r many p r a c t i c a l c a s e s of i n t e r e s t . Our c o n s t r u c t i o n a l g o r i t h m i s a p a r a m e t r i c l e a s t - s q u a r e s s o l u t i o n to o b t a i n a "best guess" f o r r ( t ) and e ( t ) , where r ( t ) . e x p [ i e ( t ) ] = Z r ;-exp[ i e • ] • 5 ( t - r-). F o l l o w i n g the f o r m a l i s m o u t l i n e d by Oldenburg (1981) c l o s e l y , we f i r s t determine, from the a v e r a g e s , the number of r e s o l v a b l e s p i k e s M and t h e i r approximate time l o c a t i o n s r , j =1,....M. Then e i t h e r the r e a l or i m a g i n a r y p o r t i o n of <m(t)> can be used t o f i n d a s e t of parameters { r j , a j , J3 j } , j - 1, ... ,M, such t h a t N M X2 = I {I a-. a ( t , -T- ) + l ; - a ( t i -T- ) -Re<m(t,i,)>} 2 (7-A.30) i s m i n i m i z e d . In e q u a t i o n (7-A.30), a j = r j • c o s ( e j ) and £j = r j s i n ( e j ) . A b e s t guess of the magnitude of the r e f l e c t i v i t y c o e f f i c i e n t s and the phase advances i s e a s i l y d e r i v e d from { a j , §_ j } . The second method f o r f i n d i n g the parameters { r j , r j , ej} uses an a p p r o x i m a t i o n i n v o l v i n g the modulus of the a v e r a g e s . We saw from 236 e q u a t i o n (7-A.20) t h a t when the data comprised a s i n g l e w avelet, |<m(t)>| = r , A ( t - T , ) , where r , was the r e f l e c t i v i t y c o e f f i c i e n t . When the data comprised m u l t i p l e w a v e l e t s , i t might be hoped t h a t |<m(t)>| = I r j A ( t - r j ) , (7-A.31) t h a t i s , the modulus of the averages would be s c a l e d and d i s p l a c e d r e p l i c a t i o n s , of A ( t ) . From an exam i n a t i o n of e q u a t i o n (7-A.19), i t takes l i t t l e e f f o r t to show t h a t e q u a t i o n (7-A..31) cannot be t r u e i n g e n e r a l . N e v e r t h e l e s s , we s t i l l expect e q u a t i o n (7-A.31) t o h o l d a p p r o x i m a t e l y whenever the r e f l e c t i v e e vents a r e s e p a r a t e d by t i m e s somewhat g r e a t e r than the width of A ( t ) . In such c i r c u m s t a n c e s the d e l a y times r j can be equated t o the times a t which |<m(t)>| a c h i e v e s a s i g n i f i c a n t maximum, and the approximate a m p l i t u d e s and phases can be d e t e r m i n e d by r j « 1 / A(0)•|<m(rj)>|, (7-A.32) and «j & t a n - ' [Im<m(rj)> /'Re<m(rj)>]. (7-A.33) V a r i a n c e s of these q u a n t i t i e s can be e v a l u a t e d by u s i n g e q u a t i o n s (7-A.27), (7-A.28), and (7-A.29). The a p p r o x i m a t i o n (31), (7-A.32), and (7-A.33) must d e t e r i o r a t e when the r e f l e c t i v e events are c l o s e r t o g e t h e r . To i l l u s t r a t e t h i s , we c o n s i d e r an example i n which s u c c e s s i v e s e t s of data are gene r a t e d from a model c o n s i s t i n g of two r e f l e c t i v e c o e f f i c i e n t s ( r , » r 2 = 1.0; e, = tr/4, e 2 = ff/2) which a r e brought c l o s e r t o g e t h e r . The d i s t a n c e between the two s p i k e s i s de c r e a s e d from 10 to 1 d i g i t i z a t i o n i n t e r v a l . A l t h o u g h no n o i s e i s added to the d a t a , the i n v e r s i o n s a r e c a r r i e d out (as i n a l l r e a l i s t i c examples) by u s i n g a nonzero v a l u e of 6 i n e q u a t i o n (7-A.16). The a v e r a g i n g f u n c t i o n s used f o r t h i s example a r e i d e n t i c a l i n c h a r a c t e r but s l i g h t l y narrower than those i n F i g u r e s 7-A.l.h, l j , and 11. Here w , the f u l l w i dth i f A ( t ) a t h a l f i t s maximum v a l u e , i s 6 d i g i t i z a t i o n i n t e r v a l s . The r e s u l t s a r e g i v e n i n T a b l e 1a shows { r j , e j , r j } o b t a i n e d by m i n i m i z i n g e q u a t i o n (30, and T a b l e 1b g i v e s the model parameters e s t i m a t e d from e q u a t i o n s (7-A..31), (7-A..32), and (7-A.33). The d e l a y times, r e f l e c t i v i t y a m p l i t u d e s , and phase advances from the l e a s t - s q u a r e s a n a l y s i s a r e seen t o be i d e n t i c a l to t h e i r t r u e v a l u e s u n t i l Ar /w «s 0.3. At t h a t .point t h e r e i s no l o n g e r any i n d i c a t i o n i n the a v e r a g e s t h a t two s p i k e s e x i s t . Rather, |<m(t)>| shows o n l y one peak; hence the l e a s t - s q u a r e s program e s t i m a t e s a r e f l e c t i v e a mplitude of a p p r o x i m a t e l y r , + - s r , 2 - 2.0. The e s t i m a t e s of { r j , c j , r j } o b t a i n e d by u s i n g e q u a t i o n s (7-A.31), (7-A.32), and (7-A.33) are r e a s o n a b l y c l o s e to the t r u e v a l u e s f o r s e p a r a t i o n s Ar/w >1, and the d e l a y times are s t i l l w e l l determined f o r even s m a l l e r s e p a r a t i o n s . T h i s i n d i c a t e s t h a t the ap p r o x i m a t i o n s i n e q u a t i o n s (7-A.32) and (7-A.33) should perform well i-n p r a c t i c e as long as Ar/w >i. 2 3 9 T a b l e 7-A.1 (a) d i s p l a y s the best guess f o r {r , e , r } o b t a i n e d by m i n i m i z i n g e q u a t i o n ( 7 - A . 3 0 ) . In the t r u e model, the f i r s t s p i k e was h e l d a t r , - 1 0 , r 2 = T , + A T , r , - T 2 = 1 . 0 , e, = 4 5 degrees, and e2 - 9 0 d e g r e e s . C a l c u l a t i o n s were made f o r T 2 = 2 0 , 1 8 , 1 6 , 1 5 , 1 4 , 1 3 , 1 2 , and 1 1 . (b) d i s p l a y s the e s t i m a t e s {r , e , r } o b t a i n e d from the approximate f o r m u l a s g i v e n by e q u a t i o n s ( 7 - A . 3 2 ) and ( 7 - A . 3 3 ) . (a) A T I i £ 2 * i * 2 T i T 2 1 0 1 . 0 0 1 . 0 0 4 5 9 0 1 0 -20 8 1 . 0 0 1 . 0 0 4 5 9 0 1 0 18 6 1 . 0 0 1 . 0 0 4 5 9 0 1 0 1 6 5 1 . 0 0 1 . 0 0 4 5 9 0 1 0 1 5 4 1 . 0 0 1 . 0 0 4 5 9 0 . 1 0 1 4 3 1 . 0 0 1 . 0 0 4 5 9 0 1 0 1 3 2 1 . 8 9 — 6 8 — 11 — 1 1 .:93 5 3 1 0 (b) A T «i T l T 2 1 0 1 . 0 6 1 . 0 6 41 9 4 1 0 .20 8 1 . 0 6 1 , 0 6 41 9 4 1 0 1 8 6 1 . 0 4 . 1 . 0 4 3 8 9 7 10 1 6 5 1 . 1 0 1 . 1 0 3 3 1 . 0 2 1 0 1 5 4 1 . 2 2 1 . 1 0 3 1 1 0 4 1 0 1 4 3 1 . 4 2 1 . 4 2 3 2 1 0 3 1 0 1 3 2 1 . 6 8 —. 6 8 — 11 1 1 . 8 6 4 9 — 1 0 240 Thus f a r we have c o n s i d e r e d v e r y s i m p l i s t i c n o i s e - f r e e examples i n v o l v i n g o n l y one or two r e f l e c t i v e c o e f f i c e n t s . More r e a l i s t i c a l l y we c o u l d expect a number of c l o s e l y spaced r e f l e c t o r s and n o i s y o b s e v a t i o n s . In the next example we deconvolve a i n a c c u r a t e data s et c o n s i s t i n g of f i v e p h a s e - s h i f t e d w a v e l e t s . The r e s u l t s are shown i n F i g u r e 7-A.2. The wavelet and the t r u e r e f l e c t i v i t y and phase f u n c t i o n s a r e shown i n F i g u r e s 7-A.2b, 2k, and 21, r e s p e c t i v e l y . The d a t a , w i t h 5 p e r c e n t white n o i s e added, a r e shown i n F i g u r e 7-A.2a. The averages and a v e r a g i n g f u n c t i o n s are shown i n F i g u r e 7-A.2c through 2h. The r e a l averages a ( t ) ,and 'a(.t) ( F i g u r e s 7-A.2c, 2d, and 2f) a r e used w i t h e q u a t i o n (7-A.30) to g e n e r a t e a best guess r ^ ( t ) and e ^ ( t ) , shown i n F i g u r e s 7-A.2i and 2 j . The r e c o n s t r u c t e d s i g n a l •x.(.t) = Re [ I r ; - e x p [ i e ; ] w(t - r j ) ] i s compared w i t h the da t a x ( t ) i n F i g u r e 7-A.2m; t h e i r d i f f e r e n c e , d i s p l a y e d i n F i g u r e 7-A.2n, i s a random s i g n a l w i t h a s t a n d a r d d e v i a t i o n of a p p r o x i m a t e l y 5 percent.. B e f o r e l e a v i n g t h i s example, i t i s of i n t e r e s t t o c a r r y out one more computation. Suppose i t were not known t h a t the wavelets were p h a s e - s h i f t e d . We would p r o c e e d by d e c o n v o l v i n g the da t a x ( t ) t o r e c o v e r a r e a l r e f l e c t i v i t y f u n c t i o n s ( t ) such t h a t x ( t ) = s ( t ) * w ( t ) . The r e s u l t s of such an a n a l y s i s a r e g i v e n i n F i g u r e 7-A.3. The averages <s(t)> i n F i g u r e 7-A.3c are e a s i l y shown t o be 241 128 I I 1 1 1 0 64 126 f I I i •64 0 ^ 1 128 -1 1 1 64 128 64 126 —I I 1 64 126 , p-64 I" I I 1 54 12B TIME o — i 1— 64 i?e TIME F i g u r e 7-A.2 In s u c c e s s i v e p a n e l s are shown (a) the d a t a ; (b) the wavelet (c) the r e a l p a r t of the averages; (d) the a v e r a g i n g f u n c t i o n a ( t ) ; (e) the imaginary a v e r a g e s ; ( f ) the H i l b e r t t r a n s f o r m of a ( t ) ; (g) the modulus of the averages; (h) the modulus A ( t ) of the a v e r a g i n g f u n c t i o n s ; ( i ) the best guess f o r the r e f l e c t i v i t y f u n c t i o n ; ( j ) the best guess f o r the phase f u n c t i o n (k) the t r u e r e f l e c t i v i t y f u n c t i o n ; (1) the t r u e phase f u n c t i o n ; (m) the s u p e r p o s i t i o n of the data x ( t ) and the data r e c o n s t r u c t e d from w(t) and r ^ ( t ) and eAt); (n) the r e s i d u a l between the -data and the r e c o n s t r u c t e d s i g n a l . 242 <s(t)> = Re<m(t)> = r ( t ) c o s e ( t ) * a ( t ) + r ( t ) s i n e ( t ) * a ( t ) , (7-A.34) where r ( t ) and e ( t ) a r e the t r u e r e f l e c t i v i t y and phase f u n c t i o n s . E q u a t i o n (7-A.34) i s t r u e i n g e n e r a l , and c o n s e q u e n t l y the n e g l e c t of p h a s e - s h i f t e d w a v e l e t s means t h a t o n l y the r e a l p a r t of a complex f u n c t i o n i s r e v e a l e d by s t a n d a r d d e c o n v o l u t i o n a l g o r i t h m s . N e v e r t h e l e s s , i f we proce e d w i t h the d e c o n v o l u t i o n , the averages i n F i g u r e 7-A.3c show f i v e w e l l - r e s o l v e d peaks, and a l e a s t - s q u a r e s f i t to f i n d the am p l i t u d e s and p o s i t i o n s of these s p i k e s produces a b e s t guess f o r the r e f l e c t i v i t y sequence s „ ( t ) ; t h a t f u n c t i o n i s shown i n o F i g u r e 7-A.3e. The r e s i d u a l e ( t ) = x ( t ) - s ^ ( t ) * w ( t ) i s shown i n F i g u r e 7-A.3f, and x ( t ) and the r e c o n s t r u c t e d s i g n a l s ^ ( t ) * w ( t ) a r e superposed i n F i g u r e 7-A.3h. I t i s somewhat s u r p r i s i n g t h a t the r e s i d u a l i s so s m a l l . I t i s a l s o d i s c o u r a g i n g , f o r i n p r a c t i c e the low-frequency d e v i a t i o n s i n the r e s i d u a l would l i k e l y be a t t r i b u t e d t o an i m p r e c i s e knowledge of the wavelet (Oldenburg et a l , 1981); t h u s s ^ t ^ would be regarded as an a c c e p t a b l e r e f l e c t i v i t y f u n c t i o n . I t s h o u l d a l s o be n o t i c e d t h a t the d e l a y times shown by s„(t) i n F i g u r e 7-A.3e a r e somewhat d i f f e r e n t from . the t r u e v a l u e s shown i n F i g u r e 7-A.3g. T h i s suggests t h a t the f i r s t - o r d e r e f f e c t of n e g l e c t i n g t r u e phase s h i f t s i n the wavelet 243 may be compensated f o r by a l t e r i n g the d e l a y t i m e s . As a f i n a l example, we show how the e f f e c t s of p h a s e - s h i f t i n g a s i g n a l can i n t r o d u c e c o m p l i c a t i o n s when a t t e m p t i n g t o c o r r e l a t e a p a r t i c u l a r peak (or trough) on s u c c e s s i v e t r a c e s . A wavelet taken from a marine s e i s m i c r e c o r d has been phase r e t a r d e d and advanced i n u n i t s of 40 d e g r e e s . The p h a s e - s h i f t e d w a v e l e t s a r e shown i n F i g u r e 7-A.4. The l a r g e r v e r t i c a l l i n e on the time a x i s denotes the p o s i t i o n of the main p o s i t i v e peak, and i t s p o s i t i o n i s seen to change by 10 time u n i t s as the phase changes from -120 to 120 degrees; a s m a l l e r v e r t i c a l r e f e r e n c e time t i c k has a l s o been p l o t t e d f o r each w a v e l e t . The second and t h i r d columns of w a v e l e t s ar e the r e s u l t s a f t e r d e c o n v o l u t i o n . The Re<m(t)> i s p l o t t e d i n the second column; t h i s would be the output from a s t a n d a r d d e c o n v o l u t i o n which n e g l e c t s the p o s s i b i l i t y of phase s h i f t s . We n o t i c e t h a t the c h a r a c t e r of the output waveform changes markedly from r e c o r d t o r e c o r d and t h a t t h e r e i s s t i l l a change i n the time p o s i t i o n of the p o s i t i v e peak on s u c c e s s i v e t r a c e s . Only i n the t h i r d column, where |<m(t)>| has been p l o t t e d , does the l o c a t i o n of the p o s i t i v e peak m a i n t a i n a f i x e d time throughout a l l the t r a c e s . 244 128 CJ -64 64 CZ .ZD . a en LU 128 6 4 128 64 TIME 128 64 TIME Ol X 128 Figure 7-A.3 Shown in successive panels are (a) the data; (b) the wavelet; (c) the averages of the r e f l e c t i v i t y function; (d) the averaging function corresponding to the averages in (c); (e) the best guess for the r e f l e c t i v i t y function; (f) the res i d u a l between the data and s(t) * w ( t ) ; (g) the true r e f l e c t i v i t y function; (h) the superposition of the data and the data and the reconstructed s i g n a l . 245 0 64 0 64 0 64 T I M E T I M E T I M E F i g u r e 7-A.4 Wavelets, p h a s e - s h i f t e d by amounts e, are d i s p l a y e d i n the f i r s t column. The Re<m(t)> and |<m(t)>| are p l o t t e d i n the second and t h i r d columns, r e s p e c t i v e l y . The l a r g e v e r t i c a l t i c k s on the time a x i s c o r r e s p o n d to the c e n t r a l time of the main p o s i t i v e l o b e . The s m a l l e r t i c k denotes a c o n s t a n t r e f e r e n c e time. The s m a l l and l a r g e t i c k s c o i n c i d e f o r the t h i r d column. 246 7-A.3 CONCLUSIONS When a s i g n a l i s composed of t i m e - d i s p l a c e d , p h a s e - s h i f t e d r e p l i c a t i o n s of a source w a v e l e t , the e f f e c t s of t h a t source wavelet can be removed by d e c o n v o l u t i o n . F i r s t , the a n a l y t i c s i g n a l can be w r i t t e n as a c o n v o l u t i o n of an a n a l y t i c source wavelet and a complex model which c o n t a i n s the r e f l e c t i v i t y f u n c t i o n r ( t ) and the phase f u n c t i o n e ( t ) . L i n e a r i n v e r s e t h e o r y can then be used t o r e c o v e r averages of the model <m(t)> = < r ( t ) e x p [ i e ( t ) ] >. But these averages, because they are composed of d e l t a - l i k e f u n c t i o n s a ( t ) and i t s H i l b e r t t r a n s f o r m a ( t ) , a r e sometimes d i f f i c u l t t o i n t e r p r e t . In s e i s m o l o g i c a l problems where r ( t ) - e x p [ i c ( t ) ] = I r - e x p [ i e r ] • 5 ( t - r-), the d e c o n v o l u t i o n r e s u l t s a r e more l u c i d l y p r e s e n t e d by d i s p l a y i n g |<m(t)>| which i s a p p r o x i m a t e l y e q u a l t o r ( t ) * A ( t ) (where *• A ( t ) = [ a ( t ) 2 + a"( t ) 2 ] 1 / 2 } i f the d i s t a n c e between the s u c c e s s i v e d e l a y t i m e s A T i s g r e a t e r than the width of A ( t ) . S i n c e A ( t ) w i l l be a peaked f u n c t i o n , the l o c a t i o n s of maximum |<m(t)>| can be used t o i n f e r v a l u e s f o r T j , and e s t i m a t e s f o r the amplitude c o e f f i c i e n t s r j and the phase c o e f f i c i e n t s ej are then e a s i l y r e c o v e r e d . E s t i m a t e s of the v a r i a n c e of r j and ej a r e a l s o a v a i l a b l e i f the s t a t i s t i c s of the o b s e r v a t i o n a l n o i s e a r e known. An a l t e r n a t i v e method f o r o b t a i n i n g r ( t ) and e ( t ) i s to r e c o v e r these f u n c t i o n s by u s i n g the averages <m(t)>, and the a v e r a g i n g 247 f u n c t i o n s a ( t ) and ' a ( t ) , to c o n s t r u c t a model which f i t s the d a t a . A p a r a m e t r i c l e a s t - s q u a r e s approach has been used here to f i n d a b e st guess f o r the parameter s e t { r j , e j , r j } . Sometimes i t i s not known a p r i o r i whether the wavelets making up the s i g n a l a r e p h a s e - s h i f t e d . What a r e the consequences of d e c o n v o l v i n g a p h a s e - s h i f t e d s i g n a l by assuming a c o n s t a n t wavelet w(t ) ? I t i s shown here t h a t o n l y Re<m(t)> w i l l be r e t u r n e d and t h a t the a v e r a g i n g f u n c t i o n w i l l be a ( t ) , A r e a l r e f l e c t i v i t y sequence s ( t ) can s t i l l be found such t h a t s ( t ) * w ( t ) i s ' an adequate r e p r e s e n t a t i o n of the o r i g i n a l d a t a . However, the d e l a y times and r e f l e c t i v i t y c o e f f i c i e n t s of s ( t ) may be d i f f e r e n t from those which g e n e r a t e d the d a t a . I t seems t h a t the f i r s t - o r d e r e f f e c t of a l t e r i n g the phase of the wavelet can be compensated f o r by a l t e r i n g the d e l a y time of a f i x e d s ource w a v e l e t . Moreover, i t i s s p e c u l a t e d t h a t the e f f e c t s of i n a c c u r a t e knowledge of the source wavelet might be i n d i s t i n g u i s h a b l e from the e f f e c t s of phase s h i f t i n g , and so s i m i l a r d e g r a d a t i o n i n the d e c o n v o l v e d output c o u l d r e s u l t e i t h e r from poor knowledge of the source wavelet or from d e c o n v o l v i n g w i t h a f i x e d s ource wavelet when the s i g n a l i s p h a s e - s h i f t e d . A l l of t h i s s e r v e s t o i l l u s t r a t e the the nonuniqueness i n h e r e n t i n the d e c o n v o l u t i o n problem and s t r e s s e s t h a t some degree of c a u t i o n w i l l have t o be used when a p p l y i n g the d e c o n v o l u t i o n t o r e a l d a t a . 248 APPENDIX 7-B: APPLICATIONS OF ANALYTIC COMMON SIGNAL ANALYSIS IN EXPLORATION GEOPHYSICS 7-B.1 INTRODUCTION The problem of e x t r a c t i n g i n f o r m a t i o n from m u l t i c h a n n e l s of c o r r e l a t e d d a t a can be approached i n many ways. One v e r y p r o m i s i n g t e c h n i q u e , however, i s to r e p r e s e n t the s i g n a l s i n terms of an o r t h o g o n a l b a s i s where the c h o i c e of b a s i s f u n c t i o n s i s determined from the i n n e r p r o d u c t m a t r i x , t h a t i s , the c o v a r i a n c e of the s i g n a l s . The approach appears under v a r i o u s names i n the p u b l i s h e d l i t e r a t u r e , f o r example: Karhunen-Loeve (KL) t r a n s f o r m a t i o n , p r i n c i p a l component a n a l y s i s , e i g e n v e c t o r a n a l y s i s , or H o t e l l i n g t r a n s f o r m a t i o n . A l t h o u g h the mathematical f o u n d a t i o n was o r i g i n a l l y d e v e l o p e d by Karhunen (1947), and Loeve ( 1 9 4 8 , 1 9 5 5 ) to r e p r e s e n t stoeha-stic s i g n a l s , the t r a n s f o r m a t i o n has found numerous a p p l i c a t i o n s i n a r e a s of data compression and s i g n a l t o n o i s e enhancement (e.g. Watanabe (1965), Ready and Wintz (1973), P e l a t (1974), Huang (1975)). T h i s t r a n s f o r m a t i o n , however, has not been w i d e l y used i n s e i s m o l o g i c a l problems even though i t s p o t e n t i a l has been e x h i b i t e d by Hemon and Mace (1978). i n an e a r l i e r paper ( U l r y c h .249 et a l . , 1.984) we have r e i n t r o d u c e d the KL t r a n s f o r m a t i o n and showed how i t c o u l d be s u c c e s s f u l l y a p p l i e d to problems i n s t a c k i n g , wavelet e s t i m a t i o n , r e s i d u a l s t a t i c s , and v e l o c i t y a n a l y s i s . The purpose of the p r e s e n t paper i s t o combine t h a t work w i t h the complex r e f l e c t i v i t y c o n v o l u t i o n a l model (Levy and Oldenburg, 1982)'. T h i s extended mathematical framework w i l l p e r m i t us t o c o n s i d e r the above problems w h i l e a d m i t t i n g the c o m p l i c a t i o n of phase s h i f t s of the w a v e l e t . In a d d i t i o n t o t h e s e a p p l i c a t i o n s , we s h a l l show how our f o r m a l i s m can be a p p l i e d t o the problems of a t t e n u a t i o n and d i s p e r s i o n , d e n s i t y and v e l o c i t y i n v e r s i o n , and ' b a d - t r a c e ' r e c o g n i t i o n . A l t h o u g h the o r i g i n a l work c o v e r s a l l the t o p i c s mentioned above, .for the purpose of t h i s t h e s i s I w i l l l i m i t the d i s c u s s i o n t o the problems of d i s p e r s i o n and the i n v e r s i o n of p h a s e - s h i f t i n f o r m a t i o n a s s o c i a t e d w i t h s u p e r c r i t i c a l r e f l e c t i o n s , . 7-B.2 MATHEMATICAL BACKGROUND As p o i n t e d out i n U l r y c h et a l . (1984), t h e r e are many ways i n which the KL t r a n s f o r m a t i o n can be d e r i v e d . The work of Kramer and Mathews (1956) i s , however, v e r y s t r a i g h t f o r w a r d and i n s i g h t f u l . We s h a l l f i r s t p r e s e n t the essence of t h e i r paper and then show i t s e x t e n s i o n t o complex s i g n a l s . From the p o i n t of view of d a t a compression, we c o n s i d e r the problem as f o l l o w s . Given a s e t of n r e a l s i g n a l s g ; ( t ) ( C = l , . . . n ) , 250 we d e f i n e a t r a n s f o r m e d s e t x_^(t) and a t r a n s f o r m a t i o n ( r o t a t i o n ) m a t r i x A (yet t o be d e f i n e d ) such that.: n i = 1 j = 1 i • • • m, m £ n ( 7 - B . 1 ) The s i g n a l s x ^ ( t ) form an o r t h o g o n a l b a s i s , and each s i g n a l g ^ ( t ) can be e x p r e s s e d ( a p p r o x i m a t e l y ) a s : where g ^ ( t ) i s the i t h r e c o n s t r u c t e d s i g n a l , B i s ,the i n v e r s e t r a n s f o r m a t i o n m a t r i x , and m i s the number of b a s i s f u n c t i o n s used i n the t r u n c a t e d e x p a n s i o n . The o b j e c t i v e a t t h i s p o i n t i s to r e c o n s t r u c t i f ^ ( t ) t o w i t h i n a gi v e n e r r o r u s i n g the s m a l l e s t p o s s i b l e number of b a s i s s i g n a l s . I f m=n, then g ^ ( t ) = g ^ ( t ) , i . e . , the o r i g i n a l s i g n a l i s reprod u c e d e x a c t l y . However., t h i s c ase i s of no i n t e r e s t s i n c e i t r e q u i r e s n b a s i s s i g n a l s i n o r d e r t o r e c o n s t r u c t the n o r i g i n a l s i g n a l s . We r e s t r i c t our a t t e n t i o n to the case where m<n, and r e q u i r e t h a t the t r a n s f o r m a t i o n m a t r i c e s A and B be those t h a t minimize the l e a s t squares e r r o r : m g - ( t ) - Z b;: x; ( t ) j-1 J i = 1 n (7-B.2) n <t>(m) = z ; ( ? £ ( t ) 7 g t - ( t ) ) 2 d t , i=1 (7-B.3) 251 Kramer and mathews (1956) showed t h a t B=A r, and t h a t the rows of the t r a n s f o r m a t i o n m a t r i x A c o n s i s t of t h e n o r m a l i z e d e i g e n v e c t o r s of V, d e f i n e d a s : r - j = J g ^ ( t ) g j ( t ) d t = (g;.,g-) (7-B..4) T i s symmetric and p o s i t i v e s e m i d e f i n i t e and hence i s decomposable. r=RAR T where A»diag('X 1 f X 2,...X^) w i t h X,£X 22:. .. X,*. and the columns of R c o n t a i n the n o r m a l i z e d e i g e n v e c t o r s r^ where r ? t --X^r- . When A=R T, the r o t a t e d s i g n a l s x j ( t ) form an n - d i m e n s i o n a l subspace of a H i l b e r t space, and w i t h t h e s e b a s i s elements, the t r u n c a t i o n e r r o r i n e q u a t i o n (7-B.3) i s n ••*(m) •- X X: (7-B.5) j=m+1 J S i n c e the e i g e n v a l u e s are a r r a n g e d i n de s c e n d i n g o r d e r i t f o l l o w s t h a t the f i r s t b a s i s f u n c t i o n can be used to r e c o n s t r u c t more of the t o t a l s i g n a l energy than any o t h e r b a s i s f u n c t i o n . For t h i s reason, i t i s c a l l e d the f i r s t p r i n c i p a l component. S i m i l a r l y , the second b a s i s f u n c t i o n w i l l sometimes be r e f e r r e d to as the second p r i n c i p a l component, e t c . The f i r s t p r i n c i p a l component has an important c h a r a c t e r i s t i c . I f g ^ ( t ) » c - - s ( t ) where c^ a r e r e a l c o n s t a n t s , and s ( t ) i s a g i v e n s i g n a l , then the f i r s t p r i n c i p a l component w i l l be x , ( t ) = (X c.2)'/s- s ( t ) • c (7-B.6) 252 That i s , the f i r s t p r i n c i p a l component w i l l be a s c a l e d v e r s i o n of the s i g n a l s ( t ) , and the complete set of input s i g n a l s g. ( t ) , can be r e c o n s t r u c t e d from t h i s b a s i s v e c t o r and an a p p r o p r i a t e s e t of weights. The remaining b a s i s v e c t o r s (ocjCt), j=2,...n), w i l l have zero e i g e n v a l u e s and are not needed i n the r e c o n s t r u c t i o n . The p r o p e r t i e s of the KL t r a n s f o r m a t i o n enumerated by Kramer and Mathews(1968) f o r r e a l s i g n a l s c a r r y over d i r e c t l y to the case when complex s i g n a l s are used. For complex s i g n a l s though, the inner product matrix i s H e r m i t i a n and p o s i t i v e s e m i - d e f i n i t e and hence a u n i t a r y m a t r i x i s r e q u i r e d f o r d i a g o n a l i z a t i o n . The e i g e n v a l u e s w i l l s t i l l be r e a l , but the e i g e n v e c t o r s are complex. N e v e r t h e l e s s , the t r u n c a t i o n e r r o r i n equation (7-B.3) i s s t i l l g i v e n by eq u a t i o n (7-B.5). Importantly, i f we c o n s i d e r the case where g ( t ) = c s ( t ) where the c are complex c o n s t a n t s , then the f i r s t p r i n c i p a l component w i l l be x , ( t ) = (I | c - J 2 ) s ( t ) (7-B.7) i T h i s l a t t e r e q u a t i o n shows t h a t complex s i g n a l s which d i f f e r o n l y be a complex s c a l e f a c t o r can be r e p r e s e n t e d by a s i n g l e p r i n c i p a l component. As an i l l u s t r a t i o n , we c o n s i d e r the f o l l o w i n g simple example. Let g i ( t ) = s ( t ) and g 2 ( t ) = e x p ( - i e ) s ( t ) . The energy i n the s i g n a l i s | |s| | 2 = (s, s * ) and T-. = (g. ,g?) where '*' denotes the complex c o n j u g a t e . Thus 253 r - I I s 1 e x p ( - i e ) e x p ( i e ) 1 The e i g e n v a l u e s of r a r e X,= 2 J |s| j 2 and X 2 » 0 , w h i l e the u n i t a r y m a t r i x i s D = e x p ( - i e) e x p ( i c ) The f i r s t b a s i s f u n c t i o n i s x , ( t ) - 0, , g , ( t ) + 0 2 , g 2 ( t ) * f i - s ( t ) where U i , are the elements of the f i r s t column of U.. The importance of t h e s e r e s u l t s to r e c o r d e d seismograms i s apparent when phase s h i f t s of a s o u r c e s i g n a l a r e c o n s i d e r e d . I f w(t) i s the i n i t i a l w a v e l e t , then a w a v e l e t , phase s h i f t e d by an amount e, i s g i v e n by w(t;e) = c o s ( e ) - w ( t ) + s i n ( e ) 'w(t) where w(t) = H[w(t)] i s the H i l b e r t t r a n s f o r m of the i n i t i a l wavelet (Aki and R i c h a r d s , 1980). In such c a s e s , i t i s e x p e d i e n t t o c o n s i d e r the a n a l y t i c s i g n a l (e.g. Taner e t a l . , 1979; Levy and Oldenburg, 1982) w(t) = w(t) - i w ( t ) 254 f o r the phase s h i f t e d s i g n a l can be w r i t t e n as w(t;e) = R e [ w ( t ) e x p ( i e ) ] L e t us now c o n s i d e r a s i g n a l g, ( t ) » w(t) and another s i g n a l g 2 ( t ) = w ( t ; e ) . The c o r r e s p o n d i n g a n a l y t i c s i g n a l s a r e g, ( t ) •- w(t) g 2 ( t ) = w ( t ) e x p ( i e ) S i n c e t h e s e s i g n a l s a r e l i k e those c o n s i d e r e d i n the n u m e r i c a l example, a p p l i c a t i o n of the Complex Karhunen-Loeve (CKL) t r a n s f o r m a t i o n w i l l produce a f i r s t p r i n c i p a l component e q u a l t o w ( t ) . Moreover, e q u a t i o n (7-.B..8) shows t h a t the phase r o t a t i o n i n the second s i g n a l i s r e c o v e r a b l e d i r e c t l y from the e i g e n v e c t o r a s s o c i a t e d w i t h the f i r s t p r i n c i p a l component. That i s , e = t a n - 1 [ I m ( U 2 , ) / R e ( U 2 , ) ] (7-B.8) In the f o l l o w i n g s e c t i o n s of t h i s paper the CKL t r a n s f o r m a t i o n w i l l be a p p l i e d t o a n a l y t i c seismograms. The e i g e n v e c t o r s of the c o v a r i a n c e m a t r i x w i l l be used as complex weights f o r s t a c k i n g and phase e s t i m a t i o n purposes w h i l e the c o r r e s p o n d i n g e i g e n v a l u e s w i l l be used i n the c o n s t r u c t i o n of a c o r r e l a t i o n measure. That i s , the 255 r a t i o m X(m) = I \-J<p(m) (7-B.9) 1 = 1 . w i l l be used as a measure of c o r r e l a t i o n between the o r i g i n a l set of s i g n a l s . X( I)>>1 i m p l i e s a good c o r r e l a t i o n , whereas x ( l ) A O ( l ) i m p l i e s a poor c o r r e l a t i o n . When X(1) >>1, each of the o r i g i n a l s i g n a l s may be expressed (to w i t h i n a s m a l l a c c e p t a b l e e r r o r ) as a s c a l e d and phase s h i f t e d verson of the f i r s t p r i n c i p a l component. Consequently, p h a s e - s h i f t s i n the v a r i o u s s i g n a l s can be e s t i m a t e d from the e i g e n v e c t o r a s s o c i a t e d with X 1 R and s ubsequently used f o r p h y s i c a l parameters i n f e r e n c e . 7-B.3 MODELLING DISPERSION BY A CONSTANT PHASE-SHIFT The p r o p e r t i e s o u t l i n e d above suggest t h a t the CKL a l g o r i t h m can be very u s e f u l i n i n v e s t i g a t i n g those p h y s i c a l phenomena that i n t r o d u c e phase s h i f t s which, to f i r s t o r d e r , can be approximated by a c o n s t a n t . I f the a p p l i c a b i l i t y of t h i s model can be e s t a b l i s h e d , then e s t i m a t i o n of the phases i n v o l v e d can y i e l d u s e f u l i n f o r m a t i o n about e a r t h p r o p e r t i e s . F o l l o w i n g Robinson (1979), we s h a l l assume a c o n s t a n t Q model f o r a t t e n u a t i o n , , a n d model d i s p e r s i o n u s i n g the F o u r i e r s c a l i n g 256 theorem. Our aim i s to use the CKL method t o determine whether an observed d i s p e r s i o n e f f e c t s can be approximated by a c o n s t a n t phase s h i f t , v i z . g ( t ) = R e [ g ( t ) e x p ( i c ] , where g ( t ) i s a measured seismogram which has undergone d i s p e r s i o n . I f a p p l i c a b i l i t y can be determined, the f i r s t e i g e n v e c t o r w i l l be used to e s t i m a t e the s h i f t a n g l e e ( E q u a t i o n 7-B.8) which w i l l r o t a t e the d i s p e r s e d t r a c e t o i t s u n d i s p e r s e d form. Given a r e f e r e n c e wavelet, the a p p l i c a b i l i t y of the c o n s t a n t phase s h i f t model i s determined by c o m p l e t i n g the f o l l o w i n g s t e p s : (1) s e l e c t a time window which c o n t a i n s a d i s p e r s e d p u l s e ; (2) c a l c u l a t e the envelope of the a n a l y t i c s i g n a l f o r the d i s p e r s e d p u l s e and r e f e r e n c e p u l s e ( i n i t i a l w a v e l e t ) , a l i g n i n g the peaks of the e n v e l o p e s so t h e r e i s no time d i s c r e p a n c y ; (3) a p p l y the CKL t r a n s f o r m a t i o n ; (4) e v a l u a t e X ( 1 ) . I f x( 1 ) i s l a r g e , then the d i s p e r s e d s i g n a l i s a p p r o x i m a t e l y a phase s h i f t e d v e r s o n of the o r i g i n a l . I f X(1)=*1 then the c o n s t a n t phase s h i f t assumption i s not v a l i d . F i g u r e 7-B.1(a) shows t h r e e time d e l a y e d R i c k e r wavelets ( c e n t e r f r e q u e n c y 25 Hz), d i s p e r s e d by c u m u l a t i v e Q v a l u e s of 1000, 60, and 30 r e s p e c t i v e l y . Large Q v a l u e s produce l i t t l e d i s t o r t i o n and so the event a t 0.5 seconds (Q=1000) i s n e a r l y i d e n t i c a l to the s o u r c e wavelet, and i s c o n s i d e r e d here as the r e f e r e n c e s i g n a l . The events, a t 0.9 seconds (Q=«60) and 1.5 seconds (Q=30) have c o n s i d e r a b l e d i s t o r t i o n and w i l l be used i n c o n j u n c t i o n w i t h the r e f e r e n c e s i g n a l to t e s t the v a l i d i t y of the c o n s t a n t phase s h i f t a p p r o x i m a t i o n . For the event at 0.9 seconds, we f i n d t h a t x ( l)=5364, w h i l e a n a l y s i s of the deeper event y i e l d s X(1)=656. These v a l u e s are INPUT SEISMOGRAM 0.0 Q.I •0.0 0. X = 5 3 6 4 X = 6 5 6 1 F i g u r e 7-B.1 (a) Dispersed Ricker wavelets for cumulative Q_ values of 1000, 60, and 30 re s p e c t i v e l y , (b) (1) The wavelet at 0.5 sees, (2) wavelet at 0.9 sees, (3) and (4) f i r s t and second p r i n c i p a l components of (1) and (2), and (5) a phase rotated version of (2). Here X(1)=5364. (c) Same as in (b) but for the wavelet at 1.5 seconds. Here X(1)=656. .258 both l a r g e and, hence, the c o r r e s p o n d i n g wavelets can be p r o p e r l y u n d i s p e r s e d by a p p l y i n g the a p p r o p r i a t e constant phase c o r r e c t i o n s as i s shown i n F i g u r e s 7-B.lb and c, A more r e a l i s t i c example i n c l u d e s the e f f e c t s of d i s p e r s i o n and a t t e n u a t i o n . In F i g u r e 7-B.2a we have regenerated the s y n t h e t i c of F i g u r e 7-B.la, and i n c l u d e d a t t e n u a t i o n . In t h i s c a s e , the r e f e r e n c e and the d i s t o r t e d s i g n a l s a r e d i f f e r e n t in both t h e i r phase and amplitude s p e c t r a . The v a l u e s of X (1 ) are s t i l l q u i t e l a r g e (x(1 ) = 195 f o r the event at 0 . 9 seconds, and X O ) = 91 f o r t h a t at 1.5 seconds, hence, the c o n s t a n t phase approximation i s s t i l l a c c e p t a b l e . T h i s i s v e r i f i e d by comparing c o r r e c t e d s i g n a l s ( F i g . 7-B.2(b) and 7-B.2(c)) to the source s i g n a l (upper t r a c e i n each p a n e l ) . Note t h a t no attempt was made to remove the e f f e c t s of a t t e n u a t i o n , and hence both c o r r e c t e d s i g n a l s are of longer time d u r a t i o n than the r e f e r e n c e s i g n a l . As expected, the c o n s t a n t phase s h i f t a pproximation to a t t e n u a t i o n r e l a t e d d i s p e r s i o n d e t e r i o r a t e s w i t h d e c r e a s i n g Q, and i n c r e a s i n g t r a v e l - t i m e . Furthermore, i t w i l l a l s o d e t e r i o r a t e with i n c r e a s i n g frequency bandwidth. However, our e x p e r i a n c e to d a t e , shows t h a t over a bandwidth of about 2-2 .5 octaves i t seems to y i e l d a r easonable approximation. 259 INPUT SEISMOGRAM X=195 X = 91 F i g u r e 7-B.2 Same as F i g u r e 7 - B . 1 . Here however, we have i n c l u d e d the e f f e c t s of a t t e n u a t i o n as w e l l as d i s p e r s o n . The valu e of xO) For F i g 7-B 2b i s 1 9 5 , w h i l e t h a t f o r F i g . 7-B.2c i s 9 1 . 3 ' . * .260 7-B.4 DENSITY AND VELOCITY ESTIMATION FROM SUPER-CRITICAL REFLECTIONS For a p l a n e wave t r a v e l l i n g i n a f l u i d l a y e r o v e r l a y i n g a f l u i d h a l f - s p a c e , the e q u a t i o n which r e l a t e s • p h a s e s h i f t S(8) w i t h the a n g l e of i n c i d e n c e 8 f o r a s u p e r c r i t i c a l r e f l e c t i o n ( R a y l e i g h , 1945) i s g i v e n by: t a n 2 e = p 2 ,/p 2 2 •(tan20 - v , 2 / v 2 2 . ( 1 / c o s 2 8 ) ) (7-B.10) where e = S ( 0 ) / 2 . Given the v e l o c i t y v,, the d e n s i t y p,, the a n g l e of i n c i d e n c e 8, and the a n g u l a r dependence of the phase s h i f t s S ( 0 ) , f o r a number of o b s e r v a t i o n s a t d i f f e r e n t o f f s e t s , we s o l v e a s e t of e q u a t i o n s ( l i n e a r i n l / p 2 2 and 1 / P 2 2 V 2 2 ) i n the form of e q u a t i o n (7-B.10) t o o b t a i n the h a l f - s p a c e d e n s i t y p 2 and i t s a c o u s t i c v e l o c i t y v 2 . In t h i s s e c t i o n , we w i l l use the CKL a l g o r i t h m t o e s t i m a t e the p h a s e - s h i f t - * a n g l e of i n c i d e n c e r e l a t i o n S{8) f o r a bottom r e f l e c t i o n i n a deep water o c e a n i c environment. The i n f o r m a t i o n c o n t a i n e d i n S(0) w i l l then be i n v e r t e d t o y i e l d both d e n s i t y and v e l o c i t y of the uppermost ocean bottom sediments l a y e r . V e l o c i t y p r o f i l e s from t r a v e l - t i m e i n v e r s i o n s w i l l then be compared t o the e s t i m a t e s o b t a i n e d from the phase i n f o r m a t i o n i n o r d e r to e s t a b l i s h the c r e d i b i l i t y of the e s t i m a t e d phases. A data s e t p r e v i o u s l y a n a l y s e d by Chapman et a l . (1984) was used f o r the purpose of d e n s i t y i n v e r s i o n . The d a t a , shown i n F i g u r e 7-B.3a, are deep water r e f l e c t i o n s from A r c t i c A b y s s a l P l a t e 261 sediments. I n the a n a l y s i s of the data we have l i m i t e d our a t t e n t i o n to the f o u r l a r g e r a m p l i t u d e events i n the second group of a r r i v a l s (the water column m u l t i p l e s ) , seen between times 1.7 sec and 2.5 sec, and o f f s e t s 25000 m and 38000 m. These a r r i v a l s were chosen because of t h e i r good s i g n a l t o n o i s e r a t i o . I t was c o n c l u d e d on the b a s i s of a p r e v i o u s a n a l y s i s , and on the times and p o l a r i t i e s of t h e s e e v e n t s , t h a t they c o r r e s p o n d e d t o r e f l e c t i o n s from a s i n g l e i n t e r f a c e i n the sediments, but w i t h d i f f e r e n t p a t h c o m b i n a t i o n s (see F i g u r e 7-B.3b), The phase s h i f t ' s a n g u l a r dependence S(5) was e s t i m a t e d from t h i s r e f l e c t i o n d a t a : t h i s a n a l y s i s began by f i n d i n g a s u i t a b l e ' r e f e r e n c e ' , i . e . a f a r - f i e l d s i g n a t u r e of a p r e - c r i t i c a l (un-phase s h i f t e d ) signal., i n t h i s case the f i r s t bottom bounce on the n e a r e s t o f f s e t t r a c e . The phase d i f f e r e n t i a l between the r e f e r e n c e and each of the p o s t - c r i t i c a l s i g n a l s was then determined u s i n g e q u a t i o n (7-B.8). These d i f f e r e n t i a l phases (summarized i n T a b l e 7-B.1) c o n s t i t u t e components of the a n g u l a r dependence f u n c t i o n .S(8) r e q u i r e d i n the s o l u t i o n of e q u a t i o n (7-B.10). An example of phase e s t i m a t i o n f o r the f i r s t of the r e f l e c t i o n e vents i s shown i n F i g u r e 7-B.3c, where the d a t a shown ar e taken from a 100 msec window e n c l o s i n g the event of i n t e r e s t f o r the d i f f e r e n t o-f-f-set t r a c e s . Shown a r e : the r e f e r e n c e s i g n a l , 6 p o s t - c r i t i c a l r e f l e c t i o n s (from o f f s e t s 25000m - 38000m), t h e e n v e l o p e s of the a n a l y t i c s i g n a l s of the 7 waveforms, and f i n a l l y , the r e f e r e n c e and the 6 p o s t - c r i t i c a l w avelets a f t e r phase r o t a t i o n to the r e f e r e n c e . U s i n g s t r a i g h t ray t r a v e l p aths through the water column we F i g u r e 7-B.3 (a) Reduced time plot of the data aligned on the water bottom r e f l e c t i o n In the inset box are the s u p e r - c r i t i c a l l y r e f l e c t e d data to be analysed. The f i r s t event on the f i r s t trace of the figure (outside the box) i s taken as the p r e - c r i t i c a l reference trace 263 TIME (seconds) o o d 6 F i g u r e 7 - B . 3 (b) Travel path for the analysed water bottom reflection event. (c) A sample window of data used for phase estimation (see text). Shown are the p r e - c r i t i c a l reference and 6 post - c r i t i c a l reflections, the envelopes of these 7 wavelets, and f i n a l l y the 7 wavelets after phase rotation to the reference. .264 FIRST MULTIPLES HORIZON # x (km) 9 1 Phase 2 Phase 3 Phase 4 Phase 28.20 1 .088 0.699 0.431 0.422 0.353 30.40 1 . 1 1 8 1 . 109 0.902 1 .000 1 .130 32.10 1 . 1 39 1 . 1.36 0.989 1 .298 1 .330 34. 10 1 . 1 6 1 1 .373 1 ..3 1 6 1 .490 1 .466 37.80 1 . 1 9 8 1 . 483 1 .390 1 .566 1 .503 TABLE 7-B.1 O f f s e t , a n g l e of i n c i d e n c e 6, and phase a n g l e ( i n r a d i a n s ) f o r the fou r r e f l e c t i o n s a s s o c i a t e d w i t h the f i r s t m u l t i p l e (see F i g u r e 7-B.1). A l l phases are r e l a t i v e t o a p r e c r i t i c a l a r r i v a l . The phases have been h a l v e d because the m u l t i p l e p a t h has two r e f l e c t i o n s from the water bottom. 265 have e s t i m a t e d the a n g l e s of i n c i d e n c e f o r each of the o f f s e t s p r e s e n t i n the a n a l y s e d data s e t s (see T a b l e 7-B.1).. We then s o l v e the s e t of e q u a t i o n s m i n i m i z i n g the square of the e r r o r s I (e' - e° ) 2 where the s u p e r s c r i p t s c and o s t a n d f o r ' c a l c u l a t e d ' and 'observed' phases r e s p e c t i v e l y . The e s t i m a t e d v e l o c i t y and d e n s i t y r a t i o s f o r the water bottom sediments o b t a i n e d from each of the a n a l y s e d d a t a s e t s a r e as g i v e n i n T a b l e 7-B.2. These l a t t e r r e s u l t s a r e c o n s i s t e n t w i t h v e l o c i t y p r o f i l e s o b t a i n e d from t r a v e l - t i m e i n v e r s i o n s (p-r i n v e r s i o n Garmany (1979), and D i x l i k e i n v e r s i o n Oldenburg e t a l . (1984)) which gave an average sediments sound speed of about I700m/sec. Consequently, the above example adds c r e d i b i l i t y t o both the CKL phase e s t i m a t i o n and the i n v e r s i o n p r o c e d u r e s . 7-B.5: CONCLUSIONS In t h i s appendix we have i n t r o d u c e d the complex Karhunen-Loeve t r a n s f o r m a t i o n and b r i e f l y reviewed some of i t s a p p l i c a t i o n s . . The a b i l i t y of the method t o d e a l w i t h phase s h i f t s and t o e x t r a c t phase .information from groups of w a v e l e t s has proven most i n s t r u c t i v e i n the d e s c r i p t i o n and a n a l y s i s of phenomena i n v o l v i n g phase changes, such as d i s p e r s i o n and s u p e r - c r i t i c a l r e f l e c t i o n . S m all time s h i f t s are t r a n s p a r a n t t o the method, as i t can approximate them as a s m a l l phase s h i f t and hence a v o i d the problems which would beset c o n v e n t i o n a l methods. The s c a l i n g p r o p e r t i e s of the a t t e n d a n t e i g e n v e c t o r s proved 266 TABLE IV FIRST MULTIPLES HORIZON # 1 2 3 4 density r a t i o 1 .23 1 , . 1 7 1 .00 1 . 1 2 sound speed r a t i o 1.17 '1.14 1 . 1 . 5 1 . 1 6 sound speed (km/s) 1 . 8 1 1.77 1 .78 1 .80 TABLE 7-B.2 Density and sound speed r a t i o s ( p 2 / P i , and v 2/v,) for the 'second layer' (see text) determined from inversion of the phases shown in Table 7-B.1. Using the estimated sound speed r a t i o s and a sound speed for the ' f i r s t layer' of 1 . 5 5 km/s, the sound speeds for the 'second layer' were also estimated. 267 u s e f u l as a d i s c r i m i n a t o r a g a i n s t u n c o r r e l a t e d s e i s m i c t r a c e s . T h i s e n a b l e s us to i s o l a t e such bad t r a c e s and omit them from f u r t h e r p r o c e s s i n g p r o c e d u r e s , thus a v o i d i n g unnecessary s i g n a l d e g r a t i o n . A u s e f u l measure of s i m i l a r i t y was o b t a i n e d from the e i g e n v a l u e s r a t i o x(m). T h i s e n a b l e d us t o perform a CKL v e l o c i t y a n a l y s i s which seems t o y i e l d b e t t e r r e s u l t s under c o n d i t i o n s of s t a t i c s c a t t e r than d i d a c o r r e s p o n d i n g semblance v e l o c i t y a n a y s i s . Given the a b i l i t y of the CKL method t o overcome the.problems of s m a l l time s h i f t s , we noted how a u s e f u l s t a c k i n g t o o l c o u l d be d e v e l o p e d . We a p p l i e d the CKL s t a c k i n g a l g o r i t h m to s e v e r a l s y n t h e t i c CDP g a t h e r s , and the r e s u l t compared f a v o u r a b l y w i t h the s e c t i o n o b t a i n e d by the a p p l i c a t i o n of the mean s t a c k r o u t i n e . We f e e l t h a t t h i s comparison j u s t i f y f u r t h e r r e s e a r c h of the s t a c k i n g p r o p e r t i e s of the .CKL a l g o r i t h m . As a c o r o l l a r y t o the above p o i n t s , the e s t i m a t i o n of the s e i s m i c wavelet i s a n a t u r a l f o l l o w - u p . The CKL e s t i m a t e of the wavelet p r o v i d e d a s h a r p e r and more a c c u r a t e r e p r e s e n t a t i o n of the i n p u t p u l s e than d i d the c o n v e n t i o n a l e s t i m a t e . In c o n c l u s i o n , we note t h a t the complex Karhunen-Loeve t r a n s f o r m a t i o n , an e x t e n s i o n of the l o n g known Karhunen-Loeve t r a n s f o r m a t i o n , has many and d i v e r s e a p p l i c a t i o n s i n the f i e l d of e x p l o r a t i o n s e i s m o l o g y . F u r t h e r a p l i c a t i o n s and e x t e n s i o n s of t h e — work d i s c u s s e d here a r e p r e s e n t l y under i n v e s t i g a t i o n . Publications and Reports Levy, S. and Clowes, R.M., 1980, Debubbling: A Generalized Linear Inverse Approach, Geophysical Prospecting 28, 840-858. Oldenburg, D.W., Levy, S., and Wh i t t a l l , K.P., 1981, Wavelet Estimation and Deconvolution, Geophysics, Vol. 46, No. 11, pp. 1528-1542. Levy, S. and Fullagar, P.K., 1981, Reconstruction of a Sparse Spike Train from a portion of i t s Spectrum, and Application to High-Resolution Deconvolution, Geophysics, Vol. 46, pp. 12 35-1243. Levy, S. and Oldenburg, D.W., 1982, The Deconvolution of Phase-Shifted Wavelets, Geophysics, Vol. 47 pp 1285-1294. Levy, S., Walker, C , Ulrych, T.J., and Fullagar, P.K., 1982, A Linear Programming Approach to the Estimation of the Power Spectra of Harmonic Processes, I.E.E.E. Transactions on Acoustic, Speech, and Signal Processing, Vol. ASSP-30, No. 4, August 1982. Oldenburg, D.W., Scheuer, T., and Levy, S., 1983, Recovery of the Acoustic Impedance from Ref l e c t i o n Seismograms, Geophysics, Vol. 48, No. 10, pp. 1318-1337. Oldenburg, D.W. , Levy, S., and Stinson, K. , 1984, RMS V e l o c i t i e s and Recovery of the Acoustic Impedance from Reflection Seismograms, Geophysics, Vol. 43, No. 10, pp. 1653-1663. Cabrera, J . J . and Levy, S., 1984, Stable Plane-Wave Decomposition and Spherical Wave Reconstruction: Applications to Converted S-Mode Separation and Trace Interpolation, Geophysics, Vol. 49, No. 11, pp. 1915-1932. Levy, S., Ulrych, T.J., Jones, I.F. and Oldenburg, D.W., 1983, Applications of Complex Common Signal Analysis i n Exploration Seismology (presented i n the 53rd SEG meeting i n Las Vegas). To be submitted to Geophysics. Levy, S. and Oldenburg, D.W. , 1985, Automatic Pliase Correction of CMP Stacked Data. Submitted to Geophysics. Stinson, K. and Levy, S., 1984, Multi-Trace Simplex Algorithm i n Seismic Data Analysis (presented i n the 54th SEG Meeting i n Atlanta). To be submitted to Geophysics. Chapman, R., Levy, S., Stinson, K., Jones, I.F., Prager, B.P. and Oldenburg, D.W. , Analysis of Deep Ocean Marine Seismograms by Inversion Techniques (submitted to the Journal of the Acoustical Society of America). Levy, S., Oldenburg, D.W. and Wang, Jiaying, 1985, Subsurface Imaging Using M.T. Data (to be submitted to Geophysics 1985). Oldenburg, D.W., Levy, S.W., Stinson, K., 1985, Inversion of Band-Limited Reflection Seismograms: Theory and Practice, Submitted to IEEE. 

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