UBC Theses and Dissertations

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UBC Theses and Dissertations

Developments in minimum entropy deconvolution Nickerson, William Alexander 1986

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DEVELOPMENTS IN MINIMUM ENTROPY DECONVOLUTION by WILLIAM ALEXANDER NICKERSON A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF GEOPHYSICS 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 q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA JANUARY,1986 © WILLIAM ALEXANDER NICKERSON, JANUARY,1986 In p resen t ing this thesis in partial fu l f i lment of the requ i rements for an a d v a n c e d d e g r e e at the Univers i ty of Brit ish C o l u m b i a , I agree that the Library shall m a k e it freely avai lable for re fe rence a n d s tudy . I further agree that p e r m i s s i o n for ex tens ive c o p y i n g o f this thesis fo r scho lar ly p u r p o s e s may be g ran ted by the h e a d of m y d e p a r t m e n t o r by his o r her representat ives . It is u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n of this thesis for f inancia l gain shall no t b e a l l o w e d w i t h o u t m y wr i t ten p e r m i s s i o n . D e p a r t m e n t T h e Un ivers i ty o f Brit ish C o l u m b i a 1956 M a i n M a l l V a n c o u v e r , C a n a d a V 6 T 1Y3 \ DE-6(3 /81) A b s t r a c t Minimum en t r o p y d e c o n v o l u t i o n (MED) i s i n v e s t i g a t e d i n l i g h t of r e c e n t work, and r e p o r t s of poor performance i n the d e c o n v o l u t i o n of r e a l r e f l e c t i o n seismograms. The problems w i t h MED f a l l i n t o two c a t a g o r i e s . . The f i r s t c o n t a i n s problems r e l a t e d t o o p e r a t o r d e s i g n such as a v o i d i n g l o c a l extrema i n the s i m p l i c i t y c r i t e r i o n , problems w i t h bandwidth, and the s e l e c t i o n of f i l t e r l e n g t h and p r e p r o c e s s i n g p arameters. The second i s the d e t e r m i n a t i o n of model v a l i d i t y f o r the d a t a a t hand. MED i s based on the i t e r a t i v e m a x i m i z a t i o n of a d a t a s i m p l i c i t y c r i t e r i o n which e x h i b i t s m u l t i p l e extrema. A 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 t h i s a l g o r i t h m l e a d s t o an i n t u i t i v e a l t e r n a t i v e a l g o r i t h m , the p r o j e c t i o n method, and t o the c o n c l u s i o n t h a t i t e r a t i o n s t a r t i n g f i l t e r governs which extremum i s reached and hence o u t p u t w a v e l e t d e l a y . C h o i c e of s t a r t i n g f i l t e r • i s t h e r e f o r e tantamount t o a w a v e l e t phase assumption and t h e r e i s no b a s i s f o r c h o o s i n g the s t a n d a r d , c e n t r e d i m p u l s e , s t a r t i n g f i l t e r i f the g l o b a l extremum i s d e s i r e d . An optimum l a g MED a l g o r i t h m i s proposed t o a c h i e v e the g l o b a l maximum. B a n d l i m i t i n g , d a t a t a p e r i n g and o t h e r a s p e c t s of MED o p e r a t o r d e s i g n a r e d i s c u s s e d . No c r i t e r i o n e x i s t s f o r a s s e s s i n g MED model v a l i d i t y . S t u d i e s w i t h v a r i o u s s y n t h e t i c r e f l e c t i v i t y s t a t i s t i c s r e v e a l t h a t " s p a r s e n e s s " and " s i m p l i c i t y " a r e poor d e s c r i p t o r s of the minimum e n t r o p y a s s u m p t i o n . P r e l i m i n a r y i i a t t e m p t s t o q u a n t i f y MED a p p l i c a b i l i t y a r e made but f a i l and i n d i c a t e t h a t any s u c c e s s f u l measure must i n c l u d e b oth u n d e r l y i n g r e f l e c t i v i t y and w a v e l e t c h a r a c t e r i s t i c s . Examples u s i n g s y n t h e t i c d a t a show the advantages of the b a n d l i m i t e d optimum l a g MED a l g o r i t h m over a c o n v e n t i o n a l s t a r t i n g f i l t e r method. T a b l e of C o n t e n t s 1 . INTRODUCTION 1 1.1 THE DECONVOLUTION PROBLEM 1 1.2 OBJECTIVES OF THIS RESEARCH 4 2. MED BACKGROUND 6 2. 1 VARIMAX MED 6 2.2 THE NORMAL EQUATIONS 9 2.3 RELATED APPROACHES 14 2.4 PROBLEMS WITH MED 26 3. A GEOMETRICAL INTERPRETATION 28 3.1 NOTATION AND INTRODUCTION 28 3.2 GEOMETRICAL INTERPRETATION OF LEAST SQUARES FILTERING ...30 3.3 GEOMETRICAL INTERPRETATION OF MED 37 4. ASPECTS OF MED OPERATOR DESIGN 42 4.1 MULTIPLE EXTREMA 42 4.2 BANDLIMITING THE SOLUTION 54 4.3 DATA TAPERING AND WINDOW SELECTION 55 4.4 SELECTION OF FILTER LENGTH 59 5. THE APPLICABILITY OF MED 62 5.1 THE MINIMUM ENTROPY ASSUMPTION 62 6. EXAMPLES USING SYNTHETIC DATA 74 6.1 INTRODUCTION 74 6.2 THE EFFECT OF OPTIMUM LAG SCANNING 75 6.3 THE EFFECT OF ADDITIVE NOISE 77 7. DISCUSSION 84 REFERENCES 87 APPENDIX A: BANDLIMITING THE FILTER OPERATION 89 i v G Table I . MED norms and 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 e s t i m a t e s 25 v L i s t of F i g u r e s 1. Varimax of some s i m p l e time s e r i e s 8 2. I t e r a t i v e d e c o n v o l u t i o n of an i s o l a t e d w a v e l e t 12 3. The g e n e r a l i z e d G a u s s i a n d i s t r i b u t i o n 16 4. R e f l e c t i v i t y e s t i m a t e s of minimum e n t r o p y norms 23 5. The time s e r i e s as a v e c t o r 29 6. The subspace of r e a l i z a b l e f i l t e r o u t p u t s 32 7. G e o m e t r i c a l l e a s t s quares f i l t e r i n g 33 8. 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 the varimax 39 9. L o c a l extrema i n the varimax 40 10. The t h r e e sample f i l t e r as a v e c t o r 43 11. C o n t o u r s on the v a r i m a x : s p a r s e seismogram 45 12. Cont o u r s on the var i m a x : i s o l a t e d w a v elet 47 13. The e f f e c t of v a r y i n g s t a r t i n g f i l t e r 49 14. I d e a l minimum phase MED 53 15. The e f f e c t of d a t a t a p e r i n g 56 16. The e f f e c t of c o n s t r u c t i v e i n t e r f e r e n c e 58 17. Cont o u r s on the varimax 64 18. C o n t o u r s on the D-norm 67 19. Cont o u r s on the parsimony 68 20. Cont o u r s on the v a r i m a x : u n i f o r m v s . e x p o n e n t i a l seismograms 69 21. The e f f e c t of r e f l e c t i v i t y p r o b a b i l i t y d i s t r i b u t i o n .71 22. The e f f e c t of optimum l a g s c a n n i n g 76 23. S y n t h e t i c d a t a w i t h c o l o u r e d random n o i s e 78 24. Optimum l a g MED output w i t h p r e w h i t e n i n g 79 25. Optimum l a g MED ou t p u t w i t h b a n d l i m i t i n g 80 v i Power s p e c t r a of MED f i l t e r s v i i Acknowledgements I am g r a t e f u l t o p r o f e s s o r Tad U l r y c h , my s u p e r v i s o r , f o r h i s support and g u i d a n c e . S p e c i a l thanks t o T o s h i Matsuoka, my s u r r o g a t e a d v i s o r i n Tad's absence, for- h i s i d e a s , encouragement and J a p a n e s e - s t y l e t u t e l a g e . C o l i n Walker has p r o v i d e d many of the computer programs used i n t h i s work and Ken W h i t t a l l has been a g r e a t h e l p i n computer m a t t e r s . T e r r y Deeming of D i g i c o n has shown me an i n d u s t r y v i e w p o i n t and c o n v e r s a t i o n s w i t h him have h e l p e d f o c u s t h i s work. T h i s r e s e a c h has been s u p p o r t e d by a g r a n t from the N a t u r a l S c i e n c e s and E n g i n e e r i n g Research C o u n c i l of Canada. v i i i 1. INTRODUCTION 1.1 THE DECONVOLUTION PROBLEM D e c o n v o l u t i o n i s a t e c h n i q u e used i n the p r o c e s s i n g of r e f l e c t i o n s e i s m i c d a t a aimed a t s h a r p e n i n g the images of deep g e o l o g i c b o u n d a r i e s . S i n c e the mid-1960's, d e c o n v o l u t i o n i n some form has been a p p l i e d t o n e a r l y a l l d i g i t a l l y r e c o r d e d d a t a i n t h e presence of a c o n t i n u i n g debate over how be s t t o f o r m u l a t e and s o l v e the problem of removing the s e i s m i c w a v e l e t . In t h i s s e c t i o n I s h a l l d i s c u s s the m a t h e m a t i c a l f o r m u l a t i o n of t h e d e c o n v o l u t i o n problem, the. two g e n e r a l approaches t o i t s s o l u t i o n and the c o n v o l u t i o n model of the seismogram on which both methods ar e based. For the d e c o n v o l u t i o n approaches c o n s i d e r e d i n t h i s s e c t i o n a d i s c r e t e l y sampled s e i s m i c t r a c e x i s r e p r e s e n t e d as a time i n v a r i a n t , one d i m e n s i o n a l l i n e a r system. x = w*r. ( 1 . 1 ) nw x . = Z w.r . i =1, . . . nx where, u s i n g a v e c t o r n o t a t i o n f o r the sampled t r a c e x = (xt,X2, . x n x ) T w = , w2, . . . wnw)T and r = (r.y, r2, . . . rnr)r nx= nw+ nr- 1. S i m p l y s t a t e d , the s e i s m i c time sequence i s m o d e l l e d as the c o n v o l u t i o n of two ele m e n t s . The f i r s t i s the wa v e l e t w, 1 2 a t r a n s i e n t p u l s e composed of the s e i s m i c source s i g n a t u r e , i n s t r u m e n t response and some e a r t h f i l t e r i n g e f f e c t s . The second i s an e a r t h impulse response s e r i e s r , the d e s i r e d s i g n a l which has been smoothed by the s e i s m i c w a v e l e t . D e c o n v o l u t i o n , which means i n v e r s e f i l t e r i n g , a t t e m p t s t o s o l v e an e q u a t i o n of the form (1.1) f o r r by d e s i g n i n g a l i n e a r f i l t e r f w hich when c o n v o l v e d w i t h the d a t a g i v e s an e s t i m a t e of r , y such t h a t y=f*x. (1.2) I t i s i n t e r e s t i n g t o note t h a t the c o n v o l u t i o n a l model of the seismogram i s j u s t a s u c c e s s f u l m a t h e m a t i c a l c o n s t r u c t i o n . Based l o o s e l y on the w a v e l e t t h e o r y of the seismogram d e v e l o p e d i n a s e r i e s of papers between 1940 and 1953 by R i c k e r , the c o n v o l u t i o n model was adopted by Robinson(1954) t o a l l o w the a p p l i c a t i o n of p r e d i c t i v e d e c o n v o l u t i o n t o seismograms. P h y s i c a l r e a s o n i n g based on a f l a t E a r t h , p l a n e wave, normal i n c i d e n c e model can be used t o j u s t i f y the c o n v o l u t i o n a l model. However, Z w i o l k o w s k i (1984) p o i n t s out i n c o n s i s t e n c i e s i n t h i s and o t h e r p h y s i c a l arguments t o emphasize t h a t model v a l i d i t y i s s t i l l open f o r debate and h i s r e f r e s h i n g book i s recommended t o any r e a d e r i n t e r e s t e d i n d e c o n v o l u t i o n . There a r e two b a s i c approaches t o f i n d i n g f depending on whether or not an e s t i m a t e of w i s a v a i l a b l e . I f the s o u r c e w a v e l e t i s i n d e e d known we may t a k e a d e t e r m i n i s t i c approach t o s o l v i n g e q u a t i o n ( 1 . 1 ) . T h i s t h e s i s however c o n c e r n s the problem of d e c o n v o l u t i o n when the w a velet i s 3 u n k n o w n . I n t h i s c a s e i f we a l l o w f o r a n a d d i t i v e n o i s e t e r m , n, o u r s e i s m o g r a m b e c o m e s x = w * r + n . ( 1 . 3 ) C l e a r l y we a r e s e e k i n g t o s o l v e o n e e q u a t i o n i n t h r e e u n k n o w n s . T h i s i s d o n e b y i n v o k i n g r e s t r i c t i v e s t a t i s t i c a l a s s u m p t i o n s o n t h e n a t u r e o f t h e u n k n o w n s e q u e n c e s . O n e p a r t i c u l a r l y s u c c e s s f u l s e t o f a s s u m p t i o n s i s t h a t 1 . n i s a g a u s s i a n , w h i t e s t a t i o n a r y p r o c e s s 2 . r i s a g a u s s i a n , w h i t e s t a t i o n a r y p r o c e s s , a n d 3 . w i s c a u s a l , m i n i m u m p h a s e a n d f i n i t e i n e n e r g y . U n d e r t h e s e a s s u m p t i o n s t h e W o l d d e c o m p o s i t i o n f o r s t a t i o n a r y t i m e s e r i e s g u a r a n t e e s a s o l u t i o n , r e g a r d l e s s o f w h e t h e r t h e a s s u m p t i o n s a r e p h y s i c a l l y j u s t i f i e d , a n d a s o l u t i o n i s o b t a i n e d u s i n g t h e c e l e b r a t e d s p i k i n g , o r u n i t g a p p r e d i c t i v e d e c o n v o l u t i o n o f R o b i n s o n ( 1 9 5 4 ) . I n r e c e n t y e a r s a f l o o d o f t e c h n i q u e s f o r s t a t i s t i c a l d e c o n v o l u t i o n h a v e a p p e a r e d i n t h e l i t e r a t u r e , e a c h o f w h i c h m a k e s i t s o w n a s s u m p t i o n s i n s o l v i n g t h e u n d e r d e t e r m i n e d p r o b l e m o f e q u a t i o n ( 1 . 3 ) a n d e a c h r a i s i n g i t s o w n o b j e c t i o n s . I t i s e v i d e n t t h a t n o o n e s t a t i s t i c a l m e t h o d i s s u i t e d t o c o p i n g w i t h t h e v a r i e t y o f p r o b l e m s m e t i n s e i s m i c d a t a . T h u s i m p r o v e m e n t i n p r e s e n t d a y d e c o n v o l u t i o n t e c h n i q u e s w i l l n o t l i k e l y c o m e f r o m s e e k i n g m o r e g e n e r a l s t a t i s t i c a l a s s u m p t i o n s . I n s t e a d we m u s t e i t h e r p u t m o r e i n f o r m a t i o n i n t o t h e s o l u t i o n o r m a k e a s s u m p t i o n s w h i c h a r e p a r t i c u l a r l y s u i t e d t o t h e t a s k a t h a n d . 4 The t e c h n i q u e t r e a t e d i n t h i s t h e s i s , Minimum E n t r o p y D e c o n v o l u t i o n which I s h a l l r e f e r t o s i m p l y as MED, f i n d s a unique e s t i m a t e of r from e q u a t i o n (1.3) based on an assumption c o m p l e t e l y d i f f e r e n t from those of s p i k i n g d e c o n v o l u t i o n . The MED f i l t e r i s t h a t one which o p t i m i z e s the s i m p l i c i t y or s p a r c e n e s s of the o u t p u t . The term 'minimum e n t r o p y ' i s s t r i c t l y g e n e r i c , e x p r e s s i n g W i g g i n s ' (1977) g o a l of i n c r e a s i n g t h e o v e r a l l apparent o r d e r of the d a t a . MED's freedom from the phase and w h i t e n e s s c o n s t r a i n t s of t r a d i t i o n a l d e c o n v o l u t i o n methods would be a tremendous advantage i f a r o b u s t and workable MED a l g o r i t h m were a v a i l a b l e . T h i s i s p a r t i c u l a r l y t r u e i n the a r e a of comp r e s s i n g nonminimum-phase marine so u r c e s i g n a t u r e s i n the absence of a u s e a b l e w a v e l e t e s t i m a t e . 1.2 OBJECTIVES OF THIS RESEARCH T h i s t h e s i s has t h r e e main o b j e c t i v e s . The f i r s t i s t o i n t r o d u c e the reader t o Minimum Enropy D e c o n v o l u t i o n and r e c e n t work i n the a r e a by o t h e r w o r k e r s , much of which remains u n p u b l i s h e d . The second i s t o d e v e l o p a b e t t e r u n d e r s t a n d i n g of the assu m p t i o n s i n h e r e n t i n MED p r o c e s s i n g i n an attempt a t s e t t i n g g u i d e l i n e s e n a b l i n g us t o a v o i d a p p l y i n g MED t o d a t a f o r which i t i s not d e s i g n e d and t h e r e f o r e d e s t i n e d t o f a i l . The t h i r d and p r i m a r y o b j e c t i v e of my work i s t o d e l i n e a t e , and where I c a n , c o n t r i b u t e t o , the o u t s t a n d i n g problems i n the c a l c u l a t i o n of MED o p e r a t o r s w i t h p a r t i c u l a r emphasis on the problem of m u l t i p l e extrema. 5 In C hapter 2 of t h i s t h e s i s I w i l l d i s c u s s r e c e n t work on MED and the problems which have l e a d t o i t s apparent f a l l from g r a c e i n the l i t e r a t u r e . Then i n C h a p t e r s 3 and 4 I d i s c u s s the c a l c u l a t i o n of MED o p e r a t o r s from a h e u r i s t i c g e o m e t r i c a l p o i n t of view and a d d r e s s what I c o n s i d e r t o be the open problems i n MED o p e r a t o r d e s i g n . Perhaps the most i m p o r t a n t problem f o r MED and o t h e r s t a t i s t i c a l t e c h n i q u e s , t h a t of when and where t o use the method t o ensure s u c c e s s i s t r e a t e d i n Chapter 5, and t h a t i s f o l l o w e d by examples of d e c o n v o l u t i o n based on r e s u l t s of the p r e v i o u s c h a p t e r s . 2 . . MED BACKGROUND 2 . 1 VARIMAX MED In 1 9 7 7 Ralphe Wiggins made what was c o n s i d e r e d by some t o be a major b r e a k t h r o u g h i n t h e d e c o n v o l u t i o n problem. H i s approach, Minimum Enropy D e c o n v o l u t i o n , d i f f e r e d from t r a d i t i o n a l t e c h n i q u e s i n t h a t no, o f t e n u n a c c e p t a b l e , assumptions r e g a r d i n g the r e f l e c t i v i t y w h i t e n e s s or wavelet phase a r e made. These a s s u m p t i o n s , r e q u i r e d f o r s p i k i n g d e c o n v o l u t i o n (Robinson, 1 9 5 4 ) , are r e p l a c e d by one s i m p l e c r i t e r i o n . That i s , t h a t MED d e s i g n s the l i n e a r f i l t e r w h i c h, when a p p l i e d t o the s e i s m i c t r a c e l e a v e s b e h i n d the s m a l l e s t number of l a r g e r e f l e c t o r s c o n s i s t e n t w i t h the c o n v o l u t i o n a l model and the d a t a . Such a f i l t e r o p e r a t o r i s d e s i g n e d by m a x i m i z i n g some d a t a s i m p l i c i t y measure w i t h r e s p e c t t o the f i l t e r c o e f f i c i e n t s . The measure chosen by W i g g i n s ( 1 9 7 7 ) i s the varimax norm, d e f i n e d f o r a m u l t i c h a n n e l time s e r i e s x as ns nx nx V(x)= L { L x) ./\ L x 2 . ] 2 } ( 1 . 4 ) /=/ j=l  3 j = l  3 where x.j i s the j ' t h sample of the i ' t h t r a c e w i t h nx time samples and ns t r a c e s i n t o t a l . For the s i n g l e c h a n n e l case t h i s becomes s i m p l y nx nx V(x)= Z x*/[ £ x 2 ] 2 . ( 1 . 5 ) j=l  3 j=J  3 The varimax was borrowed from the f i e l d of f a c t o r a n a l y s i s where i t i s used ( K a i s e r , 1 9 5 8 ) t o q u a n t i f y the 6 7 s i m p l i c i t y of a m a t r i x of f a c t o r l o a d i n g s . In time s e r i e s a p p l i c a t i o n the b e h a v i o r of the varimax norm i s shown i n F i g u r e 1. A s i n g l e t r a c e c o n s i s t i n g of a l l z e r o e s and one non-zero s p i k e g i v e s V=l, The more non-zero s p i k e s of c o n s i s t e n t a m p l i t u d e t h e r e a r e , the s m a l l e r the varimax becomes. The varimax i s u n a f f e c t e d by the s p a c i n g or p o l a r i t y of the s p i k e s . W i g g i n s c o i n e d the term "minimum e n t r o p y " s i n c e the varimax d i s c r i m i n a t e s a g a i n s t randomness i n the d e c o n v o l u t i o n outcome. Donohoe (1981) v e r i f i e d t h a t f o r the s p e c i a l case where the outp u t i s a random w h i t e n o i s e p r o c e s s w i t h time p o i n t s b e i n g i d e n t i c a l l y d i s t r i b u t e d random v a r i a b l e s , t he varimax and r e l a t e d norms do indeed r e f l e c t the s t a t i s t i c a l e n t r o p y of the trace.. I t s h o u l d be noted t h a t , as d e f i n e d by e q u a t i o n (1.5) the varimax i s c l o s e l y r e l a t e d t o the sample k u r t o s i s Kf a s t a t i s t i c r e p r e s e n t i n g the peakedness of the p r o b a b i l i t y d e n s i t y f u n c t i o n from which the d a t a a r e sampled. I n t h i s case K={[Z(x.-~x) Vn]/[Z(x.-~x)2/n]2} i i =nV(x) f o r a z e r o mean p r o c e s s x. where n i s the number of p o i n t s i n the sample and we may s t a t e t h a t varimax MED d e s i g n s t h a t f i l t e r whose ou t p u t has the l a r g e s t k u r t o s i s a t t a i n a b l e by f i l t e r i n g . 1 1 1 1 . 1 1 1 1 . » • " 1 ¥ V = 1 . 0 0 V = 0 . 6 8 V = 0 . 5 0 V = 0 . 2 5 V = 0 . 1 2 5 F i g u r e 1. Varimax norm e v a l u a t e d f o r the s i m p l e time s e r i e s shown. 9 2.2 THE NORMAL EQUATIONS We have assumed a c o n v o l u t i o n a l model f o r the seisinogram (1.3) x=w*r+n. I t i s our aim t o o b t a i n some d e c o n v o l v e d seismogram y thr o u g h c o n v o l u t i o n of the seismogram w i t h a d e c o n v o l u t i o n f i l t e r f y=f*x. To f i n d t h a t f i l t e r which when c o n v o l v e d w i t h the da t a as i n e q u a t i o n (2.1) y i e l d s an output y w i t h maximum varimax we maximize V(y) w i t h r e s p e c t t o a l l f i l t e r c o e f f i c i e n t s /jc, k= 1, . . . nf s i m u l t a n e o u s l y . S e t t i n g 9/ t " ° k=l nf (2.1 ) we o b t a i n the MED normal e q u a t i o n s ( W i g g i n s , 1977) f o r the s i n g l e c h a n n e l case V(y) 1 VlTyJ y xJ-l+lxJ-h+l = Ti^F • yJxJ-k+l k=l, . nf o r , A(y) * x x ( 0 ) * x x ( 1 ) ' ^ x x ( n ^ ] ) fx fz f nf = B(y) 4y3x(0) *y*x<» <l>y3x(nf-l) (2.2) where $ (T) i s the a u t o c o r r e l a t i o n of i n p u t x a t l a g T , <p 3 (T) i s the c r o s s c o r r e l a t i o n of x w i t h y 3 d e f i n e d y x J as y 3 = (y3, y\, . . . y3ny)r 10 A(y) = V(y)/Zy2 , a s c a l a r , and / B(y) = l / [ L y 2 ] 2 , a s c a l a r . / We must s o l v e e q u a t i o n (2.2) f o r the /. g i v e n the d a t a x^ . but u n f o r t u n a t e l y the r i g h t hand s i d e of (2.2) i t s e l f depends n o n - l i n e a r l y on f s i n c e we need i t t o form y and hence <J>„3Y(T) . In p r a c t i c e a symmetric s t a r t i n g f i l t e r such as of = (0,0, . . . 1... .0, 0)T (2.3) i s assumed (Wiggins 1977, Ooe & U l r y c h 1979, C h a c i n 1982, Wiggins and J u r k e v i c s 1985), y, y 3 and hence < l > y 2 x ( T ) i A and B a r e c a l c u l a t e d based on t h i s e s t i m a t e and e q u a t i o n (2.2) s o l v e d f o r £ u s i n g L e v i n s o n r e c u r s i o n . T h i s new f i l t e r , ,f i s then used t o r e g e n e r a t e y and the sequence i s re p e a t e d r e c u r s i v e l y u n t i l i t converges on a s o l u t i o n , a p r o c e s s which u s u a l l y r e q u i r e s 8 t o 12 i t e r a t i o n s . The r e a d e r f a m i l i a r w i t h Weiner f i l t e r i n g w i l l note t h a t t h e MED o p e r a t o r d e s i g n e q u a t i o n (2.2) i s i d e n t i c a l i n form t o the normal e q u a t i o n s f o r d e s i g n of a l e a s t squares waveshaping f i l t e r (Robinson and T r e i t e l 1980) *xx<°> *xx (1)' / , *dx(0> *xx<» • *xx<°>- f l • = * d x ( l ) • (2 • <t> (nf- 1) . . . *xx }dx(nf'D_ .4) Here ^ ^ X^ T^ i s t n e c r o s s c o r r e l a t i o n of a d e s i r e d o u t p u t d w i t h t h e d a t a x. 11 The s o l u t i o n f of e q u a t i o n (2.4) i s t h a t f i l t e r whose output when c o n v o l v e d w i t h x as i n e q u a t i o n (1.2) best a p p r o x i m a t e s d i n a l e a s t square sense. We can see t h e r e f o r e , t h a t the s i n g l e c h a n n e l MED e q u a t i o n s (2.2) d i f f e r o n l y by s c a l a r w e i g h t i n g f a c t o r s A and B, from e q u a t i o n s (2.4) above w i t h d e s i r e d output y 3 . T h i s f a c t a l l o w s us t o make the s i m p l e i n t e r p r e t a t i o n of the MED o p e r a t o r d e s i g n p r o c e s s d i s p l a y e d i n F i g u r e 2. S t a r t i n g w i t h the d a t a x we form a f i r s t e s t i m a t e of the output ,y by c o n v o l u t i o n w i t h the s t a r t i n g f i l t e r . Then the i t e r a t i o n proceeds by c u b i n g elements of ,y and d e s i g n i n g the wave-shaping f i l t e r which b e s t e s t i m a t e s ,y 3 from the d a t a . I t s o u tput 2 y i s cubed and wave-shaping a g a i n i s a p p l i e d t o r e f i n e t h e e s t i m a t e . T h i s p r o c e s s i s r e p e a t e d as shown by the p r o g r e s s i o n of o u t p u t s p l o t t e d i n F i g u r e 2 where the p r e s c r i p t s / = 1,...5 denote the i t e r a t i o n . Thus f a r I have c o n s i d e r e d MED f i l t e r d e s i g n f o r the s i n g l e c h a n n e l case o n l y . That i s the case where we can w r i t e x i n the form X = (Xi,X?,...X ) . 1 ' i ' nx I t i s t h i s case which i s most a d a p t a b l e t o the 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 p a r t 3. In p r a c t i c e MED i s u s u a l l y a p p l i e d to f i n d a s i n g l e d e c o n v o l u t i o n o p e r a t o r f o r a m u l t i c h a n n e l d a t a s e t x c o n s i s t i n g of ns segments of d a t a each of l e n g t h nx and i n t h i s case I w i l l w r i t e the m u l t i c h a n n e l d a t a s e t x "NO a s : 0 12 Wiggins MED 0 f = ( 0 , . . . 0 , 1 , 0 , . . . 0 ) I n p u t x O u t p u t y • in n - P s u 0) u Q, 01 >i <u 4-) (-1 i a cn c cr>.n cn •r-i 0) Vj 4-> o o iw C >1 to to a) c +J o e -p •i - t (0 4-> i-l cn a) • 0) 4-> CM ' H >i <U *4 Cn-• r-l 4-> M-4 fa* U a> i -H <u U-l 1-1 a) ro T> e c o CO 4-> 4-> o tO O 4-1 - P 4-> a 3 O O 4-> •rH 0) > > 6 o u • H U « 0 10 "4-1 • in 0) u c • U O 3 ,c II to C/J ~ 13 x = •^11 ^1 2 • • • X 2 1 2 2 1, ns nx, I nx, ns S i m i l a r l y I s h a l l w r i t e m u l t i c h a n n e l f i l t e r o utput y as **** and the MED normal e q u a t i o n s become: ns 2 A(y ) i=l Kx(0) Kx(1)- • • K x ( n f - ] ) xx XX / l f t f nf ns 2 B(y.) i=l 1 <t> 3 (0) * yi x % 3 (1) *y x <P 3 (nf-1) where My,) = V(y.)/Lyll (2.5) XX l 1,1 l , l - T B(yJ = l/[Zy2 . V and, i I ' <l> 3 (T) = Z y3 . x. , *y3xK t * i , l 1,1-T These e q u a t i o n s , which a r e used i n p r a c t i c e i n any W i g g i n s - t y p e a l g o r i t h m a r e s i m p l y t r a c e averaged v e r s i o n s of e q u a t i o n s ( 2 . 2 ) , the a u t o c o r r e l a t i o n s l $ x x ( T ) a n ^ c r o s s c o r r e l a t i o n s ^ ^ f r j f o r each t r a c e b e i n g w e i g h t e d by t r a c e 14 dependent s c a l a r s A and B. L i k e the s i n g l e c h a n n e l c a s e , e q u a t i o n (2.5) i s s o l v e d r e c u r s i v e l y by f o r m i n g y and hence A, B and <t>y3x(T) • Then s o l v i n g f o r f , we renew our e s t i m a t e of y and so on. Wiggins (1978) suggests t h a t w e i g h t s A and B need o n l y be c a l c u l a t e d once s i n c e they change l i t t l e i n p r a c t i c e from one i t e r a t i o n t o the n e x t . 2.3 RELATED APPROACHES In the p e r i o d 1977 t h r o u g h 1981 t h e r e was a g r e a t d e a l of a t t e n t i o n f o c u s e d on MED. T h i s r e s e a r c h d e a l t m a i n l y w i t h the s e a r c h f o r a l t e r n a t i v e , t h e o r e t i c a l l y j u s t i f i e d s p a r s e n e s s c r i t e r i a . Some of t h i s work undertaken by the S t a n f o r d E x p l o r a t i o n P r o j e c t has remained u n p u b l i s h e d due t o the 4 year d e l a y i n p u b l i c r e l e a s e of SEP r e p o r t s . Other work, such as t h a t of Deeming (1981), has been p r e s e n t e d a t SEG meetings and workshops but not i n the l i t e r a t u r e . MED e n j o y e d modest i n d u s t r i a l a p p l i c a t i o n a t t h a t time due t o i t s freedom from the cumbersome phase and w h i t e n e s s c o n s t r a i n t s of c o n v e n t i o n a l methods. In p r a c t i c e however i t f a i l e d t o l i v e up t o the h i g h e x p e c t a t i o n s of i t s proponents and more r e c e n t r e s e a r c h such as t h a t of Deeming (1984) and C a b r e l l i (1985) has f o c u s s e d on MED s h o r t f a l l s and how they might be b r i d g e d . In the f i r s t paper r e l e a s e d a f t e r the work of Wiggins (1977), Ooe and U l r y c h (1979) p r o v i d e a c l e a r statement of W i g g i n s work and i t s r e l a t i o n t o the f i e l d of f a c t o r a n a y s i s . They propose the use of a m o d i f i e d varimax norm 15 which i n v o l v e s an e x p o n e n t i a l t r a n s f o r m and a s l i g h t l y d i f f e r e n t method of summing the varimax over the t r a c e s . T h i s norm, the k u r t o s i s norm w i t h e x p o n e n t i a l t r a n s f o r m , K£, i s shown i n Ta b l e 1. T h i s norm and the varimax w i t h e x p o n e n t i a l t r a n s f o r m c o n t a i n an a d j u s t a b l e parameter s, which i f p r o p e r l y s e l e c t e d improves MED performance i n n o i s y d a t a . Gray, i n a 1979 Ph.D. t h e s i s a t S t a n f o r d c o n s i d e r s the varimax norm as j u s t one i n a f a m i l y of s t a t i s t i c s d e s c r i b e d by Hogg (1972) f o r t e s t i n g p r o b a b i l i t y d e n s i t y f u n c t i o n s of wh i t e random p r o c e s s e s . To b e s t u n d e r s t a n d Gray's r e a s o n i n g I must f i r s t i n t r o d u c e the g e n e r a l i z e d G a u s s i a n p r o b a b i l i t y d e n s i t y f u n c t i o n f ( x , a ) , g i v e n by f i x , a . ) = [a/(2T(l/a))]expi-\x\a) -»<*<» (2.6) where T(z) = (z-l)l i s the gamma f u n c t i o n . T h i s d i s t r i b u t i o n and some r e a l i z a t i o n s of independent and i d e n t i c a l l y d i s t r i b u t e d ( i . i . d . ) random p r o c e s s e s h a v i n g t h i s p r o b a b i l i t y d e n s i t y f u n c t i o n a r e p r e s e n t e d i n F i g u r e 3. When a = l , f ( x , a ) becomes the t w o - s i d e d e x p o n e n t i a l d i s t r i b u t i o n . When a=2,f(x,a) becomes the g a u s s i a n , and as a tends t o i n f i n i t y f ( x , a ) t e n d s towards the u n i f o r m . Thus the g e n e r a l i z e d g a u s s i a n d i s t r i b u t i o n d e s c r i b e s a wide range of random v a r i a b l e s from the c e r t a i n event a=0, t o the u n i f o r m l y d i s t r i b u t e d . I f we c o n s i d e r our sampled time s e r i e s of l e n g t h n t o be a s t r i n g of n i . i . d . random v a r i a b l e s , then we can model 3 in Q> tr w-TJ 3 ro c 3 TJ fD 3 rt *-l O ro tr cu >-i M fD W fD N 0) ro rt X W Q ) O 3 3 TJ 0) l - J fD O W P • - w rt • tr « Qj fD tr • (-•• iQ h->T! fD fD I " ' O Hi O o 3 fD fD 0) in 0) fD < fD N 10 fD Cu — *-] •t O M-Q > Ifl 3 l Q C C tr M i it in (D < W W Q ) « 0) OJ r-t 3 • O) Q) TJ 3 i-t • n H-a fD 3 • » Q ) l-h X • trs H fD O fD t-t l-h fD rt O CD w 0) 0) C fD W w cn 3 in 0) 3 Qi rt n tr o ^ CL r-l p fD fD II O 01 ro I-I TJ wfD o 0) 3 tr w-0) 3 3 I CO I ro i o In ro CO 01 o I n c r e a s i n g V a r i m a x > 9T 17 seismograms and r e f l e c t i v i t y s e r i e s as r e a l i z a t i o n s of s t a t i o n a r y random p r o c e s s e s w i t h a m p l i t u d e d i s t r i b u t i o n s taken from the g e n e r a l i z e d g a u s s i a n f a m i l y . T h i s i s the model f o r the seismogram and r e f l e c t i v i t y s e r i e s adapted by Gray (1979). Gray notes t h a t the s i n g l e c h a n n e l varimax norm i s a s p e c i a l case of the s t a t i s t i c t/rx,a 1,a 2^ g i v e n by [(1/n) Z \x. | a M n / a i U(x,a, ,a2) = . (2.7) [(1/n) Z \x. | a 2 ] " / a 2 i=l 1 U(x,a^,a2) i s the "most p o w e r f u l i n v a r i a n t t e s t s t a t i s t i c " t o d e t e r m i n e whether a random sample x i s drawn from the d i s t r i b u t i o n f ( x , a ^ ) as opposed t o f ( x , a 2 ) . The l a r g e r U becomes, the more p r o b a b l e i t i s t h a t the sample x i s drawn from the d i s t r i b u t i o n f ( x , a ^ ) and not f ( x , a 2 ) . L i n e a r f i l t e r s d e s i g n e d t o maximize the varimax norm a r e d e s i g n e d t h e r e f o r e t o reshape the a m p l i t u d e d i s t r i b u t i o n of the seismogram away from t h a t d e s c r i b e d by a-4 i n the d i r e c t i o n of a=2. Thus MED i s i n t e r p r e t e d as an attempt t o y i e l d t h e output w i t h the s m a l l e s t p o s s i b l e shape parameter a. Under Gray's model the s m a l l e r the shape parameter, the more p a r s i m o n i o u s or s p i k y the outcome. Gray recommends a MED-like scheme " V a r i a b l e norm r a t i o d e c o n v o l u t i o n " which i n v o l v e s a d o p t i n g the norm V(y,ax,a2) shown i n T a b l e 1 and c h o o s i n g a^-a, a2=2 where a i s a shape parameter e s t i m a t e d from the d a t a . A l t e r n a t i v e l y he p r e s e n t s arguments i n f a v o r of a,=2, a 2=7. Both schemes a l l o w f o r adjustment of the a parameters between i t e r a t i o n s t o c o n t r o l 18 convergence. Furthermore both schemes form the g e o m e t r i c , r a t h e r than the a r i t h m e t i c mean of the norm over s e v e r a l t r a c e s . An i n t e r e s t i n g outcome of Gray's model i s the l i n k between the varimax and Shannon's s t a t i s t i c a l e n t r o p y . S i n c e among p r o b a b i l i t y d i s t r i b u t i o n s of f i x e d v a r i a n c e the G a u s s i a n has maximum e n t r o p y , we must maximize the k u r t o s i s ( varimax) t o m i n i m i z e e n t r o p y when the data a r e drawn from a d i s t r i b u t i o n which i s more peaked than g a u s s i a n ( l e p t o k u r t i c ) . However, we must m i n i m i z e the varimax when the pdf i s l e s s peaked than g a u s s i a n ( p l a t y k u r t i c ) . T h i s c u r i o u s r e s u l t was o b s e r v e d by Donohoe (1981) and i s demonstrated i n s e c t i o n 5.1. A l s o a t S t a n f o r d , and a t the same time as Gray, Godfrey (1979) completed a Ph.D. t h e s i s e n t i t l e d "A s t o c h a s t i c model f o r seismogram a n a l y s i s " a l s o p u b l i s h e d as S.E.P. r e p o r t number 17. Godfrey approaches the problem of unknown wavelet d e c o n v o l u t i o n by u s i n g a s t o c h a s t i c model f o r the r e f l e c t i v i t y e x p l i c i t l y i n the problem f o r m u l a t i o n . The model adapted i s a r e f l e c t i v i t y sequence i n which the r e f l e c t i o n c o e f f i c i e n t s a r e a g a i n i . i . d . . However the r e f l e c t i v i t y pdf r e p r e s e n t s the sum of two p o p u l a t i o n s — i s o l a t e d l a r g e r e f l e c t o r s w i t h a P o i s s o n d i s t r i b u t i o n i n time p l u s G a u s s i a n n o i s e . In t h i s case t h e pdf d e s c r i b i n g the r e f l e c t i v i t y s e r i e s i s fr(r) = \b(r) + (l-\)G(o2r,r2) (2.8) where G(o2,r2) i s the G a u s s i a n pdf of z e r o mean and v a r i a n c e 19 a2. The P o i s s o n parameter X s e t s the mean w a i t i n g time between s p i k e s o c c u r i n g i n the r e f l e c t i v i t y model a t i/a-v. Based on t h i s r e f l e c t i v i t y model the au t h o r b u i l d s a model f o r the d e c o n v o l v e d (or p a r t l y deconvolved) s e i s m i c t r a c e which i s a l s o w h i t e but w i t h a pdf known as a 'normal m i x t u r e ' . fx(x) = \G(a2n,x2) + (l-\)G(o2 , x2). (2.9) Here a2 i s the v a r i a n c e of the s e i s m i c t r a c e and a2 the n v a r i a n c e of the " n o i s e " or r e s i d u a l w a v e l e t c o n v o l v e d w i t h the t r u e r e f l e c t i v i t y . Based on t h i s p r o b a b i l i s t i c model of the seismogram a MED-type d e c o n v o l u t i o n a l g o r i t h m was proposed which i s c a l l e d "Zero memory n o n - l i n e a r d e c o n v o l u t i o n " (Godfrey and Rocca 1981). W h i l e no norm such as the varimax i s e x p l i c i t l y d e f i n e d i n t h i s scheme, d e c o n v o l u t i o n i s a c h i e v e d by i t e r a t i v e l y s h a p i n g the d a t a by l i n e a r f i l t e r i n g t o match a r e f l e c t i o n c o e f f i c i e n t e s t i m a t e r f o r each i t e r a t i o n . That r e f l e c t i v i t y e s t i m a t e i s o b t a i n e d from the d a t a i n one of two ways: e i t h e r as a l e a s t s quares e s t i m a t e of the ex p e c t e d v a l u e of r g i v e n the seismogram or as the c o n d i t i o n a l median of r g i v e n x and the model. The two r e f l e c t i v i t y e s t i m a t e s proposed f o r n o n - l i n e a r d e c o n v o l u t i o n a r e g i v e n i n T a b l e 1. To c a l c u l a t e them, the p r o b a b i l i s t i c model parameters a2, a2n and X must f i r s t be s p e c i f i e d . Perhaps the be s t known of a l l the MED-type d e c o n v o l u t i o n ' a l g o r i t h m s i s p a r s i m o n i o u s d e c o n v o l u t i o n due 20 t o C l a e r b o u t (1977a). (See a l s o P o s t i c e t a l . 1980). A f t e r e x p e r i m e n t i n g w i t h a norm of the form U(x,l,2) which was a p p e a l i n g on the b a s i s of the g e n e r a l i z e d g a u s s i a n seismogram model the a u t h o r abandoned i t i n f a v o u r of the parsimony norm S2 g i v e n by: S2 =-Z \x. | 2 1 n \ x t | 2 (2.10) i s u b j e c t t o the c o n s t r a i n t T h i s norm i s i d e n t i c a l i n form t o an e x p r e s s i o n f o r parsimony g i v e n by Ferguson (1954). L i k e the v a r i m a x , the parsimony was i n t e n d e d f o r the f a c t o r a n a l y s i s problem of f i n d i n g t h a t r e f e r e n c e frame i n which a system of n - d i m e n s i o n a l v e c t o r s i s most s i m p l y d e s c r i b e d . C l a e r b o u t d e r i v e s S2 from the l i m i t as e tends t o z e r o of U(x,2,2-e). The method proposed f o r c a l c u l a t i n g a MED-type d e c o n v o l u t i o n o p e r a t o r based on the S2 norm i s an i t e r a t i v e s t e e p e s t d e s c e n t a l g o r i t h m i n which convergence can be hastened i n the f i r s t few i t e r a t i o n s by u s i n g another norm: S2 =-Z \x. |2'5/n\x. |2 ' 5 i s u b j e c t t o Lx2 ' 5 = 1. i P r o m i s i n g examples and a computer program t o implement p a r s i m o n i o u s d e c o n v o l u t i o n a r e i n c l u d e d i n C l a e r b o u t (1977b). A l s o noteworthy f o r t h i s t h e s i s a r e comments by the a u t h o r t h a t the S2 norm has many l o c a l extrema and t h a t 21 output of the parsimony r o u t i n e depends on i t e r a t i o n s t a r t i n g p o i n t . Deeming (1981) i n a p r e s e n t a t i o n t o the S.E.G. annual meeting i n Los Ange l e s drew t o g e t h e r a l l MED work t o t h a t date i n h i s " g e n e r a l i z e d minimum e n t r o p y p r i n c i p l e " f o r m u l a t i o n . That i s , he demonstrated t h a t a l l of the norms d i s c u s s e d above can be w r i t t e n i n the form. In t h i s f o r m u l a t i o n , W i g g i n s varimax c o r r e s p o n d s t o c h o o s i n g F(z^^nx* z . . Grays v a r i a b l e norm d e c o n v o l u t i o n c o r r e s p o n d s r o u g h l y t o F(z.)=z?. Ooe and U l r y c h ' s e x p o n e n t i a l t r a n s f o r m method c o r r e s p o n d s r o u g h l y t o F ( z . ) = l - e a z i and C l a e r b o u t ' s p a r s i m o n i o u s d e c o n v o l u t i o n c o r r e s p o n d s r o u g h l y t o F ( z j ) = l n ( z . Deeming shows t h a t W i g g i n s - t y p e normal e q u a t i o n s can be d e v e l o p e d f o r any norm of the form ( 2 . 1 1 ) . These e q u a t i o n s a r e s i m i l a r t o e q u a t i o n s ( 2 . 2 ) e x c e p t t h a t the d e s i r e d o u t p u t e s t i m a t e d a t each i t e r a t i o n i s g i v e n by the more g e n e r a l Deeming r e f e r s t o d as the " r e f l e c t i o n c o e f f i c i e n t e s t i m a t e " , formed from the seismogram a t each i t e r a t i o n . A l i s t of the norms d i s c u s s e d above and t h e i r r e s p e c t i v e r e f l e c t i o n c o e f f i c i e n t e s t i m a t e s i s g i v e n i n V = (1/nx) L z.F(z.) ( 2 . 1 1 ) and d. = nx-p./CL $.z.) where/3. = i i . i i H i dz 22 Ta b l e 1.. I t i s i n t e r e s t i n g t o p l o t t h e s e r e f l e c t i v i t y e s t i m a t e s d. as a f u n c t i o n of the p r e v i o u s i t e r a t i o n output y. at a g i v e n time sample. F i g u r e 4 shows the remarkable s i m i l a r i t y among e s t i m a t e s . In a l l c a s e s the r e f l e c t i v i t y e s t i m a t e s u r p r e s s e s s m a l l r e f l e c t i o n s more than l a r g e ones. Only the parsimony i s s i g n i f i c a n t l y d i f f e r e n t i n c h a r a c t e r , i t s u r p r e s s e s and negates s m a l l r e f l e c t i o n s . Two more r e c e n t a d d i t i o n s t o the MED l i t e r a t u r e a re those of U l r y c h and Walker (1982) and C a b r e l l i (1985). U l r y c h and Walker a p p l y MED w i t h e x p o n e n t i a l t r a n s f o r m t o the a n a l y t i c s i g n a l . By a d o p t i n g a complex approach t h e i r a l g o r i t h m i s a b l e t o cope w i t h a wa v e l e t which undergoes phase s h i f t s w i t h i n the t r a c e . In h i s paper, C a b r e l l i p r o v i d e s a 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 the varimax norm and proposes an a l t e r n a t i v e , the D-norm, which i s supposed t o a c h i e v e the same g o a l s but u s i n g a n o n - i t e r a t i v e a l g o r i t h m . The D-norm i s a l s o shown i n T a b l e 1. I t does not however f i t i n t o Deeming's g e n e r a l i z e d minimum e n t r o p y f o r m a l i s m and thus r e q u i r e s i t s own a l g o r i t h m and has no c o r r e s p o n d i n g r e f l e c t i o n c o e f f i c i e n t e s t i m a t e . F u r t h e r d i s c u s s i o n of C a b r e l l i MED o p e r a t o r d e s i g n w i l l be saved f o r s e c t i o n 3 as i t i s b e s t d e s c r i b e d i n terms of a 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 the varimax norm. S e v e r a l o t h e r m i s c e l l a n e o u s papers on the minimum e n t r o p y p r i n c i p l e have appeared. In "Seven E s s a y s on Minimum E n t r o p y " (1980) C l a e r b o u t proposes t h a t minimum e n t r o p y approaches c o u l d be pursued not o n l y f o r the d e b u b b l i n g 2 3 1.0 , - 0.51 F i g u r e 4 . MED r e f l e c t i v i t y e s t i m a t e s d( p l o t t e d as f u n c t i o n s of yi , the seismogram a m p l i t u d e a t the I ' t h time sample. 24 problem but a l s o i n the s e l e c t i o n of a b s o r p t i o n parameters i n time v a r i a n t d e c o n v o l u t i o n , a s p e c t s of the m i g r a t i o n problem, s p e c t r a l a n a l y s i s , and the i t e r a t i v e adjustment of r e f l e c t i o n c o e f f i c i e n t s so as t o s e l e c t and s u b t r a c t m u l t i p l e s . The a u t h o r f u r t h e r comments on an a l t e r n a t i v e f o r m u l a t i o n of h i s parsimony norm, the " e x t r i n s i c power", comments on the need f o r s p e c t r a l b a l a n c i n g and e x p o n e n t i a l g a i n r e c o v e r y b e f o r e MED p r o c e s s i n g and s p e c u l a t e s on the r e l a t i o n s h i p between parsimony and thermodynamic e n t r o p y . The a p p l i c a t i o n of minimum e n t r o p y methods i n d e c i s i o n a n a l y s i s i s c o n s i d e r e d by G o n z a l e z - S e r a n o (1980) and i n the v e l o c i t y a n a l y s i s problem by D e V r i e s and Berkhout (1984). 25 TABLE I MED NORMS AND CORRESPONDING REFLECTIVITY ESTIMATES NORM 1.VARIMAX V ( x ) REFLECTIVITY ESTIMATE ns = Z / =1 nx r Z . nx [ Z x2 . ] 2 J d. . = x3 . 'J iJ 2.KURTOSIS WITH EXPONENTIAL TRANSFORM K (x) ns nx = Z Z ns nx [ Z Z z ? . ] d. . = x. . iJ ij r(z. . - z? .) i=lj=l lJ where z,. = 1-exp(-x? ./2s2) 3 .VARIABLE RATIO NORM V(x,a^,a2) a 1 -j nx/a 1 nx L Z z . . +A: z . . e ij nx - " " sir [ f 7 / n x ; Z | x . J " 2 ] / = 7 7 4.ZERO MEMORY NONLINEAR DECONVOLUTION a 2 -x. " 7 L 2 FORMULATION: </. = — \l + c exp{ - | — -—1 }1 x . ' a 2 L 2 L a 2 a2-! -I 1 (no e x p l i c i t norm) where c = \ o / ( l - \ ) o , L , FORMULATION: dt = { raj/a2;*. where x = \I 2a {In c n x. \ <x l 1 c I x . | > x a X n — ^ — •a r/-x;J 7/2 26 TABLE I ( c o n t . ) 5.PARSIMONY S (x) n nx = - L \x. \nl n\x. \n nx where L x2 = 1 i=l 1 6.D-NORM D(x) n-1 d. = \x. \ n ~ J s g n ( x . ) [ l - l n\x. \n] = max(\x. \/\\x\\) i, J J (no e q u i v a l e n t r e f l e c t i v i t y ) 2.4 PROBLEMS WITH MED The i n i t i a l p o p u l a r i t y of MED i n t h e g e o p h y s i c a l l i t e r a t u r e l a s t e d o n l y a few y e a r s . What a t f i r s t seemed a p r o m i s i n g s o l u t i o n t o a p e r e n i a l l y u n s o l v e d problem i n g e o p h y s i c s proved t o MED u s e r s unworkable. C o m p l a i n t s i n c l u d e d poor performance i n n o i s y d a t a compared w i t h c o n v e n t i o n a l methods, ( J u r k e v i c k s & W i g g i n s , 1984) problems whenever h i g h f r e q u e n c y and h i g h a m p l i t u d e a r e w e l l c o r r e l a t e d i n the d a t a ( M o r l e y 1980) and poor r e s u l t s when f i l t e r l e n g t h i s chosen too l o n g ( M o r l e y 1980, Deeming 1985). C a b r e l l i (1985) shows i m p l i c i t l y t h a t the t r a d i t i o n a l W i g g i n s - t y p e a l g o r i t h m does not succeed i n a c h i e v i n g the g l o b a l maximum of the varimax f u n c t i o n a l w h i l e C h a c i n (1983) r e p o r t s g r o s s phase and a m p l i t u d e e r r o r s i n s i m p l e w a v e l e t e s t i m a t i o n problems u s i n g a W i g g i n s - t y p e a l g o r i t h m w i t h symmetric s t a r t i n g f i l t e r g i v e n by e q u a t i o n ( 2 . 3 ) . I t has 27 been f u r t h e r contended by Deeming (1984) t h a t MED's s p a r s e s p i k e t r a i n model of the seismogram i s n e i t h e r w e l l u n d e r s t o o d nor r e a l i s t i c . The problems w i t h MED, i f not the c o m p l a i n t s , can be d i v i d e d i n t o two d i s t i n c t c a t e g o r i e s . On the one hand we have u n r e s o l v e d d i f f i c u l t i e s a s s o c i a t e d the c a l c u l a t i o n of MED o p e r a t o r s . These i n c l u d e the c h o i c e of s t a r t i n g f i l t e r and f i l t e r l e n g t h , p r e p r o c e s s i n g i f any, and what type of a l g o r i t h m t o choose. On the o t h e r hand t h e r e i s the problem of model v a l i d i t y , a c o n c e r n which must be a d d r e s s e d each time MED i s a p p l i e d . I f , f o r example, the r e f l e c t i v i t y u n d e r l y i n g the s e i s m i c t r a c e w i t h i n our d e s i g n window does not have the "minimum e n t r o p y " p r o p e r t y , then MED w i l l d e s i g n a. f i l t e r which w i l l g i v e a s p a r s e y e t m e a n i n g l e s s o u t p u t . In t h a t case MED has been m i s a p p l i e d t o a d a t a s e t f o r which the s t a t i s t i c a l model i s i n v a l i d and i t would be u n f a i r t o c r i t i c i z e MED on the grounds t h a t i t does not produce s e n s i b l e r e s u l t s . I t i s not c l e a r i n every case which type of problem i s r e s p o n s i b l e f o r each of the c o m p l a i n t s mentioned a t the s t a r t of t h i s s e c t i o n . I t i s however p o s s i b l e t h a t p r o g r e s s can be made on both f r o n t s , t h a t of o p e r a t o r d e s i g n and s e l e c t i o n of a p p l i c a b i l i t y c r i t e r i a . Such p r o g r e s s i s the g o a l of t h i s t h e s i s . P r a c t i c a l and v a r i e d r e a l d a t a a p p l i c a t i o n w i l l then show whether t h i s work l e a d s t o a renewed f a i t h i n MED as a workable scheme. 3 . A GEOMETRICAL INTERPRETATION 3.1 NOTATION AND INTRODUCTION There i s an i n t u i t i v e , 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 MED, and l e a s t squares f i l t e r i n g i n g e n e r a l , which I hope w i l l a i d the r e a d e r ' s u n d e r s t a n d i n g as i t c e r t a i n l y has mine. T h i s h e u r i s t i c approach i s the s u b j e c t of t h i s s e c t i o n and i t s development r e q u i r e s the i n t r o d u c t i o n of a few b a s i c c o n c e p t s . C o n s i d e r a segment from a d i s c r e t e time s e r i e s y y = (y\, y-i, • • • yny)r' I have w r i t t e n y i n terms of i t s sampled v a l u e s a t e q u a l l y spaced time i n c r e m e n t s , i = l , . . . n y i n a column v e c t o r n o t a t i o n . Indeed as an o r d e r e d n y - t u p l e of r e a l numbers we may c o n s i d e r the time s e r i e s y as a v e c t o r i n an « 7 ~ d i m e n s i o n a l E u c l i d e a n space, t h a t i s , a r e a l v e c t o r space over which an i n n e r p r o d u c t i s d e f i n e d . I f we c o n s i d e r the s e t of ny b a s i s v e c t o r s f o r such a v e c t o r space: 'e = (1, 0,0 0)r 2 e = (0, 1,0,. . .0)T (3 .1 ) • nye = (0,0,0,...iy then we can t h i n k of t h e s e v e c t o r s as f o r m i n g a c o o r d i n a t e system. F i g u r e 5 shows the time s e r i e s y i n the case ny=3, our day t o day E u c l i d e a n 3-space. Note t h a t the time samples y^r y2 and y* a r e s i m p l y the c o o r d i n a t e s d e f i n i n g the v e c t o r and i n t h i s s e c t i o n I s h a l l d e a l as much as p o s s i b l e w i t h 28 29 F i g u r e 5 . A time s e r i e s y i s c o n s i d e r e d a v e c t o r i n an /zy-dimensional E u c l i d e a n space Rny. I f ny=3, time samples yy, y2 and y3 a r e c o o r d i n a t e s d e s c r i b i n g the v e c t o r i n R 3. 30 the e a s i l y v i s u a l i z e d ny=3 c a s e . An i m p o r t a n t concept i n l e a s t squares f i l t e r i n g , the sum of squared e r r o r s between two t i m e s s e r i e s , say y and d i s g i v e n by ny e2 = L (y -d.)2. (3.2) i = 1 T h i s i s r e c o g n i z e d as s i m p l y the squared v e c t o r d i s t a n c e between y and d so e = ||y-d||. (3.3) In p a r t i c u l a r the v e c t o r d i s t a n c e g i v e n by 'e = | |y- z e| |. (3.4) t h a t i s the L2 p r o x i m i t y of the y v e c t o r t o the i ' t h a x i s i n F i g u r e 5 i s the squared e r r o r between the time s e r i e s y and a u n i t s p i k e a t the i ' t h l a g , the q u a n t i t y m i n i m i z e d by a Weiner s p i k i n g f i l t e r . 3.2 GEOMETRICAL INTERPRETATION OF LEAST SQUARES FILTERING As mentioned i n s e c t i o n 2.2 the d e s i g n of MED o p e r a t o r s i s a r e c u r s i v e a p p l i c a t i o n of Weiner wavelet s h a p i n g . F o r t u n a t e l y Weiner s h a p i n g or l e a s t squares f i l t e r i n g has a s i m p l e g e o m e t r i c a l a n a l o g , one which has been e x p l o r e d e x t e n s i v e l y by N. de Voogd (1972) and which I d i s c u s s here b e f o r e t a c k l i n g MED i t s e l f . The Weiner w a v e l e t s h a p i n g f i l t e r as d i s c u s s e d by Robinson and T r e i t e l (1980) i s t h a t f i l t e r £ which when c o n v o l v e d w i t h an i n p u t time s e r i e s x b e s t r e p r o d u c e s a p r e d e t e r m i n e d d e s i r e d o u t p u t , d i n a l e a s t squares sense. 31 We may w r i t e the c o n v o l u t i o n as y = x * f . A l t e r n a t i v e l y we may w r i t e output y as the m a t r i x p r o d u c t of f w i t h a c y c l i c m a t r i x X. For the case nx=2,nf=2 and ny=3 we may w r i t e X, 0 yz y* 0 *2 / 1 / 2 (3.5) (3.6) The columns of X a r e themselves v e c t o r s g i v e n by ! v = (x,,x2,0)r 2 v = (0, x i, x2)J . Now i t can e a s i l y be seen t h a t the c o n v o l u t i o n y i s s i m p l y a l i n e a r c o m b i n a t i o n of t h e s e  lv v e c t o r s nf y = z f v. i=J 1 (3.7) What then i s the g e o m e t r i c a l s i g n i f i c a n c e of the c o n v o l u t i o n ? F i g u r e 6 shows t h a t the v e c t o r s 'v d e f i n e a subspace 0 of the ou t p u t v e c t o r space R 3. S i n c e the *v a r e l i n e a r l y independent (see e.g. C a b r e l l i , 1985) they form a b a s i s f o r t h a t subspace and any l i n e a r c o m b i n a t i o n of t h e s e v e c t o r s w i l l be c o n f i n e d t o B, i n t h i s case the shaded p l a n e of F i g u r e 6. For a g i v e n d a t a s e t x t h e n , l i n e a r f i l t e r o u t p u t s a re c o n f i n e d t o the 0 h y p e r p l a n e . G i v e n a d e s i r e d o u t p u t d f o r a wave s h a p i n g f i l t e r , the f i l t e r o u t p u t i s t h a t v e c t o r y i n fl which i s c l o s e s t t o d. As shown i n F i g u r e 7 t h i s means y i s s i m p l y the p r o j e c t i o n 32 2 V = ( 0 , x 1 f x 2) F i g u r e 6 . The v e c t o r s 'v, i = l , . . . n f span a subspace fl of Rny t o which a l l r e a l i z a b l e f i l t e r o u t p u t s , f * x are c o n f i n e d . In the case nx=2 , nf=2, nf=3, 0 i s the shaded p l a n e shown. 33 F i g u r e 7 . The l e a s t squares f i l t e r a p p r o x i m a t i o n t o d i s y, the o r t h o g o n a l p r o j e c t i o n of d onto subspace — i n t h i s case the shaded p l a n e . For p r o j e c t i o n MED the v e c t o r y i s used t o form a new v e c t o r d=y 3 and hence, by p r o j e c t i o n , y f o r the next i t e r a t i o n . 34 of d onto the p l a n e . T h i s l e a d s us i n t u i t i v e l y t o the p r o j e c t i o n method of wa v e l e t s h a p i n g . W h i l e Robinson and T r e i t e l (1980) d e s c r i b e a d e s i g n p r o c e s s f o r wave sha p i n g f i l t e r s which i n v o l v e s s o l v i n g the T o e p l i t z system of e q u a t i o n s (2.4) f o r f and then c o n v o l v i n g f w i t h the d a t a t o get y, the s i m p l e a l t e r n a t i v e of p r o j e c t i n g the d e s i r e d output v e c t o r onto the " p l a n e " of p o s s i b l e f i l t e r o u t p u t s y i e l d s the same o u t p u t . No f i l t e r i s a c t u a l l y c a l c u l a t e d . The above d i s c u s s i o n s u g g e s t s a q u i c k c o m p u t a t i o n a l method f o r p e r f o r m i n g MED u s i n g the p r o j e c t i o n method. T h i s method, c o n c e i v e d by T o s h i Matsuoka, i s p a r t i c u l a r l y s u i t e d t o . comparison of t h e v a r i o u s MED norms of T a b l e 1, s i n c e they can each be r e p r e s e n t e d by a r e f l e c t i v i t y e s t i m a t e t o be p r o j e c t e d onto the common subspace fi. The p r o j e c t i o n a l g o r i t h m proceeds as f o l l o w s : 1. Compute from the d a t a x, v e c t o r s 'v., i = l , . . . n f and hence o r t h o n o r m a l b a s i s v e c t o r s 1 a, i = I , . . . n f s p a n n i n g £2. T h i s i s done u s i n g the m o d i f i e d Gram Schmidt p r o c e s s d e s c r i b e d by de Voogd (1974). 2. Compute the d e s i r e d o u t p u t d. For i n s t a n c e , t o p e r f o r m Varimax MED we would choose d=y 3 based on the ou t p u t y of the p r e c e d i n g i t e r a t i o n 3. C a l c u l a t e the p r o j e c t i o n of d onto each 1 a, a^ g i v e n by a. = d•'a. (3.8) 4. Form the f i l t e r o u t p u t y from a l i n e a r c o m b i n a t i o n of the 1 a 35 nf Y = I a. a. ( 3 . 9 ) i=l 1 The o r t h o g o n a l i z a t i o n of the 'v v e c t o r s need o n l y be done once and s t e p s 2 t h r o u g h 4 a r e r e p e a t e d u n t i l y co n v e r g e s . Very s i m i l a r t o the w e l l known Gram Schmidt p r o c e d u r e , the method of de Voogd has t h r e e a dvantages: 1. I t t a k e s advantage of the c y c l i c n a t u r e of m a t r i x X t o speed c a l c u l a t i o n . 2. I t a l l o w s f o r s t a b i l i z a t i o n of the a l g o r i t h m t h rough a p r o c e s s e q u i v a l e n t t o the a d d i t i o n of a r i d g e parameter i n t he a u t o c o r r e l a t i o n m a t r i x of ( 2 . 4 ) . 3. I t a l l o w s c a l c u l a t i o n of the e q u i v a l e n t f i l t e r f i f r e q u i r e d . A s i m p l e example of t h i s method i s g i v e n below. Take the case w i t h d a t a x=(l,2)T and s t a r t i n g f i l t e r ot=(0,l)T. Our r e f l e c t i v i t y e s t i m a t e d i s g i v e n by d = y 3 = f x * f ; 3 = (1 3 , 23, 03)J = (l,8,0)r. The next MED i t e r a t i o n t h e r e f o r e seeks t o shape the f i l t e r o u tput y i n t o (1,8,0). The subspace of a l l p o s s i b l e l i n e a r f i l t e r o u t p u t s i s spanned by the v e c t o r s 1 v and 2 v which a r e i n t h i s case g i v e n by 'v = (1,2,or and 2 v = (0, 1, 2)T . We o r t h o g o n a l i z e the : v , i-1,2 t o get an o r t h o g o n a l b a s i s 36 f o r n such a s : 1 a = l/y/5(l, 2, 0)r and 2a = 1 /y/10 5 (- 2, 1, 10)r . Then the f i l t e r o u t put y which i s c l o s e s t t o d i n a l e a s t squares sense i s s i m p l y the p r o j e c t i o n onto the p l a n e of f i l t e r o u t p u t s . y = (6-'a)'a+(6-2a)xa - (3. 3, 6. 9, . 57)J T h i s r e s u l t would be used t o r e v i s e the r e f l e c t i v i t y e s t i m a t e d, p r o j e c t i t onto the p l a n e and so on. Armed w i t h an u n d e r s t a n d i n g of the p r o j e c t i o n method one can e a s i l y demonstrate i t s e q u i v a l e n c e t o the W i g g i n s MED a l g o r i t h m . C o n s i d e r the a c t i o n of f o r m i n g a r e f l e c t i v i t y e s t i m a t e d=y 3, p r o j e c t i n g t h i s onto .the S2 p l a n e , and c u b i n g the p r o j e c t i o n t o o b t a i n y e t another v e c t o r t o be p r o j e c t e d . Such an a l g o r i t h m w i l l converge when the cube of the v e c t o r y p r o j e c t e d onto fi y i e l d s y i t s e l f . In t h i s case y-y 3 i s o r t h o g o n a l t o the JJ ( h y p e r ) p l a n e and from t h i s s i m p l e c r i t e r i o n we may d e r i v e the MED normal e q u a t i o n s w i t h o u t an e x p l i c i t norm and w i t h o u t t a k i n g any d e r i v a t i v e s . Set (y-y 3; 1 a ( 3 . 1 0 ) then (y-y 3; 1 * v k=l,...nf. k k S i n c e the v e c t o r s v have components v^ . = x^ ._^ + ^ I may w r i t e ny z (yry.^i-k+i  = °-  ( 3 - 1 1 ) i=l J 1 J K 1 Now s i n c e y i s e q u i v a l e n t t o a f i l t e r o u t p u t w i t h components 37 nf (3.11) becomes ny nf j -l 1=1 J J (3.12) R e a r r a n g i n g I o b t a i n 2 / ^ 1 x._l+1x._k+I) = Zy*x._k+1 k=l,...nf. (3.13) E q u a t i o n s (3.13) above a r e i d e n t i c a l t o the s i n g l e c h a n n e l MED normal e q u a t i o n s (2.2) t o w i t h i n a s c a l a r m u l t i p l i e r . 3.3 GEOMETRICAL INTERPRETATION OF MED R e t u r n i n g now t o the f i e l d of MED we w i l l l o o k a t the varimax norm. The Varimax has been c o n s i d e r e d by C a b r e l l i (1985) t o be a two sta g e t r a n s f o r m a t i o n a c t i n g on the v e c t o r y where y = (y\, y2, • • • yny)T• The f i r s t s t a g e i s T,(y) = (yW\\y\\2,y22/\\Y\\2,...y2n/\\y\\2) = y and the second stage s i m p l y T2(y') = l l y ' l l 2 so t h a t V(y) = T2(T,(y)). The f i r s t s t a g e or 2", t r a n s f o r m a t i o n , maps the v e c t o r from d a t a space R " ^ t o a s u b s e t , H d e f i n e d by 38 H = {y 6 R 7 1' I Zy.=l, yt>0, i=1, 2,... ny]. i The case f o r ny=3 i s shown i n F i g u r e 8. Here H i s the h a t c h e d a r e a of the p l a n e g i v e n by (y,-l/3) + (y2-l/3) + (y2-l/3) = 0 and f a l l i n g i n t h e f i r s t q u a d r a n t . The varimax norm t h e r e f o r e r e p r e s e n t s the l e n g t h of the v e c t o r T,(y) shown i n the f i g u r e as y'. The varimax t h e n , i n c r e a s e s w i t h the d i s t a n c e between y' or y, and the c e n t e r of H, C a b r e l l i then proposed a s i m p l e r norm which measures not the d i s t a n c e of y' from the c e n t r e of H but the p r o x i m i t y of y i t s e l f t o the axes le of F i g u r e 8. To o b t a i n the s p a r s e s t o u t p u t v e c t o r y, k C a b r e l l i m i n i m i z e s t h e q u a n t i t y | | y / | | y | | - e|| over a l l axes k=l,...ny. I t can be shown t h a t m i n i m i z i n g the above q u a n t i t y i s e q u i v a l e n t t o m a x i m i z i n g |^^./||y|| | so t h a t C a b r e l l i d e f i n e s h i s new norm as f o l l o w s D(y) = max \yk\/\ |y||. l<k<ny T h i s new norm, where |^^/||y|| I i s maximized over a l l p o s s i b l e v e r t i c e s i s c a l l e d the "D-norm". J u s t as the s e t of r e a l i z a b l e f i l t e r o u t p u t s 0 formed a subset of R 3 so does i t form a s u b s e t of the H p l a n e . For the ny=3, nx=2 case c o n s i d e r e d so f a r , I have found t h i s s ubset t o be an e l l i p s e which changes i t s c h a r a c t e r depending on the i n p u t d a t a x. What then o c c u r s when we choose a s t a r t i n g f i l t e r 0t and p r o c e e d w i t h a W i g g i n s - t y p e MED a l g o r i t h m ? I t i s q u i t e easy t o v i s u a l i z e i n terms of the t r a n s f o r m e d v e c t o r y'. R e f e r r i n g t o F i g u r e 9 we see t h a t as the i t e r a t i o n proceeds 39 F i g u r e 8 . For the case ny=3 the h a t c h e d p l a n e i s the subset H on which a l l v e c t o r s y'=r,f y; l i e . V a r i m a x ( y ) i s the l e n g t h of y' and hence i n c r e a s e s toward the axes. i Start ing f i l ter=(0 .0 , 1.0) ITERATION VARIMAX I 2 3 4 5 6 .5148 . 5428 . 6043 . 6250 . 6257 . 6257 X=U. 000 ,1 .190 ) F i l t e r output=(-.4599,.3406,1.0567) Start ing f i l t e r= (1 .0 ,0 .0 ) ITERATION VARIMAX 1 2 3 4 5 6 .5148 .5251 . 5301 . 5307 . 5308 . 5308 X=U. 000 ,1 .190 ) F i l t e r output=(.9689,1.4003,.2943) F i g u r e 9 . I t e r a t i o n p a t h s and f i n a l 3-point o u t p u t of Wiggin s MED a f t e r 6 i t e r a t i o n s . C r o s s e s i n d i c a t e the T, t r a n s f o r m a t i o n s of y a f t e r s u c c e s s i v e i t e r a t i o n s . S t a r t i n g f i l t e r s g i v e n by (top) 0£=(0,Or, and (bottom) o t = ( i , 0 ) r l e a d t o two d i s t i n c t extrema. 41 the v e c t o r y ' a t t e m p t s t o maximize i t s l e n g t h w h i l e r e m a i n i n g on the e l l i p s e . Shown i n F i g u r e 9 a r e two examples of the W i g g i n s MED p r o c e d u r e . At each i t e r a t i o n an 'X' has been p l o t t e d r e p r e s e n t i n g the e n d p o i n t of the c o r r e s p o n d i n g y ' v e c t o r . A f t e r s i x i t e r a t i o n s s t a r t i n g w i t h ot=(0,1)J the a l g o r i t h m has converged t o the t o p of the e l l i p s e shown on the l e f t hand s i d e of F i g u r e 9. However, when we choose ot=(I,0)T as shown on the r i g h t , q u i t e a d i f f e r e n t extremum i s r e a c h e d . I t i s c l e a r from the diagram t h a t t h e r e a r e i n t h i s case two p o s s i b l e extrema i n the varimax f u n c t i o n a l and whether the g l o b a l extremum i s reached depends on our c h o i c e of s t a r t i n g f i l t e r . T h i s and the work i n subsequent c h a p t e r s shows t h a t i n g e n e r a l : 1. Minimum e n t r o p y norms have m u l t i p l e extrema, and 2. There i s no r e a l b a s i s f o r c h o o s i n g a s t a r t i n g f i l t e r of the symmetric form 0 f = (0,0,. . .1 0, 0)J (2.3) i f the g l o b a l extremum i s d e s i r e d . 4. ASPECTS OF MED OPERATOR DESIGN 4.1 MULTIPLE EXTREMA In the p r e v i o u s s e c t i o n i t was shown i n s i m p l e g e o m e t r i c a l terms t h a t the Wiggins MED a l g o r i t h m may r e a c h l o c a l extrema and how th e s e come about. The g o a l of t h i s s e c t i o n i s t o ensure t h a t the g l o b a l extremum i s reached. To e n v i s i o n the wo r k i n g s of the MED a l g o r i t h m I have been u s i n g the u n r e a l i s t i c a l l y s i m p l e c a s e of nx=2, nf=2 and ny=3. To a l l o w more r e a l i s t i c examples I w i l l now adopt a scheme i n which nf=3, nx can tak e on any v a l u e , and ny=nf+nx-1. F i g u r e 10 shows the t h r e e p o i n t f i l t e r v e c t o r f = f / i , / 2 , / 3 ; T . I f we s p e c i f y the c o n s t r a i n t II*II -That i s , l e t f l i e on the u n i t sphere, then we can adopt a s p h e r i c a l c o o r d i n a t e r e p r e s e n t a t i o n f o r f , / • i = si n6cos4> f 2 = si nds i n<j> f3 = cosd. Now any t h r e e p o i n t f i l t e r f can be r e p r e s e n t e d i n terms of two parameters 6 and 0 , f o r i n s t a n c e f = (l.o,oy becomes (6,4>) = (it/2,0). More i m p o r t a n t l y any f u n c t i o n a l of f can be c o n t o u r e d i n t h e (8,4>) p l a n e and the p a r t i c u l a r f u n c t i o n a l we a r e i n t e r e s t e d i n i s V ( i ) where 42 43 F i g u r e 10. The t h r e e p o i n t f i l t e r f can be p a r a m e t e r i z e d i n terms of 6 and 4>. 44 V(t) = V(t*x). Thus f o r any i n p u t d a t a s e t x a c o n t o u r map r e p r e s e n t i n g the shape of the varimax as a f u n c t i o n a l of the f i l t e r a p p l i e d can be produced. F u r t h e r m o r e , by c o n s t r u c t i n g the d a t a s e t x so as t o c o n t a i n a wavelet w i t h a r e a s o n a b l e t h r e e p o i n t i n v e r s e , i t i s p o s s i b l e t o see which peak i n the vari m a x , i f any, c o i n c i d e s w i t h the " r i g h t " answer. F i g u r e 11 i s an example of such a p l o t . In t h i s c a s e the i n p u t d a t a i s a spa r s e s p i k e s e r i e s c o n v o l v e d w i t h a s i m p l e t h r e e p o i n t w a v e l e t . To t h e s e d a t a ' I have a p p l i e d a l l p o s s i b l e t h r e e p o i n t f i l t e r s , a t l e a s t 1250 of them, s a m p l i n g the e n t i r e range of 6 from 0 t o i and <t> from 0 t o 2n. The varimax of the r e s u l t i n g f i l t e r o utput has been c o n t o u r e d and i t i s c l e a r l y seen t h a t t h e r e a r e s e v e r a l extrema. One of the f i r s t t h i n g s we n o t i c e about F i g u r e 11 i s i t s symmetry. The r i g h t hand s i d e of the p l o t c o r r e s p o n d i n g t o <p=it t o 2ir can oe g e n e r a t e d by a 180 degree r o t a t i o n of the l e f t hand s i d e . T h i s symmetry a r i s e s from the i n s e n s i t i v i t y of the varimax t o s i g n a l p o l a r i t y . That i s , f o r each maximum i n the varimax f u n c t i o n a l t h e r e i s a c o r r e s p o n d i n g maximum which w i l l g i v e an o utput r e v e r s e d i n p o l a r i t y . MED a l g o r i t h m s may t h e n , induce a p o l a r i t y r e v e r s a l i n t h e f i l t e r e d t r a c e , a f a c t a l l too f a m i l i a r t o MED u s e r s . A W i g g i n s - t y p e MED a l g o r i t h m was run on the seismogram used t o g e n e r a t e F i g u r e 11. The e v o l u t i o n of the f i l t e r s over the f i v e t o s i x i t e r a t i o n s r e q u i r e d f o r convergence i s t r a c e d w i t h a b o l d arrow. I t i s obser v e d t h a t the Wigg i n s I N P U T x : C C O N T O U R S O N T H E V A R I M A X C O z: CE 1—1 Q CE cr CE h— UJ IE 2.8 3.8 PHKRflDIflNS) F i g u r e 11. R e l a t i v e varimax of the f i l t e r e d t r a c e c o n t o u r e d f o r a l l 3 - p o i n t f i l t e r s t{8,<t>). Input data s et i s shown above. Heavy arrows show MED i t e r a t i o n paths f o r (A): ot = ( l , 0 , 0 ) T , (B) : ot=(0,l,0)T, and (C) ot=(0,0,l)T. C i r c l e d d o t s denote g l o b a l maxima. 46 MED a l g o r i t h m does indeed f i n d f i l t e r s which maximize the va r i m a x , but when the i t e r a t i o n s t a r t i n g f i l t e r i s v a r i e d , d i f f e r e n t l o c a l extrema a r e reached. Three d i s t i n c t maxima are sampled by s t a r t i n g f i l t e r s of t h e form: ot=(l,0,0)T o£=(0,l,0)r (3.14) and 0t=(0,0,l)7. Those not reached a r e s i m p l y p o l a r i t y r e v e r s a l s of the f i r s t t h r e e and a r e found by n e g a t i n g the s t a r t i n g f i l t e r . S t i l l a more i n t e r e s t i n g example i s g i v e n i n F i g u r e 12. In t h i s case the i n p u t d a t a s e t i s the s i m p l e minimum phase w a v e l e t g i v e n by w = (. 64, . 80, . 24)T . A l t h o u g h t h i s w a v elet has the minimum phase p r o p e r t y note t h a t i t s l a r g e s t v a l u e does not occur a t the f i r s t time sample. W h i l e the c o n t o u r map p r e s e n t e d i s f o r a l o n e w a v e l e t , the c o n t o u r s f o r a s u f f i c i e n t l y s p a r s e seismogram c o n t a i n i n g t h i s w a v elet have the same c h a r a c t e r . When the W i g g i n s - t y p e MED a l g o r i t h m i s run on t h i s example f o r s t a r t i n g f i l t e r s of the form ( 3 . 1 4 ) , an i n t e r e s t i n g and unexpected r e s u l t s u r f a c e s . The g l o b a l maximum of the varimax i s not reached w i t h any of the s t a r t i n g f i l t e r s ! What i s the reason f o r t h i s unexpected f i n d i n g ? The s i g n i f i c a n c e of the v a r i o u s extrema l i e s i n the output w a v e l e t l a g . In f a c t the f o u r independent extrema of F i g u r e 12 c o i n c i d e almost e x a c t l y w i t h Weiner f i l t e r s d e s i g n e d t o s p i k e t h e t h r e e p o i n t w a v e l e t a t l a g s 1, 2, 3, and 4 of the I n p u t x : C O N T O U R S O N T H E V A R I M A X PHI(RADIflNS) Figure 12. R e l a t i v e varimax of the f i l t e r e d t r a c e c o n t o u r e d f o r a l l 3 - p o i n t f i l t e r s l{6,4>). Input data set i s shown above. Heavy arrows show MED i t e r a t i o n p aths f o r (A): 0t=( 1, 0, 0)J , {B)x ot = (0, 1, 0)J , and (C) : ot=(0, 0, J ) J . In t h i s case the g l o b a l extrema ( c i r c l e d d o t s ) a re not reached. 48 f i l t e r o u t put f * w. R e c a l l t h a t the MED a l g o r i t h m seeks t o shape the wa v e l e t i n t o a l a g g e d v e r s i o n of i t s cube as demonstrated i n F i g u r e 2. That l a g i s d e t e r m i n e d by the s t a r t i n g f i l t e r and f o r the f i r s t i t e r a t i o n of the MED o p e r a t o r d e s i g n p r o c e s s we have a d e s i r e d o u t p u t g i v e n by: d = [ ('. 64, . 80, . 24, 0, 0) 3 ]T f o r ot=(l,0,0)r d = [ (0, . 64, . 80, . 24, 0) 3 ] T f o r ot=(0,l,0)T or 6 = [(0, 0, . 64, . 80, . 24)3 T f o r Qt = ( 0 , 0 , l ) r . The e f f e c t of c u b i n g y i s t o a c c e n t u a t e the major peaks i n t he wa v e l e t a t the expense of the s m a l l e r ones. I f , as i n t h i s example, the g r e a t e s t v a l u e i n the w a v e l e t o c c u r s a t w2, t h a t i s | W 2 | > | M * . | f o r a l l o t h e r /, then MED f i l t e r i n g w i l l t e n d t o p l a c e a s p i k e a t l a g two of the ou t p u t w a v e l e t f o r a s t a r t i n g f i l t e r of the form o t = ( l , 0 , 0 ) J . S i m i l a r l y i t w i l l p l a c e a s p i k e a t l a g 3 f o r o f = ( 0 , J , 0 ) r or l a g 4 f o r 0£=(0,0,l)r. Weiner f i l t e r t h e o r y however, (Robinson and T r e i t e l 1980, p.184) s u g g e s t s the o p t i m a l o u t p u t s p i k e would l i e a t the f i r s t l a g f o r a minimum phase w a v e l e t . T h i s f i r s t l a g , which a l s o r e p r e s e n t s the g l o b a l maximum of v a r i m a x , remains t o be reached by any of the s t a r t i n g f i l t e r s ( 3 . 1 4 ) . I t i s easy t o show t h a t t h i s phenomenon i s not unique t o t h e s i m p l e example of nf=3. F i g u r e 13 shows a minimum phase w a v e l e t t o which MED has been a p p l i e d w i t h t h r e e d i f f e r e n t s t a r t i n g f i l t e r s . I n t h i s case the wa v e l e t i s a p p r o x i m a t e l y 34 time samples i n l e n g t h and t h e f i l t e r has l e n g t h nf=22. I t i s obser v e d t h a t f o r f i l t e r s of t h i s l e n g t h , s t a r t i n g f i l t e r p r e d e t e r m i n e s o u t p u t s p i k e l a g hence W i g g i n s M E D W i g g i n s M E D W i g g i n s M E D ,f = ( 0 , . . . 0 , 1 ) f = ( 0 , . . . 0 , 1 . 0 . . . . 0 ) I n p u t x O u t p u t 5 y 0 f = ( 1 . 0 . . . . 0 ) 0 2 4 6 [ l l l l M l l t [ t t l l f l t l t | l t l l l l l t t | l l l I n p u t x O u t p u t 5 y F i g u r e 13. MED a p p l i e d t o a m i n i m u m p h a s e w a v e l e t w i t h t h r e e s t a r t i n g f i l t e r s 0 f , n o n e o f w h i c h l e a d s t o a n o u t p u t s p i k e a t t h e w a v e l e t o n s e t . 50 performance of the s h a p i n g f i l t e r and varimax of the output w a v e l e t . A g a i n none of the f i l t e r s used a r e a b l e t o s p i k e the w a v e l e t a t the f i r s t and optimum l a g so the g l o b a l maximum of the varimax remains unreached. I t s h o u l d be noted t h a t f u r t h e r i t e r a t i o n f o r any of the examples shown i n F i g u r e 13 does not r e s u l t i n any s i g n i f i c a n t i n c r e a s e i n va r i m a x , and cannot r e s u l t i n a change i n output s p i k e placement. S i n c e the i n c r e a s e i n varimax w i t h i t e r a t i o n i s monotonic (Waldren, 1985), once a l o c a l extremum has been re a c h e d , t h e r e i s no p o s s i b i l i t y f o r the Wiggins a l g o r i t h m t o s k i p t o a n o t h e r , and hence no p o s s i b i l i t y the g l o b a l maximum w i l l be r e a c h e d . I propose t o overcome t h i s problem by s t a r t i n g the Wiggi n s MED a l g o r i t h m not w i t h a s t a r t i n g f i l t e r but w i t h a s h i f t e d v e r s i o n of the cubed d a t a . By padding the da t a w i t h z e r o s and a l l o w i n g backward as w e l l as f o r w a r d s h i f t s i n the r e f l e c t i o n c o e f f i c i e n t e s t i m a t e as a s t a r t i n g p o i n t , the a l g o r i t h m can be d e s i g n e d t o s p i k e the w a v e l e t a t z e r o or any o t h e r l a g . For an i n p u t d a t a s e t x c o n t a i n i n g a w a v e l e t w of l e n g t h nw, t h e g l o b a l maximum i s reached by t e s t i n g t he varimax f o r extrema a t a l l nw+nf-1 " m e a n i n g f u l " o u t p u t s p i k e l a g s of the f i l t e r e d w a v e l e t , w*f. My a l g o r i t h m proceeds as f o l l o w s : 1. E s t i m a t e the w a v e l e t l e n g t h nw v i s u a l l y from the d a t a . A generous guess i s a c c e p t a b l e . 2. E s t i m a t e the w a v e l e t . r i s e t i m e , L, t h a t i s the number of time samples between the onset of the wavelet and i t s 51 peak v a l u e . A g a i n , i t i s okay t o o v e r e s t i m a t e . 3. Pad the d a t a t o be f i l t e r e d , x, w i t h L l e a d i n g z e r o s and (nw-L-1) t r a i l i n g z e r o s t o o b t a i n a new d a t a v e c t o r x' of l e n g t h nx'=(nx+nw-1). 4. . For i-1 t o (nw+nf-1) s t a r t the MED i t e r a t i o n u s i n g d a t a x' and i n i t i a l r e f l e c t i v i t y e s t i m a t e d. w i t h elements d.. where lJ 0 j<i , nx+i<j<nx'+nf-1 </.. = { 1 J x3. . , , / </'<nx+i j -i +1 J Stop the i t e r a t i o n upon convergence. 5. Choose as the g l o b a l l y optimum MED o p e r a t o r t h a t one which has a c h i e v e d the h i g h e s t varimax over a l l output l a g s /. Example: A seismogram x i s composed of a w a v e l e t , w = (-. 4,1, . 2,-. 2 ; T c o n v o l v e d w i t h r e f l e c t i v i t y r = (1, 0, 0, 0, . 5)T. That i s x = (-. 4, 1, . 2,-. 2,-. 2,. 5,. 1,-. l ) r . I f we t a k e f i l t e r l e n g t h nf=3, wavelet l e n g t h , «n'=4 and w a v e l e t r i s e time L=l t h e n , we would form x' = (0, -. 4, 1, . 2, -. 2, - . 2, . 5, . 1,-. 1, 0, 0)J and s t a r t the MED a l g o r i t h m (nf+nw-1), i n t h i s case s i x , t i m e s u s i n g d e s i r e d o u t p u t s 52 d, = [ (-. 4, 1, . 2,-. 2,-. 2, . 5, . 1,-. 1, 0, 0, 0, 0, 0)3V d 2 = I (0,~. 4, 1, . 2,~. 2, -. 2, . 5, . 1, ~. 1,0, 0,0, 0) 3 ]r • d 6 = [(0,0,0,0,0,-. 4, 1, . 2,-. 2,-. 2, .5,. 1,-. I ) 3 ] 1 . The "optimum l a g " MED output i s chosen as t h a t one of the s i x MED runs which y i e l d s the g r e a t e s t varimax o u t p u t . F i g u r e 14 shows the s u c c e s s of t h i s a l g o r i t h m when a p p l i e d t o an i s o l a t e d w a v e l e t . The optimum l a g MED i t e r a t i o n i s shown and a c h i e v e s b oth a h i g h e r varimax output and v i s u a l l y s u p e r i o r d e c o n v o l u t i o n than "MED u s i n g the c o n v e n t i o n a l symmetric s t a r t i n g f i l t e r or any of the f a m i l y of s t a r t i n g f i l t e r s g i v e n by ( 2 . 3 ) . I t i s f u r t h e r noted t h a t t h i s a l g o r i t h m , when a p p l i e d t o the t h r e e p o i n t f i l t e r example of F i g u r e 12 does ind e e d s e l e c t the g l o b a l extremum as marked. In c l o s i n g t h i s s e c t i o n i t i s n o t e d t h a t t h i s optimum l a g a l g o r i t h m i s q u i t e burdensome c o m p u t a t i o n a l l y . However, the optimum l a g i t s e l f depends on w a v e l e t phase. P r o v i d e d the phase does not change to o d r a s t i c a l l y from one r e c o r d t o the next we need o n l y p e r f o r m the f u l l l a g s c a n n i n g on one or two r e c o r d s and then r e s t r i c t the s e a r c h t o l a g s i n the immediate v i c i n i t y of the known optimum. In t h i s way c o m p u t a t i o n c o s t i s h e l d t o a few t i m e s t h a t of W i g g i n s MED. S t i l l e x p e n s i v e , but f o r some p r a c t i c a l a p p l i c a t i o n s t h e r e may w e l l be no s a t i s f a c t o r y a l t e r n a t i v e . F i g u r e For a minimum phase w a v e l e t the opt i t e r a t i o n s t a r t i n g p o i n t , ty which l e a d s w a v e l e t ' s z e r o l a g sample. 14. imum l a g MED a l g o r i t h m s e l e c t s t h a t t o a s p i k e output at t h e o u t p u t 54 4.2 BANDLIMITING THE SOLUTION The d i g i t a l s e i s m i c t r a c e we wish t o d e c o n v o l v e i s b a n d - l i m i t e d and c o n t a i n s n o i s e as w e l l as s i g n a l . I s the MED a l g o r i t h m I have d i s c u s s e d ready t o t a c k l e such data? No. I t i s w e l l u n d e r s t o o d t h a t l e a s t squares f i l t e r c a l c u l a t i o n s r e q u i r e the a d d i t i o n of a r i d g e parameter i n the a u t o c o r r e l a t i o n m a t r i x of e q u a t i o n (2.4) t o s t a b i l i z e the s o l u t i o n of the T o e p l i t z system. T h i s i s done by a d d i n g a " p r e w h i t e n i n g " c o n s t a n t t o the d i a g o n a l elements -t y p i c a l l y l e s s than one p e r c e n t of the d i a g o n a l element i t s e l f . We t h e r e f o r e e xpect the MED a l g o r i t h m t o r e q u i r e a r i d g e parameter. There i s a second problem though, p e c u l i a r t o MED which must be a d d r e s s e d . D e V r i e s and Berkhout (1984) observe t h a t varimax and r e l a t e d norms i n c r e a s e w i t h bandwidth. T h i s i s a problem i n the p r e s ence of h i g h f r e q u e n c y n o i s e where a MED a l g o r i t h m c a l l e d upon t o maximize the varimax or s i m i l a r norm may boost n o i s e a t unwanted f r e q u e n c i e s t o a c h i e v e i t s g o a l . To overcome t h i s p roblem I have adopted the b a n d - l i m i t i n g a l g o r i t h m d e s c r i b e d by Deeming (1980) and o u t l i n e d i n Appendix A. By a d d i n g t o the t r a c e a u t o c o r r e l a t i o n not j u s t a c o n s t a n t a t z e r o l a g but a b a n d - l i m i t e d w e i g h t i n g f u n c t i o n , a t r a d e o f f i s i n t r o d u c e d between m a x i m i z i n g the varimax and s u r p r e s s i n g f i l t e r energy o u t s i d e a s p e c i f i e d pass band. Examples d e m o n s r a t i n g the e f f e c t i v e n e s s of t h i s t e c h n i q u e w i l l be p r o v i d e d i n s e c t i o n 6. 55 4.3 DATA TAPERING AND WINDOW SELECTION I t has been observed i n t h i s work, t h a t poor c h o i c e of the d a t a segment over which the MED o p e r a t o r i s d e s i g n e d can y i e l d d i s a s t r o u s r e s u l t s . Such d i s a s t e r s a r i s e from e i t h e r time c l i p p e d w a v e l e t s a t the b e g i n n i n g or end of the d e s i g n window, or the c o n s t r u c t i v e i n t e r f e r e n c e of c l o s e l y spaced r e f l e c t o r s g o v e r n i n g f i l t e r d e s i g n . The f i r s t problem a r i s e s from the f a c t t h a t a s e i s m i c w a v e l e t , c u t s h o r t a t some p o i n t and r e p l a c e d w i t h z e r o s i s v e r y s p i k y i n d e e d . F i g u r e 15(b) shows the r e s u l t of a f a i l e d MED run which s u c c e s s f u l l y s p i k e d a c l i p p e d w a v e l e t a t the edge of t h e d e s i g n window shown i n F i g u r e 1 5 ( a ) . The u n a c c e p t a b l e d e c o n v o l u t i o n output has a much h i g h e r varimax than the c o r r e c t l y d e c o n v o l v e d t r a c e would y i e l d . T h i s problem i s not unique t o MED, L a z e a r (1984) r e p o r t s t h a t such w a v e l e t fragments pose problems f o r some o t h e r d e c o n v o l u t i o n t e c h n i q u e s as w e l l . Two approaches a r e recommended t o d e a l w i t h the problem d i s c u s s e d above. 1. S e l e c t the time gate c a r e f u l l y w i t h o u t t r u n c a t i n g any l a r g e e v e n t s a t i t s edges. 2. Taper the data w i t h i n the time g a t e . In t h i s paper I have a p p l i e d a t a p e r i n g f u n c t i o n due t o C o l i n Walker ( p e r s o n a l communication) i n which the / ' t h sample i n the d e s i g n gate i s w e i g h t e d by B ( i ) d e f i n e d as B(i) = [4i (m-i)/m2]a where m=ny-l, and a i s s e l e c t e d so t h a t B e q u a l s .5 a t a ( b ) M E D O U T P U T : N O T A P E R I N G F i g u r e 15. ( a ) . A m u l t i c h a n n e l s y n t h e t i c data s e t 'showing normal moveout w i t h t r u n c a t e d r e f l e c t i o n s a t the edge of the time gate ( c i r c l e d ) , ( b ) . MED o u t p u t w i t h no data t a p e r i n g y i e l d s a l a r g e s p i k e on t r a c e 2 ( c i r c l e d ) . ( c ) . MED output w i t h d a t a t a p e r e d as d e s c r i b e d . Known r e f l e c t i v i t i e s r a r e shown a t r i g h t . 57 p o i n t nf/2 samples from the window edge. The e f f e c t i v e n e s s of t h i s window i s shown i n F i g u r e 1 5 ( c ) , a r e p e a t run of the da t a from F i g u r e 15(a) w i t h t a p e r i n g . The second problem I d i s c u s s i n t h i s s e c t i o n i s c o n s t r u c t i v e i n t e r f e r e n c e . C o n s i d e r the two r e f l e c t i v i t y s e r i e s shown i n F i g u r e 16, c o n v o l v e d w i t h t h e same w a v e l e t . S e r i e s B, y i e l d s an e x c e l l e n t MED w a v e l e t i n v e r s e w h i l e s e r i e s A does n o t . Why? Because the c l o s e l y spaced r e f l e c t o r s i n . A l e a d t o c o n s t r u c t i v e i n t e r f e r e n c e among a d j a c e n t w a v e l e t s l e a d i n g MED t o s p i k e the combined w a v e l e t . S i n c e MED a l g o r i t h m s base t h e i r r e f l e c t i v i t y e s t i m a t e s s o l e l y on da t a a m p l i t u d e , the l a r g e a m p l i t u d e e v e n t s t e n d t o c o n t r o l the d e c o n v o l u t i o n . A c c o r d i n g t o M o r l e y (1979) t h i s i s p a r t i c u l a r l y t r u e when l a r g e a m p l i t u d e and h i g h f r e q u e n c y a r e w e l l c o r r e l a t e d . To combat t h i s e f f e c t we must have a d e s i g n window c o n t a i n i n g s e v e r a l l a r g e e v e n t s . Some ways of e n s u r i n g t h i s a r e : 1. A p p l y MED as a m u l t i c h a n n e l d e c o n v o l u t i o n i n c l u d i n g s e v e r a l t r a c e s of a shot r e c o r d i n the f i l t e r design.. I f normal moveout i s p r e s e n t the c o n s t r u c t i v e i n t e r f e r e n c e f o r a g i v e n event w i l l not remain c o n s t a n t a c r o s s the r e c o r d and the a l g o r i t h m i s l e s s l i k e l y t o nominate a d o u b l e t as the t r u e s e i s m i c w a v e l e t . 2. Deeming (1984) has sug g e s t e d a p p l y i n g f a s t a u t o m a t i c g a i n c o n t r o l t o the d a t a b e f o r e MED. He a t t r i b u t e s the apparent s u c c e s s of t h i s t o another argument but i t would c l e a r l y g i v e s m a l l e r , i s o l a t e d r e f l e c t o r s an e q u a l 58 M E D O U T P U T : F i g u r e 16. MED a p p l i e d to two s p a r s e s y n t h e t i c t r a c e s A and B d e r i v e d from r e f l e c t i v i t i e s r , and r 2 r e s p e c t i v e l y . A y i e l d s a poor r e f l e c t i v i t y e s t i m a t e due t o c o n s t r u c t i v e i n t e r f e r e n c e . 59 say i n c o n t r o l l i n g the f i l t e r d e s i g n . 3. Deeming (1981) has f u r t h e r m o r e s u g g ested a p p l y i n g a 1/varimax t r a c e w e i g h t i n g t o the i n p u t d a t a . T h i s would g i v e h i g h varimax t r a c e s l e s s weight i n the MED f i l t e r d e s i g n . The second p o i n t above i s u n d e s i r a b l e i n terms of the c o n v o l u t i o n a l model of the seismogram, and the t h i r d i n t h a t i t d i s c o u n t s what may be v a l u a b l e i n f o r m a t i o n i n the h i g h varimax t r a c e s . Indeed i f the t r a c e varimax i s h i g h due t o an i s o l a t e d r e f l e c t o r i t i s d e s i r a b l e t o l e t i t c o n t r o l the f i l t e r d e s i g n . These l a s t two p o i n t s have however, l e a d t o a more r o b u s t a l g o r i t h m a c c o r d i n g t o Deeming but have not been i n c l u d e d i n t h i s a l g o r i t h m . For the p r e s e n t I have a p p l i e d MED t o d a t a which have had s p h e r i c a l d i v e r g e n c e c o r r e c t i o n a p p l i e d and windows which a r e c a r e f u l l y s e l e c t e d u s i n g the c r i t e r i a d i s c u s s e d . 4.4 SELECTION OF FILTER LENGTH Most of the d i s c u s s i o n thus f a r has assumed t h a t the f i l t e r l e n g t h nf i s known. In p r a c t i c e , i t i s up t o the u s e r t o d e c i d e upon a f i l t e r l e n g t h knowing i t w i l l l i k e l y a f f e c t the q u a l i t y of the d e c o n v o l v e d o u t p u t . Deeming (1984) i n a paper e n t i t l e d "Why Minimum E n t r o p y D e c o n v o l u t i o n Doesn't Work", p r e s e n t s the f o l l o w i n g argument a g a i n s t the use of v e r y l o n g f i l t e r s . Suppose the MED f i l t e r l e n g t h nf were a l l o w e d t o tend t o i n f i n i t y . In t h a t case the l e a s t s q uares f i l t e r o u t p u t y 60 tends t o d u p l i c a t e e x a c t l y i t s d e s i r e d o u t p u t d. We would have y = d = y 3 f o r the va r i m a x . Then y. = y3 i=1,... ny and y. =0,1, or -1 i=l,...ny. A few o u t p u t s t h a t might s a t i s f y t h i s c r i t e r i o n a r e y = ( . . . , 0 , 0 , 0 , 1 , 0 , 0 , 0 , . . . ) y = (... ,i, o.-i, 1,0,0,1 ; or y = (. . . , - 1 , - 1 , 1,-1, 1, 1,1 A l l of the above d e c o n v o l u t i o n o u t p u t s have two t h i n g s i n common, they a r e a l l extrema of the varimax f u n c t i o n a l , and they a r e a l l g e o l o g i c a l l y u n r e a s o n a b l e . I f we i n t e r p r e t the o u t p u t y as an e s t i m a t e of the e a r t h r e f l e c t i v i t y t hen a l l g e o l o g i c b o u n d a r i e s have the same r e f l e c t i o n c o e f f i c i e n t , u n i t y . T h i s i s not t r u e of any b o r e h o l e d e r i v e d r e f l e c t i v i t y sequence and i s seldom t r u e even of s y n t h e t i c d a t a ! Deeming (1981) s u g g e s t s t h a t t h i s i s a major f a i l i n g of the MED t e c h n i q u e o n l y i f v e r y l o n g f i l t e r s a r e used. On the o t h e r hand, v e r y s h o r t f i l t e r s have l a r g e r squared e r r o r s e 2 as d e f i n e d i n ( 3 . 2 ) . I f chosen t o o s h o r t the f i l t e r may not a d e q u a t e l y r e p r e s e n t the wa v e l e t i n v e r s e . U l r y c h e t a l . (1978) have s u g g e s t e d c h o o s i n g a f i l t e r l e n g t h which m i n i m i z e s a c r i t e r i o n of the form 61 ns E(L) = Z {[y.(L)-y.(L-J)r[yi(L)-y.(L-l)]} i=l 1 1 1 1 where yt(L) i s the /' t h t r a c e o utput f o r a MED f i l t e r of l e n g t h L. J u r k e v i c s and Wiggins (1984) s e l e c t f i l t e r l e n g t h by v a r y i n g nf and c h o o s i n g t h a t v a l u e which sees the varimax of the output t r a c e l e v e l o f f as a f u n c t i o n of i n c r e a s i n g f i l t e r l e n g t h . In my examples i n s e c t i o n 6 I use f i l t e r l e n g t h s between 60 and 100 p e r c e n t of the w a v e l e t l e n g t h w i t h good r e s u l t s . The c a u t i o n s of Deeming (1984) whose paper might b e t t e r be e n t i t l e d "Why MED Doesn't Work For Long F i l t e r s " a r e v a l i d , and s e l e c t i o n of a s h o r t f i l t e r of a l e n g t h g i v i n g the most s u b j e c t i v e l y p l e a s i n g outcome seems a s a f e a l t e r n a t i v e . As a f i n a l comment, the n e c e s s i t y f o r u s i n g the s h o r t e s t p o s s i b l e f i l t e r makes optimum l a g s c a n n i n g e s s e n t i a l s i n c e , by a n a l o g y w i t h optimum l a g Weiner f i l t e r i n g , the dependence of f i l t e r performance on o utput l a g i n c r e a s e s w i t h d e c r e a s i n g f i l t e r l e n g t h . 5. THE APPLICABILITY OF MED 5.1 THE MINIMUM ENTROPY ASSUMPTION It was noted in chapter 2.4 that besides poor operator design there can be another, fundamentally d i f f e r e n t , cause for MED f a i l u r e . This cause i s the misapplication of MED to data for which i t i s inherently unsuited. Recall that a r e s t r i c t i v e assumption was adopted to allow us to estimate a r e f l e c t i v i t y from the seismic trace, a highly underdetermined problem. We have assumed that the desired r e f l e c t i v i t y estimate coincides with the global extremum of the varimax or another entropy measure as a functional of f . The r e f l e c t i v i t y estimate y, our seismic deconvolution output, i s v a l i d only i f that asssumption i s true. Like the minimum phase assumption of spiking deconvolution the minimum entropy assumption i s true for some data and not for others. However, unlike the spiking deconvolution case we cannot express, or at least have not yet expressed, the minimum entropy assumption in terms of measurable properties of the seismic trace components. No sat i s f a c t o r y c r i t e r i o n for MED a p p l i c a b i l i t y has yet been proposed. One c r i t e r i o n which I have attempted to apply has f a i l e d and I have opted to try MED on seismograms formed from random r e f l e c t i v i t y series to gain insight into just what "minimum entropy" means. The parameter i n i t i a l l y tested was V(y), the output trace varimax, which would increase with r e f l e c t i v i t y 62 63 s p a r s e n e s s perhaps y i e l d i n g a s a t i s f a c t o r y d e c o n v o l u t i o n above a c e r t a i n t h r e s h o l d v a l u e . I t was found t h a t t h i s parameter does in d e e d i n c r e a s e w i t h the s p a r s e n e s s as ex p e c t e d and r e f l e c t s the q u a l i t y of d e c o n v o l v e d output i n many c a s e s . However, the p r e s e n c e of c l o s e l y spaced, s t r o n g r e f l e c t o r s such as the ones i n F i g u r e 16 can y i e l d o u t p u t s which a r e both h i g h i n varimax and p h y s i c a l l y m e a n i n g l e s s . T h i s and the fr e q u e n c y dependent n a t u r e of the varimax make i t unworkable as a t h r e s h o l d c r i t e r i o n . Our g o a l i n minimum e n t r o p y d e c o n v o l u t i o n i s t o d e s i g n a f i l t e r from the s e i s m i c t r a c e which i s i t s e l f t he MED wa v e l e t i n v e r s e . By MED w a v e l e t i n v e r s e I mean t h a t f i l t e r w hich would best maximize the varimax of an i s o l a t e d w a v l e t . In a l l c a s e s s t u d i e d here t h i s has r o u g h l y , but not e x a c t l y , c o i n c i d e d w i t h the optimum l a g Weiner s p i k i n g f i l t e r f o r t h a t w a v e l e t . I f our d a t a do not s a t i s f y the minimum e n t r o p y a s s u m p t i o n , then we w i l l be moving f u r t h e r away from the MED wa v e l e t i n v e r s e w h i l e s e a r c h i n g f o r a f i l t e r which maximizes the t r a c e v a r i m a x . The f i l t e r s f and f , t h a t i s , w x ' ' the maxima of the varimax f u n c t i o n a l s f o r both seismogram and l o n e w a v e l e t , w i l l c o i n c i d e o n l y when the minimum e n t r o p y assumption i s s a t i s f i e d by the d a t a . C o n s i d e r the f a m i l i a r case w i t h f i l t e r l e n g t h «/=3. The c o n t o u r diagram mapping the varimax s u r f a c e f o r a s i m p l e w a v e l e t w i s shown i n F i g u r e 1 7 ( a ) . Below i t a r e varimax c o n t o u r maps c o n s t r u c t e d u s i n g the same w a v e l e t c o n v o l v e d w i t h (b) some G a u s s i a n i . i . d . n o i s e and (c) a s p a r s e s p i k e s e r i e s . The THETA(RADIPNS) THETR(RflDIRNS) THETR(RfiOIRNS) F i g u r e 17. Varimax of a l l t h r e e p o i n t f i l t e r s c o n t o u r e d f o r the d a t a s e t s x shown, (a) wavelet a l o n e ; (b) g a u s s i a n seismogram; (c) g a u s s i a n a m p l i t u d e s w i t h p o i s s o n ( X = . 3 ) time d i s t r i b u t i o n . tw i s the M E D w a v e l e t i n v e r s e from ( a ) , f j i s the M E D f i l t e r d e s i g n e d from each data s e t . 65 s i m i l a r i t y between the maps f o r lone w a v e l e t and s p a r s e seismogram marks MED's a p p l i c a b i l i t y i n t h i s example. To maximize the varimax of the s e i s m i c t r a c e x i s t o f i n d a f i l t e r which a l s o maximizes the varimax of the s e i s m i c w a v e l e t . The varimax s u r f a c e p o r t r a y e d f o r the G a u s s i a n seismogram (b) however, i s markedly d i f f e r e n t and i t s g l o b a l extremum does . not c o i n c i d e w i t h an a c c e p t a b l e w a v e l e t i n v e r s e as marked by the peaks on the w a v e l e t c o n t o u r . Hence the f i l t e r o u t put i s a poor e s t i m a t e of t h e r e f l e c t i v i t y . In t h i s l a t t e r case MED i s t h e r e f o r e not a p p l i c a b l e . For the case nf=3 t h e n , one can t e s t MED a p p l i c a b i l i t y f o r a v a r i e t y of s y n t h e t i c d ata s e t s by comparing the c o n t o u r e d varimax s u r f a c e s f o r w a v e l e t and seismogram. Furthermore we a r e not r e s t r i c t e d t o the varimax norm i n t h i s case and I have p r e p a r e d c o n t o u r diagrams of the varimax w i t h and w i t h o u t e x p o n e n t i a l t r a n s f o r m , parsimony, D-norm and V ( y , 2 , l ) f o r co m p a r i s o n . The d a t a s e t s used were s y n t h e t i c seismograms formed by c o n v o l v i n g a s i m p l e t h r e e p o i n t w a velet w i t h r e a l i z a t i o n s of the f o l l o w i n g random p r o c e s s e s . 1. U n i f o r m i . i . d . 2. G a u s s i a n i . i . d . 3. Two s i d e d e x p o n e n t i a l i . i . d . 4. G a u s s i a n a m p l i t u d e s P o i s s o n d i s t r i b u t e d i n time w i t h a P o i s s o n parameter X=. 3 5. G a u s s i a n a m p l i t u d e s P o i s s o n d i s t r i b u t e d i n time w i t h a P o i s s o n parameter X=.7. 66 R e a l i z a t i o n s of th e s e random p r o c e s s e s were 250 p o i n t s i n l e n g t h and some examples of these a re shown i n F i g u r e s 18 t o 20. F i g u r e 18 shows the D-norm s u r f a c e s f o r known w a v e l e t , G a u s s i a n seismogram and sp a r s e seismogram (X=.i).Note t h a t the c h a r a c t e r of the s u r f a c e i s q u i t e d i f f e r e n t from t h a t of the varimax norm i n F i g u r e 17 i n d i c a t i n g t h a t the norms are not e q u i v a l e n t . Furthermore n e i t h e r of the g l o b a l maxima c o i n c i d e w i t h the w a v e l e t i n v e r s e f and D-norm MED i s not a p p l i c a b l e i n e i t h e r c a s e . F i g u r e 19 shows the b e h a v i o u r of the parsimony f o r the same seismograms. Note t h a t the parsimony s u r f a c e i s v e r y s i m i l a r i n shape t o the var i m a x , b e i n g i n v e r t e d w i t h peaks r e p l a c i n g v a l l e y s and v a l l e y s , peaks. T h i s i s t o be ex p e c t e d s i n c e the parsimony i s m i n i m i z e d and the rough s i m i l a r i t y of parsimony and varimax s u r f a c e s was observ e d w i t h a l l the d a t a s e t s t e s t e d . A r a t h e r i n c r e d i b l e r e s u l t i s shown i n F i g u r e 2 0 ( b ) . The varimax s u r f a c e f o r a seismogram c o n s t r u c t e d from a u n i f o r m r e f l e c t i v i t y sequence i s i n v e r t e d w i t h r e s p e c t t o the s u r f a c e f o r wavelet a l o n e , F i g u r e 2 0 ( a ) . T h i s r e s u l t i s p r e d i c t e d by Donohoe (1981), as mentioned i n a p r e v i o u s c h a p t e r . Indeed i f our a l g o r i t h m were m o d i f i e d t o m i n i m i z e the varimax norm then MED would s a t i s f a c t o r i l y d e c o n v o l v e t h i s d e c i d e d l y non-sparse u n i f o r m seismogram. F i g u r e 20(c) shows the varimax b e h a v i o u r f o r an e x p o n e n t i a l l y d i s t r i b u t e d seismogram, a g a i n f i l t e r s f and £ f a i l t o c o i n c i d e . The co n t o u r diagrams d i d not suggest t h a t any norm i s F i g u r e 18. Contours on the C a b r e l l i D-norm f o r the d a t a s e t s of F i g u r e 17 The d i s t i n c t l y d i f f e r e n t c h a r a c t e r of the s u r f a c e from t h a t of the varimax i n d i c a t e s the n o n - e q u i v a l e n c e of the norms. o THETR(RRDIPNS) THETR(RRDIflNS) THETR(RRDIRNS) F i g u r e 19. Contours on the parsimony Sz f o r the dat a s e t s of F i g u r e 17. F i g u r e 20 . Contours on the varimax f o r (a) lone w a v e l e t , (b) u n i f o r m l y d i s t r i b u t e d seismogram and (c) e x p o n e n t i a l l y d i s t r i b u t e d seismogram. Note t h a t (b) i s an i n v e r t e d v e r s i o n of ( a ) . 70 u n i v e r s a l l y more a p p l i c a b l e than any other-. In the c o u r s e of s t u d i n g the varimax s u r f a c e s f o r the 3 sample f i l t e r case i t became apparent t h a t the shape of the s u r f a c e and t h e r e f o r e the a p p l i c a b i l i t y of the method depends on the example chosen. In some c a s e s the c o n t o u r diagram would change w i t h d a t a s e t l e n g t h or the p a r t i c u l a r r e a l i z a t i o n of random n o i s e used t o g e n e r a t e the r e f l e c t i v i t y . I t was t h e r e f o r e d e c i d e d t o i n v e s t i g a t e MED b e h a v i o u r over many r e a l i z a t i o n s of the random p r o c e s s e s and f u r t h e r m o r e t o do t h i s u s i n g more r e a l i s t i c w a v e l e t and f i l t e r l e n g t h s . F i g u r e 21 shows one r e a l i z a t i o n of each of the r e f l e c t i v i t i e s , which a r e the same as t h o s e used b e f o r e except P o i s s o n parameters X=. 05', . 1 and . 25 have been used t o account f o r the l o n g e r w a v e l e t . The seismograms c r e a t e d by c o n v o l v i n g t h e s e r e f l e c t i v i t i e s w i t h a 38 sample minimum phase w a v e l e t , and the d e c o n v o l v e d t r a c e s f o r optimum l a g MED w i t h nf=22 a r e a l s o shown. A f t e r r u n n i n g t e n r e a l i z a t i o n s of each r e f l e c t i v i t y sequence I t became e v i d e n t t h a t MED a p p l i c a b i l i t y b e a r s no s i m p l e r e l a t i o n t o r e f l e c t i v i t y pdf f o r samples of t h i s s i z e . In t h i s c a s e , s i n c e the varimax s u r f a c e cannot be v i s u a l i z e d f o r «/=22 I have judged the a p p l i c a b i l i t y by i n s p e c t i o n of the r e s i d u a l w a v e l e t f . MED worked most o f t e n on the d a t a c o n t a i n i n g the s p a r s e s t r e f l e c t i v i t y and l e a s t o f t e n f o r the G a u s s i a n . However, f o r none of the r e f l e c t i v i t y s e r i e s d i d i t e i t h e r succeed or f a i l c o n s i s t e n t l y . F u r t h e r m o r e , i n an attempt t o [01 INPUT R E F L E C T I V I T I E S (CI MED OUTPUTS RES10UOL unvELETS F i g u r e 21. The e f f e c t of r e f l e c t i v i t y p r o b a b i l i t y d i s t r i b u t i o n i s shown w i t h (a) segments of 1. g a u s s i a n , 2. u n i f o r m , 3. e x p o n e n t i a l , 4 . g a u s s i a n / p o i s s o n (X=.05), 5. g a u s s i a n / p o i s s o n (X=.10), and 6. g a u s s i a n / p o i s s o n (X=.25) r e f l e c t i v i t i e s . (b) s y n t h e t i c seismograms and MED o u t p u t s w i t h and w i t h o u t AGC are shown i n (c) and ( d ) . R e s i d u a l w a v e l e t s f^*w from (c) and (d) show no improvement w i t h AGC. 72 q u a n t i f y the a p p l i c a b i l i t y of MED t o a c e r t a i n d a t a s e t i t was found t h a t none of the f o l l o w i n g parameters proved a dependable guide t o MED performance: 1. Input seismogram varimax 2. Deconvolved seismogram varimax 3. R e s i d u a l w a v e l e t varimax 4. The q u a n t i t y f •£ which i s u n i t y when f =f and i s a measure of t h e d i s t a n c e beween f and f on the c o n t o u r x w diagrams of the «/=3 c a s e . A l l of the above were i n d i c a t o r s of good MED performance f o r some d a t a and not f o r o t h e r s . AGC was a p p l i e d t o one d a t a s e t w i t h a v a r i e t y of gate w i d t h s i n hopes t h a t i t might have a p o s i t i v e e f f e c t on the d e c o n v o l u t i o n as Deeming (1984) s u g g e s t s . F i g u r e 21(d) shows an example of one t r a c e from each of the v a r i o u s p d f s d e c o n v o l v e d w i t h an AGC gat e w i d t h of t w i c e the wa v e l e t l e n g t h . AGC has p r o v i d e d no apparent improvement i n t h i s example. Deeming's p o s i t i v e r e s u l t may have been due t o the AGC l e n d i n g s t a t i o n a r i t y t o h i s data which my s y n t h e t i c t r a c e s a l r e a d y e x h i b i t . One c o n c l u s i o n t h a t can be drawn from t h i s work i s t h a t MED a p p l i c a b i l i t y a p p a r e n t l y does not depend s o l e l y on the s t a t i s t i c a l p r o p e r t i e s of the u n d e r l y i n g r e f l e c t i v i t y f u n c t i o n . G i v e n seismograms c o n s t r u c t e d from two r e a l i z a t i o n s of the same random p r o c e s s one may w e l l f i n d t h a t MED works on one and not the o t h e r . T h i s i s due t o w a v e l e t s i n t e r f e r i n g i n such a way as t o s h i f t the g l o b a l 73 varimax extremum away from the t r u e r e f l e c t i v i t y r e g a r d l e s s of the s p a r s e n e s s , s i m p l i c i t y or non-Gaussian n a t u r e of t h a t r e f l e c t i v i t y . The minimum e n t r o p y a s s u m p t i o n , t h a t the t r u e r e f l e c t i v i y l i e s a t the extremum of an e n t r o p y measure such as the varimax i s not j u s t a r e s t r i c t i o n on the r e f e c t i v i t y s t a t i s t i c s or even the r e f l e c t i v i t y model. I t i s a l s o a r e s t r i c t i o n on the s e i s m i c w a v e l e t t h r o u g h the way i n which the two i n t e r a c t c o n s t r u c t i v e l y t o g i v e v e r y l a r g e e v e n t s . F u r t h e r m o r e , as e v i d e n c e d by F i g u r e 21, i n which MED works on some w h i t e n o i s e r e f l e c t i v i t i e s i t can be added t h a t the concept of " s p a r c e n e s s " or even " s i m p l i c i t y " does not a d e q u a t e l y e x p r e s s the minimum e n t r o p y a s s u m p t i o n . 6. EXAMPLES USING SYNTHETIC DATA 6.1 INTRODUCTION In t h i s s e c t i o n I s h a l l t e s t the optimum l a g MED r o u t i n e d e s c r i b e d i n the p r e v i o u s s e c t i o n s and compare r e s u l t s w i t h Wiggins MED u s i n g symmetric s t a r t i n g f i l t e r . The d a t a used i n t h e s e t e s t s i s a m u l t i c h a n n e l s y n t h e t i c d a t a s e t k i n d l y s u p p l i e d by Andy J u r k e v i c s and were i n f a c t used i n a r e c e n t paper f o r comparison of d e c o n v o l u t i o n t e c h n i q u e s ( J u r k e v i c s and Wiggins,, 1984). The d a t a c o n s i s t of a r e f l e c t i v i y f u n c t i o n whose e v e n t s have a m p l i t u d e s drawn from a G a u s s i a n p r o b a b i l i t y d i s t r i b u t i o n and occur P o i s s o n d i s t r i b u t e d i n time w i t h a P o i s s o n parameter X=.7. A t w e l v e t r a c e d a t a s e t has been g e n e r a t e d from the r e f l e c t i v i t y w i t h s i m u l a t e d normal moveout and i s c o n v o l v e d w i t h a minimum phase w a v e l e t . In the n o i s y examples random G a u s s i a n n o i s e has been g e n e r a t e d , b a n d l i m i t e d t o l e s s than 60 Hz and added t o the t r a c e s . In a l l c a s e s a f i l t e r l e n g t h of n/=22 was used. In f i l t e r l e n g t h t e s t s on n o i s e - f r e e d a t a t h i s was judged the s h o r t e s t o p e r a t o r y i e l d i n g s a t i s f a c t o r y w a v e l e t e s t i m a t e s . I had o r i g i n a l l y p l a n n e d t o i n c l u d e the C a b r e l l i a l g o r i t h m i n t h e s e t e s t s and an a l g o r i t h m was w r i t t e n f o r t h a t p urpose. As p u b l i s h e d however, C a b r e l l i MED r e q u i r e s (ns x nx) t i m e s the c o m p u t a t i o n a l e f f o r t of s p i k i n g d e c o n v o l u t i o n . Computing c o s t was p r o h i b i t i v e f o r a n y t h i n g but s i m p l e s i n g l e t r a c e examples, and u n l e s s s i g n i f i c a n t 74 75 o p t i m i z a t i o n of t h a t a l g o r i t h m can be a c h i e v e d i t s use as an a l t e r n a t i v e t o i t e r a t i v e MED i s i n doubt. 6.2 THE EFFECT OF OPTIMUM LAG SCANNING An example of optimum l a g MED v e r s u s Wiggins MED w i t h symmetric s t a r t i n g f i l t e r i s shown i n F i g u r e 22. F i g u r e 22(a) shows the da t a s e t used f o r the t e s t , a n o i s e - f r e e m u l t i c h a n n e l s y n t h e t i c shot r e c o r d . The minimum phase s e i s m i c w a v e l e t a p p r o x i m a t e l y 34 samples l o n g i s d i s p l a y e d below the r e c o r d . MED o u t p u t s f o r nf=22 w i t h and w i t h o u t l a g sc a n n i n g f o r the g l o b a l extremum a r e shown i n 22(b) and 22(c) r e s p e c t i v e l y . Wavelet e s t i m a t e s from each MED f i l t e r a r e p l o t t e d below w i t h optimum l a g MED y e i l d i n g a s u p e r i o r r e s u l t a l t h o u g h i t does c o n t a i n a s m a l l p r e c u r s o r . No s i g n i f i c a n t improvement i n the w a v e l e t e s t i m a t e i s o b t a i n e d by i n c r e a s i n g the f i l t e r l e n g t h , a t l e a s t up t o NF=55 the l o n g e s t f i l t e r t e s t e d . In the example shown, the optimum l a g MED ou t p u t i s normal i n p o l a r i t y w h i l e the p o l a r i t y of the example on the r i g h t i s r e v e r s e d . T h i s i s not the case i n g e n e r a l however , s i n c e even the optimum l a g a l g o r i t h m can come t o r e s t i n e i t h e r of the g l o b a l extrema c o r r e s p o n d i n g t o the two p o l a r i t i e s . F i n a l varimax of the optimum l a g output i s s i g n i f i c a n t l y h i g h e r than t h a t of the c o n v e n t i o n a l MED a l g o r i t h m . ( a ) I n p u t d a t a ^ / - A ^ f — — f — * ^ h ^ — V V - ^ ^ ^ ^ W f ( b ) O p t i m u m l a g M E D — ^ v ^ V — — ^ — * M > Y p " ^ *y^ "^ ii>*"^ Y^  1^**^"—t^^y^—If1—^HTYp1 - X L - y J . py\ J^Xy ' V = . 0 2 1 ( c ) W i g g i n s M E D 0 f = ( 0 , . . . 1 , . . . 0 ) r: V = . 0 1 8 F i g u r e 22 . The e f f e c t of optimum l a g sca n n i n g MED i s demonstrated u s i n g ( l e f t ) s y n t h e t i c m u l t i c h a n n e l data s e t c o n t a i n i n g the w a v e l e t shown, ( c e n t r e ) optimum l a g MED output w i t h r e f l e c t i v i t y p l o t t e d a t t o p and e x t r a c t e d wavelet below and ( r i g h t ) c o r r e s p o n d i n g o u t p u t f o r MED w i t h symmetric s t a r t i n g f i l t e r . 77 6.3 THE EFFECT OF ADDITIVE NOISE The e f f e c t of n o i s e on the b a n d l i m i t e d optimum l a g MED a l g o r i t h m i s demonstrated below ,again u s i n g the d a t a s e t s from the p r e v i o u s s e c t i o n . To t h e s e d a t a I have added 5,10 and 20 p e r c e n t G a u s s i a n n o i s e b a n d l i m i t e d between 0 and 60 Hz. The n o i s y d a t a s e t s and the s e i s m i c w a v e l e t they c o n t a i n are shown i n F i g u r e 23. To t h e s e d a t a I have a p p l i e d optimum l a g MED w i t h .01 p e r c e n t p r e w h i t e n i n g . The r e s u l t i n g o u t p u t s a l o n g w i t h w a v e l e t s e s t i m a t e d from the MED f i l t e r s a r e shown i n F i g u r e 24 w i t h the r e f l e c t i v i t y p l o t t e d a t the t o p of each s e c t i o n . I t can be seen t h a t f o r both the 5 and 10 p e r c e n t n o i s e c a s e s good r e s u l t s a r e a c h i e v e d and a r e a s o n a b l e wavelet e s t i m a t e , a l t h o u g h the p r e c u r s o r i s s t i l l p r e s e n t and the e s t i m a t e seems t o be d e g r a d i n g w i t h i n c r e a s i n g n o i s e . For the 20 p e r c e n t n o i s e case however, MED has opted t o p l a c e a s p i k e a t the b e g i n n i n g of the d a t a s e t d e s p i t e the d a t a t a p e r i n g a p p l i e d as d i s c u s s e d i n s e c t i o n 4.3. The f i l t e r , r i c h i n h i g h f r e q u e n c i e s does l i t t l e t o a t t e n u a t e n o i s e i n the d a t a and f u r t h e r m o r e y i e l d s a poor wa v e l e t e s t i m a t e . Next I a p p l i e d optimum l a g MED t o the same d a t a s e t but r e p l a c e d the p r e w h i t e n i n g or r i d g e parameter w i t h the b a n d l i m i t i n g a l g o r i t h m d e s c r i b e d i n Appendix A. Output seismograms a r e d i s p l a y e d i n F i g u r e 25. The parameters used were: 1. Passband 0-50 Hz 2. X=.025 and 5% 1 0 % 2 0 % 3 4 5 6 0.0 0.2 0.4 —I 1 1 — H 1 h 0.0 0.2 0.4 —I 1 1— H h F i g u r e 23. S y n t h e t i c data s e t s and w a v e l e t s used i n the n o i s e examples. oo 79 5% r: 3 4 5 6 -t • 1-——— -I———-—-I— -H H 1 0 % r: 3 * 5 6 -I— I— [• .— —h-H —I —— 1 I 2 0 % 3 * 5 6 _£ , 1 —+-—, f——. 1 • (-niNinun PHflSE-NEUtlEO LG*22 ITER=I0 NOISE=0.05 n iN inun PHASE-NEUtlEO LG-22 ITER-IO NOISE-OJO n iN inun PHASE-NEUMEO IG-22 ITER-IO NOISE-0.20 F i g u r e 25. optimum l a g MED ou t p u t s and e x t r a c t e d w a v e l e t s f o r the dat a s e t s of F i g u r e 23. u s i n g the b a n d l i m i t i n g a l g o r i t h m d e s c r i b e d i n Appendix A. oo o 81 3. Q=.001. The d e c o n v o l v e d o u t p u t s f o r the 5 and 10 p e r c e n t added n o i s e c a s e s are comparable t o the p r e v i o u s f i g u r e , however the w a v e l e t e s t i m a t e s a r e not u s e a b l e . The h i g h frequency n o i s e l e v e l i n t h e s e output t r a c e s i s lower than i n the p r e v i o u s example and the 20% n o i s e example i s d e c i d e d l y s u p e r i o r t o t h a t of the p r e v i o u s f i g u r e a l t h o u g h s e r i o u s l y degraded compared t o the 10% c a s e . Why has the b a n d l i m i t i n g a l g o r i t h m a p p a r e n t l y f a i l e d ? F i r s t i t s h o u l d be noted t h a t the w a v e l e t e s t i m a t e here i s not a good i n d i c a t o r of d e c o n v o l u t i o n performance. Simply e s t i m a t e d by frequency domain i n v e r s i o n of the f i l t e r i n the e s t i m a t e d passband of the w a v e l e t , i t assumes a d e l t a - t y p e output w a v e l e t . T h i s i s i n a p r o p r i a t e s i n c e I have sought a b a n d l i m i t e d o u t p u t . The b a n d l i m i t i n g a l g o r i t h m has a c h i e v e d i t s g o a l of s u r p r e s s i n g f i l t e r energy o u t s i d e the s p e c i f i e d f r e q u e n c y band. T h i s i s demonstrated i n F i g u r e 26. Here power s p e c t r a of the MED f i l t e r s f o r the 10 p e r c e n t n o i s e case a r e p l o t t e d , ( a ) : w i t h .01 p e r c e n t w h i t e n i n g and ( b ) : w i t h b a n d l i m i t i n g as d e s c r i b e d . As e x p e c t e d , the b a n d l i m i t i n g has done i t s j o b of e l i m i n a t i n g f r e q u e n c i e s above 50 Hz and would t h e r e f o r e reduce the n o i s e of the output t r a c e . However, the shape of t h e s p e c t r a w i t h i n the passband d i f f e r s . T h i s i s a r e s u l t of the b a n d l i m i t i n g a l g o r i t h m m i n i m i z i n g the sum of the squared e r r o r and the f i l t e r energy o u t s i d e the passband. T h i s i n e f f e c t c r e a t e s a . 0 1 % P r e w h i t e n e d M E D 3 FREQUENCY B a n d l i m i t e d M E D 4 0 20 40 60 80 IQO I20 140 FREQUENCY F i g u r e 26. Power s p e c t r a of MED f i l t e r s d e s i g n e d from the same da t a s e t w i t h and w i t h o u t b a n d l i m i t i n g . Shown are the f i l t e r s p e c t r a from the 10% n o i s e examples of f i g u r e s 23 and 24. 83 t r a d e o f f between m a x i m i z i n g t h e varimax and b a n d l i m i t i n g the f i l t e r and i n some examples caused the MED a l g o r i t h m t o d i v e r g e . I f such an a l g o r i t h m were t o s i m u l t a n e o u s l y maximize varimax w h i l e m i n i m i z i n g the unwanted f i l t e r f r e q u e n c i e s we would have a d i f f e r e n t , and p r e f e r a b l e r e s u l t . As f o r m u l a t e d , the b a n d l i m i t i n g scheme of Deeming (1981), when a p p l i e d t o MED i s q u i t e dependent on the amount of w e i g h t i n g X g i v e n the h i g h f r e q u e n c y f i l t e r energy and l e a v e s the s u c c e s s of any MED a l g o r i t h m dependent upon X. 7. DISCUSSION Minimum e n t r o p y d e c o n v o l u t i o n has the p o t e n t i a l t o answer some of the p e r e n n i a l l y u n s o l v e d problems i n d e c o n v o l u t i o n such as how t o d e a l e f f e c t i v e l y w i t h nonminimum phase w a v e l e t s and non-white r e f l e c t i v i t i e s . MED i s also-known t o break down i n p r a c t i c a l a p p l i c a t i o n . MED i s not r o b u s t and t h i s p r o p e r t y i s o f t e n c o n s i d e r e d t o be i n h e r e n t i n the method. However, i n t h i s work I have s t r e s s e d t h a t t h e r e a r e two d i s t i n c t t y p e s of problem t h a t may be r e s p o n s i b l e f o r t h i s f a i l i n g . The f i r s t problem i s method of MED o p e r a t o r c a l c u l a t i o n ; the second i s j u d g i n g when t o a p p l y MED. These two problems must be kept s e p a r a t e i n the minds of the s t a t i s t i c a l d e c o n v o l u t i o n u s e r , f o r no t e c h n i q u e , no m a t t e r how r o b u s t , w i l l y i e l d the r i g h t outcome i f a p p l i e d i n a s i t u a t i o n f o r which i t s assumptions do not h o l d . Where then do the assumptions of MED h o l d ? For which a p p l i c a t i o n s was MED d e s i g n e d ? These d i s a r m i n g l y s i m p l e q u e s t i o n s have y e t t o be answered, and u n t i l they a r e we can n e i t h e r s e t a c r i t e r i o n f o r when t o a p p l y MED nor d e c l a r e t h a t i t i s the wrong model f o r our s e i s m i c t r a c e . My examples suggest t h a t a s p a r s e , or even non-Gaussian r e f l e c t i v i t y does not a d e q u a t e l y d e s c r i b e the seismogram t h a t MED r e q u i r e s . We must f u r t h e r i n v o k e r e s t r i c t i o n s i n v o l v i n g the w a v e l e t and i t s c o n s t r u c t i v e i n t e r a c t i o n w i t h the r e f l e c t i v i t y f u n c t i o n . 8 4 85 Some p r o g r e s s has been made i n the a r e a of MED o p e r a t o r c a l c u l a t i o n . I t i s now u n d e r s t o o d t h a t f i l t e r l e n g t h s s h o u l d be kept as s h o r t as p o s s i b l e , t h a t c a r e must be taken t o a v o i d end e f f e c t s and t h a t v a r y i n g the t r a c e w e i g h t i n g and method of summation can improve r e s u l t s . B a n d l i m i t i n g i s c l e a r l y a n e c e s s a r y component of a MED f i l t e r d e s i g n but i t i s not y e t c l e a r i f the a l g o r i t h m proposed by Deeming (1981) i s the b e s t way t o implement t h i s . I have d e a l t , t h r o u g h a h e u r i s t i c g e o m e t r i c a l approach, w i t h the problem of m u l t i p l e extrema. T h i s work s u g g e s t s t h a t the c h o i c e of s t a r t i n g f i l t e r a f f e c t s the output s p i k e l a g and i s t h e r e f o r e tantamount t o making a wa v e l e t phase as s u m p t i o n . T h i s can be overcome w i t h the optimum l a g MED r o u t i n e proposed i n s e c t i o n 4.1. T e r r y Deeming ( p e r s o n a l communication) makes use of sideways r e c u r s i o n i n s o l u t i o n of the MED normal e q u a t i o n s i n an a l t e r n a t i v e attempt t o a c h i e v e the g l o b a l extremum. Comparison of h i s t e c h n i q u e , which i s u n p u b l i s h e d , w i t h optimum l a g MED would be i n f o r m a t i v e . The q u e s t i o n f i n a l l y a r i s e s whether MED i s y e t a p r a c t i c a l a l t e r n a t i v e t o the d e c o n v o l u t i o n t e c h n i q u e s i n use today. The answer seems t o be, "Only f o r the d a t a t o which MED i s p a r t i c u l a r l y s u i t e d . " U n f o r t u n a t e l y , t o t h i s d a t e , we can o n l y d e t e r m i n e t o which d a t a MED i s s u i t e d by t r y i n g i t on t h a t d a t a , h a r d l y an a c c e p t a b l e t e s t g i v e n the o b s e r v a t i o n t h a t MED i s a p p a r e n t l y not s u i t e d t o most d a t a . The key then i s i n g a i n i n g an u n d e r s t a n d i n g of the minimum e n t r o p y a s s u m p t i o n . Then, w i t h a c r i t e r i o n f o r a p p l i c a t i o n 86 and t h e advances i n o p e r a t o r d e s i g n t h a t w i l l o n l y come from i s o l a t i n g the c o m p u t a t i o n a l problems, MED may r e a c h i t s p o t e n t i a l . REFERENCES C h a c i n , O.E., 1985, Wavelet e s t i m a t i o n i n normal i n c i d e n c e seismograms: Ph.D. T h e s i s , U n i v e r s i t y of T u l s a . C l a e r b o u t , j . , 1980 ,Seven e s s a y s on minimum e n t r o p y : S t a n f o r d E x p l o r a t i o n P r o j e c t Report ,24,157-187. C l a e r b o u t , J . , 1977a, P a r s i m o n i o u s d e c o n v o l u t i o n : S t a n f o r d E x p l o r a t i o n P r o j e c t R e p o r t , 1 5,1-9. C l a e r b o u t , J . , 1977b ,Examples of p a r s i m o n i o u s d e c o n v o l u t i o n : S t a n f o r d E x p l o r a t i o n P r o j e c t R e p o r t , 1 5,10-32. Deeming, T.J., 1981, D e c o n v o l u t i o n and r e f l e c t i o n c o e f f i c i e n t e s t i m a t i o n , u s i n g a g e n e r a l i z e d minimum e n t r o p y p r i n c i p l e : 51st Annual I n t e r n a t i o n a l SEG M e e t i n g , October 13, 1981, Los A n g e l e s . Deeming, T . J . , 1984, Why minimum e n t r o p y decon doesn't work: SEG Decon Workshop, J u l y , 1984, V a i l C o l o r a d o . De Voogd, N., 1974, Wavelet sh a p i n g and n o i s e r e d u c t i o n : Geophys. P r o s p . , 2 2,354-369. D e V r i e s , D. and A . J . B e r k h o u t , 1984, V e l o c i t y a n a l y s i s based on minimum e n t r o p y : G e o p h y s i c s , 4 9,2132-2142. Donohoe, D., 1981, On minimum e n t r o p y d e c o n v o l u t i o n , i n F i n d l e y , D., Ed., A p p l i e d Time S e r i e s A n a l y s i s I I , Academic P r e s s . F e r g u s o n , G.A., 1954, The concept of parsimony i n f a c t o r a n a l y s i s : P s y c h o m e t r i k a , 1 9,281-290. G o d f r e y , R.J., 1979, A s t o c h a s t i c model f o r seismogram a n a l y s i s : Ph.D. T h e s i s , S t a n f o r d U n i v e r s i t y . G o d f r e y , R.J., and F.Rocca, 1981, Zero memory n o n - l i n e a r 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 , 2 9,189-228. G o n z a l e z - S e r r a n o , A., 1980, Minimum-entropy d e c i s i o n a n a l y s i s : S t a n f o r d E x p l o r a t i o n P r o j e c t R e p o r t , 2 4,199-226. Gray, W.C., 1979, V a r i a b l e norm d e c o n v o l u t i o n : Ph.D. T h e s i s , S t a n f o r d U n i v e r s i t y . Hogg, Robert V., 1972, More l i g h t on the k u r t o s i s and r e l a t e d s t a t i s t i c s : - J . of the Am. S t a t i s t i c a l A s s o c . 67,422-424. 87 8 8 Hosken, J.W.J., 1980, A s t o c h a s t i c model of s e i s m i c r e f l e c t i o n s : 50th Annual I n t e r n a t i o n a l SEG M e e t i n g , November 16-20, 1980, Houston. J u r k e v i c s , A. and R.Wiggins, 1984, A c r i t i q u e of s e i s m i c d e c o n v o l u t i o n methods: G e o p h y s i c s , 49,2109-2116. K a i s e r , H.F., 1958, The varimax c r i t e r e o n f o r a n a l y t i c r o t a t i o n i n f a c t o r a n a l y s i s : P s y c h o m e t r i k a , 23,187-200. L a z a e r , G.D., 1984, An e x a m i n a t i o n of the e x p o n e n t i a l decay method of mixed phase w a v e l e t e s t i m a t i o n : G e o p h y s i c s , 49,2094-2099. M o r l e y , L., 1980, Minimum e n t r o p y d e c o n v o l u t i o n w i t h the e x t r i n s i c power norm: S t a n f o r d E x p l o r a t i o n P r o j e c t R e p o r t , 24 , 189-198. Ooe, M. and T . J . U l r y c h , 1979, Minimum e n t r o p y d e c o n v o l u t i o n w i t h an e x p o n e n t i a l t r a n s f o r m a 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 , 27 ,458-473. P o s t i c , A., J.Fourmann and J . C l a e r b o u t 1980, P a r s i m o n i o u s d e c o n v o l u t i o n : 50th Annual I n t e r n a t i o n a l SEG M e e t i n g , November 16-20, 1980, Houston. R o b i n s o n , E.A. and S . T r e i t e l , 1980, G e o p h y s i c a l S i g n a l A n a l y s i s : P r e n t i c e - H a l l I n c . . Rob i n s o n , E.A. 1954, P r e d i c t i v e d e c o m p o s i t i o n of time s e r i e s w i t h a p p l i c a t i o n t o s e i s m i c e x p l o r a t i o n : Ph.D. d i s e r t a t i o n , M.I.T.: R e p r i n t e d i n G e o p h y s i c s , 32,418-484. Schoenberger, M., 1985, S p e c i a l r e p o r t on the s e i s m i c d e c o n v o l u t i o n workshop, sponsored by the SEG Res e a r c h Committee, J u l y 18 - 1 9 , 1984, V a i l C o l o r a d o : G e o p h y s i c s , 50,715-735. U l r y c h , T . J . , C.J.Walker and A . J . J u r k e v i c s , 1978, A f i l t e r l e n g t h c r i t e r i o n f o r minimum e n t r o p y d e c o n v o l u t i o n : CSEG J o u r n a l , 14,21-28. U l r y c h , T . J . , 1982, A n a l y t i c minimum e n t r o p y d e c o n v o l u t i o n : G e o p h y s i c s , 47,1295-1302. Walden, A.T., 1895, Non-gaussian r e f l e c t i v i t y , e n t r o p y and d e c o n v o l u t i o n : G e o p h y s i c s , 50,2862-2888. W i g g i n s , Ralphe A., 1978, Minimum e n t r o p y 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 , 16,21-35. Z i o l k o w s k i , Anton, 1984, D e c o n v o l u t i o n : IHRDC P r e s s . APPENDIX A; BANDLIMITING THE FILTER OPERATION In t h i s appendix I d i s c u s s the t h e o r y and p r a c t i c a l a p p l i c a t i o n of pass-band c o n s t r a i n t s on the d e c o n v o l u t i o n o p e r a t o r . T h i s scheme i s based on an as y e t u n p u b l i s h e d d i s c u s s i o n by Deeming (1981) and can be a p p l i e d i n s t e a d of a w h i t e n i n g parameter i n any weiner f i l t e r d e s i g n p r o c e s s . The commonly used t e c h n i q u e of a d d i n g a r i d g e parameter, or w h i t e n o i s e t o s t a b i l i z e the f i l t e r d e s i g n p r o c e s s i s shown to be a s p e c i a l case of t h i s t e c h n i q u e . The w e l l known Weiner l e a s t squares f i l t e r d e s i g n p r o c e s s f i n d s the i n v e r s e f i l t e r f f o r a w a v e l e t w which m i n i m i z e s the squared e r r o r e 2 between the f i l t e r o u t put and a d e s i r e d output d. T h i s squared e r r o r i s d e f i n e d as nd nf e 2 = Z (d.~ Z f.w,.)1. (A1 ) i =1 j = 1 M i n i m i z i n g e 2 w i t h r e s p e c t t o the f i l t e r c o e f f i c i e n t s /£,k=l,... nf l e a d s t o the normal e q u a t i o n s yk Z »t-j»t-k - ? »t-j J-i.--.nf. (A2) k i i J These e q u a t i o n s a r e s o l v e d f o r the f i l t e r c o e f f i c i e n t s u s i n g L e v i n s o n r e c u r s i o n . In the p r a c t i c a l case however, when w may be n o i s y and band l i m i t e d t h e a u t o c o r r e l a t i o n m a t r i x of the w a v e l e t may be i l l c o n d i t i o n e d . In t h i s c a s e we s o l v e the s t a b i l i z e d normal e q u a t i o n s : nd Zfk( Z Vt-jVi-k-Mjk? = L t d w .. k i J J i = 1 J Here the a d d i t i o n of a " r i d g e parameter" e t o the z e r o l a g 89 90 a u t o c o r r e l a t i o n s t a b i l i z e s the r e c u r s i o n and i s e q u i v a l e n t t o " p r e w h i t e n i n g " the w a v e l e t by the a d d i t i o n of a s m a l l amount of w h i t e n o i s e t o ensure t h a t i t s spectrum i s nowhere z e r o . W h i l e the normal e q u a t i o n s (A2) m i n i m i z e e 2 i t i s easy t o show t h a t normal e q u a t i o n s (A3) m i n i m i z e a r e l a t e d q u a n t i t y s where nf s = e 2 + X Z f\. (A4) k=l K T h i s can be shown by d i f f e r e n t i a t i n g s w i t h r e s p e c t t o the f i l t e r c o e f f i c i e n t s : dfk b f k I 1 nd = 2t l / d r y j w j - t ) w k - i + 2 V k = [Zd.w, .-Zf.Z w. .w, . ] + 2 X / i 5 . . . . i k-i . j . j -1 k-i k j k ' J i Now by s e t t i n g the d e r i v a t i v e e q u a l t o z e r o t o m i n i m i z e s we o b t a i n the d e s i r e d r e s u l t , e q u a t i o n s ( A3). J  1 J i An i n t e r e s t i n g i n t e r p r e t a t i o n of t h i s r e s u l t i s p o s s i b l e i f we i n t r o d u c e the d e s c r e t e f o u r i e r t r a n s f o r m (DFT) F of f d e f i n e d as F. = "z f_e-^nk^jM ( A 5 ) * j = l J w i t h c o r r e s p o n d i n g i n v e r s e t r a n s f o r m 91 /. = (J/n/z F r2***"'*. (A6) J k=0 K P a r s e v a l ' s theorem then s t a t e s t h a t the average f i l t e r power may be decomposed i n t o c o n t r i b u t i o n s from each harmonic. ( 7 / 1 1 / 2 P. - " E | F , | 2 y = 7 J k=0 K where | F , | 2 = F , F * the f i l t e r energy a t f r e q u e n c y kAv. K K K T h i s a l l o w s us t o r e w r i t e (A4) as n s = e 2 + Xn Z | F . | 2 k=0 K and c o n c l u d e t h a t m i n i m i z i n g t h i s q u a n t i t y too w i l l l e a d t o the s t a b i l i z e d normal e q u a t i o n ( A 3 ) . We can now i n t e r p r e t the r o l e of the r i d g e parameter X i n e q u a t i o n s ( A 3 ) . By d e s i g n i n g an o p e r a t o r f w i t h t h e s e e q u a t i o n s we a r e s e e k i n g t h a t o p e r a t o r which m i n i m i z e s the squared e r r o r X 2 p l u s some f r a c t i o n of the f i l t e r energy a t a l l f r e q u e n c i e s . That f r a c t i o n i s X, the t r a d e o f f or r i d g e parameter. There a r e two extreme cases 1. X=0: In t h i s case the f i l t e r o u t p u t w i l l b e s t r e p r o d u c e the d e s i r e d o u t p u t . However, the f i l t e r e nergy, and hence the d e c o n v o l v e d o u t p u t v a r i a n c e i s u n c o n s t r a i n e d . I f w i s b a n d l i m i t e d and n o i s y the f i l t e r may a m p l i f y n o i s e o u t s i d e t h i s f r e q u e n c y band r e n d e r i n g the out p u t n o i s y i f not u s e l e s s . 2. X>>7: In t h i s case the f i l t e r power i s the dominant term i n t he e r r o r measure s and by m i n i m i z i n g t h i s a f i l t e r 92 w i t h the l e a s t energy a t a l l f r e q u e n c i e s , w i l l be d e s i g n e d independent of the w a v e l e t . Between the s e extreme c a s e s a s o l u t i o n f i s found which tends t o m i n i m i z e e2 w h i l e l i m i t i n g the f i l t e r power a t a l l f r e q u e n c i e s . The q u e s t i o n then a r i s e s , "Why not weight c e r t a i n f r e q u e n c i e s i n the f i l t e r power of e q u a t i o n A so t h a t the f i l t e r energy i s s u p r e s s e d o n l y o u t s i d e the s p e c t r a l passband of i n t e r e s t ? " Our new e r r o r c r i t e r i o n would be n s' = e 2 + Xn S f i j F . | 2 (A8) k=0K K where Q^, k=0,.. . n i s a w e i g h t i n g f u n c t i o n t a k i n g on v a l u e s of u n i t y o u t s i d e , and near z e r o i n s i d e , t h e f r e q u e n c y band of i n t e r e s t . In o r d e r t o implement t h i s c r i t e r i o n we must f i r s t r e w r i t e the weigh t e d f i l t e r energy i n (A8) n I G J F . | 2 = n I Q . F , * F , ^ /v Iv IV /V /V = LQA(Z f . e i 2 i r k A v j A t ) ( L f . e i 2 l ( k L v l L t ) } k K j J I 1 ^ J X f f i q i j I where q . , = n Z Q i ^ k L v ( j - I ) L t ( g ) J1 k k q i s an nfxnf m a t r i x whose j ' t h row c o n s i s t s of n2 t i m e s 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 the window f u n c t i o n c e n t e r e d on the d i a g o n a l . Now r e w r i t i n g the new e r r o r c r i t e r i o n as s' = e 2 +XLZ f . f . q , , j l J 1 J1 and d i f f e r e n t i a t i n g w i t h r e s p e c t t o the f i l t e r c o e f f i c i e n t s : 93 ljk--bk[Z (diJjWi-J)2 + ^ «Jlf/l] i j  j J J "ds Now s e t t i n g T ?  = 0 *=/,...«/ gives a j k 0 = Z d.w. ,-ZZ w. .w. ,/.+XZ o,./,/. / i - k . . i -1 i — k I k i k i i i j J j or Z f (Z "i-jVi-j+Mk,) = Z dtw._k k=l,...nf. (A10) j i i These a r e the new b a n d - l i m i t e d normal e q u a t i o n s which a g a i n must be s o l v e d f o r f i l t e r c o e f f i c i e n t s / . Note the s i m i l a r i t y between the s e and e q u a t i o n s (A3) w i t h r i d g e p arameter. I n s t e a d of ad d i n g a c o n s t a n t t o the 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 m a t r i x we add the m a t r i x q. Indeed i f our s p e c t r a l w e i g h t s a r e chosen t o be c o n s t a n t the b a n d - l i m i t e d normal e q u a t i o n s r e v e r t t o e q u a t i o n s ( A 3 ) . I t i s w e l l known (Robinson and T r e i t e l , 1980) t h a t the a u t o c c o r r e l a t i o n m a t r i x i s p o s i t i v e s e m i d e f i n i t e . I w i l l show below t h a t the m a t r i x q and hence t h e i r sum a l s o has t h i s p r o p e r t y e n a b l i n g the s o l u t i o n of e q u a t i o n (A9) by L e v i n s o n r e c u r s i o n . A M a t r i x A i s p o s i t i v e , s e m i d e f i n a t e , i f f o r any r e a l v e c t o r x we may w r i t e x TAx > 0 E v a l u a t i n g 94 : Tqx = LL xrqr x s s r s = LI x x n*L Q t 2 w k L v ( r - s ) L t rs r s k k = n2L Q.(Z x e i 2 n k L v r L t ) ( J . x gi 2*kLvsLt} k r s = n2Z Q,X,X* i A- / v / v k = n2L Qk\Xk\2 k > 0 i f Qk>0 f o r a l l it. That i s , the m a t r i x q i s and hence the b a n d - l i m i t e d a u t o c o r r e l a t i o n m a t r i x i s p o s i t i v e s e m i d e f i n i t e as l o n g as the w e i g h t i n g f u n c t i o n Qk i s nowhere n e g a t i v e . 9 5 In t h i s thesis a subroutine was used which implemented a spectral weighting function of the form. 1 v, <\v ,\<vu. / i / i r I o ' k' hi c otherwise (c«l). The bandlimiting matrix q can then be calculated using equation (A9) In t h i s case however, i t was calculated a n a l y t i c a l l y as i f sampled from a continuous spectrum. Then equation (A9) becomes ~2 r " rs/ \ ' 2 i t v ( j - l ) At , 1-1 = n S Q(v)e KJ ' dv J - v n = n2 [ v ns i nc ( 2itv (j -1) At) + (c- 1) s i nc(2irv ^. ( j - l ) At) - ( c - l ) s i nc(2itvl Q ( j - l ) A t ) ] Appropriate values for the nyquist frequency v high and low bandlimits v^ and are chosen, then the values q., are calculated and added to the autocorrelation matrix. J' Typical values selected might be: v, = 0 Hz I o vhi = 50 Hz c = . 01 X = . 05(<j>xx(0)) where <t>xx(0) i s the data autocorrelation at zero lag. In practice values of q. ^  are normalized to unity on the diagonal. In i t s application to MED t h i s bandlimiting algorithm l i m i t s the frequency content of the MED f i l t e r at each i t e r a t i o n . This i s a d i s t i n c t l y d i f f e r e n t process than bandpassing the f i n a l output which in simple weiner 96 f i l t e r i n g might achieve similar results,. In MED the frequency content of the MED f i l t e r i s controlled each time i t attempts to match the non-linear r e f l e c t i v i t y estimate. A high-variance noisy f i l t e r at any stage may disrupt the MED algorithms path to an extremum so that frequency control i s ess e n t i a l at each i t e r a t i o n in noisy data ap p l i c a t i o n . The regretable practice of bandpassing the data after deconvolution comes too late in the case of MED. for by then a useless Varimax extremum may have already been reached. 

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