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

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THE RECOVERY OF SUBSURFACE 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 . F., and M u i r , F., 1973, w i t h E r r a t i c D a t a : G e o p h y s i c s , V.38,  Robust Modelling P. 826-844.  Deconvolut ion, Geophysics R e p r i n t S e r i e s 1, V o l u m e s I II, ed. by G. M. Webster, 1978, Society E x p l o r a t i o n G e o p h y s i c i s t s , T u l s a , Oklahoma.  and of  Galbraith, J. M., and G. F. Millington, 1979, Low Frequency Recovery i n the I n v e r s i o n of Seismograms: J o u r n a l of The CSEG, V.15, No.1, P. 30-39. K r a m e r , F. S., R. A. Peterson, and W. C. Walter, 1 9 6 8 , S e i s m i c E n e r g y S o u r c e s 1968 Handbook: P r e s e n t e d at the 38th Annual ISEG M e e t i n g , D e n v e r , C o l o r a d o , 1968 and P r i v a t e l y P u b l i s h e d by U n i t e d G e o p h y s i c a l C o r p . K r a m e r , H. P., and M. V. Mathews, 1956, A L i n e a r Coding for Transmitting a S e t of C o r r e l a t e d S i g n a l s : IRE Trans. Inform. T h e o r y , V . I T - 2 , P. 41-46. Lavergne, M., and C. Willm, 1977, S e i s m o g r a m s and P s e u d o v e l o c i t y L o g s : • V.25, P. 231-250. Levy,  Inversion of Geophys. Prosp.,  S., and P. K. F u l l a g a r , 1 9 8 1 , R e c o n s t r u c t i o n of a Sparse S p i k e T r a i n from a P o r t i o n of i t s Spectrum and Application to High Resolution Deconvolution: G e o p h y s i c s , V.46, N.9, P. 1235-1243.  L i n d s e t h , R. 0., 1979, S y n t h e t i c S o n i c Logs - A Process for Stratigraphic Interpretation: Geophysics, V.44, P.3-26. L i n e s , L. R., and R. W. C l a y t o n , 1977, t o V i b r o s e i s D e c o n v o l u t i o n : Geophys. 417-433.  A New Approach P r o s p . , V.25, P.  Lines, L. R., and T. J . U l r y c h , 1977, The O l d and t h e New i n S e i s m i c D e c o n v o l u t i o n and Wavelet E s t i m a t i o n : Geophys. P r o s p . , V.25, P. 512-540. Mayne, W. H., S i g n a t u r e s of 1162-1173.  and Large  R. G. Airguns:  Quay, 1971, Seismic Geophysics, V.36, P.  Oldenburg, D. W., 1981, A C o m p r e h e n s i v e S o l u t i o n To The L i n e a r D e c o n v o l u t i o n Problem: Geophys. J . R. Astr. Soc. V.65, P. 331-357. Oldenburg, D. W., S. Levy, and K. P. Whittall, 1981, Wavelet Estimation and Deconvolution: Geophysics, ( t o be p u b l i s h e d i n V.46, N.11, November, 1 9 8 1 ) .  105 Otis,  R. M., a n d R. B. Smith, 1977, Deconvolution by L o g S p e c t r a l Averaging: V . 4 2 , N.6, P.1146-1157.  Homomorphic Geophysics,  Parker, R. L.,1977, Understanding Inverse Rev. Earth Plant. S c i . , V . 5 , P. 35-64.  Theory:  P e a c o c k , K. L., 1979, D i s c r e t e O p e r a t o r s G e o p h y s i c s , V.44, N.4, P. 722-729.  Integration:  Peterson, R. A., W. R. Fillipone, 1955, The S y n t h e s i s o f S e i s m o g r a m s G e o p h y s i c s , V.20, P. 516-538.  for  Ann.  and F. B. Coker, from W e l l Log D a t a :  Ready, R. J., and P. A. Wintz, 1973, I n f o r m a t i o n Extraction, SNR Improvement, and D a t a C o m p r e s s i o n i n Multispectral Imagery: IEEE Transactions On C o m m u n i c a t i o n s , COM. 2 1 ( 1 0 ) , P. 1123-1130. Ricker, N. , 1940, The Form and N a t u r e o f S e i s m i c Waves and t h e S t r u c t u r e o f S e i s m o g r a m s : G e o p h y s i c s , V.5, P. 348-366. Ricker, N., 1953, The Form a n d Laws o f P r o p o g a t i o n o f S e i s m i c W a v e l e t s : G e o p h y s i c s , V.18, P. 10-40. R o b i n s o n , E. A., 1967, P r e d i c t i v e D e c o m p o s i t i o n of Time Series with Application to Seismic Exploration: G e o p h y s i c s , V.32, N.3, P. 418-484. T a y l o r , H. L., S. Deconvolution 39-52.  C. B a n k s , and J . F. McCoy, 1979, w i t h t h e L i Norm: G e o p h y s i c s , V.44, P.  T r e i t e l , S., and E. A. R o b i n s o n , 1966, The D e s i g n o f H i g h Resolution Filters: IEEE Transactions, Geoscience E l e c t r o n i c s , V.4, P. 25-38. 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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