THE RECOVERY OF SUBSURFACE R E F L E C T I V I T Y AND IMPEDANCE STRUCTURE FROM REFLECTION SEISMOGRAMS BY TIM E L L I S SCHEUER B.Sc, The U n i v e r s i t y Of U t a h , 1979 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in 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 a s c o n f o r m i n g to the required standard THE UNIVERSITY OF B R I T I S H COLUMBIA O c t o b e r 1981 (c) Tim E l l i s Scheuer, 1981 In presenting this requirements of British it freely agree for f o r an available that understood that financial his Library shall for reference and study. I for extensive or be her shall copying of granted by the not be of this Geoplysi'cs The U n i v e r s i t y o f B r i t i s h 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 Date DF-6 (2/79) C U . I<j . ?/ further this It and A&Wonomy Columbia thesis my is thesis a l l o w e d w i t h o u t my permission. Department of make head o f representatives. publication the University the copying or gain the that p u r p o s e s may by f u l f i l m e n t of degree at I agree permission scholarly in partial advanced Columbia, department or for thesis written ii Abstract This t h e s i s i s concerned with broadband acoustic reflection se i sinograms. the linear Oldenburg inverse This The of subsurface Then an to unique b a n d l i m i t e d bandlimited two broadband The method small f u n c t i o n may first uses portion of spectral values most v e r s a t i l e unique of which on L e v y and and has models. at method that Fullagar spectral provides a ( recovering with and as a is which is a source performed This utilized in provide a a broadband small seismic maximum portion. i s the (1981) entropy representation spectrum information to The linear which broadband a minimum 1 norm. real (1977) from w h i c h a reflectivity outside following i s modelled i s then autoregressive the by computed. c o n s t r u c t i o n method s y n t h e t i c and success an bandlimited appraisal be incidence information. algorithms reflectivity; estimating Parker seismogram information construction which approach of by reflectivity construction estimate impedance i s covered seismogram the of normal reflectivity a p p r a i s a l of reflecitivity different from topic measured wavelet. obtain problem formalisms described (1980). convolution impedance the second interpretable and programming utilizes obtained reflectivity and have broadband a predict the from an function B o t h methods have been data of shown tested good impedance Errors i n the data reflectivity models and t h e uniqueness play important impedance f u n c t i o n a n d i n assessing Karhunen-Loeve transformation real data to stabilize presence of noise. frequency impedance log of or The seismic data has r o l e s i n e s t i m a t i n g the i t s uniqueness. discussed construction trace, in from When an impedance model may be r e c o v e r e d algorithms presented in this interpretable, using thesis. well nature accurate, i n f o r m a t i o n c a n be r e c o v e r e d then the i d e a t h a t low because of the b a n d l i m i t e d challenged. The and a p p l i e d on results g e n e r a l l y accepted been reflectivity seismic constructed i n f o r m a t i o n must be s u p p l i e d v e l o c i t y analyses bandlimited the the is of from broadband t h e two c o n s t r u c t i o n iv TABLE OF CONTENTS Page Abstract i i L i s t Of I l l u s t r a t i o n s , vi Acknowledgements ix B a c k g r o u n d 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 9 And Impedance CHAPTER 2 The C o n v o l u t i o n a l M o d e l 15 CHAPTER 3 I n v e r s e T h e o r y 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 F r e q u e n c y Domain A p p r a i s a l D e c o n v o l u t i o n 21 . . . .- 27 Impedance C o m p u t a t i o n And A p p r a i s a l 29 An E x a m p l e 32 Of A p p r a i s a l D e c o n v o l u t i o n CHAPTER 4 R e f l e c t i v i t y Construction AR E x t e n s i o n Of The R e f l e c t i v i t y Broadband Reflectivity Spectrum 40 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 P r o g r a m m i n g General 37 Algorithm LP F o r u m u l a t i o n 50 52 The C o n s t r a i n e d L P S o l u t i o n 55 D a t a E r r o r s And The L P S o l u t i o n 60 Inequality Constraints 60 Equality Constraints 61 Stability CHAPTER 5 And E f f i c i e n c y Multitrace Real Seismic Data Considerations Reflectivity 67 C o n s t r u c t i o n Using 78 V The Real Data 78 The Karhunen-Loeve T r a n s f o r m a t i o n 87 Conclusion 99 References A p p e n d i x A: D e r i v a t i o n Of The 1 Common T r a c e 04 106 vi List of Illustrations Page Fig. 1 Seismic reflection Fig. 2 Common m i d p o i n t s o u r c e - r e c e i v e r resulting Fig. 3 Processing Fig. 4 Normal geometry from m u l t i f o l d s e q u e n c e on incidence compositional 3 pairs coverage a CDP 4 gather reflection from 5 a boundary 9 Fig. 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 Fig. 6 Error 13 Fig. 7 Fig. 8 Fig. 9 The c o n v o l u t i o n a l m o d e l of reflection seismograms a) t h e 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 the convolutional 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 S y n t h e t i c models for an example, of appraisal deconvolution Fig. Table Fig. Fig. Fig. Fig. Fig. 10 1 11 12 13 14 15 i n the linear approximation Appraisal deconvolution 18 20 34 35 Relative resolution and measures f o r the a p p r a i s a l s f i g . 10 variance shown in The nonuniqueness inherent r e f l e c t i v i t y measurements Autoregressive reconstruction of reflectivity function containing reflectors 36 in 39 a few 48 Autoregressive reconstruction of a reflectivity f u n c t i o n c o n t a i n i n g many reflectors 51 Flow diagram f o r algorithm 58 the constrained LP Deconvolved seismic section; represents subsurface structure in p a r t s of w e s t e r n A l b e r t a .63 vii Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. 16 17 18 19 20 21 22 23 24 25 26 27 28 Zero phase wavelet f u n c t i o n ) e s t i m a t e d from t h e box i n f i g u r e 15 (averaging data inside 63 L i n e a r p r o g r a m m i n g 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 reflectors 64 L i n e a r p r o g r a m m i n g r e c o n s t r u c t i o n of a reflectivity f u n c t i o n c o n t a i n i n g many reflectors 66 S y n t h e t i c t e s t d a t a . W a v e l e t has r a n g e of 1 0 - 5 0 H Z 68 Performance of t h e AR and LP of r e f l e c t i v i t y construction p r e s e n c e of a d d i t i v e n o i s e band methods in the 70 Performance of the construction methods u s i n g an i n n a c c u r a t e w a v e l e t 72 Stability of r e s p e c t to the reconstructions with f r e q u e n c y band u s e d 74 Stability of the AR method with r e s p e c t t o o r d e r and t h e s t a b i l i t y of order with respect t o t h e number of reflection coefficients; using 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 The a c c u r a c y and e f f i c i e n c y of t h e 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 e x p o n e n t q, u s i n g a c c u r a t e d a t a 77 D a t a 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 data along with a geological, interpretation 79 D a t a s e c t i o n B; data 79 deconvolved explosion AR reflectivity and impedance reconstructions from s e c t i o n A; b) using unsmoothed, and c) using smoothed s p e c t r a l e s t i m a t e s 82 LP reflectivity and impedance reconstructions from s e c t i o n A; b) using unsmoothed, and c) using smoothed s p e c t r a l e s t i m a t e s 83 VI Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. 29 30 31 32 33 34 35 36 37 38 39 AR reflectivity and r e c o n s t r u c t i o n s from s e c t i o n smoothed s p e c t r a l e s t i m a t e s impedance B using LP reflectivity and r e c o n s t r u c t i o n s from s e c t i o n smoothed s p e c t r a l e s t i m a t e s impedance B using 11 85 86 A p p l i c a t i o n o f t h e 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 h e c h o i c e o f AR order. Also showing the p r i n c i p l e of t h e mixing algorithm 92 A p p l i c a t i o n o f t h e K-L and m i x i n g algorithm section B 93 transformation t o the noisy A p p l i c a t i o n o f t h e K-L a l g o r i t h m t o L P reflectivity and impedance reconstructions 95 The final r e f l e c t i v i t y and r e c o n s t r u c t i o n s from s e c t i o n with the original interpretation 96 Comparison of final reconstructions with d e c o n v o l v e d s e c t i o n (A) Comparison of reconstructions averages obtained data impedance A along geologic reflectivity the original 97 final impedance with impedance from the deconvolved . 98 The f i n a l r e f l e c t i v i t y and impedance reconstructions f r o m 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 inserted for comparison 100 Comparison of final reflectivity reconstructions with the original d e c o n v o l v e d s e c t i o n (B) 101 Comparison of reconstructions averages obtained data 102 i final impedance with impedance from the deconvolved ix Acknowledgements I a c k n o w l e d g e Ms. K a t e A a s e n f o r p r o v i d i n g interesting diversions t o t h e s i s w o r k ; E l l i s and D o r o t h y ; 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 Geophysics House - Rob C o n r a a d s , J i m H o r n , 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 t i m e 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 t h e AR p a r t s o f this thesis; Shlomo L e v y f o r l e a v i n g s e v e r a l of h i s t h e n (1979) u n p u b l i s h e d p a p e r s l y i n g a r o u n d the department which inspired p a r t s of t h i s t h e s i s - Shlomo s u g g e s t e d use of t h e K-L t r a n s f o r m a t i o n ; Mr. C o m p u t e r , C o l i n W a l k e r ; Ken ' W i z a r d ' 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 t h a n k s go to 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 t h e l a n g u a g e of t h i s t h e s i s . I wish to acknowledge a l l members of this department f o r p r o v i d i n g a dynamic environment i n which t o study. 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 f r o m 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 t h e t i m e and effort they have s p e n t i n p r o v i d i n g a s u i t a b l e d a t a s e t . I am a l s o g r a t e f u l t o MRO Associates Inc. for p r o v i d i n g a second data s e t . Finally, I would l i k e t o t h a n k and a c k n o w l e d g e my a d v i s o r P r o f . Doug O l d e n b u r g . Doug has provided me with endless motivation and support during t h e p a s t 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 didactic guidance as well as his strong 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 e n e r g y t o a l l a s p e c t s of t h i s work. 1 B a c k g r o u n d and are Reflection s e i s m o l o g i s t s and generally interested subsurface acoustic seismograms. steps. The First resemble the result The Introduction in the impedance estimation the subsurface i n t o an seismogram requires impedance i n f o r m a t i o n well (Galbraith common midpoint Lindseth, because the inherent recordings. the Because information frequency information In acoustic Gilbert two processed to then is typically 10-60 t o an of low e i t h e r from nearby out or on and from Willm, The of seismic accurate only low 1977; necessary limitations, recoverable a a s u i t e of nature Hz. this impedance. been c o n s i d e r e d experimental of is in 1979), bandlimited band, r a n g e incidence seismic within frequency interpretation impedance l o g . this 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 i m p e d a n c e be philosophy; obtained s t e p has of normal introduction (Lavergne (0-lOHz) i s v i t a l of a r e c o v e r e d the carried seismograms This estimating of a c o u s t i c Millington, analysis 1979). of reflection and of f u n c t i o n ; and estimate frequency velocity from reflectivity second s t e p normally particular problem geologists i s u s u a l l y c a r r i e d out measured i s converted logs stratigraphic in a way framework of c a r r i e d out w i t h a somewhat which e s s e n t i a l l y linear inverse different f o l l o w s the theory of Backus- (Oldenburg, 2 1981). First, appraised t h e measured seismogram i s modelled t o o b t a i n unique i n f o r m a t i o n about the reflectivity function. 'reflectivity averages' it is called This f u n c t i o n i s then information along broadband with The physical this by approaching inverse formalism; procedures which reflectivity have unique an and a interpretable impedance may be o b t a i n e d . primarily impedance in broadband the in recovery efficacy the insights using a of the been r e f i n e d t o c o n s t r u c t linear various broadband and impedance models. A brief reflection and A obtain constraints result, of s u b s u r f a c e i m p o r t a n c e o f t h i s work l i e s gained to constructed using t h i s some From estimate unique i n f o r m a t i o n i s c a l l e d deconvolution'. reflectivity suitable algorithm. subsurface a n d t h e method u s e d h e r e 'appraisal and then review method o f some i m p o r t a n t aspects and terminology associated of the seismic is now presented. The basic subsurface equally (shot These spaced distance from disturbance reflection reflection survey boundary r e f l e c t i o n s a r i s i n g disturbance surface. seismic or reflections receivers the be c a l l e d are from created an near recorded at recording artificial the a at point the (see figure geophone the source wavelet from (shown ground line (geophones) which extend source arriving will source) entails a short 1). a of A single i n the 3 seismic to t r a c e s as a coherent wiggle - f i g u r e the simple laws of r e f l e c t i o n the r e f l e c t i o n p o i n t source-receiver Each (for horizontal According boundaries) (RP) i s l o c a t e d v e r t i c a l l y beneath disturbance created and recorded overlapping the single source will in of horizontal 1. geometry a t s u c c e s s i v e r e c e i v e r o f f s e t s have a common midpoint case have a figure s h o t s , i t i s p o s s i b l e t o arrange such that d i f f e r e n t the the midpoint. s i n g l e source geometry s i m i l a r t o that shown By 1). source- (see f i g u r e 2 ) . In boundaries, these common midpoint source- receiver .Surface Resulting Seismic Traces depth Source Wavelet Boundary F i g . 1. D i a g r a m s h o w i n g a t y p i c a l s o u r c e - r e c e i v e r g e o m e t r y 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 will be a s s o c i a t e d with a common depth p o i n t (CDP) as shown i n f i g u r e 2 . A c o l l e c t i o n of s e i s m i c the o f f s e t s have a common midpoint i s c a l l e d source-receiver a CDP gather gather (see f i g u r e 3 ) . The number of traces d e f i n e s the magnitude of m u l t i f o l d coverage by the o v e r l a p p i n g provides subsurface will t r a c e s f o r which be N-fold shots. CDP A coverage by NxlOO p e r c e n t . described make use of this CDP in gather and with i s said The s t a c k i n g i n the obtained N traces t o cover the procedure which the f o l l o w i n g paragraph attempts t o redundant information to increase Midpoint Resulting Traces Normal Incidence Ray Path COP 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 from m u l t i f o l d c o v e r a g e a n d t h e r e c o r d e d receiver. pairs resulting t r a c e from each the 5 s i g n a l t o n o i s e r a t i o of the d a t a . Observing the CDP gather i n f i g u r e 3 i t i s e v i d e n t that t h e r e i s d i f f e r e n t i a l t r a v e l time each reflective event source-receiver o f f s e t s . t r a v e l time normal incidence gather for After correcting this ray, a l l t r a c e s (i.e., of the r e s u l t stacked the i n each t r a c e t o match the t r a v e l algebraically ratio across (or normal moveout-NMO) t o trace is stacked) at A**^"* CDP Gather of the may be summed together enhancing plotted differential time the s i g n a l t o (assuming random n o i s e ) . then different the noise The r e s u l t i n g source-receiver Midpoint N M O Corrected Stacked Result F i g . 3P r o c e s s i n g s e q u e n c e on a common d e p t h p o i n t g a t h e r , 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 which reflective the source-receiver form producing incidence the unique processing seismogram steps. energy losses transmission layered normal and of the of: geometrical absorption, losses across thesis. from- a treatment through anelastic suitable in this effects in the assumptions proper the beneath in information many A include spreading, seismogram is methods d e s c r i b e d involves seismogram must (1) This reflectivity seismogram sequential observed by incidence is located vertically midpoint. f o r treatment Obtaining event a normal and boundaries in a media (2) d i s p e r s i o n (3) source type (4) field (5) multiple and coupling g e o m e t r i e s and array reflections and responses other coherent noise (6) the mathematical model imposed on. the seismogram (7) The this other s o l u t i o n s of thesis. processed so these complications I t i s assumed t h a t that the the will not seismic be adressed data s e i s m o g r a m , d e n o t e d by have in been s ( t ) , may be 7 considered to result from the c o n v o l u t i o n of t h e reflectivity f u n c t i o n r ( t ) w i t h the random n o i s e n(t). That i s , the is described by f o l l o w i n g c o n v o l u t i o n a l model the 5CO-ra)* wet) where t h e seismogram complete normal i n c i d e n c e will be reflectivity 'incidence, of wave model simple experimental is grounds problem. of d e c o n v o l v i n g wavelet are from a n o r m a l from linear acoustic plane the seismogram operation. basic of t h e p r o b l e m b e g i n s and convolutional estimation developed treatment plus +• nco impedance developement establish wavelet w(t) symbol * denotes the c o n v o l u t i o n Acoustic the source subsurface concepts. according theory. A then provided in 3, normal of the 2 chapter understanding In c h a p t e r between to discussion A 1 with i n chapter relationships impedance for incidence the to complete the t r a d i t i o n a l method s e i s m o g r a m when p r o p e r t i e s of t h e source known i s v i e w e d u n d e r t h e umbrella of inverse theory. Two be distinct presented synthetic maximum entropy representation data. method of predict a reflectivity 4 and i n chapter seimic spectrum to methods of The their first which small spectral construction efficacy shown will using c o n s t r u c t i o n method i s a uses portion values an of autoregressive the outside reflectivity that small 8 portion. is The s e c o n d and most a linear programming spectral information provides a reflectivity norm, leading versatile construction approach which u t i l i z e s from an unique appraisal deconvolution function naturally the method which has a and minimum t o an impedance model with 1| blocky character. The final construction chapter methods Karhunen-Loeve procedure impedance real real seismic a a strong the data. A l s o , the weighted role i s described in above averaging s t a b l i z i n g the and t h e n utilized on data. i n the data acoustic impedance this frequency nature or of the bandlimited analyses seismogram, model may in this then in assessing i n some is must challenged. information an be its detail idea supplied that from because of the b a n d l i m i t e d If accurate, c a n be r e c o v e r e d interpretable, be c o n s t r u c t e d thesis. constructed roles in estimating be d e v e l o p e d information velocity trace, and of The g e n e r a l l y a c c e p t e d reflectivity seismic presented important function thesis. impedance logs impedance play uniqueness These f e a t u r e s w i l l throughout well and t h e models w i l l uniqueness. the plays reconstructions, reflectivity low using t o examples o f transformation, which Errors the i s devoted using t h e two from broadband algorithms 9 R e f l e c t i v i t y and Impedance A review of the b a s i c plane wave relationship be theory will a in normal boundary, incident wave amplitudes are r e l a t e d by a reflection which p on impedance to thesis. compositional depends incidence, now be p r e s e n t e d t o e s t a b l i s h the between r e f l e c t i v i t y and a c o u s t i c used throughout t h i s At concepts the density and and r e f l e c t e d coefficient r v e l o c i t y v of the adjacent media I.I where A; i s the amplitude incident and A r is the reflected wave (see f i g u r e 4 ) . Aj A r Transmitted Energy F i g . 4. Normal i n c i d e n c e boundary. r e f l e c t i o n from a compositional 10 The product of density i m p e d a n c e o f t h e medium and this concept and velocity will be f o r the r e f l e c t i v i t y which is there a reflection d e s c r i b e d by (see f i g u r e Rearranging this computation where z, obtain boundaries, series of a c o u s t i c i m p e d a n c e results i s the a c o u s t i c impedance i n the an and 1.3 acoustic impedance i n f o r m a t i o n may be Defining impedance, where represent seismologist. accurate so is obtained in a t each boundary 5) First reflectivity measurement i s t h e n c o n v e r t e d initial w i t h many simple reflection acoustic Extending coefficient expression, 1.2 the z. a Expressions of t h e denoted to a layered subsurface an e x p r e s s i o n is that formula initial the layer. the main o b j e c t i v e s t h e s e i s m o l o g i s t must measurement. i n t o an e s t i m a t e o f rock for t y p e s and This subsurface perhaps other inferred. as t h e l o g a r i t h m the of normalized acoustic n o r m a l i z a t i o n i s with respect to a c o u s t i c impedance Z j , e x p r e s s i o n 1.3 becomes the 11 i i i i i i i i i r i Layer 1 II i i i i r •1 i /1 > z Layer 2 2 t- 13 • • « • ' K-l i i i i i t i /1 Layer k //; i r " '2 Layer 3 II Acoustic Impedance Reflectivity Series Surface r *k / i Layer k»l z k»l 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 s h o w i n g t h e 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 t y p e o f model. In o r d e r t o s i m p l i f y f u r t h e r from here on referred discussion of i t will be 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. F o r | r | <1 , 0 0 I F N - 1 12 This approximation 6. Using this i s valid f o r | r | <.25 a s shown simplification, expression in figure 1.4 r e d u c e s t o 1= 1 which gives a reflectivity. linear relationship between i m p e d a n c e a n d 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 continuous domain i t c a n be shown t h a t r e l a t i o n analogue 4^ (eg. Peterson 1.6 h a s t h e e t a l , 1955) = 2 ret) , ^(O = 1 $ rcu) du i.g 7 or t 0 where t h e r e f l e c t i v i t y coefficients function rj a r e r e l a t e d by rv rc-O-Z r^a-r,) 1-1 r ( t ) and the reflection 13 w h e r e N i s t h e number o f b o u n d a r i e s a n d Dirac each delta function reflection written located coefficient. as ( e g . , Peacock, S(t- ) is at the time occurence Equation 1.8 may be u(t) reflectivity of is the unit function with of also 1979) ^(O = 2 ret) * uit) where the step 1.1 function. the unit Convolving the step performs the role integration. F i g . 6. The e r r o r between f and t h e fx f o r v a l u e s o f | r | l e s s t h a n .9. ( linear approximation 14 In this reflectivity thesis, the and i m p e d a n c e i n p l a c e of the n o n l i n e a r simplify further remembered restriction than .25. relationships (equations relationship that a l l linear reflection impedance s h o u l d (equation be c a l c u l a t e d relations be u s e d 1.4), to I t must be have the | r | must be less e x c e e d s .25 t h e n the coefficients If a reflection coefficient between 1.5-1.9) w i l l mathematical transformations. however, that linear v i a equation 1.4. 15 2 The The Convolutional purpose of t h i s c h a p t e r c o n v o l u t i o n a l model i n s e i s m i c clear the / spectral trace is source reflecting boundaries about the wavelets subsurface amplitudes subsurface reflectivity and function. from The a recovering reflection can partially fulfilled technique. be conceal ultimate conventional deconvolution techniques Deconvolution, Geophysics Reprint Series good can 1, the goal of wavelet representative shown t h a t Several of traveltimes i s , they using of Information the coefficient It will A variety 1953). in that boundary. deconvolution (inverse i s to r e p l a c e each o v e r l a p p i n g corresponding only at t o make upon as a s e q u e n c e i s hidden of a s u b s u r f a c e be and the from a s e i s m i c t r a c e . looked wavelets, theory techniques ( R i c k e r , 1940 of t h e s e explain deconvolution that r e s u l t geology seismic deconvolution a information generally overlapping with reflection division) broadband r e f l e c t i v i t y and i s to b r i e f l y l i m i t a t i o n s of c o n v e n t i o n a l filtering seismic Model a this goal conventional examples of be in found volumes I and II . The simple c o n v o l u t i o n a l model w i t h o u t additive noise s t a t e s t h a t t h e m e a s u r e d s e i s m o g r a m s ( t ) ( w h e t h e r i t be result the before or a f t e r reflectivity the function wavelet w ( t ) , that i s , stacking procedure), r(t) convolved with the i s equal a to source 16 The source wavelet i s d e f i n e d t o t a k e on a v a r i e t y o f c h a r a c t e r i s t i c s depending used and the .physical source wavelet of a single on the effects experimental o f wave p r o p o g a t i o n . i s sometimes d e f i n e d as t h e p a r t i c 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 geophone. However, i t effects i n the wavelet stacked seismic data. source wavelet i s necessary Probable i n time by a deconvolution effects contributing of to the below: effects f i e l d geometries (d) g h o s t i n g (f) velocity type and c o u p l i n g (b) t r a n s m i s s i o n (e) n e a r The to include processing when c o n s i d e r i n g a a r e summarized (a) s o u r c e (c) procedures a n d geophone a r r a y s effect surface instrument weathering response (g) p r o c e s s i n g e f f e c t s (eg., stacking) (h) e t c . By following effects a c o n v o l u t i o n a l model listed source engaged set reflectivity Reflection wavelet in a l l experimental above a r e a t t a c h e d t o t h e s o u r c e w a v e l e t , a strong foundation i s subsurface where for from recovering t h e measured an estimate seismogram. s e i s m o l o g i s t s r e a l i z e the importance i n s e i s m i c d a t a p r o c e s s i n g and devising improved of methods w h i c h are of t h e actively provide source 17 wavelet estimates. three These methods may be classified knowledge the source into groups: (1) d i r e c t (eg., v i b r a t i n g of source) (2) d i r e c t measurement o f t h e s o u r c e (Mayne and Quay, 1971; (3) e s t i m a t i o n of properties function and and U l r y c h , from the measured on assumed s t a t i s t i c a l physical 1971; signature Kramer e t a l , 1968) a wavelet seismogram based Ulrych, signature of wavelet O t i s and 1977, and the or reflectivity (Robinson, 1967; Smith, Lines 1977; Oldenburg, et a l . , 1981). For a l l discussions relating will be assumed t h a t the source wavelet determination of a t o the c o n v o l u t i o n a l model, i t broadband is known reflectivity and that function is desired. This section representation Figure the may which be u n i q u e l y 7a shows t h e r e s u l t data conclude with a pictorial of t h e c o n v o l u t i o n a l model and an e x a m p l e o f spectral deconvolution information will indicates what r e c o v e r e d from the of a a r e measured i n time) reflectivity seismogram. seismic experiment (where i n which the source wavelet 18 is a delta function. with s u b s u r f a c e recovery The c o n v o l u t i o n of a d e l t a reflection coefficients results of the t r u e r e f l e c t i v i t y subsurface structure with function. reflectivity observed as shown i n the frequency theorem t o f i g u r e 7b. becomes frequency in a figure domain as shown of in in spectral figure Subsurface Function source the wavelet measurement of T h i s e f f e c t may be domain by a p p l y i n g the c o n v o l u t i o n The c o n v o l u t i o n multiplication Reflectivity 7b. in a perfect Convolving a more r e a l i s t i c leads to a poorly resolved, uninterpretable function 8a the time domain components i n the (where only the Measured Source Wavelet Reflectivity Function F i g . 7 . a ) The c o n v o l u t i o n model i n t h e t i m e domain u s i n g an i d e a l w a v e l e t a n d , b) u s i n g a more r e a l i s t i c w a v e l e t . 19 amplitude effects spectra have bandlimited the been positive left character bandlimited reflectivity seen that a for the measured of the is recovery inhibited frequencies out function complete is of nature the spectrum and by the are simplicity). source not of shown: Note reflectivity wavelet. bandlimited the original spectral nulling that is In and phase the due to general, thus it is reflectivity effect of the wavelet. Unique the measured of the words, reflectivity data wavelet a rough by removing within a figure Since the low and at both is frequencies which wavelet can wavelet high be recovered represents information recoverable The the other be made by division as shown in insignificant no the figure only through nonuniqueness is infinitely satisfy (see In may Only bandwidth 8b). unique spectral energy reflectivity frequencies. within effects bandrange. frequencies, these from those of This the band of reflectivity division or inverse techniques. reflectivity are contains obtained smoothing uniqueness contained frequencies filtering at be spectral spectral recoverable are may nonzero of bandlimited information the its appraisal performing 8b. information the now inherent apparent many from in the reflectivity resultants of figures recovering above analysis. functions 8a broadband and which 8b. The There will only 20 information that a l l such f u n c t i o n s must have i n common i s the band of f r e q u e n c i e s figure 8 b . information the However, shown on recovering from s e i s m i c data missing the provided in the side can be a t r a c t a b l e problem technique, involve Alternatively, which utilizes of reflectivity frequencies are accurately replaced. analysis. construction hand broadband methods f o r r e p l a c i n g low f r e q u e n c i e s velocity right Standard well a if l o g or reflectivity the information by a s i n g l e deconvolved t r a c e , may be used t o f i l l unknown frequencies. c o n s t r u c t i o n methods a r e p r e s e n t e d Reflectivity. Spectrum Measured Reflectivity Spectrum Wavelet Spectrum Wavelet Two such i n chapter reflectivity 4. Measured Reflectivity Spectrum 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 t h e domain a n d , 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 . frequency 21 3 I n v e r s e T h e o r y and The with p u r p o s e of t h i s the n o t a t i o n be of this section The s e c t i o n i s to basic concepts to deconvolution. Appra i s a l Deconvolut i o n of This h e u r i s t i c can be the i n v e r s e t h e o r y and somewhat s i m p l i f i e d ; topic provide found reader i t s relation approach r e q u i r e s that a more r i g o u r o u s i n Oldenburg, the treatment (1981). This i s i n most p a r t a p r e c i s o f t h a t p a p e r . c o n v o l u t i o n a l model of a s e i s m o g r a m n e g l e c t i n g the additive noise term-is written 3.1 When the wavelet either t h e . time approaches w i l l under s i m i l a r i s known, e q u a t i o n or frequency 3.1 may in Each of these 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 results c o n d i t i o n s are domain. be a p p r a i s e d identical. A) Time d o m a i n a p p r a i s a l d e c o n v o l u t i o n In the t i m e d o m a i n , an t h a t c o n t r a c t s the wavelet inverse f i l t e r into a delta v(t) i s desired function; i.e., 3.2 But i n the seismic case, 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 filter which function. wavelet will Finding contract an the inverse wavelet filter i n t o a z e r o phase a p p r o x i m a t i o n By z e r o to a delta the 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 h a s i t s e n e r g y peak its alternative. delta function p h a s e , i t i s meant energy peak. T h i s c a n be done by f i n d i n g a f i l t e r the approximation optimum may i s , the f i l t e r position depend c h a r a c t e r i s t i c s of the wavelet Oldenburg, which 1981). Obtaining t minimizes 0 of on the phase ( T r e i t e l and Robinson, 1966: best length t o perform the a p p r a i s a l spike or the the desired v ( t ) by , b o t h s i d e s o f 3.1 a r e t h e n c o n v o l v e d w i t h filter about t h e i n t e g r a t e d s q u a r e d d i f f e r e n c e b e t w e e n t h e two s i d e s o f 3.2, t h a t where that located t h e 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 minimizes 0 a which c o n t r a c t s the is at best into minimizing the inverse deconvolution soo*vet) - r c o * M O * VCO = r c o * a CO. = <ra)> 3,4 23 where <r(t)> denotes the a p p r a i s a l might reflectivity VCO convolution of the <rco> wavelet produces an a v e r a g i n g f u n c t i o n which localized than the o r i g i n a l WCO The because resolving wavelet. with the i n v e r s e i s i n some power of the sense more QCt) f u n c t i o n p l a y s a fundamental i t s character filter For example: VCO averaging theory Pictorially, look l i k e : SCO The averages. immediately wavelet. role in inverse indicates the T h i s type of a v e r a g i n g 24 function data is called a zero phase wavelet by seismic processors. From with often 3.4, the true averages. the convolution reflectivity For model of the averaging will produce reflectivity example: Ail rco aco Knowledge about seismogram, <r(t)> is and information the true completely the is inverse is unacceptably utilizing seldom filter useful will large summarized about values. recoverable by t h e function the form the on n o i s y enhance A rA I < r ( t ) > reflectivity, averaging provided Deconvolution function function the most noise unless resolved data in extra averaging because the seismogram set) = ret) * wet) * rut) values r(t). seismic The n o i s y average a(t) of from the is data the to written 25 Convolving this with an inverse filter v ( t ) produces SCO * VCO = rCO* wet) * vft) + nco * vet) = rot)* act) + net)* VCt) = < r a ) 7 -h £ < r f t ) > where £<r(t)> i s an e r r o r t e r m a r i s i n g noise. In t h i s case i t w i l l filter which and a l s o produces keeps be d e s i r a b l e by m i n i m i z i n g statistical m e a s u r e 3.3, By to the where 0 and O averaging minimizing which inverse norm o f form inverse function filter £<r(t)>. The is mathematically of the r e s o l u t i o n i s denoted the e r r o r v a r i a n c e r e s o l u t i o n measure (f) , an optimum i n v e r s e derived. additive t o f i n d an the term which functional i s i t s variance simultaneously in a statistical form of the e r r o r complimentary the t e r m minimum. D a t a e r r o r s c a n be a c c o m o d a t e d solution from t h e most r e s o l v e d the e r r o r 3,5" filter Y and v ( t ) may the be Let is called IT/2. the t r a d e o f f A tradeoff parameter and v a r i e s b e t w e e n r e s o l u t i o n and between statistical 26 error produced by the o b t a i n e d by m i n i m i z i n g © = 0, When the v(t) resolved averaging derived As © from and which will inverse f i l t e r filter values may of be 0 . 3 . 6 y i e l d s the most reflectivity averages have the g r e a t e s t u n c e r t a i n t y . resolution accuracy that one different minimizes s(t)*v(t) will p h i l o s o p h y , an © 3.6 using inverse f u n c t i o n , but the increases, statistical resulting will be may be lost achieved. but By greater using this be d e r i v e d at any value of which p r o v i d e s the best compromise between r e s o l u t i o n 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 d e f i n e a r e s o l u t i o n measure 'L' as being the i n v e r s e h e i g h t of averaging as to Q function at i t s energy peak. i n c r e a s e s from 0 to monitor resolution look something increasing variance * y we may Then, f o r example, plot a L and v a r i a n c e Y. tradeoff The result like: * eso Tradeoff Curve \ \ the \e >o xe=*/ 2 decreasing resolution curve would 27 In practical derived better appraisal deconvolution, a t a n o n z e r o v a l u e of statistical dramatic sacrifice initial decline B) accuracy w i l l decrease a small i n the of the F r e q u e n c y domain a p p r a i s a l Fourier Fourier pair of be achieved with by convolution A spectral division only steep deconvolution i n the f r e q u e n c y domain W(f) s ( t ) , r ( t ) , and is defined are t o 3.1 by respectively w(t), as, theorem a p p l i e d gives a a curve. -oo then the that Very often indicated I f S ( f ) , R ( f ) , and transforms transform as i s c a r r i e d out spectral division. can f i l t e r is p a r a m e t e r so attained. variance tradeoff inverse tradeoff be in resolution Deconvolution the the the yields where the 28 where W*(f) known that frequencies so 3.7 the has c o m p l e x c o n j u g a t e of W ( f ) . t h i s equation where W ( f ) denominator. been d e r i v e d domain be i s small For in © formulation, well i f the data are inaccurate, adding a s t a b l i z i n g data, frequency the domain term stablizing by the to term Oldenburg, f r e q u e n c y domain a p p r a i s a l time equation written in the data i n the is sufficiently Equation where is I n 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 t h e instabilities if It produce erroneous r e s u l t s at inaccurate the where o ( i s a c o n s t a n t noise will i s n e c e s s a r i l y a l t e r e d by (1981). may i s the V(f) T a k i n g the 3.8 is can d e p e n d e n t on . and i s the random t r a d e o f f parameter. spectral division will now The be overcome in curly brackets. large. be defined written as the q u a n t i t y inverse Fourier transform where v ( t ) i s t h e t h e m a g n i t u d e of inverse filter of i n the 3.9 time yields domain. Thus 29 the results of appraisal completely analagous appraisal carried reflectivity in t o those in averages, a appraisal appraisal function in the that An meaningful averages useful equation by 3.1 because of working work of i n that the domain. knowledge when t h e w a v e l e t the output an convolving known. However, f r o m an a p p r a i s a l i s used t o be taken the used and alone the a reflectivity to because they l a c k i t with provides general, integrating model. m o d e l i s one w h i c h data In to broadband of t h e r e f l e c t i v i t y reflectivity estimate. example, of t h e t r u e interpretable, f i t to function is some c a r e must be s u c c e s s f u l l y For domain in this our impedance frequencies. presented summarize adequate impedance those averaging between cannot of associated model, and a v e r a g e s an variance the i n t e r p r e t a b l e , broadband produces unique e r r o r , ' and compute an impedance f u n c t i o n , reflectivity yields value Q averages, that differentiate Each reflectivity function event domain. the completely reflectivity are and a p p r a i s a l shows statistical domain function. frequency C) Impedance c o m p u t a t i o n time statistical 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 their the deconvolutions were c a r r i e d o u t i n t h e An frequency out a t a t r a d e o f f a v e r a g e s , a n d an a v e r a g i n g All the provide a i m p o r t a n t low basic twice the u n i t model step 30 function u ( t ) , gives Denoting the left hand side s^t), 1.9 seismogram and making use of e q u a t i o n Next c o n v o l v i n g wavelet b o t h s i d e s by t h e as best the integrated gives inverse filter of t h e produces $&) * vct) = 7 a ) * woo *va) - '^tt)*atO = <7<0? This result indicates that by a p p r a i s i n g the s e i s m o g r a m , impedance a v e r a g e s , < ^ ( t ) > , a r e convolution of the integrated data p r o d u c e s an o u t p u t e q u a l w i t h the to the convolution integrated obtained. inverse of The filter the true 31 impedance model with the same from the r e f l e c t i v i t y averaging appraisal. function obtained For example: act) The r e s o l v i n g c a p a b i l i t y of the a v e r a g i n g f u n c t i o n , i n t h i s case, i s not as important Since the frequency as i t s low content frequency of the a v e r a g i n g dependent on t h a t of the wavelet, averaging function exist and peaks in frequencies in which l e a d s t o the u n i n t e r p r e t a b l e impedance estimate < ^ ( t ) > troughs function i s i t i s m i s s i n g both low and high f r e q u e n c i e s ; and i t i s the l a c k of low the content. shown above. Clearly the impedance averages i n the true impedance model and the there are which do not major structural generally known t h a t trends a r e absent. Reflection seismologists' have b a n d l i m i t e d impedance averages, when c o n s i d e r e d as model, produce will interpretations. usually not Thus, techniques of correct recovering a final structural the low 32 frequency impedance i n f o r m a t i o n from w e l l analyses have 1979; been developed Lindseth,1979). operating (Galbraith In a d d i t i o n (or philosophy l o g or velocity and M i l l i n g t o n , alternatively), of the s e i s m o l o g i s t should include the idea of c o n s t r u c t i n g broadband models seismic the i n t e r p r e t a t i o n s . next data, to help c l a r i f y chapter, appraisal reflectivity are used in broadband r e f l e c t i v i t y from averages two (and thus the obtained algorithms the appraised In the from which an construct impedance) e s t i m a t e s . A comment a b o u t u n i q u e n e s s may be made by r e f l e c t i n g on the basic model infinitely many equation is There f u n c t i o n s r ( t ) which s a t i s f y (given the bandlimited number s(t)=r(t)*w(t). further nature enlarged of the this wavelet) i f we a c c e p t functions may t o those well give data. this permissible t o a wide range of computed i m p e d a n c e f u n c t i o n s , u n l e s s some nature and statistically T h i s group of rise equation that the data are i n a c c u r a t e and,permit f u n c t i o n s which generate allowable misfits are way of 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 c o n s t r a i n i n g the functions is.devised. D) An e x a m p l e o f a p p r a i s a l d e c o n v o l u t i o n The derived impedance from represents an a shown i n f i g u r e interpretation reasonable of picture impedance i n p a r t s of A l b e r t a . This well of 9a was l o g data general and subsurface impedance model w i l l be 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 deconvolution. A reflectivity the impedance computed from convolved w i t h a wavelet seismogram max|s(t)| s(t). was noise this s e i s m o g r a m shown i n p a n e l d. this example model only i s 4msec. the via having a amplitude spectrum and STD o f 20% o f sampling shown 1.7 synthetic The t i m e The c o r r e s p o n d i n g was a providing is b) equation result, r e p r e s e n t a t i o n of t h i s p r o c e d u r e where (panel (panel c ) , t o produce Random added t o function the interval for frequency in noisy domain panels of p o s i t i v e e-g; frequencies are 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 o u t i n t h e f r e q u e n c y seven in different figure 10. 1 (where L Q normalized plotted v a l u e s o f t r a d e o f f p a r a m e t e r Q, The d e f i n e d by L / L ~ relative are l i s t e d and at a r e shown r e s o l u t i o n and v a r i a n c e measures in table simply r e f e r s t o the f i r s t v a l u e computed). The averaging each v a l u e of © are 10. The normalized amplitude spectra are i n the f i r s t functions for column of f i g u r e reflectivity averages plotted c o l u m n 2; and t h e n o r m a l i z e d in along with t h e i r a r e shown i n c o l u m n 3. averages and parameter i s increased. where domain in Note the change their spectral Stability impedance averages i n c h a r a c t e r of content as the t r a d e o f f i s achieved around f r e q u e n c i e s o u t s i d e t h e bandrange of the w a v e l e t been s i g n i f i c a n t l y reduced in importance. the have 3 ^ 1-1 (D TIME (sec) i 2 5 6 1OO 1 0.0 1 1 1 O.B a ret") o oo Y— i i i _ o o wa) iwcoi f - 1 0.0 i i 4.0 i c i 1RCOI (D b i CO a 3 F i g . 9. Synthetic deconvolution. models f o r a n example FREQUENCY iscol of appraisal 125Hz 35 2 1 3 k A. A A A — JV— ) norm a(l-©) norm <ttt;e)> FHEOUCMCT lRtf)l 125 H a norm <^(t;e)> F i g . 10. A p p r a i s a l de c o n v o l u t i o n s shown u s i n g s e v e n v a l u e s of t r a d e o f f parameter 36 Although the r e s u l t i n g averages impedance interpretation, utilized in broadband forms of r e f l e c t i v i t y reflectivity a they construction construction A are procedure and will L e are undesireable unique and which impedance. may be leads to The t o p i c of be c o v e r e d n e x t . 0 .001 1,0 ,01 1 .ST 1 .0 .1 7.0 .014 .sr 2.5" .002) i.o 3.5- .QOOiX 1.4 5.IS .0002S 1 S2 .00006Z 10.0 T a b l e 1. Relative resolution a p p r a i s a l s shown i n f i g . 10. for and variance measures f o r the 37 Reflectivity 4 Seismic reflection formation of function based It has unique been obtainable an shown information that the methods only is in the deconvolution. unique f u n c t i o n as above. unacceptable as recoverable form where r information where a ( t ) by appraisal impedance the averages same averaging I t was n o t e d t h a t i m p e d a n c e a v e r a g e s a r e a model because of t h e b a n d l i m i t e d nature (i.e., of a(t) is final of i s missing subsurface of a ( t ) . inverse f i l t e r i n g g e n e r a t e an a c c e p t a b l e wavelet reflectivity I t was a l s o shown t h a t t h e o n l y information < *Jit)>= v£(t)*a(t) , to the averages <r(t)> = r ( t ) * a (t) , function. impedance methods about t h r o u g h use of t h e c o n v o l u t i o n a l model i s i n t h e averaging unique a n a l y s i s , thus f a r , has l e d t o t h e upon t h e method o f a p p r a i s a l f o r m of r e f l e c t i v i t y is Construct ion structure Since appraisal m e t h o d s ) must a l w a y s impedance f u n c t i o n i f t h e low frequencies, i t is fail source worthwhile i n v e s t i g a t i n g w h e t h e r a p a r t i c u l a r c o n s t r u c t i o n method m i g h t successfully r(t). fill In f a c t , recoverable bandlimited recovery i n the missing since by the appraisal frequency unique of missing of c o n s t r u c t i o n information methods p o r t i o n of i t s spectrum (eg., p o r t i o n s of R ( f ) w i l l technique. information about about r ( t ) represents only a f i g u r e s 8&10), the r e q u i r e some sort 38 The problem of described h e r e i n may appraisal of the overlapping averaging each broadband be present function. phase corresponding subsurface thought of Then by as First an follows. boundary. of a Two w h i c h have shown g o a l , but first function in each phase method, replaced by a representative of a c o n s t r u c t i o n procedures w i l l be success is in accomplishing example i s p r e s e n t e d inherent with a zero construction coefficient described an i n the data use averaging reflection nonuniqueness as recovery seismogram i s performed which r e p l a c e s wavelet zero reflectivity any this which emphasizes reflectivity the construction procedure. Consider (a) A the dual i s a blocky ( o r minimum s t r u c t u r e ) more r e a l i s t i c of t h e b l o c k y Directly e x a m p l e shown i n f i g u r e impedance each model) impedance is shown model reflectivity spectrum. r^(t) The shown function bandlimited in panels i t i s reasonable algorithm carried <r^(t)>, by each, u t i l i z i n g only would lead its (10-50HZ) (h) Therefore, out is and in panel function. Adjacent associated a v e r a g e s of ( i ) , are on the bandlimited to similar both (b). to amplitude rj ( t ) very and similar. construction <rj(t)> information broadband the reflectivity 1.6. t o e x p e c t t h a t any identically panel i s the f u n c t i o n computed from each v i a e q u a t i o n each In i m p e d a n c e f u n c t i o n (a n o i s y v e r s i o n o f impedance below 11. and supplied reflectivity Fig. 11. The n o n u n i q u e n e s s inherent in reflectivity measurements. 40 models. Further, either ( t ) or ^ ( t ) ^ of t h e constructed reflectivity however, be models s i m i l a r of Both tend to rj(t), of It will a to those space may, particular discussed models with as low method frequency of t h e reflectivity spectrum i s w e l l known t h a t t h e maximum e n t r o p y method resolved conventional conventional outside assumes behaves this power s p e c t r u m a n a l y s i s , when a p p l i e d t o a s h o r t more first makes spectrum R(f) missing that series, will spectral Fourier the data estimators are than method. known i n t e r v a l , the i n some c a s e s , estimate transform spectral of few The technique, shown t h a t the in the c o n s t r u c t i o n (AR) be many spectrum. extension of a s t a t i o n a r y d a t a zero is, reflectivity reconstruct A) A u t o r e g r e s s i v e only methods in theory, that p o r t i o n s of a r e f l e c t i v i t y It considering form infinitely s i z e of m o d e l autoregressive t h a t the accurately the c o e f f i c i e n t s as p o s s i b l e . i n a p r e d i c t a b l e manner. can are on resemble averages s i m i l a r construction reflection assumption by not i s placed n t have to favour, t h e s e m e t h o d s , an the i- ( i ) . The restricted thesis c o n s t r a which (h) and c l a s s of models. significant n o m o d e l s may output because there models shown i n p a n e l s this these constructed from is assume t h a t t h e whereas 'continued the MEM portion provide that This (MEM) a any because data are estimate in a sensible way' 41 outside that known i n t e r v a l . way', it series i s modelled i n a manner w h i c h For a need wave be i n d i c a t e d by known f r a c t i o n the which only only t o d e r i v e an a d e q u a t e (AR) spectral estimator data must requires autoregressive s e r i e s t o be The AR with zero allows known i n o r d e r We series in example, o n l y a f r a c t i o n unending form. process 'continued a sensible i s meant t h a t t h e p r o c e s s w h i c h p r o d u c e d t h e unknown d a t a . sine By examined I t can AR model the be the process A general an shown t h a t t h e MEM for of whole data is ( e g . , U l r y c h and representation of i t s complete that for of period s m a l l p o r t i o n of the model assumes an o f one i s monochromatic. a process. prediction to i n f e r specify data the stationary Bishop, a s t a t i o n a r y data 1975). series x mean, i s P where e i s a w h i t e n o i s e OQ . And order p. can be X K-1 ' k~2 For x this the are This expression predicted x reason, k-p' to t h -at equation s e r i e s w i t h z e r o mean and prediction filter variance coefficients i n d i c a t e s that a future within i s , from 4.1 an 'p' error past value e^, values i s c a l l e d a forward of x^ from of x. prediction 42 representation. A reverse prediction expression may be written P where t h e e r are reverse prediction (J^. a l s o has v a r i a n c e past x H.2 c o e f f i c i e n t s and T h i s r e p r e s e n t a t i o n shows t h a t a v a l u e x ^ c a n 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 £ , f r o m X + i ' K * l ' * * " ' K*p' ^ It filter x x follows, then, t i a t * ' s £ r o m 'P' f u t u r e v a l u e s o f t h a t f o r an AR p r o c e s s x. x, p a s t a n d f u t u r e v a l u e s c a n be p r e d i c t e d f r o m any s e t o f N>p known v a l u e s of x. If the p r e d i c t i o n filter c o e f f i c i e n t s a r e d e t e r m i n e d by m i n i m i z i n g t h e v a r i a n c e of t h e e r r o r s e r i e s , t h e r e v e r s e and forward a single filter the solutions are identical. Thus, c o e f f i c i e n t s may be f o u n d by m i n i m i z i n g forward estimated and reverse by t h e f o l l o w i n g N error forumula variances. the This s e t of sum of sum i s 43 Differentiating l e a d s t o the 4.3 with respect following matrix t o the filter coefficients equation 7.7 C ^ r q where 4 .-.p - and r Co( o( . . . . o^p ) Vj The o('s may be 2} obtained Alternatively, the determined e f f i c i e n t l y Levinson (1975). prediction by m i n i m i z i n g 'p' first to t o the data method 4.3 using can be d e s c r i b e d by U l r y c h and Bishop s q u a r e s f i t of an AR s e r i e s x. of values coefficients Burg- constructing f u n c t i o n u s e s an AR predict 4.4. the Both s o l u t i o n s e f f e c t a l e a s t reflectivity spectrum i n v e r t i n g equation r e c u r s i o n technique m o d e l of o r d e r The by m o d e l of the a broadband reflectivity of R ( f ) o u t s i d e the frequency 44 band r a n g e o f t h e s o u r c e equation wavelet. s(t)=r(t)*w(t) S(f)=R(f)W(f). and Consider its the basic Fourier I f W(f) c o n t a i n s s i g n i f i c a n t band of f r e q u e n c i e s between f j and f 2 > then spectrum that R ( f ) = S ( f ) / w ( f ) c a n be a d e q u a t e l y frequency represents band. This band order, be the recovered choice frequencies of order is For the r e f l e c t i v i t y shown l a t e r how t h e c h o i c e of m a j o r s u b s u r f a c e Attempting first the r e f l e c t i v i t y recovered as boundaries an AR process only i n of some p o r t i o n o f R ( f ) may be u s e d t o fj allow a sensible extension and f^. necessary Importantly, c o n s t r u c t i o n problem, of p i s r e l a t e d the w r i t t e n i n terms of a sampling i t will to the or r e f l e c t i o n a s an AR p r o c e s s , a f o r good p r e d i c t i o n number coefficients. t o show t h a t t h e r e f l e c t i v i t y may be r e p r e s e n t e d is the series R(f). t h a t AR model w h i c h w i l l results. in a s m a l l p o r t i o n of t h e complex data of R ( f ) o u t s i d e proper energy frequencies then estimate transform recovered T h u s , i f R ( f ) c a n be m o d e l l e d of model spectrum R(f)' reflectivity r(t) function n-o where At samples, r interval, i s the sampling n i n t e r v a l , N i s t h e number o f d a t a i s the r e f l e c t i o n coefficient and ^ ( t - n a t ) i s t h e D i r a c sampling at each sample function. The 45 d i s c r e t e F o u r i e r t r a n s f o r m o f 4.5 i s 16 Rj=| r c^ ^;M. .... -. , r , where the a n d where n J W s u b s c r i p t j r e f e r s t o the j t h frequency values of j l a r g e r frequencies (N/2 being than N/2 the Nyquist refer fy=j/N4t to index). negative The i n v e r s e d i s c r e t e F o u r i e r transform i s d e f i n e d as Expression 4.6 may a l s o be w r i t t e n KM r u o rV-l These relations constraint From of will referred to as reflectivity equations. equation complex be 4.6, t h e r e f l e c t i v i t y sinusoids. s p e c t r u m i s a sum U l r y c h (1973), and U l r y c h and C l a y t o n ( 1 9 7 6 ) have shown t h a t h a r m o n i c f u n c t i o n s may be modelled sinusoid as AR processes. adequately The o r d e r o f a s i n g l e i s p=1, a n d i n g e n e r a l , the order of a complex complex 46 harmonic process of different number of i s e q u a l t o t h e number o f c o m p l e x s i n u s o i d s period complex included i n the data. sinusoids of From 4 . 6 , t h e different (T^=N&t/n) i n c l u d e d i n t h e r e f l e c t i v i t y spectrum t h e number o f n o n z e r o r e f l e c t i o n c o e f f i c i e n t s order 'p' of the discrete reflectivity 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 computational t h e number o f s p e c t r a l f2« This puts reflection a limit coefficients R a r e known, t h e n reflection frequencies spectrum to insure must on the number may exist of For be less significant within I f 20 complex frequency contributing a and t h e are directly related. t h e r e s h o u l d be l e s s coefficients Thus, the v a l u e s r e c o v e r e d b e t w e e n f j and which window b e i n g c o n s i d e r e d . i s equal to r^. c o n s i d e r a t i o n s , t h e o r d e r of R than period proper than energy the data values of 20 s i g n i f i c a n t to those determination known o f t h e AR model. The AR r e p r e s e n t a t i o n o f t h e r e f l e c t i v i t y be spectrum may written P where a l l terms extend v a l u e s of R f^ and f ^ . are complex. T h i s e x p r e s s i o n i s used t o t o frequencies o u t s i d e the known range P a r t o f t h e s u c c e s s o f t h i s AR e x t e n s i o n method d e p e n d s on a c c u r a t e l y determining the order 'p' of the 47 reflectivity of spectrum, significant constructed reflection result Optimum c r i t e r i a have and t h e optimum number coefficients expected Clayton f o r determining (1976). i f t h a t number e x c e e d s reflection coefficients averaging procedure stablizes the equation choice extend 4.8 of of the in to a significant solution chapter Having coefficients i s implemented 2/3 (an which c h o s e n a good are determined forward manner and to e n d o f R , and i n a r e v e r s e manner i n t h e m i s s i n g low f r e q u e n c i e s . The synthetic the a b i l i t y impedance e x a m p l e shown i n f i g u r e o f t h e AR t e c h n i q u e structure. An to ideal The 20 e s t i m a t e s of R (a), r e p r e s e n t an a c c u r a t e in panel corresponding functions. between a demonstrates low frequency reflectivity spectrum. 10 and 50 Hz shown i n p a n e l portion ( j ) . Panels bandlimited recover 12 s t a r t i n g p o i n t f o r t h e AR method i s an a c c u r a t e p o r t i o n o f shown in AR o r d e r ) . filter the high frequency fill number expected of s a m p l e s p r o v i d e s good i s d e s c r i b e d i n the next order, the p r e d i c t i o n then the series ( 1 9 7 5 ) , and U l r y c h an o r d e r frequency the values. the order of a data However, c h o o s i n g t h e number o f r e c o v e r e d results in g i v e n M known c o m p l e x f r e q u e n c y been d i s c u s s e d by U l r y c h a n d B i s h o p 3/4 to i . e . , determining of the true (b) and (c) reflectivity and spectrum show the impedance The b a n d l i m i t e d a n d t r u e i m p e d a n c e f u n c t i o n s a r e compared i n p a n e l ( c ) . 48 1 CD O ~ . AA A (L In \l \ \ v o o 10 Jo 1 1 1 v .. u 1 b) < r a ) > . CD o ~ rl o • • 1 1 1 w — 1 1 d) CD o 1 — CJ I o 1 1 1 1 1 1 i ; CD O ~ „ O CD i i i i fREQUENCY |RCe)| 1 . i , y - 3 6 TIME i i — 0 — • < TIME 25 IO r < ° / .36 I) 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 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 . reflectivity 49 Panel into (d) shows t h e r e s u l t o f the an order in the low frequency o f 15 ( w h i c h true illustrates i s g r e a t e r than the resulting spectrum improved resulting impedance f u n c t i o n ( p a n e l information in of (k)). constructed has frequency t h e number shown i n p a n e l function low not the r a n g e o n l y , u s i n g AR p r e d i c t i o n and reflectivity that extending appearance, spikes Panel (e) reflectivity however, the ( f ) ) c o n t a i n s the proper needed to give i t geologic meaning. Panels low (g)-(i) and h i g h m i s s i n g (panel has frequencies. of p r e d i c t i n g both the The r e f l e c t i v i t y (h)) i s p l o t t e d as a spike s e r i e s simply a complete spectrum w i t h i n the Nyquist resulting impedance s t r u c t u r e ( p a n e l predicted high This show t h e r e s u l t is frequency values a common o c c u r r e n c e of are somewhat among t h e v a r i o u s s m a l l u n c e r t a i n t i e s i n 'determining coefficients. large errors in prediction filter begins predict to predicted low reflectivity It appears frequencies because the proper p r e d i c t i o n spectral value as into the s l i d e s o f f t h e edge o f t h e known d a t a a n d spectral previously predicted values. missing The erroneous. T h e s e u n c e r t a i n t i e s may p r o p a g a t e the i t ( i ) ) i n d i c a t e s that the to p r e d i c t the very high filter because frequency. m o d e l s t h a t we have t e s t e d u s i n g t h e AR method. less profitable function frequencies values exclusively from F o r t u n a t e l y , t h e s m a l l band o f c a n , i n many c a s e s , be a c c u r a t e l y 50 recovered. Figure to 13 shows t h e r e s u l t s o f a p p l y i n g t h e AR a more c o m p l i c a t e d results of adding example. 10% w h i t e impedance model p r e s e n t e d spectral and estimates low missing high (shown i n p a n e l are (d)-(f). frequencies impedance The very 18 ( a ) ) , between encouraging 14 the a s has been The r e s u l t s o f a l s o p r e d i c t i n g t h e (panels (g)-(i)), show estimate has not a simpler f u n c t i o n but the been i m p r o v e d . An of 12 was c h o s e n f o r t h i s e x a m p l e . order More e x a m p l e s of t h i s method a r e p r e s e n t e d C) the synthetic example. p e r h a p s 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 resulting show The r e s u l t s o f p r e d i c t i n g o n l y frequencies shown i n p a n e l s (j)-(l) noise to the simple i n the previous R 50Hz, a r e a c c u r a t e . missing and of Panels method of this applied chapter to real and programming Broadband minimizing programming i n the f i n a l chapter section where i tis data. B) B r o a d b a n d r e f l e c t i v i t y a linear in c o n s t r u c t i o n using the ljnorm algorithm reflectivity the norm recovery leads formulation i n the time naturally and a linear 1 973; T a y l o r e t a l , 1 9 7 9 ) , b u t l j norm m i n i m i z a t i o n h a s a l s o been for reflectivity (Claerbout to d o m a i n by Muir, used (LP) l j and c o n s t r u c t i o n i n the frequency L e v y and F u l l a g a r (1981) have formulated a domain. procedure to 51 o 114 1 J I 1 1 b) o~l I I I I I <rC0> rco c 1 i 1 1 1 r Ct) c 9) 0 36 i 6 j) 1 f REQUENCt IRCO| F i g . 13. function r—i 125 ret) ; ?c(0 .36 0 Autoregressive r e c o n s t r u c t i o n of a c o n t a i n i n g many r e f l e c t o r s . reflectivity 52 construct a number of out reflection i n the the reflectivity information portion solution This is values not take full An outline be followed of has of by L e v y and an the s e i s m o g r a m and utilizing how bandlimited has in reliably LP the known it reflectivity their been that formulated, approach from an from v e l o c i t y does averages. approach information information a most of reflectivity spectrum the unique of carried i.e., F u l l a g a r ' s general explanation method, unique However, as of minimum advantage from o n l y R(f). advantage an a that processing reflectivity method of in averages), true more e f f i c i e n t the resulted constructed spectral made has the recovered. Their domain, assumes o n l y (reflectivity of containing coefficients. frequency seismogram function can will be appraisal or w e l l log analyses. (a) G e n e r a l An LP formulation outline reflectivity of this f u n c t i o n given method begins previously with the sampled as N-l The objective coefficients is r n then to find which s a t i s f y the that set of reflection c o n s t r a i n t equations 4.7, 53 and w h i c h have a minimum l llrcOU, - 2 f norm, t h a t i s , lr | is minimum n Minimization of t h e coefficients w i t h a minimum number of n o n z e r o As an 1| norm w i l l aside, u t i l i z i n g p r o d u c e a s e t of reflection values. 1 . 7 , equation expression 4 . 9 may be r e w r i t t e n vo Minimizing the equivalent to minimizing This 1 norm of ( minimization minimum of One obtaining spectral to the will structural way the division of 1| norm of produce an function is thus impedance g r a d i e n t s . impedance model w i t h of the the the LP solution reflectivity is by first spectrum R seismogram those spectral are used. to the j t h frequency of R o t h e r methods of z e r o they may be as components, where W Estimates obtained may phase from also before. But is sufficiently be obtained deconvolution. an by III Rj=Sj/Wj where j r e f e r s a variations. formulate estimates reflectivity appraisal only large, through For many example, deconvolution. 54 R e l i a b l y determined into the constraint selected to find satisfy i.e., then substituted 4 . 7 and an LP a l g o r i t h m i s coefficients and have a minimum l j norm. which 4 . 7 are s a t i s f i e d the c o n s t r u c t e d r e f l e c t i v i t y should c 4.7 are reflection i f b o t h 4 . 1 1 and procedure, values equations those equations Note t h a t r (t), spectral adequately reproduce in r^ the above f u n c t i o n , denoted the observed by seismogram s(t)2V (t)*w(t). c M o s t LP a l g o r i t h m s a r e d e s i g n e d t o l o c a t e o n l y unknowns, w h e r e a s coefficients to which both positive are expected. e x p r e s s each The and negative solution r ^ as t h e d i f f e r e n c e positive reflection to t h i s problem between two is positive quant i t i e s , i . e . , This 4.7, relation and unknowns an a^ is LP and t h i s procedure, However, if no then substituted algorithm b . n thereby extra will The search for the positive number o f unknowns i s d o u b l e d increasing the i n f o r m a t i o n c a n be t h e LP a l g o r i t h m , t h e f o r m u l a t i o n followed. i n t o e q u a t i o n s 4 . 9 and outlined computing by costs. incorporated into above must be 55 No that attempt unique information appraisal of analyses. This valuable the algorithm LP information, averages leads the when the and be by w e l l l o g or solution. impedance and to utilize supplied available, LP construction by an velocity can make a Assimilating estimates reliable into the s o l u t i o n of problem. LP s o l u t i o n constrained LP find that the c o n s t r a i n t equations, 1^ and t o a more e f f i c i e n t constrained In to formulation could seismogram reflectivity (b) The which contribution reflectivity the i s made i n t h e above s e t of r e f l e c t i o n approach, coefficients and which the o b j e c t i v e r^ i s to which satisfy have a minimum weighted norm, t h a t i s , N-l where t h e are a r e w e i g h t s meant t o e m p h a s i z e prominent weighting inverse Thus, the average function for this of because tend in the reflectivity t h e LP a l g o r i t h m to if reduce the an r^ average using values approach <r(t)>. is the l j norm <r(t)> which n A natural arithmetic averages, <r(t)>~'. This i s minimization, i f i t has a l a r g e values those . r relative indicate will weight. that a 56 particular 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 n e a r z e r o , interpretation a large is assisted i n t h e LP a l g o r i t h m weight r e l a t i v e t o other p o s s i b l e f o r m o f 4.13 that by g i v i n g r 's. n A it natural i s then where <r^> r e f e r s t o the d i g i t i z e d averages, weighting exponent 0 (no w e i g h t i n g ) t o 2 o r more (0=1 or 2 ranging is from usually sufficient to and q provide is a good results). The polarity polarity reflectivity information function a v e r a g e s may to the LP a l s o be u s e d t o algorithm. as I -l if i( *r > n 2 O provide Defining a 57 the c o n s t r a i n t equations Real [Rj J - 1r may 1 be r e w r i t t e n r nC<v) coiC2TTi n/w) 3 OM r\=0 where r ^ =sgn ( < r > ) r ^ n reliable search and j ranges f r e q u e n c i e s as b e f o r e . for t h es t r i c t l y over only Then t h e LP positive the most a l g o r i t h m may unknowns r ^ , e l i m i n a t i n g need f o r t h e d i f f e r e n c e e q u a t i o n 4.12. and reflection coefficients magnitude of t h e c o n s t r u c t e d be r e c o v e r e d or A constraint may A constraint equations 4 . 1 6 i f impedance e s t i m a t e s well polarity by a p p l y i n g r = sgn ( < r > ) r ^ . Additional in The t r u e log analyses. equations may may be a d d e d t o t h o s e are a v a i l a b l e From equation from 1.5, velocity impedance be w r i t t e n - ^ 1 S9h«r >)r ' n-o n where k+1 i s the index equation 4.17 i s i n t h e 4,17 n of t h e impedance t h a t for input to 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 t h e basement impedance i s known ( i . e . , relation ), this proper form i s known a n d becomes essentially a 58 constraint on following t h e dc two e q u a t i o n s frequency provide R^. identical For example, the constraints: rV-t i\=o The constrained LP a l g o r i t h m i s summarized 14. REFLECTIVITY Weighting Information Polarity Objective Function Information AVERAGES LP Reliable Spectral Estimates R Constraint Algorithm Constructed Reflectivity Equations Impedance Estimates F i g . 1^-. F l o w d i a g r a m f o r t h e c o n s t r a i n e d L P algorithm. in figure 59 Before presenting noted that by using the construction automatically sand For formation the might of reflectivity problem, placed coefficients. examples on the certain thin in the LP averages method, to i t is constrain restrictions r e s o l u t i o n of example, a result the shale dipole wedged following are reflection in a porous reflectivity funct ion rCti Experimentally, recovered the following (from e q u a t i o n reflectivity a v e r a g e s would be 3.4) <fCO> Therefore, appraisal are this or dipole construction sufficiently seperated can not methods. t o be be resolved However, resolved by by either i f such dipoles the appraisal, 60 then these features, information, may complete be with recovered the low through frequency construction procedures. (c) Data e r r o r s and t h e LP s o l u t i o n In p r a c t i c a l certain amount processing. each solve This briefly spectral presented by equations value R^' equations of subsequent an the error for reflectivity i t i s undesireable exactly. Two have been Levy and F u l l a g a r to ways discussed of for (1981) and w i l l be constraints constraint specified affect a here. Most L P a l g o r i t h m s a will translates into i n e s t i m a t e s of R approach Inequality each which data noise constraint errors LP noise In t h e presence of n o i s e , the handling 1) of recovered spectrum. this c a s e s , t h e observed seismogram c o n t a i n s equation error may allow i s required bound. be inequality constraints, easily The t o be s a t i s f i e d reflectivity converted into 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) where within constraint inequality 61 *C>j a r e s p e c i f i e d where E.j a n d uncertainties at the j t h frequency. In this required inequality be most s i m p l y level Error this. t e r m s E^ a n d c h o s e n e m p i r i c a l l y by a s s u m i n g a Note t h a t in this Equality In t h e number o f data noise the at equality constraint equations is constraints process may a l s o d i r e c t l y of determining each f r e q u e n c y . constraint J ljnorm, those the LP reflection algorithm d e t e r m i n e an e s t i m a t e o f t h e e r r o r s E-j and This J may be done by r e t a i n i n g t h e relations, u n c e r t a i n t i e s as a d d i t i o n a l but introducing the unknowns r\=o I m q c j L R ^ - £ j - I r^sinUTfjn/AO Also, c-j c a n procedure. c o e f f i c i e n t s w h i c h have a minimum £^ satisfying a n d t h e n r e q u i r i n g t h e 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 doubled 2) t h e LP a l g o r i t h m i s t o m i n i m i z e t h e 1| norm o f t h e r ^ , w h i l e inequality relations 4.20. the of formulation, an a d d i t i o n a l e q u a t i o n w h i c h c o n s t r a i n s r 4,21 a statistical 62 norm of t h e u n c e r t a i n t i e s i s necessary of t h e LP noise recovered constraint L e v y and for F u l l a g a r (1981). r^ which a l s o determine error minimization. positive 4.12 terms at each Of course, frequency the e r r o r s covered by search i n so which doing favour and can the form be of N o t e t h a t t h e number of 4 at each frequency, equations levels d e t a i l s of a is o r n e g a t i v e , so t h e y must be c a s t i n constraint but t h e number needed i n t h i s a p p r o a c h i s nearly t h e number u s e d i n t h e i n e q u a l i t y f o r m u l a t i o n . For the inequality following examples approach was of u s i n g a z e r o phase wavelet estimated from figure 15 p r o v i d e d by amplitude The spectrum first (a). phase panel (b) The was data (an a v e r a g i n g example 17. wavelet The of The computed to was produce data and wavelet The was shown estimate wavelet an function) stacked seimic data a r e shown i n f i g u r e reflectivity (c). the construction, Synthetic I m p e r i a l O i l of C a n a d a ) . shown i n f i g u r e panel t h e c o r r e l a t e d and (both LP adopted. generated in terms the Then t h e LP a l g o r i t h m w i l l f o r i n p u t t o t h e LP a l g o r i t h m . half The h a v e a minimum l j norm and unknowns i s i n c r e a s e d by of solution. on t h e l j n o r m of t h e e r r o r those the in to c o n t r o l and were its 16. 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 reflectivity from then the the function shown in impedance m o d e l i n p a n e l convolved with the zero synthetic seismic trace in 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 panel (d) 63 W 79E39 62921 F i g . 15. 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 ; r e p r e s e n t s subsurface structure i n parts o f western A l b e r t a . IWCOI \f 256sec w(t) 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 f r o m t h e d a t a i n s i d e t h e box i n f i g . 15, a n d i t s amplitude 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 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 . reflectivitv 65 indicates the little zero phase function. (e) improvement over t h e o r i g i n a l wavelet already recovered by supplying used to weight the polarity constraints impedance f u n c t i o n , shown i n p a n e l the included true were also impedance. in this ideal the LP panel with The a v e r a g e s objective function The ( f ) , compares and resulting very well impedance c o n s t r a i n t s were example. 18 shows t h e r e s u l t s o f a p p l y i n g t h e Figure in algorithm provided. No because averaging function 12 and 35 Hz. s p e c t r a l v a l u e s between <r(t)>'were with an The c o n s t r u c t e d r e f l e c t i v i t y was accurate is data LP method u s i n g a n o i s y v e r s i o n of t h e impedance l o g from t h e p r e v i o u s example. The complexity of t h e r e s u l t i n g (panels s i n c e t h e method reflectivity Accurate LP (panel is shown in panel reflectivity and (a). deconvolution a d i f f i c u l t p r o b l e m f o r t h e LP i s best suited to find v a l u e s of R between simple along with 10 and 35 Hz, s u p p l i e d t o polarity constraints ( f r o m <r(t)>""') l e a d t o t h e r e c o v e r e d i n panel (e). ( f ) ) shows The that resulting the construction By including two midway a n d one on t h e basement impedance of and reflectivity impedance s u c c e s s f u l at r e c o v e r i n g the major t r e n d s impedance. The functions. algorithm weighting function log ( b ) and ( d ) ) p r e s e n t algorithm the noisy was the function reasonably original constraints, one i m p e d a n c e , a good r e c o v e r y o f 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 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 . reflectivity 67 the original panel impedance (h). s t r u c t u r e was Further obtained e x a m p l e s of t h i s as shown method a r e in presented next. C) S t a b i l i t y and Before real efficiency considerations a p p l y i n g t h e AR seismic performance data, on it and is synthetic LP construction worthwhile examples methods investigating under the to their following complications. 1) the presence (both 2) i n a c c u r a t e knowledge of 3) 4) variance of f * 2 (both methods) reflection effects of (LP method Figure The 19 point sampling For constructions spectral shows the the source known f r e q u e n c y of AR wavelet different estimates. be range to the (AR number method weighting f j to of only) functions only) the s y n t h e t i c data of the which w i l l above be the considerations. i s 4msec. first will order coefficients f o r examples interval the data methods) sensitivity 5) i n the methods) (both starting of a d d i t i v e n o i s e and performed Since the second complications u s i n g two wavelet is above., slightly different known, estimates 68 Fig. 19. Synthetic o f 10-50Hz. of R test data. W a v e l e t has b e t w e e n f r e q u e n c i e s f j and f j may a bandlimited spectral division; R may be S(f)V(f). from taken More the trace be appraisal because frequencies Recalling may an have equation of of R 3.4, of the seismogram; may winnowed the through Or e s t i m a t e s insignificant been range be o b t a i n e d S(f)/W(f). r e l i a b l e estimates appraisal contaminated results. from a band be obtained or noise from these appraised seismic written In t h e f r e q u e n c y domain t h i s becomes DC0=RC0A(0 = I f A ( f ) i s z e r o p h a s e and 5(0V«) approximately unity within the 69 bandrange may to then the d i s c r e t e be e q u a t e d t o R.J b e t w e e n construction For and for t h e f i r s t e x a m p l e , v a r i o u s amounts synthetic trace 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 at © =1. The line. shown Fourteen between s(t) panel carried in (a) of out 11. (b). The 14 a n d 40Hz were u s e d t o The estimates AR r i g h t were obtained from recovered predict .S(f)/W(f) using Straight spectral division sharper reflectivity on spectral the appraisal by t h e known w a v e l e t has c o n s t r u c t i o n s , however, the right. low order spectral S(f)V(f). provided resulting o b t a i n e d by u s i n g estimates. between 14 from S(f)V(f) Clearly, (c). Spectral a n d 40Hz were d e r i v e d f r o m S ( f ) / W ( f ) t o o b t a i n t h e r e s u l t s on t h e l e f t ; derived are values missing estimates 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 values each and t h e r e s u l t s on t h e impedance m o d e l s a r e n o t b e t t e r t h a n t h o s e appraisal figure constructions r e s u l t s on t h e l e f t were o b t a i n e d u s i n g obtained noise figure f r e q u e n c i e s and t h e h i g h f r e q u e n c i e s t o 125Hz u s i n g an of the amount o f n o i s e added t o t h e d a t a i s i n d i c a t e d on t h e c e n t e r in in o f random The n o i s y s e i s m o g r a m s a r e shown i n p a n e l trace use D^', algorithms. were a d d e d t o t h e o r i g i n a l 19. fj spectral estimates using b e t t e r LP c o n s t r u c t i o n s . to the and spectral values were o b t a i n t h e c o n s t r u c t i o n s on t h e appraisal estimates provides 7 0 >5 •p -p o co rH CH (1) J-i CH o • CO CD CO "O o o -p C CD E CD > H! -P •H S -£o * $ cd cc CC CH < O OJ 0) o -p c «H O CD O CD to CD U C CD rt .C e -p j-i o c < CH - H CD PH C O • H -P • O O CM 3 t-i -P • CO -Ol WC •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 t h e w a v e l e t . in (a) panel shows generate the o r i g i n a l version of wavelet. this An inaccurate trace the are is the spectral division somewhat better S(f)V(f) with the true panel (b). data wavelet). Again, 'q' reflectivity frequencies estimates S(f)/W(f) to 125Hz using shown on t h e l e f t the r e s u l t s using a p p r a i s a l range were was used tested. constructions, and Also (c). different for the are estimates S(f)V(f), p r o v i d e s much b e t t e r L P c o n s t r u c t i o n s . third known f r e q u e n c y Again, example The weighting bottom using tests the the appraisal 2 using accurate of R ( f ) i n t h e ranges i n d i c a t e d . output e f f e c t s of v a r y i n g the r a n g e f| t o f 2 « The AR a n d L P r e s u l t s f i g u r e 22 were o b t a i n e d same a c o n s t r a i n t was p l a c e d on t h e impedance. in recovered The r e s u l t s basement The AR shown on t h e r i g h t . frequency exponents 14 The 14 and 40Hz were u s e d t o p r e d i c t t h e 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 known phase shown i n t r a c e 6 (compare t h i s o r d e r s as i n d i c a t e d . than zero the shown different used t o u s e d a s an i n a c c u r a t e original low f r e q u e n c i e s a n d t h e h i g h 5 a 1 using using s p e c t r a l v a l u e s between using which of is 3; an a p p r a i s a l missing and t r a c e 5 shows wavelet wavelet constructions t r u e mixed phase wavelet data appraisal Trace spectral shown estimates The L P c o n s t r u c t i o n s shown 1 2 V — wet) — 3^ ^ ^Y u 6 - J W - v — ^ ^ W ret) s C ^ t) <rct;d2i)> wet) - - A ^ v A ^ v V - <rCt;© = 0 ? 1 Q) AR —*—13 LP —»Y* ~vy~ A 1 orcier 0 .4 c) F i g . 2 1 . P e r f o r m a n c e o f t h e c o n s t r u c t i o n methods u s i n g innaccurate wavelet. an 73 on the r i g h t r e m a i n s t a b l e , b u t t h e AR r e s u l t s shown on left fail close t o , and hence, t h e o r d e r of s p i k e s The when t h e number o f known f r e q u e n c y (9) i n t h e o r i g i n a l data. Twenty s p e c t r a l obtained left are from with using accurate values between a spectral division using accurate shown on t h e r i g h t . spectral vs order, The n o r m a l i z e d values models (i.e., c o n s t r u c t i o n may be o b t a i n e d of s i g n i f i c a n t a constructions between t h e t r u e and c o n s t r u c t e d reconstructions. by d a t a and (NSE), a long p r e d i c t i o n f i l t e r spanned AR were squared e r r o r shown a t t h e b o t t o m o f f i g u r e 2 3 . Because a g r e a t e r 50Hz inaccurate spectral are number and v a l u e s a r e shown on t h e d e f i n e d by the 10 The and stable number and i n a c c u r a t e o f : 1) a c c u r a t e 10% added random n o i s e . and t h e c o n s t r u c t i o n s u s i n g plotted the f o u r t h e x a m p l e shown i n f i g u r e 23 d e m o n s t r a t e s t h e o f v a r y i n g t h e AR o r d e r 2) d a t a becomes l e s s t h a n becomes reflectivity r(t). effect obtained values the range of longer In a The NSE i n d i c a t e t h a t high other order) words, i f the order spikes expected reliable filter, is a leads to good AR greater than i n the s o l u t i o n . frequency values are a more r e l i a b l e p r e d i c t i o n o f 74 —>^X-^j-/- roo 15 — K ^ J i ^ A ^ ^ \0-SOH - 12—ibv^vX^^/u |0-«fO 7—V^-^N^^^-VSA- 1 0 - 3 0 - 15—AAVl—A^yu^ IS-^O - 10—Aj\ wwvw-~p^ /r-HcO - 5—-A^VwV^ly-N^ |S"-30 - r I J T T -»-1v- 1 LP WW o Fig. 22. S t a b i l i t y of reconstructions f r e q u e n c y band u s e d . with respect to the 75 ret) H 7 1 —h/t^A—A— 10 I I M —U^A-^Y^ fNyVwA r^^yAvA~ 11 IM I S" ./L-J^-JU 17 IS order Ax f/OlSE" FREE" \0% NOISE order F i g . 23. S t a b i l i t y o f t h e AR method w i t h r e s p e c t t o o r d e r or t h e 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 t h e 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 long data values ( c l o s e to the will also results. This possible order spectral data. very may high, results. result. l e n g t h of the be example t h a t , out The last different suggests signal choosing boundaries are a NSE of for significant example weighting weighting exponents. but ones' good using The 24 r e q u i r e d w i t h no w e i g h t i n g the udesireable largest the r a t i o of t h e d a t a is should levels l e a d t o good constructions true to the ^ ( t ) ,only s e c t i o n compares the the o b j e c t i v e use function and 40Hz c o n s t r u c t i o n s are and 3 a l l in 1/3 (q=0). of the of in shows LP c o n s t r u c t i o n s u s i n g 14 6 order. 14 different very objective function weighting solution on the impedance with respect e x p o n e n t s on between a the in this spectral values order i s too in 9 boundaries i n the Figure provide choosing to noise large t h e LP a l g o r i t h m . the perhaps g i v i n g c o n s i s t e n t w i t h assumed n o i s e I f the filter known d a t a ) , n o i s e predicted, A l s o , the indicate However, i f t h e good (q=l,2) computing time 7? CPU TIME JU- /I i 0 r<CO o I.Z Sec WO 1 .H 2 .H 1 .4 F i g . 24. The a c c u r a c y a n d e f f i c i e n c y o f t h e L P 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 solution accural 78 5 Multitrace Reflectivity Construction using Real Seismic Data A) The Real Data D a t a s e t A, shown i n f i g u r e 25 a l o n g interpretation, consists the seismic deconvolved data), Sonic at where the logs are traces in part of that The raw data energy with method energy log energy the seismic D a t a s e t B, line, shown but in the box. they figure are 26, data (38 v e l o c i t y log i s available for Thus, in if desired, each subsurface a then i t i n the log Energy constrained LP approach constraints, obtained from The standard nearby is or w e l l l o g or v e l o c i t y total energy necessary impedance A velocity then the i s matched t o the to energy available well logs. i f a v a i l a b l e ) and if and necessary trace with from a scaling represent reflection reflectivity. deconvolved trace synthetic. set is deconvolved scaling utilizes density data absolute seismogram i s c o n s t r u c t e d (and (vibrator i s i n d i c a t e d by t r a c e a m p l i t u d e s f o r each consistent synthetic 15 from t i m e window n e a r a t r a c e l o c a t i o n . impedance m a g n i t u d e s a r e of figure taken .5sec window o f d e c o n v o l v e d e x p l o s i o n voltages. the in location removed. geological adjacent traces section a l l ) . ^ A c l o s e by measured match 24 a v a i l a b l e near the least 1 mile c o n s i s t s of a of with a in in the spectral analyses, F i g . 25. D a t a s e c t i o n A; d e c o n v o l v e d with a geological i n t e r p r e t a t i o n . vibrator Fig. explosion data. 26. Data section B; deconvolved data along 80 are included Since i n the algorithm. each s e i s m i c undergone an reflectivity appraisal each data original may be set are source averaging in The A deconvolution, and it plus B has represents noise error s p e c i f i c a p p r a i s a l techniques unknown, b u t w a v e l e t has it is assumed been l o c a l i z e d function, a(t)=w(t)*v(t) . used that the into a zero phase Each deconvolved trace written cl(O rCO*ac.o + ^rc^> r In the sections averages, <r(t)>=r(t)*a(t) , £ <r(t)>=n(t)*v(t). on trace frequency domain t h i s is 5". I Normally, A(f) i s a smooth f u n c t i o n and from its 16). Therefore, spectral i d e a l value i t may generally of u n i t y between f be n e c e s s a r y } to and f ^ (eg., f i g . first remove s m o o t h i n g o f R ( f ) p r o d u c e d by A ( f ) b e f o r e spectral estimates to deviates supplying a construction algorithm. i s a v a i l a b l e for data sensitivity of t h e c o n s t r u c t i o n m e t h o d s w i t h r e s p e c t using the of t h e a v e r a g i n g following two A function slightly (fig. Since average A ( f ) smoothing e f f e c t set will different any 16), be to an the the tested estimates of 81 R(f ) : 1) R(f) ^ D ( f ) / A ( f ) 2) R(f) ~ D(f) for Figure using f j <. f >. 27 shows c o n s t r u c t i o n r e s u l t s t h e AR a l g o r i t h m . averages obtained The o r i g i n a l results estimates using frequencies D(f)/A(f); estimates D(f). an o f 24. obtained 28. from major sections, and panel (a). (c) In both cases, A Panel shows 33 to differences the predict exist results the recovered f r e q u e n c i e s t o 62 Hz however, by u s i n g e s t i m a t e s in the using the two i n panel ( b ) , D ( f ) / A ( f ) a r e somewhat noisier. 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 A a r e shown i n f i g u r e The r e s u l t s estimates shown i n p a n e l shown No set and t h e impedance between 13 a n d 46 Hz were u s e d low f r e q u e n c i e s and h i g h constructed data data and i m p e d a n c e c o n s t r u c t i o n s u s i n g missing order for f r o m them a r e shown i n p a n e l (b) shows t h e r e f l e c t i v i t y spectral ( s m o o t h i n g removed) in using spectral D(f)/A(f) ( b ) ; and t h e r e s u l t s u s i n g e s t i m a t e s panel (c). In both cases, 12 t o 35 Hz were u s e d and two are D(f) are reliable frequencies impedance constraints were i n c l u d e d ; one midway and one on t h e b a s e m e n t i m p e d a n c e . Clearly, using the constructed s e c t i o n i n panel spectral estimates, D ( f ) , provides ( c ) , obtained by t h e most c o n s i s t e n t results. The a b o v e AR a n d L P c o n s t r u c t i o n r e s u l t s a r e generally P i g . 27. AR r e f l e c t i v i t y a n d i m p e d a n c e r e c o n s t r u c t i o n s f r o m s e c t i o n 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 . A; 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 f r o m s e c t i o n 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 . A; 84 trace by trace reliability estimates, and comparable, e s t a b l i s h i n g confidence of both methods. less noisy AR will constructions. be Autoregressive f i g u r e 29. from t h e a p p r a i s e d The o r i g i n a l f r o m them a r e between 12 13 Thus, c o n s t r u c t i o n s f o r data obtained of spectral further c a r r i e d out u s i n g smoothed estimates, D ( f ) , obtained amplitude smoothed 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 constructions in Using i n the data shown spectral data. set B are shown and t h e i m p e d a n c e a v e r a g e s in panel (a). a n d 40 Hz were deemed r e l i a b l e - b y s p e c t r a of s e v e r a l deconvolved Frequencies examining the traces. An order was u s e d f o r t h e AR c o n s t r u c t i o n shown i n p a n e l ( b ) , where t h e m i s s i n g l o w f r e q u e n c i e s a n d 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 e a c h 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 30. Frequencies one impedance between s e t B a r e shown i n f i g u r e 12 a n d 40 Hz were u s e d constraint (midway) to along with 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 set B provide information. noisy similar However, b o t h character character is obtained from and of also data the original set localized A. in data. the Although s e c t i o n s , the data reflectivity constructed sections retain observed c o u l d be made f r o m t h e s e more from the A somewhat n o i s y constructed sections good i n t e r p r e t a t i o n s consistency of the F i g . 29. AR r e f l e c t i v i t y a n d i m p e d a n c e r e c o n s t r u c t i o n s f r o m 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 r e c o n s t r u c t i o n s smoothed s p e c t r a l e s t i m a t e s . from s e c t i o n B u s i n g CO ON 87 r e s u l t s must be i m p r o v e d b e f o r e t h e AR a n d L P a l g o r i t h m s c a n be data. routinely applied to seismic B) The K a r h u n e n - L o e v e The recovered inconsistent structural noise b r o a d b a n d s e c t i o n s a b o v e were changes i n the data. but more l i k e l y A linear transformation (K-L) t r a n s f o r m a t i o n similarities (eg. i n a s e t of adjacent the purpose of imroving Consider SJ ( t ) lateral of due t o t h e p r e s e n c e o f known as the Ready a n d W i n t z , Kramer a n d Mathews, 1956) c a n be u s e d t o major laterally i n some p l a c e s , n o t b e c a u s e o f t h e p r e s e n c e Karhunen-Loeve 1973; Transformation extract the seismic traces f o r consistency. N t r a c e s of s e i s m i c data, where each trace i s composed o f 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 signal n;(t); i.e., 5,2 If the n j ( t ) represent find in noise signals, then i ti s useful to an x ( t ) w h i c h s u m m a r i z e s t h e most c o r r e l a t e d c o m p o n e n t s the N traces. In general, w r i t t e n as a l i n e a r combination t h e common s i g n a l may be o f t h e s,'(t) a s f o l l o w s 88 The sj(t), space and i = 1 , . . . N , s p a n a t most N if the t r a c e s are dimensions the common orthonormal basis trace functions Hilbert h i g h l y c o r r e l a t e d , they 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 rewriting in as space. a linear This will suggests c o m b i n a t i o n of M Y,-(t) M 0-i where t h e d i r e c t i o n s of the eigenvectors The % * ( t ) may be of t h e these basis f u n c t i o n s are defined inner product or c o v a r i a n c e defined as follows (see appendix by matrix A) r.6 where s ^ c^co, \\is the s cOj ....s cty) t j t h eigenvalue N arranged i n descending order i.e., 89 X\ > >^-X|s/ in 3 a n < b-j i s t h e j t h e i g e n v e c t o r 5.4 may be f o u n d by m i n i m i z i n g squared differences of t h e sum o f P. The integrated between x ( t ) and t h e s j ( t ) , t h a t i s by minimizing 4 i-.i The r e s u l t r- 1 d e r i v e d i n appendix N I — - ;fr N where Jjfj A is 1 i s the eigenvector Substituting matrix defined i n appendix A. 5.6 a n d 5.8 i n t o 5.4 g i v e s M 1 where I-1 Each of term i n e x p a n s i o n 5.9 i s c a l l e d a t h e common t r a c e . F o r example, principal 01 b, * £ i s the f i r s t p r i n c i p a l component a n d s o o n . I f the N traces correlated, will then \\ most o f t h e i n f o r m a t i o n and Y v about • t h e component are be r e l a t i v e l y common trace highly l a r g e and will be 90 contained i n the f i r s t principal component. 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 contains In fact principal ifa l l component a l l o f t h e i n f o r m a t i o n a b o u t t h e common t r a c e a n d the o t h e r components a r e t r i v i a l . of the 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 , 5.9 A eigenvector components optimum t r a c e M=N, principal the provide if first At best, i.e., i f a l l principal gives t h e mean t r a c e components a r e then ( a s shown i n a p p e n d i x 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 d e t e r m i n e how many principal c o m p o n e n t s s h o u l d be i n c l u d e d i n t h e common t r a c e . However, i f o n l y t h e major traces are desired, similarities then only c o m p o n e n t s need be u s e d t o p r o v i d e In t h i s t h e s i s , L summed, transformation transformation The K-L expression and i scalled transformation manner t o d e c o n v o l v e d impedance help seismic constructed the trace be formed 5 principal by this component o r K-L t r a c e . a p p l i e d i n an o v e r l a p p i n g data, to r e f l e c t i v i t y and and t o t h e c h o i c e reliability two or 5.9 i s r e f e r r e d t o a s t h e K- common can or 3 a common t r a c e a principal constructions, improve For the one between and o f AR o r d e r , t o interpretability of r e f l e c t i v i t y and impedance s e c t i o n s . example, t h e l e f t hand s i d e o f f i g u r e 31 shows t h e 91 results of applying reflectivity orders. AR prediction function, The seventeen frequencies known f r e q u e n c i e s different only first shown on the consistent of AR Figure Traces are over and are the slightly using g i v e s t h e K-L traces These traces are indicating very that the 31 a l s o shows t h e p r i n c i p l e of t h e m i x i n g algorithm. are used to o b t a i n the used to o b t a i n the 32 n e x t K-L shows t h e the deconvolved data component. t o p K-L trace, traces t r a c e and results so of a p p l y i n g the s e t B u s i n g N=3 and The principal resulting character original data has been 2-6 on. only preserved algorithm the first component shown b e l o w , i n d i c a t e s t h a t s i g n i f i c a n t the information and the noisy i s absent. Application construction of t h e K-L (from figure original constructed shown on the obtained 46Hz t r a c e s and a w i d e r a n g e of o r d e r s order 12 and approach. principal in N=5 several u s i n g t h e K-L 1-5 section side. between results component hand using i s less c r i t i c a l Figure to The bandlimited frequencies Considering principal right low 125Hz. f o r each o r d e r . the choice to the (12-46HZ), <r(t)> were u s e d t o p r e d i c t t h e m i s s i n g high to left using component, are 28) the traces shown on the LP reflectivity i s shown i n f i g u r e 3 3 . reflectivity and N=3 a l g o r i t h m t o an and impedance s e c t i o n s principal and only right. The component the first The are sections, principal interpretability of 92 —IULJLJL^^L (t) — r 1 A ^ v J — A ^ ^ ^ L v 16—j 2 — ^-A-^Avv*rA^--^V^' 15 > ^ J ^ - y - vy 10 9 9 ^A^~Ar^-^Y^^ — K j ^ j ^ f \ — y ^ J ^ 10 7 trace order [f0; 0 K-L 8 - L-K^A—^—Y^7^ 11 8 ^ ^w] .4 F i g . 31* A p p l i c a t i o n o f t h e 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 h e c h o i c e o f AR o r d e r . A l s o showing t h e p r i n c i p l e o f the mixing algorithm. 93 K-L F i g . 32. A p p l i c a t i o n o f t h e K-L t r a n s f o r m a t i o n and a l g o r i t h m t o t h e n o i s y s e c t i o n B. mixing 94 both sections algorithm, has noisy preserving been m a r k e d l y i m p r o v e d . Using t r a c e s c a n be d i s c r i m i n a t e d t h e most c o r r e l a t e d i n f o r m a t i o n this K-L against while recovered by t h e construct ions. The final sections the for original shown on data the l e f t , constructions LP The positive logs) easier impedance shown t h e K-L t r a n s f o r m a t i o n previously i n the mixing provide impedance (which using nearly a N=3 impedance s e c t i o n s . t o the raw t r a c e s and t h e algorithm. at the The AR PreCretaceous dominant f e a t u r e R e e f shows up a s an i m p e d a n c e l o w . t o make s t r u c t u r a l results, identical interpretations. contrast is with shown on t h e r i g h t , i n nearby w e l l 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 t h e Redwater o i l Overall, i ti s i n t e r p r e t a t i o n s from the C o m p a r i s o n s between t h e bearing recovered original data t h e 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 a n d 36. The sections these final constructed f o r data results algorithm the and B o t h t h e AR and t h e L P r e s u l t s , by a p p l y i n g sections unconformity and interpretation. p r i n c i p a l component and reflectivity s e t A a r e shown i n f i g u r e 34 a l o n g geologic were o b t a i n e d first constructed first set B are were reflectivity shown obtained in by p r i n c i p a l component. h a s been i n s e r t e d i n t o the figure applying t o AR a n d L P c o n s t r u c t i o n s and using impedance 37. Again, t h e K-L mixing N=3 traces and The a v a i l a b l e v e l o c i t y l o g impedance reconstructions for 95 Fig. 33A p p l i c a t i o n o f t h e K-L a l g o r i t h m t o r e f l e c t i v i t y and i m p e d a n c e r e c o n s t r u c t i o n s . LP If A PARK CoLOKArDO • W A &NM SUICROP BrAveR HILL LAKE" ELK POINT AR LP LEA P A R K COLORADO ZZZZZZ WABAMU* SUBCROP 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 along with the o r i g i n a l geologic interpretation. from s e c t i o n A ON F i g ' 3 5 ' Comparison o f 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 deconvolved section (A). 98 F i g . 36. C o m p a r i s o n o f 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 a v e r a g e s o b t a i n e d from t h e d e c o n v o l v e d data. 99 comparison. adjacent LP The well log confirms the first .3sec o f r e c o n s t r u c t i o n s a n d c o n s i s t e n c y b e t w e e n t h e AR results models. provides Again, i t is interpretations section. additional evident would be confidence that aided C o m p a r i s o n s between the subsurface by the in using final structural the original and impedance data 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 38 and 39. Conclusion The p r o b l e m of e s t i m a t i n g broadband a c o u s t i c from r e f l e c t i o n linear inverse formalism. deconvolution unique should construction unknown presented frequency in precede (which technique (which information use of attempts Two thesis, methods impedance constraints from a to recover construction missing spectral in the presence the particular the methods have information. 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 both a recovers t h e AR and t h e L P m e t h o d s , at recovering stabilize using shown t h a t a c o n v e n t i o n a l the bands). this investigated reflectivity algorithm shown s u c c e s s LP I t was / inverse f i l t e r i n g bandlimited seismogram) The s e i s m o g r a m s h a s been impedance of examples t o noise provided additional s t a b i l i t y and t o the solution. The AR (computationally) method way represents to obtain an reconnaissance inexpensive broadband AR 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 The g i v e n v e l o c i t y l o g has b e e n i n s e r t e d f o r c o m p a r i s o n . LP from s e c t i o n B, o o 101 P i g . 38. w i t h the Comparison o f f i n a l r e f l e c t i v i t y o r i g i n a l deconvolved s e c t i o n (B). reconstructions 102 F i g . 39. C o m p a r i s o n o f 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 a v e r a g e s • o b t a i n e d from t h e d e c o n v o l v e d data. 103 impedance i n f o r m a t i o n . on no synthetic A l t h o u g h t h e method w o r k s v e r y and r e a l data examples, a t t h i s time there i s way t o c o n s t r a i n t h e o u t p u t u s i n g from other analyses. available well stability Other of logs AR Absolute should be the predicted frequencies The to dc efficiency both only by of the l i n e a r construction i n t e r p r e t a t i o n of performing the programming the given for both include polarity may, many an i m p e d a n c e c o n s t r u c t i o n . room for performed. data was n o t known. sets until approach improvement; of t h e d e s i g n e r and t h e algorithm. obtained greatly cases, For The be from aid the problem of solved by example, until s e t A were p e r f o r m e d a n d t h e impedance c o n t r a s t was i d e n t i f i e d , data of Robust i n t h e presence structure. f o r data ways versatile could in the considerations. impedance s e c t i o n s methods improve or s o l v i n g f o r the missing much subsurface constructions positive 0 the c r e a t i v i t y i s concluded that reflection might t h e most allows from 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 R provides available the presence of noise. broadband impedance. t h i s method restricted in to u s i n g maximum e n t r o p y LP a l g o r i t h m of n o i s e , It frequency directly recovering utilized method obtaining the p r e d i c t i o n f i l t e r information impedance i n f o r m a t i o n , constructions improvements t o t h i s well the p o l a r i t y of I n f a c t , p o l a r i t y was unknown preliminary constructions were 104 REFERENCES C l a e r b o u t , J . 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Ulrych, T. J., 1971, A p p l i c a t i o n of Homomorphic Deconvolution 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 E n 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. R e s . , V.77 P. 13961 400. Ulrych, T. J . , and T. N. B i s h o p , 1975, Maximum E n t r o p y Spectral Analysis 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. a n d S p a c e P h y s i c s , V . 1 3 , P. 183-200. Ulrych, T. J., a n d R. W. C l a y t o n , 1976, Time S e r i e s Modelling and Maximum E n t r o p y : Phys. Earth Planet. I n t e r . , V . 1 2 , P. 188-200. 106 A p p e n d i x A: D e r i v a t i o n o f t h e Common Consider N traces of seismic data, Trace where s;(t) i s composed o f a common s i g n a l x ( t ) p l u s signal nj(t); If each trace a difference that i s , t h e nj ( t ) r e p r e s e n t noise signals, then i t i s u s e f u l to f i n d an x ( t ) w h i c h s u m m a r i z e s t h e most s i m i l a r c o m p o n e n t s o f the N traces. as a l i n e a r In general, combination of t h e 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 trace t h e common t r a c e may be w r i t t e n which forward procedure minimizes the sum i s to find of that integrated common squared d i f f e r e n c e s between x ( t ) a n d t h e S j ( t ) , i . e . , N is minimized minimization > z with leads respect to x ( t ) Obviously, this t o t h e mean t r a c e 4 107 where the P| discriminate because is in against are a l l l/N. an e r r o n e o u s i t i s simply averaged desireable emphasize to very inconsistent common 2 find traces. trace trace in i n with equal that similar There set traces of c a n be done as a l i n e a r combination this procedure importance. weights and This i s no way t o fii rejects by which noisy formulating of o r t h o r n o r m a l It or the basis functions. Orthogonal The s- ( t ) span ( If the s'-(t) preferred the common a t most N d i m e n s i o n s are highly o r i e n t a t i o n i n that trace as a l i n e a r which a r e orthonormal the Recomposition projection correlated, space. between Hilbert they This combination and c o n t a i n distances in space. will suggests of b a s i s one member which have a rewiting functions minimizes i t and t h e N t r a c e s . The o r t h o g o n a l r e c o m p o s i t i o n o f t h e common t r a c e by forming the inner product or c o v a r i a n c e matrix begins 108 ii s s y m m e t r i c , i t may P Since r = BA8 De w r i t t e n as T where A0 is a diagonal descending matrix of the e i g e n v a l u e s of P order arranged in anc Kb,.. - b N B i s a matrix composed of t h e e i g e n v e c t o r s eigenvectors directions P of (analogous obtained from following property a BS -B B-I T define T to stress a set of p r i n c i p a l the principal matrix). and I =B. A set of orthonormal . The orthogonal stress directions The matrix 3 has the -^K J~ i s the identity matrix, bi'; P • lo^A.hki J where \jj o f Xj^is basis the Kronecker functions ^f^(t) delta may be 109 obtained each by p r o j e c t i n g the vector eigenvector and of normalizing seismic by the traces onto corresponding e i g e n v a l u e as f o l l o w s where and .b a 7 IT.-CO ^CO i t = i The o r i g i n a l t r a c e s may be r e c o n s t r u c t e d from the basis follows u k=i T h i s may be checked by s u b s t i t u t i n g 6 i n t o 8 as M IV U,. - Si CO L follows as 110 The common combination trace Ms of M o r t h o n o r m a l then basis written as functions - M The c ^ j may be f o u n d Q-Z.) Minimizing by m i n i m i z i n g ( x C t ) - S , ( 0 ) <it this, with IV r e s p e c t to ^>C £gives < V / b /- W IV 1 i^b 0 i K <f co^} K at a linear Thus N Substituting in 10 and 6 i n t o 9 gives Mr- N I f M=N, then 11 r e d u c e s t o t h e mean I f M<N, then 11 c a n be tit) where - £. Y^j written bj •S trace
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The recovery of subsurface reflectivity and impedance structure from reflection seismograms Scheuer, Tim Ellis 1981-12-31
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Title | The recovery of subsurface reflectivity and impedance structure from reflection seismograms |
Creator |
Scheuer, Tim Ellis |
Date | 1981 |
Date Issued | 2010-03-26T02:47:14Z |
Description | This thesis is concerned with the problem of estimating broadband acoustic impedance from normal incidence reflection seismograms. This topic is covered by following the linear inverse formalisms described by Parker (1977) and Oldenburg (1980). The measured seismogram is modelled as a convolution of subsurface reflectivity with a source wavelet. Then an appraisal of the seismogram is performed to obtain unique bandlimited reflectivity information. This bandlimited reflecitivity information is then utilized in two different construction algorithms which provide a broadband estimate of reflectivity; from which a broadband impedance function may be computed. The first construction method is a maximum entropy method which uses an autoregressive representation of a small portion of the reflectivity spectrum to predict spectral values outside that small portion. The second and most versatile construction method is the linear programming approach of Levy and Fullagar (1981) which utilizes the unique bandlimited spectral information obtained from an appraisal and provides a broadband reflectivity function which has a minimum 1( norm. Both methods have been tested on synthetic and real seismic data and have shown good success at recovering interpretable broadband impedance models. Errors in the data and the uniqueness of constructed reflectivity models play important roles in estimating the impedance function and in assessing its uniqueness. The Karhunen-Loeve transformation is discussed and applied on real data to stabilize the construction results in the presence of noise. The generally accepted idea that low frequency impedance information must be supplied from well log or velocity analyses because of the bandlimited nature of seismic data has been challenged. When accurate, bandlimited reflectivity information can be recovered from the seismic trace, then an interpretable, broadband impedance model may be recovered using the two construction algorithms presented in this thesis. |
Subject |
Seismic reflection method Acoustic impedance |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2010-03-25 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0052871 |
URI | http://hdl.handle.net/2429/22549 |
Degree |
Master of Science - MSc |
Program |
Geophysics |
Affiliation |
Science, Faculty of Earth, Ocean and Atmospheric Sciences, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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