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The recovery of subsurface reflectivity and impedance structure from reflection seismograms Scheuer, Tim Ellis 1981

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THE RECOVERY OF SUBSURFACE REFLECTIVITY AND IMPEDANCE STRUCTURE FROM REFLECTION SEISMOGRAMS BY TIM ELLIS SCHEUER B . S c , The U n i v e r s i t y Of Utah, 1979 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 co 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 October 1981 (c) Tim E l l i s Scheuer, 1981 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of Geoplysi'cs and A&Wonomy The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date C U . I<j . ?/  DF-6 (2/79) i i A b s t r a c t T h i s t h e s i s i s concerned with the problem of e s t i m a t i n g broadband a c o u s t i c impedance from normal i n c i d e n c e r e f l e c t i o n se i sinograms. T h i s t o p i c i s covered by f o l l o w i n g the l i n e a r i n v e r s e formalisms d e s c r i b e d by Parker (1977) and Oldenburg (1980). The measured seismogram i s modelled as a c o n v o l u t i o n of subsurface r e f l e c t i v i t y with a source wavelet. Then an a p p r a i s a l of the seismogram i s performed to o b t a i n unique b a n d l i m i t e d r e f l e c t i v i t y i n f o r m a t i o n . T h i s b a n d l i m i t e d r e f l e c i t i v i t y i n f o r m a t i o n i s then u t i l i z e d in two d i f f e r e n t c o n s t r u c t i o n algorithms which provide a broadband estimate of r e f l e c t i v i t y ; from which a broadband impedance f u n c t i o n may be computed. The f i r s t c o n s t r u c t i o n method i s a maximum entropy method which uses an a u t o r e g r e s s i v e r e p r e s e n t a t i o n of a small p o r t i o n of the r e f l e c t i v i t y spectrum to p r e d i c t s p e c t r a l values outside that small p o r t i o n . The second and most v e r s a t i l e c o n s t r u c t i o n method i s the l i n e a r programming approach of Levy and F u l l a g a r (1981) which u t i l i z e s the unique b a n d l i m i t e d s p e c t r a l i n f o r m a t i o n obtained from an a p p r a i s a l and provides a broadband r e f l e c t i v i t y f u n c t i o n which has a minimum 1( norm. Both methods have been t e s t e d on s y n t h e t i c and r e a l s e i s m i c data and have shown good success at r e c o v e r i n g i n t e r p r e t a b l e broadband impedance models. E r r o r s i n the d a t a and the uniqueness of c o n s t r u c t e d r e f l e c t i v i t y models p l a y i m p o r t a n t r o l e s i n e s t i m a t i n g the impedance f u n c t i o n and i n a s s e s s i n g i t s u n i q u e n e s s . The Karhunen-Loeve t r a n s f o r m a t i o n i s d i s c u s s e d and a p p l i e d on r e a l d a t a t o s t a b i l i z e the c o n s t r u c t i o n r e s u l t s i n the presence of n o i s e . The g e n e r a l l y a c c e p t e d i d e a t h a t low frequ e n c y impedance i n f o r m a t i o n must be s u p p l i e d from w e l l l o g or v e l o c i t y a n a l y s e s because of the b a n d l i m i t e d n a t u r e of s e i s m i c data has been c h a l l e n g e d . When a c c u r a t e , b a n d l i m i t e d r e f l e c t i v i t y i n f o r m a t i o n can be r e c o v e r e d from the s e i s m i c t r a c e , then an i n t e r p r e t a b l e , broadband impedance model may be r e c o v e r e d u s i n g the two c o n s t r u c t i o n a l g o r i t h m s p r e s e n t e d i n t h i s t h e s i s . i v TABLE OF CONTENTS Page A b s t r a c t i i L i s t Of I l l u s t r a t i o n s , v i Acknowledgements i x Background And I n t r o d u c t i o n 1 CHAPTER 1 R e f l e c t i v i t y And Impedance 9 CHAPTER 2 The C o n v o l u t i o n a l Model 15 CHAPTER 3 I n v e r s e Theory And A p p r a i s a l D e c o n v o l u t i o n .. 21 Time Domain A p p r a i s a l D e c o n v o l u t i o n 21 Frequency Domain A p p r a i s a l D e c o n v o l u t i o n . . . .- 27 Impedance Computation And A p p r a i s a l 29 An Example Of A p p r a i s a l D e c o n v o l u t i o n 32 CHAPTER 4 R e f l e c t i v i t y C o n s t r u c t i o n 37 AR E x t e n s i o n Of The R e f l e c t i v i t y Spectrum 40 Broadband R e f l e c t i v i t y C o n s t r u c t i o n U s i n g The L Norm And A L i n e a r Programming A l g o r i t h m 50 G e n e r a l LP F o r u m u l a t i o n 52 The C o n s t r a i n e d LP S o l u t i o n 55 Data E r r o r s And The LP S o l u t i o n 60 I n e q u a l i t y C o n s t r a i n t s 60 E q u a l i t y C o n s t r a i n t s 61 S t a b i l i t y And E f f i c i e n c y C o n s i d e r a t i o n s 67 CHAPTER 5 M u l t i t r a c e R e f l e c t i v i t y C o n s t r u c t i o n U s i n g R e a l S e i s m i c Data 78 V The R e a l Data 78 The Karhunen-Loeve T r a n s f o r m a t i o n 87 C o n c l u s i o n 99 R e f e r e n c e s 104 Appendix A: D e r i v a t i o n Of The Common Trace 106 v i L i s t of I l l u s t r a t i o n s Page F i g . 1 S e i s m i c r e f l e c t i o n geometry 3 F i g . 2 Common m i d p o i n t s o u r c e - r e c e i v e r p a i r s r e s u l t i n g from m u l t i f o l d coverage 4 F i g . 3 P r o c e s s i n g sequence on a CDP g a t h e r 5 F i g . 4 Normal i n c i d e n c e r e f l e c t i o n from a c o m p o s i t i o n a l boundary 9 F i g . 5 S u b s u r f a c e p l a n e l a y e r e d model 11 F i g . 6 E r r o r i n the l i n e a r a p p r o x i m a t i o n 13 F i g . 7 The c o n v o l u t i o n a l model of r e f l e c t i o n seismograms 18 F i g . 8 a) the frequency domain r e p r e s e n t a t i o n of the c o n v o l u t i o n a l model, and b) d e c o n v o l u t i o n by s p e c t r a l d i v i s i o n 20 F i g . 9 S y n t h e t i c models f o r an example, of a p p r a i s a l d e c o n v o l u t i o n 34 F i g . 10 A p p r a i s a l d e c o n v o l u t i o n 35 T a b l e 1 R e l a t i v e r e s o l u t i o n and v a r i a n c e measures f o r the a p p r a i s a l s shown i n f i g . 10 36 F i g . 11 The nonuniqueness i n h e r e n t i n r e f l e c t i v i t y measurements 39 F i g . 12 A u t o r e g r e s s i v e r e c o n s t r u c t i o n of a r e f l e c t i v i t y f u n c t i o n c o n t a i n i n g few r e f l e c t o r s 48 F i g . 13 A u t o r e g r e s s i v e r e c o n s t r u c t i o n of a r e f l e c t i v i t y f u n c t i o n c o n t a i n i n g many r e f l e c t o r s 51 F i g . 14 Flow diagram f o r the c o n s t r a i n e d LP a l g o r i t h m 58 F i g . 15 Deconvolved s e i s m i c s e c t i o n ; r e p r e s e n t s s u b s u r f a c e s t r u c t u r e i n p a r t s of western A l b e r t a .63 v i i F i g . 16 Zero phase w a v e l e t ( a v e r a g i n g f u n c t i o n ) e s t i m a t e d from d a t a i n s i d e the box i n f i g u r e 15 63 F i g . 17 L i n e a r programming r e c o n s t r u c t i o n of a r e f l e c t i v i t y f u n c t i o n c o n t a i n i n g a few r e f l e c t o r s 64 F i g . 18 L i n e a r programming r e c o n s t r u c t i o n of a r e f l e c t i v i t y f u n c t i o n c o n t a i n i n g many r e f l e c t o r s 66 F i g . 19 S y n t h e t i c t e s t d a t a . Wavelet has band range of 1 0 - 5 0 H Z 68 F i g . 20 Performance of the AR and LP methods of r e f l e c t i v i t y c o n s t r u c t i o n i n the p r e s e nce of a d d i t i v e n o i s e 70 F i g . 21 Performance of the c o n s t r u c t i o n methods u s i n g an i n n a c c u r a t e w a velet 72 F i g . 22 S t a b i l i t y of r e c o n s t r u c t i o n s w i t h r e s p e c t t o the f r e q u e n c y band used 74 F i g . 23 S t a b i l i t y of the AR method w i t h r e s p e c t t o o r d e r and the s t a b i l i t y of o r d e r w i t h r e s p e c t t o the number of r e f l e c t i o n c o e f f i c i e n t s ; u s i n g a c c u r a t e and i n n a c c u r a t e d a t a 75 F i g . 24 The a c c u r a c y and e f f i c i e n c y of the LP s o l u t i o n w i t h r e s p e c t t o the w e i g h t i n g exponent q, u s i n g a c c u r a t e d a t a 77 F i g . 25 Data s e c t i o n A; • d e c o n v o l v e d v i b r a t o r d a t a a l o n g w i t h a g e o l o g i c a l , i n t e r p r e t a t i o n 79 F i g . 26 Data s e c t i o n B; d e c o n v o l v e d e x p l o s i o n d a t a 79 F i g . 27 AR r e f l e c t i v i t y and impedance r e c o n s t r u c t i o n s from s e c t i o n A; b) u s i n g unsmoothed, and c) u s i n g smoothed s p e c t r a l e s t i m a t e s 82 F i g . 28 LP r e f l e c t i v i t y and impedance r e c o n s t r u c t i o n s from s e c t i o n A; b) u s i n g unsmoothed, and c) u s i n g smoothed s p e c t r a l e s t i m a t e s 83 V I 1 1 F i g . 29 AR r e f l e c t i v i t y and impedance r e c o n s t r u c t i o n s from s e c t i o n B u s i n g smoothed s p e c t r a l e s t i m a t e s 85 F i g . 30 LP r e f l e c t i v i t y and impedance r e c o n s t r u c t i o n s from s e c t i o n B u s i n g smoothed s p e c t r a l e s t i m a t e s 86 F i g . 31 A p p l i c a t i o n of the K-L t r a n s f o r m a t i o n t o s t a b i l i z e the c h o i c e of AR o r d e r . A l s o showing the p r i n c i p l e of the m i x i n g a l g o r i t h m 92 F i g . 32 A p p l i c a t i o n of the K-L t r a n s f o r m a t i o n and m i x i n g a l g o r i t h m t o the n o i s y s e c t i o n B 93 F i g . 33 A p p l i c a t i o n of the K-L a l g o r i t h m t o LP r e f l e c t i v i t y and impedance r e c o n s t r u c t i o n s 95 F i g . 34 The f i n a l r e f l e c t i v i t y and impedance r e c o n s t r u c t i o n s from s e c t i o n A a l o n g w i t h the o r i g i n a l g e o l o g i c i n t e r p r e t a t i o n 96 F i g . 35 Comparison of f i n a l r e f l e c t i v i t y r e c o n s t r u c t i o n s w i t h the o r i g i n a l d e c o nvolved s e c t i o n (A) 97 F i g . 36 Comparison of f i n a l impedance r e c o n s t r u c t i o n s w i t h impedance averages o b t a i n e d from the d e c o n v o l v e d data . 98 F i g . 37 The f i n a l r e f l e c t i v i t y and impedance r e c o n s t r u c t i o n s from s e c t i o n B. The g i v e n v e l o c i t y l o g has been i n s e r t e d f o r comparison 100 F i g . 38 Comparison of f i n a l r e f l e c t i v i t y r e c o n s t r u c t i o n s w i t h the o r i g i n a l d e c o n v o l v e d s e c t i o n (B) 101 F i g . 39 Comparison of f i n a l impedance r e c o n s t r u c t i o n s w i t h impedance averages o b t a i n e d from the d e c o n v o l v e d data 102 i i x Acknowledgements I acknowledge Ms. Kate Aasen f o r p r o v i d i n g i n t e r e s t i n g d i v e r s i o n s t o t h e s i s work; E l l i s and Dorothy; Mr. P a t i e n c e , my o f f i c e mate, K e r r y S t i n s o n ; the members of G e o p h y s i c s House - Rob Conraads, J i m Horn, and Don P l e n d e r l e i t h ; P r o f . Tad U l r y c h f o r b e i n g an i n s p i r a t i o n i n time s e r i e s a n a l y s i s and i n p a r t i c u l a r I a p p r e c i a t e h i s h e l p w i t h the AR p a r t s of t h i s t h e s i s ; Shlomo Levy f o r l e a v i n g s e v e r a l of h i s then (1979) u n p u b l i s h e d papers l y i n g around the department which i n s p i r e d p a r t s of t h i s t h e s i s - Shlomo suggested use of the K-L t r a n s f o r m a t i o n ; Mr. Computer, C o l i n Walker; Ken 'Wizard' W h i t t a l l ; and P e t e r F u l l a g a r . A d d i t i o n a l thanks go t o Kate and K e r r y a l o n g w i t h Don K i n g f o r i m p r o v i n g the language of t h i s t h e s i s . I wish t o acknowledge a l l members of t h i s department f o r p r o v i d i n g a dynamic environment i n which t o s t u d y . I have been f o r t u n a t e i n r e c e i v i n g f i n a n c i a l a s s i s t a n c e from I m p e r i a l O i l of Canada and I d e e p l y a p p r e c i a t e the time and e f f o r t they have spent i n p r o v i d i n g a s u i t a b l e data s e t . I am a l s o g r a t e f u l t o MRO A s s o c i a t e s I n c . f o r p r o v i d i n g a second d a t a s e t . F i n a l l y , I would l i k e t o thank and acknowledge my a d v i s o r P r o f . Doug Oldenburg. Doug has p r o v i d e d me w i t h e n d l e s s m o t i v a t i o n and support d u r i n g the past two y e a r s . Dougs' e n t h u s i a s t i c and p r e c e p t i v e s t y l e of d i d a c t i c guidance as w e l l as h i s s t r o n g d e s i r e t o seek f u r t h e r knowledge have c o n t r i b u t e d much energy t o a l l a s p e c t s of t h i s work. 1 Background and I n t r o d u c t i o n R e f l e c t i o n s e i s m o l o g i s t s and s t r a t i g r a p h i c g e o l o g i s t s a r e g e n e r a l l y i n t e r e s t e d i n the problem of e s t i m a t i n g s u b s u r f a c e a c o u s t i c impedance from normal i n c i d e n c e seismograms. The e s t i m a t i o n i s u s u a l l y c a r r i e d out i n two s t e p s . F i r s t the measured seismogram i s p r o c e s s e d t o resemble the s u b s u r f a c e r e f l e c t i v i t y f u n c t i o n ; and then t h i s r e s u l t i s c o n v e r t e d i n t o an e s t i m a t e of a c o u s t i c impedance. The second s t e p n o r m a l l y r e q u i r e s the i n t r o d u c t i o n of low f r e q u e n c y impedance i n f o r m a t i o n o b t a i n e d e i t h e r from nearby w e l l l o g s ( G a l b r a i t h and M i l l i n g t o n , 1979), or from a p a r t i c u l a r v e l o c i t y a n a l y s i s c a r r i e d out on a s u i t e of common m i d p o i n t seismograms (Lavergne and W i l l m , 1977; L i n d s e t h , 1979). T h i s s t e p has been c o n s i d e r e d n e c e s s a r y because of the i n h e r e n t b a n d l i m i t e d n a t u r e of s e i s m i c r e c o r d i n g s . Because of e x p e r i m e n t a l l i m i t a t i o n s , s e i s m i c r e f l e c t i o n i n f o r m a t i o n i s t y p i c a l l y r e c o v e r a b l e o n l y w i t h i n the f r e q u e n c y band, range of 10-60 Hz. The low f r e q u e n c y i n f o r m a t i o n (0-lOHz) i s v i t a l t o an a c c u r a t e i n t e r p r e t a t i o n of a r e c o v e r e d impedance l o g . In t h i s t h e s i s , i t i s proposed t h a t the e s t i m a t i o n of a c o u s t i c impedance be c a r r i e d out w i t h a somewhat d i f f e r e n t p h i l o s o p h y ; i n a way which e s s e n t i a l l y f o l l o w s the Backus-G i l b e r t framework of l i n e a r i n v e r s e t h e o r y (Oldenburg, 2 1981). F i r s t , the measured seismogram i s mo d e l l e d and then a p p r a i s e d t o o b t a i n unique i n f o r m a t i o n about the s u b s u r f a c e r e f l e c t i v i t y f u n c t i o n . T h i s unique i n f o r m a t i o n i s c a l l e d ' r e f l e c t i v i t y a v e rages' and the method used here t o o b t a i n i t i s c a l l e d ' a p p r a i s a l d e c o n v o l u t i o n ' . A broadband r e f l e c t i v i t y f u n c t i o n i s then c o n s t r u c t e d u s i n g t h i s unique i n f o r m a t i o n a l o n g w i t h some p h y s i c a l c o n s t r a i n t s and a s u i t a b l e a l g o r i t h m . From t h i s r e s u l t , an i n t e r p r e t a b l e broadband e s t i m a t e of s u b s u r f a c e impedance may be o b t a i n e d . The importance of t h i s work l i e s p r i m a r i l y i n the i n s i g h t s g a i n e d by a p p r o a c h i n g impedance r e c o v e r y u s i n g a l i n e a r i n v e r s e f o r m a l i s m ; and i n the e f f i c a c y of the v a r i o u s p r o c e d u r e s which have been r e f i n e d t o c o n s t r u c t broadband r e f l e c t i v i t y and impedance models. A b r i e f review of some i m p o r t a n t a s p e c t s of the s e i s m i c r e f l e c t i o n method and a s s o c i a t e d t e r m i n o l o g y i s now p r e s e n t e d . The b a s i c s e i s m i c r e f l e c t i o n s u r v ey e n t a i l s r e c o r d i n g s u b s u r f a c e boundary r e f l e c t i o n s a r i s i n g from an a r t i f i c i a l d i s t u r b a n c e (shot or sou r c e ) c r e a t e d near the ground s u r f a c e . These r e f l e c t i o n s a re r e c o r d e d a t a l i n e of e q u a l l y spaced r e c e i v e r s (geophones) which ex t e n d a s h o r t d i s t a n c e from the source p o i n t (see f i g u r e 1 ) . A d i s t u r b a n c e a r r i v i n g a t the geophone from a s i n g l e r e f l e c t i o n w i l l be c a l l e d the source wavelet (shown i n the 3 seismic traces as a coherent wiggle - figure 1 ) . According to the simple laws of r e f l e c t i o n (for horizontal boundaries) the r e f l e c t i o n point (RP) i s located v e r t i c a l l y beneath the source-receiver midpoint. Each disturbance created and recorded w i l l have a single source geometry similar to that shown in figure 1. By overlapping the single source geometry at successive shots, i t i s possible to arrange such that d i f f e r e n t source-receiver o f f s e t s have a common midpoint (see figure 2 ) . In the case of horizontal boundaries, these common midpoint source- receiver depth Boundary Source Wavelet .Surface Resulting Seismic Traces F i g . 1. Diagram showing a t y p i c a l s o u r c e - r e c e i v e r geometry used i n s e i s m i c r e f l e c t i o n p r o s p e c t i n g and the r e c o r d e d s e i s m i c t r a c e from each r e c e i v e r . 4 pairs w i l l be associated with a common depth point (CDP) as shown in figure 2. A c o l l e c t i o n of seismic traces for which the source-receiver o f f s e t s have a common midpoint i s c a l l e d a CDP gather (see figure 3 ) . The number of traces in the gather defines the magnitude of mult i f o l d coverage obtained by the overlapping shots. A CDP gather with N traces provides N-fold CDP coverage and i s said to cover the subsurface by NxlOO percent. The stacking procedure which w i l l be described in the following paragraph attempts to make use of thi s redundant information to increase the Midpoint Resulting Traces Normal Incidence Ray Path COP F i g . 2. Common midpoint s o u r c e - r e c e i v e r p a i r s r e s u l t i n g from m u l t i f o l d coverage and the recorded t r a c e from each r e c e i v e r . 5 signal to noise r a t i o of the data. Observing the CDP gather in figure 3 i t i s evident that there i s d i f f e r e n t i a l t r a v e l time (or normal moveout-NMO) to each r e f l e c t i v e event across the gather for d i f f e r e n t source-receiver o f f s e t s . After correcting t h i s d i f f e r e n t i a l t r a v e l time in each trace to match the t r a v e l time of the normal incidence ray, a l l traces may be summed together a l g e b r a i c a l l y ( i . e . , stacked) enhancing the signal to noise r a t i o of the resu l t (assuming random noise). The resu l t i n g stacked trace i s then plotted at the source-receiver A * * ^ " * Midpoint Stacked Result CDP Gather NMO Corrected F i g . 3- P r o c e s s i n g sequence on a common depth p o i n t gather, r e s u l t i n g i n a stacked normal i n c i d e n c e s e i s m i c t r a c e . 6 midpoint, thereby producing a normal i n c i d e n c e seismogram i n which the r e f l e c t i v e event i s l o c a t e d v e r t i c a l l y beneath the s o u r c e - r e c e i v e r midpoint. T h i s seismogram i s i n s u i t a b l e form f o r treatment by the methods d e s c r i b e d i n t h i s t h e s i s . O b t a i n i n g unique r e f l e c t i v i t y i n f o r m a t i o n from- a normal i n c i d e n c e seismogram i n v o l v e s many assumptions and s e q u e n t i a l p r o c e s s i n g s t e p s . A proper treatment of the observed seismogram must i n c l u d e the e f f e c t s o f : (1) energy l o s s e s through g e o m e t r i c a l spreading, a n e l a s t i c a b s o r p t i o n , and t r a n s m i s s i o n l o s s e s a c r o s s boundaries in a l a y e r e d media (2) d i s p e r s i o n (3) source type and c o u p l i n g (4) f i e l d geometries and a r r a y responses (5) m u l t i p l e r e f l e c t i o n s and other coherent noise (6) the mathematical model imposed on. the seismogram (7) other The s o l u t i o n s of these c o m p l i c a t i o n s w i l l not be adressed i n t h i s t h e s i s . I t i s assumed that the seismic data have been processed so that the seismogram, denoted by s ( t ) , may be 7 c o n s i d e r e d t o r e s u l t from the c o n v o l u t i o n of the s u b s u r f a c e r e f l e c t i v i t y f u n c t i o n r ( t ) w i t h the source w a v e l e t w(t) p l u s random n o i s e n ( t ) . That i s , the normal i n c i d e n c e seismogram i s d e s c r i b e d by the f o l l o w i n g c o n v o l u t i o n a l model 5CO-ra)* wet) +• nco where the symbol * denotes the c o n v o l u t i o n o p e r a t i o n . A c o u s t i c impedance e s t i m a t i o n from a normal i n c i d e n c e seismogram w i l l be d e v e l o p e d from b a s i c c o n c e p t s . A complete t r e a t m e n t of the problem be g i n s i n c h a p t e r 1 w i t h the developement of l i n e a r r e l a t i o n s h i p s between r e f l e c t i v i t y and a c o u s t i c impedance a c c o r d i n g t o normal ' i n c i d e n c e , p l a n e wave t h e o r y . A d i s c u s s i o n of the c o n v o l u t i o n a l model i s then p r o v i d e d i n c h a p t e r 2 t o e s t a b l i s h s i m p l e grounds f o r u n d e r s t a n d i n g the complete e x p e r i m e n t a l problem. In c h a p t e r 3, the t r a d i t i o n a l method of d e c o n v o l v i n g the seismogram when p r o p e r t i e s of the source w a v elet a r e known i s viewed under the u m b r e l l a of i n v e r s e t h e o r y . Two d i s t i n c t methods of r e f l e c t i v i t y c o n s t r u c t i o n w i l l be p r e s e n t e d i n c h a p t e r 4 and t h e i r e f f i c a c y shown u s i n g s y n t h e t i c s e i m i c d a t a . The f i r s t c o n s t r u c t i o n method i s a maximum e n t r o p y method which uses an a u t o r e g r e s s i v e r e p r e s e n t a t i o n of a s m a l l p o r t i o n of the r e f l e c t i v i t y s pectrum t o p r e d i c t s p e c t r a l v a l u e s o u t s i d e t h a t s m a l l 8 p o r t i o n . The second and most v e r s a t i l e c o n s t r u c t i o n method i s a l i n e a r programming approach which u t i l i z e s the unique s p e c t r a l i n f o r m a t i o n from an a p p r a i s a l d e c o n v o l u t i o n and provi d e s a r e f l e c t i v i t y f u n c t i o n which has a minimum 1| norm, l e a d i n g n a t u r a l l y to an impedance model with blocky c h a r a c t e r . The f i n a l chapter i s devoted to examples of the above c o n s t r u c t i o n methods using r e a l seismic data. A l s o , the Karhunen-Loeve t r a n s f o r m a t i o n , a weighted averaging procedure which p l a y s a strong r o l e in s t a b l i z i n g the impedance r e c o n s t r u c t i o n s , i s d e s c r i b e d and then u t i l i z e d on r e a l data. E r r o r s in the data and the uniqueness of c o n s t r u c t e d r e f l e c t i v i t y models w i l l p l ay important r o l e s i n e s t i m a t i n g the a c o u s t i c impedance f u n c t i o n and i n a s s e s s i n g i t s uniqueness. These f e a t u r e s w i l l be developed i n some d e t a i l throughout t h i s t h e s i s . The g e n e r a l l y accepted idea that low frequency impedance i n f o r m a t i o n must be s u p p l i e d from w e l l l o g s or v e l o c i t y a n a l y ses because of the b a n d l i m i t e d nature of the seismogram, i s c h a l l e n g e d . I f a c c u r a t e , b a n d l i m i t e d r e f l e c t i v i t y i n f o r m a t i o n can be recovered from the seismic t r a c e , then an i n t e r p r e t a b l e , broadband impedance model may be c o n s t r u c t e d using the two al g o r i t h m s presented i n t h i s t h e s i s . 9 R e f l e c t i v i t y and Impedance A review of the basic concepts in normal incidence, plane wave theory w i l l now be presented to e s t a b l i s h the re l a t i o n s h i p between r e f l e c t i v i t y and acoustic impedance to be used throughout t h i s t h e s i s . At a compositional boundary, incident and re f l e c t e d wave amplitudes are related by a r e f l e c t i o n c o e f f i c i e n t r which depends on the density p and v e l o c i t y v of the adjacent media where A; i s the incident and A r i s the re f l e c t e d wave amplitude (see figure 4 ) . I.I Aj A r Transmitted Energy F i g . 4. Normal i n c i d e n c e r e f l e c t i o n from a co m p o s i t i o n a l boundary. 10 The p r o d u c t of d e n s i t y and v e l o c i t y i s the a c o u s t i c impedance of the medium and w i l l be denoted z. E x t e n d i n g t h i s concept t o a l a y e r e d s u b s u r f a c e w i t h many b o u n d a r i e s , an e x p r e s s i o n f o r the r e f l e c t i v i t y s e r i e s i s o b t a i n e d i n which t h e r e i s a r e f l e c t i o n c o e f f i c i e n t a t each boundary d e s c r i b e d by (see f i g u r e 5) R e a r r a n g i n g t h i s e x p r e s s i o n , a s i m p l e f o r m u l a f o r the c o m p u t a t i o n of a c o u s t i c impedance r e s u l t s where z, i s the a c o u s t i c impedance i n the i n i t i a l l a y e r . E x p r e s s i o n s 1.2 and 1.3 r e p r e s e n t the main o b j e c t i v e s of the r e f l e c t i o n s e i s m o l o g i s t . F i r s t the s e i s m o l o g i s t must o b t a i n an a c c u r a t e r e f l e c t i v i t y measurement. T h i s measurement i s then c o n v e r t e d i n t o an e s t i m a t e of s u b s u r f a c e a c o u s t i c impedance so t h a t rock t y p e s and perhaps o t h e r i n f o r m a t i o n may be i n f e r r e d . D e f i n i n g as the l o g a r i t h m of n o r m a l i z e d a c o u s t i c impedance, where the n o r m a l i z a t i o n i s w i t h r e s p e c t t o the i n i t i a l a c o u s t i c impedance Z j , e x p r e s s i o n 1.3 becomes 11 Surface Layer k»l z k» l Reflectivity Series i i i i i i i i i r i Layer 1 r II i i i i i /1 > Layer 2 z 2 •1 r Layer 3 " '2 t-• • « • - 13 i i i i i t i /1 Layer k I I / / ; i / i *k ' r K - l Acoustic Impedance F i g . 5. Subsurface plane l a y e r e d model showing the r e f l e c t i v i t y s e r i e s and a c o u s t i c impedance f u n t i o n d e f i n e d f o r t h i s type of model. In o r d e r t o s i m p l i f y f u r t h e r d i s c u s s i o n of i t w i l l be from here on r e f e r r e d t o as impedance, d i s t i n g u i s h i n g i t from 2 which i s the a c o u s t i c impedance. For | r | <1 , 0 0 F I N - 1 12 T h i s a p p r o x i m a t i o n i s v a l i d f o r |r| <.25 as shown i n f i g u r e 6. U s i n g t h i s s i m p l i f i c a t i o n , e x p r e s s i o n 1.4 reduces t o 1= 1 which g i v e s a l i n e a r r e l a t i o n s h i p between impedance and r e f l e c t i v i t y . T h i s e x p r e s s i o n may a l s o be w r i t t e n or In the time domain i t can be shown t h a t r e l a t i o n 1.6 has the c o n t i n u o u s analogue (eg. P e t e r s o n et a l , 1955) 4 ^ = 2 ret) , 7 or ^(O = 1 $trcu) du i.g 0 where the r e f l e c t i v i t y f u n c t i o n r ( t ) and t h e r e f l e c t i o n c o e f f i c i e n t s rj a r e r e l a t e d by rv rc-O-Z r^a-r,) 1-1 13 where N i s t h e number o f b o u n d a r i e s and S ( t - ) i s the D i r a c d e l t a f u n c t i o n l o c a t e d a t t h e t i m e o c c u r e n c e of each r e f l e c t i o n c o e f f i c i e n t . E q u a t i o n 1.8 may be a l s o w r i t t e n as ( e g . , P e a c o c k , 1979) ^(O = 2 ret) * uit) 1.1 where u ( t ) i s t h e u n i t s t e p f u n c t i o n . C o n v o l v i n g the r e f l e c t i v i t y f u n c t i o n w i t h the u n i t s t e p p e r f o r m s the r o l e of i n t e g r a t i o n . F i g . 6. The e r r o r between f ( and the l i n e a r approximation fx f o r values o f |r| l e s s than .9. 14 In t h i s t h e s i s , the l i n e a r r e l a t i o n s h i p s between r e f l e c t i v i t y and impedance ( e q u a t i o n s 1.5-1.9) w i l l be used i n p l a c e of the n o n l i n e a r r e l a t i o n s h i p ( e q u a t i o n 1.4), t o s i m p l i f y f u r t h e r m a t h e m a t i c a l t r a n s f o r m a t i o n s . I t must be remembered however, t h a t a l l l i n e a r r e l a t i o n s have the r e s t r i c t i o n t h a t r e f l e c t i o n c o e f f i c i e n t s | r | must be l e s s than .25. I f a r e f l e c t i o n c o e f f i c i e n t exceeds .25 then the impedance s h o u l d be c a l c u l a t e d v i a e q u a t i o n 1.4. 15 2 The C o n v o l u t i o n a l Model The purpose of t h i s c h a p t e r i s t o b r i e f l y e x p l a i n t h e c o n v o l u t i o n a l model i n s e i s m i c r e f l e c t i o n t h e o r y and t o make c l e a r the l i m i t a t i o n s of c o n v e n t i o n a l d e c o n v o l u t i o n ( i n v e r s e f i l t e r i n g / s p e c t r a l d i v i s i o n ) t e c h n i q u e s a t r e c o v e r i n g broadband r e f l e c t i v i t y i n f o r m a t i o n from a s e i s m i c t r a c e . A s e i s m i c t r a c e i s g e n e r a l l y l o o k e d upon as a sequence of o v e r l a p p i n g source w a v e l e t s t h a t r e s u l t from a v a r i e t y of r e f l e c t i n g b o u n d a r i e s ( R i c k e r , 1940 and 1953). I n f o r m a t i o n about the s u b s u r f a c e geology i s h idden i n the t r a v e l t i m e s and a m p l i t u d e s of t h e s e w a v e l e t s , t h a t i s , they c o n c e a l the s u b s u r f a c e r e f l e c t i v i t y f u n c t i o n . The u l t i m a t e g o a l of s e i s m i c d e c o n v o l u t i o n i s t o r e p l a c e each o v e r l a p p i n g w a v e l e t w i t h a 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 r e p r e s e n t a t i v e of a s u b s u r f a c e boundary. I t w i l l be shown t h a t t h i s g o a l can be o n l y p a r t i a l l y f u l f i l l e d u s i n g a c o n v e n t i o n a l d e c o n v o l u t i o n t e c h n i q u e . S e v e r a l good examples of c o n v e n t i o n a l d e c o n v o l u t i o n t e c h n i q u e s can be found i n D e c o n v o l u t i o n , G e o p h y s i c s R e p r i n t S e r i e s 1, volumes I and I I . The s i m p l e c o n v o l u t i o n a l model w i t h o u t a d d i t i v e n o i s e s t a t e s t h a t the measured seismogram s ( t ) (whether i t be the r e s u l t b e f o r e or a f t e r the s t a c k i n g p r o c e d u r e ) , i s e q u a l t o the r e f l e c t i v i t y f u n c t i o n r ( t ) c o n v o l v e d w i t h a s o u r c e w a v e l e t w ( t ) , t h a t i s , 1 6 The s o u r c e wavelet i s d e f i n e d t o ta k e on a v a r i e t y of c h a r a c t e r i s t i c s depending on the e x p e r i m e n t a l p r o c e d u r e s used and the . p h y s i c a l e f f e c t s of wave p r o p o g a t i o n . The sourc e w a v e l e t i s sometimes d e f i n e d as the p a r t i c l e v e l o c i t y of a s i n g l e r e f l e c t e d d i s t u r b a n c e as r e c o r d e d i n time by a geophone. However, i t i s n e c e s s a r y t o i n c l u d e p r o c e s s i n g e f f e c t s i n the wavelet when c o n s i d e r i n g a d e c o n v o l u t i o n of s t a c k e d s e i s m i c d a t a . P r o b a b l e e f f e c t s c o n t r i b u t i n g t o the source wavelet a re summarized below: (a) s o u r c e type and c o u p l i n g (b) t r a n s m i s s i o n e f f e c t s (c) f i e l d g e o m e t r i e s and geophone a r r a y s (d) g h o s t i n g e f f e c t (e) near s u r f a c e w e a t h e r i n g ( f ) i n s t r u m e n t response (g) p r o c e s s i n g e f f e c t s ( eg., s t a c k i n g ) (h) e t c . By f o l l o w i n g a c o n v o l u t i o n a l model where a l l e x p e r i m e n t a l e f f e c t s l i s t e d above are a t t a c h e d t o the sour c e w a v e l e t , a s t r o n g f o u n d a t i o n i s s e t f o r r e c o v e r i n g an e s t i m a t e of s u b s u r f a c e r e f l e c t i v i t y from the measured seismogram. R e f l e c t i o n s e i s m o l o g i s t s r e a l i z e the importance of the sourc e w a v e l e t i n s e i s m i c d a t a p r o c e s s i n g and a r e a c t i v e l y engaged i n d e v i s i n g improved methods which p r o v i d e source 17 w a v e l e t e s t i m a t e s . These methods may be c l a s s i f i e d i n t o t h r e e groups: (1) d i r e c t knowledge of the s o u r c e s i g n a t u r e (eg., v i b r a t i n g s o u r c e ) (2) d i r e c t measurement of the source s i g n a t u r e (Mayne and Quay, 1971; Kramer et a l , 1968) (3) e s t i m a t i o n of a w a v e l e t from the measured seismogram based on assumed s t a t i s t i c a l or p h y s i c a l p r o p e r t i e s of the r e f l e c t i v i t y f u n c t i o n and w a v e l e t ( R o b i n s o n , 1967; U l r y c h , 1971; O t i s and Smith, 1977; L i n e s and U l r y c h , 1977, and Oldenburg, et a l . , 1981). For a l l d i s c u s s i o n s r e l a t i n g t o the c o n v o l u t i o n a l model, i t w i l l be assumed t h a t the s o u r c e wavelet i s known and t h a t d e t e r m i n a t i o n of a broadband r e f l e c t i v i t y f u n c t i o n i s d e s i r e d . T h i s s e c t i o n w i l l c o n c l u d e w i t h a p i c t o r i a l r e p r e s e n t a t i o n of the c o n v o l u t i o n a l model and an example of s p e c t r a l d e c o n v o l u t i o n which i n d i c a t e s what r e f l e c t i v i t y i n f o r m a t i o n may be u n i q u e l y r e c o v e r e d from the seismogram. F i g u r e 7a shows the r e s u l t of a s e i s m i c experiment (where the d a t a a r e measured i n time) i n which the source wavelet 1 8 i s a delta function. The convolution of a delta function with subsurface r e f l e c t i o n c o e f f i c i e n t s r e s u l t s in a perfect recovery of the true r e f l e c t i v i t y function. Convolving the subsurface structure with a more r e a l i s t i c source wavelet leads to a poorly resolved, uninterpretable measurement of r e f l e c t i v i t y as shown in figure 7b. This e f f e c t may be observed in the frequency domain by applying the convolution theorem to figure 7b. The convolution in the time domain becomes a m u l t i p l i c a t i o n of sp e c t r a l components in the frequency domain as shown in figure 8a (where only the Subsurface Measured Reflectivity Source Reflectivity Function Wavelet Function F i g . 7 . a) The c o n v o l u t i o n model i n the time domain u s i n g an i d e a l wavelet and, b) u s i n g a more r e a l i s t i c wavelet. 19 a m p l i t u d e s p e c t r a and p o s i t i v e f r e q u e n c i e s a r e shown: p h a s e e f f e c t s h a v e been l e f t o u t f o r s i m p l i c i t y ) . N o t e t h a t t h e b a n d l i m i t e d c h a r a c t e r o f t h e m e a s u r e d r e f l e c t i v i t y i s due t o t h e b a n d l i m i t e d n a t u r e o f t h e s o u r c e w a v e l e t . In g e n e r a l , t h e r e f l e c t i v i t y f u n c t i o n i s n o t b a n d l i m i t e d and t h u s i t i s s e e n t h a t a c o m p l e t e r e c o v e r y o f t h e o r i g i n a l r e f l e c t i v i t y s p e c t r u m i s i n h i b i t e d by t h e s p e c t r a l n u l l i n g e f f e c t o f t h e w a v e l e t . U n i q u e r e f l e c t i v i t y i n f o r m a t i o n may be o b t a i n e d f r o m t h e m e a s u r e d d a t a by r e m o v i n g t h e s p e c t r a l s m o o t h i n g e f f e c t s o f t h e w a v e l e t w i t h i n i t s n o n z e r o b a n d r a n g e . In o t h e r w o r d s , a r o u g h a p p r a i s a l o f u n i q u e n e s s may be made by p e r f o r m i n g a b a n d l i m i t e d s p e c t r a l d i v i s i o n as shown i n f i g u r e 8b. S i n c e t h e w a v e l e t c o n t a i n s i n s i g n i f i c a n t e n e r g y a t b o t h low and h i g h f r e q u e n c i e s , no r e f l e c t i v i t y i n f o r m a t i o n i s r e c o v e r a b l e a t t h e s e f r e q u e n c i e s . O n l y t h o s e f r e q u e n c i e s w h i c h a r e c o n t a i n e d w i t h i n t h e b a n d w i d t h o f t h e w a v e l e t c a n be r e c o v e r e d ( s e e f i g u r e 8 b ) . T h i s band o f f r e q u e n c i e s r e p r e s e n t s t h e o n l y u n i q u e r e f l e c t i v i t y i n f o r m a t i o n r e c o v e r a b l e t h r o u g h s p e c t r a l d i v i s i o n o r i n v e r s e f i l t e r i n g t e c h n i q u e s . The n o n u n i q u e n e s s i n h e r e n t i n r e c o v e r i n g b r o a d b a n d r e f l e c t i v i t y i s now a p p a r e n t f r o m t h e a b o v e a n a l y s i s . T h e r e a r e i n f i n i t e l y many r e f l e c t i v i t y f u n c t i o n s w h i c h w i l l s a t i s f y t h e r e s u l t a n t s o f f i g u r e s 8a a n d 8 b . The o n l y 20 information that a l l such functions must have in common i s the band of frequencies shown on the right hand side of figure 8 b . However, recovering broadband r e f l e c t i v i t y information from seismic data can be a tractable problem i f the missing frequencies are accurately replaced. Standard methods for replacing low frequencies involve well log or v e l o c i t y a n a l y s i s . A l t e r n a t i v e l y , a r e f l e c t i v i t y construction technique, which u t i l i z e s the information provided by a single deconvolved trace, may be used to f i l l in the unknown frequencies. Two such r e f l e c t i v i t y construction methods are presented in chapter 4 . Reflectivity. Spectrum Wavelet Spectrum Measured Reflectivity Spectrum Measured Reflectivity Spectrum Wavelet Recovered Reflectivity Spectrum F i g . 8. a) The c o n v o l u t i o n model ( f i g . 7b) i n the frequency domain and, b) d e c o n v o l u t i o n by s p e c t r a l d i v i s i o n . 21 3 I n v e r s e Theory and Appra i s a l Deconvolut i o n The purpose of t h i s s e c t i o n i s t o p r o v i d e the reader w i t h the b a s i c c o n c e p t s of i n v e r s e t h e o r y and i t s r e l a t i o n t o d e c o n v o l u t i o n . T h i s h e u r i s t i c approach r e q u i r e s t h a t the n o t a t i o n be somewhat s i m p l i f i e d ; a more r i g o u r o u s t r e a t m e n t of t h i s t o p i c can be found i n Oldenburg, (1981). T h i s s e c t i o n i s i n most p a r t a p r e c i s of t h a t paper. The c o n v o l u t i o n a l model of a seismogram n e g l e c t i n g the a d d i t i v e n o i s e t e r m - i s w r i t t e n When the wavelet i s known, e q u a t i o n 3.1 may be a p p r a i s e d i n e i t h e r t h e . time or f r e q u e n c y domain. Each of t h e s e approaches w i l l be d i s c u s s e d s e p e r a t e l y a l t h o u g h the r e s u l t s under s i m i l a r c o n d i t i o n s are i d e n t i c a l . A) Time domain a p p r a i s a l d e c o n v o l u t i o n In the time domain, an i n v e r s e f i l t e r v ( t ) i s d e s i r e d t h a t c o n t r a c t s the wavelet i n t o a d e l t a f u n c t i o n ; i . e . , 3.1 3.2 But i n the s e i s m i c c a s e , w(t) i s b a n d l i m i t e d , so t h e r e i s no 22 f i l t e r which w i l l c o n t r a c t the wa v e l e t i n t o a d e l t a f u n c t i o n . F i n d i n g an i n v e r s e f i l t e r which c o n t r a c t s the w a v e l e t i n t o a z e r o phase a p p r o x i m a t i o n t o a d e l t a f u n c t i o n i s the best a l t e r n a t i v e . By z e r o phase, i t i s meant t h a t the a p p r o x i m a t e d d e l t a f u n c t i o n has i t s energy peak l o c a t e d a t the t r u e d e l t a f u n c t i o n p o s i t i o n and i s s y m m e t r i c a l about i t s energy peak. T h i s can be done by f i n d i n g a f i l t e r which m i n i m i z e s the i n t e g r a t e d squared d i f f e r e n c e between the two s i d e s of 3.2, t h a t i s , the f i l t e r m i n i m i z e s where the optimum p o s i t i o n t 0 of the d e s i r e d s p i k e a p p r o x i m a t i o n may depend on the l e n g t h or phase c h a r a c t e r i s t i c s of the wavelet ( T r e i t e l and Robinson, 1966: Oldenburg, 1981). O b t a i n i n g the best v ( t ) by m i n i m i z i n g 0 , both s i d e s of 3.1 are then c o n v o l v e d w i t h the i n v e r s e f i l t e r t o p e r f o r m the a p p r a i s a l d e c o n v o l u t i o n soo*vet) - r c o * M O * VCO = r c o * a CO. = <ra)> 3 ,4 23 where <r(t)> denotes r e f l e c t i v i t y averages. P i c t o r i a l l y , the appraisal might look l i k e : SCO VCO <rco> The convolution of the wavelet with the inverse f i l t e r produces an averaging function which i s in some sense more l o c a l i z e d than the o r i g i n a l wavelet. For example: WCO VCO QCt) The averaging function plays a fundamental role in inverse theory because i t s character immediately indicates the resolving power of the wavelet. This type of averaging 24 f u n c t i o n i s o f t e n c a l l e d a z e r o p h a s e w a v e l e t by s e i s m i c d a t a p r o c e s s o r s . F r o m 3 . 4 , t h e c o n v o l u t i o n o f t h e a v e r a g i n g f u n c t i o n w i t h t h e t r u e r e f l e c t i v i t y m o d e l w i l l p r o d u c e r e f l e c t i v i t y a v e r a g e s . F o r e x a m p l e : Ail A rA I rco aco < r ( t ) > K n o w l e d g e a b o u t t h e t r u e r e f l e c t i v i t y , r e c o v e r a b l e f r o m t h e s e i s m o g r a m , i s c o m p l e t e l y s u m m a r i z e d by t h e a v e r a g e v a l u e s < r ( t ) > a n d t h e a v e r a g i n g f u n c t i o n a ( t ) u n l e s s e x t r a i n f o r m a t i o n i s p r o v i d e d a b o u t t h e f o r m o f r ( t ) . D e c o n v o l u t i o n u t i l i z i n g t h e most r e s o l v e d a v e r a g i n g f u n c t i o n i s s e l d o m u s e f u l on n o i s y s e i s m i c d a t a b e c a u s e t h e i n v e r s e f i l t e r w i l l e n h a n c e t h e n o i s e i n t h e d a t a t o u n a c c e p t a b l y l a r g e v a l u e s . The n o i s y s e i s m o g r a m i s w r i t t e n set) = ret) * wet) * rut) 25 C o n v o l v i n g t h i s w i t h an i n v e r s e f i l t e r v ( t ) produces SCO * VCO = rCO* wet) * vft) + nco * vet) = rot)* act) + net)* VCt) = < r a)7 -h £< r f t )> 3,5" where £<r(t)> i s an e r r o r term a r i s i n g from the a d d i t i v e n o i s e . In t h i s case i t w i l l be d e s i r a b l e t o f i n d an i n v e r s e f i l t e r which produces the most r e s o l v e d a v e r a g i n g f u n c t i o n and a l s o keeps the e r r o r term minimum. Data e r r o r s can be accomodated i n the i n v e r s e f i l t e r s o l u t i o n by m i n i m i z i n g a s t a t i s t i c a l norm of £<r(t)>. The s t a t i s t i c a l form of the e r r o r term which i s m a t h e m a t i c a l l y complimentary t o the f u n c t i o n a l form of the r e s o l u t i o n measure 3.3, i s i t s v a r i a n c e which i s denoted By s i m u l t a n e o u s l y m i n i m i z i n g the e r r o r v a r i a n c e Y and the r e s o l u t i o n measure (f) , an optimum i n v e r s e f i l t e r v ( t ) may be d e r i v e d . L e t where O i s c a l l e d the t r a d e o f f parameter and v a r i e s between 0 and IT/2. A t r a d e o f f between r e s o l u t i o n and s t a t i s t i c a l 26 error produced by the r e s u l t i n g inverse f i l t e r may be obtained by minimizing 3.6 using d i f f e r e n t values of 0 . When © = 0 , the v(t) which minimizes 3 .6 y i e l d s the most resolved averaging function, but the r e f l e c t i v i t y averages derived from s ( t ) * v ( t ) w i l l have the greatest uncertainty. As © increases, resolution w i l l be l o s t but greater s t a t i s t i c a l accuracy w i l l be achieved. By using t h i s philosophy, an inverse f i l t e r may be derived at any value of © and that one which provides the best compromise between resolution and accuracy must be chosen. Since (f) i s not c a l c u l a b l e , i t i s convenient to define a resolution measure 'L' as being the inverse height of the averaging function at i t s energy peak. Then, for example, as Q increases from 0 to we may plot a tradeoff curve to monitor resolution L and variance Y . The result would look something l i k e : increasing * eso Tradeoff Curve variance \ * \ \e >o y xe=*/2 decreasing resolution 27 In p r a c t i c a l a p p r a i s a l d e c o n v o l u t i o n , the i n v e r s e f i l t e r i s d e r i v e d a t a nonzero v a l u e of the t r a d e o f f parameter so t h a t b e t t e r s t a t i s t i c a l a c c u r a c y w i l l be a t t a i n e d . Very o f t e n a d r a m a t i c d e c r e a s e i n the v a r i a n c e can be a c h i e v e d w i t h o n l y a s m a l l s a c r i f i c e i n r e s o l u t i o n as i n d i c a t e d by a s t e e p i n i t i a l d e c l i n e of the t r a d e o f f c u r v e . B) Frequency domain a p p r a i s a l d e c o n v o l u t i o n D e c o n v o l u t i o n i s c a r r i e d out i n the f r e q u e n c y domain by s p e c t r a l d i v i s i o n . I f S ( f ) , R ( f ) , and W(f) a r e r e s p e c t i v e l y the F o u r i e r t r a n s f o r m s of s ( t ) , r ( t ) , and w ( t ) , where the F o u r i e r t r a n s f o r m p a i r i s d e f i n e d a s , -oo then the c o n v o l u t i o n theorem a p p l i e d t o 3.1 y i e l d s A s p e c t r a l d i v i s i o n g i v e s 28 where W*(f) i s the complex c o n j u g a t e of W ( f ) . I t i s w e l l known t h a t t h i s e q u a t i o n w i l l produce e r r o n e o u s r e s u l t s a t f r e q u e n c i e s where W(f) i s s m a l l i f the d a t a a r e i n a c c u r a t e , so 3.7 i s n e c e s s a r i l y a l t e r e d by a d d i n g a s t a b l i z i n g term t o the denominator. For i n a c c u r a t e d a t a , the s t a b l i z i n g term has been d e r i v e d i n the f r e q u e n c y domain by Oldenburg, ( 1 9 8 1 ) . In a manner q u a n t i t a t i v e l y c o n s i s t e n t w i t h the time domain f o r m u l a t i o n , the f r e q u e n c y domain a p p r a i s a l e q u a t i o n may be w r i t t e n where o( i s a c o n s t a n t dependent on the magnitude of random n o i s e i n the d a t a . and i s the t r a d e o f f parameter. The i n s t a b i l i t i e s i n the s p e c t r a l d i v i s i o n w i l l now be overcome i f © i s s u f f i c i e n t l y l a r g e . E q u a t i o n 3.8 can be w r i t t e n where V ( f ) i s d e f i n e d as the q u a n t i t y i n c u r l y b r a c k e t s . T a k i n g 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 3.9 y i e l d s where v ( t ) i s the i n v e r s e f i l t e r i n the time domain. Thus 29 the r e s u l t s of a p p r a i s a l i n the fr e q u e n c y domain are c o m p l e t e l y analagous t o t h o s e i n the time domain. Each a p p r a i s a l c a r r i e d out a t a t r a d e o f f v a l u e Q y i e l d s unique r e f l e c t i v i t y a v e r a g e s , a s t a t i s t i c a l v a r i a n c e of those a v e r a g e s , and an a v e r a g i n g f u n c t i o n . A l l a p p r a i s a l d e c o n v o l u t i o n s p r e s e n t e d i n t h i s work were c a r r i e d out i n the fr e q u e n c y domain because of the c o m p u t a t i o n a l and c o n c e p t u a l ease of w o r k i n g i n t h a t domain. C) Impedance computation and a p p r a i s a l An a p p r a i s a l shows t h a t the r e f l e c t i v i t y a v e r a g e s , t h e i r s t a t i s t i c a l e r r o r , ' and the a s s o c i a t e d a v e r a g i n g f u n c t i o n c o m p l e t e l y summarize our knowledge of the t r u e r e f l e c t i v i t y f u n c t i o n when the wavelet i s known. However, i n the event t h a t the output from an a p p r a i s a l i s used t o compute an impedance f u n c t i o n , some c a r e must be t a k e n t o d i f f e r e n t i a t e between an i n t e r p r e t a b l e , broadband r e f l e c t i v i t y model, and averages of the r e f l e c t i v i t y model. An i n t e r p r e t a b l e , broadband r e f l e c t i v i t y model i s one which produces an adequate f i t t o the d a t a and p r o v i d e s a m e a n i n g f u l impedance e s t i m a t e . In g e n e r a l , r e f l e c t i v i t y a v erages cannot be s u c c e s s f u l l y used a l o n e t o p r o v i d e a u s e f u l impedance f u n c t i o n because they l a c k i m p o r t a n t low f r e q u e n c i e s . For example, i n t e g r a t i n g the b a s i c model e q u a t i o n 3.1 by c o n v o l v i n g i t w i t h t w i c e the u n i t s t e p 30 f u n c t i o n u ( t ) , g i v e s D e n o t i n g the l e f t hand s i d e s ^ t ) , as the i n t e g r a t e d seismogram and making use of e q u a t i o n 1 . 9 g i v e s Next c o n v o l v i n g both s i d e s by the b e s t i n v e r s e f i l t e r of the wavelet produces $&) * vct) = 7 a ) * woo *va) - ' ^ t t ) * a t O = < 7 < 0 ? T h i s r e s u l t i n d i c a t e s t h a t by a p p r a i s i n g the i n t e g r a t e d seismogram, impedance a v e r a g e s , < ^ ( t ) > , a r e o b t a i n e d . The c o n v o l u t i o n of the i n t e g r a t e d d a t a w i t h the i n v e r s e f i l t e r p roduces an output e q u a l t o the c o n v o l u t i o n of the t r u e 3 1 impedance model with the same averaging function obtained from the r e f l e c t i v i t y a p p r a i s a l . For example: act) The resolving c a p a b i l i t y of the averaging function, in t h i s case, i s not as important as i t s low frequency content. Since the frequency content of the averaging function i s dependent on that of the wavelet, i t i s missing both low and high frequencies; and i t i s the lack of low frequencies in the averaging function which leads to the uninterpretable impedance estimate < ^ ( t ) > shown above. Cl e a r l y there are troughs and peaks in the impedance averages which do not exist in the true impedance model and the major s t r u c t u r a l trends are absent. Ref l e c t i o n seismologists' have generally known that bandlimited impedance averages, when considered as a f i n a l model, w i l l usually not produce correct s t r u c t u r a l i n t e r p r e t a t i o n s . Thus, techniques of recovering the low 32 f r e q u e n c y impedance i n f o r m a t i o n from w e l l l o g or v e l o c i t y a n a l y s e s have been d e v e l o p e d ( G a l b r a i t h and M i l l i n g t o n , 1979; L i n d s e t h , 1 9 7 9 ) . In a d d i t i o n (or a l t e r n a t i v e l y ) , the o p e r a t i n g p h i l o s o p h y of the s e i s m o l o g i s t s h o u l d i n c l u d e the i d e a of c o n s t r u c t i n g broadband models from the a p p r a i s e d s e i s m i c d a t a , t o h e l p c l a r i f y the i n t e r p r e t a t i o n s . In the next c h a p t e r , r e f l e c t i v i t y a verages o b t a i n e d from an a p p r a i s a l a re used i n two a l g o r i t h m s which c o n s t r u c t broadband r e f l e c t i v i t y (and thus impedance) e s t i m a t e s . A comment about uniqueness may be made by r e f l e c t i n g on the b a s i c model e q u a t i o n s ( t ) = r ( t ) * w ( t ) . There a r e i n f i n i t e l y many f u n c t i o n s r ( t ) which s a t i s f y t h i s e q u a t i o n ( g i v e n the b a n d l i m i t e d n a t u r e of the w a v e l e t ) and t h i s number i s f u r t h e r e n l a r g e d i f we a c c e p t t h a t the da t a a re i n a c c u r a t e and,permit f u n c t i o n s which generate s t a t i s t i c a l l y a l l o w a b l e m i s f i t s t o those d a t a . T h i s group of p e r m i s s i b l e f u n c t i o n s may w e l l g i v e r i s e t o a wide range of computed impedance f u n c t i o n s , u n l e s s some way of c o n s t r a i n i n g the n a t u r e of the c o n s t r u c t e d r e f l e c t i v i t y f u n c t i o n s i s . d e v i s e d . D) An example of a p p r a i s a l d e c o n v o l u t i o n d e r i v e d from an i n t e r p r e t a t i o n of w e l l l o g d a t a and r e p r e s e n t s a r e a s o n a b l e p i c t u r e of g e n e r a l s u b s u r f a c e impedance i n p a r t s of A l b e r t a . T h i s impedance model w i l l be The impedance shown i n f i g u r e 9a was 33 used as a s t a r t i n g p o i n t f o r s y n t h e t i c examples of a p p r a i s a l d e c o n v o l u t i o n . A r e f l e c t i v i t y f u n c t i o n ( p a n e l b) was computed from the impedance model v i a e q u a t i o n 1.7 and c o n v o l v e d w i t h a w a v e l e t ( p a n e l c ) , t o produce a s y n t h e t i c seismogram s ( t ) . Random n o i s e h a v i n g a STD of 20% of m a x | s ( t ) | was added t o t h i s r e s u l t , p r o v i d i n g the n o i s y seismogram shown i n p a n e l d. The time sampling i n t e r v a l f o r t h i s example i s 4msec. The c o r r e s p o n d i n g f r e q u e n c y domain r e p r e s e n t a t i o n of t h i s p rocedure i s shown i n p a n e l s e-g; where o n l y the a m p l i t u d e spectrum of p o s i t i v e f r e q u e n c i e s a r e d i s p l a y e d . The a p p r a i s a l s , c a r r i e d out i n the frequency domain a t seven d i f f e r e n t v a l u e s of t r a d e o f f parameter Q, are shown i n f i g u r e 10. The r e l a t i v e r e s o l u t i o n and v a r i a n c e measures 1 (where LQ s i m p l y r e f e r s t o the f i r s t v a l u e computed). The n o r m a l i z e d a v e r a g i n g f u n c t i o n s f o r each v a l u e of © are p l o t t e d i n the f i r s t column of f i g u r e 10. The n o r m a l i z e d r e f l e c t i v i t y a verages a l o n g w i t h t h e i r a m p l i t u d e s p e c t r a are p l o t t e d i n column 2; and the n o r m a l i z e d impedance averages a r e shown i n column 3. Note the change i n c h a r a c t e r of the averages and i n t h e i r s p e c t r a l c o n t e n t as the t r a d e o f f parameter i s i n c r e a s e d . S t a b i l i t y i s a c h i e v e d around where f r e q u e n c i e s o u t s i d e the bandrange of the w a v e l e t have been s i g n i f i c a n t l y reduced i n i m p o r t a n c e . d e f i n e d by L/L~ and are l i s t e d i n t a b l e 3 ^ 1-1 ( a D TIME (sec) 2 i 5 6 1OO ret") b (D 0.0 O.B 1 1 1 1 i i i i i 1RCOI Y— wa) c o oo _ o o f iwcoi a CO 0.0 4.0 - 1 i i i 3 FREQUENCY 125Hz iscol F i g . 9. S y n t h e t i c models f o r an example o f a p p r a i s a l d e c o n v o l u t i o n . 35 1 2 3 k A . A A A — JV— ) FHEOUCMCT 125 H a n o r m a(l-©) n o r m <ttt;e)> l R t f ) l n o r m < (^t;e)> F i g . 10. A p p r a i s a l de convolutions shown u s i n g seven values of t r a d e o f f parameter 36 A l t h o u g h the r e s u l t i n g a v e r a g e s are u n d e s i r e a b l e f o r impedance i n t e r p r e t a t i o n , t hey a r e unique and may be u t i l i z e d i n a c o n s t r u c t i o n procedure which l e a d s t o broadband forms of r e f l e c t i v i t y and impedance. The t o p i c of r e f l e c t i v i t y c o n s t r u c t i o n w i l l be c o v e r e d n e x t . e L A 0 .001 1,0 1 .0 ,01 1 .ST .1 7.0 . 0 1 4 .sr 2.5" .002) i.o 3 . 5 - .QOOiX 1 . 4 5.IS .0002S 1 S2 10.0 .00006Z Table 1. R e l a t i v e r e s o l u t i o n and v a r i a n c e measures f o r the a p p r a i s a l s shown i n f i g . 10. 37 4 R e f l e c t i v i t y C o n s t r u c t i o n S e i s m i c r e f l e c t i o n a n a l y s i s , thus f a r , has l e d t o the f o r m a t i o n of unique i n f o r m a t i o n about the r e f l e c t i v i t y f u n c t i o n based upon the method of a p p r a i s a l d e c o n v o l u t i o n . I t has been shown t h a t the o n l y unique i n f o r m a t i o n o b t a i n a b l e through use of the c o n v o l u t i o n a l model i s i n the form of r e f l e c t i v i t y averages < r ( t ) > = r ( t ) * a ( t ) , where a ( t ) i s an a v e r a g i n g f u n c t i o n . I t was a l s o shown t h a t the o n l y unique impedance i n f o r m a t i o n r e c o v e r a b l e by a p p r a i s a l methods i s i n the form of impedance averages < r*Jit)>= v£(t)*a(t) , where a ( t ) i s the same a v e r a g i n g f u n c t i o n as above. I t was noted t h a t impedance averages a re u n a c c e p t a b l e as a f i n a l model of s u b s u r f a c e s t r u c t u r e because of the b a n d l i m i t e d n a t u r e of a ( t ) . S i n c e a p p r a i s a l methods ( i . e . , i n v e r s e f i l t e r i n g methods) must always f a i l t o g e n e r a t e an a c c e p t a b l e impedance f u n c t i o n i f the source w a v elet i s m i s s i n g low f r e q u e n c i e s , i t i s w o r t h w h i l e i n v e s t i g a t i n g whether a p a r t i c u l a r c o n s t r u c t i o n method might s u c c e s s f u l l y f i l l i n the m i s s i n g f r e q u e n c y i n f o r m a t i o n about r ( t ) . In f a c t , s i n c e the unique i n f o r m a t i o n about r ( t ) r e c o v e r a b l e by a p p r a i s a l methods r e p r e s e n t s o n l y a b a n d l i m i t e d p o r t i o n of i t s spectrum (e g . , f i g u r e s 8 & 1 0 ) , the r e c o v e r y of m i s s i n g p o r t i o n s of R ( f ) w i l l r e q u i r e some s o r t of c o n s t r u c t i o n t e c h n i q u e . 38 The problem of broadband r e f l e c t i v i t y r e c o v e r y as d e s c r i b e d h e r e i n may be thought of as f o l l o w s . F i r s t an a p p r a i s a l of the seismogram i s performed which r e p l a c e s each o v e r l a p p i n g w a v e l e t p r e s e n t i n the d a t a w i t h a z e r o phase a v e r a g i n g f u n c t i o n . Then by use of a c o n s t r u c t i o n method, each z e r o phase a v e r a g i n g f u n c t i o n i s r e p l a c e d by a 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 r e p r e s e n t a t i v e of a s u b s u r f a c e boundary. Two c o n s t r u c t i o n p r o c e d u r e s w i l l be d e s c r i b e d which have shown s u c c e s s i n a c c o m p l i s h i n g t h i s g o a l , but f i r s t an example i s p r e s e n t e d which emphasizes the nonuniqueness i n h e r e n t i n any r e f l e c t i v i t y c o n s t r u c t i o n p r o c e d u r e . C o n s i d e r the d u a l example shown i n f i g u r e 1 1 . In p a n e l (a) i s a b l o c k y (or minimum s t r u c t u r e ) impedance f u n c t i o n . A more r e a l i s t i c impedance f u n c t i o n (a n o i s y v e r s i o n of the of the b l o c k y impedance model) i s shown i n p a n e l ( b ) . D i r e c t l y below each impedance model i s the r e f l e c t i v i t y f u n c t i o n computed from each v i a e q u a t i o n 1 . 6 . A d j a c e n t t o each r e f l e c t i v i t y f u n c t i o n i s i t s a s s o c i a t e d a m p l i t u d e spectrum. The b a n d l i m i t e d ( 1 0 - 5 0 H Z ) averages of rj ( t ) and r ^ ( t ) shown i n p a n e l s (h) and ( i ) , are v e r y s i m i l a r . T h e r e f o r e , i t i s r e a s o n a b l e t o expect t h a t any c o n s t r u c t i o n a l g o r i t h m c a r r i e d out i d e n t i c a l l y on both < r j ( t ) > and < r ^ ( t ) > , u t i l i z i n g o n l y the b a n d l i m i t e d i n f o r m a t i o n s u p p l i e d by each, would l e a d t o s i m i l a r broadband r e f l e c t i v i t y F i g . 11. The nonuniqueness inherent i n r e f l e c t i v i t y measurements. 40 models. F u r t h e r , t h e s e c o n s t r u c t e d models may not resemble e i t h e r ( t ) or ^ ( t ) ^ n o c o n s t r a i - n t i s p l a c e d on the form of the c o n s t r u c t e d o utput because t h e r e are i n f i n i t e l y many r e f l e c t i v i t y models which have averages s i m i l a r t o t h o s e shown i n p a n e l s (h) and ( i ) . The s i z e of model space may, however, be r e s t r i c t e d by c o n s i d e r i n g o n l y a p a r t i c u l a r c l a s s of models. Both c o n s t r u c t i o n methods d i s c u s s e d i n t h i s t h e s i s tend t o f a v o u r , i n t h e o r y , the c o n s t r u c t i o n of models s i m i l a r t o r j ( t ) , t h a t i s , models w i t h as few s i g n i f i c a n t r e f l e c t i o n c o e f f i c i e n t s as p o s s i b l e . The f i r s t of t h e s e methods, an a u t o r e g r e s s i v e (AR) t e c h n i q u e , makes the assumption t h a t the r e f l e c t i v i t y spectrum R ( f ) behaves i n a p r e d i c t a b l e manner. I t w i l l be shown t h a t t h i s method can a c c u r a t e l y r e c o n s t r u c t the m i s s i n g low frequency p o r t i o n s of a r e f l e c t i v i t y spectrum. A) A u t o r e g r e s s i v e e x t e n s i o n of the r e f l e c t i v i t y spectrum I t i s w e l l known t h a t the maximum e n t r o p y method (MEM) of power spectrum a n a l y s i s , when a p p l i e d t o a s h o r t p o r t i o n of a s t a t i o n a r y d a t a s e r i e s , w i l l i n some c a s e s , p r o v i d e a more r e s o l v e d s p e c t r a l e s t i m a t e than t h a t from any c o n v e n t i o n a l F o u r i e r t r a n s f o r m method. T h i s i s because c o n v e n t i o n a l s p e c t r a l e s t i m a t o r s assume t h a t the d a t a a r e z e r o o u t s i d e the known i n t e r v a l , whereas the MEM e s t i m a t e assumes t h a t the d a t a a r e ' c o n t i n u e d i n a s e n s i b l e way' 41 o u t s i d e t h a t known i n t e r v a l . By ' c o n t i n u e d i n a s e n s i b l e way', i t i s meant t h a t the p r o c e s s which produced the d a t a s e r i e s i s m o d e l l e d i n a manner which a l l o w s p r e d i c t i o n of unknown d a t a . For example, o n l y a f r a c t i o n of one p e r i o d of a s i n e wave need be known i n o r d e r t o i n f e r i t s complete unending form. We must o n l y s p e c i f y t h a t the p r o c e s s i n d i c a t e d by the known f r a c t i o n i s monochromatic. A g e n e r a l p r o c e s s which r e q u i r e s o n l y a s m a l l p o r t i o n of the d a t a s e r i e s t o d e r i v e an adequate model f o r the whole i s an a u t o r e g r e s s i v e (AR) p r o c e s s . I t can be shown t h a t the MEM s p e c t r a l e s t i m a t o r assumes an AR model f o r the s t a t i o n a r y data s e r i e s t o be examined (eg . , U l r y c h and B i s h o p , 1975). The AR r e p r e s e n t a t i o n of a s t a t i o n a r y d a t a s e r i e s x w i t h z e r o mean, i s P where e i s a w h i t e n o i s e s e r i e s w i t h z e r o mean and v a r i a n c e OQ . And the are p r e d i c t i o n f i l t e r c o e f f i c i e n t s of o r d e r p. T h i s e x p r e s s i o n i n d i c a t e s t h a t a f u t u r e v a l u e x ^ can be p r e d i c t e d t o w i t h i n an e r r o r e^, from X K - 1 ' x k ~ 2 x k-p' t h - a t i s , from 'p' p a s t v a l u e s of x. For t h i s r e a s o n , e q u a t i o n 4.1 i s c a l l e d a f o r w a r d p r e d i c t i o n 42 r e p r e s e n t a t i o n . A r e v e r s e p r e d i c t i o n e x p r e s s i o n may be w r i t t e n p a s t v a l u e x ^ can be p r e d i c t e d t o w i t h i n an e r r o r e£ , from xX+i ' x K * l ' * * " ' x K*p' t ^ i a t * s ' £ r o m 'P' f u t u r e v a l u e s of x. I t f o l l o w s , t h e n , t h a t f o r an AR p r o c e s s x, p a s t and f u t u r e v a l u e s can be p r e d i c t e d from any s e t of N>p known v a l u e s of x. I f the p r e d i c t i o n f i l t e r c o e f f i c i e n t s are de t e r m i n e d by m i n i m i z i n g the v a r i a n c e of the e r r o r s e r i e s , the r e v e r s e and f o r w a r d s o l u t i o n s a r e i d e n t i c a l . Thus, a s i n g l e s e t of f i l t e r c o e f f i c i e n t s may be found by m i n i m i z i n g the sum of the f o r w a r d and r e v e r s e e r r o r v a r i a n c e s . T h i s sum i s e s t i m a t e d by the f o l l o w i n g forumula P H.2 where the a r e r e v e r s e p r e d i c t i o n f i l t e r c o e f f i c i e n t s and e r a l s o has v a r i a n c e (J^. T h i s r e p r e s e n t a t i o n shows t h a t a N 43 D i f f e r e n t i a t i n g 4.3 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 l e a d s t o the f o l l o w i n g m a t r i x e q u a t i o n C ^ r q 7.7 where 4 - .-.p and r Co(Vjo(2 }. . . . o^p ) The o('s may be o b t a i n e d by i n v e r t i n g e q u a t i o n 4.4. A l t e r n a t i v e l y , the p r e d i c t i o n c o e f f i c i e n t s can be d e t e r m i n e d e f f i c i e n t l y by m i n i m i z i n g 4.3 u s i n g the Burg-L e v i n s o n r e c u r s i o n t e c h n i q u e d e s c r i b e d by U l r y c h and B i s h o p (1975). Both s o l u t i o n s e f f e c t a l e a s t squares f i t of an AR model of o r d e r 'p' t o the d a t a s e r i e s x. The f i r s t method of c o n s t r u c t i n g a broadband r e f l e c t i v i t y f u n c t i o n uses an AR model of the r e f l e c t i v i t y spectrum t o p r e d i c t v a l u e s of R ( f ) o u t s i d e the f r e q u e n c y 44 band range of the source w a v e l e t . C o n s i d e r the b a s i c model e q u a t i o n s ( t ) = r ( t ) * w ( t ) and i t s F o u r i e r t r a n s f o r m S ( f ) = R ( f ) W ( f ) . I f W(f) c o n t a i n s s i g n i f i c a n t energy i n the band of f r e q u e n c i e s between f j and f 2 > then the r e f l e c t i v i t y spectrum R ( f ) = S ( f ) / w ( f ) can be a d e q u a t e l y r e c o v e r e d o n l y i n t h a t f r e q u e n c y band. T h i s band of r e c o v e r e d f r e q u e n c i e s r e p r e s e n t s a s m a l l p o r t i o n of the complex d a t a s e r i e s R ( f ) . Thus, i f R ( f ) can be m o d e l l e d as an AR p r o c e s s of some o r d e r , then the r e c o v e r e d p o r t i o n of R ( f ) may be used t o e s t i m a t e t h a t AR model which w i l l a l l o w a s e n s i b l e e x t e n s i o n of R ( f ) o u t s i d e f r e q u e n c i e s f j and f ^ . I m p o r t a n t l y , a proper c h o i c e of order i s n e c e s s a r y f o r good p r e d i c t i o n r e s u l t s . For the r e f l e c t i v i t y c o n s t r u c t i o n problem, i t w i l l be shown l a t e r how the c h o i c e of p i s r e l a t e d t o the number of major s u b s u r f a c e b o u n d a r i e s or r e f l e c t i o n c o e f f i c i e n t s . A t t e m p t i n g t o show t h a t the r e f l e c t i v i t y spectrum R ( f ) ' may be r e p r e s e n t e d as an AR p r o c e s s , the r e f l e c t i v i t y r ( t ) i s f i r s t w r i t t e n i n terms of a sampl i n g f u n c t i o n n-o where A t i s the sampling i n t e r v a l , N i s the number of data samples, r n i s the r e f l e c t i o n c o e f f i c i e n t a t each sample i n t e r v a l , and ^ ( t - n a t ) i s the D i r a c s a m p l i n g f u n c t i o n . The 4 5 d i s c r e t e F o u r i e r t r a n s f o r m of 4.5 i s Rj= | , r n c^ , r ^ ;M. J . . . . W - . 16 where the s u b s c r i p t j r e f e r s t o the j t h freq u e n c y fy=j/N4t and where v a l u e s of j l a r g e r than N/2 r e f e r t o n e g a t i v e f r e q u e n c i e s (N/2 b e i n g the N y q u i s t i n d e x ) . The i n v e r s e d i s c r e t e F o u r i e r t r a n s f o r m i s d e f i n e d as E x p r e s s i o n 4.6 may a l s o be w r i t t e n KM r u o rV-l These r e l a t i o n s w i l l be r e f e r r e d t o as r e f l e c t i v i t y c o n s t r a i n t e q u a t i o n s . From e q u a t i o n 4.6, the r e f l e c t i v i t y spectrum i s a sum of complex s i n u s o i d s . U l r y c h (1973), and U l r y c h and C l a y t o n (1976) have shown t h a t harmonic f u n c t i o n s may be a d e q u a t e l y m o d e l l e d as AR p r o c e s s e s . The o r d e r of a s i n g l e complex s i n u s o i d i s p=1, and i n g e n e r a l , the o r d e r of a complex 46 harmonic p r o c e s s i s e q u a l t o the number of complex s i n u s o i d s of d i f f e r e n t p e r i o d i n c l u d e d i n t h e d a t a . From 4.6, the number of complex s i n u s o i d s of d i f f e r e n t p e r i o d (T^=N&t/n) i n c l u d e d i n the r e f l e c t i v i t y spectrum i s e q u a l t o the number of nonzero r e f l e c t i o n c o e f f i c i e n t s r ^ . Thus, the o r d e r 'p' of the d i s c r e t e r e f l e c t i v i t y spectrum and the number of 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 d i r e c t l y r e l a t e d . For c o m p u t a t i o n a l c o n s i d e r a t i o n s , the o r d e r of R must be l e s s than the number of s p e c t r a l v a l u e s r e c o v e r e d between fj and f2« T h i s p u t s a l i m i t on the number of s i g n i f i c a n t r e f l e c t i o n c o e f f i c i e n t s which may e x i s t w i t h i n the data window b e i n g c o n s i d e r e d . I f 20 complex frequency v a l u e s of R a r e known, then t h e r e s h o u l d be l e s s than 20 s i g n i f i c a n t r e f l e c t i o n c o e f f i c i e n t s c o n t r i b u t i n g energy t o t h o s e known f r e q u e n c i e s t o i n s u r e a prop e r d e t e r m i n a t i o n of the AR model. The AR r e p r e s e n t a t i o n of the r e f l e c t i v i t y spectrum may be w r i t t e n P where a l l terms a r e complex. T h i s e x p r e s s i o n i s used t o exte n d v a l u e s of R t o f r e q u e n c i e s o u t s i d e the known range f^ and f ^ . P a r t of the s u c c e s s of t h i s AR e x t e n s i o n method depends on a c c u r a t e l y d e t e r m i n i n g the orde r 'p' of the 47 r e f l e c t i v i t y spectrum, i . e . , d e t e r m i n i n g the optimum number of s i g n i f i c a n t r e f l e c t i o n c o e f f i c i e n t s e x p e c t e d i n the c o n s t r u c t e d r e s u l t g i v e n M known complex fre q u e n c y v a l u e s . Optimum c r i t e r i a f o r d e t e r m i n i n g the o r d e r of a data s e r i e s have been d i s c u s s e d by U l r y c h and B i s h o p (1975), and U l r y c h and C l a y t o n (1976). However, c h o o s i n g an o r d e r of 2/3 t o 3/4 the number of r e c o v e r e d f r e q u e n c y samples p r o v i d e s good r e s u l t s i f t h a t number exceeds the number of s i g n i f i c a n t r e f l e c t i o n c o e f f i c i e n t s e x p e c t e d i n the s o l u t i o n (an a v e r a g i n g p r o c e d u r e i s d e s c r i b e d i n the next c h a p t e r which s t a b l i z e s the c h o i c e of AR o r d e r ) . Having chosen a good o r d e r , the p r e d i c t i o n f i l t e r c o e f f i c i e n t s a re det e r m i n e d and then e q u a t i o n 4.8 i s implemented i n a f o r w a r d manner t o extend the h i g h frequency end of R , and i n a r e v e r s e manner to f i l l i n the m i s s i n g low f r e q u e n c i e s . The s y n t h e t i c example shown i n f i g u r e 12 demonstrates the a b i l i t y of the AR t e c h n i q u e t o r e c o v e r low freq u e n c y impedance s t r u c t u r e . An i d e a l s t a r t i n g p o i n t f o r the AR method i s an a c c u r a t e p o r t i o n of a r e f l e c t i v i t y spectrum. The 20 e s t i m a t e s of R between 10 and 50 Hz shown i n p a n e l ( a ) , r e p r e s e n t an a c c u r a t e p o r t i o n of the t r u e spectrum shown i n p a n e l ( j ) . P a n e l s (b) and (c) show the c o r r e s p o n d i n g b a n d l i m i t e d r e f l e c t i v i t y and impedance f u n c t i o n s . The b a n d l i m i t e d and t r u e impedance f u n c t i o n s a r e compared i n p a n e l ( c ) . 48 CD O ~ o o 10 1 Jo 1 1 1 . AA A (L In . . 1 \l \ \ v v u b) < r a ) > . CD o ~ • • w — r l o d) 1 1 1 1 1 CD o — 1 CJ o I 1 1 1 1 1 1 i ; CD O ~ CD „ — . i , y - 3 6 TIME IO r < ° O i i i i i i fREQUENCY 1 25 |RCe)| 0 I) —•< / .36 TIME F i g . 12. A u t o r e g r e s s i v e r e c o n s t r u c t i o n o f a r e f l e c t i v i t y f u n c t i o n c o n t a i n i n g a few r e f l e c t o r s . 4 9 P a n e l (d) shows the r e s u l t of e x t e n d i n g the spectrum i n t o the low frequ e n c y range o n l y , u s i n g AR p r e d i c t i o n and an o r d e r of 15 (which i s g r e a t e r than the number of s p i k e s i n the t r u e r e f l e c t i v i t y shown i n p a n e l ( k ) ) . P a n e l (e) i l l u s t r a t e s t h a t the r e s u l t i n g c o n s t r u c t e d r e f l e c t i v i t y f u n c t i o n has not improved i n appearance, however, the r e s u l t i n g impedance f u n c t i o n ( p a n e l ( f ) ) c o n t a i n s the proper low f r e q u e n c y i n f o r m a t i o n needed t o g i v e i t g e o l o g i c meaning. P a n e l s ( g ) - ( i ) show the r e s u l t of p r e d i c t i n g both the low and h i g h m i s s i n g f r e q u e n c i e s . The r e f l e c t i v i t y f u n c t i o n ( p a n e l (h)) i s p l o t t e d as a s p i k e s e r i e s s i m p l y because i t has a complete spectrum w i t h i n the N y q u i s t f r e q u e n c y . The r e s u l t i n g impedance s t r u c t u r e (panel ( i ) ) i n d i c a t e s t h a t the p r e d i c t e d h i g h f r e q u e n c y v a l u e s a r e somewhat e r r o n e o u s . T h i s i s a common o c c u r r e n c e among the v a r i o u s r e f l e c t i v i t y models t h a t we have t e s t e d u s i n g the AR method. I t appears l e s s p r o f i t a b l e t o p r e d i c t the v e r y h i g h f r e q u e n c i e s because of s m a l l u n c e r t a i n t i e s i n 'determining the proper p r e d i c t i o n f i l t e r c o e f f i c i e n t s . These u n c e r t a i n t i e s may propagate i n t o l a r g e e r r o r s i n the p r e d i c t e d s p e c t r a l v a l u e as the p r e d i c t i o n f i l t e r s l i d e s o f f the edge of the known da t a and b e g i n s t o p r e d i c t s p e c t r a l v a l u e s e x c l u s i v e l y from p r e v i o u s l y p r e d i c t e d v a l u e s . F o r t u n a t e l y , the s m a l l band of m i s s i n g low f r e q u e n c i e s can, i n many c a s e s , be a c c u r a t e l y 50 r e c o v e r e d . F i g u r e 13 shows the r e s u l t s of a p p l y i n g the AR method t o a more c o m p l i c a t e d example. P a n e l s ( j ) - ( l ) show the r e s u l t s of ad d i n g 10% w h i t e n o i s e t o the s i m p l e s y n t h e t i c impedance model p r e s e n t e d i n the p r e v i o u s example. The 18 s p e c t r a l e s t i m a t e s of R (shown i n p a n e l ( a ) ) , between 14 and 50Hz, a r e a c c u r a t e . The r e s u l t s of p r e d i c t i n g o n l y the m i s s i n g low f r e q u e n c i e s a r e v e r y e n c o u r a g i n g as has been shown i n p a n e l s ( d ) - ( f ) . The r e s u l t s of a l s o p r e d i c t i n g the m i s s i n g h i g h f r e q u e n c i e s ( p a n e l s ( g ) - ( i ) ) , show a s i m p l e r and perhaps more i n t e r p r e t a b l e r e f l e c t i v i t y f u n c t i o n but the r e s u l t i n g impedance e s t i m a t e has not been improved. An o r d e r of 12 was chosen f o r t h i s example. More examples of t h i s method a r e p r e s e n t e d i n s e c t i o n C) of t h i s c h a p t e r and i n the f i n a l c h a p t e r where i t i s a p p l i e d t o r e a l d a t a . B) Broadband r e f l e c t i v i t y c o n s t r u c t i o n u s i n g the l j n o r m and a l i n e a r programming a l g o r i t h m Broadband r e f l e c t i v i t y r e c o v e r y i n the time domain by m i n i m i z i n g the l j norm l e a d s n a t u r a l l y t o a l i n e a r programming (LP) f o r m u l a t i o n ( C l a e r b o u t and M u i r , 1 973; T a y l o r e t a l , 1979), but l j norm m i n i m i z a t i o n has a l s o been used f o r r e f l e c t i v i t y c o n s t r u c t i o n i n the f r e q u e n c y domain. Levy and F u l l a g a r (1981) have f o r m u l a t e d a p r o c e d u r e t o 51 o 114 I J 1 1 1 b) <rC0> o ~ l I I I I I rc o 9) i 1 1 1 1 rcCt) 0 ; ? c ( 0 36 .36 i 1 r — i 6 f REQUENCt 125 j ) IRCO| ret) 0 F i g . 13. A u t o r e g r e s s i v e r e c o n s t r u c t i o n of a r e f l e c t i v i t y f u n c t i o n c o n t a i n i n g many r e f l e c t o r s . 52 c o n s t r u c t a r e f l e c t i v i t y f u n c t i o n c o n t a i n i n g a minimum number of r e f l e c t i o n c o e f f i c i e n t s . T h e i r method, c a r r i e d out i n the frequency domain, assumes only that p r o c e s s i n g of the seismogram has r e s u l t e d i n unique r e f l e c t i v i t y i n f o r m a t i o n ( r e f l e c t i v i t y averages), i . e . , a b a n d l i m i t e d p o r t i o n of the true r e f l e c t i v i t y spectrum has been recovered. T h i s method has an advantage in that the s o l u t i o n i s c o n s t r u c t e d from only the most r e l i a b l y known s p e c t r a l v a l u e s of R ( f ) . However, as formulated, i t does not take f u l l advantage of the unique r e f l e c t i v i t y averages. An o u t l i n e of Levy and F u l l a g a r ' s general LP approach w i l l be f o l l owed by an e x p l a n a t i o n of how t h e i r approach can be made more e f f i c i e n t u t i l i z i n g i n f o r m a t i o n from an a p p r a i s a l of the seismogram and i n f o r m a t i o n from v e l o c i t y or w e l l l o g a n a l y s e s . (a) General LP f o r m u l a t i o n An o u t l i n e of t h i s method begins with the sampled r e f l e c t i v i t y f u n c t i o n given p r e v i o u s l y as The o b j e c t i v e i s then to f i n d that set of r e f l e c t i o n c o e f f i c i e n t s r n which s a t i s f y the c o n s t r a i n t equations 4.7, N-l 53 and which have a minimum l f norm, that i s , llrcOU, - 2 lr n| is minimum M i n i m i z a t i o n of the 1| norm w i l l produce a set of r e f l e c t i o n c o e f f i c i e n t s with a minimum number of nonzero v a l u e s . As an a s i d e , u t i l i z i n g e x p r e s s i o n 1 . 7 , equation 4 . 9 may be r e w r i t t e n M i n i m i z i n g the 1( norm of the r e f l e c t i v i t y f u n c t i o n i s thus e q u i v a l e n t to minimizing the 1| norm of impedance g r a d i e n t s . This m i n i m i z a t i o n w i l l produce an impedance model with a minimum of s t r u c t u r a l v a r i a t i o n s . One way to formulate the LP s o l u t i o n i s by f i r s t o b t a i n i n g estimates of the r e f l e c t i v i t y spectrum R by s p e c t r a l d i v i s i o n of the seismogram where j r e f e r s to the j t h frequency as b e f o r e . But only those s p e c t r a l components, where W i s s u f f i c i e n t l y l a r g e , are used. Estimates of R may a l s o be obtained through many other methods of zero phase d e c o n v o l u t i o n . For example, they may be obtained from an a p p r a i s a l d e c o n v o l u t i o n . vo Rj=Sj/Wj III 5 4 R e l i a b l y d e t e r m i n e d s p e c t r a l v a l u e s are then s u b s t i t u t e d i n t o the c o n s t r a i n t e q u a t i o n s 4 . 7 and an LP a l g o r i t h m i s s e l e c t e d t o f i n d those r e f l e c t i o n c o e f f i c i e n t s r ^ which s a t i s f y e q u a t i o n s 4 . 7 and which have a minimum l j norm. Note t h a t i f both 4 . 1 1 and 4 . 7 a r e s a t i s f i e d i n the above p r o c e d u r e , the c o n s t r u c t e d r e f l e c t i v i t y f u n c t i o n , denoted by r c ( t ) , s h o u l d a d e q u a t e l y reproduce the o b s e r v e d seismogram i . e . , s ( t ) 2 V c ( t ) * w ( t ) . Most LP a l g o r i t h m s are d e s i g n e d t o l o c a t e o n l y p o s i t i v e unknowns, whereas both p o s i t i v e and n e g a 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 s a r e e x p e c t e d . The s o l u t i o n t o t h i s problem i s t o e x p r e s s each r^ as the d i f f e r e n c e between two p o s i t i v e quant i t i e s , i . e . , T h i s r e l a t i o n i s then s u b s t i t u t e d i n t o e q u a t i o n s 4 . 9 and 4 . 7 , and an LP a l g o r i t h m w i l l s e a r c h f o r the p o s i t i v e unknowns a ^ and b n . The number of unknowns i s d o u b l e d by t h i s p r o c e d u r e , t h e r e b y i n c r e a s i n g the computing c o s t s . However, i f no e x t r a i n f o r m a t i o n can be i n c o r p o r a t e d i n t o the LP a l g o r i t h m , the f o r m u l a t i o n o u t l i n e d above must be f o l l o w e d . 55 No attempt i s made i n the above f o r m u l a t i o n to u t i l i z e that unique i n f o r m a t i o n which c o u l d be s u p p l i e d by an a p p r a i s a l of the seismogram and by w e l l l o g or v e l o c i t y a n a l y s e s . T h i s i n f o r m a t i o n , when a v a i l a b l e , can make a v a l u a b l e c o n t r i b u t i o n to the LP s o l u t i o n . A s s i m i l a t i n g r e f l e c t i v i t y averages and impedance estimates i n t o the a l g o r i t h m l e a d s to a more e f f i c i e n t and r e l i a b l e s o l u t i o n of the LP r e f l e c t i v i t y c o n s t r u c t i o n problem. (b) The c o n s t r a i n e d LP s o l u t i o n In the c o n s t r a i n e d LP approach, the o b j e c t i v e i s to f i n d that set of r e f l e c t i o n c o e f f i c i e n t s r^ which s a t i s f y the c o n s t r a i n t equations, and which have a minimum weighted 1^  norm, that i s , N-l where the are weights meant to emphasize those . r n which are prominent i n the average values <r(t)>. A n a t u r a l weighting f u n c t i o n f o r t h i s approach i s the a r i t h m e t i c i n v e r s e of the r e f l e c t i v i t y averages, <r(t)>~'. T h i s i s because the LP a l g o r i t h m using l j norm m i n i m i z a t i o n , w i l l tend to reduce an r ^ i f i t has a l a r g e r e l a t i v e weight. Thus, i f the average values <r(t)> i n d i c a t e that a 56 p a r t i c u l a r r e f l e c t i o n c o e f f i c i e n t s h o u l d be near z e r o , t h a t i n t e r p r e t a t i o n i s a s s i s t e d i n the LP a l g o r i t h m by g i v i n g i t where <r^> r e f e r s t o the d i g i t i z e d a v e r a g e s , and q i s a w e i g h t i n g exponent r a n g i n g from 0 (no w e i g h t i n g ) t o 2 or more (0=1 or 2 i s u s u a l l y s u f f i c i e n t t o p r o v i d e good r e s u l t s ) . The r e f l e c t i v i t y averages may a l s o be used t o p r o v i d e p o l a r i t y i n f o r m a t i o n t o the LP a l g o r i t h m . D e f i n i n g a p o l a r i t y f u n c t i o n as a l a r g e weight r e l a t i v e t o o t h e r p o s s i b l e r n ' s . A n a t u r a l form of 4.13 i s then -l i( I if *rn> 2 O 57 the c o n s t r a i n t e q u a t i o n s may be r e w r i t t e n Real [Rj J - 1 r 1 coiC2TTi n/w) r 3 n C < v ) OM r \ = 0 where r^ =sgn (<r n> ) r ^ and j ranges over o n l y the most r e l i a b l e f r e q u e n c i e s as b e f o r e . Then the LP a l g o r i t h m may se a r c h f o r the s t r i c t l y p o s i t i v e unknowns r^ , e l i m i n a t i n g need f o r the d i f f e r e n c e e q u a t i o n 4.12. The t r u e p o l a r i t y and magnitude of the c o n s t r u c t e d r e f l e c t i o n c o e f f i c i e n t s may be r e c o v e r e d by a p p l y i n g r A = sgn (<r A> ) r ^ . A d d i t i o n a l c o n s t r a i n t e q u a t i o n s may be added t o those i n 4 . 1 6 i f impedance e s t i m a t e s a r e a v a i l a b l e from v e l o c i t y or w e l l l o g a n a l y s e s . From e q u a t i o n 1.5, impedance c o n s t r a i n t e q u a t i o n s may be w r i t t e n - ^ 1 S9h«rn>)rn ' 4,17 n-o where k+1 i s the index of the impedance t h a t i s known and eq u a t i o n 4.17 i s i n the p r o p e r form f o r i n p u t t o the c o n s t r a i n e d LP a l g o r i t h m . I f o n l y the basement impedance i s known ( i . e . , ), t h i s r e l a t i o n becomes e s s e n t i a l l y a 58 c o n s t r a i n t on the dc f r e q u e n c y R^. For example, the f o l l o w i n g two e q u a t i o n s p r o v i d e i d e n t i c a l c o n s t r a i n t s : rV-t i\=o The c o n s t r a i n e d LP a l g o r i t h m i s summarized i n f i g u r e 14. REFLECTIVITY Weighting Information Polarity Information AVERAGES Reliable Spectral Estimates R Impedance Estimates Objective Function Constraint Equations LP Algorithm Constructed Reflectivity F i g . 1^-. Flow diagram f o r the c o n s t r a i n e d LP a l g o r i t h m . 59 Before p r e s e n t i n g examples of the LP method, i t i s noted that by using the r e f l e c t i v i t y averages to c o n s t r a i n the c o n s t r u c t i o n problem, c e r t a i n r e s t r i c t i o n s are a u t o m a t i c a l l y p l a c e d on the r e s o l u t i o n of d i p o l e r e f l e c t i o n c o e f f i c i e n t s . For example, a t h i n shale wedged i n a porous sand formation might r e s u l t i n the f o l l o w i n g r e f l e c t i v i t y funct ion E x p e r i m e n t a l l y , the f o l l o w i n g r e f l e c t i v i t y averages would be recovered (from equation 3.4) rCti <fCO> T h e r e f o r e , t h i s d i p o l e can not be r e s o l v e d by e i t h e r a p p r a i s a l or c o n s t r u c t i o n methods. However, i f such d i p o l e s are s u f f i c i e n t l y seperated to be r e s o l v e d by the a p p r a i s a l , 60 then t h e s e f e a t u r e s , complete w i t h the low f r e q u e n c y i n f o r m a t i o n , may be r e c o v e r e d t h r o u g h c o n s t r u c t i o n p r o c e d u r e s . (c) Data e r r o r s and the LP s o l u t i o n In p r a c t i c a l c a s e s , the observed seismogram c o n t a i n s a c e r t a i n amount of n o i s e which w i l l a f f e c t subsequent p r o c e s s i n g . T h i s d a t a n o i s e t r a n s l a t e s i n t o an e r r o r f o r each r e c o v e r e d s p e c t r a l v a l u e R^ ' of the r e f l e c t i v i t y spectrum. In the presence of n o i s e , i t i s u n d e s i r e a b l e t o s o l v e the c o n s t r a i n t e q u a t i o n s e x a c t l y . Two ways of h a n d l i n g e r r o r s i n e s t i m a t e s of R have been d i s c u s s e d f o r t h i s LP approach by Levy and F u l l a g a r (1981) and w i l l be b r i e f l y p r e s e n t e d h e r e . 1) I n e q u a l i t y c o n s t r a i n t s Most LP a l g o r i t h m s a l l o w i n e q u a l i t y c o n s t r a i n t s , where each c o n s t r a i n t e q u a t i o n i s r e q u i r e d t o be s a t i s f i e d w i t h i n a s p e c i f i e d e r r o r bound. The r e f l e c t i v i t y c o n s t r a i n t e q u a t i o n s may be e a s i l y c o n v e r t e d i n t o i n e q u a l i t y c o n s t r a i n t s as f o l l o w s ±Real[R.i] + Ej > fL^COLbxtin/H) 61 where E.j and *C>j a r e s p e c i f i e d u n c e r t a i n t i e s a t the j t h f r e q u e n c y . In t h i s i n e q u a l i t y f o r m u l a t i o n , the LP a l g o r i t h m i s r e q u i r e d t o m i n i m i z e the 1| norm of the r ^ , w h i l e s a t i s f y i n g the i n e q u a l i t y r e l a t i o n s 4 . 2 0 . E r r o r terms E^ and c-j can be most s i m p l y chosen e m p i r i c a l l y by assuming a d a t a n o i s e l e v e l and then r e q u i r i n g the u n c e r t a i n t i e s t o be a f r a c t i o n of t h i s . Note t h a t the number of c o n s t r a i n t e q u a t i o n s i s doubled i n t h i s p r o c e d u r e . 2) E q u a l i t y c o n s t r a i n t s In the p r o c e s s of d e t e r m i n i n g those r e f l e c t i o n c o e f f i c i e n t s which have a minimum l j n o r m , the LP a l g o r i t h m may a l s o d i r e c t l y d e termine an e s t i m a t e of the e r r o r s E-j and £^ at each f r e q u e n c y . T h i s may be done by r e t a i n i n g the e q u a l i t y c o n s t r a i n t r e l a t i o n s , but i n t r o d u c i n g the u n c e r t a i n t i e s as a d d i t i o n a l unknowns J J r \ = o I m q c j L R ^ - £ j - I r^sinUTfjn/AO 4 , 2 1 A l s o , an a d d i t i o n a l e q u a t i o n which c o n s t r a i n s a s t a t i s t i c a l r 62 norm of the u n c e r t a i n t i e s i s n e c e s s a r y t o c o n t r o l the l e v e l s of n o i s e r e c o v e r e d i n the LP s o l u t i o n . The d e t a i l s of a c o n s t r a i n t on the l j n o r m of the e r r o r terms i s c o v e r e d by Levy and F u l l a g a r (1981). Then the LP a l g o r i t h m w i l l s e a r c h f o r t h o s e r ^ which have a minimum l j norm and i n so d o i n g a l s o determine e r r o r terms at each fre q u e n c y which f a v o u r the m i n i m i z a t i o n . Of c o u r s e , the e r r o r s and can be p o s i t i v e or n e g a t i v e , so they must be c a s t i n the form of 4.12 f o r i n p u t t o the LP a l g o r i t h m . Note t h a t the number of unknowns i s i n c r e a s e d by 4 at each f r e q u e n c y , but the number of c o n s t r a i n t e q u a t i o n s needed i n t h i s approach i s n e a r l y h a l f the number used i n the i n e q u a l i t y f o r m u l a t i o n . For the f o l l o w i n g examples of LP c o n s t r u c t i o n , an i n e q u a l i t y approach was adopted. S y n t h e t i c data was g e n e r a t e d u s i n g a z e r o phase w a v e l e t (an a v e r a g i n g f u n c t i o n ) e s t i m a t e d from the c o r r e l a t e d and s t a c k e d s e i m i c d a t a shown i n f i g u r e 15 (both the d a t a and wavelet e s t i m a t e were p r o v i d e d by I m p e r i a l O i l of Canada). The w a v e l e t and i t s a m p l i t u d e spectrum are shown i n f i g u r e 16. The f i r s t example of c o n s t r a i n e d LP c o n s t r u c t i o n i s shown i n f i g u r e 17. The r e f l e c t i v i t y f u n c t i o n shown i n p a n e l (b) was computed from the impedance model i n p a n e l ( a ) . The r e f l e c t i v i t y was then c o n v o l v e d w i t h the z e r o phase wavelet t o produce the s y n t h e t i c s e i s m i c t r a c e i n p a n e l ( c ) . The a p p r a i s a l d e c o n v o l u t i o n shown i n p a n e l (d) W 79E39 62921 63 F i g . 15. Deconvolved s e i s m i c s e c t i o n ; r e p r e s e n t s s u b s u r f a c e s t r u c t u r e i n p a r t s o f w e s t e r n A l b e r t a . \f 256sec w(t) IWCOI I25h; F i g . 16. Zero phase w a v e l e t ( a v e r a g i n g f u n c t i o n ) e s t i m a t e d from the d a t a i n s i d e the box i n f i g . 15, a n d i t s a m p l i t u d e spectrum. 64 <0 b) <r(t;©=0> 0 T .72 72 F i g . 1?. L i n e a r programming r e c o n s t r u c t i o n o f a r e f l e c t i v i t v f u n c t i o n c o n t a i n i n g a few r e f l e c t o r s . 65 i n d i c a t e s l i t t l e improvement over the o r i g i n a l d a t a because the z e r o phase w a v e l e t i s a l r e a d y an i d e a l a v e r a g i n g f u n c t i o n . The c o n s t r u c t e d r e f l e c t i v i t y f u n c t i o n i n p a n e l (e) was r e c o v e r e d by s u p p l y i n g the LP a l g o r i t h m w i t h a c c u r a t e s p e c t r a l v a l u e s between 12 and 35 Hz. The averages <r(t)>'were used t o weight the o b j e c t i v e f u n c t i o n and p o l a r i t y c o n s t r a i n t s were a l s o p r o v i d e d . The r e s u l t i n g impedance f u n c t i o n , shown i n p a n e l ( f ) , compares v e r y w e l l w i t h the t r u e impedance. No impedance c o n s t r a i n t s were i n c l u d e d i n t h i s example. F i g u r e 18 shows the r e s u l t s of a p p l y i n g the LP method u s i n g a n o i s y v e r s i o n of the impedance l o g from the p r e v i o u s example. The n o i s y l o g i s shown i n p a n e l ( a ) . The c o m p l e x i t y of the r e s u l t i n g r e f l e c t i v i t y and d e c o n v o l u t i o n ( p a n e l s (b) and ( d ) ) p r e s e n t a d i f f i c u l t p roblem f o r the LP a l g o r i t h m s i n c e the method i s b e s t s u i t e d t o f i n d s i m p l e r e f l e c t i v i t y f u n c t i o n s . A c c u r a t e v a l u e s of R between 10 and 35 Hz, s u p p l i e d t o the LP a l g o r i t h m a l o n g w i t h p o l a r i t y c o n s t r a i n t s and w e i g h t i n g (from <r(t)>""') l e a d t o the r e c o v e r e d r e f l e c t i v i t y f u n c t i o n i n p a n e l ( e ) . The r e s u l t i n g impedance f u n c t i o n ( p a n e l ( f ) ) shows t h a t the c o n s t r u c t i o n was r e a s o n a b l y s u c c e s s f u l a t r e c o v e r i n g the major t r e n d s of the o r i g i n a l impedance. By i n c l u d i n g two impedance c o n s t r a i n t s , one midway and one on the basement impedance, a good r e c o v e r y of 66 F i g . 18. L i n e a r programming r e c o n s t r u c t i o n o f a r e f l e c t i v i t y f u n c t i o n c o n t a i n i n g many r e f l e c t o r s . 67 the o r i g i n a l impedance s t r u c t u r e was obtained as shown i n panel (h). F u r t h e r examples of t h i s method are presented next. C) S t a b i l i t y and e f f i c i e n c y c o n s i d e r a t i o n s Before a p p l y i n g the AR and LP c o n s t r u c t i o n methods to r e a l seismic data, i t i s worthwhile i n v e s t i g a t i n g t h e i r performance on s y n t h e t i c examples under the f o l l o w i n g c o m p l i c a t i o n s . 1) the presence of a d d i t i v e n o i s e i n the data (both methods) 2 ) i n a c c u r a t e knowledge of the source wavelet (both methods) 3) v a r i a n c e of the known frequency range f j to f * 2 (both methods) 4) s e n s i t i v i t y of AR order to the number of r e f l e c t i o n c o e f f i c i e n t s (AR method only) 5) e f f e c t s of d i f f e r e n t weighting f u n c t i o n s (LP method only) F i g u r e 19 shows the s y n t h e t i c data which w i l l be the s t a r t i n g p o i n t f o r examples of the above c o n s i d e r a t i o n s . The sampling i n t e r v a l i s 4msec. For the f i r s t and second c o m p l i c a t i o n s above., c o n s t r u c t i o n s w i l l be performed using two s l i g h t l y d i f f e r e n t s p e c t r a l e s t i m a t e s . Since the wavelet i s known, estimates 68 F i g . 19. S y n t h e t i c t e s t d a t a . Wavelet has a band range of 10-50Hz. of R between f r e q u e n c i e s f j and f j may be o b t a i n e d through a b a n d l i m i t e d s p e c t r a l d i v i s i o n ; S ( f ) / W ( f ) . Or e s t i m a t e s of R may be taken from an a p p r a i s a l of the seismogram; S ( f ) V ( f ) . More r e l i a b l e e s t i m a t e s of R may be o b t a i n e d from the a p p r a i s a l because i n s i g n i f i c a n t or n o i s e c o n t a m i n a t e d f r e q u e n c i e s have been winnowed from th e s e r e s u l t s . R e c a l l i n g e q u a t i o n 3 . 4 , the a p p r a i s e d s e i s m i c t r a c e may be w r i t t e n In the f r e q u e n c y domain t h i s becomes D C 0 = R C 0 A ( 0 = 5 ( 0 V « ) I f A ( f ) i s z e r o phase and a p p r o x i m a t e l y u n i t y w i t h i n the 6 9 bandrange t o then the d i s c r e t e s p e c t r a l e s t i m a t e s D^ ', may be equated t o R.J between f j and f o r use i n the c o n s t r u c t i o n a l g o r i t h m s . For the f i r s t example, v a r i o u s amounts of random n o i s e were added t o the o r i g i n a l s y n t h e t i c t r a c e s ( t ) i n f i g u r e 19. The n o i s y seismograms are shown i n p a n e l (a) of f i g u r e 20 a l o n g w i t h a p p r a i s a l d e c o n v o l u t i o n s c a r r i e d out on each t r a c e a t © =1. The amount of n o i s e added t o the d a t a i s i n d i c a t e d on the c e n t e r l i n e . The AR c o n s t r u c t i o n s a r e shown i n p a n e l ( b ) . F o u r t e e n r e c o v e r e d s p e c t r a l v a l u e s between 14 and 40Hz were used t o p r e d i c t the m i s s i n g low f r e q u e n c i e s and the h i g h f r e q u e n c i e s t o 125Hz u s i n g an o r d e r of 11. The r e s u l t s on the l e f t were o b t a i n e d u s i n g s p e c t r a l e s t i m a t e s o b t a i n e d from .S(f)/W(f) and the r e s u l t s on the r i g h t were o b t a i n e d u s i n g a p p r a i s a l e s t i m a t e s S ( f ) V ( f ) . S t r a i g h t s p e c t r a l d i v i s i o n by the known wavelet has p r o v i d e d s h a r p e r r e f l e c t i v i t y c o n s t r u c t i o n s , however, the r e s u l t i n g impedance models a r e not b e t t e r than those o b t a i n e d by u s i n g a p p r a i s a l e s t i m a t e s . The LP c o n s t r u c t i o n s a r e shown i n p a n e l ( c ) . S p e c t r a l v a l u e s between 14 and 40Hz were d e r i v e d from S ( f ) / W ( f ) t o o b t a i n the r e s u l t s on the l e f t ; and s p e c t r a l v a l u e s were d e r i v e d from S ( f ) V ( f ) t o o b t a i n the c o n s t r u c t i o n s on the r i g h t . C l e a r l y , u s i n g the a p p r a i s a l e s t i m a t e s p r o v i d e s b e t t e r LP c o n s t r u c t i o n s . 7 0 S -£o * $ < >5 •p -p o co r H CH (1) J-i CH o • CO CD "O CO o -p CD E o C CD > H ! - P • H cd cc CC CH < O OJ 0) -p «H O o c CD to CD U -Ol CD O C CD rt .C e -p j-i o c CH - H CD C PH O • H - P • O O 3 CM t-i -P • CO W C •H O O 71 The second example, shown i n f i g u r e 21, demonstrates the e f f e c t of i n a c c u r a t e knowledge of the w a v e l e t . Trace 1 i n p a n e l (a) shows the t r u e mixed phase wavelet used t o g e n e r a t e the o r i g i n a l d a t a and t r a c e 5 shows a z e r o phase v e r s i o n of t h i s w a v e l e t which i s used as an i n a c c u r a t e w a v e l e t . An a p p r a i s a l of the o r i g i n a l d a t a u s i n g the i n a c c u r a t e wavelet i s shown i n t r a c e 6 (compare t h i s w i t h t r a c e 3; an a p p r a i s a l u s i n g the t r u e w a v e l e t ) . The AR c o n s t r u c t i o n s are shown p a n e l ( b ) . A g a i n , 14 r e c o v e r e d s p e c t r a l v a l u e s between 14 and 40Hz were used t o p r e d i c t the m i s s i n g low f r e q u e n c i e s and the h i g h f r e q u e n c i e s t o 125Hz u s i n g 5 d i f f e r e n t o r d e r s as i n d i c a t e d . The r e s u l t s u s i n g s p e c t r a l d i v i s i o n e s t i m a t e s S ( f ) / W ( f ) shown on the l e f t a r e somewhat b e t t e r than the r e s u l t s u s i n g a p p r a i s a l e s t i m a t e s S ( f ) V ( f ) shown on the r i g h t . The LP c o n s t r u c t i o n s a re shown i n p a n e l ( c ) . The same known frequency range was used and d i f f e r e n t w e i g h t i n g exponents 'q' were t e s t e d . A l s o f o r the bottom 2 r e f l e c t i v i t y c o n s t r u c t i o n s , a c o n s t r a i n t was p l a c e d on the basement impedance. A g a i n , u s i n g the a p p r a i s a l o u t p u t S ( f ) V ( f ) , p r o v i d e s much b e t t e r LP c o n s t r u c t i o n s . The t h i r d example t e s t s the e f f e c t s of v a r y i n g the known freq u e n c y range f| t o f 2 « The AR and LP r e s u l t s shown i n f i g u r e 22 were o b t a i n e d u s i n g a c c u r a t e s p e c t r a l e s t i m a t e s of R ( f ) i n the ranges i n d i c a t e d . The LP c o n s t r u c t i o n s shown 1 V wet) 2 — — - v — ret) 3 ^ ^ ^ ^ s C t ) u - J W W ^ < r c t ; d 2 i ) > ^ Y wet) 6 - - A ^ v A ^ v V 1 - <rCt;© = 0 ? Q) A R — * — 1 3 —»Y* A~vy~ orcier 0 .4 L P 1 c) F i g . 2 1 . Performance of the c o n s t r u c t i o n methods us i n g an innaccurate wavelet. 73 on the r i g h t remain s t a b l e , but the AR r e s u l t s shown on the l e f t f a i l when the number of known fr e q u e n c y v a l u e s becomes c l o s e t o , and hence, the o r d e r becomes l e s s than the number of s p i k e s (9) i n the o r i g i n a l r e f l e c t i v i t y r ( t ) . The f o u r t h example shown i n f i g u r e 23 demonstrates the e f f e c t of v a r y i n g the AR o r d e r u s i n g a c c u r a t e and i n a c c u r a t e d a t a . Twenty s p e c t r a l v a l u e s between 10 and 50Hz were o b t a i n e d from a s p e c t r a l d i v i s i o n o f : 1) a c c u r a t e data and 2) da t a w i t h 10% added random n o i s e . The AR c o n s t r u c t i o n s o b t a i n e d u s i n g a c c u r a t e s p e c t r a l v a l u e s are shown on the l e f t and the c o n s t r u c t i o n s u s i n g i n a c c u r a t e s p e c t r a l v a l u e s are shown on the r i g h t . The n o r m a l i z e d squared e r r o r (NSE), p l o t t e d vs o r d e r , between the t r u e and c o n s t r u c t e d models and d e f i n e d by are shown a t the bottom of f i g u r e 23. The NSE i n d i c a t e t h a t a l o n g p r e d i c t i o n f i l t e r ( i . e . , a h i g h o r d e r ) l e a d s t o c o n s t r u c t i o n may be o b t a i n e d i f the o r d e r i s g r e a t e r than the number of s i g n i f i c a n t s p i k e s e x p e c t e d i n the s o l u t i o n . Because a g r e a t e r range of r e l i a b l e f requency v a l u e s a re spanned by a l o n g e r f i l t e r , a more r e l i a b l e p r e d i c t i o n of s t a b l e r e c o n s t r u c t i o n s . In o t h e r words, a good AR 74 —>^X-^j-/- roo -1 5 — K ^ J i ^ A ^ ^ \0-SOH -12—ibv^vX^^/u |0-«fO -7 — V ^ - ^ N ^ ^ ^ - V S A - 1 0 - 3 0 -1 5 — A A V l — A ^ y u ^ I S -^O -1 0 — A j \ r w w v w - ~ p ^ /r-HcO -5 — - A ^ V w V ^ l y - N ^ |S"-30 - 1 LP I T T J - » - 1 v -WW o F i g . 2 2 . S t a b i l i t y o f r e c o n s t r u c t i o n s with r e s p e c t to the frequency band used. 75 —h/t^A—A— — U ^ A - ^ Y ^ ./L-J^-JU ret) H 7 1 10 I I M 11 IM I S" 17 IS order fNyVwA r^^yAvA~ Ax f/OlSE" FREE" \0% NOISE order F i g . 23. S t a b i l i t y o f the AR method w i t h r e s p e c t t o o r d e r o r the s t a b i l i t y o f o r d e r w i t h r e s p e c t t o the number o f r e f l e c t i o n c o e f f i c i e n t s ; u s i n g a c c u r a t e and i n n a c c u r a t e d a t a . 76 unknown v a l u e s may r e s u l t . However, i f the f i l t e r i s too l o n g ( c l o s e t o the l e n g t h of the known d a t a ) , n o i s e on the d a t a w i l l a l s o be p r e d i c t e d , perhaps g i v i n g u d e s i r e a b l e r e s u l t s . T h i s example suggests c h o o s i n g the l a r g e s t p o s s i b l e o r d e r c o n s i s t e n t w i t h assumed n o i s e l e v e l s i n the s p e c t r a l d a t a . I f the s i g n a l t o n o i s e r a t i o of the d a t a i s v e r y h i g h , c h o o s i n g a l a r g e o r d e r s h o u l d l e a d t o good r e s u l t s . A l s o , the NSE f o r the impedance c o n s t r u c t i o n s i n d i c a t e t h a t , out of 9 b o u n d a r i e s i n the t r u e ^ ( t ) , o n l y 6 b o u n d a r i e s a r e s i g n i f i c a n t w i t h r e s p e c t t o the o r d e r . The l a s t example i n t h i s s e c t i o n compares the use of d i f f e r e n t w e i g h t i n g exponents on the o b j e c t i v e f u n c t i o n i n the LP a l g o r i t h m . F i g u r e 24 shows LP c o n s t r u c t i o n s u s i n g 14 s p e c t r a l v a l u e s between 14 and 40Hz and 3 d i f f e r e n t w e i g h t i n g exponents. The c o n s t r u c t i o n s are a l l v e r y good but the ones' u s i n g o b j e c t i v e f u n c t i o n w e i g h t i n g (q=l,2) p r o v i d e a good s o l u t i o n i n 1/3 of the computing time r e q u i r e d w i t h no w e i g h t i n g (q=0). 7? JU- /I CPU TIME r<CO o I.Z Sec WO 1 .H 2 .H i 1 0 .4 F i g . 24. The a c c u r a c y and e f f i c i e n c y o f the LP s o l u t i o n w i t h a s p e c t t o the w e i g h t i n g exponent q, u s i n g accural 78 5 M u l t i t r a c e R e f l e c t i v i t y C o n s t r u c t i o n u s i n g R e a l S e i s m i c  Data A) The R e a l Data Data s e t A, shown i n f i g u r e 25 a l o n g w i t h a g e o l o g i c a l i n t e r p r e t a t i o n , c o n s i s t s of 24 a d j a c e n t t r a c e s t aken from the d e c o n v o l v e d s e i s m i c s e c t i o n i n f i g u r e 15 ( v i b r a t o r d a t a ) , where the d a t a l o c a t i o n i s i n d i c a t e d by the box. S o n i c l o g s are a v a i l a b l e near the s e i s m i c l i n e , but they are a t l e a s t 1 m i l e removed. Data s e t B, shown i n f i g u r e 26, c o n s i s t s of a .5sec window of d e c o n v o l v e d e x p l o s i o n d a t a (38 t r a c e s i n a l l ) . ^ A c l o s e by v e l o c i t y l o g i s a v a i l a b l e f o r p a r t of t h a t time window near a t r a c e l o c a t i o n . The raw t r a c e a m p l i t u d e s f o r each d a t a s e t r e p r e s e n t measured v o l t a g e s . Thus, i f a b s o l u t e r e f l e c t i o n and impedance magnitudes a r e d e s i r e d , then i t i s n e c e s s a r y t o match the energy i n each d e c o n v o l v e d t r a c e w i t h energy c o n s i s t e n t w i t h s u b s u r f a c e r e f l e c t i v i t y . The s t a n d a r d method of energy s c a l i n g u t i l i z e s a v a i l a b l e w e l l l o g s . A s y n t h e t i c seismogram i s c o n s t r u c t e d from a nearby v e l o c i t y l o g (and a d e n s i t y l o g i f a v a i l a b l e ) and then the t o t a l energy i n the d e c o n v o l v e d t r a c e i s matched t o the energy i n the s y n t h e t i c . Energy s c a l i n g i s n e c e s s a r y i n the c o n s t r a i n e d LP approach i f impedance or s p e c t r a l c o n s t r a i n t s , o b t a i n e d from w e l l l o g or v e l o c i t y a n a l y s e s , F i g . 25. Data s e c t i o n A; deconvolved v i b r a t o r data along with a g e o l o g i c a l i n t e r p r e t a t i o n . F i g . 26. Data s e c t i o n B; deconvolved e x p l o s i o n data. 80 a r e i n c l u d e d i n the a l g o r i t h m . S i n c e each s e i s m i c t r a c e i n s e c t i o n s A and B has undergone an a p p r a i s a l d e c o n v o l u t i o n , i t r e p r e s e n t s r e f l e c t i v i t y a v e r a g e s , < r ( t ) > = r ( t ) * a ( t ) , p l u s n o i s e e r r o r £ < r ( t ) > = n ( t ) * v ( t ) . The s p e c i f i c a p p r a i s a l t e c h n i q u e s used on each d a t a s e t are unknown, but i t i s assumed t h a t the o r i g i n a l s o u r c e wavelet has been l o c a l i z e d i n t o a z e r o phase a v e r a g i n g f u n c t i o n , a ( t ) = w ( t ) * v ( t ) . Each d e c o n v o l v e d t r a c e may be w r i t t e n N o r m a l l y , A ( f ) i s a smooth f u n c t i o n and g e n e r a l l y d e v i a t e s from i t s i d e a l v a l u e of u n i t y between f } and f ^ ( e g . , f i g . 16). T h e r e f o r e , i t may be n e c e s s a r y t o f i r s t remove any s p e c t r a l smoothing of R ( f ) produced by A ( f ) b e f o r e s u p p l y i n g s p e c t r a l e s t i m a t e s t o a c o n s t r u c t i o n a l g o r i t h m . S i n c e an average A ( f ) i s a v a i l a b l e f o r d a t a s e t A ( f i g . 16), the s e n s i t i v i t y of the c o n s t r u c t i o n methods w i t h r e s p e c t t o the smoothing e f f e c t of the a v e r a g i n g f u n c t i o n w i l l be t e s t e d u s i n g the f o l l o w i n g two s l i g h t l y d i f f e r e n t e s t i m a t e s of c l ( O r r C O * a c . o + ^ r c ^ > In the f r e q u e n c y domain t h i s i s 5". I 81 R(f ) : 1) R ( f ) ^ D ( f ) / A ( f ) (smoothing removed) 2) R ( f ) ~ D ( f ) f o r f j <. f >. F i g u r e 27 shows c o n s t r u c t i o n r e s u l t s f o r d a t a s e t A u s i n g the AR a l g o r i t h m . The o r i g i n a l data and the impedance averages o b t a i n e d from them a r e shown i n p a n e l ( a ) . Pa n e l (b) shows the r e f l e c t i v i t y and impedance c o n s t r u c t i o n s u s i n g s p e c t r a l e s t i m a t e s D ( f ) / A ( f ) ; and p a n e l (c) shows the r e s u l t s u s i n g e s t i m a t e s D ( f ) . In both c a s e s , 33 r e c o v e r e d f r e q u e n c i e s between 13 and 46 Hz were used t o p r e d i c t the m i s s i n g low f r e q u e n c i e s and h i g h f r e q u e n c i e s t o 62 Hz u s i n g an o r d e r of 24. No major d i f f e r e n c e s e x i s t i n the two c o n s t r u c t e d s e c t i o n s , however, the r e s u l t s i n p a n e l ( b ) , o b t a i n e d by u s i n g e s t i m a t e s D ( f ) / A ( f ) a re somewhat n o i s i e r . The LP c o n s t r u c t i o n s f o r d a t a set A a r e shown i n f i g u r e 28. The r e s u l t s u s i n g s p e c t r a l e s t i m a t e s D ( f ) / A ( f ) a re shown i n p a n e l ( b ) ; and the r e s u l t s u s i n g e s t i m a t e s D ( f ) are shown i n p a n e l ( c ) . In both c a s e s , r e l i a b l e f r e q u e n c i e s from 12 t o 35 Hz were used and two impedance c o n s t r a i n t s were i n c l u d e d ; one midway and one on the basement impedance. C l e a r l y , the c o n s t r u c t e d s e c t i o n i n p a n e l ( c ) , o b t a i n e d by u s i n g s p e c t r a l e s t i m a t e s , D ( f ) , p r o v i d e s the most c o n s i s t e n t r e s u l t s . The above AR and LP c o n s t r u c t i o n r e s u l t s a r e g e n e r a l l y P i g . 27. AR r e f l e c t i v i t y and impedance r e c o n s t r u c t i o n s from s e c t i o n A; b) u s i n g unsmoothed, and c) u s i n g smoothed s p e c t r a l e s t i m a t e s . F i g . 28. LP r e f l e c t i v i t y and impedance r e c o n s t r u c t i o n s from s e c t i o n A; b) u s i n g unsmoothed, and c) u s i n g smoothed s p e c t r a l e s t i m a t e s . 84 t r a c e by t r a c e comparable, e s t a b l i s h i n g c o n f i d e n c e i n the r e l i a b i l i t y of both methods. U s i n g smoothed s p e c t r a l e s t i m a t e s , D ( f ) , p r o v i d e d more c o n s i s t e n t LP c o n s t r u c t i o n s and l e s s n o i s y AR c o n s t r u c t i o n s . Thus, f u r t h e r c o n s t r u c t i o n s w i l l be c a r r i e d out u s i n g smoothed s p e c t r a l e s t i m a t e s , D ( f ) , o b t a i n e d from the a p p r a i s e d d a t a . A u t o r e g r e s s i v e c o n s t r u c t i o n s f o r data s e t B are shown i n f i g u r e 29. The o r i g i n a l d a t a and the impedance averages o b t a i n e d from them a r e shown i n p a n e l ( a ) . F r e q u e n c i e s between 12 and 40 Hz were deemed r e l i a b l e - b y examining the a m p l i t u d e s p e c t r a of s e v e r a l d e c o n v o l v e d t r a c e s . An o r d e r of 13 was used f o r the AR c o n s t r u c t i o n shown i n p a n e l ( b ) , where the m i s s i n g low f r e q u e n c i e s and h i g h f r e q u e n c i e s t o 65 Hz were p r e d i c t e d f o r each t r a c e . The LP c o n s t r u c t i o n s f o r d a t a s e t B a r e shown i n f i g u r e 30. F r e q u e n c i e s between 12 and 40 Hz were used a l o n g w i t h one impedance c o n s t r a i n t (midway) t o o b t a i n the r e s u l t s shown i n p a n e l ( b ) . The AR and LP c o n s t r u c t e d s e c t i o n s o b t a i n e d from d a t a s e t B p r o v i d e s i m i l a r and more l o c a l i z e d r e f l e c t i v i t y i n f o r m a t i o n . However, both c o n s t r u c t e d s e c t i o n s r e t a i n the n o i s y c h a r a c t e r of the o r i g i n a l d a t a . A somewhat n o i s y c h a r a c t e r i s a l s o o b s e r v e d i n the c o n s t r u c t e d s e c t i o n s o b t a i n e d from d a t a s e t A. A l t h o u g h good i n t e r p r e t a t i o n s c o u l d be made from t h e s e s e c t i o n s , the c o n s i s t e n c y of the F i g . 29. AR r e f l e c t i v i t y and impedance r e c o n s t r u c t i o n s from s e c t i o n B u s i n g smoothed s p e c t r a l e s t i m a t e s . CO F i g . 30. LP r e f l e c t i v i t y and impedance reconstructions from section B using smoothed spectral estimates. CO ON 87 r e s u l t s must be improved b e f o r e the AR and LP a l g o r i t h m s can be r o u t i n e l y a p p l i e d t o s e i s m i c d a t a . The r e c o v e r e d broadband s e c t i o n s above were l a t e r a l l y i n c o n s i s t e n t i n some p l a c e s , not because of the presence of s t r u c t u r a l changes but more l i k e l y due t o the presence of n o i s e i n the d a t a . A l i n e a r t r a n s f o r m a t i o n known as the Karhunen-Loeve (K-L) t r a n s f o r m a t i o n (eg. Ready and W i n t z , 1973; Kramer and Mathews, 1956) can be used t o e x t r a c t the major s i m i l a r i t i e s i n a s e t of a d j a c e n t s e i s m i c t r a c e s f o r the purpose of i m r o v i n g l a t e r a l c o n s i s t e n c y . C o n s i d e r N t r a c e s of s e i s m i c d a t a , where each t r a c e SJ ( t ) i s composed of a common s i g n a l x ( t ) p l u s a d i f f e r e n c e s i g n a l n ; ( t ) ; i . e . , I f the n j ( t ) r e p r e s e n t n o i s e s i g n a l s , then i t i s u s e f u l t o f i n d an x ( t ) which summarizes the most c o r r e l a t e d components i n the N t r a c e s . In g e n e r a l , the common s i g n a l may be w r i t t e n as a l i n e a r c o m b i n a t i o n of the s,'(t) as f o l l o w s B) The Karhunen-Loeve T r a n s f o r m a t i o n 5,2 88 The s j ( t ) , i=1,...N, span at most N d i m e n s i o n s i n H i l b e r t space and i f the t r a c e s are h i g h l y c o r r e l a t e d , they w i l l have a p r e f e r r e d o r i e n t a t i o n i n t h a t space. T h i s s u g g e s t s r e w r i t i n g t h e common t r a c e as a l i n e a r c o m b i n a t i o n of M o r t h o n o r m a l b a s i s f u n c t i o n s Y,-(t) where the d i r e c t i o n s of t h e s e b a s i s f u n c t i o n s are d e f i n e d by the e i g e n v e c t o r s of the i n n e r p r o d u c t or c o v a r i a n c e m a t r i x M 0-i The % * ( t ) may be d e f i n e d as f o l l o w s (see appendix A) r.6 where s ^ c^ co, stcOj ....sNcty) \ \ i s the j t h e i g e n v a l u e a r r a n g e d i n d e s c e n d i n g o r d e r i . e . , 89 X\ > >^-X|s/ a n < 3 b-j i s t h e j t h e i g e n v e c t o r of P . The i n 5.4 may be found by m i n i m i z i n g the sum of i n t e g r a t e d squared d i f f e r e n c e s between x ( t ) and the s j ( t ) , t h a t i s by m i n i m i z i n g i-.i 4 r-1 The r e s u l t d e r i v e d i n appendix A i s I — - N N ;fr 1 where Jjfj i s the e i g e n v e c t o r m a t r i x d e f i n e d i n appendix A. S u b s t i t u t i n g 5.6 and 5.8 i n t o 5.4 g i v e s M 1 where I-1 Each term i n e x p a n s i o n 5.9 i s c a l l e d a p r i n c i p a l component of the common t r a c e . For example, 01 b, * £ i s the f i r s t p r i n c i p a l component and so on. I f the N t r a c e s a r e h i g h l y c o r r e l a t e d , then \\ and Y v w i l l be r e l a t i v e l y l a r g e and most of the i n f o r m a t i o n about • the common t r a c e w i l l be 90 c o n t a i n e d i n the f i r s t p r i n c i p a l component. In f a c t i f a l l N t r a c e s a r e i d e n t i c a l , then the f i r s t p r i n c i p a l component c o n t a i n s a l l of the i n f o r m a t i o n about the common t r a c e and the o t h e r components a r e t r i v i a l . At b e s t , the components of the f i r s t p r i n c i p a l e i g e n v e c t o r p r o v i d e optimum t r a c e w e i g h t s t o c o n s t r u c t t h e d e s i r e d common t r a c e . I n any c a s e , i f M=N, i . e . , i f a l l p r i n c i p a l components are summed, then 5.9 g i v e s the mean t r a c e (as shown i n appendix A) A c r i t e r i o n c o u l d be e s t a b l i s h e d t p h e l p determine how many p r i n c i p a l components s h o u l d be i n c l u d e d i n the common t r a c e . However, i f o n l y the major s i m i l a r i t i e s between 3 or 5 t r a c e s a r e d e s i r e d , then o n l y one or two p r i n c i p a l components need be used t o p r o v i d e a common t r a c e In t h i s t h e s i s , e x p r e s s i o n 5.9 i s r e f e r r e d t o as the K-L t r a n s f o r m a t i o n and the common t r a c e formed by t h i s t r a n s f o r m a t i o n i s c a l l e d a p r i n c i p a l component or K-L t r a c e . The K-L t r a n s f o r m a t i o n can be a p p l i e d i n an o v e r l a p p i n g manner t o dec o n v o l v e d s e i s m i c d a t a , t o r e f l e c t i v i t y and impedance c o n s t r u c t i o n s , and t o the c h o i c e of AR o r d e r , t o h e l p improve the r e l i a b i l i t y and i n t e r p r e t a b i l i t y of c o n s t r u c t e d r e f l e c t i v i t y and impedance s e c t i o n s . For example, the l e f t hand s i d e of f i g u r e 31 shows the 91 r e s u l t s of a p p l y i n g AR p r e d i c t i o n t o the b a n d l i m i t e d r e f l e c t i v i t y f u n c t i o n , < r ( t ) > ( 1 2-46H Z), u s i n g s e v e r a l o r d e r s . The seventeen known f r e q u e n c i e s between 12 and 46Hz were used t o p r e d i c t the m i s s i n g low f r e q u e n c i e s and the h i g h f r e q u e n c i e s t o 125Hz. The r e s u l t s are s l i g h t l y d i f f e r e n t f o r each o r d e r . C o n s i d e r i n g N=5 t r a c e s and u s i n g o n l y the f i r s t p r i n c i p a l component g i v e s the K-L t r a c e s shown on the r i g h t hand s i d e . These t r a c e s a r e v e r y c o n s i s t e n t over a wide range of o r d e r s i n d i c a t i n g t h a t the c h o i c e of AR o r d e r i s l e s s c r i t i c a l u s i n g the K-L approach. F i g u r e 31 a l s o shows the p r i n c i p l e of the m i x i n g a l g o r i t h m . T r a c e s 1-5 are used t o o b t a i n the t o p K-L t r a c e , t r a c e s 2-6 are used to o b t a i n the next K-L t r a c e and so on. F i g u r e 32 shows the r e s u l t s of a p p l y i n g the a l g o r i t h m to the d e c o n v o l v e d d a t a s e t B u s i n g N=3 and o n l y the f i r s t p r i n c i p a l component. The r e s u l t i n g p r i n c i p a l component s e c t i o n shown below, i n d i c a t e s t h a t s i g n i f i c a n t i n f o r m a t i o n i n the o r i g i n a l d a t a has been p r e s e r v e d and the n o i s y c h a r a c t e r i s a b s e n t . A p p l i c a t i o n of the K-L a l g o r i t h m t o an LP r e f l e c t i v i t y c o n s t r u c t i o n (from f i g u r e 28) i s shown i n f i g u r e 3 3 . The o r i g i n a l c o n s t r u c t e d r e f l e c t i v i t y and impedance s e c t i o n s a r e shown on the l e f t and the p r i n c i p a l component s e c t i o n s , o b t a i n e d u s i n g N=3 t r a c e s and o n l y the f i r s t p r i n c i p a l component, are shown on the r i g h t . The i n t e r p r e t a b i l i t y of 92 11 10 8 ^A^~Ar^-^Y^^ 9 9 — K j ^ j ^ f \ — y ^ J ^ 10 trace 8 7 order [f0; ^w] — I U L J L J L ^ ^ L r(t) — ^ J ^ - y - ^ -1 A ^ v J — A ^ ^ ^ L v 16—j > L-K^A—^—Y^7^ 2 — -^A-vy^ Avv*rA -^-^ V^ ' 15 K-L 0 .4 F i g . 31* A p p l i c a t i o n o f the K-L t r a n s f o r m a t i o n t o s t a b i l i z e t he c h o i c e o f AR o r d e r . A l s o showing the p r i n c i p l e o f the m i x i n g a l g o r i t h m . 93 K-L F i g . 32. A p p l i c a t i o n o f the K-L t r a n s f o r m a t i o n and m i x i n g a l g o r i t h m t o the n o i s y s e c t i o n B. 94 both s e c t i o n s has been markedly improved. U s i n g t h i s K-L a l g o r i t h m , n o i s y t r a c e s can be d i s c r i m i n a t e d a g a i n s t w h i l e p r e s e r v i n g the most c o r r e l a t e d i n f o r m a t i o n r e c o v e r e d by the c o n s t r u c t i o n s . The f i n a l c o n s t r u c t e d r e f l e c t i v i t y and impedance s e c t i o n s f o r d a t a s e t A a r e shown i n f i g u r e 34 a l o n g w i t h the o r i g i n a l g e o l o g i c i n t e r p r e t a t i o n . Both the AR r e s u l t s , shown on the l e f t , and the LP r e s u l t s , shown on the r i g h t , were o b t a i n e d by a p p l y i n g the K-L t r a n s f o r m a t i o n t o the raw c o n s t r u c t i o n s shown p r e v i o u s l y u s i n g N=3 t r a c e s and the f i r s t p r i n c i p a l component i n t h e m i x i n g a l g o r i t h m . The AR and LP s e c t i o n s p r o v i d e n e a r l y i d e n t i c a l i n t e r p r e t a t i o n s . The p o s i t i v e impedance c o n t r a s t a t the P r e C r e t a c e o u s u n c o n f o r m i t y (which i s a dominant f e a t u r e i n nearby w e l l l o g s ) c l e a r l y c o r r e l a t e s i n a l l t r a c e s and the o i l b e a r i n g Redwater Reef shows up as an impedance low. O v e r a l l , i t i s e a s i e r t o make s t r u c t u r a l i n t e r p r e t a t i o n s from the r e c o v e r e d impedance s e c t i o n s . Comparisons between the o r i g i n a l d a t a and the r e c o n s t r u c t i o n s a r e shown i n f i g u r e s 35 and 36. The f i n a l c o n s t r u c t e d r e f l e c t i v i t y and impedance s e c t i o n s f o r d a t a s e t B a r e shown i n f i g u r e 37. A g a i n , t h e s e r e s u l t s were o b t a i n e d by a p p l y i n g the K-L m i x i n g a l g o r i t h m t o AR and LP c o n s t r u c t i o n s u s i n g N=3 t r a c e s and the f i r s t p r i n c i p a l component. The a v a i l a b l e v e l o c i t y l o g has been i n s e r t e d i n t o the impedance r e c o n s t r u c t i o n s f o r 95 F i g . 3 3 - A p p l i c a t i o n o f the K-L a l g o r i t h m t o LP r e f l e c t i v i t y and impedance r e c o n s t r u c t i o n s . AR L P If A PARK CoLOKArDO • W A & N M SUICROP BrAveR HILL LAKE" ELK POINT LEA PARK COLORADO WABAMU* SUBCROP ZZZZZZ RfD^ ATR REEF Bwvep. HUL LAKE" ELK fO'NT F i g . 3>4-. The f i n a l r e f l e c t i v i t y and impedance r e c o n s t r u c t i o n s from s e c t i o n A along with the o r i g i n a l g e o l o g i c i n t e r p r e t a t i o n . ON F i g ' 3 5 ' Comparison of f i n a l r e f l e c t i v i t y -r e c o n s t r u c t i o n s with the o r i g i n a l deconvolved s e c t i o n (A). 98 F i g . 36. Comparison o f f i n a l impedance r e c o n s t r u c t i o n s with impedance averages obtained from the deconvolved data. 99 c o m p a r i s o n . The w e l l l o g c o n f i r m s the f i r s t .3sec of a d j a c e n t r e c o n s t r u c t i o n s and c o n s i s t e n c y between the AR and LP r e s u l t s p r o v i d e s a d d i t i o n a l c o n f i d e n c e i n the f i n a l models. A g a i n , i t i s e v i d e n t t h a t s u b s u r f a c e s t r u c t u r a l i n t e r p r e t a t i o n s would be a i d e d by u s i n g the impedance s e c t i o n . Comparisons between the o r i g i n a l d a t a and the r e c o n s t r u c t i o n s are shown i n f i g u r e s 38 and 39. C o n c l u s i o n The problem of e s t i m a t i n g broadband a c o u s t i c impedance from r e f l e c t i o n seismograms has been i n v e s t i g a t e d u s i n g a l i n e a r i n v e r s e f o r m a l i s m . I t was shown t h a t a c o n v e n t i o n a l d e c o n v o l u t i o n / i n v e r s e f i l t e r i n g t e c h n i q u e (which r e c o v e r s unique b a n d l i m i t e d r e f l e c t i v i t y i n f o r m a t i o n from the seismogram) s h o u l d precede the use of a p a r t i c u l a r c o n s t r u c t i o n a l g o r i t h m (which a t t e m p t s t o r e c o v e r the unknown f r e q u e n c y bands). Two c o n s t r u c t i o n methods p r e s e n t e d i n t h i s t h e s i s , the AR and the LP methods, have shown s u c c e s s a t r e c o v e r i n g m i s s i n g s p e c t r a l i n f o r m a t i o n . The K-L t r a n s f o r m a t i o n was a p p l i e d i n r e a l d a t a examples t o s t a b i l i z e b o t h methods i n the presence of n o i s e and impedance c o n s t r a i n t s p r o v i d e d a d d i t i o n a l s t a b i l i t y t o the LP s o l u t i o n . The AR method r e p r e s e n t s an i n e x p e n s i v e ( c o m p u t a t i o n a l l y ) way t o o b t a i n r e c o n n a i s s a n c e broadband A R LP F i g . 37. The f i n a l r e f l e c t i v i t y and impedance r e c o n s t r u c t i o n s from s e c t i o n B, The g i v e n v e l o c i t y l o g has been i n s e r t e d f o r comparison. o o 101 P i g . 38. Comparison of f i n a l r e f l e c t i v i t y r e c o n s t r u c t i o n s with the o r i g i n a l deconvolved s e c t i o n ( B). 102 F i g . 39. Comparison of f i n a l impedance r e c o n s t r u c t i o n s with impedance averages•obtained from the deconvolved data. 1 0 3 impedance i n f o r m a t i o n . A l t h o u g h the method works v e r y w e l l on s y n t h e t i c and r e a l d a t a examples, a t t h i s time t h e r e i s no way t o c o n s t r a i n the output u s i n g i n f o r m a t i o n a v a i l a b l e from o t h e r a n a l y s e s . A b s o l u t e impedance i n f o r m a t i o n , from a v a i l a b l e w e l l l o g s s h o u l d be u t i l i z e d t o improve the s t a b i l i t y of AR c o n s t r u c t i o n s i n the presence of n o i s e . Other improvements t o t h i s method might i n c l u d e ways of o b t a i n i n g the p r e d i c t i o n f i l t e r c o e f f i c i e n t s by c o n s t r a i n i n g the p r e d i c t e d dc fr e q u e n c y R 0 or s o l v i n g f o r the m i s s i n g f r e q u e n c i e s d i r e c t l y u s i n g maximum e n t r o p y c o n s i d e r a t i o n s . The LP a l g o r i t h m p r o v i d e s the most v e r s a t i l e approach t o r e c o v e r i n g broadband impedance. Robust i n the presence of n o i s e , t h i s method a l l o w s much room f o r improvement; r e s t r i c t e d o n l y by the c r e a t i v i t y of the d e s i g n e r and the e f f i c i e n c y of the l i n e a r programming a l g o r i t h m . I t i s c o n c l u d e d t h a t impedance s e c t i o n s o b t a i n e d from both c o n s t r u c t i o n methods c o u l d g r e a t l y a i d the i n t e r p r e t a t i o n of s u b s u r f a c e s t r u c t u r e . The problem of r e f l e c t i o n p o l a r i t y may, i n many c a s e s , be s o l v e d by p e r f o r m i n g an impedance c o n s t r u c t i o n . For example, u n t i l the c o n s t r u c t i o n s f o r da t a s e t A were performed and the p o s i t i v e impedance c o n t r a s t was i d e n t i f i e d , the p o l a r i t y of the g i v e n d a t a was not known. In f a c t , p o l a r i t y was unknown f o r b oth d a t a s e t s u n t i l p r e l i m i n a r y c o n s t r u c t i o n s were performed. 104 REFERENCES C l a e r b o u t , J . F., and M u i r , F., 1973, Robust M o d e l l i n g w i t h E r r a t i c D a t a : G e o p h y s i c s , V.38, P. 826-844. De c o n v o l u t i o n , G e o p h y s i c s R e p r i n t S e r i e s 1, Volumes I and I I , ed. by G. M. Webster, 1978, S o c i e t y of E x p l o r a t i o n G e o p h y s i c i s t s , T u l s a , Oklahoma. G a l b r a i t h , J . M., and G. F. M i l l i n g t o n , 1979, Low Frequency Recovery i n the I n v e r s i o n of Seismograms: J o u r n a l of The CSEG, V.15, No.1, P. 30-39. Kramer, F. S., R. A. P e t e r s o n , and W. C. W a l t e r , 1 9 6 8 , S e i s m i c Energy Sources 1968 Handbook: P r e s e n t e d a t the 38th Annual ISEG M e e t i n g , Denver, C o l o r a d o , 1968 and P r i v a t e l y P u b l i s h e d by U n i t e d G e o p h y s i c a l Corp. Kramer, H. P., and M. V. Mathews, 1956, A L i n e a r Coding f o r T r a n s m i t t i n g a Set of C o r r e l a t e d S i g n a l s : IRE T r a n s . Inform. Theory, V.IT-2, P. 41-46. Lavergne, M., and C. W i l l m , 1977, I n v e r s i o n of Seismograms and P s e u d o v e l o c i t y Logs: Geophys. P r o s p . , • V.25, P. 231-250. Levy, S., and P. K. F u l l a g a r , 1 9 8 1 , R e c o n s t r u c t i o n of a Sparse S p i k e T r a i n from a P o r t i o n of i t s Spectrum and A p p l i c a t i o n t o High R e s o l u t i o n D e c o n v o l u t i o n : G e o p h y s i c s , V.46, N.9, P. 1235-1243. L i n d s e t h , R. 0., 1979, S y n t h e t i c S o n i c Logs - A P r o c e s s f o r S t r a t i g r a p h i c I n t e r p r e t a t i o n : G e o p h y s i c s , V.44, P.3-26. L i n e s , L. R., and R. W. C l a y t o n , 1977, A New Approach t o V i b r o s e i s D e c o n v o l u t i o n : Geophys. P r o s p . , V.25, P. 417-433. L i n e s , L. R., and T. J . U l r y c h , 1977, The O l d and the New i n S e i s m i c D e c o n v o l u t i o n and Wavelet E s t i m a t i o n : Geophys. P r o s p . , V.25, P. 512-540. Mayne, W. H., and R. G. Quay, 1971, S e i s m i c S i g n a t u r e s of Large A i r g u n s : G e o p h y s i c s , V.36, P. 1162-1173. Olde n b u r g , D. W., 1981, A Comprehensive S o l u t i o n To The L i n e a r D e c o n v o l u t i o n Problem: Geophys. J . R. A s t r . Soc. V.65, P. 331-357. Oldenburg, D. W., S. Levy, and K. P. W h i t t a l l , 1981, Wavelet E s t i m a t i o n and D e c o n v o l u t i o n : G e o p h y s i c s , ( t o be p u b l i s h e d i n V.46, N.11, November, 1981). 105 O t i s , R. M., and R. B. S m i t h , 1977, Homomorphic D e c o n v o l u t i o n by Log S p e c t r a l A v e r a g i n g : G e o p h y s i c s , V.42, N.6, P.1146-1157. P a r k e r , R. L.,1977, U n d e r s t a n d i n g I n v e r s e Theory: Ann. Rev. E a r t h P l a n t . S c i . , V.5, P. 35-64. Peacock, K. L., 1979, D i s c r e t e O p e r a t o r s f o r I n t e g r a t i o n : G e o p h y s i c s , V.44, N.4, P. 722-729. P e t e r s o n , R. A., W. R. F i l l i p o n e , and F. B. Coker, 1955, The S y n t h e s i s of Seismograms from W e l l Log D a t a : G e o p h y s i c s , V.20, P. 516-538. Ready, R. J . , and P. A. W i n t z , 1973, I n f o r m a t i o n E x t r a c t i o n , SNR Improvement, and Data Compression i n M u l t i s p e c t r a l Imagery: IEEE T r a n s a c t i o n s On Communications, COM. 2 1 ( 1 0 ) , P. 1123-1130. R i c k e r , N. , 1940, The Form and Nature of S e i s m i c Waves and the S t r u c t u r e of Seismograms: G e o p h y s i c s , V.5, P. 348-366. R i c k e r , N., 1953, The Form and Laws of P r o p o g a t i o n of S e i s m i c W a v e l e t s : G e o p h y s i c s , V.18, P. 10-40. Robin s o n , E. A., 1967, 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 : G e o p h y s i c s , V.32, N.3, P. 418-484. T a y l o r , H. L., S. C. Banks, and J . F. McCoy, 1979, D e c o n v o l u t i o n w i t h the L i Norm: G e o p h y s i c s , V.44, P. 39-52. T r e i t e l , S., and E. A. Robi n s o n , 1966, The Design of H i g h R e s o l u t i o n F i l t e r s : IEEE T r a n s a c t i o n s , G e o s c i e n c e E l e c t r o n i c s , V.4, P. 25-38. U l r y c h , T. J . , 1971, A p p l i c a t i o n of Homomorphic D e c o n v o l u t i o n t o S e i s m o l o g y : V.36, P. 650-660. U l r y c h , T. J . , 19-72, Maximum En t r o p y Power Spectrum of T r u n c a t e d S i n u s o i d s : J . Geophys. Res., V.77 P. 1396-1 400. U l r y c h , T. J . , and T. N. B i s h o p , 1975, Maximum E n t r o p y S p e c t r a l A n a l y s i s and A u t o r e g r e s s i v e D e c o m p o s i t i o n : Rev. Geophys. and Space P h y s i c s , V.13, P. 183-200. U l r y c h , T. J . , and R. W. C l a y t o n , 1976, Time S e r i e s M o d e l l i n g and Maximum E n t r o p y : Phys. E a r t h P l a n e t . I n t e r . , V.12, P. 188-200. 106 Appendix A: D e r i v a t i o n of the Common Trace C o n s i d e r N t r a c e s of s e i s m i c d a t a , where each t r a c e s ; ( t ) i s composed of a common s i g n a l x ( t ) p l u s a d i f f e r e n c e s i g n a l n j ( t ) ; t h a t i s , I f t he nj ( t ) r e p r e s e n t n o i s e s i g n a l s , then i t i s u s e f u l t o f i n d an x ( t ) which summarizes the most s i m i l a r components of the N t r a c e s . In g e n e r a l , the common t r a c e may be w r i t t e n as a l i n e a r c o m b i n a t i o n of the N t r a c e s as f o l l o w s u The most s t r a i g h t f o r w a r d p r o c e d u r e i s t o f i n d t h a t common t r a c e which m i n i m i z e s the sum of i n t e g r a t e d squared d i f f e r e n c e s between x ( t ) and the S j ( t ) , i . e . , N > z O b v i o u s l y , t h i s 4 i s m i n i m i z e d w i t h r e s p e c t t o x ( t ) m i n i m i z a t i o n l e a d s t o the mean t r a c e 107 where the P| in 2 are a l l l/N. There i s no way to d i s c r i m i n a t e a g a i n s t an erroneous t r a c e i n t h i s procedure because i t i s simply averaged i n with equal importance. I t i s d e s i r e a b l e to f i n d that set of weights fii which emphasize very s i m i l a r t r a c e s and r e j e c t s noisy or i n c o n s i s t e n t t r a c e s . T h i s can be done by for m u l a t i n g the common t r a c e as a l i n e a r combination of orthornormal b a s i s f u n c t i o n s . Orthogonal Recomposition The s(- (t) span at most N dimensions in H i l b e r t space. If the s'-(t) are h i g h l y c o r r e l a t e d , they w i l l have a p r e f e r r e d o r i e n t a t i o n in that space. T h i s suggests r e w i t i n g the common t r a c e as a l i n e a r combination of b a s i s f u n c t i o n s which are orthonormal and c o n t a i n one member which minimizes the p r o j e c t i o n d i s t a n c e s between i t and the N t r a c e s . The orthogonal recomposition of the common t r a c e begins by forming the inner product or co v a r i a n c e matrix 108 S i n c e P i i s symmetric, i t may De w r i t t e n as r = B A 8 T w h e r e A -0 i s a d i a g o n a l m a t r i x o f t h e e i g e n v a l u e s o f P a r r a n g e d i n d e s c e n d i n g o r d e r a n c B -K b , . . - b N i s a m a t r i x c o m p o s e d o f t h e e i g e n v e c t o r s \jj o f P . The e i g e n v e c t o r s o f P d e f i n e a s e t o f p r i n c i p a l o r t h o g o n a l d i r e c t i o n s ( a n a l o g o u s t o t h e p r i n c i p a l s t r e s s d i r e c t i o n s o b t a i n e d f r o m a s t r e s s m a t r i x ) . The m a t r i x 3 h a s t h e f o l l o w i n g p r o p e r t y B S T - B T B - I • lo^A.hki - ^ K J J~ w h e r e I i s t h e i d e n t i t y m a t r i x , Xj^is t h e K r o n e c k e r d e l t a and bi'; =B. A s e t o f o r t h o n o r m a l b a s i s f u n c t i o n s ^f^(t) may be 109 obtained by projecting the vector of seismic traces onto each eigenvector and normalizing by the corresponding eigenvalue as follows where and .b IT.-CO ^CO i t = i 7 a The o r i g i n a l traces may be reconstructed from the basis as follows u k=i This may be checked by substituting 6 into 8 as follows M IV U,. L - Si CO 110 The common t r a c e M s then w r i t t e n as a l i n e a r combination of M orthonormal basis - f u n c t i o n s M The c ^ j may be found by minimizing Q-Z.) ( x C t ) - S , ( 0 ) <it M i n i m i z i n g t h i s , with respect to < ^>C V £gives IV / b / - W IV 1 i ^ b i K < f K c o ^ } at 0 Thus N i n S u b s t i t u t i n g 10 and 6 i n t o 9 g i v e s M r - N I f M=N, then 11 reduces t o the mean t r a c e I f M<N, then 11 can be w r i t t e n tit) - £. Y^ j bj • S where 

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