Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Applications of the Karhunen-Loève transform in reflection seismology Jones, Ian Frederick 1985-12-31

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
UBC_1985_A1 J66.pdf [ 10.83MB ]
[if-you-see-this-DO-NOT-CLICK]
Metadata
JSON: 1.0052986.json
JSON-LD: 1.0052986+ld.json
RDF/XML (Pretty): 1.0052986.xml
RDF/JSON: 1.0052986+rdf.json
Turtle: 1.0052986+rdf-turtle.txt
N-Triples: 1.0052986+rdf-ntriples.txt
Original Record: 1.0052986 +original-record.json
Full Text
1.0052986.txt
Citation
1.0052986.ris

Full Text

APPLICATIONS  OF  IN  THE  KARHUNEN-LOEVE  REFLECTION  TRANSFORM  SEISMOLOGY  by  IAN  FREDERICK  JONES  M.Sc. The U n i v e r s i t y of Western O n t a r i o , B.Sc.(Hon) The U n i v e r s i t y of Manchester,  A  THESIS  IN PARTIAL  SUBMITTED  THE  FOR  REQUIREMENTS DOCTOR  THE  1980 1976  FULFILMENT DEGREE  OF  OF  PHILOSOPHY  OF  in THE  FACULTY  (Department  We  accept to  THE  Of  OF  GRADUATE  Geophysics  t h i s  t h e s i s  t h e required  UNIVERSITY  OF  A p r i l  ©  Ian F r e d e r i c k  STUDIES  And  as  Astronomy)  conforming  standard  BRITISH  "COLUMBIA  1985  Jones,  1985  (  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of  requirements f o r an advanced degree a t the  the  University  o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make it  f r e e l y a v a i l a b l e f o r reference  and  study.  I further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying, o f t h i s t h e s i s f o r s c h o l a r l y purposes may  be granted by the head o f  department o r by h i s o r her r e p r e s e n t a t i v e s .  my  It is  understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain  s h a l l not be  allowed without my  permission.  Department o f Q ^ C ^ W ^ ' S l o A The  U n i v e r s i t y of B r i t i s h Columbia  1956  Main M a l l  Vancouver, Canada V6T 1Y3 Date  DE-6  (3/81)  A  AcV^  written  ABSTRACT  The  Karhunen-Loeve  information  transform,  from  problems  in  derived  by  principal  multichannel  reflection a  components  and  eigenvalues.  Data  the  components.  The which are  are  mathematical  render  algorithms  and  are  'production-1ine' programs  A  have  new  successful  residual the  to  An  as  in  new  to  orthogonal  seismic  algorithms data  versions  is  set  of  corresponding  linear  real  several  transform  combination  Karhunen-Loeve  viable  seismic  data.  by  anomalous  misfit  velocities to  anomaly  basis  sections,  data  of  transform processing  developed.  Most  examples,  and  of  the  some  technique, has  features shows  of  of in  some  less stacked  promise.  to  with  And,  the  success.  coherent of  a  identification  i i  new  information multiple  scheme,  based  to  on be  coherent seismic  Diffraction  used  some  based  proved  r e c o n s t r u c t i o n are  segregate  the  enhancement  Reconstruction  reconstruction)  isolated  successfully technique.  of  a  coherent  developed.  emphasize  transform  the  s y n t h e t i c and  stacked  migration  as  problems  number  real  ("misfit"  hyperbolae  of  of  on  information data  with  signal-to-noise ratio  reconstruction  The  eigenvectors,  industrially  been  applied  an  p r o p e r t i e s of  on  been  processing.  reconstructed  a  tested  data  has  extracts  c o n s t r u c t i o n of  i t a p p l i c a b l e to  reviewed,  optimally  data,  seismic  least-squares  principal  which  estimate ability is  used  suppression on  cluster  a n a l y s i s  of  the  e i g e n v e c t o r s  s y n t h e t i c  data  r e a l  d a t a .  A  the  e i g e n v a l u e s ,  and  o f f e r s  s t u d i e s .  a  Use  new  of  r e l a t i v e of  the  e f f e c t  r e p r e s e n t s  an  works  the  to  l e a d i n g phase  w e l l f o r  f o r  data  r e s u l t s  at  e v a l u a t i o n  s y n t h e t i c  when  data  e a r l y  of  a  development,  on  of  t h i s  V i b r o s e i s ©  i n v e s t i g a t i o n ,  i s  data  r a t i o  of  a  t r a v e l  times,  i n v e r s i o n  constant  d i s p e r s i o n  i n an  phase  promising terms  of  a n a l y s i s  a c q u i s i t i o n ,  p r e s e n t e d .  the to  p r o v i d e s  As  on  a p p l i e d  v e l o c i t y  of  p a r t  w e l l  u t i l i z i n g  r e s o l u t i o n i n  works  q u a n t i f i c a t i o n  d i s p e r s i o n  o r i g i n a l  method,  good  h i g h  f o r  t r a n s f o r m ,  promising  a n a l y s i s  d i s p e r s i o n to  the  g i v e s  e i g e n v a l u e s  s h i f t s . of  and  v e l o c i t y  p o t e n t i a l  approximation r e s u l t s ,  used,  of  which  Table of Contents  ABSTRACT  ..  TABLE  CONTENTS  LIST  OF OF  .  ..  .  ..  i  i  . i v  FIGURES  i x  ACKNOWLEDGEMENTS  x i i i  DEDICATION  x i v  CHAPTER 1. 1  INTRODUCTION SECTION  1.1:  GENERAL  SECTION  1.11: REVIEW  INTRODUCTION OF  PREVIOUS  1 WORK.  a.  Orthogonal  Expansion  o f  a  b.  Orthogonal  Expansion  of  M u l t i c h a n n e l Data  c.  G e o p h y s i c a l  A p p l i c a t i o n s  8  d.  The  t h e U.B.C.  9  SECTION  Work  of  1. I I I :  SEISMIC  a.  R e c o n s t r u c t i o n  b.  M i s f i t  o f  DATA  F u n c t i o n  4  Group RECONSTRUCTION  Coherent  I n f o r m a t i o n  R e c o n s t r u c t i o n  4 5  11 11 13  c . D i f f r a c t i o n s  13  d.  14  M u l t i p l e  SECTION  S u p p r e s s i o n  1.IV: SIMILARITY  a.  C l u s t e r  b.  V e l o c i t y  MEASURES  A n a l y s i s  15 16  A n a l y s i s  17  c . <2-Inversion  17  iv  CHAPTER  2. .  THEORY AND BACKGROUND SECTION  2 . 1 : THE  KARHUNEN-LOEVE  TRANSFORMATION  .20 20  a.  I n t r o d u c t i o n  20  b.  Theory  20  c.  S i n g u l a r  d.  O u t e r - P r o d u c t  SECTION  V a l u e  Image  I d e n t i c a l  b.  S i m i l a r  c.  Winnowing  28  Summation  2 . I I : RECONSTRUCTION  a.  SECTION  Decomposition  AND  30  ASSOCIATED  S i g n a l s  ERRORS  31  ...  31  S i g n a l s  2 . I l l :  and  ..31  Truncation  APPLICATION  TO  E r r o r  33  SEISMIC  DATA  34  a.  I n t r o d u c t i o n  34  b.  F l a t  35  c.  D i p p i n g  d.  Less  e.  The  SECTION  L y i n g  Events  E v e n t s  Coherent  35 Information  S i g n i f i c a n c e 2 . I V : THE  a.  I n t r o d u c t i o n  b.  The  c.  I d e n t i c a l  d.  Phase  e.  The  f.  Complex  Complex  of  COMPLEX  36  t h e E i g e n v e c t o r KARHUNEN-LOEVE  Elements TRANSFORMATION.  37 ..39 39  Trace  40  S i g n a l s  41  Recovery  Complex  KL  42 T r a n s f o r m a t i o n  E i g e n v e c t o r s  and  Time  S h i f t s  43 46  v  CHAPTER DATA  3. RECONSTRUCTION.  SECTION  51  3.1: RECONSTRUCTION  a.  I n t r o d u c t i o n .  b.  KL  c.  S y n t h e t i c  d.  R e c o n s t r u c t i o n  of  e.  S y n t h e t i c  Data  Examples  f.  Real  Data  Examples  g.  Data  Compression  h.  D i s c u s s i o n  SECTION  Stack  Data  t h e Mean  S y n t h e t i c  Data  c.  Real  Examples  d.  D i s c u s s i o n  Stacked  ..54 55 57  Examples  79  79 80 81  SEPARATION  I n t r o d u c t i o n  b.  S y n t h e t i c  Data  c.  M i g r a t i o n  of  d.  R e a l  Examples  e.  D i s c u s s i o n  OF  DIFFRACTIONS  89 89  Examples  D i f f r a c t i o n  89 S e c t i o n s  91 92 93  MULTIPLE  a.  I n t r o d u c t i o n  b.  S y n t h e t i c  Data  c.  R e a l  Examples  d.  D i s c u s s i o n  Data  Data  79  a.  3.I V :  .51 53  RECONSTRUCTION  b.  SECTION  Stack  62  I n t r o d u c t i o n  Data  ......51  62  3 . I I : MISFIT  3 . I II:  INFORMATION.  Examples  a.  SECTION  COHERENT  ...51  Versus  Data  OF  SUPPRESSION  105 105  Examples  106 109 110  v i  CHAPTER  4.  SIMILARITY SECTION  119  MEASURES. 4.1:  TRACE  CLUSTER  a.  I n t r o d u c t i o n .  b.  S y n t h e t i c  Data  c.  Real  Examples  d.  D i s c u s s i o n  SECTION  Data  Examples.  I n t r o d u c t i o n  b.  The  c.  S y n t h e t i c  Data  d.  Real  Examples  e.  D i s c u s s i o n  E f f e c t s  4 . I II:  b.  An  I t e r a t i v e  The  d.  D i s p e r s i o n  e.  D i s c u s s i o n  138  S t a t i c  S h i f t s  138  Examples  ....139 142  INVERSION  I n t r o d u c t i o n  c.  ANALYSIS.  146 Q  Q  125  138 of  a.  E f f e c t i v e  122  127  a.  SECTION  119 119  4.I I : VELOCITY  Data  ANALYSIS  169 169  D i s p e r s i o n  Removal  Scheme  U s i n g  Value  Constant  Phase  an 170  A p p r o x i m a t i o n  Q u a n t i f i c a t i o n O b j e c t i v e  173 F u n c t i o n s  180 185  v i i  CHAPTER 5 . 200  CONCLUSIONS SECTION  5.1: REVIEW  SECTION  5 .I I :  THE  GOALS  OF  THIS  WORK.  200  REMARKS  202  p r o c e s s i n g  202  a.  Image  b.  S i m i l a r i t y  SECTION  OF  .  5 .I l l : RECOMMENDATIONS.  204 206  a.  S t a c k i n g  206  b.  R e c o n s t r u c t i o n  206  c.  S i m i l a r i t y  207  d.  C o n c l u s i o n  C r i t e r i a  209  211  REFERENCES APPENDIX 1. ATTENUATION a.  Theory  b.  Numerical  c.  Cumulative  APPENDIX  21 9  RELATED DISPERSION  219 Methods t o  I n t e r v a l  223 Q  I n v e r s i o n  225  2.  THE EFFECT OF DISPERSION ON VIBROSEIS DATA PROCESSING  22 6  a.  I n t r o d u c t i o n  226  b.  Theory  227  c.  The  d.  S y n t h e t i c  e.  Comparison  f.  C o n c l u s i o n s  Vibroseis® Data w i t h  Technique  229  Examples  230  P r e v i o u s  Work  233 235  v i i i  List  Percentage  of Figures  2.1:  Cumulative  2.2:  E f f e c t . o f  2.3:  Rotated  3.1:  Time  3.2:  S y n t h e t i c  F a u l t e d  3.3:  S y n t h e t i c  Noisy  3.4:  Procedure  f o r S l a n t  3.5a:  Real  3.5b:  95%  R e c o n s t r u c t i o n  69  3.5c:  85%  R e c o n s t r u c t i o n  70  3.6a:  Real  3.6b:  95%  R e c o n s t r u c t i o n  72  3.6c:  85%  R e c o n s t r u c t i o n  73  3.7:  Stacked  3.8:  95%  3.9a:  Dipping  3.9b:  Slant-KL  95%  R e c o n s t r u c t i o n  77  3.9c:  Slant-KL  85%  R e c o n s t r u c t i o n  78  3.10:  Dipping  3.11:  S y n t h e t i c  3.12a:  Real  F a u l t e d  3.12b:  Grey  Shade  Time  Time  and  S h i f t s  S h i f t e d  Phase  Stacked  Stacked  Data  Data  CKL  and  Phase  ...  KL  with  w i t h of  S t a c k i n g and  64 R e c o n s t r u c t i o n  R e c o n s t r u c t i o n  66 67  "Lens"  68  B i f u r c a t i o n  Caldera  and  C a l d e r a  Block  Data  71  F a u l t s  74 75 ..76  M i s f i t  R e c o n s t r u c t i o n  M i s f i t  R e c o n s t r u c t i o n  Data  M i s f i t  65  R e c o n s t r u c t i o n  Data  "Lens"  ..49 50  Noise,  and  w i t h  Data  48  Wavelets  Data,  R e c o n s t r u c t i o n  Noise  on  S h i f t s  S e c t i o n  Real  Energy  83 84 85  Overlay  86  ix  3.13a:  Real  Braided  3.13b:  Grey  Shade  3.14a:  Stream  Data  87  M i s f i t  Overlay  .88  S y n t h e t i c  A r r i v a l s  from  3.14c:  F l a t t e n e d  Data  3.15:  D i f f r a c t i o n  3.16:  S y n t h e t i c  3.17:  D i f f r a c t i o n  3.18:  Migrated  3.19a:  Real  3.19b:  92%  3.19c:  M i s f i t  3.l9d:  Migrated  M i s f i t  R e c o n s t r u c t i o n  103  3.l9e:  M i g r a t e d  M i s f i t  R e c o n s t r u c t i o n  104  3.20:  S y n t h e t i c  3.21:  Time  3.22:  Unstreched  3.23:  Moveout  3.24:  Real  3.25:  VA  3.26:  Stacks  4.1:  The  4.2:  C l u s t e r  Groups  f o r  F a u l t e d  4.3:'  C l u s t e r  Groups  f o r  Phase  4.4:  C l u s t e r  Groups  f o r  D i p p i n g  a  Common  Source  95  R e c o n s t r u c t i o n  A r r i v a l s  for  96  Zero  O f f s e t  97  R e c o n s t r u c t i o n  M i s f i t  F a u l t e d  98  R e c o n s t r u c t i o n s  of  D i f f r a c t i o n  Data  Streched  w i t h Data  M i s f i t  Data  and  Before  Data  of  to  I s o l a t e  M u l t i p l e and  D i f f r a c t i o n s  Events  M i s f i t  M u l t i p l e w i t h  R e c o n s t r u c t i o n  A f t e r and  Hagen  M u l t i p l e  A f t e r  113 114  Suppressed  M u l t i p l e  102  112  R e c o n s t r u c t i o n  C o r r e c t e d  Marine  ..99  101  R e c o n s t r u c t i o n  Data  Events  100  R e c o n s t r u c t i o n  Before  94  Data  115  Events  116  Suppression  117  M u l t i p l e  (1982)  Suppression  118  129 Layers  D r i f t s Layers  x  130 131 132  4.5:  C l u s t e r  Groups  f o r a  4.6:  C l u s t e r  Groups  f o r D i s c o n t i n u o u s  4.7:  C l u s t e r  Groups  f o r F a u l t e d  4.8:  C l u s t e r  Groups  f o r a  4.9:  C l u s t e r  Groups  f o r Real  4.10:  S i n g l e  Layer  4.11:  E f f e c t  of  Time  S h i f t s  on  RKL  V e l o c i t y  A n a l y s i s  148  4.12:  E f f e c t  of  Time  S h i f t s  on  CKL  V e l o c i t y  A n a l y s i s  149  4.13:  E f f e c t  of  Noise  L e v e l  on  V e l o c i t y  4.14:  S y n t h e t i c  4.15:  V e l o c i t y  4.16:  S y n t h e t i c  Seismograms  4.17a:  Semblance  V e l o c i t y  4.17b:  RKL  V e l o c i t y  A n a l y s i s  R e s u l t s  155  4.17c:  CKL  V e l o c i t y  A n a l y s i s  R e s u l t s  156  4.18:  Time  4.19a:  Semblance  4.19b:  RKL  V e l o c i t y  A n a l y s i s  R e s u l t s  159  4.19c:  CKL  V e l o c i t y  A n a l y s i s  R e s u l t s  160  4.20:  Real  4.21:  Sample  4.22:  RKL  V e l o c i t y  A n a l y s i s  R e s u l t s  163  4.23:  CKL  V e l o c i t y  A n a l y s i s  R e s u l t s  164  4.24:  Real  4.25:  Sample  R e a l  S y n t h e t i c  A n a l y s i s  Windows  Marine  and  f o r Ten  A n a l y s i s  133  Data  .......134  data  135  C h a r a c t e r Stream  Change  Data  147  A n a l y s i s  151  Layer  Model  Layers  154  f o r Ten  V e l o c i t y  Layers  A n a l y s i s  V e l o c i t y  A n a l y s i s  157 158  A n a l y s i s  xi  152 ....153  R e s u l t s  Semblance  f o r V e l o c i t y  150  Layers  Seismograms  Semblance  136 137  R e s u l t s  A n a l y s i s  and  Real  f o r Three  f o r V e l o c i t y  Data  Windows  Data  f o r Three  S y n t h e t i c  Data  Real  B r a i d e d  R e s u l t s  V e l o c i t y  ..  Seismograms  Seismograms  S h i f t e d  Land  "Reef"  161 ....162  A n a l y s i s  165 166  4.26:  RKL  V e l o c i t y  A n a l y s i s  R e s u l t s  167  4.27:  CKL  V e l o c i t y  A n a l y s i s  R e s u l t s  ..168  4.28:  D i s p e r s e d  4.29:  A t t e n u a t e d ,  4.30:  e Versus  Centre  Frequency  4.31:  e Versus  Centre  Frequency  4.32:  I n t e r c e p t  4.33:  e  4.34:  I n t e r c e p t  4.35:  KL  4.36:  Model  4.37:  O b j e c t i v e  4.38:  Example  4.39:  O b j e c t i v e  F u n c t i o n s  f o r  Four  L a y e r s  4.40:  O b j e c t i v e  F u n c t i o n s  f o r  Four  L a y e r s  A.1:  Versus  and  Rotated  D i s p e r s e d  Versus Centre  and  and  Centre  187  Rotated  Wavelets  189 (Attenuated)  190  Frequency  191 192  Centre  193  Frequency  F u n c t i o n s  194  S y n t h e t i c  Data  F u n c t i o n s  f o r  195 Hyperbolae  196  Wavelets  Correlograms  f o r  f o r  188  Frequency  Versus  O b j e c t i v e  Wavelets  197  Three  V a r i o u s  Vibroseis®  A.2:  S p e c t r a  A.3:  Correlograms  f o r  D i s p e r s e d  A.4:  Correlograms  f o r  A t t e n u a t e d  A.5:  Correlogram  A.6:  Uni-Octave  A f t e r Versus  Sweep  High  198 ( A t t e n u a t e d )  Methods  Methods  239  D i s p e r s e d  Frequency  x i i  237 238  data  M u l t i - O c t a v e  199  Data  Recovery  S t a c k s  240 241 242  ACKNOWLEDGEMENTS. I Clowes at  express  a n d R.M.  U.B.C,  Y e d l i n ,  and  My  U l r y c h , i n p u t  t o D r s . P.  committee  Brad  w h i l s t  past  thanks  four  s u p e r v i s o r s D r s . R.M. support  D.W.  H.  d u r i n g  Oldenburg,  and  a d v i c e  LeBlond  member  P r a g e r ,  who's  o f  and  my  time  and  M.J.  d u r i n g  a n d K.  t h i s  Larner  e x t e r n a l  f o r  examiner,  J u l i a n  was  Petro-Canada  Theory  and  t o keep  supported  computing  L t d . ,  and  a l s o  f o r t h i s  my  l e a d i n g  t o  C a b r e r a , h e l p i n g head  who  input  i n i t i a l l y  guided  me  i t s present  Clowes,  Mat  and me  above  f a c i l i t i e s ,  M u i r ,  of Chevron p r o j e c t .  M o b i l  d a t a ,  t o D r . J . den  by  ( g r a n t s  (IT&A)  f o r t h e c o m p l e t i o n  data  c o n t i n u e d  i n p a r t  C o u n c i l  A p p l i c a t i o n s  J .  i n  Levy,  past  form.  Y e d l i n ,  David  through t h e  Don  Waldron, p e r i o d s  waves  of  d u r i n g  y e a r s .  Research  Mr.  Shlomo  Ron  i n s t r u m e n t a l  A1804),  thanks  and  t o  i t s e x e c u t i o n ,  s t r i v i n g  work  E n g i n e e r i n g  necessary  goes  t o p i c ,  a l l been  T h i s  My  t o my  c o n t i n u e d  T . J .  f r i e n d s h i p s  P l e n d e r l e i t h ,  the  a l s o as  d u r i n g  The  the  D r s .  thanks  t h i s  h u r d l e s  t u r m o i l  f o r t h e i r  c o n s i d e r a b l e  s p e c i a l  suggested  have  thanks  i v e l y .  My  many  t o  p a r t i c i p a t i o n  respect  deepest  E l l i s ,  f o r t h e i r  p r o j e c t . t h e i r  my  Boer,  Canada  A7707, O i l  I n c . , a n d  o f much  t h e N a t u r a l  o f  S c i e n c e s  A2617,  Canada  generously  A4270,  L t d . made  programming t h e work  o f M o b i l Resources,  O i l  Canada,  and  Inverse  a v a i l a b l e e x p e r t i s e  p r e s e n t e d  f o r k i n d l y  and  here.  and  t o  s u p p l y i n g  DEDICATION  GEMMA:  A  causa  Y  por  con Yo  de  e s t e  tiempo  ensuenos d i g o  Lo  mal no  t r a b a j o , pasado, r e a l i z a d o s ,  ahora:  demas  es  para t i .  x i v  1  CHAPTER  1.  INTRODUCTION.  SECTION  1.1:  The  INTRODUCTION.  Karhunen-Loeve  u t i l i z e d  i n  however, have  GENERAL  i n  been  the  image  geophysical  s c a n t .  exemplify  the  In  t h i s  problems  m u l t i c h a n n e l  i n  behind  t h i s  seismic  data  p r o j e c t e d  data'  i n t o  t r a c e s .  f e a t u r e  to  the  a  p a r t s  improve b a s i s  of  of  f i r s t  e x i s t i n g c o - a u t h o r .  where  time,  l i t e r a t u r e ,  the  the  d a t a .  I  or  w h i l e i n  others recent  to  The  data  b a s i c  tenet  t r a n s f o r m , may  be  i n f o r m a t i o n  number  of  the  c o r r e l a t e d  t h i s  t h e s i s  have  been on  from  s e p a r a b i l i t y  data  t h i s  i s  ' a l t e r n a t i v e  t e c h n i q u e s ,  a r t i c l e s  and  s e v e r a l  coherency  e x p l o i t  date  seismic  Karhunen-Loeve  seismic i n  1975);  a d d r e s s i n g  p r o c e s s i n g  t o p i c s  Rao,  w e l l  i n t r o d u c e  the  for  separates  s e i s m i c  the  to  to  coherent  m u l t i c h a n n e l  of  i s  p r o c e s s i n g .  p o s s i b l e  input  and  and  a p p l i c a t i o n s  t r a c e - t o - t r a c e  compression  new  here  data  known  (Ahmed  method  using  l i n e a r  e x i s t i n g  Several  the  that  the  f i e l d  o b j e c t i v e  v i a b l e  s m a l l e s t  This  been  t r a n s f o r m a t i o n  a  space  the  my  seismic  i s  t e c h n i q u e s . for  t h e s i s ,  as  possessing i n t o  u n c o r r e l a t e d  form  work  long  p r o c e s s i n g ,  Karhunen-Loeve  community  has  p r o c e s s i n g data  p r o c e s s i n g  compressed  t r a n s f o r m  and  to  p r o c e s s i n g  are  p r e s e n t e d  mentioned which  I  i n am  2  My S.  Levy  been in  a t t e n t i o n and  was  T.J.  U l r y c h ,  i n v e s t i g a t i n g the  data  framework  p r o c e s s i n g .  the of  ( r e f e r r e d  t r a n s f o r m ,  or  seismic  the  i n  the  the  However,  evidence  i n  s e c t i o n  of  e f f e c t s data  As  i n d u s t r i a l  of  the  on  of  the  i n i t i a l  4,  given  and on  the  of  i t s  t h i s  a b i l i t y  to  that  i n  the  the  c h o i c e  of  of  my  i n  m a j o r i t y  s e i s m i c  to  produce  each  of  the  a  of  data  i n t e r e s t  p r o c e s s i n g  v i a b l e  i n  t h i s  community,  i n v e s t i g a t e d .  i n  c o r r e c t i o n  of more  problems. i s  to  be  Q - i n v e r s i o n  f o r  the  Vibroseis®  d a t a ,  proved  p o w e r f u l  f o r  of  Karhunen-Loeve  as  a p p l i c a t i o n s myself  on  i s  w i t h i n  the  my  goals  has  package  f o r  of  behalf  proved  the of  my  to  However,  they  by  d i s p e r s i o n  c o n c e r n i n g  one  although  presented  was  work  a p p l i c a t i o n s ,  Some  I  ' p r o d u c t i o n - l i n e ' p r o c e s s i n g  a p p l i c a t i o n s  not  as  the  bandwidth  KL  2 ) .  been  have  the  q u a n t i f y  other  developments of  as  to  t h e s i s  problem  s e i s m i c  s h i f t e d  v a r i o u s  t h r u s t  to  arose  i n t e r e s t  to  have  t r a n s f o r m  work  subsequent my  myself  by  Karhunen-Loeve  i n t e r e s t  the  KLT  and  t o p i c  approach  q u a n t i f i c a t i o n  Subsequently, of  i n  remainder  and  t h i s  Karhunen-Loeve  i n v e r s i o n  T h i s  the  d a t a ,  d i s p e r s i o n  the  the  to  Oldenburg  i n t e r e s t  waveforms.  coverage  (Appendix  of  focussed  problem  Chapter  of  my  a p p l i c a t i o n  the  D.W.  general  i n  KLT)  e f f e c t s .  towards  found  to  body-wave  d i s p e r s i v e  more  d i r e c t e d  w i t h  u s e f u l n e s s  a  between  i n t e r e s t e d  who  I n i t i a l l y ,  t r a n s f o r m  s i m i l a r i t y  o r i g i n a l l y  be  r e a l  some  promising data  co-authors  on  of  the  s y n t h e t i c  examples.  at  t r a n s f o r m the  1983  were Annual  3  I n t e r n a t i o n a l (SEG)  in  Las  m u l t i c h a n n e l the of  1984  Meeting Vegas,  Annual  Levy.  A  elsewhere  by  at  the  To  I  and  C a l g a r y ,  in  the and  body  of  of  of  the  the  given  S o c i e t y  Canada,  reader  to  in  the  by  by  V a i l ,  t h e s i s  my  of  c o - a u t h o r , here  of  E x p l o r a t i o n  the  G e o l o g i s t s SEG  Seismic  C o l o r a d o ,  of  the  i s  d i v i d e d  work  using  and  Oldenburg,  P e t r o l e u m  background  at  A s s o c i a t i o n  D.W.  at  on  p r e s e n t e d  c o n s i d e r e d  a l s o  a p p l i c a t i o n  t h i s  was  S o c i e t y of  and  p r e p a r a t i o n  European  work  Canadian  1984,  1985)  the  G e o p h y s i c i s t s  in  England,  was  J u l y  the  Levy,  London,  myself of  E x p l o r a t i o n  manuscript  and  Canadian  Workshop  the  A  some  A l b e r t a ,  c o n s i d e r ,  p r o b l e m s ,  and  of  Meeting i n  of  meeting  introduce  problems  sect  review  (CSEG)  D e c o n v o l u t i o n  USA.  (Jones  G e o p h y s i c i s t s  j o i n t  in  Nevada,  co-workers  G e o p h y s i c i s t s  S o c i e t y  I n t e r n a t i o n a l  b r i e f  1984  (CSPG),  the  a p p l i c a t i o n s  E x p l o r a t i o n  S.  of  USA.  the  KLT,  the  KLT  to  those  i n t o  four  major  i o n s : 1.  An  f i e l d s ,  and  i n t r o d u c t i o n to  the  to  previous  subsequent  s e c t i o n s  the  c o n s i d e r e d  KLT  i n  here  other  (Chapter  D; 2.  A  t h e o r e t i c a l  KLT  o u t l i n i n g  and  i n t r o d u c i n g  a p p l i c a t i o n s 3. s e i s m i c to  the  the  w e l l  the  s e t s ,  r e l e v a n t  here  of  and  the to  to  e s t a b l i s h e d  p h y s i c a l  c o n s i d e r e d  A p p l i c a t i o n s data  i n t r o d u c t i o n  the  theory  i n s i g h t s  (Chapter method  from  a s p e c t s the  behind  of  the  l i t e r a t u r e ,  each  of  the  2); to  m u l t i c h a n n e l  t h e o r e t i c a l  v a r i o u s  development  s y n t h e t i c  f i e l d  data,  (Chapters  m u l t i c h a n n e l with  3  and  reference 4);  and  4  4. method  An  assesment  i n each  SECTION  of  of  t h i s  l i t e r a t u r e  on  expansion  of  u s e f u l n e s s  the a p p l i c a t i o n s  REVIEW  In  t h e  OF  PREVIOUS  s e c t i o n ,  I  orthogonal  l i m i t a t i o n s  d i s c u s s e d  (Chapter  of  the  i n  the  5 ) .  WORK.  review  t h e  t r a n s f o r m s ,  m u l t i c h a n n e l s  and  of  p r e v i o u s  and  data  more i n  work  s p e c i f i c a l l y ,  terms  of  the  orthogonal  t r a n s f o r m s .  a.  Orthogonal  The  best  transforms which  Expansion  met  known,  was  s i n u s o i d s .  T h i s  r e p r e s e n t a t i o n orthogonal  a  other  f u n c t i o n s ,  p r i n c i p a l  t h e  s e t  an  F o u r i e r ,  be  of  (1933) more  t e s t  t r e a t i s e who  was  as  a  l a t e r  a l s o  by  For  square-wave-like  by  on  orthogonal  that  a  l i n e a r  l i n e a r  t o  Walsh  of  (1923)  f u n c t i o n s , t h e  of the  combination  orthogonal ( e . g .  see  combination  example,  d e v i s e d  f u n c t i o n  D i r i c h l e t ;  g e n e r a l i z e d  f u n c t i o n  f u n c t i o n s .  were  a s s e r t e d  ( s p e c i f i e d  expressed  a r b i t a r y  f u n c t i o n s  component  p s y c h o l o g i c a l  e a r l i e s t  approach  Beauchamp,  H o t e l l i n g u t i l i z e  c o u l d  of  F u n c t i o n .  c o n d i t i o n s  n o n - s i n u s o i d a l  i n t r o d u c e d s e v e r a l  of  broad  1982)  a  and  that  c e r t a i n  Kanasewich,  of  and  Rademacher  1975).  was  one  general  of  t h e  e a r l i e s t  approach  a n a l y s i s . s c o r e s ;  H i s  i n t h e  i n t e r e s t hence  h i s  workers guise was data  of  i n  t o  begin  t o  f a c t o r ,  or  c l a s s i f y i n g  c o n s i s t e d  of  5.  m u l t i c h a n n e l s  These  of  d i s c r e t e  methods  l i t e r a t u r e  when  s i n g l e - c h a n n e l  v a l u e s .  f i r s t  appeared  Karhunen  s t e l l a r  (1947)  f u n c t i o n s . H i s approach  1955),  t r e a t e d  s t a t i s t i c a l appeared  b.  sense,  i n t h e  p r i n c i p a l  The  as  Expansion  c o r o l l a r y  t o  an  s i m u l t a n e o u s l y s e t of  c o v a r i a n c e  of  of  a  expanding  sum  of  s u i t e Loeve  i n a  of (1948,  r i g o r o u s  t r a n s f o r m  then  H o t e l l i n g  and  t h e data  or  d i s c r e t e )  t h e m u l t i c h a n n e l  d e f i n e d  orthogonal e x p l o r e d  In  of  input be  of  As  p o s s i b i l i t y  data  i n terms  and  n  represented  by  a  a of  t h e t h e  channels orthogonal  I f t h e o r i g i n a l l i n e a r l y  of  case,  f o r n  ( a t most)  a r e not  s i n g l e  f u n c t i o n s .  t h e m u l t i c h a n n e l  need  a  t h e  i s decomposed, we  t o s t a t i s t i c a l l y  r e a l i z a t i o n s  r e p r e s e n t a t i o n .  presumably  f u n c t i o n s .  Data.  m u l t i c h a n n e l s  of  exact  of  workers  m a t r i x  c o u l d  by  t h e terms  was  of  f u n c t i o n s .  comprising  orthogonal  s u i t e  orthogonal  f o r an  they  a  Karhunen-Loeve  t r a n s f o r m  s e v e r a l  (continuous  i n t o  t h e t r a n s f o r m  M u l t i c h a n n e l  f u n c t i o n s  then  term  p r o c e s s i n g  r e p l i c a t i o n s  f o r m a l i z e d  t o complement  i n f i n i t e  t h i s ,  data  was of  s i g n a l  t r a n s f o r m s .  t h e expansion  f u n c t i o n  data  t h e  Karhunen-Loeve  d e s c r i b e  of  and  l i t e r a t u r e  component  Orthogonal  same  t h e d e r i v a t i o n  t h e  expanded  l i n e - s p e c t r a l  orthogonal who  i n  f u n c t i o n s  independent, subset  of  t h e  6  The  f i r s t  t r a n s f o r m were  f o r  Kramer  speech  workers  Mathews  a n a l y s i s In  communications and  e x p l o i t  Young (1979)  and  and  Huggins  C a l v e r t  form  as  I n t r i n s i c  i n v e s t i g a t e d  of  the  use  f i r s t  KL  1963) the  r e f e r  m a t r i x  Haar  a l g e b r a  t h e i r  groups  the  approach  p r o c e d u r e s  p r i o r  and  t o  t o  t e l e g r a p h i c  and  t e l e g r a p h i c  t o  C h r i s t e n s e n  m u l t i c h a n n e l forms  and  t o  of  f u l l y  e x p l o r e  t r a n s f o r m .  the  and  from  method A l s o ,  Walsh  and  data  Huggins  A n a l y s i s .  of  development  p r o c e s s i n g  the  Young  Component  a  a p p l i e d  s i g n a l  s i g n a l s .  the  who  were  (1962,  a l s o  using  speech  u n d e r l y i n g  (1974)  the  compression  the  u s e f u l n e s s  the  and  Hirschman  approach  t o  a  of  s u i t e  (1962) i n  and  Young  i t s m u l t i c h a n n e l  Gubbins  et  f u n c t i o n s  a l .  i n  (1971)  f i l t e r i n g  i q u e s .  Watanabe t r a n s f o r m ,  (1965)  showing  l e a s t - s q u a r e s Mathews  o b j e c t i v e  developed  how  but  w i t h f o r  or  a l s o  a  way  i n  m i n i m i z e d  component  f u n c t i o n .  not  i n d e p e n d e n t l y  which  mind)  overview  d e r i v e d  e l u c i d a t e d  p r i n c i p a l  c o n t i n u o u s  be  done  a l s o  data  comprehensive  c o u l d  was i n  He  d i s c r e t e a  a  i t as  f u n c t i o n .  f a c t o r ,  d e r i v e d  gave  sense,  (1956),  between  was  data  f a c t ,  e l e c t r o c a r d i o g r a p h  an  (1956)  communities  the  data  s u b s e q u e n t l y u t i l i z e d  e x t r a c t  techn  demonstrate  m u l t i c h a n n e l  and  t r a n s m i s s i o n .  t o  and  the  the  only by  the  s a l i e n t  a n a l y s i s KL  of  the i n  a  Kramer  and  e n t r o p y  of  d i f f e r e n c e s (which  t r a n s f o r m ,  was which  7  A reads  s u c c i n c t as  d e f i n i t i o n  K-L-expansion  m i n i m i z e s f i n i t e  the  number  or  s e l e c t i o n method  y e a r ,  of  (Chien  p r o c e s s i n g  and  Ahmed o r t h o g o n a l above  d e f i n i t i o n  t o  F u ,  and f i e l d  a ,  Murthy,  a p p l i c a t i o n  i n i t i a l l y  work  P e l a t ,  1976  b,  i n t o  on  a  Watanabe  by  of  one  which  t a k i n g  only  s e r i e s  f u n c t i o n s  some  a  of  an  i s t o  be  s e t  of  complete  1967)  and  Koontz,  1974;  Ahmed  and  J a i n ,  1976,  and  c;  A c c o r d i n g  Chien  and  Fu  (1967)  (1975),  t h e  t o  i n  t h e i r  d i g i t a l  t e c h n i q u e s  and  t o t o  Karhunen-Loeve  Rao,  f e a t u r e t o  the  i n  the  Ready  and  Andrews  and  L o w i t z ,  and  C a l v e r t  1978; (1974),  e x t r a c t i o n  was  due  a p p l i c a t i o n s  of  (1965).  t h e  p r o c e s s i n g ,  a p p l i c a t i o n s . t r a n s f o r m  method  1975;  f e a t u r e  on  and  1970;  1977;  Young  book  t h e  ( P r a t t ,  Watanabe  image  and  of  a  s e t .  m o d i f i c a t i o n  use  widespread  t r a n s f o r m  mentioned  proposed  as  t r a n s f o r m e d  r e c o g n i t i o n  1969),  method  o r t h o g o n a l  became  1984).  t r a n s f o r m s  a  the  ' o p t i m a l l y '  p a t t e r n  KL  Rao  presented  complete  the  of  of  (1965)  of  and  as  i n f i n i t e  terms  P a p o u l i s  v e c t o r s  and  (Fukunaga  P a t t e r s o n ,  i n  the  l e a s t - s q u a r e s sense,  f u r t h e r  1973;  committed  c o l l e c t i o n  a  i n a  s u i t e  A f t e r  M a l l i c k  by  f u n c t i o n s . "  same  which,  e r r o r  g i v e n  s e r i e s  W i n t z ,  c o i n e d  known  i n  as  the  u s u a l l y  terms  expressed  image  t r a n s f o r m  of a  v e c t o r  the  average  when  o r t h o g o n a l  the  i s  expansion  that  the  f o l l o w s :  "The  In  of  as  review They  most use  a p p l i e d  t o  of the the  8  decomposition as  a  s e r i e s  o f of  row  Throughout d a t a ,  and w i l l  mean  t h e  t h i s use  i n v e s t i g a t i o n  by  CDP  noise  troublesome method.  of  t r a c e s  and  However,  of  be  d e a l i n g  data  v i a  a  and  a s  an  w i t h  when  d e s c r i b e d  m u l t i c h a n n e l s  (KL)  of  t r a n s f o r m  t o  d i a g o n a l i z a t i o n of  t h e  (1978)  of  from  a  a l t e r n a t i v e  approach  examples,  Sandvin  than  paper  i t s  was  t h e a b i l i t y s e t  o f  how  t h e  time  s h i f t s  p u b l i s h e d  being  i n  overlooked  e f f e c t s  by  l a r g e  a  community.  methods,  t o d i s t i n g u i s h  e x p l o s i o n s  and  t h e  o f  b a s i s  orthogonal  f u n c t i o n s  needed  earthquake  a s  t o a  opposed  s t a c k i n g which  a u t o - r e g r e s s i v e  on  l e s s  French  t h e KL  n u c l e a r  o f  were  employed  underground  of t h e  s t a c k i n g .  (1979)  w i t h  t h e  move-out  t o t h e c o n v e n t i o n a l  was  work  method  t o c o n v e n t i o n a l  t r a c e - t o - t r a c e s t a t i c  t h e i r  c o n j u n c t i o n  success  of  s i g n a l  s y n t h e t i c  t h e g e o p h y s i c a l  and  Mace  common  w i t h of  a p p l i c a t i o n  l e d t o t h e i r  Tjostheim  claimed  image  'Karhunen-Loeve'  such  Hemon  t o t h e KL  u n f o r t u n a t e l y segment  I w i l l  g e o p h y s i c a l  demonstrated,  Gaussian  work  t h e term  t o e x t r a c t  c o r r e c t e d They  an  A p p l i c a t i o n s .  f i r s t  transform  of  m a t r i x .  Geophysical  The  m a t r i x  v e c t o r s .  expansion  c o v a r i a n c e  c.  t h e c o v a r i a n c e  t o  nuclear  l a r g e  t r a n s f o r m ,  represent e x p l o s i o n .  between  earthquakes.  t h e d i f f e r e n c e t h e  i n  They  i n number  records  of  of an  9  ,  M i l l i g a n  records  i n  et  an  on  the  the  a l i g n e d  attempt  b a s i s  of  bottom  r e p r e s e n t a t i v e orthogonal  l o c a l e ,  proved  to  be  the  method  to  v e l o c i t y  (1982), of  h i s  of  the  i n t o  an  the  used  the  More of  of  i n  the  l i n e  the  a p p l i c a t i o n s  i n  the  and  s i m i l a r i t i e s  amplitude  areas  U.B.C.  w i t h  at would  o i l and  which  t h e i r  subset  i n t o  r e g i o n s  type.  M i d d l e t o n ,  d a t a ,  i n t o  ping  of  from  In  which  a  1980)  a  l a t e r  used  the  underwater  sound  a n a l y s i s ,  Hagen  components phase  c l u s t e r  (Taner  hope  et  a n a l y s i s  c o e f f i c i e n t s )  H i s  these  of  the  between  i n s t a n t a n e o u s of  ms  type  s c a l e .  p r i n c i p a l  glance  for  l o c a l e s  bottom-sediment  c l u s t e r s .  method,  group  1.8  upon  Vpinger'  sediment  decomposed  r e p r e s e n t  (LeBlanc  c o n t r o l ,  l o c a t i o n s  Work  of  technique  ' n a t u r a l '  a  to  Using  Depending  to  a c o u s t i c  water-bottom  were  ocean-wide  w i t h  u n d r i l l e d  l i k e l y  used  a b l e  the  of  adjacent  aspects  were  g u i s e  w e l l - l o g  The  be  140,000  c h a r a c t e r .  v e c t o r s .  group  Hence,  d.  records  r e p r e s e n t a t i v e  p o r o s i t y .  the  'ping'  t r a n s f o r m a t i o n ' s  t r a c e s w i t h  pulse  p r o f i l e s ' -on  d a t a ,  d i s c r i m i n a t e  c o u l d  same  a n a l y s e d  s e i s m i c  they  working  the  to  i n v e s t i g a t e  Under  (1978)  orthogonal  v e c t o r s  given  paper,  a l .  was  that  a l . ,  (on  to  the  group  b a s i s s e i s m i c  i n  c o n j u n c t i o n  be  r e l a t e d  c l u s t e r s  c o u l d  a  c l u s t e r i n g  'map'  i n t e r p r e t e r  to  t r a c e  enable gas  an  1979)  to for  d i s c e r n  a c c u m u l a t i o n s .  Group.  the  U l r y c h  s i g n a l et  g e o p h y s i c a l  a l .  e x t r a c t i o n (1983)  s i g n a l s  and  presented  p r o c e s s i n g  coherency a  s e r i e s  f i e l d .  of  They  10  concerned  themselves  hyperbolae s t a c k i n g  and and  t r a n s f o r m  c o n v e n t i o n a l  method  to  address  a n a l y s i s  semblance  deal  w i t h  complex  s i g n a l s .  problem  d a t a ,  of  phase  s t a c k s and by  c o e f f i c i e n t s  (as  'dead-trace'  d e t e c t i o n  when,  Extending  f u r t h e r ,  they  r e f l e c t i o n s phase  and  a c r o s s  an  In these  a  s y n t h e t i c  Taner,  paper  the  c o r r e c t e d the  a n a l y s i s ) , to  and  the  for  i n  the  KL  .  a  u n u s u a l l y the  f i n a l v e l o c i t y  method  s e t s  of  of  even  common  s u p e r - c r i t i c a l  were  recover  the  amplitude  of  from  case  i n t e r f a c e , to  a n a l y s i s  developed  complex  to  gathers  performing  i n f o r m a t i o n  f l u i d - f l u i d i n f o r m a t i o n  of  or  KL  They  U t i l i z i n g  d e g r a d a t i o n  s t a c k i n g  the  wavelet.  v e l o c i t y  they  the  authors  d i s c r i m i n a t e a g a i n s t  a v o i d  phase  the  (1983).  the  1971).  CDP  new  of  extended  s e i s m i c  out  r e p l a c e  t r a n s f o r m a t i o n ' s  v e r s a t i l i t y  d a t a ,  a  v e l o c i t y  able  to  d e n s i t y  i n v e r t  c o n t r a s t s  i n t e r f a c e .  more  recent  procedures  r e l a t i v e  to  example, the  i n  c a r r i e d  to  a l l o w e d  a l .  the  r o u t i n e  e x t r a c t e d  from  et  c l u s t e r  t r a c e s  a n a l y s i s .  for  U l r y c h  and  T h i s  p a r a l l e l l e d  of  i n  for  changes  a l s o  move-out  e i g e n v a l u e s  companion  move-out  a l s o  f e a t u r e s  s e i s m i c  of  a  They  both  measure  ( N e i d e l l  i n  from  the  s i m i l a r i t y  c r i t e r i o n  d i a g n o s t i c  s i g n a l s  a  d a t a .  using  (1983)  i n t r o d u c e d  r e s u l t s  of  a l .  s y n t h e t i c  d i f f e r e n t  e x t r a c t i o n  channels  c o n s t r u c t  enhanced  approach  wavelet  et  the  produced  s i n g l e  v e l o c i t y  to  Levy  w i t h  phase  to  work,  Chapman  et  e x t r a c t  d e n s i t y  c o n t r a s t  changes  observed  i n  a l .  ( i n  press)  u t i l i z e d  i n f o r m a t i o n  s u p e r - c r i t i c a l  from  the  r e f l e c t i o n s  11  w i t h i n  A r c t i c  On (1983),  the and  a b y s s a l  p l a i n s  papers  by  Levy  a n a l y s i s  and  components  from  t h e s i s .  I t  As was  S.  my  the  was  devoted  c e n t r a l to  of  a  ( i n  T.J.  p r e s s ) ,  U l r y c h  c o - a u t h o r .  a p p r o x i m a t i o n papers  The  these  two  U l r y c h  and  D.W.  et  a l .  v e l o c i t y  for  comprise  on  the  e i g e n v a l u e s  measure.  An  overview  model, i s  i v e l y .  given  the  and  of the  in  o r i g i n a l  d i s p e r s i o n  t h i s  p o s s i b i l i t y  the  respect  two  measures.  theme  the  of  u t i l i z i n g  s i g n a l s  am  phase  p r e v i o u s l y ,  assuming  Q,  constant-Cj  I  a l .  d i s p e r s i o n  part  papers  of  and  t h i s f u r t h e r  Oldenburg,  that  evolved.  s i m i l a r i t y  be  et  work  Levy,  i n v e s t i g a t i o n  s e n s i t i v e  (1983),  l a t t e r  from  mentioned  an  f a c t o r  the  with  t h e s i s  a l .  Chapman  c o n s t a n t  i s  d i s c u s s i o n s t h i s  et  sediments.  From work. of  the the  e f f e c t s  S e c t i o n  t h i s , A  KL  the  KL  model method  the  t h e s i s  to  and  of  seismic  d e f i n e of on  a Q  using  grew  amount  (Futtermann,  d i s p e r s i o n  4 . I l l  t h i s  method  s i g n i f i c a n t  q u a n t i f i c a t i o n of  of  i t s q u a n t i f i c a t i o n  e s t i m a t i n g  constant-Q of  and  t h r u s t  to time  q u a l i t y 1962)  and  s i m i l a r i t y v a l u e s ,  seismic  Appendices  the  source 1  and  2,  12  SECTION  a.  1.III:  R e c o n s t r u c t i o n  Given  a  f l a t - l y i n g t r a c e  o r  and  of  those  coherency  from  w i l l  be  t h e most  coherent  p a r t s .  an  equal  (known of  number as  coherent method  o f  a l t e r n a t i v e  c o n t a i n s  seismic  e i t h e r  be  that  s e c t i o n .  viewed  as  ' c o r r e c t i o n  terms'  the  s i g n a l s .  input from  those  s i g n a l The  t h e  t o be  i n  part  w i l l  seen  enquire  that  from  horizons  t o w i l l  we  c a n  t r a c e - t o - t r a c e  of  t h e a  t h e KLT data  s e t of  (and  c o n t e n t ) . which  from  seismic  o r t h o g o n a l )  which  enables  us t o  t h e  l e s s  t r a c e s  w i t h  data  a r e arranged The  f i r s t  i s most  common  t r a c e s i n  across  p r i n c i p a l components  next  common  most  t o t h e common  R e c o n s t r u c t i n g  s i g n a l s ,  s i g n a l  t h e o r i g i n a l  p r i n c i p a l components  which  t h e c a n  or as t h e  t o  stacked  account  order  p r i n c i p a l  subsequent  added  t h e  t r a c e  whether  d i s p l a y  from  s t r u c t u r a l l y  assume  t h e g e o l o g i c a l  which  r e l a t i v e l y  t h e changes  c h a p t e r s ,  r e p l a c e s  ( o r energy  of  do n o t .  p a r t s  The  v a r i a n c e  stacked  s e c t i o n  i n t h e next  I  I  t h e data which  region  a r e n o t c o r r e l a t e d  t h e p r i n c i p a l components),  decreasing  component  p a r t s  a  a r e  n o i s e .  assumptions,  shown  separate  s e c t i o n  events  o f  from  l a y e r s ,  r e p r e s e n t i n g  p a r t s those  s e c t i o n  random  random  these  I n f o r m a t i o n .  d i p p i n g t h e  t h e data  separate  As  seismic  part  from  RECONSTRUCTION.  Coherent  across  whereas  I n view  DATA  p a r a l l e l  i n  c o n t r i b u t i o n s t r a c e ,  of  stacked  t o t r a c e  r e l a t e d  be.  SEISMIC  reproduce seismic f o r  t h e  13 g r e a t e s t l e a s t  variance  coherent  This  w i l l , p r o d u c e  i n f o r m a t i o n  truncated  has  seismic  been  p r e s e n t e d  p r e v i o u s l y  forms  t h e b a s i s  o f  b.  M i s f i t  As  we  a  n a t u r a l  of  may a  superimposed  some  p i n c h - o u t ,  we  may  h i g h l i g h t  t h e  d i s p l a y p l o t t e d  The  answer  t h e  t r y t o  which  method  l i t e r a t u r e ,  t h e  has  n o t  a l t h o u g h  i t  t e c h n i q u e s .  r e c o n s t r u c t i o n  which t h e  o f  on  t o p  t h e  common  that from  where  What  we  have  a  s t r u c t u r e  such  a  warrant  o f  o r i g i n a l  such  f u r t h e r  p a r t s a  f e a t u r e s  of  i n  have  those  t r a c e .  which  sand i n  t h i s  i s  l e n s , order  would  of  or t o  draw  i n v e s t i g a t i o n . F o r t h e  v a r i a b l e  data  as  have  background  upon  as  we  coherent  t o  events  t e c h n i q u e  do  then  dipping)  t h e common  coherent  l e a s t  t r a c e  we  of  data  area  order  could  p l o t  of  t o produce  be a an  s e c t i o n .  t e c h n i q u e , has  may  using  anomalous  t h e  ' a n o m a l y - h i g h l i g h t e d '  A  q u e s t i o n :  i s simply  case  remove  a n o m a l i e s .  i n grey-shades  r e c o n s t r u c t i o n ,  t h e  t h e f o l l o w i n g  s m a l l - s c a l e  regions  r e c o n s t r u c t i o n  t o  ( o r p a r a l l e l  p u r p o s e s ,  This  seismic  enhancement  a r e l e s s  i n  f l a t - l y i n g  t o  a s k  which  example,  a t t e n t i o n  from  r e c o n s t r u c t i o n  i n t h e  image  s e c t i o n  omitted.  r e c o n s t r u c t i o n  p r e s e n t ?  t h e data  For s i m i l a r  been  data  a d j u n c t  we  perform  i n f o r m a t i o n p a r t s  seismic  R e c o n s t r u c t i o n .  i n f o r m a t i o n , if  many  a  which n o t  been  I  c a l l  p r e s e n t e d  anomaly p r e v i o u s l y  or  m i s f i t i n  t h e  14  s e i s m o l o g i c a l  c.  n o t i c e d  p r o c e s s i n g  a  performing most  m i s f i t  l i t e r a t u r e .  s e c t i o n  as  s e p a r a t i o n  gather  i n  areas  may  f a c t  with  much  that  a  d a t a ,  energy  hyperbolae stacked  i n d i c a t e s  sources:  s e c t i o n t h i s  that  m i g r a t i n g  r e c o n s t r u c t i o n a  I  f a u l t i n g ,  m i g r a t e d  F u r t h e r ,  produce  t h e d i f f r a c t i o n  s e i s m i c  t h e d i f f r a c t i o n  m i s f i t  t h e  showing  w i t h  c o u l d  events  serve  f o r  i n d i c a t o r .  events  d i f f r a c t i o n (Levin  e t  i s  from  new,  stacked  a l . , 1983, Harlan  although  seismic e t  t h e  s e c t i o n s  a l . ,  by  1983).  Suppression.  marine  The  m u l t i p l e  v e l o c i t y , a  and  constant  common  using  I n  stacked  o r d e r .  of  i s n o t  of  d i f f r a c t i o n  a p p l i c a t i o n  Performing i n f e s t e d  (a  shallow  move-out  some be  f a u l t - e d g e  M u l t i p l e  problem.  The  s e r e n d i p i t o u s  means  In  may  i t s e l f near  a  This  edges.  s e c t i o n  concentrated  f e a t u r e .  h i g h l i g h t e d  i n c o r p o r a t e s  d i f f r a c t i o n s )  other  f a u l t  m i g r a t i o n  d i f f r a c t i o n  example  i n t e r e s t i n g  w i t h  s t i l l  r e s i d u a l  r e c o n s t r u c t i o n s  r e c o n s t r u c t i o n  a s s o c i a t e d  d.  image  D i f f r a c t i o n s .  W h i l s t  the  or  depth  s e i s m i c  d a t a ,  events o f t e n  d i s t o r t  v e l o c i t y p o i n t  t h e v e l o c i t y  of  a l l  m u l t i p l e s have  t h e  move-out  (CDP)  o r  o f t e n  approximately  i n f o r m a t i o n c o r r e c t i o n  common  t h e m u l t i p l e ,  we  t h e  o f on  shot produce  pose  same  i n t e r e s t . a  m u l t i p l e  p o i n t a  a  (CSP)  gather  i n  .15  which  t h e  words in  ' m u l t i p l e '  they  t h e  w i l l  then  appear  KL  r e c o n s t r u c t i o n  w i l l  produce  then  be  a  gather  un-move-out  N a t u r a l l y ,  t h e  event  not  so  as  This  f o r t h e  to  that  SECTION  In  t o  of  t h e most  f l a t t e n e d .  coherent  I n  other  i n f o r m a t i o n  a  p r i n c i p a l input  time.  Ryu  s e t of As  we  t r a c e s  s e c t i o n ,  w i l l  e i g e n v e c t o r s  of  would  t o  common  a r r i v a l s .  t h e  same  commence  who  The  i n f o r m a t i o n gather  c o n s t a n t  j u s t  a f t e r  would  v e l o c i t y . t h e  primary  of  s e c t i o n  an  i s  presented  a l t e r n a t i v e  proposed move-out  i t  was  p r i n c i p a l  i n t h e t h e o r y  a  approach  procedure  a l t e r e d  u s i n g  g a t h e r s .  (and v i c e  w i l l  these  v e r s a  components  from  a  m a t r i x  (Chapter l i n e a r  f o r data  a r e of  t h e  t h e 2 ) ,  KLT input these of  r e c o n s t r u c t i o n ) .  l i n e a r t h e  t h e  combination  t h e p h y s i c a l  i n these  weights  t h e c o v a r i a n c e  that  as  i n t r o d u c e used  mentioned  s e c t i o n s  a r e c o n s t r u c t e d  I  s u p p r e s s i o n  MEASURES.  c o e f f i c i e n t s s e e ,  m u l t i p l e  I t c o n s t i t u t e s  o r t h o g o n a l  shown  t h e w e i g h t i n g  w i t h  f i l t e r i n g  components  t h i s  most  m u l t i p l e  (1982)  SIMILARITY  data  of  t h e  suppress i t .  t h e p r e v i o u s  t r a c e s .  In  free  approach  f i r s t  1.IV:  produces  As  be  c o r r e c t e d  frequency-wavenumber  of  t o  o m i t t i n g  p r o c e s s i n g  novel  here  the  a r e e s s e n t i a l l y  g a t h e r .  A  data  events  s i g n i f i c a n c e c o m b i n a t i o n s .  elements  input  data  of  t h e  t r a c e s .  16  a.  C l u s t e r  A n a l y s i s .  When  c o n s t r u c t i n g ,  component, much  of  we  w i l l  each  have  input  a  f o r  example,  s e t of  t r a c e  weights  goes  i n t o  t h e  which  making  f i r s t w i l l  t h e  p r i n c i p a l  t e l l  f i r s t  us  how  p r i n c i p a l  component.  If  t h e  conclude the  that  f i r s t  s i m i l a r dozen, of  f i r s t  input  input  c o u l d  i n f e r  t r a c e s ,  a s s o c i a t e d  the  major  groups noted  of from  I  w i t h  had  aims  input  input  we  same  t r a c e s  v a l u e ,  had  c h a r a c t e r i z e d  dozen  r a d i c a l l y  two  t h e  we  weights  and  t o  were  t h e  s i m i l a r i t y  f i r s t  would  e q u a l l y  from  ' n a t u r a l '  by  then  c o n t r i b u t e d  I f t h e second  d i f f e r e d  that  t h e t h e  of  t h e  f i r s t groups  second  dozen  r e s p e c t i v e l y .  c l u s t e r s ,  groupings KLT,  of  such  t r a c e s  a t t e n d a n t  review  but  t r a c e s ,  s t u d y i n g  or  dozen  other  data  groups,  f i r s t  each  data  By  weights  p r i n c i p a l component.  t o we  t h e  dozen  input an  w e l l  t h e  a p p l i c a b i l i t y  seismic  e x p l o r a t i o n .  can  t h e form  s i g n a l s  a n a l y s i s  t o a  t h e approach  a s s e s s  we  of  l o g  be  b a s i s were t o  f o r  e l e m e n t s i d e n t i f y i n g  s i m i l a r .  l i n k  p a r a m e t e r ,  t h e e g .  One  of  r e s u l t i n g p o r o s i t y ,  i n f o r m a t i o n .  taken of  a  which  would  p h y s i c a l  weighting  by  Hagen  c l u s t e r  (1982)  a n a l y s i s  and  c r i t i c a l l y  i n t h e context  o f  b.  V e l o c i t y  The j.th  A n a l y s i s .  energy  e i g e n v a l u e  of of  that  s i g n a l  most  l a r g e  energy,  then  very  the  remaining t o t a l  d a t a . of  a  T h i s sum  we  can  p r i n c i p a l data  to say  be  component  c o v a r i a n c e  the  input  that  the  i s . g i v e n  m a t r i x .  data  input  can  be  used  has  a  data  by  the  F u r t h e r ,  i f  c o m p a r a t i v e l y  t r a c e s  time  s h i f t s  under  c o n s i d e r a t i o n .  a n a l y s i s . p r e s e n t e d  i n  I  l a t e r to  the  t h i s  are  a l l  the  (1983)  to  the  sum  of  d i s t r i b u t i o n  p a r t s to  of  of  of  the  input  be  the  r a t i o  the  remaining  measure.  s i m i l a r i t y  i n  of  the  presence  s u i t e  t h i s  and  Levy  of  proves c r i t e r i o n  of  i n p u t  e i g e n v a l u e  c r i t e r i o n  of  measure  semblance  the  m o d i f i e d  semblance  a l .  sum  c o n v e n t i o n a l  members  r e s u l t s  the  modify  the  s i m i l a r i t y  p r e s e n t  the  et  I  of  uncommon  e s p e c i a l l y  the  i n i t i a l  U l r y c h  which  a  than  1973),  between  The  as  e i g e n v a l u e  measure and  c i r c u m s t a n c e s ,  Taner,  to  a  e i g e n v a l u e s  r e l i a b l e  a l t e r n a t i v e  f i r s t  common  r a t i o , m  the  g i v e s  between  f i r s t  c e r t a i n  and  of  e i g e n v a l u e s  the  more  ( N e i d e l l  r a t i o  e i g e n v a l u e  of  Under  an  input  common  a  v a r i a n c e  e i g e n v a l u e s ,  to  the  j . t h  s i m i l a r .  Consequently,  the  the  used  a l .  s i g n a l s  r a t i o i n  i n v e s t i g a t i o n et  small  (1983).  as  v e l o c i t y were  18  c.  Q-Inversion.  In  a  procedure  a n a l y s i s ,  I  u t i l i z e  s i m i l a r i t y  between  U t i l i z i n g  t h e  change  i s ,  a t  t h e  each  frequency  t h e  p a i r  e i g e n v a l u e  (1962)  of  s i g n a l the  an  s i g n a l  t h e processed  a  i n t h e  f u n c t i o n  minimum  Q-value, s e i s m i c  In  I  wavelet  of  t o  d i s p e r s i o n  1  s i g n a l  has  e i g e n v a l u e  Knopoff That  s u b j e c t e d t h e  t o  e f f e c t s  magnitude  of  how  r a t i o  of  r a t i o  of  s i m i l a r  a  w i t h  eigenvalue  become.  of  t h e  u n d i s p e r s e d .  t h e  i n  i n d i c a t i o n  r e l a t e s  i n c o n j u n c t i o n  of  change  s i g n a l .  medium;  negate  i s used,  t h e Q-value  t h e a p p l i c a t i o n  q u a n t i f i c a t i o n ,  t h e e f f e c t s  (Appendices  source  i d e n t i f y  a  t h e  t h e  t o t h e  By  i d e n t i f y i n g  versus  u n d i s p e r s i n g  t h e medium  i n which  t h e  t r a v e l l e d .  a d d i t i o n  s i m i l a r i t y and  t r y t o  of  t o  q u a n t i f y  (which  f i r s t  t h e value  The  r e f e r e n c e  i s  v e l o c i t y  r e f e r e n c e  i s .i t e r a t i v e l y  s i g n a l  give  a  f o r  designed  compute  w i l l  and  t o  t o  r e l a t i o n s h i p  s i g n a l  s i g n a l s .  r a t i o  r a t i o  t h e Q-value  undispersed t o  a p p l i c a t i o n  s i g n a l  s t r e t c h i n g  s i g n a l ,  t h e  eigenvalue  t o  d i s p e r s e d  dependent  r e f e r e n c e  for  v e l o c i t y  This  t o  d i s p e r s e d  i t e r a t i o n ,  d i s p e r s i o n . a  a  the  Futtermann  i n phase  (1964)),  s i m i l a r  and  removal  f u n c t i o n .  of  an  d i s p e r s i o n 2 ) . In  technique T h i s  of  t h e KL  overview on  S e c t i o n  of  t h e constant  Vibroseis® 4 . I l l ,  f o r s y n t h e t i c  extends  e i g e n v a l u e  t h e work  of  I  data  present  data  from  Robinson  i s  r a t i o Q  t o  model,  presented  an  i t e r a t i v e  an  i m p u l s i v e  (1979,  1982)  19 who  attempted  data  i n an  W i t h i n the  end  t e x t .  of  ad  t o hoc  the t h e  remove  the  e f f e c t s  of  d i s p e r s i o n  from  s e i s m i c  f a s h i o n .  body  of  p e r t i n e n t  t h i s  t h e s i s ,  s e c t i o n ,  so  a l l f i g u r e s as  not  t o  a r e  c o l l a t e d  i n t e r r u p t  a t the  20 CHAPTER 2. THEORY AND BACKGROUND.  SECTION  a.  2.1:  THE KARHUNEN-LOEVE  I n t r o d u c t i o n .  In  t h i s  and  o u t l i n e  to  problems  KL  c h a p t e r ,  I present  t h e p h y s i c a l  (1975),  Levy  (1984))  t h i s  work.  s e i s m o l o g y .  a s a p p l i e d  U l r y c h  e t a l .  a n d begin  t h e theory  i n t e r p r e t a t i o n  i n r e f l e c t i o n  t r a n s f o r m a t i o n  Rao  b.  TRANSFORMATION.  from  o f t h i s I  w i l l  by d e f i n i n g  Levy  theory  a s a p p l i e d  be d e a l i n g  t o m u l t i c h a n n e l s (1983),  t h e l i t e r a t u r e ,  o f data  e t a l .  (Ahmed  (1983),  t h e t e r m i n o l o g y  w i t h t h e a n d  Jones  a n d  t o be used  i n  Theory.  Given  a  s e to f  n  t r a n s f o r m a t i o n  m a t r i x  of  data  a l t e r n a t i v e  r e a l  A  a s a  w i t h  elements  l i n e a r  n i// .(\<) = L a. . x. (t) J i=l 1J  C o n v e r s e l y , combination  we  x (t.) =  ii.(t)  n L b. ^.(t) j= l J  J  jj>  combination  r e c o n s t r u c t  o f t h ev e c t o r s  a  0 < t < T, w  e  m  a  Y  a n d an  c o n s t r u c t  o f t h e input  [nx.n]  a s e t  d a t a :  j = J, . . . , n; 0 < t < T  1  may  x.(t),  s i g n a l s  ,  t h e input  w i t h  weights  1=1,..., n  s i g n a l s  b. .  (I)  a s a  o f a  l i n e a r  m a t r i x  B:  (2)  21  We  may  d e r i v e  the  c o e f f i c i e n t s  a. . a n d  ways  ( U l r y c h  et  l e a s t - s q u a r e s (l)  ,  s i g n a l This  method  a r r a n g e d  i n  w i t h  l a r g e s t  the  approach  that  the  packs  the  the  was  B  =  R,  data  i s  R  m a t r i x my  A  an  Kramer  of  of  be  can  be  input  data  use  of  c o n t e n t ,  of  (1956) that  the  v e c t o r s  i.e.  ,  \p, (t)  coherent  and  so  who  on.  showed which  i n f o r m a t i o n  i n t o  are  A = R  simply  m a t r i x  d e r i v e d  from  ( i n  n o t a t i o n  of  the  the  t r a n s f o r m  ii.(i),  v e c t o r s  number  o r t h o g o n a l  on  for  B  e i g e n v e c t o r  the  of  Mathews  and  amount  number  would  r e s u l t  set  appear  and A  However,  energy  w i l l  m a t r i c e s  of  a  d e c r e a s i n g  the  Kramer  t r a n s p o s e d ) . shown  as  f o l l o w s .  D e f i n i n g  a  t r u n c a t e d  as:  ii (l) J  i =1,  m < n  (3)  3  a s s o c i a t e d  m i s f i t  (or  t r u n c a t i o n  (x. (l) - x. ( I ) )  T = ; 0  n i (x. (t) i=l  T = J 0  n m Z [x. (t) - Z i=l . j=l  t h a t  <p(m)  be  a  d e r i v a t i v e s  of  4>(m)  Demanding  For  by  the  T n <i>(m) = / Z 0 i=l  p a r t i a l  of  to  a  ij  p r e p . ) .  v a r i a n c e  p o s s i b l e  r e c o n s t r u c t i o n  have  r i s e  p o s s i b l e  m x (t) = Z b j =l we  g i v e s  taken  where  Mathews, T h e i r  i n  order  g r e a t e s t  c o v a r i a n c e and  1985,  c o e f f i c i e n t  s m a l l e s t  and  a l . ,  i n  b. .  ij  -  m z Ji=l  1  minimum w i t h  dt  2  b.  (4)  ^.(t))  dt  2  n Z k=l f o r  e r r o r ) :  b. . a,. x , ( t ) ) l J  1  r e s p e c t  < to  K }  K  m  < a.j  n, and  2  we b^.,  dl set to  the z e r o .  22  d<p(m)/db  ,  we  have:  T j 0  d(j>(m)/db = -2 PQ  \x (t) p  m Z j=l  n I k=l  b . a, . x, (t)} P J  k j  k  n • { T. a. x. (t)} dt 1=1 ' l  T S 0  = -2  - {  m I  J = J  n .[ . {  I  1=1 n Z k=l  a, l  x (t)x. (t)} '  q  n I 1=1  q  P  a. b . a,. x,(t) x,(t)} lq PJ kj k I  ] dt  = 0 But  t h e  i n t e g r a l  c o v a r i a n c e  7.. 'U So,  we  over  m a t r i x ,  T =  f  0  p a i r s  of  input  v e c t o r s  symmetric  x . (t ) x . (t ) dt i J  have:  nn  =1  l a,} pi l q nl  m  n  J = J  over  /  summing  K  =  n /=;  J  kj  pj kl r  lq  g i v e s :  d<j>(m)/db = -2 [ ( T A ) - { PQ Pq  And  t h e  i . e . :  n d<p(m)/db = -2 [ { L pq i  Summing  g i v e s  over  k and  j  g i v e s :  m Z j =1  n Z k  =  1  a,, b . ( T A ) , } kj pj kq  23  d<p(m)/db  -2  =  (r  [  -  A)  A  (B  r  T  A)  pq  =0 Dropping  the  r=  A  r  L i k e w i s e ,  r To  element  A  T  from  r  B =  A  T  A B  From  r = c  r  c  the  =  C T  c  l a t t e r  T  C C,  T  C  =  C ,  C  =  C  other  in  some  (5) d<j>(m)/da  =  we  0,  o b t a i n :  (6) p r o p e r t i e s  S e t t i n g  of  C = B A,  and  A  equations  J  B,  we  perform  (5)  and  (6)  the  become:  7  equation,  or  C C,  In  g i v e s :  = r cT c  T  C =  t r a n s p o s i n g  T  the  r  and  B  R  s t e p s : .  c  B  s e t t i n g  i n v e s t i g a t e  f o l l o w i n g  s u b s c r i p t s  see  t h a t :  t r a n s p o s i n g :  thus: and  hence:  2  words, as  we  yet  C  diagonal  unknown  Proceeding d e c o m p o s i t i o n  i s  with of  T  i n  order  a  w i t h along  e i t h e r the  p a r a l l e l  terms  of  R  and  ones  zeroes  arranged  d i a g o n a l .  argument A,  or  we  f o r  have  the  the  s p e c t r a l  f o l l o w i n g :  24  r = where the  R A R  i s the  R  d i a g o n a l  ^ 2 r ' ' - , ^  As  C  d i a g o n a l  m a t r i x m a t r i x  arranged  ,  n  (7)  R  i s a l s o m a t r i x  of A  orthogonal  c o n t a i n s  t h e  i n d e c r e a s i n g  symmetric,  w i t h  rank  we  equal  column  e i g e n v e c t o r s  c o r r e s p o n d i n g  ,  r  and  e i g e n v a l u e s  A,,  s i z e .  can t o  w r i t e the  C  rank  =  R  of  Q C  R  where  J  i s a  Q  ( i . e . rank  <  m).  Now:  Q  2  = R  1  C R R  = R  T  C  = R  T  C R  C R  r  R  2  = Q C o n s e q u e n t l y , be  u n i t y  In  or  elements  of  t h e  d i a g o n a l  m a t r i x  Q  must  (4)  i n  terms  from  t h e  e i t h e r  z e r o .  a d d i t i o n ,  d e v i a t i o n d a t a ,  the  of  a  we  can  forward  a l s o and  w r i t e  equation  i n v e r s e  t r a n s f o r m  of  the  o r i g i n a l  i . e . :  4>(m) = Trace[(I = Trace[(I but  as  t h i s  w e l l  r o t a t e  only  t h e  - B A)  (/ - A B )  r  7  T]  - C) T ] looks  argument  a t  t h e  i n t o  a  d i a g o n a l d i a g o n a l  e l e m e n t s , form:  we  can  e q u a l l y  25  4>(m) = Trace[R  = Tracel(I  - R  = TraceKl  - Q)A]  P L j =m+1 where  p  g i v e  a  i s  the  r  C R) A]  J  rank  of  r.  when  and  T R]  T  X.  minimum  e i g e n v a l u e s , o r d e r e d  (I - C) R R  T  as  the  the  e i g e n v e c t o r s  But  of  t h i s sum  ordered R,  we  r e p r e s e n t a t i o n extends  over  e i g e n v a l u e s  must  have  the  of  <p(m)  w i l l  s m a l l e s t  (p-m)  correspond  t o  the  t h a t :  Q = M M  where  i s an  i d e n t i t y  C = R Q  R  = R M  R  m a t r i x  of  rank  m,  but:  of  A  r  J  = B A  r  In  other  f i r s t  m  words,  columns  (1956)  that  R  m<n.  f o r  A  =  the  of B  R. =  Consequently, problem  f,  of  i s  the  r = x  t o data  x  r  f i r s t  R  a  f i r s t  m  columns  Hence  the  f o r m=n,  c o n v e n i e n t compute  the  r e s u l t and  way  A  =  of  or  of B  =  B  correspond  Kramer the  s o l v i n g  o u t e r - p r o d u c t  f i r s t  t h i s  and m  t o  the  Mathews  columns  of  l e a s t - s q u a r e s  c o v a r i a n c e  m a t r i x  x.(t): (8)  26  where  and  X' =  {x . (t ) ,  proceed  o b t a i n that  the a  t o  e i g e n v e c t o r s  of  the  A  With and  R.  row  B  thus  d e f i n e  (8)  (Throughout may  x.(t)  i.'th  and  (2)  equation  d i a g o n a l i z e  f u n c t i o n  elements  (1)  i = 7, . . . , n] ,  of  a  be  the  work  i t  d i s c r e t i z e d  matrix  chosen,  t h i s  the  equation  v i a  so  i s  as  (7)  t o  understood t o  form  the  X) .  t r a n s f o r m a t i o n  m u l t i c h a n n e l  p a i r  equations  Karhunen-Loeve  (KL)  transform:  n (t)  =  r  I  x. (l) ,  i-7  or  *  j = J, . . . , n,  '  R  =  and  x (t) =  X  or  where  A . ,  L j=l  J  i s  the  the  j.th  and  components  F u r t h e r ,  the  p r i n c i p a l  element  i.th  ^ = { ^ j ( l ) , j = l,...,n], p r i n c i p a l  i=l, . . . , n,  J  R *  =  iJ  r Ai (t) ,  j.th  of  the  the  the  v e c t o r s  f i r s t  column a  ^j^ ^ 1  r  e  vector  known  as  of  R,  '  the  t r a n s f o r m a t i o n .  eigenvalue  component,  of  X  i . e . :  g i v e s  the  energy  content  of  27  T = j 0  X J  In  other  matrix  =  t o t a l  input  a l t e r n a t i v e  t h e t r a n s f o r m a t i o n  Due  m a t r i x  A  i s simply  t h e  c o v a r i a n c e  \\i.(\)\  data  (12)  energy  T = / 0  Trace[T]  n I x.(i) i=l  words,  i s  2  and  a  we  s p a t i a l  r o t a t i o n ,  t h e  have:  dt = Trace[A]  (13)  1  i n which  component  which  i s merely  i s conserved  t o t h e way  p r i n c i p a l  other  t h e eigenvalue  «1> T  *  as  s i g n a l  (11)  t h e o r t h o g o n a l  A l s o ,  the  dl  2  J  words,  of  A  f.(t)  w i t h  most  t h e f i r s t  t h e c o v a r i a n c e t h e l a r g e s t  h i g h l y  i s c a l c u l a t e d ,  v a r i a n c e  c o r r e l a t e d  p r i n c i p a l  matrix  from  component  c o n t a i n s  that  t r a c e - t o - t r a c e . I n  r e p r e s e n t s  t h e  'common  s i g n a l ' .  The  u n d e r l y i n g  s i g n a l  which  e x t r a c t  that  predominant number  of  If x.(t)  where  s i g n a l  t h e r e  \p.(t)  common  s i g n a l .  input  were  J  i s  assumption  t o each  I n matrix  component  input  t e r m s ,  w i t h  work  a  i s t h a t  t r a c e :  we  much  i s  o u r problem  a s s e r t  rank  there  that  s m a l l e r  a  i s t o has  X than  a  n, t h e  v e c t o r s .  were  no  o r t h o g o n a l ) ,  = x.(t), 1  i n t h i s  t r a c e - t o - t r a c e r  would  and  be  s i m i l a r i t y  d i a g o n a l ,  X. = / J  o  and  x.(t)  we  2  ( i . e .i ft h e would  have:  d t .  1  t h e  28  are  x(t)  \p.(t))  c.  t o  meet  S i n g u l a r  For per  r e - a r r a n g e d  d e c r e a s i n g  s e i s m o l o g i c a l a p p l i c a t i o n s  we  have  have  t r a c e s .  proceed  number  t r a c e  ( M i l l i g a n  of  KL  t r a n s f o r m  the  c o m p u t a t i o n a l  the  as  t r a c e s  et  the  (as  n  Decomposition.  we  the  X,  content  > X  >  where  that  energy  X2  to  Given  of  >  than  convenient  order  requirement  Value  most  t r a c e  the  i n  t h i s  o u t l i n e d f a r  a l . , 1978)  and  In  case  above.  exceeds we  the  u t i l i z e  s i n g u l a r - v a l u e  many  i t  i s  However, number  the  more simple  i n  of  p o i n t s  (SVD)  to  the  i s  data  m a t r i x  X,  we  may  w r i t e :  (14)  i n  g e n e r a l  s i n g u l a r  v a l u e s  m a t r i x ,  which  reduce  expense.  r  fl  per  between  X = R fl V where  and  s i t u a t i o n s  r e l a t i o n s h i p  decomposition  p o i n t s  i s  a  r e c t a n g u l a r of  l a t e r  seen  t h i s to  d i a g o n a l  m a t r i x  d e c o m p o s i t i o n , c o n t a i n  the  and  V  c o n t a i n i n g i s  n o r m a l i z e d  the  post  p r i n c i p a l  components.  Now, for  our  we data  r =  and  as  note  x  i s  X  x  equation  from g i v e n  by  the  (8)  that  the  c o v a r i a n c e  V  o u t e r - p r o d u c t :  (15)  r  = R n V  T  both  and  R  m a t r i x  V fl R , 1  V  are  o r t h o g o n a l ,  t h i s  becomes:  29  r = so  we  see  The t r a c e s  fl2  R  A  1  ^  each  of  a t  most  Now,  i f  the  number  the  f o l l o w i n g  = x  of  we  r e c a l l  where  fl  both  p o i n t s ,  we  have  n < N,  t h e  n non-zero f u l l y  had  a  2  =  [nxn]. that  system  i n a  where  t r a c e ,  i n n e r - p r o d u c t  X  i s  However,  fl  [nxN],  =  which  i . e . n >  of  we  i s t h e t h a t  = R  J  we  Hence  t r a c e s  may  w i s h  exceeded t o  form  m a t r i x :  (16) 1  e i g e n v e c t o r R  c o n t a i n e d  t h e p r i n c i p a l  m a t r i x t h e  of  (both  V  and  V  e i g e n v e c t o r s  components  *  V  a r e  d e r i v e d  (equation  (9))  from  [Nx/V]); T.  We  a s :  (17)  X [nxN],  i s  and  and fl.  n  [nxN],  i s  N,  our  = [nxN],  i n t h e m a t r i x  t h e number  c o v a r i a n c e  fl  f o r  o v e r d e t e r m i n e d  e i g e n v a l u e s  r e p r e s e n t s  case  A.  x  r  c o n s t r u c t e d  *  A  (7)  a r e  p o i n t s  2  V  and  [nxn]  = V fl V Here  N  f o r  i s  r  R  but  which  equation  from  m a t r i c e s  have  2  J  t h a t  V = [NxN], only  R,  S u b s t i t u t i n g  equation  (14)  i n t o  equation  (17)  y i e l d s :  = R  T  R fl V  7  = fl V  T  (18)  30  So  we  see  t h a t  transposed Given  and  So,  we  (17)  i f  fl"  e i g e n v e c t o r  d e r i v e  the  (18)  and  of  r  i s  m a t r i x  e i g e n v e c t o r  simply  a  of  r.  V,  matrix  R,  of  V:  g i v e s :  p r i n c i p a l  I  simply  SVD  and  Andrews  or  R,  Image  approach, and  N a r e n d r a ,  (1969)).  T h i s  images,  each  is, from X =  i f N  present  c o n s t i t u t i n g  L :-J  want  decompose  r, >>  may  image  Hunt  image  can which  equation X / 1  J  2  use  i f  J  we  components, are  decompose  T  the  (1976  and be  i n t e r e s t e d  or  i f  i n  the  r.  X  i s  a  l i n e a r  weights.  t e r m i n o l o g y a,  b,  c)  (Davenport  K u b l e r ,  decomposed  (14):  r.v J  we  n,  m a t r i x  1975;  of  whereas  a p p r o p r i a t e  P a t t e r s o n  an  p r i n c i p a l  r e c o n s t r u c t i o n as w i t h  we  the  Summation.  a  components  the  becomes:  (19)  we we  N,  Here  That  the  *,  1  Outer-Product  and  may  rearranged  e i g e n v e c t o r s  d.  fl  of  m a t r i x  1  when  »  v e r s i o n  component  X = fl V ,  r  R = X V  n  p r i n c i p a l  equations  equating  which  s c a l e d  V  R  the  formed  by  and  an  of  the  Root,  1958;  Fukunaga a  sum  of  outer-product  of  f o l l o w i n g  Andrews  t h i n k  or  i n t o  However, of  and  1984;  combination  and  (1970) data  as  Huang Koontz rank-one  expansion.  31  = v/ where  each  s i z e  as  images  a.  an  2. I I :  a  have  c o n s t a n t  the  the  s i g n a l  {x.(t),  y = 2 , . . . , « } ,  a  rank-one such  image  rank-one  of  the  same  d e c o m p o s i t i o n  (1970).  AND  ASSOCIATED  of  f a c t o r ,  w i l l  c. ) 2  f i r s t  n},  w i l l  r e c o n s t r u c t i o n .  of  1  a  ERRORS.  /  2  the  can and  R) .  have  s i g n a l s  which  the  f i r s t  I f  x.(t) = c s(t)  are  p r i n c i p a l  i  given  s i g n a l ,  i d e n t i c a l component  where  the  then  the  t o has  c.  a r e  f i r s t  be:  s(t)  p r i n c i p a l and  s(t),  row  i s  s(t)  L  component  f i r s t  s e v e r a l  s u i t e  s c a l e  and  i=],...,  p r i n c i p a l  of  c h a r a c t e r i s t i c .  = (  i s ,  forms  Andrews  a  component  *i(t)  n n  S i g n a l s .  c o n s t a n t s ,  That  i n  T  vT  RECONSTRUCTION  we  p r i n c i p a l  the  found  i m p o r t a n t  r e a l  of  Photographs  be  When  r v  l / 2  n  X.  I d e n t i c a l  w i t h i n  r , v j .+ ... + X  o u t e r - p r o d u c t ^  can  SECTION  2  the The  zero  (20) component  complete be  set  w i l l of  r e c o n s t r u c t e d  a p p r o p r i a t e r e m a i n i n g  s c a l e d s i g n a l  e x a c t l y  weights  and  a  input  p r i n c i p a l  e i g e n v a l u e s  be  are  from  (the  v e c t o r s t h i s  one  elements  of  components not  v e r s i o n  needed  {yp.(t), i n  the  32  b.  S i m i l a r  If  S i g n a l s .  t h e  c o n t a m i n a n t s where  r e p r e s e n t  energy mind,  them,  can  I  s u i t e  d e f i n e  t h e  =  of  some  (11)  a  m Z j=l  w i t h  X J  l a r g e .  x(m) other  c o v a r i a n c e  p h y s i c a l and  t h e  by  f a c t o r s .  a  t h e  p r i n c i p a l  i d e a l  s i n g l e  Rather,  case  p r i n c i p a l  we  component  other  may  s t i l l  and  accept  noted  that  j.th  t h e  p r i n c i p a l  e i g e n v a l u e  component.  g i v e s  With  t h e  t h i s  i n  measure:  (21)  X J  w i t h  X  2  ,  X  3  ,  X^  =  0  we  w i l l  have:  m = 1, 2, . . . , n- 1.  s i g n i f i c a n c e  contaminated  C o n v e r s e l y ,  w i t h  have  or  e r r o r .  t h e j.th  case  noise  longer  represented  s i n g l e  n Z j=m+l  /  random  no  s c a l e  s i m i l a r i t y  = »,  more  we  a  had we  e x a c t l y  w i t h  ' i d e n t i c a l '  s i m i l a r ,  noise)  be  a p p r o p r i a t e  a s s o c i a t e d  Of  In  t o  equation  X(m)  be  added  w i t h t h e  X(m)  are  s i g n a l s  i n t r o d u c t i o n  In  For  of  t h e s u i t e  component  the  s u i t e  same  w i t h  i s t h e case n o i s e .  f o r i n h e r e n t l y mean  energy,  we  I n  when  would  waveforms  case,  x(m)  w i l l  s i g n a l s  ( o r  white  t h i s  d i s s i m i l a r  t h e  have:  0{m/(n-m) } words, m a t r i x  i f  a l l  would  t h e be  s i g n a l s  mostly  were  d i s s i m i l a r ,  d i a g o n a l ,  and  then a l l  t h e t h e  33  e i g e n v a l u e s  would  would  a  y i e l d  T h i s  c o n d i t i o n  work  as  a  must  the  r e s u l t  of  order  and  the  of  upon  t o  represent)  the  d a t a .  given  the  bulk  p r i n c i p a l an  i n  t e l l s  m i s f i t us  how  i s  used  analagous  to  throughout  energy  of  {equation  determine  how  p r i n c i p a l  i s  the  packed  t h i s  t r u n c a t i o n )  much  and  s a t i s f a c t o r i l y  However,  (or  to  d i s c a r d  the  i s  \(m),  t r a n s f o r m a t i o n  to  may  of  (21)  t h i s  E r r o r .  order We  (EVR)  1980),  c r i t e r i o n  components.  a s s o c i a t e d  which the  use  r a t i o  i n v e r s e  a  equation  hence  c o h e r e n c y .  T r u n c a t i o n  components  t h a t  s i g n a l  magnitude,  {m/(n-m)}.  ( S t r a n g ,  t r u n c a t e d  decide  same  e i g e n v a l u e  number  measure  Winnowing  For  of  m o d i f i e d  the  c.  be  many  input  p r i n c i p a l (or  components  m+1  i n t o  f i r s t  m  r i s e  to  <t>(m)  energy  we  r e c o n s t r u c t  the  approximation e r r o r ,  (3))  we  g i v e s  to  {equation  have  n  (4)),  omitted  from  r e c o n s t r u c t i o n .  Given the  t h a t  energy  (equation input  the  j.th  content  of  (11)),  and  the a  (equation  energy  e i g e n v a l u e  sum  (13)),  j.th  X.  simply  g i v e s  p r i n c i p a l  a  measure  i>.(t)  component  of  the  e i g e n v a l u e s  we  may  r e w r i t e  t h i s  y i e l d s  of  the  t o t a l  as:  n 4>(m) = So, the  to  Z  decide  (22)  X.  j=m+1 upon  J the  r e c o n s t r u c t i o n ,  number I  simply  of  p r i n c i p a l  request  t h a t  components a  to  s p e c i f i e d  use amount  i n of  34  the  input  energy  requirement  i n  be  terms  of  m •n(m) = 100 Z j=l Conversely,  u(m)  we  the  i n  of  F i g u r e  versus see  2.1,  t h e  n n Z X. / Z j=m+1 j=1  we  number  see  of  manifest  here  t h e  v i r t u a l l y  f i r s t  the  of  100  Furthermore, the  KL  be  that  the  t r a c e s .  a.  a s s o c i a t e d  2 . I l l :  the  APPLICATION  stacked  packing s i g n a l  o p t i m a l  f e a t u r e s  TO  percentage s e i s m i c  p r o p e r t y  energy  percentage  w i t h  (24) energy  s e c t i o n . of  the  i s contained  We KL  i n  t h e  components.  the  t h i s  cumulative  f o r a  t h e  t o  m i s f i t :  packing  of  i n f o r m a t i o n  of  the  input  which  a r e  most  SEISMIC  energy s i m i l a r  by w i l l  a c r o s s  DATA,  I n t r o d u c t i o n  From the  due  (23)  X. = 100 4>(m) / (p(0)  of  p r i n c i p a l  t h i s  1  energy  a l l  t r a n s f o r m a t i o n ,  SECTION  p l o t  express  energy:  percentage  e i g e n v a l u e s ,  transform: 15  a  I  J  J  In  output.  X.  J  speak  the  r e c o n s t r u c t i o n  n X. / Z j=l  may  = 100  present  the  simple  d e s c r i p t i o n  the o f t e n  most  of  p a r t ,  i n c o r p o r a t e  case  of  waveforms I  w i l l  s i m i l a r w i t h i n  d i s c u s s  f l a t - l y i n g  or  waveforms, a  s e t  stacked  d i p p i n g  of  I  seismic  s e i s m i c  events  now  progress d a t a .  s e c t i o n s ,  w i t h  a  t o For  which  p a r t i c u l a r  35  waveform  b.  F l a t  across  L y i n g  From  t h e data  Events.  t h e way  waveform  which  time  be  w i l l  i n which occurs  seen  as  s i g n a l  c h a r a c t e r  dominant  and  coherent, then  we  w i l l  r e c o n s t r u c t  c.  across  few  only  l y i n g  emphasize  p r e f e r r e d r a t h e r  h o r i z o n t a l  events.  end  i n t o of  s h i f t e d  i s  t r a c e s ,  f l a t  l y i n g  p r i n c i p a l  i s d e f i n e d ,  t h e  t h i s  which  same  a r r i v a l  s i m i l a r i t y  of  c o n s t i t u t e s  I f t h e data  stacked  a r e  s e i s m i c  components  t o  a  t h e  h i g h l y  s e c t i o n , adequately  d a t a .  d i r e c t i o n ,  zeroes  a  I t  c h a r a c t e r i s t i c .  need  t h e c o v a r i a n c e  T h i s  t h e  coherent  r  t r a c e - t o - t r a c e a t  coherent.  t h e data  events  t h e c o v a r i a n c e  t o  represent  event  most  m a t r i x  Events.  i n which  a d j u s t  from  i n a  f l a t  wished  t h e c o v a r i a n c e  f o r example  t h e  D i p p i n g  The way  most  as  s e c t i o n .  a  f i e l d .  m a t r i x  s e t of  T  as  'most  p a r a l l e l  i n  c o r r e s p o n d i n g  events,  such t o  a  z e r o - l a g  which  most  e a s i l y  i n t r o d u c i n g  t h e d a t a .  d i p appear  The t o  be  wedge  t o t h e i f  we  way  some  t h e  by  due  However,  d i p p i n g  l a g s ,  of  common'  i s d e f i n e d .  c a l c u l a t i o n  i s achieved  s p e c i f i e d  seen  m a t r i x  than  t h e s t a r t  a r e  we  c o u l d as  t o  d e s i r e d  d i p  corresponds  a  i s chosen  h o r i z o n t a l  i n  t o  wedge  of  t o make  an  t h e  new,  36  In  e f f e c t ,  we  are  producing  y (t) = b { t - V ( i - l ) ) 8  i s  the  D i r a c  expressed  i n  V  p o s i t i v e  can  r e f e r  be  l a t e r  analogy  I  to  time-samples  to  a  the  d.  Coherent  i s  over  that  we  l e a s t  If  c o n s t r u c t coherent e x h i b i t e d  we  p r i n c i p a l  have  w i t h  i n  d a t a ,  our  a  of  a  s e c t i o n  e v e n t s . an  r e c o n s t r u c t i o n  and  t r a c e ;  V  i s  '.*'  depending on  of  the  d i p  denotes on  the as  Y  data  components  at  t h i s  t h a t :  the  beds  c o n v o l u t i o n .  d i p a  of  d i r e c t i o n .  ' s l a n t - K L '  r e c o n s t r u c t i o n  i n  As a  noted  r e s i d u e  d a t a .  Chapter  few  stacked which  u n d u l a t i n g from  to  3,  I by  from  S e c t i o n  random  the  i n c o r p o r a t e d events  a 3.1.  the the  that  c o n s i s t (white)  what  i s  f i r s t  m  answer  i s  which  i s  p r i m a r i l y  of  n o i s e .  p r i n c i p a l  components  s e c t i o n ,  then  we  would  most  f l a t  l y i n g  only which  s t r u c t u r e , next  1.111b,  w i l l  again  from  r e p r e s e n t s  s e i s m i c  the  set  S e c t i o n  and  of  c o n s i d e r  data  r e s i d u e  below,  very  i n  a  which  This  Consequently,  forged  stage  r e c o n s t r u c t e d  d e s c r i b e d  u t i l i z e d  r e c o n s t r u c t i o n  per  u s e f u l n e s s  i n t e r e s t  l e f t  we  f u n c t i o n ,  n e g a t i v e ,  components.  common  such  Y,  I n f o r m a t i o n .  of  are  anomalous  of  when  p r i n c i p a l  set  (25)  t r a n s f o r m a t i o n  the  number  It  data  s l a n t - s t a c k .  l i m i t e d  l e f t  KL  c o n s i d e r  Less  d e l t a  or  new  * x. (t)  t  where  a  few  the  were  would  be  p r i n c i p a l  i n  d i p p i n g , o m i t t e d .  a  or A  components  37  would  comprise  p r i n c i p a l  So,  we  y i e l d  data.  see  The  r e f l e c t i o n  The  i n  of  a  3.IV.  denote  i n t o  component.  s e c t i o n ,  remaining  n o i s e .  d e f i n e d  as:  i =1,. . . , n; m < p < n the  of  s u b t l e r  m i s f i t s e c t i o n  our  a l s o  the  while  been  (26)  c h a r a c t e r i s t i c s of  r e c o n s t r u c t i o n i s  i n  demonstrated  the  l a t t e r  i n  by  r  ,  the  from  of  which  the  the  o u t l i n i n g  in  Chapter  C o n s e q u e n t l y ,  s i m i l a r ,  s i z e  except  then  for  r3 ,  a l l  g a i n e d  the  which  each  i f  Sections  d i f f r a c t i o n  separate  from  m u l t i p l e s  normal  moveout  (our  d e s i r e d  events  a l l  to  elements would  the  seismic the  be  Elements.  from  e i g e n v e c t o r s  of  I  for  d a t a .  correspond  c o n t r i b u t i o n  were  be  separate  primary  E i g e n v e c t o r  can  b a s i s  c o n s t a n t - v e l o c i t y  s i g n a l )  of  the  I  common-depth-point  i n s i g h t  forms  former,  a l i g n e d  common  p r o p e r t i e s  the  seismic  The  3  seismic  In  e v e n t s ,  U s e f u l  e i g e n v e c t o r  incoherent  ii .(\)  J  i n s i g h t  S i g n i f i c a n c e  p h y s i c a l  r  r e c o n s t r u c t i o n  have  c o r r e c t i o n :  e.  m i s f i t  e v e n t s .  11.  and  s i g n a l )  P Z • j=m+l  p a r t s  M i s f i t  (which  a  u n d u l a t i n g  represent  e f f e c t i v e n e s s  Sect ion  3 . I l l  or  would  that  u s e f u l  anomalous 3,  d i p p i n g  components  x t - (t) = can  the  in  much  i n v e s t i g a t i o n The  R.  t r a c e  r,  elements  p r i n c i p a l  j.th  input of  an  to  t r a c e s would  smaller  of  the  p r i n c i p a l  except  (see  the  component,  that  be  of  the  of Levy  t h i r d  s i m i l a r et  a l . ,  38  1983,  F i g u r e  We  4 ) .  may  a l s o  i d e n t i c a l ,  t h e  i n f o r m a t i o n  that  X  2  ,  X  f o r X  3 >  e i g e n v e c t o r  i s p r o j e c t e d  l i g h t  t h i s ,  I  the  ' d i s t a n c e '  given  group  use  t h e  between  of  , J =1  where  a. . =  m  =  A ^  t h e  of of  same  coherent i n f e r  that  t h e  us  t h e j.th  how  much  of  t h e  i.th  p r i n c i p a l  s i m i l a r i t y of  component.  measure  t h e  input  t o  In  examine  t r a c e s  onto  a  (27)  now  t h i s  only  and  So,  l a r g e ,  energy, t r a c e s  /  t h e  then  over  a l l m  and  f i n d  and  k  seismic  t h e  0,  we  t h e  i n f e r  as  f o l l o w s :  f o r  w i l l  be  a m p l i t u d e s onto  that  1.st p r i n c i p a l  e i g e n v e c t o r s  have  seen  t r a c e s  t h e converse  that  be  e i g e n v e c t o r ,  between  = t o  can  f i r s t  k.th  i f  c o n t r i b u t i o n s  sum  measure  d i f f e r e n c e  t h e i.th  component.  we  and  components:  of  t h e  i s r e l a t i v e l y  If  a l l t h e  J  s i g n i f i c a n c e  p r i n c i p a l  c a r r y  a r e  J  J  p r o j e c t i o n s  w i l l  V2  J  1  square  have  tj -  s a y , i . e . using  1,  the  a  alone  t r a c e s  r . . / j/X.  U The  (  onto  input  z e r o .  t e l l  r.  f o l l o w i n g  p r i n c i p a l  L  be  t h e  r e c o n s t r u c t i o n  t h e p r o j e c t i o n s  m =  s i g n a l  elements  t r a c e  a l l  component  w i l l  seismic of  when  1.st p r i n c i p a l  necessary  e i g e n v a l u e s  The  note  A. ^  s i m i l a r  i s  t h e  t r a c e s  t h e 1.st  /  and  component.  k I f  t r u e .  a s s o c i a t e d  i s s t i l l  of  with  near  zero,  p r o j e c t i o n s  onto  t h e  then a l l  we m  39  p r i n c i p a l  components,  s i m i l a r .  We  would  and  d e t e r m i n e  i n c o r p o r a t e  t h e e f f e c t s  input  energy  data  T j o s t h e i m e v e n t s  and  groups  of  SECTION  a.  and  o f  nuclear  t r a c e s  from  t h e  t h e most  (1979) b l a s t s  (22)).  t o  on  /  and  p e r c e n t a g e  This  method  d i s c r i m i n a t e  t h e b a s i s  of  a r e  k  r e q u i r e m e n t  s i g n i f i c a n t  equation  (using  Sandvin  m  that  was  between  very  that  we  o f t h e used  by  seismic  t h e e i g e n s t r u c t u r e  of  e v e n t s .  2 . I V : THE  COMPLEX  KARHUNEN-LOEVE  TRANSFORMATION.  I n t r o d u c t i o n .  The  important  enumerated over  by  to  e t  of  be  change)  by  complex  KLT  comparison  In g r e a t e r  Mathews  t h e  t h e  a l . ,  Most c a n  t o  and  case  f o r c o n s i d e r i n g  a d d r e s s  (Levy  c h a r a c t e r i s t i c s  Kramer  d i r e c t l y  r a t i o n a l e  KLT  f u r t h e r ,  (1956)  when  complex  problem  o f  o f  t h e f o r  complex  s i g n a l s phase  KL  t r a n s f o r m a t i o n  r e a l  s i g n a l s  s i g n a l s  i s that  a r e used.  we  changes  c a r r y  w i l l  i n  t h e  be  The able  s i g n a l  1983).  t h e a p p l i c a t i o n s  given  a  f u r t h e r  u t i l i z i n g i s used between  g e n e r a l ,  i n t h i s  work  degree  of  freedom  t h e complex  KL  t r a n s f o r m a t i o n .  as t h e  w e l l  of  as  t h e  r e a l  ( i . e . that  KLT,  I  using o f  t h e phase  Whenever w i l l  t h e  draw  a  r e s u l t s .  t h e complex  c o m p r e s s i o n  considered  data  KL  ( o r CKL)  than  transform  t h e r e a l  KL  w i l l  ( o r RKL)  a c h i e v e  a  t r a n s f o r m .  40  T h i s  i s  due  s i g n a l s  even  us  deal  to  s i g n a l band  the  more  common  of  w i t h  to  a  w i t h i n can  we  of  p r o c e e d .  T h i s  complex  t r a c e  Taner  a l . , 1977;  b.  et  The  Given  w(t) where  (or  a  i n i t i a l  To  complex  s h i f t ,  domain at  each  t o  make  c e r t a i n  t e c h n i q u e  a l l o w s  data.,  from  which  frequency  i n  a the  e x t r a c t e d .  a  r e a l  s i g n a l ,  c o n s t r u c t i n g r e a d i l y  a  we  are  complex  a c c o m p l i s h e d s i g n a l )  B r a c e w e l l ,  1978).  analogue by  of  f a c e d  the  i n i t i a l l y i n  order  c o n s i d e r i n g data  to the (eg.  x.(t)  wavelet  and  i t s complex  w(t),  t r a c e  i s  w(t)  d e f i n e d  as:  - iw(t) i s  (28) the  R i c h a r d s ,  H i l b e r t  t r a n s f o r m  1980;  Levy  and  phase  change  (eg.  Oldenburg,  B r a c e w e l l , 1982)  of  the  w a v e l e t .  i n t r o d u c e  complex  wavelet  the  change  T r a c e .  = w(t)  Aki  phase  a n a l y t i c  w(t) = H[w(t)]  1978;  i t s  i s most  Complex  be  a  frequency  phase  r e c o r d  problem  of  F u r t h e r ,  d i r e c t l y  that  the  a b i l i t y  s i m i l a r .  i n t e r e s t ,  Given w i t h  to  t r a c e  phase  w(t;e)  a  pure  by  a  s h i f t e d  by  = Re[w(t) = cos(e)  complex an  amount  to  a  w a v e l e t ,  e x p o n e n t i a l . e,  i s  g i v e n  we  m u l t i p l y  Consequently,  a  by:  exp(-ie)] w(t.)  + sin(e)  w(t)  (29)  41  For  complex  p o s i t i v e for  s i g n a l s ,  the  s e m i - d e f i n i t e and  hence  d i a g o n a l i z a t i o n ( S t r a n g ,  r =  X X  X  where  = U A  H  i s  c o v a r i a n c e a  matrix  u n i t a r y  1980),  i s  H e r m i t i a n  U,  matrix  i s  and  r e q u i r e d  i.e.:  U, H  complex;  '//'  denotes  the  H e r m i t i a n ,  or  c o n j u g a t e ,  t r a n s p o s e .  The  e i g e n v a l u e s  complex.  N e v e r t h e l e s s ,  r e c o n s t r u c t i o n  c.  I d e n t i c a l  As x.(t)  =  i s  c  s(t),  component  w i l l  by  l a t t e r a  elements  As example.  are  but  e r r o r  equation  by  the  e i g e n v e c t o r s  a s s o c i a t e d  w i t h  are a  (22).  Z i=l  Let  we  c o n s i d e r  the  a l s o  |c. |  s c a l e  the  an  i f  are  c.  the  now  complex,  case  complex  then  the  where c o n s t a n t s , f i r s t  and  p r i n c i p a l  be:  p r i n c i p a l of  given  where  equation  complex  complex  r e a l ,  t r u n c a t i o n  2.11a,  but  s(t)  1>y(t) = ( This  be  S i g n a l s .  S e c t i o n  i  s t i l l  the  s t i l l  i n  and  x.(t)  w i l l  2  ;  shows  ,  /  that  f a c t o r  can  component  f i r s t  row  = s(t)  (30)  complex be  and  of  i l l u s t r a t i o n ,  x^(t)  s(t)  2  s i g n a l s  represented  a s s o c i a t e d  which  e x a c t l y  weights  d i f f e r by  a  only s i n g l e  (the  complex  f o l l o w i n g  simple  s(t).  energy  U).  we  c o n s i d e r  and  x (t) 2  the  = exp(-ie)  The  42  in  the  s i g n a l i s :  | | s ||  where  '*'  = J  2  s(t)  denotes  the  r  r = I  U t )  complex  of  m a t r i x  r  2  a r e  X,=2||s|| ,  exp(-i  1  e)  f i r s t  p r i n c i p a l  =0.  The  u n i t a r y  =  u^j  a r e  x  the  e)  component i s ;  a , , x (t)  tyy(l)  Phase  Let  + M  elements  2  1  of  x (t)  = /2  2  u,  ,  the  us  now  c o n s i d e r  =  w(t)  ,  and  x  =  w(t;  e)  .  x  ( t )  the  two  s(t)  f i r s t  R e c o v e r y .  x (i )  2  2  i  U.  d.  X  l/}/2 exp(i  where  e)  i s :  r  The  e)  \s  e i g e n v e c t o r  =  Thus:  i  e i g e n v a l u e s  U  c o n j u g a t e .  exp(-i  1  e xp(i  The  dt  0  s i g n a l s :  (31)  column  e i g e n v e c t o r  of  43  Their  corresponding  x\(t  )  w(t  =  a n a l y t i c  s i g n a l s  are:  )  x2(t)=w(l)exp(ie)  and  hence  these  s i g n a l s  preceding  example.  produce  complex  a  Furthermore,  equation  i n t o  are  l i k e  A p p l i c a t i o n f i r s t  those  of  the  p r i n c i p a l  s u b s t i t u t i o n  of  (31)  that  demands  the  considered  CKL  the  t r a n s f o r m a t i o n  w i l l  phase  shows  that  d i r e c t l y  e  e.  The  the  from  component. =  phase the  That  tan"  1  Complex  of  c o n s i d e r  a  x(t) Then:  e i g e n v e c t o r  2  KL  u.  c o n s t a n t s  a  s i g n a l  <  the  In  other  second  a s s o c i a t e d  and  x2(t)  . are  such  that  y  words, s i g n a l  w i t h  the  equation i s  (31)  r e c o v e r a b l e  f i r s t  p r i n c i p a l  (32)  Transformation  wavelet  by  i n  x\(t)  ,)  s t a t i c s ' .  accommodated  c a n c e l l e d .  r o t a t i o n  seismic  the  ' r e s i d u a l  i s  w(t).  i s :  (u  Adjacent s h i f t s  e  s h i f t  to  for  1  the  equal  component  e x p r e s s i o n s  the  i n  s i g n a l s but  a l s o  I f  these  phase  x(t)  >  and  s h i f t and  X(f)  Time  d i f f e r because  of  time the  S h i f t s .  not of  only  time  s h i f t s  s h i f t s ,  are  waveform.  i t s F o u r i e r  because  transform  small To  phase  that they  see  A'(f).  of  can  t h i s I f :  i s , be we  44  x(t-t0)  e  where  =  A l t h o u g h as  a  2i:ft0r  =  2  ^fc f  an  r o u t i n e ) between c e n t r e  However, the  the  the  time  phase  s u f f i c i e n t l y  s h i f t .  s h i f t band  cannot  l i m i t e d  t r u l y  be  s i g n a l  can  be  n e g l e c t e d  x(t-l0)  can  be  w r i t t e n  (29),  regarded  w i t h  dependence  w i t h KL  minimize  angle  the  s t a t i c  (ergo  over  5,  15,  which  p r o g r e s s i v e l y ,  l i n e a r i t y  so  the  Each  the as  center  and  using  x(t;e)  as  of  35,  The  i s  t h i s  a  phase  s h i f t  by  ( i n f o r  a  and  s i n c e more  As  s t a t i c  a  55  the  between  the  s e r i e s  of  complex  KL  a the  p o i n t s )  d i f f e r e n t  wavelet  shown Hz,  are  l i n e  decreases  waveform  the  i n c r e a s e s .  not  i s  then  e f f e c t .  d i a g n o s t i c  waveforms We  for  r e s p e c t i v e l y .  pronounced i s  time  wavelet  sample  s t r a i g h t  r e l a t i o n s h i p  r o t a t i o n .  see  f i g u r e s  45,  e x p e c t e d , has  i s  a  m i s f i t  we  t0  by  d i s p l a c e d  ( r e t u r n e d  p l o t  l o c u s  s h i f t  phase  the  25,  e  2.2,  s h i f t  b a n d w i d t h ) .  s t a t i c  of  F i g u r e  time  w a v e l e t s .  the  l e a s t - s q u a r e s  r o t a t i o n  the  u s e f u l n e s s  the  o f f s e t  r o t a t e  phase  f r e q e n c i e s ,  the  w i l l  In  f r e q u e n c i e s :  and  r o u t i n e  s i g n a l s  s i g n a l s .  R i c k e r  s h i f t  two  two  versus  higher  narrower  the  frequency  to  frequency  time  the  o  of  two  c e n t r e  that a  complex  of  i s  e) ,  as:  attempt  p a r t  t0  presented  the  expd  equation  of  e  p l o t s  for  the  r e d e f i n e d  s h i f t ,  r e a l  f o r  e  When  The  ,  n o t a t i o n  with  in  shows  c o n s t a n t , /  X(f)  and  t h i s  frequency the  >  <-  are  of  s h i f t  maximizing  45  the  commonality  t h i s  does  not  l i n e a r i t y  does w i l l  at  a c t u a l  present  t h e  when  there  Up  not  f i r s t  a  stacked versus down  t o about  Here  we  (3)  t h e  see  5  (1) t h e  three  d i f f e r e n t  seen  v i s u a l l y s h i f t e d  and  a r e  from  s h i f t s  wavelets  as  and  from  a r e very  l i k e l y  t h e percentage w i l l  we  look  of  energy  equal  100%  f a l l - o f f  t o  of  equation  by  a r e small  a c t u a l  wavelet  w e l l  t h i s ,  when  t o  50%  enough  t o  t h e s t r a i g h t  u n i f o r m a l l y waveforms  s i m i l a r  stack  x(l)  t o  frequency  wavelet,  and  r o t a t i o n  by  As  e, f o r can  t h e  than  be  r o t a t e d  r e f e r e n c e  c o n t r u c t i v e l y  e  2.3.  v a l u e s ,  t h e  of  i n F i g u r e  f r e q u e n c i e s .  t h e l a r g e  good  segments  s h i f t e d  phase  a  c e n t r e  a r e shown  (2) a f t e r  below, a l l  produce l i n e  w i t h  (2) t h e time  c e n t r e  (3 3)  p u l s e ,  a r e  t h e i r  I  found  c o u n t e r p a r t s .  For  t h e  e m p i r i c a l l y approximation  0  as  more  unrotated  p r e s c r i b e d  r e f e r e n c e ,  time  a t  and  t h e  wavelet,  a s c e r t a i n  T h i s  but  Thus,  To  a r e c o i n c i d e n t ,  changes  The  or  s i m i l a r .  t h e s h i f t e d  component.  t h e slope  2.2)  Hz.  s h i f t e d  time  s h i f t  and  look  wavelets,  a ta l l .  d i s t o r t i o n s  ( F i g u r e  0  t  o v e r l a p  r e s u l t , t  waveforms,  waveforms  time  rotated-wavelet  them  t h e complex  whether  p r i n c i p a l  i d e n t i c a l  t o  make  of  t h e r e f e r e n c e .  r o t a t e d  i s no  p a r t s  i n d i c a t e  resemble  i n t h e  when  t h e r e a l  n e c e s s a r i l y  r o t a t e d , t h e  of  <  R i c k e r t h a t  t h e  was  v a l i d  (0.4//  )  wavelets phase over  seconds  of s h i f t  time  c e n t r e  frequency  versus  s h i f t s  given  s t a t i c  / time  s h i f t  by:  (33)  46  and  that  t h e r e l a t i o n s h i p  r o t a t i o n  e  Equal  p e r m i l l i s e c o n d  =  ms  i on  33  wavelet. is  0.006  of  simple lobe  i n t h e f i r s t  should  be ^  E i g e n v e c t o r s .  Complex  here  s t r u c t u r e  e i g e n v e c t o r a s  lobe  t o t h e shape  breadth  o f a  elements  o f a  R i c k e r  A.,= i k  t h e complex  n Z  o f <  2.0  g e n e r a t e d  a s  case  case  we  r e f l e c t e d  ( b y which  by  R i c k e r  p u l s e ,  b,  I  keep noted  equation  90 %  that  t h e  (33).  (27)  t o deal  by t h ecomplex would  be  at an{1m(r  KL  s e e i n g  i n t h e 'phase'  I mean  85 -  This  r a d i a n s .  o f equation  an e x t e n s i o n  I n t h i s  which  component,  a s e x e m p l i f i e d  s h i f t s  i n t h eamplitudes  For  r o t a t i o n s ,  p r i n c i p a l  e i g e n v e c t o r s  e t a l . , 1983).  phase  where  phase  i n t r o d u c e  complex  w e l l  r e l a t e d  phase  b/2,  f.  in  o r i g i n  a c c e p t a b l e  i n t o  (Levy  was:  t o s i d e  t r a n s l a t e s  the  phase  )  v i s u a l l y  s h i f t  I  a n d  c  t h e energy  time  frequency  b y :  b = y/6/(nf  f o r  s h i f t  c e n t r e  c  h a s a  The s i d e  given  and  /  between  .  w i t h  t r a n s f o r m  d i f f e r e n c e s  o f t h e complex  )/Re(r  .  ) } ) ,  a s  o f t h ep r o j e c t i o n s .  equation  (27)  becomes:  * j  '*' denotes  (a. . - a . . ) ij kj t h e complex  (34)  (a. . - a , . ) ij kj c o n j u g a t e ,  a n d  t h e  a's  a r e  now  47  complex,  i . e . :  a. . = r 1. . / i/X . tJ J J for  the  I I of  complex  u t i l i z e  review the  the  r  •  •  both  equations  approach  a p p l i c a b i l i t y  of of  (27) Hagen  c l u s t e r  (34)  and  (1982)  and  a n a l y s i s  i n  S e c t i o n  assess t o  the  s e i s m i c  4.1,  where  l i m i t a t i o n s d a t a .  48  FIGURE The the  c u m u l a t i v e 100  t r a c e s  v i r t u a l l y p r i n c i p a l  a l l  percentage of of  s e i s m i c the  components.  energy  data  s i g n a l  2.1 from  shown  i n  energy  i s  the  RKL  F i g u r e c o n t a i n e d  decomposition 3.5a. i n  the  Note f i r s t  of how 15  49  0  TIME SHIFT  °\  0  •  TIME SHIFT  FIGURE Eight p l minimize d i s p l a c e d wavelets 75 Hz, given i n  o t s each showing t h e l e a s t - s q u a r e s s i g n a l s . I n t h i s w i t h c e n t r e freque r e s p e c t i v e l y , sam sample p o i n t s .  t h e m i s f c a s ncy pled  ^  0  TIME SHIFT  <?  2.2  r e q u i s i t e phase i t between t h e r e , t h e s i g n a l s a 5, 15, 25, 35, a t 2 ms. The  s h i f t necessary t o e a l part of time r e p a i r s o f R i c k e r 45, 55, 65, and time s h i f t a x i s i s  50  16 ms  8 ms  4 ms  TV  0.0  0.0 0.1  0.0  32 ms  0.1  0.0  OJ  1 2 3  15Hz  Xfl):  0.0 0.1  0.1.  o.o.  0.0  (  1  ^£  2 3  —v—  Xd):  35Hz  —vI"  11  5  0.0 0.1  0.0 0.1  0.0 0.1  0.0 0.1.  0.0 0.1  0.1  I-  44  15  27  60  2 4 0  1 2 3  55Hz  Xd):  18  FIGURE Three  sets  of  w a v e l e t ,  (2)  wavelet  a f t e r  35,  and  each  time  below  phase  2.3  showing  s h i f t e d  each  r o t a t i o n  a r e  i , 0  (1)  w a v e l e t ,  by e, x(l), and a r e p a i r s of R i c k e r wavelets 55 H z , r e s p e c t i v e l y .  I n d i c a t e d s i g n a l s  the  p l o t s  the and  f o r e. w i t h  r e f e r e n c e (3)  t h r e e In  t h i s  c e n t r e  R i c k e r  the  s h i f t e d  time  s h i f t s .  case,  frequency  the 15,  51  CHAPTER DATA  SECTION  a.  I  t r a n s f o r m . w i l l  one  o t h e r .  The  o f  F u r t h e r m o r e , a  COHERENT  INFORMATION.  s e t of  s i m i l a r  component  p r i n c i p a l  components,  a f f o r d  incoherent t h e  t h e  t h a t  accuracy,  input  c o n t a i n s  when  present  more  ordered  a l s o  a l l o w s  from  a  i n  of  KL  there  energy  than  i n  decreasing  and  subsequent  t h e  input  r e c o n s t r u c t i o n  r e l a t i v e l y  t h e  t r a c e s ,  i d e n t i f i c a t i o n  energy  method  p r o g r e s s i n g  s e i s m i c  s e c t i o n s ,  presence  o f  sum  t h e most  time  d e c o m p o s i t i o n  method  f o r a  p r o p e r t y  d a t a .  o f  small  data  t o  number  of  components.  Before  of  energy-packing  p r i n c i p a l  d e s i r e d  p r i n c i p a l  t h e  that  c o n t e n t ,  e l i m i n a t i o n  w i t h i n  employ  R e c a l l  be  energy  the  OF  I n t r o d u c t i o n .  Here  any  RECONSTRUCTION.  RECONSTRUCTION  3.1;  3.  may  transformat  be i o n .  I  and  t o  examples  i n t r o d u c e phase  of  a  by  r e c o n s t r u c t i o n example  e i t h e r  o f  stacked  s t a c k i n g a  components  c o r r e c t e d  using  o f  r e p r e s e n t i n g  p r i n c i p a l  move-out  performed  an  s h i f t s  s i g n i f i c a n t  of  gather. t h e r e a l  i n t h e  stack  by  r e s u l t i n g T h i s or  a  from  s t a c k i n g complex  KL  52  b.  KL  Stack  A which  Versus  complex  KL  a r e random  t h e Mean  a n a l y s i s and  have  i s  l i k e  component  that  immediate  a p p l i c a t i o n  s t a t i c  s h i f t s  extent  f o r  U l r y c h  e t  component gather that  r e a l  a l . from  c a n be  a f t e r  which  a  i s  (Levy  a  KL  as  an  contaminated  The  same  Hemon  t o  there  a l l t h e  of  common i s a  step,  which  was  t h e  a b i l i t y  using  a  a  w i l l  be  t r a c e  l e a s t  by  p r i n c i p a l  t h i s  u s u a l l y  F o r  c o n s t r u c t i n g  t h e  e s t i m a t e  i s  t r a c e  i s  ensemble.  squares  CDP  a s s e r t i o n  unknown  t r a c e s :  t h e input  and  c o r r e c t e d  The  an  l e s s e r  (1978)  t h e f i r s t  moveout  an with  t o a  Mace  t h e stack.  seismic  t h e mean  which  o f  how  of  suggests  i s true  and  demonstrate  common  t h e f i r s t  of  t h e  s i g n a l .  f u r t h e r i s  a s s o c i a t e d  p a r t i c u l a r l y  with  s i g n a l s  c o r r e c t i o n ,  r e p r e s e n t a t i o n  f u l l y  p r i n c i p a l  s t a c k i n g  moveout  component,  f i r s t  s h i f t s  This  e s t i m a t e  that  a  time  s i g n a l .  d e c o m p o s i t i o n  used  have  having  t r u e  a n a l y s i s ;  KL  e s t i m a t e  are  t h e  s i g n a l s  w i l l  r e a l  we  p a p e r s ,  mean  both  KL  A  zero  s e t of  (1983)  e x t r a c t i n g  common  a  e t a l . , 1983).  by  p r i n c i p a l  of  i n  e s t i m a t e d case  S t a c k .  t o number  t h e l a r g e s t  i n t e r e s t i n g  account r e s i d u a l  with  small  n o t f u l l y  s t a t i c s  p r o b l e m s ) .  i n  t h e  r e c o n s t r u c t  t h e  of  components  p r i n c i p a l  e i g e n v a l u e s .  f o r cases  f o r t h e observed  e x p l o r e d  where  a  This phase  above  z e r o - o f f s e t which  p o s s i b i l i t y s h i f t  phenomenon  ( a s would  I n e f f e c t ,  I  replace  be  i s  cannot t h e  t h e  case mean  53  s t a c k :  x(t)  w i t h  a  =  sum  I  . e  have  t r u n c a t e d  c.  of  1  Data  t h e  i n F i g u r e  randomly  s h i f t e d  o t h e r  hand,  i n i t i a l  s i g n a l s  has  very  r e s u l t i n g reproduces  s t a c k ,  t h e  b e t t e r RKL,  and  equation  i n  (35)  with  i t s  (3).  time  provide  mean  ( o r phase)  a  This  i s i l l u s t r a t e d  i n t h e  3.1(a)  15  s i g n a l s  up  t o ± 20  ms.  been  computed  and  i t  seen  I n  phase  input  CKL  because  I  phase  t h e mean o f  of The  e x a c t l y , s t a c k .  s i g n a l s  CKL  that  been  stack  s i g n a l .  On  represent  The  stack t h e RKL  3.1(c)  a r e phase  t h e t h e  r e s u l t s  when  mean  stack e f f e c t s  of  course  stack  shows  of  3.1(a)  c a n c e l l a t i o n  w h i l e  F i g u r e  mean  a n a l a g o u s  s h i f t s .  have  (Figure  c l o s e l y  show  s h i f t s .  The  t h e i n i t i a l  because  s i g n a l  s t a c k s  o f  i s  more  3.1(b)  amplitude  t h e  of  s t a c k s  F i g u r e  only  than  CKL  f i r s t  o f t h e  amounts  and  t h e  r e p r e s e n t a t i o n  s t a c k .  r e p r e s e n t a t i o n  s h i f t s ,  b e t t e r  by  small from  small  3.1. I n F i g u r e  t h e RKL  d i f f e r  o f  t h e  poor  s i g n a l .  the  somewhat  has a  (36)  equation  from  should  than  t o be  components:  ij  x.(t)  replaced  component  s i g n a l s  r  p r i n c i p a l  Examples.  examples  bottom)  m Z  p r e s e n c e  s i g n a l  these  e n e r g e t i c  n Z  )  (35)  x.(i) i  r e c o n s t r u c t i o n  p r i n c i p a l t r u e  n Z i=l  )  simply  S y n t h e t i c  In  1  t h e most  = ( n -  x(t)  I  ( n -  f a r e s  t h e  mean  s h i f t e d  and  54  contaminated  In the  s u r p r i s e  in  s t a t i c s .  a l l cases,  input  stack  by  s i g n a l  us  since  one)  t h e  t h e KL than t h e CKL  a d d i t i o n a l  mean  components,  s t a c k . each  the  stack  s h i f t  so  each  r o t a t i o n o f f s e t ' )  t r a c e t h e  Unless of  assumption  that  we  of  t h e z e r o - o f f s e t  d.  R e c o n s t r u c t i o n  Moving  on  i s with  s p e c i f i e d ,  f o r a  o u t  throughout  S t a c k e d  t h e s i t u a t i o n  how  r e c o n s t r u c t i o n  with  f l a t  and  found  r e c o n s t r u c t i o n s  t h a t  u n d u l a t i o n s  l y i n g  i n t e r e s t e d  u t i l i z e d  a l s o ,  t h e  p r i n c i p a l phase  i n t h e r e a l KL  part  i n t h e  i s  a n a l y s i s ,  t h e uppermost work  of f o r  constant  t h i s  RKL  t h e degree  complex  of  t h e  ('near  under  t h e  r e c o n s t r u c t i o n  Data.  demonstrate  d i p s  a  not  t r a c e .  c o n s i d e r  small  o f  by  t h e phase  t o  of  p r i n c i p a l  s i g n a l s ;  c o n s t r u c t i o n  t o  (and  t h e  a c c o r d i n g input  should  a r e not  of  energy  I  I t was  which  r o t a t e d  This  two  c o r r e l a t e d  s e c t i o n s ,  e v e n t s .  freedom  t h e other  a r e g e n e r a l l y  r e p r e s e n t a t i o n  s t a c k .  possess  weighted  i s f u r t h e r t o t a l  b e t t e r  c o n s t r u c t i o n  t h e  c a r r i e d  o f  t o  t h e  s i g n a l s  was  of  a  mean  a l g o r i t h m  o t h e r w i s e  t h e  t r a c e s  t h e  before  that  i n c r e a s e d .  does  degrees  t r a c e  i t e x h i b i t s  components,  y i e l d s  I n  c o r r e l a t i o n CKL  stack  phase  shown  t h e  where  f o l l o w i n g i s  t h e CKL  able  s h i f t e d  have  seismic  examples  segregate  tended  t o  d i f f e r e n t  t o o b l i t e r a t e  t h e  r e p l a c i n g  them  s e c t i o n s ,  waveforms.  Chapter  stacked  s y n t h e t i c t o  method  i n seismic  i n t h i s  we  C o n s e q u e n t l y ,  a r e d e r i v e d  from  t h e  a l l RKL  55  transform  It f i g u r e s  which  should I  i n a l l  S y n t h e t i c  s h a l l o w l y  motion,  or  (Larner  e t  introduced  as  a  a  added  g e o l o g i c a l h o r i z o n s s e c t i o n ,  r o l l ,  l a r g e s t  value;  a l l  t h e  hence  s e t t o  f o l l o w i n g  t h e  maximum  u n i t y .  model:  i n t o  a  o f f s e t  by  a  shows  t h e contaminated  above  ( i . e . t h e sum  r e c o n s t r u c t i o n  o f  of was  2a  and  t h e  marine  t r a c e  ms  seismic  c a b l e  s e c t i o n  i n t r o d u c e d  t o t h e  a  w a v e l e t s  h o r i z o n s , 10.th  and  t r a c e .  a r e expressed  I  here  a m p l i t u d e ) .  r e p r e s e n t a t i o n  f a u l t .  i n  F i g u r e  p l u s  s y n t h e t i c  3.2b  seismic  F i g u r e  data:  t h e  o f  t h e  s h a l l o w l y  random  2 b ) . I n  t h e r e c o n s t r u c t i o n  I  t h e  l e v e l s  w a v e l e t s  events  a t  coherent  streamer  g e o l o g i c a l 36  t h e s e i s m i c  i . e . s t e e p l y - d i p p i n g  by  r e p r e s e n t i n g  d i p p i n g  radians)  t h e  ( a l l noise  v e r t i c a l  s t e e p l y  stacked  ir/3  o f f s e t  s h i f t e d  t r a c e s  c o n s t r u c t e d . t o  0  of  noise  shows  3.2c  f i n a l  were  (from  f a u l t  phase  on  i n t r o d u c e d  t h e maximum  3.2a  seismic  those  r e p r e s e n t i n g  white o f  s y n t h e t i c  1983)  v e r t i c a l  r e c o n s t r u c t i o n  p l o t t i n g  superimposed  change  events  percentage  F i g u r e  a s  a l . ,  10%  i n  i s a r b i t a r i l y  four  such  phase  t h e  t h e  events  ground  p r o g r e s s i v e  t o  that  r e s u l t s .  Examples.  d i p p i n g  events,  then  noted  Twenty  'noise'  e x c e l l e n t  f i g u r e s  Data  1.  across  be  normalize  amplitude  e.  produced  3.2d  n o i s e .  we  d e f i n e d  'noise' F i g u r e  d e s c r i b e d  see  c r i t e r i o n  energy,  d i p p i n g  i s t h e  s e c t i o n  b a s i c  a  75%  governing i n  equation  56  We  (23).  note  c h a r a c t e r  of  that the  r e p r e s e n t a t i o n  the  events  model  of  the  have  which  been  d e v i a t e  s e v e r e l y  u n d e r l y i n g  from  the  f l a t - l y i n g  a t t e n u a t e d ,  ' g e o l o g i c a l '  l e a v i n g  s t r u c t u r e  t h e  b a s i c a l l y  i n t a c t .  has  The  phase  c h a r a c t e r  of  the  s i g n a l s  has  the  d i s t i n c t i v e n e s s  of  the  ' f a u l t '  edge.  i n c r e a s e T h i s  i n  i s  a  components noise  the  r e s i d u a l which  and  the  corresponding The  f i r s t  events, data show  a  form  5  and  of  few  2.  combine  (shaded  waveforms  would  have  a  second  a l s o  that  the  t h i s  example.  In  n o i s y  s e c t i o n the  model  the  noise f a u l t  the  energy.  both  a r e  seen  weights  f i g u r e )  the  input  the  components  F i g u r e of  t o  3.2e.  f l a t  l y i n g  produce  which d a t a .  12)  combines  t o  The  (6  the -  p r i n c i p a l  otherwise  i s s i m i l a r f i r s t  i s between  and  i s  d i p p i n g  components  the  3.2a.  the  i n  an  only  the  amplitude.  t o  n o i s e ,  note  p r i n c i p a l  p r i n c i p a l  p r i n c i p a l  b,  f i l t e r i n g  As  24  as  F i g u r e  c h a r a c t e r i s t i c  c o n s i d e r  3.3a  t o  we  d i s c a r d i n g  f o r p l o t t i n g ,  l o c a t i o n  bandpass  from  t o t a l  I  added  F i g u r e  a f t e r  i n  d i s c e r n a b l e  example  white  few  of  normalized  Note  of  next  been  The  The  3.2c  a r e  by from  a p p r o p r i a t e  have  50%  72%  F i g u r e  the  The  p a t t e r n  w i t h  to  waveforms. i n  w i t h  about  preserved,  However,  comparison  brought  components  3.2d.  but  i s o l a t e  data  p r i n c i p a l  i n  c o n t r i b u t i o n s  l y i n g  the  d i p p i n g  components  e f f e c t  f l a t  t o  sawtooth  noise  contained  F i g u r e  the  f i r s t  background  been  I  we (0  see -  7  and  the 50  i n  the  f i r s t ,  l a s t  8  7  and  t r a c e s  s e i s m i c  H z ) .  r e c o n s t r u c t e d seen  t o  8  data  Attempting  the  F i g u r e  t r a c e s . i n and t o  seismograms 3.3c,  t h i s  57  attempt  has  been  c o n s i d e r a b l y been  w e l l  reasonably  reduced,  p r e s e r v e d .  smeared  over  3.  I f  used  For  adjacent  the  s t e e p l y - d i p p i n g have  and  the  the the  p a r a l l e l  'slant  KL'  favour  those  with  the  A  i n  data  set  s i m i l a r  c o n s t r u c t e d ,  but  the  i n c r e a s e d  by  a  we  see  s e c t i o n ;  the  s e c t i o n  a f t e r  a  d i p  r e c o n s t r u c t i o n the  8  and  d i p p i n g  n o i s e  f.  Data  Real  The  ms  f a u l t events  the  d e f i n i t i o n have  been  are  c,  w e l l  s e v e r e l y  I  could  ( S e c t i o n events  i n This  ( 5 ) ) .  example  same d,  1  was  h o r i z o n s procedure  and  e,  the  n o i s e ' ;  been  the  and  r e - i n s t a t e d . preserved,  were as  seismic  removed;  s t r u c t u r e ,  d i p  more  then  d i p p i n g  i n  has  u n d e r l y i n g the  not  e a r l i e r  'streamer  t r a c e  w i t h  i s  edge  beds,  the  b,  and  have  ' g e o l o g i c a l '  3.4a,  n o i s e  edge  example.  used  F o l l o w i n g  per  data  data  the  75%  f i n a l l y , The  phase  whereas  the  a t t e n u a t e d .  Examples.  f o l l o w i n g  r e c o n s t r u c t i o n  p l u s  f a u l t  (equation  d i p  the  been  comprised  of  f o l l o w i n g  on  5.  had  d e s c r i b e d  that  F i g u r e s  d e p i c t i n g  r e c o n s t r u c t e d  c h a r a c t e r  of  i n  of  to  has  d i s t i n c t .  s e g r e g a t i o n  the  d i p s  f a c t o r  b e f o r e ,  the  l e v e l  f a u l t  s u b - p a r a l l e l )  s p e c i f i e d  demonstrated  and  the  remains  procedure  b i a s  noise  c h a r a c t e r  model  (or  w i l l  i s  but  the  p a r t ,  g e o l o g i c a l  T h i s  procedure  phase most  t r a c e s  2 . I I I c ) . of  s u c c e s s f u l :  to  f i g u r e s  enhance  e x e m p l i f y  coherency  i n  r e a l  the  a b i l i t y  stacked  d a t a .  of In  KL a l l  58  r e c o n s t r u c t i o n s background has a  been  n o i s e  w i l l  l e v e l  somewhat  background  a s s o c i a t e d  geology,  of  and  the  as  are  r e a l  c o n c e r n i n g or  been  the  eye  the  l o n g e r  coherency  d i s t r a c t e d  by  requirements  e n e r g i e s  were  l a t e r .  s u p p l i e d  s p a c i n g ,  p o t e n t i a l of  u n c o r r e l a t e d  s t o r a g e  examples  t r a c e  d e s c r i p t i o n  no  d i f f e r e n t  mentioned  the  the  T r a c e - t o - t r a c e  The  of  data  how  i s  energy.  hydrocarbon  Consequently,  see  reduced.  r e c o n s t r u c t i o n s  reduced  i n f o r m a t i o n  has  i n c o h e r e n t  w i t h  most  immediately  enhanced,  of  c o n s i d e r a b l y  In  we  the  to  me,  s e c t i o n  no  l o c a t i o n ,  was  made  a v a i l a b l e .  data  p r e s e n t e d  here  i s  s k e t c h y .  For  p r o c e s s i n g  o v e r l a p p i n g  t r a c e  seismic  and  time  the  u n d e r l y i n g  F i r s t  f l a t  d i p p i n g )  e v e n t s ,  an  at  windows  (eg.  we  look  the  c o m p u t a t i o n a l  u s i n g  SVD  (Strang  s m a l l  i s  segments  i n t o  of  CPU  r e c o n s t r u c t i o n l a r g e  more  of  530  o v e r l a p p i n g  minutes  s e c t i o n s  The  t r a c e s  segments  time  on  took of  than  data  a a  of of  by  we  do  d a t a , 85  100 each t r a c e s  I  data  i n t o  necessary  f o r  two  that  by  we  have  breaks 1  of  not  Elmer  minutes.  t h i s  i s  the  s e c o n d ) .  the  to  t r a c e s . of by  500  3220. r e f e r  to  m a t r i x  of  t r a c e s  decompose  For  p o i n t s ,  The  example, broken took  70  c o r r e s p o n d i n g  r e c o n s t r u c t i o n  o v e r l a p p i n g t e c h n i q u e  i s  a  p o i n t s ,  150  u n l e s s  Second  of  number  d e s i r e  p a r a l l e l  down  d e c o m p o s i t i o n  cube  about  P e r k i n few  f o r  the  break  which  t r a c e s  time to  I  assumption  100  Consequently,  w i t h  d e c o m p o s i t i o n  expense.  T h i s  assumption  p r o p o r t i o n a l  ( 1 9 8 2 ) ) .  s e c t i o n s ,  segments.  r e a s o n s . (or  i s  l a r g e  as  of  compound  59  r e c o n s t r u c t i o n .  1. from  a  In  F i g u r e  3.5a,  c o n v e n t i o n a l l y  decomposed  i n t o  components  and  f i r s t  the  other  words,  components F i g u r e s  12  for to  -  1.5  data  I  need  the  for  85%  of  for  only  95%  12%  c o n s i s t s  of  100  t r a c e s  The  t r a c e s  were  the  f i r s t  f i v e  of  input the  of  the  energy,  energy.  the  the  show in  of  the  r e c o n s t r u c t  i n c r e a s e  which  window  components:  r e s p e c t i v e l y  Note  a  s e c t i o n .  components  p e r f e c t l y  3.5c  seconds,  s t a c k e d  accounted  p r i n c i p a l  and  r e c o n s t r u c t i o n s . 1.1  alone  t h i s  s e l e c t e d  p r i n c i p a l  almost  3.5b  have  p r o c e s s e d  t h e i r  p r i n c i p a l  I  p r i n c i p a l  input 95%  data.  and  coherency  in  i n t e r b e d d e d  In  85%  the  zone  sand-shale  sequences.  In  the  c o h e r e n c y ,  in  box the  For  at 95%  w i t h in  as  1.1  i s  l o o k i n g  examining  the  done  very  s c a l e  i n t a c t the  i n  g e n e r a l  increase  in  s m a l l - s c a l e  features  are  depicted  in  l e n s - l i k e  between  the  and  f e a t u r e s .  waveforms  the  small  However,  smooth  both  note time  for  The  we  same  seconds  r e c o n s t r u c t i o n .  remains  example.  the  example,  about  small  t h i s  at  r e c o n s t r u c t i o n .  components s e c t i o n  r e c o n s t r u c t i o n ,  while  p r e s e r v e d . the  95%  t r a c e s  85%  the  75,  s t r u c t u r e  f e a t u r e of  edges  the  this of  the  i s  more  e s p e c i a l l y  r e c o n s t r u c t i o n s ;  a l o n g  -  r e c o n s t r u c t i o n  l e n s - l i k e  phase  55  d i s c a r d i n g  c o n t i n u o u s ,  The  f e a t u r e  i s  enhanced p r i n c i p a l  leaves over  the areas  o b l i t e r a t e d  major  horizons  can  seen  be  s e c t i o n ,  by for  60  2.  The  second  c o n v e n t i o n a l l y decomposed  processed i n t o  components  f i r s t  p r i n c i p a l  only  about  30%  r e c o n s t r u c t 95%  and  In in  box  t h e  85%  account  data  we  see  a  I  r e f l e c t i o n  c o e f f i c i e n t ,  hydrocarbon  i n  background  noise  j u s t  second  above  p r e s e r v e d d e f i n i t i o n  3. 3.7  As  part  i d e n t i f i e d window my  both  of  95%  of  t h e  an  example  of by t h e  a  trough  of  i n t e r p r e t i v e  of  two  a t  3.6c  a  strong  of  a  need  show  t h e  centre  of  p o s i t i v e  l o c a l i z e d  h i g h  i n d i c a t i v e  of  same  box  b u t , by  i n t e r e s t ,  data  F i r s t l y ,  f e a t u r e  b i f u r c a t i o n  g e o l o g i c a l  over  i n Figure  f i g u r e .  The  of  B.  85%  i s  time  w e l l t h e  t h e  This we  r e f l e c t o r f e a t u r e  see  a  l o s s  i s of  winnowing.  we  i n c o r p o r a t i n g  r e c o n s t r u c t i o n t h e  I  p e r f e c t l y  f e a t u r e s .  t h e  of  compound  sketch  on  i n d i c a t i v e  of  and  t h e  reduced.  i s t h e  i n t e r p r e t e r s , and 95%  3.6b  12  energy; Here  a  were  f i r s t  almost  p o s i t i v e - n e g a t i v e  i n t h e  s e c t i o n  The  sometimes  r e c o n s t r u c t i o n ,  event,  t o  from  t r a c e s  t h e energy.  under  i s  i s g r e a t l y  noted  96  input  i s i n d i c a t i v e  and  seconds  i n t h e  a t t e n t i o n  This  feature  t h e of  r e c o n s t r u c t i o n s ;  l e v e l  1.7  of  components  c o n t r a s t  p o t e n t i a l .  p r e s e r v e d  The  sharp  i s a l s o  r e s p e c t i v e l y .  negative  This  The  s e t . Figures  focus  r e f l e c t i o n .  3.6a)  components.  f o r 95%  the p r i n c i p a l  example,  s e c t i o n .  f o r 85%  components  input  (Figure  p r i n c i p a l  r e c o n s t r u c t i o n s ,  t h i s A,  of  example  s t a c k e d  t h e i r  p r i n c i p a l 32  r e a l  3.8, of  see a  t h e  b u r i e d  Figure c a l d e r a ,  c o r r e s p o n d i n g  t h e data,  c r a t e r ,  i n  as  seen  w e l l i n  as t h e  61  sketch,  spans  o r i g i n a l  d a t a .  incoherent f e a t u r e s d r a p i n g more  over  of  This  seen  normal i s  block  which  hydrocarbons,  and  i n t e r p r e t a t i o n  of  c o n s i d e r  To an  t h i s  t r a c e s ,  and  the  to  accommodate  1.0  s  in  the  much is  of the  the  t h i s  small  i n d i c a t i v e  i s  p r e s e r v e d  the  85%  i n  of  of  A l s o  c e n t r a l  a  the  and  the the  the  are  of  c r a t e r ,  i n t e r e s t of  the  impact.  may  i s  the  s e c t i o n .  I t  p o t e n t i a l  p r e s e n t a t i o n  the  d e m o n s t r a t i n g  part  m e t e o r i t e  the  to  of The  (a)  i s  such  t r a p  a for  enable  a the  r e c o n s t r u c t i o n .  A  v e l o c i t y 95%  c o v a r i a n c e p a i r  the  85%  b e t t e r  i n  anomaly  Figure  F i g u r e  0.1  m a t r i x  of  s  the  l o s s  in  of  a s s o c i a t e d  with  has  The  the  seen  This  but  I  96  a d j u s t e d  A.  r e c o n s t r u c t i o n ,  box  3.6.  was  r e c o n s t r u c t i o n of  3.9  over  a r r i v a l s  markedly  example  event  i n  i n  about  c l a r i f i e d  f u r t h e r type  shown  l e f t  strong  are  p r o c e d u r e ,  data  the  the  However,  d e t a i l .  of  d r a s t i c a l l y ,  edge  as  in  r e c o n s t r u c t i o n ,  s t r u c t u r e  s l a n t - K L  d i p  d i p .  ' p u l l - u p '  perhaps well  the  s e c t i o n  data  (b).  o f f  v i s i b l e  d a t a .  computation  f i n e r  the  enhanced  s e c t i o n  input  r e c o n s t r u c t i o n  to  t h i s the  compound  the  the  i n t e r e s t  e n l a r g e d  in  i n  of  demonstrate  events  over  b a r e l y  r e c o n s t r u c t i o n .  r e l a t e d i s  95%  f i n e - s c a l e  s t r a t a the  a  i s  f a l l e n  f a u l t i n g  p r o b a b l y  s t r u c t u r e  4.  i n  has the  sedimentary  and  a f t e r  l e v e l  i n t e r e s t ,  c l e a r l y  severe  t r a c e s ,  However,  noise of  80  above  the (c)  95% l o s e s  r e s o l u t i o n  been  f e a t u r e , a  r e e f , l o s t  i n  62  g.  Data  Compression.  As  a  f o l l o w i n g how  the  subset  c o r o l l a r y important  data of  storage  being for  be  p r i n c i p a l  t r a n s m i s s i o n  that  we  space  ( t h i s  be  a  In  the  p o i n t  data  the  h.  o r i g i n a l  of  computer  examples  i n  F i g u r e  energy  i t  i s of  p r i n c i p a l  when  data  of  l i n e a r  components  a u x i l i a r y to  our  2.1,  that  r e q u i r e d  concern  a s s o c i a t e d  noted the  90%  the  small  the  p r o c e s s e d  any  f i n d storage  shows  the  themselves  the  p r i n c i p a l  p r o c e s s i n g  c o n v o l u t i o n a l  are  I  r e q u i r e d which  data  machine  date,  w i t h  saw  a  w i l l  an  the  we  as  to  e x e m p l i f i e d  c o n d i t i o n s the  i s  -  F u r t h e r ,  on  p o i n t  note  from  words,  reduced,  70  components).  performed  other  I  c o n s i d e r e d ,  r e c o n s t r u c t e d  w i t h  percentage  the  In  main-frame  cumulative  assumes  examples  p e r f e c t l y  l a t t e r  d i s p e n s e i s  r e c o n s t r u c t i o n ,  d r a s t i c a l l y  This  from  t y p i c a l l y  a l l the  components.  speed.  p r o c e s s i n g .  In  almost  w i l l  shunted  p a r s i m o n i o u s  p o i n t .  c o u l d  space  to  model  rather  which may  be  than  on  i n t r o d u c e  the  s e c t i o n .  D i s c u s s i o n .  The  main  a p p l i c a t i o n for  image  for  the  to  moveout  of  t h r u s t the  KLT  enhancement  f i r s t  of  t i m e ) .  to of  The  c o r r e c t e d  t h i s the  s e c t i o n  recovery  stacked s t a c k i n g gathers  has  been  of  coherent  seismic of by  s e c t i o n s  s e i s m i c Hemon  to  and  data Mace  i n f o r m a t i o n  (presented ( f i r s t (1978)  here  a p p l i e d for  the  63  r e a l an  KLT  a l o n e )  i n t r o d u c t i o n  was  mentioned  t o t h e  f o r t h e RKL  s i g n a l  e x t r a c t i o n  and CKL  t r a n s f o r m s ,  c a p a b i l i t i e s  as  of  t h e  proved  very  method.  The  s t a c k e d  s u c c e s s f u l in  an  i t s  s o  f a r ,  i n d u s t r i a l a b i l i t y  n o i s e ,  r e c o n s t r u c t i o n i n many  storage data  i s a  of  seem energy  of  of  below  a p p l i e d  of  I t smain  a t e a r l i e r  s u i t a b l e t h e data  give 90%  I  has  e x t e n s i v e l y  t h e background  t o  have  r e a s o n a b l e r e s u l t s  l e v e l  times  of i n  i n t e r m e d i a t e t h e t r a n s f o r m  r e s u l t s . of  p r e s e n t e d  i s i n  incoherent a  seismic  90  -  Dropping  here..  i s 95% t h e  r e s o l u t i o n ,  p r o c e s s i n g i s a l s o  data  energy  examined,  i n l o s s  r e a l  advantage  r e c o n s t r u c t i o n  r e c o n s t r u c t i o n s  during  aspect  technique  environment.  a  t o  t h e 85%  problem  compression  been  reduce  f o r most  r e c o n s t r u c t i o n s  seen  has  p r e v a l e n t  c h o i c e  b u t  r e c o n s t r u c t i o n  p r o c e s s i n g  i s often  The  s u b j e c t i v e ,  and  t o g r e a t l y  which  s e c t i o n .  s e c t i o n  as  When  d i s c  s t a g e s ,  t h e  v a l u a b l e .  64  FIGURE (a) ampl (1) used (b) are:  F i f t e e n R i itude) ti t h e mean, one p r i n c The same R (1) t h e m  (c) The s h i f t s , (2) Note  wavelets AND up  RKL,  and  how  reproduce w e l l  c k e r me s h (2) R i p a l i c k e r ean,  with t o  (3) CKL  i n  3.1  w a v e l e t s , p l u s noise (up t o i f t e d by u p t o ± 20 ms. The KL, and (3) CKL s t a c k s ( t h e component). wavelets w i t h phase s h i f t s (2) RKL, and (3) CKL s t a c k s <100% n o i s e , and ±rr phase s h i f t s .  40% o f t h e maximum lower panel shows: RKL and CKL s t a c k s of  ±TT. B e l o w  s t a c k s  a l lt h r e e  examples  t h e CKL  stack  wavelet,  and  t h e RKL  l a r g e  t o  up t o ±20 ms s t a t i c Below a r e : (1) t h e mean,  t h e u n d e r l y i n g  u n l e s s  up  phase  s h i f t s  a r e  how  i n v o l v e d .  i s b e t t e r stack  a b l e  t o  performs  65  0-5  0  c  0-5  o  d FIGURE  o-5  e  e 3.2  (a) Seismic r e p r e s e n t a t i o n of the g e o l o g i c a l model: phase s h i f t e d w a v e l e t s i n s h a l l o w l y d i p p i n g h o r i z o n s (up t o 2 ms per t r a c e d i p ) , o f f s e t by a v e r t i c a l f a u l t a t t h e 10th t r a c e . (b) The "noise" s e c t i o n : s t e e p l y d i p p i n g coherent events (dips between 16 and 24 ms per t r a c e , such as could be p r o d u c e d by marine streamer noise of ground r o l l ) and 10% random n o i s e . (c) A sum of the previous two d a t a s e t s : t h i s i s the input f o r the p r o c e s s i n g . (d) A 75% r e c o n s t r u c t i o n of 2c, r e q u i r i n g the f i r s t 5 o f t h e 24 p r i n c i p a l components. Note the p r e s e r v a t i o n of phase i n f o r m a t i o n , and the c l a r i t y of the f a u l t edge. (e) The 24 p r i n c i p a l c o m p o n e n t s c o r r e s p o n d i n g t o 2 c . The f i r s t 5 p r i n c i p a l components a r e dominated by e n e r g e t i c peaks c h a r a c t e r i s t i c of f l a t l y i n g s t r u c t u r e . Note the smaller a m p l i t u d e peaks i n the band of p r i n c i p a l components 6 - 1 7 (shaded i n the f i g u r e ) . I t i s these waveforms which combine t o form the s t e e p l y d i p p i n g e v e n t s .  66  FIGURE 3.3 (a) S y n t h e t i c data s i m i l a r t o v e r t i c a l f a u l t between t r a c e s (10 - 50 H z ) . (b) and  The l a s t  8  7  that and  of F i g u r e 3.2a, 8, a f t e r bandpass  data a f t e r a d d i t i o n of 50% random t r a c e s , and bandpass f i l t e r i n g as  noise t o above.  the  w i t h a f i l t e r i n g f i r s t  7  (c) A 72% r e c o n s t r u c t i o n r e q u i r i n g 4 of the 24 p r i n c i p a l components. Note the marked r e d u c t i o n i n the background noise l e v e l and the p r e s e r v a t i o n of the e s s e n t i a l f e a t u r e s .  67  FIGURE  3.4  (a) S t e e p l y d i p p i n g (between 7 and 1 0 ms p e r phase s h i f t e d events w i t h a v e r t i c a l f a u l t . (b) The d a t a p l (c) The f l a t t 8 ms per t r a c e (d) A 75% r e components. (e) The the phase whereas The b a c k g omitted d e s i r e d s  t r a c e )  sub-parallel  u s the d i p p i n g "noise" events of F i g u r e 3.2b. e n e d contaminated d a t a , i . e . 3.4b a f t e r a d i p of has been removed. c o n s t r u c t i o n r e q u i r i n g 5 of the 24 p r i n c i p a l  r e c o n s t r u c t i o n c h a r a c t e r and the d i p p i n g no round n o i s e l e v some p r i n c i p a l i g n a l and the d  a f t f a ise e l c i p p  e r the d i p has been r e i n s t a t e d . Again, u l t d e f i n i t i o n are w e l l preserved, events have been s e v e r e l y a t t e n u a t e d . i n p l a c e s has i n c r e a s e d as we have omponents which contained both the i n g noise events.  68  IIIIIIJIl|lllllltll|lllllllll|lllllllll|lllllllll|lh  lllllllll|lllllllll|llllllHl|lllllllll|lllllllll|lllllllll|llllHHl|m  FIGURE  One hundred s e c t i o n . Note s e c t i o n (2.1 the box at 1.1  3.5a  t r a c e s from a c o n v e n t i o n a l l y p r o c e s s e d stacked the d i s c o n t i n u o u s events i n the lower part of the 2.7 s e c o n d s ) and the small l e n s - l i k e f e a t u r e i n s.  69  o  .  —  J  J  r  \  J  |<»MtMM|lMIMlM|MM  j J  m  ^ J  m  <  J  o J  r  J  c  o J  a  >  o  «  *  «  M  <  o  *  w  t  o  f  *  -  n j ( M ( \ i i \ i ( M i \ i ( \ i i M  J  [lltlHlll|«»l M* III |l IH IH »l |tH*t***l | H1 »l 111 > | >* 111 *l 1 M > 111 * 11111»ltt**4  tllllllll[lllllllll|lllHlin|llHMin|lH»Mll|lllllllll|lllllllll|llllMMl|lllllllll|lllllllll|  11111111111111111111111 III 11 l|l II111H11  |lllllllll|lllllllll|lllllllll|lHlllln|lllllllll|llHMIll|  FIGURE 3 . 5 b The  95%  c o m p o n e n t s . e n e r g y the  r e c o n s t r u c t i o n I n  h a s been  l e n s  t h i s  r e d u c e d  h i g h l i g h t e d  r e q u i r i n g  r e c o n s t r u c t i o n , g r e a t l y ,  i n t h e box  a n d a r e  12  o f  t h e  t h e b a c k g r o u n d s m a l l  s c a l e  p r e s e r v e d .  100 o f  p r i n c i p a l i n c o h e r e n t  f e a t u r e s  s u c h  a s  70  FIGURE 3.5c The  85%  r e c o n s t r u c t i o n  components. energy  has  such  as  the  because  by  f l a t t e s t  In  t h i s  a l s o lens  been  reduced  h i g h l i g h t e d  t h i s  h o r i z o n s .  r e q u i r i n g  r e c o n s t r u c t i o n , i n  5  the  of  the  background  g r e a t l y ,  but  the  have  been  we  have  r e c o n s t r u c t i o n  box energy  small  100 of  p r i n c i p a l i n c o h e r e n t  s c a l e  f e a t u r e s  o b l i t e r a t e d , l e f t  only  the  71  FIGURE Ninety  s i x  t r a c e s  s e c t i o n .  Note  the  strong  box  A,  r e f l e c t i o n  i n  from, and  a  negative the  3.6a  c o n v e n t i o n a l l y trough  b i f u r c a t i o n  of  p r o c e s s e d  below the  the  h o r i z o n  stacked prominent i n  box  B.  72  80  60  FIGURE The 95% components in box A s e v e r e l y 1.7 s c e n t  r e c o n s t r u c t i o n . In t h i s r e c o n s i s well p r e s e r attenuated. A l s r a l t o box B i s  U0  20  3.6b  r e q u i r i n g 32 of the 96 p r i n c i p a l t r u c t i o n the p o s i t i v e - n e g a t i v e c o n t r a s t v e d , while the background noise i s o , the b i f u r c a t i o n of the horizon a t w e l l p r e s e r v e d .  73  FIGURE 3.6c The 85% r e c o n s t r u c t i o n r e q components. A l s o i n t h i s r e c o n c o n t r a s t i n box A i s w e l l p r e is s e v e r e l y a t t e n u a t e d . However at 1.7 s . c e n t r a l t o box B has l  u s s , o  i r i n g 12 of the 96 p r i n c i p t r u c t i o n , the p o s i t i v e - n e g a t i e r v e d , while the background noi the b i f u r c a t i o n of the horiz s t d e f i n i t i o n .  a l v e se on  74  3.7  FIGURE (a) .  A  window  f e a t u r e , input  of  p i c k e d  d a t a ,  w i t h  (b) .  i s an  d a t a .  by  which  a s s o c i a t e d Below  140  t r a c e s  from  i n t e r p r e t e r s , i s n o i s y ,  the  impact  and  a  s e c t i o n  i s not e x h i b i t s  w i t h  a  c a l d e r a .  p a r t i c u l a r l y severe  c l e a r  f a u l t i n g  This i n  the  perhaps  f e a t u r e .  i n t e r p r e t i v e  sketch  based  on  the  r e c o n s t r u c t e d  220  200  180  220  200  180  1  1  _!  160  140  120  160  140  120  1  1  1  .  o.o  FIGURE 3.8 (a) . 95% f i l l  A  compound a r e  normal (b) .  window much  block  The  of  140  t r a c e s  r e c o n s t r u c t i o n c l e a r e r . f a u l t  sketch  edges  from  a  s e c t i o n  (see t e x t )  A l s o  p r e s e r v e d  c l e a r l y  i s reproduced  the  v i s i b l e  below.  with  a  c r a t e r  and  c a l d e r a . and  c l a r i f i e d  i n t h e  In  t h e  sedimentary  s e c t i o n .  a r e  t h e  76  FIGURE Ninety  s i x  t r a c e s  s e c t i o n .  Note  d i p p i n g  events  box  A.  the  from  a  background  above  1.0  3.9a  c o n v e n t i o n a l l y s.  noise  A l s o  note  p r o c e s s e d  stacked  surrounding  the  the  r e e f - l i k e  seismic p a i r  s t r u c t u r e  of i n  FIGURE S l a n t - K L c l a r i f i e d ,  95% and  3.9b  r e c o n s t r u c t i o n : the d i p p i n g events t h e r e e f - l i k e s t r u c t u r e has been l e f t  have i n t a c t .  been  FIGURE 3 . 9 c S l a n t - K L  85%  r e c o n s t r u c t i o n :  r e e f - l i k e  s t r u c t u r e  has  been  r e s o l u t i o n smeared-out.  h$s  been  l o s t ,  and  t h e  79  SECTION  a.  3 . I I :  RECONSTRUCTION.  I n t r o d u c t i o n .  equation  Using data  set  l e s s  o b v i o u s ,  b.  MISFIT  and  In  ( F i g u r e to  perhaps  I  used  the  5  c o n s t r u c t the  ( i . e . I  events  r e c o n s t r u c t i o n components before in  2.  The  i n  the  the  of  data  the  F i g u r e  been  coherent  (before i n  example  over  o m i t t e d .  n o r m a l i z a t i o n ,  F i g u r e  with  sum  would  of  to  the of  y i e l d  the  was  a b l e  from  In  3.2c  i n c l u d i n g 24).  A  the  F i g u r e  F i g u r e  and  sum  of  by the the  n o r m a l i z a t i o n )  3.2c.  more 3.10b  when A  6  a  example  events  p l o t t i n g  F i g u r e  I  s t r u c t u r e .  components  shown  n o i s e  energy,  of  of  data  t h e r e ,  s e c t i o n  p a r t  remain.  s y n t h e t i c  components;  the  3.4.  l e f t  the  seismic  3.10a  of  i s  of  common  which  g e o l o g i c a l  m i s f i t  and  which  f e a t u r e s  d i p p i n g  75%  t r u e  have  p l o t t i n g  F i g u r e  s i g n i f i c a n c e  p r i n c i p a l  3.2d  i s  shown  a  sum  e x a c t l y  same  of  f i r s t  F i g u r e s  The  and  p r i n c i p a l  s t e e p l y  I  y i e l d  appearance  r e c o n s t r u c t i o n  3.10a,  would  most  the  r e p r e s e n t a t i o n  i n  the  d i a g n o s t i c  s e i s m i c  data  the  d i s c a r d  Examples.  separate  remainder  may  Data  3.2d),  l e a v i n g - o u t  I  i n v e s t i g a t e  S y n t h e t i c 1.  (23),  s t e e p l y shows  f i r s t  F i g u r e s e x a c t l y  d i p p i n g  the 5  3.4d  p r i n c i p a l and  the  m i s f i t  3.10b,  data  shown  l a y e r s  w i t h  3.4c.  second  example  shows  p a r a l l e l  f l a t - l y i n g  80  a  c e n t r a l  the  l e n s - l i k e  data,  a  94%  r e s p e c t i v e l y .  The  94%  i.e.  s t r u c t u r e .  However, f e a t u r e ,  the  3  c.  to  Real  a  (b)  In  components.  Figure f a u l t .  event  which  t r a c e I  show  and 85  the  In  to  the  f a u l t  m i s f i t  m i s f i t  over  the  strong edge.  values  f a u l t .  In  box  the  2.  An  to  the  shows  see  the  t h i s  b a s i c  a g a i n s t  the  shows  the  c l e a r l y  of  we  r e c o n s t r u c t i o n ,  c o n t r i b u t i n g  d i s p l a y  box  at  1.45  at  about  whole  as B,  the  i t  dips  we  again  i n  the  r i g h t  p l o t  example  from  of  f a c t o r  kind  of  the  to  would  q u i t e see  f a u l t  of  the  f a u l t .  a  braided  the  o r i g i n a l  zone  tends  to  the  In  to  near  the  box.  stream  w i t h  a  B,  the  as  In  to  give  r i g h t ,  due  system,  grey  box  e v e n t s  c o n t r a s t ,  A, at  l a r g e to  t h i s  white  of  In  m i s f i t  energy)  The  50.  a  the  we  r i g h t  t r a c e  data.  a  strong  box  p l o t t i n g  black-white of  68.  h i g h l i g h t i n g  s t e e p l y  the  by  area  see  e x t e n d i n g  the  upper  an  we  t r a c e  70%  c o n t r a s t  from  s  the  f i r s t  c e n t r e at  by  data  produced  the  t r a c e  show  h o r i z o n  o v e r l a y  black-white  The  hump  t e r m i n a t e s  major  I  i t s t e r m i n a t i o n  w i g g l e  a  c,  d i s c r i m i n a t e s  (a)  d i s c o n t i n u o u s  shades  note  a A  t e r m i n a t e s  ( o m i t t i n g  we  m i s f i t  e s s e n t i a l l y and  and  f e a t u r e s .  3.12  r e c o n s t r u c t i o n over  not  b,  Examples.  sinuous  from  a  r e c o n s t r u c t i o n  was  anomalous  3.11a,  and  e v e n t s ,  m i s f i t  t h i s  d i p p i n g  negative see  the as  h i g h l i g h t  1. s t e e p l y  l y i n g  p r i n c i p a l  Data  F i g u r e s  r e c o n s t r u c t i o n  f l a t  l e n s - l i k e  serve  In  r e c o n s t r u c t i o n ,  f e a t u r e s ,  f i r s t  f e a t u r e .  the time  p a t c h  sand  and  81  g r a v e l see  a  i n  and  m i s f i t  having  many  we  p o s i t i v e  anomaly  f e a t u r e  (box  t r a c e s  d.  1 and  t o o l  noted  t h i s  do  j u s t  see  l e n s e s  above  0.7  0.73  I n F i g u r e  s .  As  terminate  s  a  t o  of  t h e  t h e  A  as  p i c t u r e  i s  i n t e r e s t .  of  a g a i n .  r e g i o n  of  s i m i l a r  apex  we  events  begin  t h e pinch-out  ( a r r o w ) .  t o  and  c e n t r a l  whole  f e a t u r e s  3.13(a)  I n d i v i d u a l  t h e e n t i r e  some  corresponds  a t  0.7  c o r r e s p o n d i n g  from  expressed of  i s  I n  box  t h e  t h i n  b l a c k - w h i t e  t h e hump  between  s .  from  t o  have  a  l o tof  may  o c c u r ,  we  be  o f  would  p r e s e n t a t i o n .  i n t r o d u c e d  areas  of  anomalous  a p p l i c a b l e .  f l a t  l y i n g  f l a t  Hence  background  ' m i s f i t s ' l i t t l e expect  data  everywhere.  see a  t o  or  many  (by  such  model,  w i l l  d a t a ,  m i s f i t  v i r t u e event  small  where  a r e  any  energy  i n data  n o t i c e a b l e  events  as  t h i s  m i s f i t s  p a r a l l e l  w i t h  With  whether  t h e  as  However,  i s computed)  incoherent  u s e . However, t o  as  Because  m a t r i x  l y i n g ,  r e c o n s t r u c t i o n  s t r u c t u r e .  ( o r p a r a l l e l )  t h e c o v a r i a n c e a  m i s f i t  i t i s q u e s t i o n a b l e  t h e m i s f i t .  t o produce  o v e r l a y  t o  i n which  d e v i a t e s  w i l l  have  g e n e r a l l y  r e l a t i v e  c o n t r i b u t e  tend  I  t h e examples,  t h e way  which  s e c t i o n ,  f o r h i g h l i g h t i n g  technique  i t  we  case.  D i s c u s s i o n .  In a  20  about  f e a t u r e s .  area  B)  a t  t h e sand  unusual  white  t h e next  (b) h i g h l i g h t s  however,  see a  forms  h o r i z o n  o u t , as  o v e r l a y  c o n f u s i n g ; A,  d e p o s i t s ,  d i s c o n t i n u o u s  fade The  type  ' k i n k s ' which t h e  i n  w i l l m i s f i t  pinch-outs  i n t h e  o v e r l a y  82  T h i s  method  d i a g n o s t i c the  of  s t a c k e d  e x p e r i e n c e d am  and t h i s  data  to  s p e c i f i c  and  and  the  which  However,  would  m i s f i t best  department  i t s e l f  of  where  access  g e o l o g i c a l background  are  draw  the  input  an  the  data  i n to  to  of  u s e f u l n e s s  assessed  not  t y p i c a l  o v e r l a y s . The be  tend  not  without  or  perhaps  may  are  knowledge  s i g n i f i c a n c e  of  i n  s t r u c t u r e s ,  zones  s p e c i f i c  examples  technique  t h e i r  to  s e c t i o n .  determine  i n t e r p r e t a t i o n  s e t s  q u i c k l y  i n t e r p r e t e r  p a r t i c u l a r  s e i s m i c  eye  s e i s m i c  unable  other  p r e s e n t a t i o n , although  h y d r o c a r b o n - t r a p p i n g  i n t e r p r e t e r ' s  the  I  of  of  a r e a s ,  of  these  u s e f u l n e s s an  i n d u s t r i a l  many  r e a d i l y  of  d i v e r s e  a v a i l a b l e .  83  FIGURE 3.10 (a) A m i s f i t r e c o n s t r u c t i o n ( p r i n c i p a l components data shown i n F i g u r e 3.2c, i s o l a t i n g t h e d i p p i n g (c) . (b) A data (c) .  m i s f i t shown  6  - 24) noise  of t h e events  r e c o n s t r u c t i o n ( p r i n c i p a l components 6 - 24) of t h e i n F i g u r e 3.4c, i s o l a t i n g t h e d i p p i n g noise events  A g a i n , because I have i n c l u d e d some o f t h e p r i n c i p a l components which c o n t a i n i n f o r m a t i o n from the h o r i z o n t a l s t r u c t u r e , t h e g e n e r a l "noise" l e v e l has been i n c r e a s e d .  84  FIGURE (a)  A  s y n t h e t i c  l e n s - l i k e (b)  The The  s e c t i o n  s t r u c t u r e .  94%  components) (c)  seismic  h i g h l i g h t i n g  t  r e c o n s t r u c t i o n i s o l a t i n g  m i s f i t t h e  3.11  d e p i c t i n g  the  ( r e q u i r i n g predominant  3  of  h o r i z o n s the  - l y i n g  24  e v e n t s .  with  a  p r i n c i p a l  f l a t components 3 24) ( p r i n c i p a l r e n o r m a l i z e d f o r p l o t t i n g . f e a t u r e ;  r e c o n s t r u c t i o n l e n s - l i k e  ,  f l a t  TSN  100  90  80  70  GO  50  FIGURE Input  data  which and  a  t e r m i n a t e s  the  r i g h t  from  a t  f a u l t e d t o  d i s c o n t i n u o u s the  f a u l t .  the  r e g i o n .  40  30  20  10  3.12a Note  the  l e f t  at  the  f a u l t  h o r i z o n  i n  box  B,  l a r g e near  which  n e g a t i v e t r a c e  68  t e r m i n a t e s  anomaly (box  A ) ,  t o  the  86  FIGURE M i s f i t f i r s t  o v e r l a y four  c o n t r a s t s  which  (see  A  boxes  ( o m i t t i n g  p r i n c i p a l and  f i r s t  components).  h i g h l i g h t B ) .  the the  3.12b 70%  Note  events  of the  the  energy;  strong  t e r m i n a t i n g  near  i . e .  the  black-white the  f a u l t  87  FIGURE Data  from  the  event  t r a c e s  1  an and  a r e a i n 20  of  box a t  b r a i d e d A,  0.73  denoted s  i n  3.13a  streams. by  box  the B.  Note  the  arrow,  and  d i s c o n t i n u i t y the  hump  i n  between  88  FIGURE M i s f i t f i r s t  o v e r l a y four  c o n t r a s t s see the the  the  p r i n c i p a l which  n e g a t i v e  arrow, apex  of  ( o m i t t i n g  w h i l e a  h i g h l i g h t i n  3.13b  f i r s t  components).  t r o u g h  hump.  the  box  the  which B  we  e v e n t s  r e p l a c e s see  60%  Note  a  of the  i n the  the  the  energy;  s t r o n g boxes.  t h i n  b l a c k - w h i t e  event p a i r  i . e .  the  b l a c k - w h i t e In  box  marked  A,  we w i t h  h i g h l i g h t i n g  89  SECTION  a.  3 . I l l :  SEPARATION  DIFFRACTIONS.  I n t r o d u c t i o n . As  noted  an  o f f s h o o t  that  i n  areas  d i f f r a c t o r s , the  f l a t It  i s  to  events  the  and  S y n t h e t i c  to  For  a  I  21  s e p a r a t e d  (M.  t r a c e  using  Y e d l i n , wavelet  model  y i e l d e d  10,  the  be  I  ' p o i n t '  s e p a r a t e d  m i g r a t i o n without  r e f l e c t i o n (r-p)  e v e n t s ,  or  from  f i l t e r i n g  to  to  estimate  i s o l a t i n g  e v e n t s .  l e a v i n g  here  t h i s  only  those  Levin  et  a l .  procedure  to  the  u t i l i z e  d i f f r a c t i o n the  m i s f i t  g o a l .  a  buried  by  50  at  4  s y n t h e t i c  example  l a r g e  extent  r i g h t  m,  ms. of  c e n t e r e d  This  i n  a  of  The The  about  with  1.0  the  5  edge  was  a  a  i s  16  Hz t h i s  h y p e r b o l i c  (Figure  event  were  of  p r o b l e m a t i c a l ,  d i f f r a c t i o n  km/s,  program  response and  sheet  d u r a t i o n  f u n c t i o n  e v e n t s , sheet  v e l o c i t y  seismogram  seismic  the  h o r i z o n t a l  second  source  p a r t i c u l a r l y of  s i n g l e  s y n t h e t i c  r e f l e c t i o n  i s  hand  a  medium  each  1984).  s e r i e s  a r r i v a l  of  i n t e g r a l  comm.,  sampled  the  numerous  r e a d i l y  h y p e r b o l a e  c o n s i s t i n g  F r e s n e l  p e r s .  Ricker  d i f f r a c t i o n  a  c o u l d  r e s i d u a l  propose  model  at  produced  or. w i t h  p r o c e d u r e ,  Examples.  t e r m i n a t i n g t r a c e s  perform  achieve  Data  f a u l t i n g  s l a n t - s t a c k  n o i s e .  r e c o n s t r u c t i o n  h o r i z o n s .  from  a  m i s f i t  events  r e f l e c t i o n  r e c o n s t r u c t i o n  1.  severe  d i f f r a c t i o n  employed  events  the  l y i n g  d i f f i c u l t from  e l i m i n a t e  of  p a r a l l e l  d i f f r a c t i o n (1983)  of  d i f f r a c t i o n  or  v e l o c i t i e s  b.  OF  3.14b). as  to  b u r i e d  a i n  90  the  r e f l e c t i o n . The  moveout  r e f l e c t i o n event same  event,  we  see  the  the  r e a l  KL  f l a t  c o r r e c t i o n 98%  s e c t i o n  event  the  and of  a m p l i t u d e s  of  d i f f r a c t i o n  event  has  except  been  where  o c c u r s  v e r s i o n s 2. 1984)  The  of  r e c o n s t r u c t i o n  f o r  was;  t h i s  changed  t o t a l l y two  be  On  the  reproduced  11  and  and  t o  i s  10%  o n l y  of  the  other both  In  as  waveforms the  fact  s l i g h t l y  i n  they  that  a  the l i n e a r  However,  a m p l i t u d e the  the  to of  the the  d i f f r a c t i o n  and  events  a m p l i t u d e ,  ran  t o g e t h e r ,  r e g i o n s  are  the  a c r o s s as  phase  these  where  comparable  hand,  r e f l e c t i o n  21.  3.15a  components.  events  ( i . e . about  shown  reproduced  p r i n c i p a l  s e p a r a b l e ,  the  e s s e n t i a l l y  two scaled  a n o t h e r .  second a  b,  r e s i d u a l  d i f f r a c t i o n  not  one  zero  h o r i z o n t a l  r e f l e c t i o n  (Figure for  of  shows  b u r i e d  set  are  and  see  event  d i f f r a c t i o n  F i g u r e s  due  the  the  s t i l l  i s  f l a t t e n  a f t e r  We  f a i t h f u l l y  t r a c e s  In  i s  m i s f i t  waveforms).  the  3.14d.  t o  i s o l a t e d  the  r e s i d u a l  energy  between  waveforms  of  these  an  and  not  f i r s t  r e f l e c t i o n  event  as  the  data  comparison)  F i g u r e  r e f l e c t i o n c o u l d  the  and  f o r  r e c o n s t r u c t i o n  combination  input  i n  d e c o m p o s i t i o n .  i n  from  3.14c),  s e p a r a t e l y  l y i n g  waveform  removed  (Figure  (produced moveout  the  was  s y n t h e t i c  o f f s e t sheet  events  d i f f r a c t i o n  example  s e i s m i c (Figure  which  s e c t i o n 3.14a).  t e r m i n a t e  a r r i v a l s  c e n t r e d  3.16a).  The  d i f f r a c t i o n  event  comparison  and  n o r m a l i z e d ,  i s  over  (M.  Y e d l i n ,  over T h i s the  about  i n  same  r e s u l t s sheet  the  i t s e l f ,  shown  the  p e r s .  comm.,  t r u n c a t e d  i n  a  s e r i e s  edge,  and  t e r m i n a t i o n  produced  F i g u r e  a  point  s e p a r a t e l y  3.16b.  F i g u r e s  91  3.17a  and  b  r e c o n s t r u c t i o n . a b l e  to  phase  and  the  was,  M i g r a t i o n the  a  medium  w i t h  t r a c e  The  d e r i v e d  of  I  other  on  in  m i s f i t  from  to  that  e a r l i e r ,  the  terms  of  c o l l a p s i n g  v e l o c i t y  of  4  f i r s t the  the  remain  in  r e f l e c t i o n  f e a t u r e  e x i s t s  m  and  I  a  s i n g l e  c r e a t e d  a  z e r o - o f f s e t  v e l o c i t y  F i g u r e  along  f e a t u r e s  w i t h  shown  i n  w e l l  p r e s e r v i n g  h o r i z o n t a l  that  s e p a r a t e l y  3.18a.  i s  on  f i r s t .  i n  frequency-wavenumber  l e s s  the  produced  the  m i s f i t  S e c t i o n s .  12.5  i s  w h i l s t  r e s i d u a l  where  the  and  r e c o n s t r u c t i o n  event  Some  v e l o c i t y ,  event,  waveforms  segregated  than  d i s c u s s e d  constant  r e c o n s t r u c t i o n  d i f f r a c t i o n  D i f f r a c t i o n  F i g u r e  i s  the  c o n t r i b u t i o n  m i g r a t e d  from  KL  s e c t i o n  s e p a r a t i o n  the  c u r v a t u r e the  some  r e s u l t s ,  Using (1978),  the  the  model  d i f f r a c t i o n  subsequent  of  of  97%  i n f o r m a t i o n .  of as  the seen,  out  components  For in  i s  amplitude  p r i n c i p a l  c.  As  separate  v i c i n i t y  f e a t u r e  show  the  case,  was  km/s.  The  events  are  case  of  i n c o r r e c t l y  a l g o r i t h m  w i t h  the i s  as  component.  seen  a f t e r  S t o l t  The  w i t h  3.18c)  d i f f r a c t i o n  more  more  achieved  of  ( F i g u r e s  as  there  h y p e r b o l a ,  event,  3.18a).  comparison  s e c t i o n  3.16b,  p r i n c i p a l  s e c t i o n  ( F i g u r e  f o r  m i g r a t i o n  t h i s  F i g u r e  km/s  b u r i e d  3.18b.  d i f f r a c t i o n In  4  sheet  d i s t o r t i o n energy  best  r e s u l t ,  the  m i g r a t i o n  i s  c o r r e c t i n  F i g u r e  3.I8d. In i n f e r r e d m i g r a t e d  the c o u l d  be  stacked  used  f o r  s e c t i o n  r e s i d u a l showing  m i g r a t e d  d a t a ,  m i g r a t i o n r e s i d u a l  of  v e l o c i t y an  d i f f r a c t i o n  thus  o r i g i n a l events  92  (Levin the  et  v e l o c i t y  r e g i o n s a  a l . , 1983, of  of  severe  f a u l t - e d g e  d.  Real  less a  d i f f r a c t i o n  t o  an  of  the  p o i n t s  f o r  the  96%)  s t i l l  the  data  r e s i d u a l  has  o n l y  p r o d u c i n g  a  shows  l y i n g  e v e n t s ,  Event  A  we  c o l l a p s e  s p e c i f i c was  we  a s c e r t a i n  F u r t h e r ,  i n  serve  the  the  a l .  1983).  as  t r a c e s  1983)  65  have  and  been  s e r i e s  of at  and  of  f i r s t c e n t r e d  3.19c). from  m i g r a t i o n s  on  Now  the  f l a t  these  data  T h i s  mind  ( i n d i c a t e d  i n  F i g u r e  w i t h  a  of  3  v e l o c i t y  92%  d a t a .  the  events  depth.  a  the  segregated  each  a l s o  d i f f r a c t i o n  ( e x c l u d i n g  ( F i g u r e  we  e s t i m a t e  shows  f e a t u r e s  75  been  q u e s t i o n  to  of  d i f f r a c t i o n  hyperbolae  c o l l a p s e d  a l .  3.19b  of  (probably  the  a l l evidence  of  limb  d i f f r a c t i o n s ,  opens  F i g u r e  are  had  migrated  et  65  hand  d i f f r a c t i o n s  r e c o n s t r u c t i o n  i n  of  was  I  and  s e c t i o n  T h i s  common  which  45  r i g h t  i n c o r r e c t l y  such  a  the  (Levin  d a t a ,  t r a c e s  evidence  u s i n g  events  perform  see  t h a t  were  s e r i e s  hyperbolae  best  s  g r o s s ,  between  may  between  e l i m i n a t e d  a  d i f f r a c t i o n  two  zone  Events  shows  et  m i s f i t  the  w i t h  to  p o s s i b l y  s t a c k e d  f u n c t i o n ) .  of  the  that  to  may  of  m i g r a t i o n  (Harlan  the  order  s i m p l y  d i f f r a c t o r .  t r a c e s  Given  but  throughout  in  the  or  procedure  1.1  v e l o c i t y  which  l e a v i n g  c l e a r l y  96  about  p o s s i b i l i t y  r e c o n s t r u c t i o n  However,  t h i s  h y p e r b o l a .  v e l o c i t y  events  at  t h a t  need  s e i s m i c  and  inadequate  to  about  m i g r a t e d .  m i g r a t e d ,  conclude  due  shows  time  c o n t i n u o u s ,  may  f a u l t i n g ,  3.19a  were  p r e v i o u s l y  m a t e r i a l s  a l . , 1983)  Examples.  F i g u r e  informed  et  i n d i c a t o r .  Data  1.  the  Harlan  km/s  was  done  3.19c). ( F i g u r e  93  3 . l 9 d ) .  Event  ( F i g u r e  3 . l 9 e ) .  used  for  v i s u a l and  r e s u l t ,  i s  both  these  f i g u r e s ,  v e l o c i t i e s were  r e s u l t  t h i s .  The  using  the  the  a  high  r a i s e s  the  p o s i b i l i t y  fact  p r e v i o u s l y  m i g r a t e d .  i n  v e l o c i t y  1  v e l o c i t y .  of  4  of  data  the  best  km/s the  The  the  h i g h e r m i g r a t e d  i n  the  a t t e n t i o n  centred f a c t  that  1984).  data  one's  km/s  of  (Berkhout,  were  are  window i.e.  f o c u s s i n g  d i f f r a c t i o n s  of  r e s u l t s :  noise  s u i t a b l e  not  the  m a j o r i t y  However,  m i g r a t i o n  v e l o c i t y  appearance  m i g r a t e d  s i n c e  n o i s e .  where  from  of  show  m i g r a t i o n  o v e r a l l  that  i s  I  a  can  a l l o w  that  these  these  p a r t i c u l a r  D i s c u s s i o n .  events  r e c o n s t r u c t i o n  o f f e r s  seismic  a  d a t a .  c o n s i d e r a b l y  new  The more  L e v i n  (1983)  a l .  i t e r a t i v e As  a  f a u l t i n g ,  may  r e s i d u a l  such  as  as  a  to  s t r u c t u r e s .  of  r e s i d u a l  c o r o l l a r y  serve  h e l p i n g  way  a  of  the  to  m i g r a t i o n  of  r e s i d u a l Sea,  fault-edge  d e l i n e a t e  Once  used  North  f a u l t  to  a  s e p a r a t i n g  d i f f r a c t i o n  d i f f r a c t i o n s  H a r l a n  m i g r a t i o n  be  to  view  i s o l a t i n g  p r o t r a c t e d .  the  et  with  method  s e g r e g a t e d ,  may  i s  with  a  M i s f i t  for  three  the  s e c t i o n  of  and  s u r p r i s i n g ,  zones  s e l e c t i o n  e.  In  s e c t i o n s  d i f f r a c t i o n  data  c o l l a p s e d  and  not  the  best  than  d i f r a c t i o n  on  was  m i g r a t i o n ,  lower  This  B  et the  a l .  d i f f r a c t i o n  estimate  events  proposed  c o r r e c t i o n  stacked  seismic  m i g r a t i o n  i n of  stacked  (1983,1984)  technique  d i s p l a y  from  are by  v e l o c i t i e s  d a t a .  areas  with  the may  i s  severe  d i f f r a c t i o n s  i n d i c a t o r .  This  be  c o n t r o l l e d  hydrocarbon  of  use  i n  t r a p p i n g  94  source, 1  € f  « c  t i o r i  ~V  cr  b  y  . .to  d)f|f4a|c|t|c|r|  20  10  (a). ( b K  e  The  model:  T h e  r e f l e c t i o n  source  w a v e l e t ,  a  thin and  FIGURE 3.14a sheet b u r i e d a t 1 km d i f f r a c t i o n  propagating  depth,  a r r i v a l s  i n the model l a ) .  f o r a  t e r m i n a t i n g 16  Hz  a t  R i c k e r  95  FIGURE (c) (d)  The  data  The  comparison  a f t e r  f l a t t e n i n g  f l a t t e n e d w i t h  l a t e r  on  d i f f r a c t i o n r e s u l t s ,  3.14c the  r e f l e c t i o n  event  produced  r e n o r m a l i z e d  f o r  event. s e p a r a t e l y p l o t t i n g .  f o r  96  JFIGURE (a) 21  A  98%  r e c o n s t r u c t i o n  p r i n c i p a l  r e c o v e r e d (b)  The  the  t h e  i s o l a t e d  where  components). f l a t t e n e d  m i s f i t  components  of'the  limb  had  i s subject  runs  r e f l e c t i o n ,  i n t o the  the  same.  of  r e n o r m a l i z e d  r e c o n s t r u c t i o n are  eye  r e f l e c t i o n  d i f f r a c t i o n  i n t e r f e r e n c e  i n  the  To  r e c o n s t r u c t i o n  3 - 2 1 )  3.15  data  i s  i n  f o r p l o t t i n g . f a i t h f u l l y  a r e  appears  2  t h e  of  t o  the have  _  data  d i s t o r t i o n .  they  ( r e q u i r i n g  t h i s  e v e n t .  the  d i s t o r t e d t o  14(b)  14(b) For  ( p r i n c i p a l  t h e  most  reproduced.  o r i g i n a l Where  i n s e p a r a b l e ,  l i m b , the as  p a r t ,  However, the  m i s f i t  d i f f r a c t i o n the  waveforms  97  F (a) A zero o f f s e t seismic s (see s k e t c h i n F i g u r e 3.14(a g i v i n g r i s e to r e f l e c t i o n e t b )  U  t  T h e  a  d i f f r a c t i o n  subsequent  r e s u l t s ;  events  I e ) v  GURE 3.16 c t i o n over a b u r i e d h o r i z o n t a l sheet ) which terminates at t r a c e 10, e n t s , and d i f f r a c t i o n events c e n t r e d  produced  r e n o r m a l i z e d  f o r  s e p a r a t e l y p l o t t i n g .  f o r  comparison  t o  98  20  10  D r  r-J  . .cn  m o m o o :z  .co  FIGURE (a) the  The 21  (b)  97%  p r i n c i p a l  The  event  2  Phase i n  -  of  3.17  the  data  i n  16(a)  components).  m i s f i t  components seen  r e c o n s t r u c t i o n  21) and  comparison  w i t h  of  the-data  s u c c e s s f u l l y  amplitude  i n f o r m a t i o n  16(b).  1  of  ,  r e c o n s t r u c t i o n has  ( r e q u i r i n g  in  i s o l a t e d are  Renormalized  w e l l f o r  ^ ( a )  ( p r i n c i p a l  Jhe diffraction conserved, p l o t t i n g .  as  i s  99  FIGURE 3.18 (a). The input s e c t i o n (see t e x t ) , ( b ) i n d e p e n d e n t l y f o r c o m p a r i s o n . ( c ) . u s i n g p r i n c i p a l components 2-128. ( d ) . (c) above, a f t e r m i g r a t i o n w i t h v e l o c i t  . The The The y 4  d i f f r a c t i o n produced m i s f i t r e c o n s t r u c t i o n d i f f r a c t i o n s e c t i o n , km/s.  100  FIGURE N i n e t y  s i x  f a u l t i n g . 45  -  65  t r a c e s Note  between  of the  0.6  s t a c k e d  time  d i s c o n t i n u o u s and  0.9  s.  3.19a migrated events  data i n  the  i n  a  area  r e g i o n from  of  t r a c e s  101  FIGURE 3.19b The 92% r e c o n s t r u c t i o n ( r e q u i r i n g 3 of the 96 p r i n c i p a l components) which h i g h l i g h t s the gross s t r u c t u r a l f e a t u r e s of the s e c t i o n . Note the loss of c o n t i n u i t y of the events between traces 45 65.  1 0 2  FIGURE The m i s f i t r e c o n s t r u c t i o n ( p r e x c l u d i n g t h e f i r s t 96%) showing a in the r e g i o n of i n t e r e s t . H y p e r b t a r g e t events f o r m i g r a t i o n . The d p l o t t i n g .  3.19c i n c i p a l components 6 t o 96, s e r i e s of d i f f r a c t i o n events o l i c events "A" and "B" a r e t h e ata have been r e n o r m a l i z e d f o r  103  Region  A  2000m/s  3000m/s  4000m/s  FIGURE 3.19d Region w i t h has  A  of  the  v e l o c i t y : been  m i s f i t 2000,  f l a t t e n e d  r e n o r m a l i z e d  f o r  r e c o n s t r u c t i o n  3000,  best  and  u s i n g  p l o t t i n g .  4000 3000  ( F i g u r e  m/s.  The  m/s.  3.19c). h y p e r b o l i c  The  data  M i g r a t i o n s event have  A been  104  Region w i t h has  B  of  t h e  v e l o c i t y : been  m i s f i t 3000,  f l a t t e n e d  r e n o r m a l i z e d  f o r  FIGURE 3.19e r e c o n s t r u c t i o n (Figure  4000,  best  and  u s i n g  p l o t t i n g .  5 0 0 0  m/s.  4000  The  m/s.  3 . 1 9 c ) . h y p e r b o l i c  The  data  M i g r a t i o n s event have  "B" been  105  SECTION  a.  3.IV:  energy to  the  purpose  packing  I  v e l o c i t y i i .  and  a  d a t a .  note  We have  the  RMS  or  more  over  A  be  KL  f o r  with  the  The  the  idea  m u l t i p l e s  r e c o n s t r u c t i o n  v e l o c i t y  here  onto  o m i t t i n g  a  that  m u l t i p l e - f r e e .  a n a l y s i s ,  a s s o c i a t e d moveout the  and  w i t h  w i l l  the  c o r r e c t i o n  a r r i v a l s  f l a t t e n e d ,  c o r r e c t e d ,  i d e n t i f y  the  m u l t i p l e .  t h a t  l e s s  u t i l i z e  s t e p s :  a  stage  I  t r a n s f o r m .  e s s e n t i a l l y  v e l o c i t y  or  r e a l  data  v e l o c i t y  t h i s  s u p p r e s s i o n ,  a s s o c i a t e d  b a s i c  time  constant at  the  standard  the  been  under  f i v e  onset  perform  should  the  Using  of  component.  i n  From  m u l t i p l e  energy  component  proceed  i .  w i l l  the  p r i n c i p a l  p r i n c i p a l  of  p r o p e r t y  segregate  s i n g l e  are  SUPPRESSION.  I n t r o d u c t i o n .  For  is  MULTIPLE  due  whereas have  on  m u l t i p l e , the  to  the  s e i s m i c  m u l t i p l e s  primary  i n c r e a s e d  events  c u r v a t u r e  i n  s e c t i o n . i i i .  moveout which  Compute c o r r e c t e d  had  v e l o c i t y component. a s s o c i a t e d a c r o s s  the  the  a  data  moveout  w i l l T h i s w i t h  KL  t r a n s f o r m s e t .  v e l o c i t y  now  appear  i s  because the  s e c t i o n ,  and  of  the  C o r r e l a t e d equal  to  energy that  predominantly coherent  m u l t i p l e w i l l  now  has  energy been  appear  constant  to  on  i n  the  seismogram  of  the  the  f i r s t  p r i n c i p a l  the  v e l o c i t y  having  m u l t i p l e ' s  f l a t t e n e d , be  v e l o c i t y  the  or  a l i g n e d  most  h i g h l y  106  c o r r e l a t e d i v .  energy. Reconstruct  seismogram  from  p r i n c i p a l  w i t h  advantageous m u l t i p l e  the  in  the  data  problem  However, one,  encountered of  t r a c e s  b.  S y n t h e t i c  s e c t i o n h a l f  (D.  In  to  the  ' m u l t i p l e '  step i  the  f i r s t  c o r r e l a t e d I t  may  energy a l s o  be  when  the  d i s t o r t e d  by  component  s e v e r e l y  a l l the  p e r s .  f o r  data  a  (1982) FK  the  new  to  addressed  t r a n s f o r m a t i o n c o r r e c t i o n .  and  a l t e r n a t i v e  a l i a s i n g  d e a l i n g  which  c o r r e c t i o n  moveout  to  comm.,  Ryu  v i a i s  when  NMO  the  here  prone  method  u s u a l  u s i n g  s t r e t c h i n g  as  r e c o n s t r u c t e d  m u l t i p l e s  a p p l i e d .  suppression  d e s c r i b e d  the  .  for  i s  FK  from  i  s t a c k i n g  3.20a,  I  show  r e f l e c t i o n  a l l o v e r l a i n t r a v e l  omit  c o r r e c t e d  w i t h  a  problems  small  number  1985).  Examples.  r e p r e s e n t i n g  space,  been  then  be  Hampson,  F i g u r e  has  i n  data  not  Data  any  p r i n c i p a l  and  approach  w i t h  but  m u l t i p l e .  second  repeated  m u l t i p l e  w i l l  the  s t r e t c h i n g  as  i s  w i t h  the  and  1.  moveout  p r i o r  c o n j u n c t i o n  out  moveout  events.  d e s i r e d ,  of  components  of  the  v e l o c i t y  i s  v e l o c i t y  leave  waveform  procedure  suppression  the  omit  the  same  i.e.  v e l o c i t y  primary  Remove  The  a l i g n  the  to  constant  p r i n c i p a l  a r r i v a l ' s  i n t e r f e r i n g  u s i n g  the  component,  a s s o c i a t e d  v.  the  paths  by i n  events  water. the  a  simple from  Included  s u r f i c i a l  s y n t h e t i c 9  seismic  f l a t  l a y e r s  over  two  events  due  are water  l a y e r  a to  ( i n d i c a t e d  107  by  a r r o w s ) .  (VA)  F i g u r e  ( N e i d e l l  energy  a t  and  1.1  3.20b  Taner,  1971)  1.65  seconds  and  1450  m/s.  F i g u r e  3.21a  NMO  c o r r e c t i o n .  Note  m u l t i p l e primary the  event  s e c t i o n  second The  e f f e c t  of  the  primary  data  t h e  w i t h  from:  the  m u l t i p l e  w i t h  m u l t i p l e s ;  m u l t i p l e  two  noise  ( F i g u r e  a t  m u l t i p l e  the  data  stack  the  a r r i v a l s ; (3)  the  using  t h e  t o  1.65 the  components  KL  3.23b s  have  s e c t i o n  ' m u l t i p l e '  o m i t t i n g  from  F i g u r e  VA  We  v e l o c i t y .  comparison which  c o n v e n t i o n a l  stack  stack  of  of  note  only  the  a by  been the 3.23a  NMO  p r i o r  the  t r a c e  never of  data  shows  F i g u r e  c o n v e n t i o n a l  data  3.21b.  have  shows  of and  the  3.22b  the  a  the  f i r s t  i n F i g u r e  that  see  a f t e r  t h e  m u l t i p l e s  a  the  d e c o m p o s i t i o n  p r o c e s s i n g .  water  of  v e l o c i t y  j u s t  the  a f t e r  v e l o c i t y  f l a t t e n e d  removed  s y n t h e t i c  the  KL  r e s u l t s  c o n v e n t i o n a l  F i g u r e s and  we of  (2)  and  the  has  3.22a).  a n a l y s i s  constant  r e a l  the  that  d a t a ,  (1)  1.1  a  i s then  w i t h  a  S t a r t i n g  suppression  3.23b,  of  a f t e r  3.21a),  3.20  see  c h a r a c t e r i s t i c  seconds.  ( F i g u r e  F i g u r e  i n t r o d u c e d  p r i n c i p a l  m/s  v e l o c i t y  We  procedure  g i v e s  suppressed  suppression  a  data  s t r e t c h i n g  from  and  data.  o u t . R e c o n s t r u c t i o n  r e f l e c t i o n  comparison  m u l t i p l e s  1.65  F i g u r e  removed  In  t h i s  1450  m u l t i p l e  s t a c k .  how  semblance  t h i s  w i t h  the  components  a f t e r  the  of  shows  and  s  moveout  bottom  r e s u l t i n g  The  1.0  v e l o c i t y  e f f e c t i v e l y  A  1.1  i s c a r r i e d  comparison  to  a t  of  the  shows  a t  p r i n c i p a l  u s i n g VA  events  shows  the  data  had data a f t e r  technique.  (2)  and  been a f t e r  ( F i g u r e  (3)  shows  s u c c e s s f u l l y removal 3.21b)  that  the  suppressed.  of  t h e  f i r s t  d i d  not  stack  108  c o n s t r u c t i v e l y .  With  a  view  t o automation,  a u t o m a t i c a l l y  d e t e c t  onset  t h i s  the is  t i m e . VA  To  map  was  r e a s o n a b l y  whereas expect  a  at  t h e  are  d e t e c t e d ,  just  p r i m a r y . 3.20b.  m u l t i p l e  1.28  s .  w e l l  of  that  shows  peak  3.20c  a  a t  i n t h e VA  a t  2000  Care  suppress whole,  t h e v e l o c i t y  a  must  primary t h e  i f  m/s  we  c h a r a c t e r i s t i c  saw  l o c u s  of  t o  mode  v e l o c i t y :  RMS  v e l o c i t y  and  time  segments  of  that  m u l t i p l e  c o n s t a n t  map  each  m u l t i p l e  t o  energy  v e l o c i t y ,  d i s p l a y  m u l t i p l e .  so  f o r a as  of  t h e f i r s t  1450  m/s  which  t o  These  maxima m u l t i p l e  suppress  h a l f  we  maxima  g i v e n  n o t  s t a c k  t h e  appears s t a c k  t h e primary  taken on  wish  t o  i s n o t . Consequently,  t h e VA  c o r r e s p o n d i n g  events  o n l y  we  i s s t a r t e d  maximum  be  automatic  o v e r l a p p i n g  o f  m o d i f i e d  of  t h e  F i g u r e  corresponds  t o t h e  t h e v e l o c i t y  of t h e  map.  c o r r e s p o n d i n g  and  of  was  i . e . t h e i r  t r a j e c t o r y  segments  (1450 m / s ) . However,  m/s s ,  a  F i g u r e  a  t h e p r o c e s s i n g  3.20c  see  t r e n d  In  of  of  3,20b  a l o n g  v e l o c i t y  t h e onset  We  s t a c k  I n F i g u r e  time  and  F i g u r e  observed  1.10  of  l o c a t i o n  a f t e r  1700  made.  v e r t i c a l  s t a c k  a l g o r i t h m  t h e m u l t i p l e s ,  e n d , a  t h e s t a c k i n g  t h e  so  as  t h e b a s i s must t o  u s u a l l y  be  a t a l s o  events t h e  not  t o a l l o w  of  these  suppress  maxima  between  t o  t r e a t e d  water  shows  p r i m a r y  event  maxima.  m u l t i p l e s  On but  w i t h  m u l t i p l e s .  and a t  t h e program  c a u t i o u s l y ,  bottom  0.85  a t  t o t h e  works one  109  c.  Real  Data  Of r e a l  g r e a t e r  s e i s m i c 1.  shot at  Examples.  i n t e r e s t  data  F i g u r e  3.24  o f f Sable  4  ms,  here.  but  3.25a  see a  dominant  at  0.5  seconds.  used  m u l t i p l e (1620  mode.  0.5  and  2400  has  been  no  longer  Twenty comparison m u l t i p l e  stack  and  0.91  s  gather  every  o f  broad  3.25b o f  a t  a  t r a c e s  t r a c e VA  of  1620  t h e VA  s t a r t i n g  t h e VA  3.24  a u t o m a t i c a l l y  p i c k e d  t h e absence  of  t h e  band  m u l t i p l e  a l s o  how  enhanced by  of  Note  i n t h e m u l t i p l e  a r r i v a l s  data  were  t h e c o n v e n t i o n a l l y  f e a t u r e s  data  i s made  which  t h e c o n v e n t i o n a l 'A')  d i f f e r stack  due  t h e event suppressed  t o m u l t i p l e  p r o c e s s e d stacked  i n  data  i n F i g u r e between  t h i s w i t h  3.26a  a  and  suppressed  energy  2.2  VA,  s  and  as  i t  p a t h s .  way,  and  stack  of  b.  t h e m u l t i p l e of  a f t e r  v e l o c i t y  t r a v e l  a r e : t h e absence  i n t h e m u l t i p l e  a t  i s  m u l t i p l e  an  s .  map,  i n f o r m a t i o n  u s i n g  3.4  used g a t h e r .  m/s  F i g u r e  o f  sampled  was  automatic  of  data  t h e  of  t h i s  f o r t h e program's shows  on  seismic  stack  t h e m u l t i p l e s :  3.25c  a l g o r i t h m  60  band  energy  shows  has t h i r d  v e l o c i t y  m u l t i p l e  t h e  marine  only  F i g u r e  gathers  (marked  of  expense, a  of  problem.  gather  F i g u r e  suppressed  n o t i c e a b l e  a  Each  trend  obscured  o f  pose  Canada.  shows  suppression  between  i s  CDP  t h e m u l t i p l e s  m/s). Note  m/s  a  t h e presence  t o d e t e c t  suppression  shows  reduce  We  h i g h l i g h t i n g  m u l t i p l e s  I s l a n d ,  t o  F i g u r e  when  i s t h e performance  The  a t h e  most  suppressed  t h e event  s e c t i o n ,  a t  which  110  appears  as  changes  part  at  1.35  of  s  suppressed  trough  1.10  S u b t r a c t i n g (Figure  3.26c)  m u l t i p l e  between  r e l a t e d  the  i n  the  energy  events  such  as  that  at  p a i r  at  2.45  and  y i e l d s  the  p r o c e s s e d are  unprocessed  S e v e r a l  s e c t i o n s  emphasizes  the  seen  a  0.44  c l e a r e r  0.86  s  and  s  the  in  the  ('C'),  the  2.55  s  ('E'). s e c t i o n  nature  unprocessed  w i t h  and  d i f f e r e n c e  l o c a t i o n and  are  d a t a ,  of  d a t a .  the  Bands  of  s p a c i n g .  D i s c u s s i o n .  M u l t i p l e water  and  for  the  are  of  a r r i v a l s  c e r t a i n  use  i n  moveout  model  On  the  s m a l l  sample  the  almost  complete  event  the  was t i m e .  no  of  r e a l  In  a d d i t i o n ,  the  VA  a c t u a l longer  map  of  data masked  problems. and  the  p r o c e s s i n g  methods  s t a c k i n g Here  and  I  v e l o c i t y  present  the  stack. has VA  which  were  energy  an  m u l t i p l e On  simple  worked maps  w e l l .  h i g h l i g h t  energy  a f t e r  were  small  g r e a t l y  c o r r e s p o n d i n g  m u l t i p l e  s h a l l o w  e f f e c t i v e  r e l a t e d  events  i n  removing  method  m u l t i p l e  h y p e r b o l a by  known  a n a l y s e d ,  primary  before  and  c o r r e c t i o n  d a t a , data  absence  of  i s o l a t i n g  s y n t h e t i c  problem  Simple  these  f i n a l  e a r t h  because  to  s e r i o u s  events  for  layered  i n  m u l t i p l e  method  p r i o r  a  e n v i r o n m e n t s .  of  events  a m p l i t u d e  pose  a l l e v i a t i n g  simple  p r o c e s s i n g .  often  land  e l i m i n a t i o n  i n t u i t i v e l y  same  and  two  which  'B').  s e c t i o n ,  ('D'),  these  d i f f e r e n c e s  d.  s  'doublet'  (marked  m u l t i p l e at  a  to  of  enhanced, the  a r r i v i n g  primary at  the  111 The very  r e s u l t s  p r o m i s i n g ,  n o t i c e a b l e number a c t u a l an  of  on  and  t h e b a s i s i n i t i a l  d i f f e r e n c e s , m u l t i p l e  waveforms  important  p o i n t  events  of  t e s t s  t h e i r have  VA  n o t e .  on  and  twenty  maps  been  a r e e s s e n t i a l l y t o  'before  a f t e r '  gathers  showing  removed. uncorrupted  t h a t I n by  VA  maps  show a  s e v e r a l  s i g n i f i c a n t  a d d i t i o n , t h e  look  t h e  p r o c e s s i n g :  1 12 WIS VEL(Ws) 1SO0  INP" 7 SECTION  10  2000  a .  FIGURE (a)  The  s e i s m i c  o v e r l y i n g muted  a  h a l f  and  f i r s t ,  (b)  The  the  m u l t i p l e s  lower (c) tend a  Two  p o l a r i t y  v e l o c i t y  p o r t i o n  Stack  space,  AGC'd.  the  o f  by  bottom a t  1.1  (VA) of  v e l o c i t y  o f  water.  m u l t i p l e  The  t h e d a t a . 1450  m/s  data  events  s , t h e second Note  9  from  r e f l e c t i o n s  a t  been  c a n be 1.65  t h e peaks  which  layers  have  seen:  s . due  dominate  to  t h e  t h e VA.  t h e f i r s t  t o l i e v e r t i c a l l y  maximum,  water  r e v e r s e d a  o f  a l l o v e r l a i n  a n a l y s i s  w i t h o f  3 . 2 0  r e p r e s e n t a t i o n  whereas  h a l f  o f  t h e VA  i n t h e VA  c o m p a r a t i v e l y ,  map,  map  ( b ) .A s  they  primary  w i l l events  m u l t i p l e  events  s t a c k t o w i l l n o t  produce  FIGURE (a)  A  segment  of  the  data  a f t e r  1450  m/s.  N o t i c e  constant  v e l o c i t y  of  m u l t i p l e  a r r i v a l s  have  (b)  A  m i s f i t  f i r s t '  2  l o c a t i o n s l a r g e l y ef  f e c t .  been  r e c o n s t r u c t i o n  p r i n c i p a l f o r m e r l y  3.21  of  the Some  occupied  the  and  the  c o r r e c t i o n  two  events  w i t h due  a t o  f l a t t e n e d .  components.  i n c o h e r e n t ,  move-out how  by w i l l  not  f l a t t e n e d r e s i d u a l m u l t i p l e stack  t o  data noise  o m i t t i n g remains  events, produce  but a  i n  t h i s  the tne i s  n o t i c e a b l e  1 1 4  V E L ^ / S )  RMS  1500 800D  SECTION AFTER MULTIPLE SUPPRESSION  FIGURE 3.22 (a)  The  v e l o c i t y the  r e c o n s t r u c t e d moveout  o r i g i n a l  d a t a ,  c o r r e c t i o n  d a t a ,  a f t e r  have  r e p l a c i n g  the  e f f e c t s  been  removed,  the  segment  of  the  embedded which  constant back  c o n t a i n e d  i n t o the  m u l t i p l e s . (b) the  The  VA  absence  c o r r e c t  VA  of of  t r e n d  the  data  a f t e r  m u l t i p l e s can  be  more  m u l t i p l e  (which r e a d i l y  were  s u p p r e s s i o n . e n e r g e t i c  d i s c e r n e d  than  Note  that  events) i n  i n the  3.20(b).  11 5  10  1  FIGURE  32 1  3.23  (a) The m u l t i p l e suppressed data a f t e r normal moveout c o r r e c t i o n using a v e l o c i t y f u n c t i o n p i c k e d f r o m 3 . 2 2 b . The r e s i d u a l events remaining from the m u l t i p l e s can s t i l l be seen, but these do not stack c o n s t r u c t i v e l y . (b) T h r e e v e r s i o n s of a s t a c k o f the data: (1) the stack of a s y n t h e t i c data set produced without m u l t i p l e s : t h i s i s our d e s i r e d , or optimum r e s u l t ; (2) the stack of the data i n 3.20(a) a f t e r NMO c o r r e c t i o n . N o t i c e the m u l t i p l e a r r i v a l s at 1.1 and 1.65 s which stacked c o n s t r u c t i v e l y ; and (3) the stack of the m u l t i p l e suppressed data i n 3.22(a) a f t e r NMO c o r r e c t i o n . In comparison w i t h ( 2 ) we note the absence of m u l t i p l e events.  1 1 6  FIGURE Every  t h i r d  bandpass  (5  t r a c e -  55  H z ) ,  from  a  muting,  3.24  marine and  gather  a p p l i c a t i o n  of  60  of  an  t r a c e s , AGC.  a f t e r  SEflB.RflS VEL(^/S) 1500  £  2500| |  1500  0.4  0.4 .  0.6  0.6.  0.8  0.8.  1.0  1.0 .  1.2  1.2 .  1.4  1.4  1.6  1.6  1.8  1.8  J  2.01  2.0  J  2.2  j  2.4  J  >  cr F 5  200^  SEHB.RHS VEL ( W i )  22.2> .4 J-  200n  1 17  2500  J  2.6 J 2.81 3.0  1  i 3.4 I 3.2  3.61  1500 FIGURE 3.25 (a) VA o f t h e d a t a i n 3.24. Note t h e v e r t i c a l trend of energy a t 1 6 2 0 m/s d u e t o w a t e r b o t t o m r e l a t e d m u l t i p l e s . (b) The stack o f t h e V A m a p a b o v e . T h e m a x i m u m a t 1 6 2 0 m/s was detected by t h e a l g o r i t h m and subsequently used as t h e m u l t i p l e suppression v e l o c i t y . (c) The VA map o f t h e s e c t i o n a f t e r m u l t i p l e the absence o f energy a t 1620 m/s a f t e r enhancement o f t h e primary a r r i v a l peak a t 2.2 2400 m/s.  s u p p r e s s i o n . Note 0.5 s , and t h e s with v e l o c i t y  118  FIGURE (a) The c o n v e n (b) The same a u t o m a t i c a l l y absence of t Also note the and the p a i r of the gathers (c) of  The the  Bands 0.44  s  t i o n a l stack gathers a l o c a t e d m u l t he event a enhancement at 2.45 and has been p l  d i f f e r e n c e  of 20 f t e r i p l e v t 0.91 of eve 2.55 o t t e d  s e c t i o n ,  d i f f e r e n c e s  between  of  a s s o c i a t e d  energy  i n t e r v a l s ,  i n d i c a t e d  to  the  emphasize  the  procesed  and  with by  3.26  g a t h e r s . m u l t i p l e suppression using the e l o c i t y of 1620 m/s. Note the s (A) and changes at 1.35 s (B). nts at 0.86 s (C), 1.10 s (D), s ( E ) . Only the processed s e c t i o n here.  the  the  l o c a t i o n  m u l t i p l e s  a r r o w s .  and  unprocessed appear  at  nature d a t a . about  119  CHAPTER SIMILARITY  SECTION  a.  4.1: TRACE  MEASURES.  ANALYSIS.  I n t r o d u c t i o n .  R e c a l l  that  simply  t h e w e i g h t s  given  p r i n c i p a l  (group  1,  then  elements  a p p l i e d  i n p u t  t o  each  component. t h e  We  same  those  t r a c e s  see another  group  of  w i t h i n  group  each  In w i t h i n  t h i s a  p a r t i c u l a r changes  2,  o t h e r  procedure  along  which  d i f f e r e d we  but  way  must  bottom  one  only  e v e n t .  may  e t  ' a c o u s t i c sediments  t h e  from  data  I n and  t h e  s i m i l a r  t o  (group  2,  group  To  narrow we  time w i l l of  group  of  p r i n c i p a l  had  were  s i m i l a r  2  a r e  s i m i l a r  1.  n a t u r a l  g r o u p i n g s  c e n t r e d  s e a r c h i n g  c e r t a i n  we  but  m e a n i n g f u l ,  be  I f  which  group  window  a  t r a c e s  o t h e r .  say)  1,  a r e  c o n s t r u c t  each  of  be  (5)  f i r s t  i n group  r e c o g n i s e  r e p e t i t i o n s  such  a  about  a  f o r  both  c h a r a c t e r i s t i c s  window.  a l . (1978) p i n g e r '  f o r  t o  i f a  t h e members  s e t . a  that  of  equation  from  t r a c e  t r a c e s  t o  e f f e c t ,  i n t h e  i n a  t h o s e  s t r i v e  c o n s i d e r  i n c h a r a c t e r  M i l l i g a n  from  d i f f e r e n t  a r e  t r a c e s  that  m u l t i c h a n n e l  t h e h o r i z o n  of  i n f e r  i n f e r  weight  then  w e i g h t i n g s  case  t h e e i g e n v e c t o r  say) has  component,  to  CLUSTER  4.  c o n s i d e r e d  data  w h i l s t  s h a l l o w  b a y .  t h e above mapping They  problem  f o r  t h e d i s t r i b u t i o n  a n a l y s e d  t h e  t h e of  bottom  120  r e f l e c t i o n w i t h  p u l s e ,  known  and  bottom  implemented  the  many  t r a c e s  more  elements  by  (1982)  (a  et  and  dominant  groupings  a of  Phase  to  t r a n s f o r m  of  the  In  (Levy  s e i s m i c  changes  the  wavelet  pulse  c h a r a c t e r  s u c c e s s .  because  d e r i v e d  i n  the  c h a r a c t e r  He  They  they  the  had  e i g e n v e c t o r  KL  the  a n a l y s i s  With  a  s i n c e  the  e.g.  we  and  I  f o r  He  KL  of  the  used  the  as  the  o b t a i n  for  t r a n s f o r m , the  the  of  of  n a t u r a l  groupings  i n t r o d u c t i o n  about he  was  Instantaneous of  the  complex  e i g e n s t r u c t u r e  d a t a .  found  c l u s t e r i n g f a u l t  value  t r a n s f o r m  as  of  p o r o s i t y  l a r g e  searched  r e a l  may  o r i g i n a l  examples,  t r a c e  the  of  d a t a .  KL  w e l l  s i n g l e  a  to  'instantaneous  decomposition  e i g e n s t r u c t u r e  a l . , 1983) of  as  to  h i s  r o u t i n e ,  e l e m e n t s ,  the  i n d i c a t i v e  c o r r e s p o n d i n g  w i t h  a p p l i c a t i o n  r e c o g n i t i o n  that  sometimes  the  for  the  noted  a s s o c i a t e d  data  d a t a ,  method  of  However,  t r a c e  dominated  was  the  et  t h e i r  p e r f o r m  l o c a t i o n s .  s y n t h e t i c  v a r i a t i o n s  and  f a c i l i t a t e  phase  e i g e n v e c t o r  complex  i n  (6),  r e p r e s e n t a t i o n  c l u s t e r  a l o n e .  KL  t o  change.  to  o b t a i n i n g  data  adapt  a l . , 1977)  Phase  d e f i n e d  l i m i t e d  to  data  e i g e n v e c t o r s  f o r  user  t r i e d  decided  Instantaneous  input  samples,  time  the  c o n s i d e r a b l e  equation  than  instantaneous  p o r o s i t y )  w i t h  v i a  c h a r a c t e r  (Taner  l a r g e  type  c o r r e l a t e  t r a n s f o r m  r e f l e c t i o n  s t r a t i g r a p h i c phase'  sediment  to  c o r r e l a t i o n .  Hagen s e i s m i c  KL  proceeded  that  the  were  the  or  steep  o f f s e t s  data  such  as  phase  d r i f t s ,  c o u l d  only  be  i s o l a t e d  when  or  f e a t u r e s gross  s t r u c t u r a l  d i p s . other  gross  which  S u b t l e r  changes  i n  s t r u c t u r a l  121  f e a t u r e s  A 2;  were  absent.  problem  reproduced  peaks  at  event  near  on  the  f o l d  i n  about  the  1.66  l a y e r  155  in  As  and  the  e x p e c t e d ,  e i g e n v e c t o r s  F i g u r e  near  t r a c e s  170  and  user  d e f i n e d  ( c o n t a i n i n g  two  method  Hagen  and  we i n  presented  examine  data  the  So,  to  and  and  which  h o r i z o n ) .  was  sound, using  the  1982,  I  the  w i l l  l y i n g and  F i g u r e  6;  simply  t e l l  the  data  extremal  essence  proceeded r e a l  changes Hagen's  the  i n  two  f i r s t  of  l i e i n  However,  a  f l a t  region  t r a c e s  and  i s  c h a r a c t e r  230  c e n t r a l  lower  l e v e l l e d  there  4.1b,  gross  the  a p p r o x i m a t i o n ,  214  one  s e t s  not  (Hagen  f i r s t  f l a t  were  otherwise  showing  t r a c e s  w i t h  F i g u r e  F i g u r e  ( p o s i t i v e  j o i n s  data  an  (1982,  event  Consequently  i n  a  Hagen  upper or  antiphase and  the  only  the  to  see  h o r i z o n s ) ,  ( c o n t a i n i n g  A l s o ,  about  l i e i n  r e g i o n s  l e a d  215.  by  pinches-out  4.1b)  c l u s t e r i n g  t r a c e s  that  a n a l y s e d .  roughly  i n  which  i s  225.  zone  c o n s i d e r e d  p e r t u r b a t i o n  reproduced  us  4.1a)  seconds)  c h a r a c t e r  s t r u c t u r e .  example  F i g u r e  t r a c e s  gross  second  w i t h  to  the  f o l l o w  h i s  complex  KL  i s o l a t i o n  of  and  t r a n s f o r m s .  I s u b t l e w i t h  i n t r o d u c e c h a r a c t e r  m i s f i t  c l u s t e r and  here  an  change  a l t e r n a t i v e from  r e c o n s t r u c t i o n  a n a l y s i s  c o n c e n t r a t e  on  on  the  This  procedure  had  phase  changes  s u b t l e  i n  dominant  ( d e f i n e d  by  e i g e n v e c t o r s  determining  a n a l y s e s .  the  approach  the  would an  be  the  s t r u c t u r e s . By  equation  (26)),  s e p a r a t e l y ,  groupings expected  otherwise  to  or  d i s p l a y e d to  f l a t  be  of  l y i n g  analogy  I  perform  i n  groups,  by use  these when  we  s t r u c t u r e .  122  The  h o r i z o n ' s  component  s t r u c t u r e  and  e i g e n v e c t o r .  The  c h a r a c t e r i z e d in  d e v i a t i o n  by  I  d i v e r g e  a  respect mode  o f  x.(t),  s l i g h t l y  membership t o  each  of  t h e given  of  That of  t h e  p r i n c i p a l  from  f o r on  i s ,  each  t h e  Hagen  t h e o f  be  seen  I  w i t h  s t a t i s t i c a l  s e i s m i c group  be  i n that  t r a c e s  t h e  i.th  f i r s t  would  a n d may  of  t o t h e j.th  t h e k.th  t h e  s t r u c t u r e  of  t h e b a s i s  p r i n c i p a l  of  components  f o r  from  f i r s t  makeup  t h e approach  belonging  obtained  t h e  t h e h o r i z o n t a l  t h e g r o u p s ,  group.  i n f o r m a t i o n  given  i n  p r o b a b i l i t y  t h e p r o b a b i l i t y  b a s i s  from  subsequent  e i g e n v e c t o r s .  a l s o  dominate  r e f l e c t e d  t h e i r  compute  is  be  would  G.,  p r i n c i p a l  t r a c e on t h e  component  by:  P{x.(t)eG.}  [1.0  =  / D{r .k;w.  k  }]/NORM  where:  D[r  ik' wjk ]  is  t h e  w.,, K  J  o f  c l u s t e r  =  I'/*  ' d i s t a n c e '  of  t h e e i g e n v e c t o r  "  w  jk\  t h e e i g e n v e c t o r elements  o f  element t h e  r . ^,  members  G. , a n d :  group  nc NORM  =  j= where  nc  [1.0  I  / D{r .k',w  k  ) ]  1  i s t h e number  o f  c l u s t e r  groups  formed.  from of  t h e t h e  mode  j.th  1 23  b.  S y n t h e t i c  To  b a s i c  f i r s t  two  l a y e r s  f a u l t .  s e p a r a t e l y  should  t h e i r The the were from each s i d e  group  r e s u l t s r e s u l t s  of  f i r s t  s i d e the  appears  on  compound  s e v e r a l groupings  h i g h l i g h t more  from a  of  the  a d v e r s e l y complex  using  f a u l t  w i t h  the  t o  as  of  o p t i o n  t r a c e s  per  the  the  were  s e p a r a t e l y a  as  equation  the  as  most  combines  (27), f i r s t  energy).  second  two I  groups  found  that can  of  to  that  t r a c e s  f a l l  i n t o  r e s u l t s . whereas  the  c l u s t e r  p a r t i c u l a r  which  t o t a l  the  data  groupings  i n d i c a t e  how  e i g e n v e c t o r . the  system,  ' e n e r g e t i c '  The and  event.  the  e f f e c t s  y i e l d e d  these  two  a  determined  o v e r a l l  phase  so  w i t h  i d e n t i c a l ,  dominates  these  so  the over  and  only  t r a c e s )  d i s p l a y e d  o f f s e t  f i r s t  general  e x p e c t s ,  a f f e c t i n g  dominates  combining 73%  one  i n  instantaneous  have  mode,  but  using  l a y e r s ,  groupings case  ('bad'  KL  e i g e n v e c t o r  c l u s t e r i n g  when  I  most  f i r s t  e i g e n v e c t o r s  f a u l t , than  the  c l u s t e r  c o n s i d e r e d  i n v o l v e d  s h i f t e d ,  t h i s  I  d i p p i n g  d a t a ,  In  d i s s i m i l a r  e i g e n v e c t o r s  the  f a u l t  ( c o r r e s p o n d i n g  the  i n f e r i o r . two  4.2c).  f i r s t  phase  the  method,  g e n t l y  these  and  groups  the  run  i.e.  shows  ( F i g u r e  without  s l i g h t l y the  4.2  t h i s  The  p r o g r e s s i v e l y  c o m p l e t e l y  of  The  to  3.2a,  4.2b),  them  request  are own  i s  of  examples.  F i g u r e  F i g u r e  from  a p p l i c a t i o n  data  ( F i g u r e  needed  which  of  wavelet  e i g e n v e c t o r s  one  the  s y n t h e t i c  the  v e r t i c a l  were  Examples.  demonstrate  four  which  Data  of same  e i g e n v e c t o r s  124 In  the  c l o s e l y  second  spaced  events  c e n t r e ,  and  dominant  s t r u c t u r e  i n d i c a t e y i e l d s  the  the  groupings  the  to  97%  phase  Again, the  d r i f t to  two  at  about  the  t o t a l  r e s u l t s  of  i s of  to  l a s t  the  t r a c e .  The  no  l a y e r s :  p a r t i c u l a r a b l e  to  phase  d r i f t ,  and  t r a c e s .  In  the  dominated  pronounced f i r s t  energy)  a l s o  l y i n g at  -n/2  r e a d i l y  c e n t r a l  the  f l a t  h o r i z o n t a l  the  from  two  shows  was  most  c o m b i n a t i o n the  the  p a r a l l e l  f e a t u r e s  the  see  i n c r e a s i n g  zero  c l u s t e r i n g  the  we  e i g e n v e c t o r  s u b t l e r  of  4.3,  e i g e n v e c t o r  i t c o n t a i n e d  (corresponding  and  of  symmetric  from  d r i f t .  phase  second  compound  as  F i g u r e  again  that  u n d e r l y i n g ,  groupings  i d e n t i c a l ,  a  f i r s t  However,  e i g e n v e c t o r , the  i s  the  the  i n s t a n c e ,  w i t h  f a i l i n g - o f f  consequently, s t r u c t u r e .  example.  phase  r e s u l t s  g e n t l y  l a y e r s  ( F i g u r e  the  second  s t r u c t u r e , two  and  e i g e n v e c t o r s  d e l i n e a t e d  complex  instantaneous  by  t h i s  RL  the  o p t i o n  were  were  s l i g h t l y  i n f e r i o r .  In and  the  do  moving  see  grouping change  tendency  these  are  However,  the  c l u s t e r s  of  f o r  across  would  a  d i p  evident second  the  ( i n  to  not  c o n t a i n I f  the  the  the  4.4)  groupings shows  example  of f o r the w i t h  expect i n  data  had  and  no  C o n s e q u e n t l y ,  absence  we  t r a c e s  the  e x i s t  p r o g r e s s i v e l y  e i g e n v e c t o r  s i m i l a r  to  s e c t i o n .  e v i d e n t .  and  i n  c l u s t e r s  would  be  s y s t e m a t i c a l l y  i n d i c a t i v e  d i p p i n g  groups  p a t t e r n s  r e s u l t s  of  a  of  p r o g r e s s i v e l y  f l a t t e n e d ,  be  case  groupings the  phase  c h a n g e s ) . f i r s t  symmetric a  been  p a r t i c u l a r  a c r o s s  the  groups  phase  which  s e c t i o n  w i l l These  e i g e n v e c t o r . d i s t r i b u t i o n  d r i f t  i n t o  the  125  c e n t r e  o f  t h e  e i g e n v e c t o r a  i s t e l l i n g  c o n s t a n t l y  time as  s h i f t s . a  d i p p i n g This  determine  a s  of us  waveforms  p a t t e r n  f l a t  d i s t i n g u i s h e d  by  methods  e s s e n t i a l l y  F i n a l l y , w i t h  an  I  c o n s i d e r  anomalous  or  sand  l e n s :  so  much  from  i s  spread  groupings  see F i g u r e t h e f l a t  over  s e v e r a l  c o n t a i n i n g  c l u s t e r  a n a l y s i s  shows  t h e  r e s p e c t i v e l y . d e l i n e a t e poor.  zone  of  r e s u l t s Both  t h i s  uses  a  Two  o f  ( i nt h i s  case  events,  t h e f i r s t f o r  f e a t u r e ,  4  t h e  r e a l  a  Because  seen second  m i s f i t s  d r i f t  i n  cases  c o u l d  n o t  t h e r e a l  t o  i ft h e  phase  g e n t l y a  hump  and complex  t h e be KL  c a n be F i g u r e  b u t t h e instantaneous  KL  seen  w i t h i t t h e  i n t h e  4.5b and  c ,  e i g e n v e c t o r s ,  methods phase  reef  d e v i a t e s  Consequently,  f o u r t h  complex  hump  a  a s s o c i a t e d  i n t h e hump  and  h o r i z o n  r e p r e s e n t i n g  t h e energy  f i r s t  d i p p i n g  t h e c e n t r a l  e i g e n v e c t o r s .  and  f o r t h e  obtained  components.  t r a c e s  i s  would  be  f o r  r e s u l t s .  t h e case  4.5a).  s t r u c t u r e  o f  both  second  ' s t a t i c '  squares  such  Again,  t h e same  t h e  antisymmetric  constant  p r i n c i p a l  t h e  t h e  t h e  s t r u c t u r e , which  groupings  r e s u l t s  e v e n t .  l y i n g  because  i n t h e r e s i d u a l  w i t h  method.  i s  of  i n t h e c l u s t e r  l a y e r s  t h i s  s e r i e s  S i m i l a r  t h e  This  t h e r e s i d u a l  i s a  t h e a l g o r i t h m  across  y i e l d  about  l a y e r  t h e groupings.  comprised  t r a c e s .  antisymmetry  symmetric  e i g e n v e c t o r ,  data  group  c l e a r l y  r e s u l t s  were  1 26  c.  Real  Data  1. from  a  Examples.  The  f i r s t  r e a l  p r e d o m i n a n t l y  dominates event  the  (2)  f l a t  and  the  s i g n i f i c a n t  g r o u p i n g s .  38  -  94.  f i r s t  shows  zones  three  f l a t  54),  and  The  t h i r d  2. f a u l t  The  and  f o u r t h  r e a l  58  the  data  (Figure  f i r s t  three  e i g e n v e c t o r s ,  d).  For  to  (B)  c l e a r l y  t h i s  c o r r e s o n d s  f a u l t ,  a  d e l i n e a t e  shows  t h i r d a  second  i s  l e f t up  the  of i n  the the  anomaly  group f a u l t  1.66  ( t r a c e s  c e n t r a l  ( F i g u r e to  (A)  the  45,  (B)  the  28,  and  73  and  d)  a l s o  18  -  t r a c e s 4.6c  and  zone.  of  r i g h t  o f f s e t  the  membership,  below  (Figure  group,  However.,  s.  namely To  brackets  t r a c e s  to  the  near  A, a  60  t r a c e  4.7b,  c,  c o n t a i n s  t r a c e s r i g h t to 80  76, at  1.6  a the  based  small  the  by  f a u l t  group  C  r i g h t  and  between  groupings; at  c o r r e s p o n d  s  f i r s t  l e s s  eigenvector  the  f a u l t .  are  p a r t i c u l a r l y  events  shown the  no  1  (1)  i n t e r m i t t e n t  comprises  c l u s t e r  e i g e n v e c t o r ,  the  to  The  to  t r a c e s  event  (3)  ( F i g u r e  d i s r u p t e d  4.7):  the  t r a c e s  0.61  100  upper  event  zone  example  s l i g h t l y .  the  at  of  weak  t r a c e s  e i g e n v e c t o r s  drop  of  the  which  uniform  h o r i z o n s  f i r s t  while  between  trough  f a i r l y  of  strong  second  groups,  deep  i s  showed  the  s t r u c t u r e  l o c a t i o n  t r a c e  the  The  4.6)  d i s c o n t i n u o u s  main  the  second  near  zone.  However,  the  (C)  the  (Figure  eigenvector  l y i n g  c o n t a i n i n g  emphasize  l y i n g  lower  The  predominant  example  s t r u c t u r a l c h a r a c t e r ,  c o h e r e n t .  4.6b),  data  on and  most  anomaly 33  -  of  40: the  and s.  may  127  The  g r o u p i n g s  e i g e n v e c t o r the  show  c e n t r a l  a  f a u l t .  a n t i s y m m e t r i c ,  d e t e r m i n e d symmetric  These  as  d i f f e r e n c e s .  denotes  the  c e n t r a l  3.  The  next  example,  a  marked  4.8).  can  be  of  i n t e r e s t  e i g e n v e c t o r  the  of  data  t r a c e s  In  the  from 29  a  members  a  of  (at  0.67  ( F i g u r e group  -  70.  4.9d)  f l a n k e d  of the  the  second  symmetry  about  rather  than  symmetric  c r i t e r i o n  was  based  e i g e n v e c t o r  t r a c e s ,  near  i n  again  t h i s i n  on  the  c l e a r l y  4.9c),  resembles members  i s the  (A) we  a l s o  r e f l e c t i o n  the of  f i r s t a  20  of  again  and  the the  near  t r a c e  (Figure  4.8b  denotes  and  which  s e c t i o n  (Figure  near  4 . 9 ) , seen f i r s t of  (Figure  becoming  the  event  t h i r d s l i g h t  t r a c e  85  36,  (these  f i g u r e ) .  region  group  changes,  the  c a s e , the  the  event  an  appearance  c l e a r l y  t r a c e  bar  of  across  the  (Figure  The  i s  e i g e n v e c t o r s  i s  the  b a r .  In  two  which  example  where  by  s  marked  (Figure  s)  are  change  here  d i f f e r e n t  85,  96  the  of  group  e i g e n v e c t o r 65  sand  i n  2.04  f i n a l  a  and  d e l i n e a t e  t r a c e s  are  of  4.8d),  t r u n c a t i o n  d i s c o n t i n u i t i e s  4.  seen  at  b e l y i n g  t h i r d  c h a r a c t e r  ( F i g u r e  d i s c o n t i n u i t i e s and  The  b a s i s  f a u l t .  undergoes  A l s o  p a t t e r n  c l u s t e r i n g  of  c ) .  the  g r o u p i n g s  the  squares  This  on  note  I  at  second  about  the  0.1  0.68  bar  (B)  4.9b).  In  a  window  s,  between  c l u s t e r i n g s b r a c k e t e d  by  second  d i s c o n t i n u i t y  between  of  The  i t  group.  s  the  c h a r a c t e r  that  a  e i g e n v e c t o r  s t r o n g e r . i n  show  the  t h i r d  denotes  upper  event  e i g e n v e c t o r a  c e n t r a l  128  d.  D i s c u s s i o n .  For  data  c l u s t e r  i n which  a n a l y s i s  e x p l o r a t i o n i s t p r e s e n t a t i o n  may  may  these prove  dominant provide  i n l o c a t i n g of  p r e s e n t a t i o n  t h e  a  unusual  data. q u i t e  s t r u c t u r e s u s e f u l  f e a t u r e s .  A  colour  u s e f u l  a r e  t o o l Much  t o  a i d  t h e  on  t h e  depends  o v e r l a y i n g  i n conveying  p a r a l l e l ,  technique  t h e  of  i n f o r m a t i o n  c l e a r l y .  F e a t u r e s a l l  f a l l  the  in  many  order  of  groups.  s e n s i t i v i t y c l u s t e r  f u l l o f  T h i s  where  t o  as  being  much  w e l l and  can  p i t f a l l  with  n o n - p a r a l l e l i s m on  problem  e a r l i e r .  t h e as  ( i f  areas  meaning  of  data  t h i s  of  However,  as  s e t s  input  be  necessary  t h e  method.  t o give seen  an  i n t h e  a c c o m p l i s h e d  i n an  a r e a v a i l a b l e .  i s l i k e l y  data.  eigenvector  t h e m i s f i t  of  i s  i s needed  technique  i n t h e  has  t h e r e s u l t s  r e a d i l y  l e n s e s ,  t h e method  study  any)  s t u d i e d  b a s i s  does  f o r which  i n t e r a c t i v e  most  and  ' d i f f e r e n t ' .  p o t e n t i a l  t h e a p p r o p r i a t e  a n a l y s i s  t h e same  p r e s e n t e d  t h e  p i n c h - o u t s ,  s t r u c t u r e s  techniques,  a n a l y s i s  major  b o u n d a r i e s , of  t h e r e l i a b i l i t y  environment  The  f a u l t  i d e n t i f y i n g  t o a s s e s s  c l u s t e r  from  of  novel  Comprehensive o u t l i n e  as  i n t h e c a t e g o r y  p o t e n t i a l  with  such  I n  t o  be i t s  t h i s  regard,  elements  s u f f e r s  r e c o n s t r u c t i o n  technique  129  ISO  160  170  180  200  190  210  a  220  230  WELL CLASSIFICATION • POROUS O NON-POROUS  • 10  — 1st COMPONENT — 2nd COMPONENT -- 3rd COMPONENT  WELL LOCATIONS  IA! - 10"  J  150  I  I  L  160  170  180  1/ J  190  200  210  220  L  230 240  FIGURE 4.1  (a) T h edata used b y Hagen ( 1 9 8 2 ) f o r h i s c l u s t e r a n a l y s i s . Note the c e n t r a l p a i r o f events which a r e f l a n k e d b y s i n g l e h o r i z o n s (A) . (b) T h e f i r s t three e i g e n v e c t o r s f o rd e c o m p o s i t i o n o f t h e data i n (a). Note t h e antiphase r e l a t i o n s h i p between t h e f i r s t a n d second e i g e n v e c t o r s , a n dhow t h e c h a r a c t e r change c o i n c i d e s w i t h t h e p i n c h - o u t s o f t h e f l a t c e n t r a l region o f t h e data (diagrams from Hagen, 1982).  1 30  1.0  6.0  1.0  6.0  11.0  16.0  21.0  26.0  11.0  16.0  21.0  26.0  FIRST EIGENVECTOR  SECOND EIGENVECTOR  FIGURE  4.2  (a) The s e i s m i c r e p r e s e n t a t i o n o f t h e upper about 2 ms p e r t r a c e , o f f s e t by a v e r t i c a l  two l a y e r s f a u l t .  d i p p i n g  The f i r s t two e i g e n v e c t o r s (a a n d b) w i t h t h e c l u s t e r memberships denoted on them. Note how t h e t r a c e s o n e i t h e r of t h e f a u l t f a l l i n t o separate g r o u p s .  a t  group s i d e  131  FIGURE (.,  The  d r i f t  seismic i n c r e a s i n g  - p r e s e n t a t i o n to  t o ^ T h e ^ l u s t e r i n g s p a t t e r n s , ( C  )  as  The out  at of  dominant  c l u s t e r i n g s  d i s t i n c t i v e and  the  ir/2  groupings  a g a i n .  tne  the  of  4.3 two  c e n t r e  f i r s t  e i g e n v e c t o r  «e secona  s t r u c t u r e f o r  ,  «l.t^"^^cAo^ero  d e l i n e a t i n g  is  the  show  f l a t .  phase  no  p a r t i c u l a r r  y d r i r r  d  i c  s e  p n  l t  a r  y e  132  FIGURE (a) t r a (b) mem c h a l a y  c b r e  The s e i s m i c r e p r e s e n t e . The f i r s t e i g e n v e c ership moving p r o g r a c t e r i s t i c of a d i p , r s .  a t i o n  of  4.4 two  l a y e r s  t o r d e l i n e a t e s e s s i v e l y across or of a c o n s t a n t  d i p p i n g  a t  2  ms  per  groupings w i t h t r a c e the s e c t i o n . This i s phase d r i f t over f l a t  (c) The second e i g e n v e c t o r d e l i n e a t e s the r e s i d u a l s t r u c t u r e , which f o r a c o n s t a n t l y d i p p i n g l a y e r i s an a n t i s y m m e t r i c f e a t u r e .  133  FIGURE 4 . 5 (a)  The  s e i s m i c  r e p r e s e n t a t i o n  of  two  f l a t  l a y e r s  with  a  c e n t r a l  hump. The  f i r s t  f e a t u r e .  (b) and  f o u r t h  ( c ) eigenvectors  c l e a r l y  d e l i n e a t e  t h i s  134  0.00  2.00  oo 4.00 G, 10 TRACE NUMBER  B.00  10.00  8.00  10.00  UJ  o -0  2.00  4.00  TRACE  ?.00  NUMBER  4.00  6.00  mIO  6.00  TRACE NUMBER «10 FIGURE  10.00  4.6  (a) One hundred t r a c e s from a r e a l stacked seismic s e c t i o n , with three c h a r a c t e r i s t i c events: the strong upper event ( 1 ) , the weak i n t e r m i t t e n t event ( 2 ) , and the lower event ( 3 ) . The f i r s t e i g e n v e c t o r i s f a i r l y f l a t , hence shows no c l e a r groupings and i s not shown. (b) The second e i g e n v e c t o r shows three major groups which correspond to (A) the f l a t l y i n g s t r u c t u r e between t r a c e s 1 and 45,(5)the deep trough at 0.61 s ( t r a c e s 18-28 and 38-54), and (C) the f a i r l y uniform zone between t r a c e s 73 and 94. (c) The t h i r d e i g e n v e c t o r appears to r e f l e c t c h a r a c t e r change i n the trough below the f i r s t event ( 1 ) , while the fourth e i g e n v e c t o r ( d ) , again demarcates events to the r i g h t and l e f t of the c e n t r a l d i s c o n t i n u o u s zone.  TSN  20  40  TRACE  GO  80  TSN  1  3 5  NUMBER » 1 0  • o O o  CM  CC — CJ UJ  I u  UJ  1  0.00  2.00  1 4.00  TRACE  1 6.00  '•—I  8.00  10.00  8.00  10.00  NUMBER « 1 0  • o O o  cc — CJ UJ  > < L U ^" — J  " 'o.oo  2.00  4 . 0 0  6.00  TRACE NUMBER a!0  FIGURE 4.7 (a) N i n e t y s i xt r a c e s from a f a u l t e d zone. (b) F o r t h e f i r s t e i g e n v e c t o r ( 1 ) ,group (A)c o n t a i n s most o f t h e t r a c e s t o t h el e f t o f t h ef a u l t . A small anomaly ( B )c a n be seen between t r a c e s 3 3t o 4 0a t 1.66 s . To t h er i g h t , a t h i r d group may d e l i n e a t e a second f a u l t ( C ) near t r a c e 7 6 . (c) T h e g r o u p i n g s from t h esecond e i g e n v e c t o r show a symmetric p a t t e r n b e l y i n g t h e symmetry about t h ec e n t r a l f a u l t , a s does t h e t h i r d e i g e n v e c t o r ( d ) .  1 36 TSN  2.0  a :: 2.1  cc —  o°  >—  CJ UJ  :> o '0l00  0  2T0O TRACE  [D  0  TToO  '0.00  NUMBER  1  1 6.00  TRACE  sloO  10.00  1 -  10.00  MIO  4.00  T 2.00  6loO  NUMBER  «10  NUMBER  »10  8.00  Oo  " '0.00  10.00 TRACE  (a) N i n e t y the e v e n t . change. (d) The s u b t l e and t r a c (marked  s i xtraces The f i r s t  c l u s t e r i changes e 36 a t #1, # 2 ,  FIGURE 4.8 which change n o t i c e a b l y i n character a c r o s s two eigenvectors (b a n d c ) b e l i e t h i s  n g s from t h e t h i r d e i g e n v e c t o r c l e a r l y denote i n t h edata: namely d i s c o n t i n u i t i e s near t r a c e 20 2.04 s , t h e t r u n c a t i o n o f t h eevent near t r a c e 85 a n d #3, r e s p e c t i v e l y ) .  FIGURE  4.9  (a) One hundred t r a c e o f 0.1 s o f data from a region of b r a i d e d sand l e n s e s , i n a p o t e n t i a l p r o d u c t i o n zone. Note t h e p e r s i s t a n c e of a t h i n p o s i t i v e e v e n t a t 0.68 s , b e t w e e n t r a c e s 29 a n d 7 0 . In t h i s c a s e , t h e c l u s e r i n g s from both t h e f i r s t and second e i g e n v e c t o r s (b and c) d e l i n e a t e a group i n t h e region of t h e b a r (B) bracketed by members o f a d i f f e r e n t group ( A ) .A l s o i n t h e second eigenvector we n o t e a d i s c o n t i n u i t y b e t w e e n t r a c e s 65-85 where t h e r e f l e c t i o n c h a r a c t e r near 0.67 s c h a n g e s , becoming s t r o n g e r ( C ) .The t h i r d e i g e n v e c t o r (d) a l s o demonstrates t h e p r e s e n c e of a c e n t r a l group, f l a n k e d by members o f a second group.  138  SECTION  a.  4 . I I :  t h i s  c o v a r i a n c e c r i t e r i o n ,  t r a c e s w h i l e  I  m a t r i x  can  used  x(m)  With  s e c t i o n  which  commonly  The  amount  equation  to  of  (33).  i m p o s s i b l e  to  p r i n c i p a l  component;  So,  for  small  the  s t a t i c  c o h e r e n t , e f f e c t s  CKL  of  and the  then  have  time the  VA  time  be  N e i d e l  note  for  from  s u p e r i o r and  the  c o r r e l a t i o n to  the  Taner,  1971).  n o r m a l i z e d  input  s i g n a l s  are  p e r f e c t l y  the  s i g n a l s  are  n e a r l y  by  a  been a  s t a t i c the  more  v e l o c i t y  (eg.  may  a  the  s h i f t  r e p r e s e n t  e s t a b l i s h  c o n d i t i o n s  we  d e r i v e d  c o r r e l a t e d  o r t h o g o n a l .  S h i f t s .  p r o v i d e  the  to  (21)  00  =  s t a t i c As  used  equation  which  e i g e n v a l u e s  c r i t e r i o n  S t a t i c  x(l)  the  c e r t a i n  - l/(n-l)  of  how  be  \(1)  w a v e l e t s  expect  the  in  i f  E f f e c t s  a l s o  under  d e f i n e d  x(1)  i f  show  semblance  that  For  if  ANALYSIS.  I n t r o d u c t i o n .  In  b.  VELOCITY  good i s  l e s s  energy  p r i n c i p a l  s h i f t  w i l l  should  be  than  i n c r e a s e s  s i g n a l  be  in  of  t h a t  a  s e n s i t i v e  the  so  c o r r e l a t i o n  to  i t  of  w i l l  as  we  p r e s c r i b e d  terms  wavelet  r o t a t e d  s h i f t  however,  components  (VA),  l e s s  s t a t i c  i n d i c a t i o n  s h i f t  a n a l y s i s  s h i f t s .  o f f s e t  w i l l  a  be  the  be  s i n g l e r e q u i r e d .  d i s p l a c e d to  by  appear  by  a more  degrading  139  c.  S y n t h e t i c  set was  Data  In  t h e f o l l o w i n g  t o  16  samples,  narrowed  VA  methods  is  t o o wide  narrow  or  a  events most  w i l l  complex  KL  t r a c e s  s t a t i c  time  s t a t i c  samples.  The  s t a t i c  f i g u r e  a r e  x(m)  v a l u e s  RKL  VA  of  we  s h i f t s  an  from  of  shown  t h e  s t i l l  be  t h e number  ( i . e . i n c r e a s i n g  and  m  1.0  s)  time  o f  an  s h i f t . even  e i g e n v a l u e s i n x(m))  f a i r l y  r e a l  t h e case  of  of  1500 m / s ) .  s u b j e c t e d of  ^  ± 8  t h e  ± 6,  ± 8  VA  l a r g e  of  s t a t i c  as  of  t h e peak  e f f e c t i v e l y  time  smearing  r i g h t  f o r each how  time  smeared  t h e v e r t i c a l t h e  t o  semblance  and  i n t h e numerator  . T h i s  a r e  very  t h e  i s q u i c k l y  Note a t  KL  which  a  v e r s u s  maximum  On  RKL  and  w i t h  v e l o c i t y  note  r e s u l t s .  window  window  c o n s i d e r  t o a  ± 2,  A l s o  i d e n t i f i e d  of  I  ± 4,  ± 0,  was  component.  <  VA  a  p r o g r e s s i v e l y  zero  t h e  they  semblance  r a t i o ,  As  w h i l e  column  r e s u l t s  s t a t i c  w i t h  i n t h e l e f t  i n c r e a s e s . semblance  of  were  see  mean  window  semblance  i n that  a s s o c i a t e d  ( a t 1500 m/s,  f o r each  c a n  i n c r e a s i n g  4.11  s h i f t  c h a r a c t e r i s t i c  r a n g i n g  time  peak  common  e i g e n v a l u e  4.10)  because  s i m i l a r ,  time  w i d t h .  d i s s i m i l a r ,  look  l a r g e  t h e VA  f o r both  e x p e c t e d ,  s , w i t h  s h i f t s ,  VA  f o r  a  (Figure  I n F i g u r e  t h e r e s u l t s  as  of  t h e wavelet  t h e performance  ( a t 1  samples.  o f  appear  m o d i f i e d  event  Twelve  (21)  have  t h e width  2/3  events  demonstrate  s i n g l e  the  about  broadened,  most  window  To  t r i a l s ,  d e t e r i o r a t e d ,  constant, and  and  Examples.  t h e  s e v e r a l of  t h e  s h i f t s  of  i n c l u d e s  by  equation more  140  of  the  s i g n a l  This  the  w i t h  the  peak  KL  as  VA  i n c r e a s i n g  (21).  method  the  T h i s  i s  of  because  white  the  c o v a r i a n c e  elements  on  the  top  are  shown  4.13).  f o r  5  noise  Although  i n c r e a s e  i n  r e s u l t s .  In  l i n e  comparison  the  noise other  noise  l e v e l s  n o i s e  l e v e l s  (<30%  are  of  the  l e v e l s  the  than by  for  KL  i s  the  the 4.12).  l o c a t i o n  case nothing  the  adds  of  of  i n c r e a s i n g  was  numerator  energy  the  semblance,  RKL  and  CKL  85,  and  100%)  70,  r e s u l t s remain VA  noise  degrade  s u p e r i o r  method  i s  semblance.  amplitude)  the  number  of  energy VA  i n  the  r e s u l t s i n  both  more  F i g u r e w i t h  the  KL  s e n s i t i v e  However, the  to  s e v e r e l y  to  by  equation  u n i f o r m a l l y  i n c r e a s i n g  55,  g a i n e d of  more  the  maximum  denote  the  i n  and  (40,  they  to  and  ( F i g u r e  i n c l u d e s  semblance l e v e l ,  VA,  again  i n s t a n c e  m a t r i x  CKL  (21).  i n t r o d u c e d .  noise  merely  words,  high  a b l e  equation  of  the  made  e i g e n v a l u e s  of  A  f o r  r e a d i l y  t h i s  diagonal  numerator.  numerator  performed  In  number  the  r e s u l t s  s h i f t s  were  l e v e l s .  i n  repeated  i s  time  t r i a l s  noise  was  semblance  s t a t i c  S i m i l a r white  energy  e x c e r c i s e  comparison Again,  r e l a t e d  KL  at  VA to  moderate  methods  are  s u p e r i o r .  To  examine  the  the  above  repeated p r e v i o u s  example,  moveout  was  I n c r e a s i n g  the  r e s o l u t i o n example  the  v e l o c i t y  pronounced  and  v e l o c i t y  reduces  of  c l o s e l y  w i t h was the the  a  spaced  four q u i t e  ' t h i n  l a y e r low  a l g o r i t h m s c u r v a t u r e ,  model.  and a l l  and  beds', In  hence worked  we  expect  I the the  w e l l . to  141  see 12  a  smearing  s y n t h e t i c  of  the  peaks  seismograms,  i n t e r f a c e  r e f l e c t i o n s  v e l o c i t y :  1800  50  m/s  m,  1950  s t a c k i n g  three  x(m)  r e s u l t  for  x(2)  the  a m p l i t u d e .  In of  a  by  the  white  the  4  e a r t h  noise  s t a b i l i z e i s  v e l o c i t y and  c,  be  t r i p l e t  that  the  4.16a  were the  performance  p r e s e n t . are  The shown  r e s p e c t i v e l y .  vague  t h i s  The  a f t e r  gain  f i r s t The  RKL x(1)  CKL  smeared,  zone  f a i l s of  be  high b e t t e r  c o n c l u s i o n  I  present  input  by  i s  not  the  the  small  at  adjacent  amount  zones  semblance,  c o n t r o l  i n  case  seismograms,  to  a  the  a m p l i t u d e ) ,  a l g o r i t h m of  the  r e s u l t s .  s i g n a l  r e s u l t s  from  VA  should  correponding  the  the  somewhat  a  c o n t a m i n a t e d  of  model,  the  i n t o  examples,  maximum  and  shown).  semblance  method  and  (the  The  However,  data  a  4.15  i s  three  m/s;  r e s u l t s  are  the  the  1900  whereas  VA  sequence.  F i g u r e  of  runs  subsequent  r e f l e c t i o n  5%  KL  KL  m,  The  F i g u r e  seen.  shows  t h i c k n e s s  such  i n  x(2)  the  s y n t h e t i c  ( b ) ,  i n  events,  CKL  of  to  a n a l y s e s  the  m a j o r i t y  model  the  can  50  l i t t l e .  t h r e e  the  ( w i t h  m/s;  that  shown  events.  ms,  (up  1850  m,  4.14  r e p r e s e n t i n g  50  p r o x i m a l  f o l l o w i n g  at  s i g n a l  b,  i n d i c a t e s  m u l t i l a y e r e d  l a y e r e d  events  ms,  l a y e r s  f o r the  4  F i g u r e  four  are  and  smears  r e s o l v e  sampled  to  and T h i s  confirmed  VA's  map.  at  very  VA's  w e l l  t h r e e  c o m p l e t e l y ,  to  CKL  separates  l e s s  although  a b l e  and  VA  Note  change  c r i t e r i a  does  m/s;  r e s p e c t i v e l y ) .  RKL  the  sampled  from  1800  v e l o c i t i e s  semblance,  VA  m,  in  of  p r i m a r i l y where  RKL,  F i g u r e  and  no CKL  4.17a,  142  We x(3)  ,  than  - see  that  the  r e s p e c t i v e l y ) does  s t a c k i n g  the  KL  give  r e s u l t s  b e t t e r  semblance.  v e l o c i t i e s  are  t o  However,  the  balanced  w i l l  y i e l d  of  the  peak  of  1.0.0  and  In  VA  maps  the  l a r g e  the  the  as  i n  s,  when  ±12  s t a r t  ms  chosen  needed  ( F i g u r e  those  i n  F i g u r e  x(3)  p l o t s  v e l o c i t i e s . m u l t i p l e  where  the  I  important  s t a c k i n g  was  from  t o  r e s u l t s  able  x(2),  t o  of  and  e v e n t s  advantages  recover a r e  when  i n t e r v a l  f a i r l y  v e l o c i t y  evenly  c u r v e .  d e l i n e a t e  i t s adjacent  demonstrate  a r e  t o  the  4.17,  but  both  peaks  semblance  (by  the  peaks  None  c e n t r a l  ( a t  other appear power  the  the  peaks  i s  s h i f t s or  more  s i g n a l  shown and  f o r  S t a t i c  d e l e t i n g  b,  RKL  hand,  performance  about  the  c)  i n are  CKL  l a r g e ) .  a d d i n g  v a l u e s zeroes  as t o  and  x(3)  was  c r i t e r i o n .  The  d a t a ,  VA's  p r i n c i p a l  4.18.  c o n s i d e r a b l y  times  VA  energy,  F i g u r e  semblance  c e r t a i n  w i t h  the  one  e x h i b i t e d  and  of  than  c o r r e l a t i o n  are  the  the  C o n s e q u e n t l y  best  s t a t i c s  the  p r e s e n t .  represent  4.19a,  f o r On  s  t r a c e ) .  as  w i t h  r e s u l t s  good  i n t r o d u c e d  each  v i s u a l l y  a  x(D,  r e s o l u t i o n  i n v e r t e d  semblance  1.06  example  are  i s  be  example  s t a t i c s  of  contaminated  at  has  f o r  r e s p e c t i v e l y ) .  next  component  and  t h i s  t r i p l e t  1.12  a l g o r i t h m s  a l s o  temporal  T h i s  v e l o c i t i e s . and  (presented  by  the  are  at  r e s u l t s ( e . g . at  The  worse x(2) the  VA than and  c o r r e c t  a r e  smeared  1.34  seconds  143  d.  Real  Data  The gather The 72  Examples.  16 shot  t r a c e s i n  o f f s e t s m,  and  s e l e c t e d VA  northern  range  the  shown  data  so  as  582 i s  to  to  at  Chevron  1662  1.6  m,  s.  p r o v i d e  4.20a  1.2  s  a  t h i s of  taken  Canada  w i t h  For  are  from  CSP  Resources  geophone  L t d .  spacing  examination,  zero  a  o f f s e t  I  time  of have  i n  the  map.  The  semblance  r e s u l t s  t r a j e c t o r y ,  but  attempt  determine  a  windows:  one  to  two  small  0.9  -  VA's  i s  shown  Both  the  RKL  The  1.1  RKL  for in  F i g u r e  A l b e r t a ' b y  from  basement  i n  s.  and  A  A  and CKL  the w e l l the  l a c k at  to  CKL  from  4.23,  problem  s t a r t  t i m e s . i n c r e a s e  s;  to  use  i n  -  0.2  the  a f t e r  the  a  to  1  KL  VA,  s,  x(D,  data  an  s e l e c t e d  one RKL,  from and  CKL  r e p e c t i v e l y . or  x(2),  sample  a p p l i c a t i o n  In  I  windows  using  smeared  s.  and  semblance,  two  complete  show  down  f o r  suggested  are  of  an  AGC  the  KL  VA  x(3). shown  o p e r a t o r ,  r e s p e c t i v e l y .  a r i s e s  down the the  the map,  x(2)  general i n  made  r e s u l t s  the  t h a t  of  The m  f o r  4.20b)  be  0.0  for  v a l u e s ,  and  0.4  of  4.21  maps  x(m)  near  x(m)  r e s u l t s  c o n s i s t e n c y  the  region  g r e a t e r need  of  r e a d i l y  good  F i g u r e  VA  4.22  major  can  comparison  i n  a l l three F i g u r e s  p i c k s  ( F i g u r e  x(™)  with  w i t h  VA  map.  but and  both  not x(3)  trends  A  c r i t e r i o n  x(1)  l a t e r .  None  c r i t e r i a  are  i n c r e a s i n g  those time,  methods  work  work  expected, as  the  i s  works w e l l  w e l l i.e.  i n at we  moveout  144  hyperbolae  a r e becoming  f l a t t e r  and  s t a t i c  s h i f t s  w i l l  be  more  d u r i n g  t h e  (von Breymann  and  n o t i c a b l e .  The  second  UBC-RECOPE Clowes, data  shot  1630  The  (the  water  To two  However,  o v e r a l l x(4)  both  an  average  o f f s e t  two  t r a v e l  c r i t e r i o n  windows  were  chosen  proved  t o  these were by  o b t a i n e d  t h e sample  and  CKL  and  From  t h e semblance  x(2)  r  =  2.95  s , V  r  =  3.50  s , V  t  =  4.25  s , V  a r e  R M S  R M S  spacing time  of  x(l)  a r e  f o r  and  and The  shown  p i c k s  =  1500  m/s;  =  1600 m/s;  =  1700  m/s.  and  2.9  from The  VA  t o 4.4  s  t h e  KL  V A ' s ,  s:  see  F i g u r e  4.0  as  x(2),  a f t e r  t h e  and  r e s u l t s  F o r t h e KL  c a n be  range  3.0 s ) .  AGC  n o t from  i n  made f o r :  best w i t h t h e  F i g u r e s  V A ' s , r e s u l t s  shown.  map,  ( s e e F i g u r e  1 2 0 m.  window  t o use  windows.  VA's  km)  m i s l e a d i n g ,  u s i n g  r e f l e c t i o n  c o n s i d e r  ( a t 3.0 be  angle  (>2  i n a t about  4.27, r e s p e c t i v e l y .  x(1)  I  r e c e i v e r  way  wide  water  subset  x(m)  RKL,  4.26, and  f o r t h e data  c o l l e c t e d  1982  of  deep  which  suggested  semblance,  I t c o n s i s t s i n  was  i n  comes  r e s u l t s  a s  4.24b,  time  R i c a  r e f l e c t i o n  determine  small  4.25).  bottom  examine,  Costa  a i r g u n  w i t h  t h e zero  s e t I  1984).  an  o f f s e t s  t o 4415 m span  i n  comm.,  w i t h  maps  data  p r o j e c t  p e r s .  4.24a).  r e a l  f o r  145  From  t h e RKL T  =  2.95  s , V  r  =  3.04  s , V  T  =  3.18  s , V  T  =  3.45  s, V  T  =  4.23  s , V  f rom  s , V  7 =  3.04  s , V  r  =  3.18  s , V  T  =  3.45  s , V  T  =  3.90  s , v  T  =  4.45  s , v  t h e RKL  and  4.43  the had  VA  s)  t h e wavelet  from  VA's  were  expected  D e c r e a s i n g  RMS RMS RMS RMS  AV  x(2)),  and  =  1500  m/s;  =  1550  m/s;  =  1580  m/s;  =  1650  =  17 10  (x(0  RMS RMS RMS RMS RMS RMS  which  m/s;  =  1550  m/s;  =  1580  m/s;  =  1640  m/s;  =  1690  m/s;  =  17 5 0  m/s;  some  t r a i l i n g  ( t h e VA  i t s o r i g i n a l  -  t h e  u n c o r r e l a t e d d e g r a d a t i o n  KL  VA's.  h i g h o f  2  t h e  85  o f  Hz  x(m)  energy,  c r i t e r i o n .  t o  shows  g e n e r a l  t h e  -  s h o u l d and  50  r e f l e c t i o n s  w a v e l e t . A l l  of 50  reduce thus  U n c o r r e l a t e d  as  I  a r r i v a l s . t o  Band Hz  a t  s t r u c t u r e  appearance,  from  r e s u l t s . 5  ( e . g .  i n t e r n a l  d e f i n i t i o n  t h e KL down  made f o r :  a r e seen  t h e  increment)  n o t  be  t h e downgoing  and  Bandpassing  frequency  t o  i n t h e i r  v e l o c i t y  data  made f o r :  can  a r r i v a l s  i . e . t h e map  c l a r i t y  b u t  be  p i c k s  due  edges  d i s s a p o i n t i n g  t h e semblance,  improve  double  a r e p r o b a b l y  g r e a t e r  can  and  x(2)),  1500  r e s o l v e d ,  and  p i c k s  m/s; and  improved from  m/s;  =  maps,  b e i n g  t h e l e a d i n g KL  RMS  CKL maps  2.95  In  of  the  =  r  3.20  (x(D  maps  25  p a s s i n g  a g a i n  d i d  t h e amount reduce h i g h  m/s t h e not of t h e  frequency  146  energy  would  c o v a r i a n c e  e.  a c t as  m a t r i x  e i g e n v a l u e  h i g h l y  s e n s i t i v e  measure  seems  gather  method  and  be  t h a t  l o c a t i o n s .  For example  r e s u l t  f a i l  adding  power  degrading  u n i f o r m a l l y  t o t h e  \(m).  r o t a t e  most  t h e  VA  w i l l  have  w i t h  4 . 2 1 ,  h i g h e r  i s c l e a r l y  f o r high  method  a r r i v a l s ,  t h e  such t h e  semblance n o i s e  near  The  d a t a .  moveout  waveforms maxima must  as KL  be  t e c h n i q u e  s u r f a c e  w i t h I n  make  s p u r i o u s  data  t h e  t h e  CKL  them r  and  so V  R M  g  c a u t i o u s l y .  i n t h e r e a l  works f o r  4.25)  v e l o c i t y  a  c o m p l i c a t e d  h y p e r b o l a e ,  seen  may  r e a l a  t r e a t e d  However, ( F i g u r e  c o n s t i t u t e s  and  a t  those method  VA.  l e v e l s  s u p e r i o r .  r e s o l u t i o n  However,  i n t e r f e r i n g  t h i s  c r i t e r i o n  f o r n o i s y  t o  F i g u r e t o  measure.  a b l e  e a r l y of  s i m i l a r i t y  s e v e r e l y  many  T h e r e f o r e ,  comparison example  t o  r a t i o  s i m i l a r i t y  w i t h  w i l l  s i m i l a r  use  d i a g o n a l  n o i s e ,  D i s c u s s i o n .  The  CDP  white  thus  very  w e l l  t h e r e a l t h e prove  i n v e r s i o n  data i n data  semblance t o be  of  problems.  147  0.6  0.8 ' •  1.0  1  1  1.2 1  1.4 i  1  r * "  i  1.6 i  1.8 i  2.0 i  —r  '  1  *L  ~  —  1  •  1  w  1  1 J  w  ir  jl  w 1  1  1  1  FIGURE  1  1  1  « 4 .-  4.10  Twelve t r a c e s r e p r e s e n t i n g r e f l e c t i o n s from 750 m t h i c k w i t h v e l o c i t y 1500 m/s. O f f s e t s 2000 m.  the base of a range from  l a y e r 0 t o  1 48  V(RMS): 1500 nws  KKI. V A  Hilli MM  HHHWti  m  mm*  r  <±2  <±4  mm stHrtew p t a  ""I  ]  mm mm  mm mm mm  mm mkm  iwttjw m$m  WW Ml  mm mm mm mm mm  •HiJHH-! ItWftttl l«H)M l i # f « (HWtottl Itftttlttttl  <±6  mm mm mm  iiiinLiiiJ fttntftTtr.  < ± 8  jSmb  X  J i  ITIITTTTT"  L e f t column: Semblance s t a t i c time s h i f t s ; sample p o i n t s ( i . e . 0, Second for  l e f t  t o  x(D , x(2)  As t h e s t i s unable a r r i v a l . in x(m), s t a t i c s c  ...,  IHHJHMI  mmm  mm\  x  IWHJWH umm  %7  Xa  FIGURE 4.11 c r i t e r i o n VA f o r p r o g r e s s i v e l y i n c r e a s i n g down t h e c o l u m n : u p t o 0 , ± 2 , ± 4 , ± 6 , ±8 ± 8 , ± 1 6 , ± 2 4 , ±32 m s ) .  f a r r i g h t : ,  x  Xi  l»ni u n  m mM mm\  RKL  eigenvalue  r a t i o  c r i t e r i o n ;  columns  s e m b l a n c e VA b r e a k s peak corresponding a p r o g r e s s i n g i n c r e a e f f e c t of an i n c r ks a r e s e p a r a t e d by  down a n d t o t h e s e i n m, e a s e i n 50 m/s.  x(S) .  a t i c s c a t t e r i n c r e a t o r e l i a b l y l o c a F o r t h e R K L V A , we c o u n t e r a c t s t h e d e a t t e r . The v e l o c i t y  s e t e no g r t  s ,  t h e t h e t e how a d i n g i c k mar  149  V(RMS):isoom/5 jljii iiiii irTTTtltn  UllllliUl." jTITIipiiT  wfym < ± 2  CKL VA  njT;"ltp!ti it;:Iiiui  k t p t l Iwnpti b - p t l  IHWHWI wttpti  «JH«J WftjttHj |WH|HH |(HfjHHj JWHH j Hj (ttWJWt  Sd  «pd  w p ) wHpt) wwpl kwp« Hfwp  ^±4  LULU JlJl, ffTTTT TTTT  < ± 6  ^2  -  » « P « w«P*f ™ r ™ w # ™ £ ± 8  EX  3 Snb  x  Xi  IHJHHI  WttAtrt  FIGURE 4 . L e f t column: Semblance c r i t e r i o n VA s t a t i c time s h i f t s ; down t h e co sample p o i n t s ( i . e . 0 , ±8, ±16,±24 Second  l e f t  t o f a r r i g h t :  for x(l), x(2), As t h e s t a t i c s c a t t e r is unable t o r e l i a b l a r r i v a l . F o r t h e CKL i n x(m), c o u n t e r a c t s s t a t i c s c a t t e r . The v  CKL  x»  x«  Xa  iHlllltllll  1 2 f o r p r o g r e s s i v e l y i n c r e a s i n g l u m n : u p t o 0 , ± 2 , ± 4 , ± 6 , ±8 , ±32 m s ) .  e i g e n v a l u e  r a t i o  c r i t e r i o n ;  columns  x(8). i n c r e a y l o c a V A , we t h e de e l o c i t y  s e s , t h e t e t h e n o t e how grading t i c k mar  s e m b l a n c e VA b r e a k s down a n d peak c o r r e s p o n d i n g t o t h e a p r o g r e s s i n g increase i n m, e f f e c t o f an increase i n ks a r e s e p a r a t e d by 50 m/s.  1 50  CKL  RKL  V(RMS):1500m/s  mm  40%  m\m mm  p»h  i X3 mm mm  55%  iii mm  p p  mm  pWHtj  Xi pt™  WWW  WKW  70%  EE mm  MHUHH M M  mm WWWH  mm mm  W«l  inffnnrn ....lmi  nUntnt U i nUi il i  W W  • nm  torn  IWM ««ww  WWW*) wwwn INHUMI WWWf  85%  100%  mm  z3z  iiml)  FIGURE 4.13 The e f f e c t o f i n c r e a s i n g random noise l e v e l s (down t h e columns: 40, 55, 7 0 , 8 5 , a n d 100% o f t h e maximum a m p l i t u d e ) on semblance (far l e f t ) and KL V A ' s . Centre a r e t h e RKL r e s u l t s , f a r r i g h t a r e the CKL r e s u l t s . An i n c r e a s e i n m i n x(m) d o e s not improve t h e r e s u l t f o r added random n o i s e . This i s because white noise adds energy t o t h e d i a g o n a l of t h e covariance matrix u n i f o r m a l l y , degrading t h e s i m i l a r i t y measure f o r a l l m ( s e e t e x t ) . As - t h e noise l e v e l i n c r e a s e s , t h e semblance VA r e s u l t s a r e able t o r e l i a b l y l o c a t e t h e maximum, b u t , due t o t h e i n c r e a s e i n energy a l l along t h e covariance d i a g o n a l , t h e KL VA m a x i m a a r e l o s t i n the background n o i s e . The v e l o c i t y t i c k m a r k s a r e a t 50 m/s.  151  FIGURE Twelve  s y n t h e t i c  c l o s e l y  spaced  2000  m.  seisinograms  i n t e r f a c e s  (50  4.14  r e p r e s e n t i n g m  a p a r t ) .  r e f l e c t i o n s  O f f s e t s  range  from from  three 0  t o  152  1800  1900m/s  4—.—»-  H—(-  RKL  .  20 .  **lUHL!'IU1'"' " •  1  mjuiruuuupw.  2.2. -\—I—f-  H—I—h  I  H—I—f-  CKL  .  20 .  -f—f-  I I  "TjTJUUUUUUuwi  uuUOTJUUUuu«»-  vOUtJUUUUUuiix rftflflnTDUllliu . . . . . .  22 ._ Xi  X2  H—I—h  H—I—h  FIGURE VA  r e s u l t s  smears  t h e events  VA's  a r e  RKL  x(2)  c e n t r a l about 1800  f o r t h e  w e l l  m/s  a t  In s ,  2.0  t i c k  t o  CKL  e v e n t .  v e l o c i t y  and  a b l e  and  2.125  t h r e e  both  s ,  marks  t o  d i s t i n g u i s h  which 1861  4.15  r e f l e c t i o n  i s unable  x(3)  4—1—h  VA's  these i s a m/s  a r e a t  e v e n t s . r e s o l v e  t h e  V A ' s , a t  2.06, m/s.  two  semblance  However, e v e n t s ,  t h e p o s s i b i l i t y  t h e  b i t l a t e . 50  o u t e r  i n d i c a t e  The  them.  outer True  and  r  1804  event and m/s  V_  and of  a  appears „  Kt  VA  both  KL t h e  t h i r d a t  v a l u e s a r e : 2.11  s .  The  153  (0  E o in  CO  CO  E E o o o  E o  CO \  0>  10 (O  (0 \  (fl  E o  O) 0>  o CM  E o o  E o m  E o E o o m  (0  \ E o o Csl  (0  E o o CM CM  E E o oE o o m  T(s)  FIGURE  4.16  Twelve s y n t h e t i c seismograms (a) r e p r e s e n t i n g r e f l e c t i o n s from ten f l a t l a y e r s , as shown i n ( b ) . O f f s e t s range from 0 t o 2000 m.  154  SEMB.RMS VEL 1800  1500 -*  1  1  1  (-  0.6  LU  0.8 1.0  CE  err  1.2  r—  cr 1.4 1.6  FIGURE For  the  data  c o n s i s t e n t v e l o c i t y t r u e  t~VD.  of  Figure  p i c t u r e  of  t r a j e c t o r y c  l o c u s .  4.16,  the  smearedcan  The  a r r i v a l s ,  r e a d i l y  v e l o c i t y  4.17a semblance  VA  (a)  gives  however,  be  p i c k e d .  t i c k  marks  The a r e  s o l i d at  a  the 50  l i n e m/s.  f a i r l y  s t a c k i n g i s  the  155  CO  X  CM  X  T  o  CO  X  cr cr cn LO  CO  CD  CD  CD  CM  CD  CD  VER.TRflV.TiriE FIGURE  4.17b x(2), a n d x(3)) giv d e f i n i t i o n of events, and a l l except x(0 have d e l i n e a t i o n of the very f i r s t event ( a t 0.6 w h e r e a s t h e s e m b l a n c e VA g i v e s a s p u r i o u s v a l u e t h i s a r r i v a l . However, t h e s e m b l a n c e VA gives a of t h e p r e s e n c e o f t h e e v e n t a t 1.05 s, whereas not. The s o l i d l i n e i s the t r u e r-Vms l o c u s . m a r k s a r e a t 50 m/s. The  RKL  method  ( f o r x(D,  es  a  much  given s,  and  of  1600  c l e a r  sharper a  1550 m/s  c l e a r m/s), f o r  i n d i c a t i o n  the  RKL  The  v e l o c i t y  VA  does t i c k  1  56  oo  CO  X  cr  CD  LO  CD  CO  cz)  CM  CD  CD  «—*  —•  CD —'  —'  VER.TRRV.TiriE FIGURE 4.17c The CKL method ( f o r x(D, x(2), a n d x(3)) g i v e s a much sharper d e f i n i t i o n of events, and a l l have g i v e n a c l e a r d e l i n e a t i o n of the very f i r s t event ( a t 0.6 s, and 1550 m / s ) , w h e r e a s t h e s e m b l a n c e VA g i v e s a s p u r i o u s v a l u e of 1 6 0 0 m/s f o r t h i s a r r i v a l . However, the semblance VA g i v e s a c l e a r i n d i c a t i o n of the o f t h e e v e n t a t 1 . 0 5 s . w h e r e a s t h e C K L V A d o e s n o t . The presence t r u e T -V l o c u s . T h e v e l o c i t y t i c k m a r k s a r e nir the s o l i d l i n e i s RMS at  50  m/s.  157  T  CM  2  1  Twelve  s y n t h e t i c  s h i f t s  of  up  "from  t o  2000  0  t o  FIGURE seismograms as  ± 12 m.  ms  i n t r o d u c e d  4.18 per F i g u r e t o  each  4.16a,  t r a c e .  but  with  O f f s e t s  time range  158  SEMB.RMS VEL  1800 r./£  1500 H  1  1  1  1-  0.6 0.8 1.0 CT  cr h— cr  1.2 1.4 1.6  FIGURE In  the  (Figure l o c u s .  case  with  4.18) The  s t a t i c  performs  v e l o c i t y  t i c k  time  4.19a  s h i f t s ,  p o o r l y . marks  The a r e  the s o l i d  a t  50  semblance l i n e m/s.  VA  i s t h e  of  the  true  data  r-V  RMS  159  CM  X  CO  cr cr  VER.TRRV.TIME  FIGURE For and  t h e x(3)  e v e n t s . t i c k  RKL  VA  of  t h e data  a r e a b l e  t o  r e s o l v e  The  marks  s o l i d  a r e a t  50  l i n e m/s.  i s  4.19b  (Figure t h e  4.18),  t h e VA  m a j o r i t y  t h e true  t-V  RMS  maps  f o r x( 2)  of t h e r e f l e c t i o n l o c u s . The v e l o c i t y  160  CM X  C3 CD  CO  CO  n  cn  CD" CD LO  CD  CO  CD  CD  CM  CD T  CD  —  VER.TRAV.TIflE  FIGURE For  t h e CKL  a b l e  t o  l i n e 50  r e s o l v e  i s m/s.  VA  t h e  of  t h e data  t h e m a j o r i t y true  r-V  ms  4.19c  (Figure of  t h e  l o c u s .  4.18),  a l lt h r e e  r e f l e c t i o n The  v e l o c i t y  VA  e v e n t s . t i c k  maps,  a r e  The  s o l i d  marks  a r e a t  161  LU  CD CD •  00 -  CO  5Z cn  cri  LU  CDCD CD  co  C\J  ^ CD  CO  00 •  CDCDCD  t  —  3UIi'Addl'd3A CO -4—  —^_  ( a ) . Three secon ( 5 8 2 t o 1 6 6 2 m) a p p l i c a t i o n of semblance VA o f p i c k e d f o r s e v e r  FIGURE 4.20 d s o f 2 ms r e f l e c t i o n s e i s m i c data f o r 16 o f f s e t s a f t e r bandpass f i l t e r i n g (8 t o 55 H z ) , m u t e , and an AGC. The basement i s a t about 1.62 s. ( b ) . The t h i s data: a s t a c k i n g v e l o c i t y l o c u s can b e a l p o i n t s . The v e l o c i t y t i c k m a r k s a r e a t 50 m/s.  V(RMS)  162  V(RMS) 2 5 0 0  2 0 0 0  I  11111  IIIIIIH  Sm b  Smb  1 0 0 . 2  44-  4+  +4-  RKL  RKL  1.0 0 . 2  J  4-f +444  444444  llllllll  4-4+44444+4  4444  C K L  +4444444  CKL  10 . 02 . X, 4444-  Sample r e s u l t s the KL p i c k s - o e q u i v a l one e v e d e l i n e a The v e l  X2  X3  X2  X3  4444-  4444-  +4+44444  +4+44+44  F I G U R E 4.21 maps f o r 0.2 s o f data i n two time were used t o determine a good c h o i c e V A ' s . A t 0.2 s ( l e f t ) , t h e RKL VA f o u t two d i s t i n c t events c o r r e s p o n d i n e n t i n t h e s e m b l a n c e VA map. The CKL VA nt. F o r t h e window a t 1.0 s ( r i g h t ) , R K L t e a s i n g l e event, whereas t h e semblance o c i t y t i c k marks a r e a t 50 m/s. VA  windows. These o f x(m) t o use i n r \(2) and x(3) g t o a smeared o n l y d e l i n e a t e s x(2) and CKL x(O i s again smeared.  163  VER.TRflV.TIflE  RKL 4.20.  A l t  semblance are  w e l l  s t a r t e d the  FIGURE 4.22 x(l), x(2), and x(3) f o r the data seen i n h o u g h i n d i v i d u a l events are sharper than those o VA map, the g e n e r a l p i c t u r e i s l e s s c o n v i n c i n g . r e s o l v e d at e a r l y t i m e s , but by 0.4 s, events o d i s a p p e a r . The x(3) map g i v e s a c l e a r p i c k at 1 d i n g events seen i n t h e s e m b l a n c e VA map a t 0.95  r e s u l t s  t  p r e c e  absent.  f o r  F i g u r e f  the Events have  s,  but  s,  i s  164  OJ  CD  CD  CD  co  CD  CD  VER.TRflV.TIME  FIGURE CKL r e s u l t s f o r As w i t h t h e RKL R e s o l u t i o n a t e a 0.3 s. O v e r a l l , the s e m b l a n c e VA  4.23  x(0, x(2), and x(3) f o r the data o VA r e s u l t s , t h e s e VA maps are p o o r l y times i s good, but d e t e r i o r a t e s these r e s u l t s seen l e s s p l a u s i b l e t map. The v e l o c i t y t i c k marks are a  f F i g u r e 4.20. r l y b a l a n c e d . markedly past han those of t 50 m/s.  o r N j ^ ( D C O o ( \ i ^ r  oo  co  ro  co* oo  VERJRflV.TINE  4  )  ..)  1  t  I  I  FIGURE  I  I  t  t  )  I  4.24  (a) . A sample o f t h e wide-angle s e i s m i c d u r i n g t h e UBC-RECOPE (Costa Rican s e i s m i c p r o j e c t (1982). O f f s e t s range  r e f l e c t i o n data c o l l e c t e d N a t i o n a l O i l Co.) j o i n t from 1630 t o 4415 m.  (b) . R e s u l t s of a semblance VA. Although r e a d i l y be made f o r t h r e e p r i m a r y a r r i v a l s .  smeared,  p i c k s  c a n  166  co  Two  sample  time  windows  c r i t e r i o n  f o r the  suggested  u s i n g  window p r a c t i c e v e l o c i t y  u  em  ( r i g h t ) , t h e t i c k  KL  VA  x(4) x(3)  x(D marks  and a r e  FIGURE  4.25  t o  a s s i s t  i n choosing  of  t h e  d a t a .  f o r both x(4)  and x(2) a t  the  VA's 50  The RKL  look proved  m/s.  an  r e s u l t s and  CKL  a t VA's.  p r o m i s i n g . t o  be  of  x(m)  a p p r o p r i a t e 3  s For  ( l e f t ) the  However,  g r e a t e r  u s e .  4 s i n The  167  o  CO  IZ  cn  CD" CD  cr  LO  C 5  co cr CD" CD  cr  LO  CM CO  CD  CO*  CO CO*  CD CO*  co*  CD  VER.TRflV.TiriE  FIGURE RKL  VA  f i v e  r e s u l t s  RMS  f o r x( O  v e l o c i t i e s  semblance  VA  these  i s n o t h e l p f u l  are  a t  maps 50  m/s.  map  and can  x(2). be  ( s e e t e x t ) . t o  4.26 A f t e r  made: However,  much  an  s c r u t i n y ,  improvement  t h e o v e r a l l  t h e p i c k i n g .  The  p i c k s  f o r  over  t h e  appearance  v e l o c i t y  t i c k  of  marks  168  6 o o r-' LU  CO ZZ  CM  X  cm  o  CD" CD LO  ? o oLU > CO  CJ  CD LO  CD 00*  oo  oo*  CD  CO  00*  00*  CD  C\I  VER.TRflV.TIHE  FIGURE CKL  VA  from 50  r e s u l t s these  m/s.  f o r x(l)  maps  ( s e e  and  x(2).  t e x t ) .  4.27 S i x RMS The  v e l o c i t i e s  v e l o c i t y  t i c k  c a n  be  marks  made a r e a t  169  SECTION  a.  4. I l l :  INVERSION.  Q  I n t r o d u c t i o n .  Both (1980)  Robinson  developed  d i s p e r s i v e the  produce  s y n t h e t i c  show  how  s y n t h e t i c v a l u e  and  t o  e f f e c t s  1 ) .  s e i s m i c  of  Robinson the  assumed  the  s t r u c t u r e  p r i o r i .  the  Beresford-Smith  degree  the  of  medium.  d i s p e r s i o n  t o  Subsequent was  d i s t o r t i o n  from  s e c t i o n  the  treatment  of  problem e x i s t i n g  Mason  used  w i t h  the  d i r e c t l y  and  of  remove the  should  c o n v e n t i o n a l l y  the  d a t a . prove  c o l l e c t e d  a  i n  i n  both  known h i s  1979  However,  on  Q  v e r t i c a l l y  s e c t i o n  t h i s  t o  proceeded  with  d i s p e r s i v e  (1979)  from  work  r e l i e d  of  (1962)  and  d e p t h ) .  r e f l e c t i o n (1980)  t o of  medium  d i s p e r s i o n  m i n i m i z a t i o n  procedure  a t t a c k e d  the  estimate  i t e r a t i v e  each  of  Robinson  removed  the  Mason  removal  d a t a ,  be  extended of  and  Futtermann  may  v a r y i n g  and  d a t a .  homogeneous  e f f e c t s  ( i . e . Q  a  of  the  d i s p e r s e d  (1982)  media Q  f o r  an  inhomogeneous that  r e f l e c t i o n  d i s p e r s i o n  f o r  Beresford-Smith  f a c i l i t a t e  developments  data  remove  and  t o  seismograms  r e a l  (Appendix  paper  from  t h e o r e t i c a l  the  1982)  techniques  e f f e c t s  u t i l i s e d  to  (1979,  a  was  he  known  measure  of  p r o p e r t i e s  of  measure c o r r e c t  amount  Both  these  most  u s e f u l  s e i s m i c  i n  an of  works i n  the  r e f l e c t i o n  d a t a .  Ganley remove  the  and  Kanasewich  d i s p e r s i v e  (1979),  e f f e c t s  from  r a t h e r the  than  d a t a ,  attempting  sought  an  t o  e s t i m a t e  170  of  Q  the  assumed a  s t r u c t u r e  that  the Q  Q  constant  t e c h n i q u e ,  the  d i s p e r s e d  s i g n a l  were  data  to  s e i s m i c  and  known. the  o b t a i n  the  r e f l e c t i o n  d i s p e r s i o n  Using  the  r e l a t i o n  complex  non-dispersed  the  s e c t i o n .  and  (again  s p e c t r a l  r a t i o  s i g n a l  from  input  c u m u l a t i v e ,  They  thence  i n t e r v a l ,  v a l u e s .  Conference  Johnson  b.  the  d i v i d e d - o u t  A t t e n t i o n Q  input  model)  they  f o r  An  was  a l s o  (1980),  and  (1981), McMechan  I t e r a t i v e  d i r e c t e d  to  by  and  Smith  and  D i s p e r s i o n  t h i s  Y e d l i n  Removal  problem  N e i d e l (1981  ) ,  Scheme  at  the  (1976), and  Toksoz  R a f i p o u r  Using  S t a n f o r d  an  and  (1981).  E f f e c t i v e - Q  Value  F o l l o w i n g Appendix segment  1, of  approach each  the the  data and  using  wavelet  members  a  maximum with  s u i t e  which  e q u i v a l e n t r e f e r e n c e  and of  a  Q  procedure  the  they and  s i g n a l s were  have  p r i o r  to  a  versus  comparison  w i t h  or  from  be the  (by to  h i s at  between  a  between of  should w i t h  the the  y i e l d Q  the the a  value  e a r t h ) .  d i s p e r s e  d i s p e r s e d  a  v a l u e s ,  I n s p e c t i o n  d i s p e r s e d would  Q  of  x(1),  x(l)  i n  undisperse  d i v e r g e  s i g n a l ,  undispersed  approach  I  o u t l i n e d  to  s u i t e  s i g n a l s .  o r i g i n a l l y  a l t e r n a t i v e  s i g n a l  been  Here  measure  undispersed  undispersed  (1979)  employed  f o r  s i m i l a r i t y  undispersing-C2 value the  was  v a l u e .  _  of when  the  Robinson  model  s p e c i f i c  computing  reference  f u n c t i o n  a  of Q  c o n s t a n t  repeat  i t e r a t i o n  of  approach  d a t a .  An the  171  For  wavelets  t r a n s f o r m i n g  many  w i t h  l a r g e  long  time  here  an  a p p r o x i m a t i o n  pulse  at  a  q u a l i t y  l a r g e  f a c t o r  to  time ,  Q  Q  w i t h  a p p r o x i m a t i o n  Q  For  a  r.  wavelet  a r r i v a l  time  T^,  centred  i n  window  the  a  waveform  A(t)  which  i s  produce a r r i v a l  I  w i l l  = A  of  r e p l a c e of  width  s u f f e r  equation  time  l o s s ,  T  of  Aft)  l o s s  a  i n t r o d u c e  medium  small  than  time  T  w i t h  the  same f o r  amplitude  given  a  I  .  v a l u e ,  c e n t r e d  T  a  w i t h  e f f e c t i v e - Q  p r o p a g a t i n g  i n  u>  where  energy,  where  T  =  w h i l e /2,  T  s  v a l u e ,  Q  s e r i e s  I  F o u r i e r  r e p r e s e n t a t i o n of  some  an  l e s s  . A f t e r a  problem.  of  about wavelet  time  T^,  by:  ,  s ' e f f e c t i v e '  time  T  from  (Al.l),  same  at  i n t r o d u c i n g  the  expense  through  l o c a t e d  d u r a t i o n  exp(-u>l/2Q)  0  the  by  a  adequate  t r a v e l l e d  p u l s e  the  poses  f a c i l i t a t e  a  t h i s  t i m e s ,  segments  which  e f f e c t  f  t r a v e l  i s  angular  frequency.  i n s e r t i n g  a  we  choose  must  small  To  apparant a  lower  w  hence:  exp(-uT /2Q )  = A  s  0  eff  so:  s Qeff  T  V2^  -  /2  or:  Qeff It  i s  = t h i s  p r o c e s s i n g  Q  T  / W  2  T  L  ' e f f e c t i v e '  (  Q  value  that  i s  used  i n  most  3  ?  )  subsequent  172  We  may  f o l l o w s :  f o r  p r o p o r t i o n a  of  s i n u s o i d  same  Q  t h i n k  =  2 5 .  energy  I  i n v o l v e d . window,  a l l o w i n g  an  as  i s i n c r e a s e d  near  d i s p e r s e d s i g n a l s  a  w i t h  may  s i m i l a r  way  be  t h e e n t i r e  and  t h e s u i t e  a s  whole  In  t h i s  w i t h  case,  i t s a p p r o p r i a t e  s u i t e  as  a  whole  when  t h e a c t u a l  the  Q  s t r u c t u r e  h o r i z o n t a l  o f  t o  u n t i l  below i s  Qefj-  s i g n a l then t o  a l l o w i n g t h e  c y c l e s  w i t h  time  i n  wavelet time  t h e  2ir  The  one  a  t h e t r a v e l  sample  may  window  would  and  has a  be  hence used. be  v a r i e d  i s  Q  a  more  s u i t e  o f  hyperbola undispersed  t o t h e  e f f e c t i v e  h a s been  paths  t h e  f o r example  moveout  s i g n a l s  way  given  (with  subsequent,  u n d i s p e r s e d ,  i n t h i s  f o r a  doubled  i t .When  and  v a l u e  approximations  i s a v a i l a b l e ,  o f  o n l y  - 47r.  ( a s along  s u i t e  best  Q  f i x e d  r e l a t i o n  t h e  i n t h e s u i t e  d e r i v e d  component  due  each  c u m u l a t i v e  as  50, incurs  t h e Futterman  compared  be  a  i n a r r i v a l  subsequently  a r r i v a l w i l l  =  f o r 50  r e c a l c u l a t e d .  paths  then  Q  w i t h  window  compared  t r a v e l  loses  Consequently,  f a l l s  a r r i v a l ,  example),  l a t t e r  2-n,  a n d £?eyy  for  a  s i n u s o i d  s h i f t  1,  @eff  r e f e r e n c e  a  approximation  v a l u e .  t h e sample  e a r l y  s i g n a l s  Q  t o  i n t h i s  r e l i a b l e o f f s e t  i n t o  i f  centred)  a  t r a v e l l e d .  i n Appendix down  of  t h i s  i t t o propagate  e f f e c t i v e  v a l u e s  again  model,  t h i s  noted  behind  f o r 100 c y c l e s  Consequently,  When c l e a n  a s  t h e l e n g t h  wavelet l e n g t h  propagate  w i t h  Q  Q  p e r c y c l e  t r a n s l a t e  However, f o r  t h e r e a s o n i n g  constant  l o s s  c o n j u n c t i o n  h o l d s  a  t o  energy  o f  r e f e r e n c e . undispersed v a l u e .  most  The  s i m i l a r ,  Consequently,  a b s o l u t e . g r e a t l y ,  I f t h e  t h e  n e t  173  e f f e c t  would  c.  Constant  The  The q u a l i t y  to  i s  I  what  to  the  the  complex  v i z . x.  d i s p e r s e d )  the  u n d e r l y i n g be  to  changes  degree  d i s t o r t i o n  of  over  be  i n  o u t - o f - p l a n e  s c a t t e r i n g .  h i g h l i g h t  r a p i d i t y  With  the  c o u l d  be  t h i n  a p p r o p r i a t e used  as  a  a  the  of  A  d i r e c t  the of  high (thence  o b j e c t  of  d i s p e r s i o n ,  we  determine  whether,  approximated  e) ] ,  x.(t)  and  where  i s  w i l l  a  x.(t)  i s  complex 1982).  I f  w i l l  be  r o t a t e  the  e i g e n v e c t o r which  by  the  Oldenburg,  and  form.  of  s e c t i o n  the  would  by  of  hydrocarbon  d a t a ,  s i g n a l .  e  values  p r o p a g a t i n g such  i n d i c a t o r :  a  map a  T h i s  such  d i s p e r s i o n ,  d e p i c t i n g the  i n d i c a t i v e  phenomena  a t t e n u a t i o n ,  i n  be  u n d e r l y i n g  about  s e c t i o n  q u a l i t y  and  w i t h  method.  be  e  s e i s m i c  change  when  f i r s t  angle  l a y e r s ,  to  (Levy  brought  i n t e r f e r e n c e  the  s i g n a l  used  e f f e c t s  can  seismogram,  d i s t o r t i o n  would  Q.  cumulative  [ x. (t. ) exp(i  i t s undispersed  e  the  method  (t ) =Re  s h i f t  the  e f f e c t i v e  KL  determined the  of  i n  be  However,  d i s p e r s i o n  (and  can  d a t a .  c o s t  may  recover  a l l e v i a t e  more  s h i f t ,  t r a c e  The  to  and  e s t i m a t e  d i s p e r s e d  the  c o n d i t i o n s ,  a p p l i c a b i l i t y  of  use  phase  of  of  simply  s i m p l e r  Here,  measured  attempt  s t r u c t u r e  Q  a  i n  changes  A p p r o x i m a t i o n .  an  seek  used  l a t e r a l  data  may  t r a c e  Phase  any  scheme  e x e r c i s e  a  average  u n d i s p e r s i o n  the  constant  to  i t e r a t i v e  i n t e r v a l )  under  be  as and  would  waveform. of  e  v a l u e s  l a r g e  e  value  174  being may  i n d i c a t i v e be  of  the  brought medium,  However, the  f o l l o w i n g  here  as  I  a  set  s e l e c t  (2)  c a l c u l a t e  a  of  of  the  the and  (3)  apply  (4)  e v a l u a t e x(D  x(0  the  i s  i n  simply  i n  the i n  gas  s i g n a l .  the  s a t u r a t e d  demonstrate  data  Such  a  d i s p e r s i v e  p r o p e r t i e s  a p p l i c a b i l i t y  of  d i s p e r s i o n  by  to  examples  change  media.  the  approximation  s y n t h e t i c  s h i f t e d  In  wavelet  d i s p e r s e d ) ; from  a  each (a  and  completing  component;  component,  p r i n c i p a l  of  t r a n s f o r m a t i o n ;  no  (equation  the  the  the  f i g u r e s of  d i s p e r s e d  c e n t r e wavelet  the  r e a l  p a r t  components  from  the  (5)  the  i s  approximately  -  the  1  then  constant  p r e s e n t e d ,  r e a l  and  a l i g n i n g  the  v a l i d .  the  waveforms,  the  d i s c r e p a n c y ;  x(1)  covers  (3)  for  w a v e l e t ) ,  s i g n a l  I f  not  wavelet  s i g n a l  time  d i s p e r s e d  i s  p u l s e ;  (32)).  o r i g i n a l .  t i m e ) ;  (4)  ( i n i t i a l  i s  examples  the  s p e c i f i e d  p u l s e  d i s p e r s e d  a n a l y t i c  KL  the  R i c k e r  (2)  the  a  there  e  of  of  c o n t a i n s  so  assumption  p r i n c i p a l  input  and  of  s e r i e s  r e f e r e n c e  r e f e r e n c e  verson  f i r s t  which  envelope  then  The  a p p r o x i m a t i o n .  window  l a r g e ,  s h i f t  two  occurs  complex  phase  a r r i v a l  i n c r e a s e  an  envelopes  constant  not  by  s h i f t  time  p u l s e  peaks  phase  change  w i l l  phase  (1)  If a  about  l a r g e  s t e p s :  d i s p e r s e d the  a  such  constant  c o n s i d e r i n g  of  of CKL  d i s p e r s e d  I  show  frequency (which p a r t the  (1)  the  35  Hz,  s i m u l a t e s  of  the  second  (as  an  f i r s t  p r i n c i p a l  decomposition waveform  phase  of i n  the (2))  175  r o t a t e d shows at  by  four  0.5  the p a n e l s ,  seconds  r e s p e c t i v e l y . case  where  examples, t r a c e  phase each  as  F i g u r e  4.29  e f f e c t s  waveforms  g e n e r a l ,  e x t r a c t  a  wavelet  ( 2 ) ,  p u r e l y  we  were  note  s i g n a l  i n c r e a s i n g  w i t h  f o r  values  Q  the  a  a l i g n e d  on  F i g u r e  wavelet  25,  analagous  a t t e n u a t i o n  phase  a  l a r g e as  r o t a t e d  55,  the  85,  and  115  for  i n c l u d e d . b a s i s  4.28  centred  r e s u l t s  are  degrades  v a l u e ,  the  d i s p e r s i v e  shown  35  the  frequency  c e n t r e  Ln(\e\)  versus  c e n t r e  e m p i r i c a l  In  r e f e r e n c e x(J).  Ln(Q yj.) e  The  of  f o r  the  In  a l l  complex  .  For  above  a 10  w e l l  and  the  the  to  d i s p e r s e d  improve  becomes  phase  able  more  w i t h  l i k e  e f f e c t s  s h i f t  w i t h  a of  r e q u i r e d  to  i n c r e a s i n g  Q  decreases.  a  4.30, 15  i s  r e s u l t s  decreases  d i s t o r t i o n  F i g u r e  The  I n c l u d i n g  a l s o  were  (1)  waveform  r e s u l t .  between  frequency  R i c k e r I  and  55  f i x e d Hz,  show  I  wavelet  the Hz,  window  r e s u l t s on  c e n t r e  of  v a r y i n g  the  l e n g t h ,  determined  of  locus  for  the  of  wavelets f o l l o w i n g  r e l a t i o n s h i p :  Ln(\e\) I  Hz.  the of  a l g o r i t h m  d i s p e r s e d  d i s p e r s i o n  frequency  CKL  value  the  the  examples  the  wavelet.  for  The  to  the  ' c o r r e c t ' as  t h a t  common  v a l u e ,  Q  a t t e n u a t i o n  Here  of  above  (32)).  e n v e l o p e .  In  w i t h  w i t h  shows  equation  (from  d e s c r i b e d  d i s p e r s e d  the  the  e  angle  = 3.7-1.0  have  d u p l i c a t e d  by  invoked a  change  Ln(Q /Tj eff  my i n  assumption Q,  hence  the  that  a  s h i f t  r e l a t i o n  i n  time  i n v o l v e s  can  only  be the  176  Q j-j->  v a r i a b l e centre below  =  Q  the  the  rather  frequency  s h i f t i n g of  e  85;  onto  i n t e r c e p t  centre on  i s  wavelet  p o i n t s  i s  does  f o r  the  wavelet  alignment  down  a  Changing  not  slope  The gives  Q  eff  e  the  form as  of  being  (the  dependence  d e t a i l  of  a  l e n g t h  i n t e r c e p t  t h i s  of  of  r e l a t i o n s h i p  i n v e r s e l y  of  and  each  e r r a t i c  (manually moved  the  4.31).  the  i s  most  were  Here,  l o c u s  sample  of  i n t e r c e p t  r e s u l t s  the  wavelet  for  mid-range  (Figure  the  the  jumped  p o i n t  window  T  of d i d  ,  c u r v e s .  i s  p r o p o r t i o n a l  the  The  for  lowest  'constant'  l a t e r ) .  problem,  t r a j e c t o r y  or  but  a t t e n u a t i o n  the  sample  for  more  m i s a l i g n m e n t .  time  ,  some  was  one  w i t h  /  the  problems  the  i n  w i t h  alignment by  For  r e l a t i o n s h i p  A l s o ,  3.7  case  the  l o c u s ) .  about  s u b p a r a l l e l  a f f e c t  alignment  common  e x p l i c i t l y .  Hz),  wavelet  change  i s  Q  and  (15  to  i n t o  d i s c u s s e d  c o n s i s t e n t  to  due  the  f r e q u e n c i e s  /  time  considered  t h i s  the  than  not to  s u r p r i s i n g ,  the  e f f e c t i v e  s i n c e Q  i t  v a l u e ,  , or: |e|  where  This  = C(2T /Q)  (38)  L  C = exp(3. 7) = 45 f o r m u l a t i o n  d i r e c t l y  To  from  see  v i e w p o i n t , wavelet  X(f),  the  the I  a l s o  r a i s e s  estimated  form  of  consider which  was  the phase  t h i s the  s h i f t  i n  about  of  e s t i m a t i n g  Q  e.  r e l a t i o n s h i p  problem  c e n t r e d  p o s s i b i l i t y  the time  from  an  frequency z e r o ,  and  a n a l y t i c a l  domain. which  For can  a be  177  d i s p e r s e d a s s e r t  a c c o r d i n g  that  s h i f t ,  e.  the  fb  i s the  For  an  than  f o r  at  i n  second  f o r  a  phase  (f/f  time  be  X(f[l  frequency  we  wavelet,  (A1.10),  by  a  I  s i n g l e  phase  a  the  t h a t  Gaussian  J/nQ))  Ln (f/f  i n  Appendix  c o n s i d e r  i n t r o d u c e d  Assuming  of  +  d i s c u s s e d  s i g n a l ,  domain.  s i g n a l by  i s  i s a  p o s i t i v e  r e a l ,  c e n t r i n g  X(f)  p u l s e ,  o n l y  1.  other  the  wavelet  R i c k e r  R i c k e r ,  wavelet  1953):  (f/fc) 2exp(-(f/fc)*)  a  wavelet  ) 2exp(-(f/f  exp(2TTifTL  [1  +  (f/fc) 2  the  c  r e a l  L  )  2  )  +  about  exp(ie)  Ln(f/fb)/nQ])  [1  Cos(2nfT [J  c e n t r e d  \l  Ln(f/fb)/7tQ] 2  p a r t s ,  +  +  =  Ln(f/fb)/itQ\  exp(-(f/f  and  Ln(f/f  T^i  exp(2nifTL )  r e a r r a n g i n g  b  )/irQ]  exp([-(f/fc) 2Ln(f/fb)/irQ}  or:  equation  and  r e p r e s e n t e d  Ln(f/fb)/nQ\  R i c k e r  d e r i v a t i v e  f o r  c  Equating  can  component  =  have,  Cos(e)=  +  r e f e r e n c e  and  the  |/  a n a l y t i c  the  X(f)  we  d i s p e r s i o n  =  f r e q u e n c i e s ,  (the  (A1.9)  That i s :  X(f)exp(ie)  where  equation  t o  -  2irfTL)\l  c  )  2  \ l  Ln(f/fb)/nQ] 2)  +  g i v e s :  +  Ln (f/f  [2  +  b  )/nQ\  3  Ln(f/fb)/irQ])  178  Cos(e)  = Cos(2T fLn(f/f L  b  ) /Q)  \1 +  Ln(f/f )/vQ]  3  b  exp([-(f/f )*Ln(f/f )/nQ] c  [2 + Ln( f/f ) / itQ})  b  b  (39) I n .  t h e  problem  d i s p e r s e d  s i g n a l  w a v e l e t .  The  by  how  much  d i s p e r s e d  which  we  must  r o t a t e  e m p i r i c a l component  of part  t h e  by  noted  from  In  c a s e ,  t h e phase  two  on  a m p l i t u d e  t h e  r e l a t e d  best  t o  be  p r o c e s s  terms  of  t e l l s  (that  b e t t e r  mimic  t h e  d i s t o r t i v e  a s s o c i a t e d  a m p l i t u d e  i s  Hence,  reproduce  phase  us t h e  waveform  component.  t h e  t h e  Ricker  resemble  e x a c t l y  t h e  t o  with  t h e  d i s t o r t i v e t h e  phase  (38).  r e l a t e d  side  and  from  able  equation  hand  t o  s e p a r a t i n g  part  simply  t h e d i s p e r s e d  equation  t h e amplitude  r i g h t  s i g n a l  second  s i g n a l )  should  w h i l e  KLT  p r i n c i p a l  r e l a t i o n s h i p  terms,  t h e complex  t h e  t h e above  that  n o n - d i s p e r s e d  on  we  c o s i n e  t o a  of  d i s p e r s i v e  of  e x t r a c t i n g  component  component,  t h i s  by  t h i s  However,  t h e  am  A  expect  r e s u l t .  I  s i m i l a r  returned  appears  not  here,  i s most  e value  and  should  complex  posed  waveform.  d i s c a r d e d we  as  f a c t o r s  r e l a t e d of  equating  a r e those  terms  equation  a r e  w i t h i n t h e  l a t t e r  (39). D r o p p i n g  t h e arguments  of  t h e  t h e  t h e  c o s i n e s  g i v e s :  e = (2T /Q)fLn(f/f ) L  which  i s of  t h e  (40)  b  same  form  as  equation  (38).  179 The  (40)  frequency  may  be  estimated  (38).  The  c e n t r e  frequency  dominant  equation  i n t o  of  (40)  base  frequency  d i f f e r e n t show  so c e n t r e of  of  C(f  agreement  i s  /,  )  C(f  to  and  case  of  be  i n t e r e s t  i s  /  ,  s u b s t i t u t i n g  the /  however,  was  f i x e d  between  the t h i s  observed  ,  and  the  the  Nyquist  C(f  with  In  F i g u r e  4.32,  )  values  from  superimposed  on  equation  from  as  at  wavelets  c o n s i s t e n t .  c a l c u l a t e d  that  i n  i n  F i g u r e s  the  shown 4.33  here  F i g u r e  narrower  r e l a t i n g  ( i . e . a  those  p e r s i s t  f requenc i e s .  equation  i n  t h i s  the  (40).  bandwidth  The  i n c r e a s e s ,  the  degrades.  waveform  )  would  a g a i n s t  good,  to  i s  of and  comparisons  values  very  s i m i l a r  s i g n a l  t h i s  magnitude  R e s u l t s  t r e n d  band  found  have:  f r e q u e n c i e s the  r e s u l t  l i n e a r  the  that  wavelet,  i n  a l l  wavelet  alignment  we  fb  cosine-Gaussian  t h i s  R i c k e r  p l o t t e d  approximation  This  i n  w i t h  equation  i n  (41)  that  (38)  l o c u s  comparison  term  fcLn(fc/fb)  p l o t s  equation  p r o p o r t i o n a l i t y  frequency  f o r  =  frequency,  by  the  C(fc)  The  I  dependent  for and  than  t h e o r e t i c a l  Q  to  c o s i n e  a l s o ,  4.33.  e  a  i n  However,  curve  of  hold  Gaussian wavelet  as  i n the  the band  t r u e  f o r  a  envelope).  are  r e s p e c t i v e l y .  r e s u l t i n g  that  a  R i c k e r  4.34  a l s o  f f  shown  for  Problems  of  d e v i a t i o n width  a  R i c k e r  wavelet,  i s  much  b e t t e r  of  from  a  t h i s  the  f i t of  at  high  180 In  c o n c l u s i o n ,  summarized  e  constant  phase  approximation  which  best  i s  by:  e = where  the  2T f Ln(f /f )/Q L  i s  c  the  d i s p e r s i v e  c  b  constant  e f f e c t  t r a v e l l i n g  f o r  phase  on  a  seconds  T^  s h i f t  wavelet i n  a  of  medium  dominant  w i t h  mimics  the  frequency  s e i s m i c  q u a l i t y  /  ,  f a c t o r  Q-  d.  D i s p e r s i o n  I  Q u a n t i f i c a t i o n O b j e c t i v e  d e a l  next  i t e r a t i v e  scheme  case  known  of  a  w i t h  d e s c r i b e d reference  wavelets  a r e  p r i o r  subsequent  t o  In  the  a l i g n e d  f i r s t  measures,  or  eigenvalue  r a t i o  d i f f e r e n c e  e,  wavelet,  a  of  or  between respect  on  set  of  one  appearance  of  sum  of  I f  the  and  b a s i s  estimate  F i r s t a  I  u s i n g  Q  present  s i n g l e  of  the  d i s p e r s e d  complex  x(0;  i . e . the  the  another,  ( F i g u r e  f u n c t i o n s ,  the  simple  p u l s e .  t r a c e  no  when the  constant  the  were  the  A l l  envelopes  i s  the  the  phase  the  undispersed  the  case  of  a  i d e n t i c a l  a r e  between This  f i r s t  i s  d i f f e r e n c e  r o t a t i o n .  s i m i l a r i t y  and  look  s i g n a l s  d i f f e r e n c e s  phase  i n  t o  phase  The  second  t r a c e  made  two  shown.  d i f f e r e n c e s  s i g n a l s then  4.35),  a r e  r e f e r e n c e  phase  However,  a  t r a c e  f i g u r e s  (EVR)  procedure,  t o  e a r l i e r .  t o  p r o c e s s i n g .  between  them.  attempt  the  o b j e c t i v e  s i g n a l s .  u n d i s p e r s i n g  an  F u n c t i o n s .  s u i t e by  would  e x i s t  d i s p e r s e d  them  take  r o t a t i o n  the  on  w i t h the  i n c r e a s e s  181 w i t h  r e l a t i v e  In  d i s p e r s i o n .  F i g u r e  Q  u n d i s p e r s i n g at  1.0s)  4.35a,  was  (2=30,  simple  case  which  o p t i m a l l y  the  f o r  of  wavelet, d e f i n e d  In  no  i s  a p p r o p r i a t e s i g n a l s measure  members  for  the  we  must  wavelet from  the  f o r  to  t h i s of  a  a l o n g  d i f f e r e n c e  can  each  case  be  Q  w i l l  ensured  seen,  In  t h i s  the  Adding  value  from  there  random and  s t i l l  the  analagous  wavelet:  e  EVR  Q  the  shown  R i c k e r  the  t h e r e .  a t t e n u a t e d  use  a  s u i t e  h y p e r b o l i c  @eff  the  i s noise  d i s p e r s e d  has  a  of  s i g n a l s  and  move-out  w e l l  the  be  members  d i f f e r e n c e O t h e r w i s e w i l l  The  the  to  f a i l .  s u i t e  A  when  the  must  work.  w i l l  wavelet  s u f f i c i e n t  of  s i m i l a r i t y  of be  to  moveout,  be  data met  F i r s t l y ,  d i s p e r s i o n  a l l wavelets  Each  (37))  the  c o n d i t i o n s  hyperbolae i n  of  However,  c e r t a i n  w i t h i n  having  x(l).  c u r v e .  (equation  value  s i m u l t a n e o u s l y .  s t i l l  by  Hz  d i s p e r s i o n  are  but  I a  method  35  versus  v a l u e .  hyperbolae,  the  i s  l o c a t e d of  the  EVR,  time,  comparisons  and  to  (a  minimum  r e s u l t s .  other  hyperbola.  o u t s e t ,  the  of  As  d i s p e r s e d  t r i a l s ,  s i g n i f i c a n t a  and  using  moveout of  4.35b  c o r r e c t  i t s a r r i v a l  a  e f f e c t s  the  members  has  f u n c t i o n s  wavelet  Q=30.  w i t h  amplitude)  undispersed  method see  the  the  s u c e s s f u l l y  the  degrades  the  i n  has  i n  o b j e c t i v e  case,  F i g u r e  f o l l o w i n g  compared  are  In  at  two  value  attenuated  s e v e r e l y  the  e  removes  maximum  minimum  t h i s  the  d i f f e r e n c e  the  the  d i s p e r s e d  a l g o r i t h m  an  r e p r e s e n t i n g wavelet  and  w a v e l e t .  e s s e n t i a l l y (20%  In  o r i g i n a l l y  at  r e s u l t s  show  v a l u e .  peaks  d i s p e r s e d  I  each s i m i l a r  wavelet such  that  182  the  i n d i v i d u a l  o f f s e t  range  e a r t h  l a y e r s  wavelets a  t r a v e l  i s t o o g r e a t , vary  r e f l e c t o r  assumption  that  each  breaks  down,  moveout  h y p e r b o l a  Q  v a l u e .  up  t o  3  km  paths  t h i s  example  w i l l  a l s o  r a y path  used  a  spanned  2  F i g u r e  4.37, I  v e r s u s  x(U)  here  a  The  t h e data  r e c e i v e r s t h i s  from  by from  case  an  Q  t h e value  i n d i v i d u a l  common  c u m u l a t i v e  4.36a) over  s y n t h e t i c were  seen  c u m u l a t i v e  Q  i ft h e  i n d i v i d u a l  v a l u e  Q  I n  ( F i g u r e  i n c u m u l a t i v e  i s two  l a y e r s  r e s p e c t i v e l y  t h e o b j e c t i v e  I n 4.37a  o f f s e t s  of  t h e v a r i o u s  seismograms  contaminated  i s  i n s u f f i c i e n t  d i f f e r e n c e  i n  t h e  waveforms.  I n  F i g u r e  h o r i z o n ,  a r r i v a l  time  t r a v e l  w i t h  of  a  each  l a y e r .  The  v a l u e s  of  times I  (hence show  sequence wavelet  t o  o f  ( t h e  peak  i n 5%  42  degrade  produce  f o rt h e  1  5 1 ,  l a t e r ) .  s e v e r e l y ,  a  n o t i c e a b l e  appearance)  o f  t h e  f o r t h e  f i r s t  u n d i s p e r s i n g Q t h e window  Q=30,  and  Table  KL  t h r e e  a t  maxima  t h e wavelets  w i t h i n  r e a l  t h e f i r s t  d e f i n e d  t h e r e s u l t s  moveout  4.38,  from  Q  w e l l  f o r  a r e t o o high-see  h y p e r b o l a ,  t h e r e  r e f l e c t i n g  see a  f o r t h i s  v a l u e s  f u n c t i o n  v a l u e  Q  c u m u l a t i v e  these  moveout  we  v a l u e  Q  g i v e  (both  t h e t h i r d  show  u n d i s p e r s i n g  t h e c o r r e c t  the  used  and  p i c k e d  t o e s t i m a t e  r e f l e c t o r .  km,  t h e same  wavelets  change  given  which  as  model  l i t t l e  h o r i z o n s .  A f t e r  be  g r e a t l y .  samples  cannot  t h e  t o d i f f e r e n t  vary  of  i n  t h e c u m u l a t i v e  r a y paths  r e f l e c t i n g  next  and  times  However,  n o i s e .  In EVR,  f o r  t h e t r a v e l  t h e s u i t e  produce  s i g n i f i c a n t l y .  along  F o r t h e simple  ray  random  and  vary  g r e a t l y ,  p r o p a g a t i n g  given  times  v a l u e s . ( p i c k e d  As from  183  the  moveout  p r o c e s s i n g each  w i l l  over  i s  30.  r e s t o r e d  Of  the  one  case  wavelet  I  summed  i s  (Figure  4.36)  y i e l d s  the  the  repeated  from  between  event l a y e r  the  top  In  (Figure  s e v e r e l y  degraded KL  was  r e s u l t s  for  the of  value  the  base  2  and  instance  were  of  the  e s s e n t i a l l y  and the  a l l  a  the  s i n g l e  This  of  the  f i r s t  i s the  the  f u n c t i o n s  same.  The t r a c e  as  compared  i n c l u d i n g  a l l  Q  comparison  r e f l e c t i o n  for  that  each  c u m u l a t i v e  noise  This  r e s u l t s  l a y e r ,  being  In  f i r s t  g a t h e r .  w i t h i n  f i r s t  of  t r a c e .  assumes  the  The  case  been  r e s u l t  the  o b j e c t i v e  r e s u l t s ,  have  under  and  1  are  are  t u r n ,  give  I n c r e a s i n g  25)  t r a c e .  the  the  and  f i r s t  the  subsequent for  the  along  4.39.  l a y e r  for  between  v a l u e s .  for  e f f e c t  45)  i s  wavelets  r e s u l t s  repeated  these  i n  the  F i g u r e  Q  4.40).  d i s p e r s i o n  region  l a y e r  t h i s  time  wavelet  the  w a v e l e t .  a r r i v a l  over  of  panel  case  t r a c e  and  Ricker  Q  i n  35  the  c u m u l a t i v e  Subsequent  d i s t i n c t  complex  the  {Q=  window,  for  (Q=20  t h i s  each  2.  procedure  e f f e c t s .  for  value  i n  r e l a t i v e  shown  from  w i t h i n  the  d i s p e r s i v e  Q  panels  i n t e r e s t  down  panels  n o n - d i s p e r s e d  comparison  are  e a r l y  l a t e r  each  l a t e r a l l y a  the  subsequent  i n t e r v a l  r e f l e c t i o n  T h i s  compute  v a r y i n g for  a  from  e s t i m a t e  r e s u l t s  that  of  more  c o r r e c t  w a v e l e t s  p r a c t i c a l  each  to  not  the  wavelet  and  procedure  The  the  form  greater  p i c k i n g  and  (2=30,  s l i g h t l y  p r o p o r t i o n a l l y  C o n s e q u e n t l y ,  At  to  i n c r e a s e s  w a v e l e t .  c o r r e c t e d ,  c o r r e c t e d .  is  remove  s u c c e s s i v e  h o r i z o n  t h i s  hyperbola)  l e v e l s  Q  the to value  h o r i z o n s . a t t e n u a t i v e are above  examples,  l e s s 10% the  184  equation  From is  r e l a t e d  (A1.12),  we  t h e c u m u l a t i v e Q  t o  d e r i v e  value  j  Q.  =  J  This to  be  t h e  data  l  i=i  )/Qcum.  scheme  adequate change  in  t h e  holds  f o r  the  -(  J  Q  t h e cumulative  shown  i n  i=  j  o f f s e t s  F i g u r e s  t  }~ 1  )/Qcum.  J  o f f s e t  d a t a ,  considered  v a l u e s .  4.37  value  by:  L  f o r zero  Q  i n t e r v a l  J - J  L  J  i n v e r s i o n  l i t t l e on  l . {{  that  and  For  but  here, the  was as  found  there  r e s u l t s  4 . 3 9 , we  note  was based  i n Table  1  t h a t :  TABLE j  Layer  Q(True)  I n t e r v a l  Thickness  1 C2(Disp)  Qcum(est)  Veloc it y  Q(Atten)  Q(est)  Qcum(est)  Q(est)  1  500  1700  30  --  30  2  500  1850  50  53  53  51  51  3  400  2000  80  62  80  66  109  4  500  21 5 0  90  69  90  70  80  5  300  2350  100  78  280  75  1 24  Where  t h e  i n t e r v a l  v e l o c i t y  s y n t h e t i c r e s u l t s , (for  columns  d a t a , and  t h e Q  t h e d i s p e r s e d  values  and  s y n t h e t i c  show:  (m/s),  a c t u a l  t h e cumulative value  The  Q  values  Q  number,  Q Q  estimated  s y n t h e t i c  e s t i m a t e d d a t a .  l a y e r  d a t a ) ,  l a y e r  value  used  values  p i c k e d  from and  f o r t h e  value  30  t h e  t o  (m),  generate  t h e  from  t h e  i n v e r s i o n  f i n a l l y ,  p l o t t e d  procedure  t h e cumulative  a t t e n u a t e d  f o r t h e f i r s t  t h i c k n e s s  l a y e r  and was  Q  d i s p e r s e d estimated  185  from  the  were  a s s o c i a t e d  e s t i m a t e d  from  moveout the  down-trace  The  agreement,  (the  l a y e r  t h i c k n e s s  i n t e r e s t ) .  The  i n t e r v a l  scheme  p i c k e d  good  not  were  assumed  a  f u n c t i o n  However,  f o r  f i v e  from  and  an  the  times  f o r  a l l  very  shown  f o r  the  a n a l y s i s  of  the  data,  and  s y n t h e t i c  d a t a ,  the  12  t r a c e s  d i s p e r s e d  more  are  i s  i n  d i s p e r s e d - o n l y  (using  values  use  the  and  a r r i v a l s  v e l o c i t y  v e l o c i t y  f o r  subsequent  e a r l i e r  RKL  For  used  f o r  i n t e r v a l  attenuated  were  whereas  c o m p a r i s o n s .  t r a v e l  estimates  the  t r a c e s  e s p e c i a l l y  p r i o r i .  o b j e c t i v e  hyperbola,  were  data  degraded  i n v e r s i o n  combined.  only  the  the  f i r s t  o b j e c t i v e  funct i o n ) .  e.  D i s c u s s i o n .  The  eigenvalue  t r a n s f o r m a t i o n measure. under  can  Here  what  r a t i o be  I  c o n d i t i o n s , by  examples,  constant  the  w e l l ,  frequency, and  a  and  and  t o  u t i l i z e d  approximated  q u i t e  used  d e r i v e d  a  from  c o n s t r u c t t h i s  phase  phase  complex  t o  a p p r o x i m a t i o n  was  e f f e c t i v e was  simple  and  can  be  s y n t h e t i c  shown Q  t o  v a l u e ,  d e r i v e d  KL  when,  d i s t o r t i o n W i t h  s h i f t  r e a l  s i m i l a r i t y  a s c e r t a i n  change.  phase  or  s e n s i t i v e  s i g n a l  r e l a t i o n s h i p between  r e p r e s e n t a t i v e  a  measure  d i s p e r s i v e  constant  the  hold c e n t r e  e m p i r i c a l l y  a n a l y t i c a l l y .  F u r t h e r , success  i n  I  used  removing  the  measure  d i s p e r s i v e  t o  e f f e c t s  q u a n t i f y from  the  s i g n a l s .  degree In  the  of case  186  of the  a  s y n t h e t i c  m u l t i c h a n n e l  cumulative  model.  T h i s  hyperbolae  and was  and  down  R e s u l t s  f o r  Q  e f f e c t s ,  and  a l s o  ampli t u d e ) .  common  shot  p o i n t  i n t e r v a l  Q  s t r u c t u r e  achieved  by  comparing  t r a c e s  w i t h i n  s t r u c t u r e by  were  a  loop  degraded  i n c l u d i n g  random  gather,  f o r  a  by  along over  i n c l u d i n g noise  estimated  l a y e r e d  waveforms i t e r a t i n g  I  Q  e a r t h moveout v a l u e s .  a t t e n u a t i v e  (above  10%  by  187  Each  panel  w a v e l e t , (4)  the  a m p l i t u d e waveform 1 1 5  shows:  (3)  t h e  r e a l  (1)  r e a l part  exaggerated as  i n  (2)  FIGURE 4.28 t h e reference  part of  of  t h e f o r  r o t a t e d  the CKL  CKL  wavelet, f i r s t  second  p l o t t i n g ) , by  e.  (2)  t h e  p r i n c i p a l  d i s p e r s e d component  p r i n c i p a l component and  Panels  a r e  (5)  the  shown  (with  d i s p e r s e d  f o r Q-  25  t o  188  0.1  0.1  o  FIGURE Each  panel  d i s p e r s e d  shows:  w a v e l e t ,  component,  (4)  component,  (with  d i s p e r s e d  waveform  <2=  25  t o  1 15  (1) (3)  the  t h e t h e  r e a l  a m p l i t u d e as  i n  4.29  r e f e r e n c e r e a l  part  part  o  wavelet, of  of  the  the  CKL CKL  (2)  t h e  f o r p l o t t i n g ) ,  (2)  by  e.  p r i n c i p a l  second  e x a g g e r a t e d r o t a t e d  attenuated  f i r s t  Panels  P r i n c i p a l and  a r e  (5)  shown  t h e f o r  189  2.71  3.323  3.935  4.548  5.16  L N ( Q - E F F )  FIGURE The e f f v e r s u s l the l o C i r c l e s =  4.30  e c t of v a r y i n g c e n t r e f r e q u e n c y on t h e l o c u s o f l o g | e | og((? f,) , f o r d i s p e r s e d R i c k e r w a v e l e t s . D e v i a t i o n s from c i i a r e caused by a l i g n m e n t problems (see t e x t ) . 1 5 H z , t r i a n g l e s = 2 5 H z , +=35Hz, X=45Hz, and diamonds=55Hz  190  LO  2 71  3.323  3.935  4.548  5.16  LN(RTT(Q-EFFjJ  FIGURE The ver D e v t e x dia  s i t m  4.31  e f f e c t o f v a r y i n g c e n t r e frequency on t h e l o c u s o f l o g | e | us log(£? , , ) , f o ra t t e n u a t e d and d i s p e r s e d R i c k e r w a v e l e t s . a t i o n s from t h e l o c i i a r e caused by alignment problems ( s e e ) . C i r c l e s = 1 5 H z , t r i a n g l e s = 2 5 H z , +=35Hz, X=45Hz, a n d onds=55Hz  1 9 1  o 0  0  — i  0  0  20.0  40.0  60.0  8.0.0  FCCHZJ  FIGURE The  v a r i a t i o n  c e n t r e  of  the  frequency  a n a l y t i c a l  r e s u l t s .  i n t e r c e p t  (shown  as  i n  4.32 F i g u r e  c i r c l e s ) .  4.30,  as  a  T r i a n g l e s  f u n c t i o n represent  of the  1  2.708  3.206  3.704  4.202  92  4.7  LN(Q-EFF)  F I G U R E 4.33 The e f f e c t o f v a r y i n g c e n t r e frequency on versus l o g ( ( 2 /•/•)/ f o r d i s p e r s e d c o s i n e - G D e v i a t i o n s from t h e l o c c i a r e caused by C i r c l e s = 2 5 H z , t r i a n g l e s = 3 5 H z , +=45Hz, X=55Hz,  t h e l o c u s o f log|e| a u s s i a n w a v e l e t s . alignment problems. and diamonds=65Hz  1 93  o 0  3  — i  o  oo'H  0.0  I 20.0  1 40.0  I 60.0  I 80.0  FC(HZ)  FIGURE  4.34  The v a r i a t i o n of the i n t e r c e p t i n F i g u r e 4.33, as a f u n c t i o n of c e n t r e frequency (shown as c i r c l e s ) . T r i a n g l e s represent the a n a l y t i c a l r e s u l t s .  1 94  FIGURE (a) The removal  EVR and e procedure  (b)  The  r e s u l t s  (c)  degraded  4.35  o b j e c t i v e f u n c t i o n s on a d i s p e r s e d R i c k e r  for  r e s u l t s  a t t e n u a t e d from  n o i s y  and  from a wavelet.  d i s p e r s e d  a t t e n u a t e d  CKL  d a t a ,  d i s p e r s e d  d i s p e r s i o n  and d a t a .  195  O O  (0  o \ > E  o  o o  o  00  o  o m  CO  o  w  CO  0)  c  u  (a)  The  \  CM  o  O 0)  o  \  V  \  CO CM  CM  o  (0 CO  o  m  T-  CM  T"  T—  o  o o  o  v N  *- \ T-  \ v \  o o m  l a y e r e d  o o  e a r t h  model  (b) D i s p e r s e d s y n t h e t i c from 0 t o 2000 m.  o o <r  o o w  FIGURE 4.36 used f o r Q  seismograms,  O O CO  \ \ \ \ \  i n v e r s i o n .  w i t h  5%  n o i s e .  O f f s e t s  range  196  FIGURE The  RKL  h y p e r b o l a e .  o b j e c t i v e The  hyperbola  as  v a l u e s  t o o  a r e  a  f u n c t i o n  4.37 f o r  s i m i l a r i t y  i s computed  whole.  t h e  h i g h .  A f t e r  the  f i r s t  between  f i r s t  t h r e e  t h e  r e s u l t ,  members  t h e  moveout of  cumulative  a Q  197  0.4  • m i m ) n n  FIGURE  Twelve wavelets picked f r o m t h e f i r in F i g u r e 4.36), shown a t d i f f e r e orocess A t Q=30, t h e w a v e l e t s a r e c o r r e s p o n d i n g value o f the o b j e c t i v  4.38 s t move n t sta c o r r e c t e f u n c t  o g l i  ut h es y u n o n e  y p e r b o l a ( a t 0.4 s i n t h e i t e r a t i v e d i s p e r s e d , and t h e x h i b i t s a maximum.  198  o  FIGURE 4.39 The RKL o b j e c t i v e f u n c t i o n f o r the l a t t e r fou (2, 3, 4, and 5 ) , f o r d i s p e r s e d d a t a . Here computed p a i r - w i s e between the members of r e s u l t s f r o m , e a c h o f t h e 12 t r a c e s were combi s i n g l e compound o b j e c t i v e f u n c t i o n f o r a given  r r e f l e c t i o n t h e s i m i l a r a s i n g l e t r a ned t o pro r e f l e c t i o n  eve i t y c e . duce e v e n  nts i s The a t .  199  (-,10  —1  25.0  60.0  42.5  0-VflLUE  FIGURE The RKL (2, 3, s i m i l a r i t r a c e . produce r e f l e c t i  o b j e c t i v e f u n c t i o n 4, and 5 ) , f o r a t y i s computed-pair The r e s u l t s from a s i n g l e compoun o n e v e n t .  1  60.0  Q-VALUE  i  77.  I  95.0  4.40  f o r the l a t t e r fou t t e n u a t e d and d i s p -wise between the e a c h o f t h e 12 t r a d o b j e c t i v e f u n c  r r e f l e c t i o n e r s e d d a t a . He members of a c e s were combi t i o n f o r a  e v e n t re t h s i n g l ned t give  s e e o n  200  CHAPTER  5.  CONCLUSIONS.  SECTION  5.1;  My  REVIEW  i n i t i a l  OF  THE  GOALS  i n t e r e s t  i n  concerned  i t s a b i l i t y  to  or  of  This  a  s u i t e  s i g n a l s .  q u a n t i f y  d i s p e r s i o n  i n t e r e s t  s h i f t e d  transform  to  i n to  THIS  the  q u a n t i f y  seismic  the  s i m i l a r i t y  arose  d a t a .  more  WORK.  Karhunen-Loeve  i n t e r e s t  the  image  OF  from  E x t e n d i n g  g e n e r a l  p r o c e s s i n g  transform  between my  p a i r ,  attempts  from  that  i n  to  work,  a p p l i c a t i o n s  problems  a  my  of  t h i s  seismic  data  p r o c e s s i n g .  I  f i r s t  used  the  p r o c e s s i n g  techniques  i n f o r m a t i o n  from  to  . data  s t a c k e d  s a t e l l i t e  image  d i f f r a c t i o n  et  t r a c e s found  the  problem.  by  N o t i n g  prompted  me  LeBlanc  and  to  I  b a s i s  paper  view  From  these  p r o c e s s i n g  i n s i g h t s  the  a  r e c o n s t r u c t i o n  a l . , 1983), on  w i t h  d a t a .  s e p a r a t i o n ,  Applying (Levy  the  t r a n s f o r m a t i o n  and  some  m u l t i p l e  from  endeavours  work  e i g e n v e c t o r  of  which  i n v e s t i g a t e  that  problem  e x i s t i n g a d d i t i o n a l  a p p l i c a t i o n s lead  of  the  r e c o n s t r u c t i o n , have  t r a c e  evolved.  d e t e c t i o n '  g r o u p i n g  w e i g h t i n g s .  p i t f a l l s  and  'dead  problem  the  (1980)  m i s f i t  on  of  M i d d l e t o n  the  the  s u p p r e s s i o n  the  (1982)  update  e x t r a c t i n g  ( f o l l o w i n g  c o n s i d e r e d  Hagen  to  community),  g a i n e d  of  to  seismic  Subsequently  I  d e a l t  w i t h  the  same  i n f e r r e d  from  t h i s  paper  f u r t h e r .  M i l l i g a n  et  a l .  The (1978)  work  by  d e a l i n g  201  w i t h  a c o u s t i c  r e l i a b l e the  i n f e r e n c e s  b a s i s  of  measurements technique has  of  p h y s i c a l  work i n  o p t i m i s t i c  of  of  be  on the  r e s u l t s  on  technique  an  I  B e r e s f o r d - S m i t h  et  complex  r e a l  versus  r e s u l t s  p o t e n t i a l  i n  the  use.  produces  a  and  I  the  examples w i t h  my  r e a l of  The  sediment i n  type  on  c o n j u n c t i o n  a p p l i c a t i o n  c h a r a c t e r  t h a t  change  w i t h  of  t h i s  r e c o g n i t i o n  I  of  study.  by  the  t h a t  extremely  data  p o i n t  work.  a k i n  s e i s m i c of  to  T h i s  to for  the the  waveform the  a n a l y t i c  d e a l t  w i t h  geometric  i n s t a n t a n e o u s  the  l i m i t a t i o n  e f f e c t As  a  to  use  on  r e s u l t , of  the  u s e f u l n e s s  of  frequency  of  the  as  a  search  d e v i s e d  by  diagrams  ( i . e . a  They  i n v e s t i g a t e d  was  t e c h n i q u e ,  the  phasor  p i c t u r e s q u e ,  enroute  I  q u a n t i f i c a t i o n  were  and  my  encouraging.  t h a t  here)  d i s p e r s i o n .  time  began  shown  profound  from  a n a l y s i s  were  examining  part  evolved  At  v e l o c i t y  (1980),  began  a l s o  n o i s e .  i n i t i a l  Mason  more  t o p i c ,  of  f u n c t i o n '  complex  a t t e n u a t i o n - r e l a t e d  procedure,  a n a l y s i s  (1983)  presence  present  t r a c e  p r a c t i c a l  phase  the  and  (not  s p e c i f i c  c o n c l u s i v e l y  bottom  a n a l y s i s  a l .  s y n t h e t i c  e f f e c t s .  the  c l u s t e r  v e l o c i t y  ' o b j e c t i v e  d i s p e r s i v e  about  s t r a t i g r a p h i c  Levy  the  F i n a l l y , for  made  showed  parameters.  d i s a p p o i n t i n g r e s u l t s the  had  s i g n i f i c a n c e .  involvement  The  c o u l d  s e i s m i c  p o t e n t i a l  the  r e c o r d s ,  e i g e n v e c t o r  to  The  as  pinger  p l o t  t r a c e ) . but  of  f o r  of  the  These l i t t l e  d i s p e r s i o n , data  of  which  than  does  I  abandoned  t h i s  KL  e i g e n v a l u e  r a t i o  the d a t a .  instantaneous N e i t h e r  of  the  202  l a t t e r t r i e d  t r a n s f o r m a t i o n s t h e eigenvalue  While s h i f t e d  appearance  This t h e  e s t i m a t e  of  c h a r a c t e r  f o r a  of  a  I  noted  f i n a l l y  I  t h e  phase  t o  model  constant  phase  a t t e m p t e d simple  a c c e p t a b l e ,  i t e r a t i v e  d i s p e r s i o n  d i s p e r s e d  t o  and  i n S e c t i o n 2 . I I .  and  w i t h  and  h e l p ,  problem,  reasonably  ( r e l a t i v e  Q  much  s i g n a l s ,  reference  of  of  d e s c r i b e d  change  proved  procedure  estimate  method  d i s p e r s e d  procedure  e  t o be  t h e d i s p e r s i o n  of  wavelet  o b v i a t e  d i r e c t  r a t i o  a d d r e s s i n g  d i s p e r s i v e change.  proved  and  removal,  s i g n a l  t h e r e f e r e n c e  may  even as  y i e l d s  s i g n a l :  an a  S e c t i o n  4 . I l l ) .  SECTION  a.  5.11:  Image  The  REMARKS.  p r o c e s s i n g .  main  a p p l i c a t i o n s  of  Karhunen-Loeve i n f o r m a t i o n d e p i c t i o n  The has  been  image  t h e core  t h e most  e x t e n s i v e l y  i n t o  t h i s  t h e r e a l  work  p r o c e s s i n g  of  ( S e c t i o n  s e c t i o n  t o  based  i n t r o d u c e on  stacked i n  s e i s m i c  those  t h e o f  r e a l  coherent  s e c t i o n s  s t a c k e d  new  and  s e c t i o n s  3.  t o  i n d u s t r i a l i n an  been  2 . 1 ) . Recovery  r e c o n s t r u c t i o n  s u c c e s s f u l  data  o f  i n f o r m a t i o n Chapter  has  techniques  enhancement  anomalous  stacked  i n t e r e s t  of  t r a n s f o r m a t i o n  f o r image o f  c o n s t i t u t e d  t h r u s t  technique  ( S e c t i o n  3.1)  and  has  a r o u s e d  some  community,  and  been  date,  i n d u s t r i a l  p r o c e s s i n g  a p p l i e d  environment.  203  I t s  main  advantage  background  l e v e l  t h i s  t h e s i s ,  most  s i g n i f i c a n t .  d a t a ,  I  of  widely  I t  t h i s  i s  a  good  drawing  a t t e n t i o n  but  m i s f i t  for  to  would  more  u s e f u l n e s s  of  s e i s m i c data  sets  to  Once  for  f a u l t  t h e i r  i s o l a t i n g  the edges.  c o n t r o l l e d  hydrocarbon  i n t e r e s t i n g  one,  l i k e l y  w i l l  f a u l t e d  i t  r e g i o n s .  but  best  zones.  best  be  the  a p p r o p r i a t e  i s  m i s f i t  from  stacked  d i f f r a c t i o n  t r a p p i n g probably used  not w i t h  be  data  area  of  an to  r e a d i l y  s e i s m i c  i n  t h i s  d i v e r s e  a v a i l a b l e .  data can  areas  T h i s of  s p e c i f i c  with  the  form  the  severe l o c a t i o n  s t r u c t u r a l l y  a p p l i c a t i o n  g e n e r a l data  a  ( S e c t i o n  w i t h  i n d i c a t e  or  i n d u s t r i a l many  d e l i n e a t i n g  f e a t u r e s .  an  r e c o n s t r u c t i o n  may  i n  i n  h y p e r b o l a e  a d d i t i o n ,  u s e f u l  the  i n  s i g n i f i c a n c e  a c c e s s are  t h i s  from  u s e f u l n e s s  where  as  a s s i s t a n c e  of  the  not  for  input  a s s e s s e d  background  In  of  knowledge  d i f f r a c t i o n s  prove  • be  i n  be  probably  be  The  The  the  w i l l  to  w i t h  i s  may  o v e r l a y .  m i g r a t i o n .  may  and  the  considered  a p p r o p r i a t e  of  c o r o l l a r y  segregated  are  e v a l u a t i o n  g e o l o g i c a l  This  which  environment  s e g r e g a t e d ,  t o p i c s  3.II)  s p e c i f i c  d i f f r a c t i o n s  r e s i d u a l  f a u l t i n g , of  would  reduce  a p p l i c a t i o n  p r e s e n t a t i o n  m i s f i t  i n t e r e s t i n g  3.111). b a s i s  and  data  and  i n t e r p r e t a t i o n  An view  technique  the  implement,  ( S e c t i o n  meaningful the  to  anomalous  i n t e r p r e t e r  Of  g r e a t l y  r e s u l t s .  o v e r l a y  experienced enable  n o i s e .  to  p a r t i c u l a r  r e c o n s t r u c t i o n  method,  a b i l i t y  simple  y i e l d s  a p p l i c a b l e ,  p a r t i c u l a r  i t s  incoherent  c o n s i d e r  g e n e r a l l y  M i s f i t  i s  use.  from  i s  an  More  s e v e r e l y  204  F i n a l l y ,  I  e l i m i n a t i o n (Section-  present  of  3.IV).  As  an  s t a c k i n g  d a t a ,  VA  maps  events  which  p r o c e s s i n g  w i t h  method  s t a c k i n g .  That  waveforms t e s t s  such  of  p r o m i s i n g .  I  produces  b.  t h a t  the  t h i s  i n c o r p o r a t e d  stacked  to  t h e s i s .  In  no  longer  time.  i s  to i t  standard  of  i s  before hyperbola  the simple  p r o c e s s i n g  by  importance  shape  not  s e c t i o n s be  of  d i s t o r t e d  degraded.  stacked  A g a i n ,  data  A l s o  does  r e a l  primary  map  wavelet  technique  On  absence  VA  a c t u a l  preserve  m u l t i p l e  a d d i t i o n ,  the  the  v e l o c i t y  s u c c e s s f u l .  the  same  for  the  complete  i n  was  technique  a  e s t i m a t e  almost  as  method s t a c k i n g  proved  r e s u l t  produced  i n t o  encouraging  the  p r o c e s s i n g  t h i s  to  p r o c e s s i n g .  event at  technique  of  the  i t s a b i l i t y  i s , the  known  a l s o  enhanced,  a r r i v i n g i s  of  amplitude  primary  c o n s i d e r  c o n t r i b u t i o n r e a d i l y  the  t h i s  map  small  e f f e c t i v e  attempt  a f t e r  g r e a t l y  to  energy  t h i s  of  were  m u l t i p l e  VA  energy  were  c o r r e s p o n d i n g  the  an  h i g h l i g h t  r e l a t e d  and  events  a s i d e ,  by  m u l t i p l e  simple  m u l t i p l e  v e l o c i t y the  a  a f t e r  d i s t o r t The  the  i n i t i a l  which  looked  second  major  enough  to  stream,  be and  r e s u l t s .  S i m i l a r i t y .  In  Chapter  e i g e n v e c t o r s  and  4,  e x t e n s i o n  d e t a i l e d  knowledge i t s  i n v e s t i g a t e d  e i g e n v a l u e s  n a t u r a l  a p p r e c i a t e  I  of of  the the  p i t f a l l s .  of  the  work  v a g a r i e s  the  p r o p e r t i e s  t r a n s f o r m a t i o n . on of  T h i s  r e c o n s t r u c t i o n , the  method  was  of  as  was a  necessary  the a more to  205  A p p l i c a t i o n 4.1)  showed  the  of same  r e c o n s t r u c t i o n .  e x p l o r a t i o n i s t  on  the  stacked  r e c o n s t r u c t i o n , an  The  a n a l y s i s  when  a p p l i e d  background  RKL  and  semblance the  data of  CKL  VA  i n  a p p l i c a t i o n  v e l o c i t y h i g h  r a t i o  phase  methods  the  However, r e a l  be  but how  m i s f i t  t o o l  t o  much  a i d  depends  non-uniform  w i t h  the  a  m i s f i t  i n v e s t i g a t e d  c o n s t i t u t e s the  the  of the  c l e a r l y  as  c o n t e x t  measure  good  f u r t h e r  a r r i v a l s t o  d e t a i l e d  a t  e a r l y  v e l o c i t y t o  f a i l  l e v e l s  t h i s  f a i l u r e .  i n c r e a s e  demonstrated  out-performed  of  h i g h l y  High  f o r  may  a  seems  d a t a .  cause  d e c o n v o l u t i o n  However,  f o r  u s e f u l  on  be  i n  d a t a ,  t o  VA  d i d  environment.  seem  KL  a  as  ( S e c t i o n  i n  F i g u r e  the  t i m e s .  the 4.21,  c o n v e n t i o n a l T h i s  p o i n t s  t o  h i g h  r e s o l u t i o n  VA  and  RMS  study  of  s u r f a c e  from  the  near  d a t a .  c r i t e r i o n  t o  s i g n a l  change, measure  best  segments  r e s o l v i n g of  As  small  success.  a d d i t i o n  d i s p e r s i v e  w i t h  a f t e r  i n v e r s i o n ,  q u a l i t y  In  the  noise  w i t h  p r o b a b i l i t y  t o  be.  c r i t e r i o n  measure.  problems  f e a t u r e s , and  would  r a t i o  4 . I I )  p r o v i d e  d a t a ,  may  technique  s i m i l a r i t y  may  t h i s  s e i s m i c  weaknesses  unusual  i n t e r p r e t a t i o n  ( S e c t i o n  Working  the  of  e i g e n v a l u e  s e n s i t i v e  of  l o c a t i n g  t o  and  a n a l y s i s  s e c t i o n  t h i s  i n d u s t r i a l  except  i n  p r e s e n t a t i o n  p a r t i c u l a r  a n a l y s i s  s t r e n g t h s  C l u s t e r  the  in  c l u s t e r  t o  v e l o c i t y  a s c e r t a i n  a n a l y s i s , when,  d i s t o r t i o n  can  I  and be  u t i l i z e d under  the  what  approximated  f o r  s y n t h e t i c  data  (Sect ion  t o  q u a n t i f y  the  degree  4.111). of  e i g e n v a l u e c o n d i t i o n s ,  be  a  constant  F u r t h e r ,  success  i n  I  used  removing  206  d i s p e r s i v e the  e f f e c t s  constant  phase  r e l a t i o n s h i p  r e s u l t  phase  s h i f t  which  5 . 1 I I ;  was  shown  v a l u e ,  Q  d e r i v e d  f o r R i c k e r  c o r r o b o r a t e d  t o  h o l d ,  centre  was  t h e e m p i r i c a l  and  frequency,  e m p i r i c a l l y .  wavelets  examples,  A  a and  t h e o r e t i c a l  presented,  and  a  f i n d i n g s .  RECOMMENDATIONS.  a p p l i c a t i o n  e x t r a c t i o n been  of  i n s t a c k i n g  e x p l o r e d  (explored  f u l l y  i n d e t a i l  r e c o n s t r u c t i o n  of  moveout  i n t h i s  by  o f  t h e Karhunen-Loeve  CKL  transforms  p r e - s t a c k e d  r e s t r i c t e d  problem  due  of  t o l i m i t a t i o n s  CDP  d a t a .  e x t r a c t I  was  t h i s  of  a  t h e s i s  s i g n a l  has  preamble  both  t o t h e  been t h e  noted  RKL  s i g n a l  and  from  i n those  e x c l u d i n g  n o t  a p p l i c a t i o n  has  coherent  by  t h i s a  I t  co-author  t o  gathers  1978) as  e t a l . , 1983) that  A l t h o u g h  t h e scope  t r a n s f o r m  i n t r o d u c e d  s e i s m i c  o p t i m a l l y  g a t h e r .  I  and Mace,  stacked  e t a l . , 1983, U l r y c h  c o r r e c t e d  work.  Hemon  (Levy  b.  was  s y n t h e t i c  S t a c k i n g .  The  I  With.simple  e f f e c t i v e  t h e problem  d e r i v e d  SECTION  a.  o f  s i g n a l s .  approximation  between  r e p r e s e n t a t i v e treatment  from  a  s t u d i e s ,  t h e  s t a c k i n g  time.  R e c o n s t r u c t i o n .  The a p p l i e d  r e c o n s t r u c t i o n e x t e n s i v e l y  environment.  F u r t h e r  o f  stacked  t o d a t a ,  w i t h  m o d i f i c a t i o n s  s e i s m i c good t o  s e c t i o n s  r e s u l t s t h e  i n t h e  e x i s t i n g  has  been  i n d u s t r i a l a l g o r i t h m  207  would  i n c l u d e  ( S e c t i o n  2 . I l l )  present, i s  the  l i m i t e d  combines  KL  the  t o p i c s ,  Hampson,  f e a t u r e s  and  m u l t i p l e  the  l a t t e r simple  s l a n t  a  of  dependent' s i n g l e  l a r g e  procedure 3 . 1 ( f ) ) .  c o v a r i a n c e  image.  seismic does  KL  ( S e c t i o n  The  not  yet  At  matrix  program  s e c t i o n s  Ryu  which  ( i . e .  the  i n c o r p o r a t e  the  f o r  other  so  far  a  g e n e r a l l y  s e p a r a t i o n i n g .  scheme  have  to  t e c h n i q u e .  of i t  the works  s u f f e r s  stacked  g r e a t e s t w e l l  a v a i l a b l e ,  introduced  h i g h l i g h t seismic Of  even  when the  both  data;  these  immediate  whereas  from  a p p l i c a t i o n s , not  been  r e c o g n i t i o n  p a r a l l e l may  attempt  I  three  v a l u e .  only  commonly  wrap-around  and  a  An few  used  problems  FK (D.  1985).  two  anomaly  an  s u p p r e s s i o n  are  comm.  i n  r e c o n s t r u c t i o n ,  i n  i s  (1982)  m i s f i t  d i f f r a c t i o n s  scheme,  gather  of  found  of  s e c t i o n  p e r s .  For  f a u l t  'dip  technique)  c a t e g o r y  new  t e c h n i q u e  with  the  r e c o n s t r u c t i o n  of  segments  seismic  per  s e c t i o n  a  decomposition  the  i n t u i t i v e l y  have  of  of  o p t i o n .  anomalous  t r a c e s  compound  r e c o n s t r u c t i o n '  m i s f i t  a l s o  to  computation to  Under the  i n c o r p o r a t i o n  o v e r l a p p i n g  'compound s l a n t  the  as  the  data  c o n v i n c i n g .  w i l l  probably  s t r u c t u r e s ,  produce  r e a l  u s e f u l  The  r e s u l t s  m i s f i t  work  whereas i n  examples  best  the regions  i n  I  stacked areas  d i f f r a c t i o n of  severe  208  c.  S i m i l a r i t y  In  the  C r i t e r i a .  f o u r t h  e i g e n v e c t o r s c l u s t e r were  of  groups  shown  changes data  i n  seen  The  q u e s t i o n  these  then  the  would  best  seismic  the  ( S e c t i o n  (1983), for  be  e a r l y  r e s o l v e d  e i g e n v a l u e s .  The  e i g e n v e c t o r s  ( S e c t i o n  changes,  the  groupings d a t a ,  may  a s s o c i a t e d  was  can  not  be  as  great through  or  denote r e a l  Groupings the  d a t a .  u s e f u l n e s s  logs  match for  F u r t h e r  i n t e r a c t i o n  4.1)  For  i n  which  w e l l  v a l u e .  to  c l e a r .  f e a t u r e s  found  from  a l s o  examples.  s i g n i f i c a n c e  o b t a i n e d  have  and  data  d i s c e r n a b l e  to  to  eigenvalue 4.111  a r r i v a l s than  be  i n  r a t i o expanded  KL  c l e a n of  to  of the  example,  development  with  a  VA  work  t e c h n i q u e .  d a t a :  the  KL  f o r  for  near  s u r f a c e  l e v e l s the  high  of  gave  events  r e s o l u t i o n  sediments.  VA  of  semblance  n o i s e , KL  T h i s  VA  c o r r e s p o n d i n g  use  4.25,  c o n s t i t u t e  the  of  s i g n i f i c a n t F i g u r e  I  the  those  s t u d i e s  of  ) ,  a p p r a i s e  i n v e r s i o n  example  I f  the  s y n t h e t i c  w i t h  performed  may  had  t h e i r the  considered  s i t u a t i o n  as  other  technique  which  and  I  experienced  i n t e r p r e t e r s .  Using measure  the  remains  technique  t h e s i s ,  for  c o r r e l a t e d  of  t h i s  from  c h a r a c t e r  c o r r e l a t i o n s .  of  s t r u c t u r a l  however,  which  i n f e r e n c e s  KLT  b e l i e  phase  were  the  determined  to  examples  chapter  as  a  s i m i l a r i t y  Levy  a l .  good  r e s u l t s  being  b e t t e r  VA. RMS  However, evidenced  technique  et  The  KL  VA  v e l o c i t y for  data  by  the  f a i l e d .  T h i s  209  i n c o n s i s t e n c y standard  i n  p r o c e s s i n g  The  f i n a l  d i s p e r s i o n  i n  found  the  that  s i g n a l  between  the  s y n t h e t i c  i t  data  i n a p p r o p r i a t e  I  a  was s h i f t  of a  CKL  as  a  e f f e c t s  of  the  KL  to  angle  e,  as  be  those  s i m i l a r i t y . i n d i c a t o r  technique  and  a  the  and  r e l a t i o n s h i p  e m p i r i c a l l y  and  c o r r o b o r a t i o n  by  of  an  d i s p e r s i n g  using  I  estimate  and  performed  presented  to  s i g n a l  d e r i v e  i n v e s t i g a t i o n  c o u l d  of  good  d i s p e r s e d  a b l e  the  s i g n a l  t h e o r e t i c a l l y  F u r t h e r  r e s u l t s  the  with  a b i l i t y  was  using  both  such  d e a l t the  x(m),  r a t i o  phase  r e c o r d i n g s ,  and  between  d a t a ,  4  q u a n t i f i c a t i o n  F u r t h e r ,  seismograms.  waveform  d.  eigenvalue  measured  for  Chapter  d a t a ,  i n  r e f e r e n c e ,  the  s y n t h e t i c the  a i d  d i f f e r e n c e  undispersed  value  to  s i m i l a r i t y .  phase  of  s y n t h e t i c  r a t i o  renders  t e c h n i q u e .  s e c t i o n  eigenvalue  the  performance  high  Ricker  Q for of  q u a l i t y  (1953).  C o n c l u s i o n .  In  t h i s  work  Karhunen-Loeve s e i s m i c  I  p i c t u r e  are  data  as  to  and  a p p l i e d  the  s e v e r a l  p r o c e s s i n g .  e s s e n t i a l l y  p r o c e s s i n g  However,  demonstrated  t r a n s f o r m  r e f l e c t i o n  techniques  have  an  to  the  r e c o n s t r u c t i o n s ) ,  the  developed  d u r i n g  t h i s  work  i n d u s t r i a l  environment,  where  t o p i c s The  e x t e n s i o n  s a t e l l i t e  are  are  in  of  of  r e c o n s t r u c t i o n  a p p l i c a t i o n s  from  t r a n s m i s s i o n of  r e s i d u a l  new.  The  c u r r e n t l y  being  a d d i t i o n a l data  and  the  m u l t i c h a n n e l  image  data  s e p a r a t i o n  techniques  a p p l i c a b i l i t y  the  f i e l d s .  data  ( m i s f i t  a l g o r i t h m s t e s t e d  e x p e r t i s e  i n  an are  210  a v a i l a b l e  t o  judge  H o p e f u l l y , given  f u l l y  in  the  e x p l o i t i n g  s e i s m i c  u s e f u l n e s s  f u r t h e r  c o n s i d e r a t i o n  to  the  data  by the  work  on  of  these  i n d u s t r i a l  procedures.  and  r e l a t e d  p r o c e s s i n g  p o t e n t i a l of  p r o c e s s i n g  the  f i e l d .  the  t o p i c s  groups,  w i t h  Karhunen-Loeve  w i l l a  be view  t r a n s f o r m  21 1  REFERENCES.  1.  Ahmed,  N.,  and  d i g i t a l  Rao, image  K.R.,  1975,  p r o c e s s i n g :  K., and R i c h a r d s , W.H. F r e e m a n , New  O r t h o g o n a l S p r i n g e r ,  transforms  B e r l i n ,  p.  2.  A k i ,  3.  Andrews, H.C, Academic  4.  Andrews, H.C, and P a t t e r s o n , C . L . ( I l l ) , product expansions and t h e i r uses p r o c e s s i n g : IEEE Trans. Comput., v.25,  5.  Andrews, H.C, and P a t t e r s o n , C . L . ( I l l ) , 1976(b), S i n g u l a r value d e c o m p o s i t i o n (SVD) image coding: IEEE T r a n s . Commun., v.24, p. 425-432.  6.  Andrews, H.C, and P a t t e r s o n , C . L . ( I l l ) , 1976(c), S i n g u l a r value d e c o m p o s i t i o n and d i g i t a l image p r o c e s s i n g : IEEE T r a n s . A c o u s t i c s , Speech, and S i g n a l P r o c e s s i n g , v.24, p. 26-53.  7.  B e a u c h a m p , K.G, Academic  Beresford-Smith, approach frequency 551-571 .  9.  10.  P.G., 1980, Q u a n t i t a t i v e York, p 1 6 7 - l 8 0 .  f o r  189-224.  1970, Computer techniques P r e s s , New York.  1975, P r e s s ,  i n  image  p r o c e s s i n g :  1976(a), Outer i n d i g i t a l image p. 140-148.  Walsh f u n c t i o n s and t h e i r London and New Y o r k .  G., and Mason, I.M., to the compression of t r a n s f o r m a t i o n : Geoph.  seismology:  a p p l i c a t i o n s :  1980, A s e i s m i c P r o s p . ,  parametric s i g n a l s by v.28, p.  B e r k h o u t , A . , J . , 1984, Seismic m i g r a t i o n : imaging of energy by w a v e f i e l d e x t r a p o l a t i o n . B. aspects: E l s e v i e r , New York.  B r a c e w e l l , R.N., a p p l i c a t i o n s :  1978, second  The F o u r i e r t r a n s f o r m e d i t i o n , M c G r a w - H i l l , New  a c o u s t i c p r a c t i c a l  and i t s York.  212  11.  C h a p m a n , N.R, L e v y , S . , S t i n s o n , K., Jones, B.T., and Oldenburg, D.W., 1984, I n v e speed and d e n s i t y p r o f i l e s i n deep oce submitted t o t h e J o u r n a l o f t h e A c o u s t i America.  12.  Chapman, W.L., Brown, G.L., Vibroseis® system: a h i g h v.46, p. 1657-1666.  13.  C h i e n , Y.T., and F u , K.S., 1 9 6 7 , On t h e Karhunen-Loeve e x p a n s i o n : IEEE Trans. Theory, v.IT-13, p.518-520.  14.  C h r i s t e n s e n , R.A., a n d H i r s c h m a n , A.D. 1 9 7 9 , A u t o m a t i c alignment f o r t h e Karhunen-Loeve expansion: IEEE Biomed. E n g . , v.BME-26.  15.  Davenport, W.B.,Jr., and Root, W.L., to t h e theory of random s i g n a l s New York.  16.  D i x ,  17.  Edelmann, H.A.K., and W e r n e r , H., 1 9 8 2 , The e n c o d e d sweep t e c h n i q u e f o rVibroseis®: G e o p h y s i c s , v . 4 7 , p. 809-818.  18.  Fukunaga, K., a n d K o o n t z , W.L.G., Karhunen-Loeve expansion t o o r d e r i n g . IEEE Trans. Comput.,  19.  Futterman, Res.,  20.  Ganley, D . C , and Kanasewich, E.R., 1980, Measurement a b s o r p t i o n and d i s p e r s i o n i n check shot s u r v e y s : Geophys. R e s . , v . 8 5 , p. 5219-5226.  21.  C.H., 1955, S e i s m i c measurements: G e o p h y s i c s ,  .Gubbins,  and F a i r , D.W. frequency t o o l :  D.,  S c h o l l a r , d i g i t a l  g e n e r a l i z e d I n f o r m a t i o n  1 9 5 8 , An and n o i s e :  phase T r a n s .  i n t r o d u c t i o n M c G r a w - H i l l ,  s u r f a c e  1969, A p p l i c a t i o n o f t h e f e a t u r e s e l e c t i o n and v.C-19, p. 311-318.  body  waves:  I., and W i s s k i r c h e n , f i l t e r i n g  e r , und ts: of  1981, The Geophysics,  v e l o c i t i e s from v . 2 0 , p. 68-86.  W.I., 1962, D i s p e r s i v e v . 6 7 , p. 5279-5291.  d i m e n s i o n a l  I.F., P r a g r s i o n of so an sedimen c a l S o c i e t y  w i t h  J .  P.,  HAAR  Geophys.  o f J .  1 9 7 1 , Two and  WALSH  213  t r a n s f o r m s :  Annales  de  Geophysique,  v.21,  p.  85-104.  22.  Gurbuz, B.M., 1982, Upsweep s i g n a l s w i t h high-frequency a t t e n u a t i o n and t h e i r use i n t h e c o n s t r u c t i o n of Vibroseis® s y n t h e t i c seismograms: Geoph. Prosp., v . 3 0 , p. 432-443.  23.  Hagen, D.C., 1982, The a n a l y s i s to seismic 93- 1 1 1 .  24.  H a r l a n , W.S., C l a e r b o u t , J . F . , and Rocca, F., 1983, E x t r a c t i n g v e l o c i t i e s from d i f f r a c t i o n s : Proceedings of the 5 3 r d A n n u a l I n t e r n a t i o n a l SEG M e e t i n g L a s Vegas, paper S22.2, p. 574-577.  25.  H a r l a n , W.S., C l a e r b o u t , J . F . , and S i g n a l / n o i s e s e p a r a t i o n and Geophysics, v . 4 9 , p. 1869-1895.  Rocca, v e l o c i t y  26.  Hemon, Ch., and M a c e , D., 1978, Use t r a n s f o r m a t i o n i n s e i s m i c data Prosp., v . 2 6 , p. 600-626.  of t h e Karhunen-Loeve p r o c e s s i n g : Geoph.  27.  H o t e l l i n g , H., 1933, A n a l y s i s of complex s t a t i s t i c a l v a r i a b l e s i n t o p r i n c i p a l components: J . Educ. P s y c h o l . , v.24, p. 417-438 and p. 498-520.  28.  Huang, T . S . , a n d N a r e n d r a , P.M., 1975, Image s i n g u l a r value decomposition: A p p l . p.2213-2216.  29.  Hunt,  30.  J a i n ,  a p p l i c a t i o n of p r i n c i p a l components data s e t s : G e o e x p l o r a t i o n , v . 2 0 , p.  F., 1984, e s t i m a t i o n :  r e s t o r a t i o n by O p t . , v.14,  B.R., and K u b l e r , 0 . , 1984, Karhunen-Loeve m u l t i s p e c t r a l image r e s t o r a t i o n , p a r t 1: t h e o r y : IEEE Trans Accous., Speech, S i g n a l P r o c , v.Assp-32, p.592-599.  A.K., c l a s s  1976, A of  v.COM-24,  f a s t  random  Karhunen-Loeve processes:  p.1023-1029.  IEEE  t r a n s f o r m Trans.  f o r  a  Commun.,  214  31.  J a i n ,  A.K.,  1977,  d i g i t a l n o i s e :  A  f a s t  r e s t o r a t i o n IEEE  T r a n s .  Karhunen-Loeve  of  images  Comput.,  by  v.C-26,  t r a n s f o r m  white  and  f o r  c o l o r e d  p.560-571.  32.  J o n e s , I.F., and L e v y , S., 1 9 8 5 , On the and coherent n o i s e : f o r submission  33.  Kanamori, H., and A n d e r s o n , D.L., 1977, Importance of p h y s i c a l d i s p e r s i o n i n s u r f a c e wave and free o s c i l l a t i o n problems: review: Rev. Geophys. Space Phys., v. 15, p. 105-112.  34.  K a n a s e w i c h , E.R., 1982, g e o p h y s i c s : U n i v e r s i t y  35.  Karhunen, K., W a h r s c h e i n l (Suomalaine ( T r a n s l a t i o p r o b a b i l i t y C a l i f o r n i a )  1947, i c h k e i t s n T i e n by I. t h e o r y .  r d S .  of  Ub e c h e a k e l i " :  Time sequence A l b e r t a P r e s s ,  er l i n e a r e n u n g : Ann. A c a t e m i a ) , v.3 n , 1960: "On l i T-131 RAND C o r  36.  K j a r t a n s s o n , E., a t t e n u a t i o n :  37.  K n o p o f f , L., 625-660.  38.  Kramer, H.P., and t r a n s m i t t i n g a Inform. Theo.,  39.  L a r n e r , K., C h a m b e r s , R., Coherent n o i s e i n v.48, p. 854-886.  40.  L e B l a n c , L.R., and M i d d l e t o n , a c o u s t i c sound v e l o c i t y data Am., v.67, p. 2055-2062.  41.  L e v i n ,  S.A.,  1979, Constant J . Geophys. Res.,  1964,  Q:  s e p a r a t i o n t o Geoph.  Rev.  a n a l y s i s Edmonton.  i n  methoden i n der a d . S c i . Fenn., 7, p. 1-79. n e a r methods i n p., Santa Monica,  Q wave v.84, p.  Geophys.  of s i g n a l Prosp.  p r o p a g a t i o n 4737-4748.  Space  Phys.,  v.2,  and  p.  M a t h e w s , M.V., 1956, A l i n e a r c o d i n g f o r s e t of c o r r e l a t e d s i g n a l s : IRE T r a n s . v.IT-2, p. 41-46.  Rothman,  D.,  Y a n g , M., l y n n , marine seismic  and  W., W a i , W., 1983, data: G e o p h y s i c s ,  F.H., 1980, model: J .  Rocca,  F.,  An u n d e r w a t e r A c o u s t . Soc.  1983,  R e s i d u a l  215  m i g r a t i o n : a p p l i c a t i o n s and l i m i t a t i o n s : Proceedings of the 53rd A n n u a l I n t e r n a t i o n a l SEG M e e t i n g , Las Vegas, p a p e r S 1 0 . 7 , p. 393-395.  42.  Levy,  S., and Oldenburg, D.W., 1982, D e c o n v o l u t i o n of s h i f t e d w a v e l e t s : G e o p h y s i c s , v.47, p. 1285-1294.  43.  Levy,  S., U l 1983, A p p e x p l o r a t i I n t e r n a t i 325-328.  44.  L o e v e , M., 1948, F u n c t i o n s a l e a t o r i e s de Chapter 8, p . 2 9 9 - 3 5 2 , H e r m a n n , P a r i s .  second  45.  L o e v e , M. , York.  N o s t r a n d ,  46.  L o w i t z , G.E., 1978, S t a b i l i t y and Karhunen-Loeve m u l t i s p e c t r a l image R e c o g n i t i o n , v.10, p. 359-363.  47.  M a l l i c k , K., and M u r t h y , Y.V.S., 1984, MSS data over Zawar l e a d - z i n c mines, F i r s t Break, v.2, p.16-21.  48.  M c M e c h a n , G., w a v e s by 869-874.  49.  M i l l i g a n , S.D., L e B l a n c , L.W., and M i d d l e t o n , F.H., 1978, S t a t i s t i c a l g r o u p i n g of a c o u s t i c r e f l e c t i o n p r o f i l e s : J . A c o u s t . S o c . Am., v.64, p. 759-807.  50.  N e i d e l l , N.S., coherency v.36, p.  51.  P a p o u l i s , A., s t o c h a s i c  r l o o  y c h , T . J . , i c a t i o n s of n s e i s m o l o g n a l SEG Meet  1955,  phase  Jones, I.F., and Oldenburg, D.W., c o m p l e x common s i g n a l a n a l y s i s i n y : Proceedings of the- 5 3 r d Annual ing i n Las Vegas, paper S6.6, p.  P r o b a b i l i t y  t h e o r y :  D.  van  o r d r e :  New  d i m e n s i o n a l i t y of expansions: P a t t e r n  P a t t e r n of R a j a s t h a n ,  Landsat I n d i a :  and Y e d l i n , M. , 1 9 8 1 , A n a l y s i s of d i s p e r s i v e w a v e f i e l d t r a n s f o r m a t i o n : G e o p h y s i c s , v.46, p.  and Taner, measures 482-497.  M.T., 1971, Semblance f o r m u l t i c h a n n e l d a t a :  and other G e o p h y s i c s ,  1965, P r o b a b i l i t y , random v a r i a b l e s , p r o c e s s e s : M c G r a w - H i l l , New York.  and  216  52.  P e l a t , D., 1974, Karhunen-Loeve s e r i e s expansion: a approach f o r s t u d y i n g a s t r o p h y s i c a l d a t a : A s t r o n . A s t r o p h y s . , v . 3 3 , p. 321-329.  53.  P r a t t , W.K., 1970, Karhunen-Loeve t r a n s f o r m c o d i n g of i m a g e s : P r o c e e d i n g s o f t h e 1.970 I E E E I n t . S y m p . Inform. Theory.  54.  R a f i p o u r , B . J . , 1981, Phase and a t t e n u a t i o n phase-matched f i l t e r i n g t e c h n i q u e : Southern M e t h o d i s t U n i v e r s i t y , D a l l a s .  55.  Ready, R . J . , and W i n t z , P.,A., 1973, I n f o r m a t i o n e x t r a c t i o n , SNR i m p r o v e m e n t , a n d d a t a c o m p r e s s i o n i n m u l t i s p e c t r a l imagery: IEEE T r a n s . Commun., v.COM-21, p. 1123-1130.  56.  R i c k e r , N.H., 1953, The form and 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 ,  57.  R i e t s c h , E., 1981, Reduction of harmonic d i s t o r t i o n v i b r a t o r y source r e c o r d s : Geoph. P r o s p . , v.29, 178-188.  58.  Robinson, J . C , 1979, r e p r e s e n t a t i o n of G e o p h y s i c s , v . 4 4 , p.  59.  Robinson,  J . C ,  1982,  laws o f v . 1 8 , p.  A  t e c h n i q u e d i s p e r s i o n 1345-1351.  T i m e - v a r i a b l e  through t h e use of "phased" v.47, p. 1 106-1110.  sine  s t u d i e s Ph.D .  p r o c e s s i n g  f u n c t i o n s :  G e o p h y s i c s ,  61.  R y u , J . V . , 1982, Decomposition (DECOM) approach f i e l d a n a l y s i s w i t h s e i s m i c r e f l e c t i o n G e o p h y s i c s , v . 4 7 , p. 869-883.  62.  Smith,  and  N e i d e l l ,  i n l a b o r a t o r y  N.S., and  i n p.  d i s p e r s i o n  Rosenbaum, J.H., and Boudreaux, G.F., convergence of some s e i s m i c p r o c e s s i n g G e o p h y s i c s , v . 4 6 , p. 1667-1672.  T.A.,  o f  t h e c o n t i n u o u s s e i s m i c d a t a :  60.  e s t i m a t i o n  u s i n g t h e s i s ,  p r o p a g a t i o n 10-40.  f o r i n  new and  1981, Rapid a l g o r i t h m s :  1976, S e i s m i c marine  t o wave r e c o r d s :  d i s p e r s i o n  r e f l e c t i o n  d a t a :  217  Proceedings Meeting, New  of the O r l e a n s .  50th  Annual  63.  S t a n f o r d Q Conference p.5171-5256.  64.  S t r a n g , G., 1980, L i n e a r a l g e b r a A c a d e m i c P r e s s , New York.  65.  S t r i c k ,  E.,  1970,  propagation 387-403.  1980,  J .  I n t e r n a t i o n a l  Geophys.  and  A  p r e d i c t e d  pedestal  i n  constant-Q  s o l i d s :  Res.,  i t s  v.85,  a p p l i c a t i o n s :  e f f e c t  for  Geophysics,  K o e h l e r , F., and S h e r i f f , t r a c e a n a l y s i s : Geophysics,  SEG  p.  66.  Taner, M.T., s e i s m i c  67.  T j o s t h a o A  68.  Toksoz, M.N., and Johnston, wave a t t e n u a t i o n : S o c i e t y Geophysics R e p r i n t S e r i e s  69.  U l r y c h , T.J., L e v y , S., O l d e n b u r g , D.W., and Jones, I.F., 1983, A p p l i c a t i o n s of the Karhunen-Loeve t r a n s f o r m a t i o n in r e f l e c t i o n seismology: Proceedings of the 53rd Annual I n t e r n a t i o n a l SEG Meeting i n Las Vegas, paper S6.5, p. 323-325.  70.  Walsh, J.L., 1923, J o u r n . Math.,  71.  Watanabe,  e i m , D., and San u t o r e g r e s s i v e f e a t u r e f m u l t i c h a n n e l wavefor n a l y s i s and Machine I n  S.,  A c l o s e d v.55, p.  1965,  d e m t  R.E., v.44,  pulse  v.35,  1979, Complex p. 1041-1057.  vin, 0, 1979, M u l t i v a r i a t e x t r a c t i o n and the r e c o g n i t i o n s: I E E E T r a n s a c t i o n s on P a t t e r n e l l i g e n c e , v . P A M I - 1 , p. 80-86.  D.H., ( e d i t o r s ) 1981, Seismic of E x p l o r a t i o n G e o p h y s i c i s t s , no.2.  set of 5-24.  orthogonal  Karhunen-Loeve  f u n c t i o n s :  expansion  and  Am.  f a c t o r  a n a l y s i s . T h e o r e t i c a l remarks and a p p l i c a t i o n s . R e p r i n t e d i n : P a t t e r n R e c o g n i t i o n , J . S k l a n s k y , ed., S t r o u d s b u r g , P e n n s y l v a n i a , 1973 (p. 146-171).  72.  Werner, to  H., sweep  and  Krey,  T.,  t e c h n i q u e s :  1979, Geoph.  Combisweep Prosp.,  -  v.27,  a  c o n t r i b u t i o n p.  78-105.  218  73.  Wuenschel,  P . C ,  e x p e r i m e n t a l  1965, study:  D i s p e r s i v e G e o p h y s i c s ,  body v.30,  waves  74.  Young, T.Y., and Huggins, W.H., 1962, The component theory of e l e c t r o c a r d i o g r a m s : B i o m e d i c a l E l e c . , v.BME-9, p. 214-221.  75.  Young, T.Y., and H u g g i n s , of e l e c t r o c a r d i o g r a m s : v.BME-10, p.86-95.  76.  Young, T.Y., and e s t i m a t i o n and  W.H., IEEE  1 9 6 3 , On Trans.  -  an  p.539-551.  i n t r i n s i c IRE T r a n s .  t h e r e p r e s e n t a t i o n B i o m e d i c a l E l e c . ,  C a l v e r t , T.W., 1974, C l a s s i f i c a t i o n p a t t e r n r e c o g n i t i o n : E l s e v i e r , New York.  219  APPENDIX 1. ATTENUATION RELATED DISPERSION.  a.  Theory.  For  a  m a t e r i a l t h e  d e f i n e d  r a t i o  of  over  a  amplitude  8A,  R i c h a r d s ,  For  t h e  1980, p.  a  Q(CJ)  From  g i v e n  where  Using v e l o c i t y ,  f a c t o r  f o r  t h e  a  t r a v e l  path  l i n e a r  s t r e s s - s t r a i n  monochromatic A  amplitude of  t o one  t h e  plane  wave  i s  decrease  wavelength  i n  ( A k i and  168.  angular  frequency  t h i s  =  d e f i n i t i o n ,  wave  a t  some  we  time  OJ,  we  have:  see t,  that  t h e  i s given  by:  amplitude  of  t h i s  A (l-n/Q) , n  0  i s t h e  n  t h e and  number  r e l a t i o n  assuming  that  of  wavelengths  X,  t  =  2irn/u,  Q  o b t a i n :  A(l ,u>) = which  a  = -vA/bA  monochromatic  A(t)  obeys  Q  r e l a t i o n s h i p , as  which  f o r l a r g e  A (J-u)t/2Qn) , n  0  n  becomes:  nX/v  =  i s c o n s t a n t  t r a v e l l e d  where  i n time  v  t .  i s t h e  f o r a l lf r e q u e n c i e s ,  we  220  equat  A(l  ,u>)  For  a  i on  (Al.  A(l)  Now,  for  S a t i s f y i n g as  to  0, tor  to  V(f)  t  <  i n v i o l a t e ,  r e q u i r e s  the  we  have  from  (Al.  2)  d i s p e r s e d ,  as  r e q u i r e :  from  that  (1962),  c a u s a l i t y  the  t h i r d  waves using  be  Robinson  of  these  (A1.2)  equation  c o n d i t i o n  p a i r s .  0  V(f)  i s  to  o b t a i n  (1979)  three  used  the  phase  v e l o c i t y  the  low  V0  the  phase  v e l o c i t y  Q  the  value  of  frequency  E u l e r ' s  the  at  frequency  chose  an  a r b i t a r y  r e l a t i o n  equation  3)  (Al.  4)  c u t - o f f ,  f0,  at  s e i s m i c  c o n s t a n t ,  expression  (Al.  /,  q u a l i t y  f a c t o r  at  /  0  ,  and  Q(f)V(f) t h i s  the  p a i r s :  0  f0  =  d i s p e r s i o n  1)  0  Futtermann  invoke  Ln(e)  render  we  V0/{l-Ln(ef/f )/7TQ }  0  frequency  remain  =  0  a t t e n u a t i o n ,  xp(-u>t/2QJ  c o n d i t i o n  r e l a t i o n  where  Robinson  undergoes  a b s o r p t i o n - d i s p e r s i o n  d i s p e r s i o n  QoV  which  xp( i u>t ) e  a t t e n u a t e d .  p o s s i b l e  (AJ.  1) :  t h i s  proceeded  To  wave  c a u s a l i t y  =  / 2Q)  xp (-ul  plane  = A0e  A(t)  w e l l  A0e  =  independent base  of  frequency  (A1.3)  w i t h  the  low and  respect  frequency r e d e f i n e d to  For  l i m i t the some  221  +  V(fb  5f)  = V(fb)  equation  S u b s t i t u t i n g  (2.3)  and  +  (A1.4)  sv(f,f )/{v(f ) b  Under which  t h e  (Futtermann, K j a r t a n s s o n , y i e l d s the  a  removal  Now,  u s e f u l  a  of  these  t h e  equation  i n t o  r e s u l t  y i e l d s :  (AI. 6)  i s independent band  1965; Kanamori  b a s i s  of  frequency,  s e i s m i c &  1980, p.167) f o r d i s p e r s i o n  s i g n a l s  Anderson,  1977;  equation  (A1.6)  m o d e l l i n g  or f o r  e f f e c t s .  p a i r  s i m u l t a n e o u s l y times  Q  R i c h a r d s ,  d i s p e r s i v e  c o n s i d e r  i n i t i a t e d a r r i v a l  of  and  from  narrow  Wuenschel,  1979; A k i &  simple  t h a t  f o r r e l a t i v e l y  1962;  (A1.5)  Ln(f/fb)/{*Q(fb)}  b  assumption  i s j u s t i f i e d  (A1.3)  bV(f,f )}  +  b  equation  and  equation  s u b t r a c t i n g  (Al. 5)  W(f,fb)  of  a t  plane  some  waves  a r b i t a r y  waves  a t  some  of  f r e q u e n c i e s  source r e c e i v e r  /  l o c a t i o n l o c a t i o n  and  X  .  X  by:  T(f)  bX/V(f) hX/V(fb)  T(fb)  where  thus,  ,  5X  =  X  T(fb)  r ~  =  and  ,  -  X  s  T(fb)  T(f)  T(fb)  -  -  bX/V(f)  T(fb)V(fb)/V(f)  b  The , i s  r given  f  222  =  By  analogy  w i t h  bT(f,fb)  =  ST(f,fb),  equal  i on  T(fb)  -  T(fb)V(fb)/{V(fb)  1  ~  bT(f,fb )/T(fb)  =  =  Equating  t h i s  equal  =  equation  p o s i t i v e a r r i v e  (A1.7) bT,  and  states  that  t o  time  f o r  a x i s  summarized  For time  our  l(f,fb)  /  have:  f r e q u e n c i e s  equation  (A1.3)  (T(fb)  +  above  fb  w i l l  t h e higher  of  a  s p e c i f i e d  frequency from  frequency. F o u r i e r  a  T h i s s c a l i n g  i n  have  a  f r e q u e n c i e s  equation  dependent  simple  x(t),  we  i n t r o d u c e  f a c t o r :  =  =  we  l i n e a r  s c a l i n g or  (A1.3)  v e l o c i t y s c a l i n g  of  o p e r a t i o n  s i m i l a r i t y  a r e t h e i s  theorem  p.101).  s i g n a l  r e s c a l i n g  bV(f,fb)}  that  o b t a i n e d  t h e  1978,  bV(f,fb)}  (A1.8)  r e l a t i o n  each  by  ( B r a c e w e l l ,  t h e  those  6)  so:  v e r s a .  t h e e f f e c t s  e q u i v a l e n t  (Al.  ),  bV(f,fb)}  +  +  see  from  v i c e  E s s e n t i a l l y ,  +  V(fb)/{V(fb)  i on  f  Ln(f/fb)/{irQ}  we  i . e .  f i r s t ,  l e t V(f)=V(fb)+W(f,  b  bT(f,fb )/T(fb)  From  (A1.5)  bv(f,f )/{v(fb)  w i t h  7)  (AI.  say  bT(f,fb ))/T(fb)  +  bT(f,fb)/T(fb)  t h i s  frequency  dependent  223  which  equation  from  l(f,fb)  =  T h e r e f o r e ,  x(t)  1. e .  f  >  upon  say,  In  d i s p e r s i o n  given  S  =  i s  t h i s  of  in  time  a  t  from  frequency  s h o r t e r  time  than  s c a l i n g  theorem,  i t  /,  for  would  in  an  t h i s  the  time  s i g n a l  by:  yX(fy)exP(i  < —  2*ft)  1),  i n  i f  df  sampled ...)  the at  y(f,fb)X(y(f,fb)f)  upon  which  d i s p e r s i o n  our  s y n t h e t i c  were data  (ALIO)  Robinson's based,  c o u l d  be  (1979),  while  and  an  introduced  our,  a t t e n u a t i v e  equation  v i a  d e s i r e d .  Numerical  As  >  expression f o r  component  2,  a r r i v e s  at  i s :  programs  b.  have:  x(t)  to  context  x(t/y(f,fb))  (Al.  we  9)  s i g n a l .  x(t/y)  It  (Al.  x(t/y(f,fb)),  now  the i s  becomes:  Ln(f/fb)/{irQ}  >  u n d i s p e r s e d  that  +  c o n t r i b u t i o n  fb  x(l/y)  1  —  the  (A1.8)  Methods.  complex  X(nhf) we  must  spectrum  (where at  hf  some  i s  X(f) the  stage  i s  r e s c a l e d  frequency introduce  to  yX(yf)  only  n=0,  1,  i n t e r p o l a t i o n .  Our  i n t e r v a l an  but and  224  approach  was  t o  l o c a t i o n s  nbf,  where  phase some  and  change  where  q u e s t i o n :  d i d t h e s p e c t r a l come  due  from  given  component,  (assuming  t o d i s p e r s i o n ) ?  [ 1 + Ln(fQ/f  /  f0  f i n d i n g f u n c t i o n  we  d e v e l o p  given t o  =  =  2  that  Knowing  t h e d i s c r e t e  with  i t s  i t was  c u r r e n t  s h i f t e d  by  t h a t :  be  . To  frequency,  and  /  Newton-Raphson  do  m i n i m i z e d ,  t h i s , and  we  i t s  need  i s t h e  0  frequency  that  i t e r a t i v e  approach  t o  e x p r e s s i o n s  f o r both  t h e  t o e v a l u a t e  t h e  d e r i v a t i v e :  [ 1 + Ln(f0/fb)/{irQ}]f0  / +  -  [/+  Having  found  complex  f u n c t i o n on  /  a  )/{*Q}]f0,  fQ  a4>,/9/ 0  =  based  b  i s t h e c u r r e n t  s h i f t e d ,  $  f o l l o w i n g  = 7/o  =  was  t h e  amplitude,  phase  fc  a s k  t h e  Ln(f0/fb)]/{nQ}  l o c a t i o n  X(f),  we  /  0  , a t  which  u t i l i z e  i n t e g r a t i o n - b y - p a r t s  an  ( a f t e r  we  wish  i n t e r p o l a t i o n Rosenbaum  and  a l g o r i t h m Boudreaux,  1981).  the T  c  .  To  invoke  data  w i t h  The  a t t e n u a t i o n a  data  window i n  of  t h e  s a t i s f y i n g width window  W,  equation  centred were  (Al.l),  about  F o u r i e r  we  sampled  t h e c u r r e n t  time  t r a n s f o r m e d  and  225  m u l t i p l i e d  by  exp(-<JT /2Q)  in  W/2  and  steps  of  s p e c t r a were  c.  then  combined  Cumulative  In produce in  transformed  an  w i t h  t h e  t o  same  RMS  i n t e r v a l  i n t e r v a l  r e s u l t i n g  back  i n t o  Q  way  i n t e r v a l  t r a v e l  time  ( D i x ,1955),  3  A  sequence  t r e a t e d  of  ={ L j=l measured  i n t h e usual  B a c k u s - G i l b e r t s t r u c t u r e .  a t t e n u a t e d  domain.  along  The  data  time  t h e  data  complex segments  s e r i e s .  we  a r e  t h e  l a y e r  Q  d a t a .  F o r  a  c o r r e s p o n d i n g  t o  averaged  v a l u e s  a r e  l a y e r e d  t h e i.th  t o seen  e a r t h  l a y e r  w i t h  have:  / \/Qcum.  of  v e l o c i t i e s  i n t h e measured  t ..  moved  I n v e r s i o n .  v a l u e s  Q  was  t h e a t t e n u a t e d  I n t e r v a l  t h a t  s e t  t h e time  t o c o n s t r u c t  sense  window  t h e  v e l o c i t i e s  averaged  . The  i ig/Q,}/ L j=I c u m u l a t i v e 'layer  i n v e r s i o n  t Q  (ALU) v a l u e s  s t r i p p i n g '  methods,  t o  may  manner, y i e l d  be or t h e  i t e r a t i v e l y  i n v e r t e d  u s i n g  i n t e r v a l  Q  226 2.  APPENDIX  THE EFFECT OF DISPERSION ON VIBROSEIS DATA  a.  I n t r o d u c t i o n .  The  m a t e r i a l  manuscript  i n  i n  by  S.  Here  we  present  a f f e c t s  constant how  the  of  the  a  Levy  Appendix  constant  regard  d i s p e r s i o n model  Q  nature  the  problem.  The  1965;  Kanamori  &  Aki  &  l i n e a r l y  w i t h  f r a c t i o n a l  s i g n a l  the  d i s p e r s i o n . as  those  1980,  l o s s  of  a p p l i e d  the  the  of  of  a  work  to  c y c l e ,  of  i s  constant  be  i n  Q  model  that of  we  for  the  d e l e t e r i o u s  p o s s i b l e  i n  V i b r o s e i s ®  adoption not  i n t o  the 1962;  a  l o c a t i o n  w i t h  t e c h n i q u e .  tuned  medium.  show  of  the  Q,  given  to  1979;  i n c r e a s e s  i.e.  proceed  overcome only  Our  a t t e n u a t i o n  band  i s  demonstrate  (Futtermann,  i n t e r e s t :  as the  i n s t r u c t i v e ,  and  c h o i c e  to  of  K j a r t a n s s o n ,  constant  Q,  l i g h t  i n s i g h t s  1977;  s t a t e s  d i s p e r s i o n  t h e m s e l v e s .  Anderson,  band  and,  u s e f u l  constant  of  seismograms  meant  the  problem  band-width  the  t e x t  c o n t i n u a t i o n  bandwidth  i t y i e l d s  over per  p a r t i a l l y Such  a  manifest  p167)  assumption  c h o i c e can  t h a t  frequency  energy  Given j u d i c i a l  b e l i e v e  R i c h a r d s ,  i s  to  to  i s  we  Wuenschel,  e s s e n t i a l l y  s y n t h e t i c  d e f i n i t i v e : of  and  background  produce  of  i s  1980.  w i t h  model,  problems  i n  a  s i g n a l  Q  t h i s  p r e p a r a t i o n ,  i n i t i a t e d  i t  PROCESSING.  how  the  a  input  e f f e c t s  of  sources,  such  Through  the  use  of  227  s y n t h e t i c  d a t a ,  d i s p e r s i o n  we  on  r a t i o n a l e  f o r  V i b r o s e i s ®  consider  i n  Vibroseis®  a  method  sweep  data  of  ( h e r e a f t e r  octave  separate  s e t s  c r o s s  sweep, and  the  correlograms  then  c o m p i l e d  s t a c k e d . i n  d i s p e r s i o n  t h i s  Combisweep  b.  I t  i s  manner  than  o v e r l a p p i n g  from  would  a  s i n g l e ,  Werner  &  Krey  as  sweep)  i n t o  r e c o r d i n g  the  the  sweeps  c o n t e n t i o n l e s s  the  m u l t i - o c t a v e  the  with  of  d e v e l o p  A f t e r  d i f f e r e n t  a p p r o p r i a t e  are  weighted  that  data  g a t h e r e d  and  from  the  e f f e c t s  of  m u l t i - o c t a v e  frequency  We  e f f e c t  usual  to  sweeps.  s u f f e r  m u l t i - o c t a v e of  our  may  the  c o r r e l a t i n g  the  the  p r o c e s s i n g .  r e f e r r e d  i n d i v i d u a l and  d e t a i l  s e p a r a t i n g  non-overlapping data  some  sweep,  band  or  sweeps  a  stack  (e.g.  of the  (1979)).  Theory.  D i s p e r s i o n s t r e t c h i n g becomes  of  the  d i s p e r s e d ,  x(t)  b  and  phase  Q  time  a x i s  (Appendix  1).  one  has,  f o l l o w i n g  equation  =l  some i s  a  For  frequency a  s i g n a l  dependent  x(t)  which  (A1.9):  + Ln(f/f )/{nQ}  (A2.1)  b  base,  the  e x p r e s s i o n  advanced,  as  b  b  i s  c o n c e p t u a l i s e d  x(t/y(f,f )),  -y(f,f )  f  t h i s  be  >  where  From  can  we  or  r e f e r e n c e  seismic note  f r e q u e n c i e s  q u a l i t y  that below  frequency, f a c t o r .  f r e q u e n c i e s /,  phase  above  delayed.  f  b  w i l l  be  228  We degree  now of  c o n s i d e r  d i s t o r t i o n  equation  (A2.1)  a t  l(fy,fb)-  For  a  =  one  2f2,  =  hi(fy,f2),  octave  =  f u n c t i o n  of  Thus,  f r e q u e n c i e s  phase  s h i f t s .  r e c e i v e d  to  even  procedures recorded  This  w i t h  g r e a t e r i s not s i g n a l  =  we  a  =  a b s o l u t e  y(f,fb)  Taking <  t h e  f i) , w e  from  have:  Ln(f2)}/{nQ)  (A2.  ITQ}  r e g a r d l e s s  Ln(2)/{nQ}  of  s i g n a l . or  problems, d i s p e r s e d  frequency,  we  have  0.22/Q.  end  t o  (A2.3)  a  of  frequency  given  i n t h a t  spectrum  e i t h e r  known  =  t h e  over  octaves  a  of  0.22n/Q  i n  leads  a r e  on  have:  y(f,f^),  w i t h a t  width  ( / 1  f2  and  -  by(f^,f2)  d i s p e r s e d  c o r r e l a t i o n  /{  nLn(2)/{nQ}  s i g n a l  band  say  octaves  f a c t o r  have  / ,  i n w i d t h ,  t h e number  a  2.)  of  d i s p e r s i o n .  {Ln(fJ  8y(2f,f)  d i f f e r e n c e  t i m e - r e s c a l i n g  t o  =  Ln (f \ /f  thus  Sy(2 nf/f)  e f f e c t  f r e q u e n c i e s  =  f o r n  T h e r e f o r e ,  The  due  7(f2,fb)  band  /,  t h e  s e v e r a l  octaves  t h e band  w i t h  Subsequent  as w h i l s t  i n c r e a s i n g l y  i s  i t s  l i n e a r  wide  l a r g e  d i s t o r t i o n  d e c o n v o l u t i o n  input  t h e  a  band.  c o n s i d e r a b l e  estimated  band  dependent  s i g n a t u r e  s i g n a t u r e  used  r e p l i c a t i o n s  d i s p e r s e d ,  and  w i l l  r e l a t i v e of  t h e  or  c r o s s  g i v e s  r i s e  i n w i t h i n  a t t e n u a t e d ,  these t h e w i t h  2)  229  g r e a t e r  c.  The  t r a v e l  Vibroseis®  One  of  d e g r a d a t i o n S e v e r a l or  of  encoded  Rrey  i n  &  we  R e c a l l i n g  to  that  been  way  i n  the  p a r t  to  suggested  so  as  to  r e s o l u t i o n  of  the  1982;  from  a  Chapman  Vibroseis®  d i s p e r s i o n whereby  the  minimize  a l .  method  and  i s  the  m o d i f i e d  n o i s e  s i g n a l  1981;  i s  a t t e n u a t i o n .  sweep  the  recorded  et  succumbed c r o s s  s t a c k i n g ,  would  d i s t o r t i o n c o l l e c t e d  than and  frequency i n  a t t e n u a t i v e c o n t e n t .  T h i s  sweep,  would a  i n ,  and  (Gurbuz  R i e t s c h  1981;  the y i e l d  c o r r e l a t i o n produce data  of  procedure  the  e x i s t i n g  prone that  sweep  sets of  1982; Werner  stack  i n i t i a l c o u l d  be  sweep  d a t a ,  data  d i s t o r t i o n be  and  would  p a r a l l e l e d  w i t h  data  sweeps, w i t h  that  set  been higher  these  high e x i s t i n g  and l e s s  had  the  i n c l u d i n g  the  three  e f f e c t s .  p a r t i a l l y of  c o n f i n e d  for  set  Repeating  the  methods.  each  r e s p e c t i v e  band-width  times  b o o s t i n g  the  of  d i s p e r s i v e  m u l t i - o c t a v e  s e v e r a l  more  of  the  comparable  to  s u c c e s s i v e l y  amount w i t h  a n a l y s i s  c o n v e n t i o n a l l y .  sweeps  by  a  i s  three  from  of  suggest  minimal  subsequent l o s s  sweep  we  Repeating  to  gleened  m o d i f i c a t i o n  processed  octave the  i n s i g h t s  a  octave  octaves  Subsequent  the  m u l t i - o c t a v e  o c t a v e .  d i f f e r e n t  sets  have  propose  s i n g l e  one  having  due  w i t h  1979).  t h e o r y ,  a  s i g n a l  Werner  F o l l o w i n g  than  problems  some  the  Edelmann  Technique.  the  methods  i n c r e a s e  &  time.  data  a l l e v i a t e frequency data  by  230  bandpassing  the  p r e - c o r r e l a t e d  correlograms  from  d.  Data  S y n t h e t i c  The the  s e v e r a l  amplitude  of  of  data  Four  the  d i s p e r s i o n  s t r u c t u r e  s y n t h e t i c  were  of  and  w i t h  p a r t i c u l a r  sampling  h a l f  at  i n t e r v a l  s y n t h e t i c  A l l  a  one  i s  sweeps  of  4  -  21.5  Hz  b.  18.5  -  43.0  Hz  c.  37.0  -  80.0  Hz  d.  10.0  -  80.0  Hz.  s l i g h t l y  to  c o r r e l o g r a m . second  of  ms.  tapered a v o i d  The  data  the  examples  d a t a .  To  i s  on  to  demonstrate  the  f u r t h e r  phase  t h i s  and  end,  the  four  c o m p r i s i n g  seconds, For  t h i s  d u r a t i o n  were  used  seconds  bandwidths  f i v e  were  sweep.  s e v e r a l The  each  i n  s p i k e s  of  c r e a t e d  and  demonstration, ( i n  p r a c t i c e  d u r a t i o n )  chosen  for  the  w i t h  a  four  were:  10.0  were  over  second  l e a s t  a.  s p e c t r a  s t a c k i n g  r e p l i c a t i o n s .  a t t e n u a t i o n  seismograms,  c o r r e l a t e d  sweep  and  s i m p l e .  spread  the  l i m i t e d  input  p o l a r i t y  only  band  of  the  kept  s y n t h e t i c  of  wavelet,  f o l l o w i n g  a l t e r n a t i n g  sweeps  and  Examples.  o b j e c t i v e  e f f e c t s  such  data  bandwidths  gaps  s p e c t r a  c e n t r e d  and  i n  shown  about  i n  the  the  a,  &  spectrum  F i g u r e p u l s e  b,  A.2,  near  c  were of  are 3.6  s.  overlapped  the of  stacked one  h a l f  231  F o l l o w i n g S t r i c k ( 1 9 7 0 ) , the  data  and  the  the programs  given base  the  s e i s m i c  to  l i e at  chosen In  F i g u r e  sweep  ( a ) ,  i s  of  three  s i n g l e  and  the  b ) .  The  This  e x p e c t e d , band  data  mimics  the  w i t h  w i t h  the  were  again  d i s p a r i t y  s e t s  t r a v e l  from  somewhat  two  e f f e c t e d .  s i m i l a r . A.2a)  the  (a  three and  a f t e r to  phase  s t a c k ,  w i t h  m u l t i - o c t a v e  from The  the  stack  stack  was  ( F i g u r e  A.2  lobes  i n  The  are  base  r e s p e c t i v e  due  to  b)  the  be  i.e.  l o s t  much  constant  the i s  Q  Gibb's  of  c r o s s  as of  octave  sweeps  the  marked  m u l t i - o c t a v e  r e s u l t s  second the  of  two  d i s p e r s i v e  the  b e t t e r ,  s i g n a l ,  c o r r e l a t e d  demonstrates The  150,  sweeps.  recorded  i n d i v i d u a l  f i r s t  s t r u c t u r e i s  a  non-dispersed  e f f e c t s ' ,  procedures.  begins  the  are  band,  the  the  the  formed  three  ( F i g u r e  The  A.3  d i s t o r t i o n  the  of  octaves  the  a t t e n u a t e  case  i n  procedure,  sweep.  the  d i s t o r t  each  of  and  frequency  ( b ) .  r e s p e c t i v e  'earth  between  i n c r e a s i n g l y r e s u l t  f i e l d  input  P o l a r i t y  the  mid-point  d i s p e r s e d  the  F i g u r e  time.  In  and  correlograms  r e s u l t s  were  s t a c k e d .  s i g n i f i c a n t  Q,  correlogram  spectrum  v a r i o u s  known  d i s p e r s e  correlogram  sweep  (1979)  1978).  a c t u a l  the  the  these  ( B r a c e w e l l ,  four  the  to  1).  the  e n e r g i e s  c r o s s - c o r r e l a t e d w i t h  complete  show  the  c o n v e n t i o n a l  to  octave  balance  As  phenomenon  and  compared  A.1  Robinson  f a c t o r ,  was  band.  a  q u a l i t y  (Appendix  frequency  to  developed  fb  fb  weighted  were  of  frequency,  frequency  the  approaches  s i g n a l .  although  way  e f f e c t s However,  i t a l s o  i s  232  R e p e a t i n g a t t e n u a t i o n i n c r e a s i n g are  of  a t t e n u a t e d .  P o l a r i t y  of  F i g u r e  In  automatic  can  d e f i n e d  be  g a i n  f r e q u e n c i e s  are the  a  s p e c t r a l  n o r m a l i z a t i o n .  e q u a l i z e d  so  is  only  implemented  not  to  a m p l i f y  and  the  a f t e r  c o n s i d e r a b l y , and used  d ) .  T h i s to  boost  n o i s e  as  problem.  mean high  noted  procedure i n t r o d u c e  i n  the the  to  In  our  proceed  i n  energy  r e s p e c t i v e band.  standard  e)  we  note  component  of  implementation  of  i n  r e s p e c t i v e  the  a t t e n u a t e d  w i t h  The  the  of  a  w e l l  form  octave  of was  n o r m a l i z a t i o n  frequency A.5  band (a  F i g u r e  s p e c t r a  so  and  A.4b  wavelets  knowledge  i n  of  each  F i g u r e  The  a t t e n u a t i o n  the  s t a c k i n g .  implement  f r e q u e n c i e s .  p r i o r  more  the  and  case,  to  r e c o v e r y .  no  once  recovery  data  assumed  d  the  by  d i s p e r s e d  the  f r e q u e n c i e s  octave  A.2  an  a t t e n u a t e d .  the  The  frequency  shows  degraded  compensated  we  and  high  the  i n t r o d u c i n g e x p e c t e d ,  time-dependent  sweeps,  o u t s i d e  as  ( F i g u r e  However,  the  as  approach  the  and  have,  b)  s e v e r e l y  to  w i t h i n  a t t e n u a t e d  h i g h  and  s e v e r e l y  g r e a t e r  octave  to  are  p a r t i a l l y  poses  as  (a  now  we depth  s p e c t r a  due  c o n t r o l .  separate  compares  A.4  f r e q u e n c i e s  d i s t o r t i o n  a t t e n u a t i o n  w i t h  c o n v e n t i o n a l  sweep.  high  The  r e s o l u t i o n  62=125  w i t h  d i s p e r s i o n  p o s i t i o n  m u l t i - o c t a v e  procedure  to  the  and  the  above  p r i o r l o s s  comparison  that  the  as b)  before  sharpen-up ( F i g u r e  of  the  o r i g i n a l  Q  A.5  c  value  s y n t h e t i c  seismograms.  T h i s r e p e a t i n g  t h e o r e t i c a l the  high  procedure  frequency  c o u l d  octave  be  mimiced  sweep  i n  s e v e r a l  the  f i e l d  times  by and  233  i n c l u d i n g  these  r e p e t i t i o n s  sweeps.  In  compound  spectrum.  in  a  more  e f f e c t  exact  a b s o r p t i v e  we In  the  boost  order  manner,  and  i n  we  the  to  w i t h  h i g h  need  a  the  other  frequency  implement  would  s c a t t e r i n g  stack  high  content  frequency  d e t a i l e d  c o n t r i b u t i o n s  octave  to  of  recovery  knowledge the  the  of  the  a t t e n u a t i o n  p r o c e s s .  e.  Comparison  w i t h  O v e r l a p p i n g (1979) but To  p r o v i d e  the  method  e x e m p l i f y  d i v i d i n g  the  P r e v i o u s  frequency  Work.  bands,  an  improvement  i s  s t i l l  t h i s 10  -.80  over  i n f e r i o r  p o i n t ,  we  Hz  as  s i n g l e  to  the  c r e a t e d  range  suggested  i n t o  by  Werner  m u l t i - o c t a v e  separate  and  procedure,  octave  three  more  three  o v e r l a p p i n g  Krey  approach.  s y n t h e t i c  t r a c e s ,  bands.  They  were: e.  10  -  52  f .  24  -  66  g.  38  -  80  A l l  bands  14  Hz,  of  t h i s  Werner  stacked are  are  and  Hz,  42  and  Hz. Hz  wide,  c h o i c e  of  Krey  shown.  the  o v e r l a p p i n g  f o r  stack  In  our  subsequent  bands  F i g u r e s  the  (2=150.  c o n v e n t i o n a l band  each  frequency  F o l l o w i n g u s i n g  and  (1979).  c o r r e l o g r a m  seismograms from  Hz,  A.1c  e a r l i e r  method ( c ) ,  f o r  &  2c  A.3  bands  d i s p e r s e d  mimics  and we  compares octave  s h i f t e d  r e s p e c t i v e l y ,  examples  ( a ) ,  i s  a p p r o x i m a t e l y  o v e r l a p p i n g  F i g u r e  band  i t s  stack  d a t a .  The  those the  spectrum  d i s p e r s e  the  by  the  correlograms ( b ) ,  and  improvement  234  of  the  same  octave  as  that  stack  of  over  the  the  l a t t e r  o v e r l a p p i n g over  a  band  stack  i s  c o n v e n t i o n a l  about  the  m u l t i - o c t a v e  c o r r e l o g r a m .  A t t e n u a t i n g r e s u l t s  s i m i l a r  d r o p - o f f  of  s i g n a l  and  d i s p e r s i n g  to  those  h i g h  f r e q u e n c i e s  s t r e n g t h  w i t h  time  et  a l .  Vibroseis®  comparing  c o n v e n t i o n a l  e l e g a n t l y the  more  show  span  i n  frequency  28  the  order  t h i s  A.2f)  show  paper  we  the  on  of  see  the  l o s s  of  o c t a v e s , back  octaves  h i g h  t h e i r  w h i l e to  use  data  sweeps.  the  However,  the  r e a l  frequency  improvement of  now  and  s e v e r a l  improvement  two  o n l y  produces  A.4).  t h e i r  R e f e r r i n g  number  to  one  h.  10  -  32  i .  24  -  80  these -  t h a t  ( F i g u r e  0=125  T h e i r  higher  be  low  method  sweeps  span  (A2.2)  and  given  the  expected  encompassed  data  frequency  equations  would  s e c t i o n s  frequency  t h e i r  high  i n  the  high  range.  c o n s t r u c t e d  Both  than  w i t h  A.3  h i g h  sweep.  o c t a v e s .  note  to  vast  l e s s  two  we  r e d u c t i o n  In  sweeps  c o n v e n t i o n a l  than  (A2.3)  the  i n  data  F i g u r e  ( F i g u r e  (1981),  frequency  sweeps  i n  the  Chapman  over  the  demonstrate more Hz,  Hz  spectrum,  and  of  c o n t e n t i o n  s y n t h e t i c  more  c l e a r l y ,  we  t r a c e s :  and  Hz.  sweeps  80  set  t h i s  c o n s t i t u t e  w i t h span  some the  same  about  1.4  o v e r l a p frequency  to  o c t a v e s ,  i.e.  a v o i d  gaps  range  as  i n d i d  10  -  the our  28  and  compound f i r s t  235  t r i a d and  of  u n i - o c t a v e  s t a c k e d  knowledge expect  of  data  as  s e t s  s l i g h t l y r e s u l t  manner  to  i s  the  the  The  of  the  sweeps  the  e a r l i e r  d i s p e r s e d ,  range  a f t e r  p a i r  stacked  of  was  Q  constant  three  However,  stacked  the  when  frequency  same.  n o i s e .  that  of  of  spanned  component d i s p e r s i o n  1.4  octave  u n i - o c t a v e  - c o r r e l a t e d  t r i a d .  Without  model  we  would  to  g i v e  s i m i l a r  by  the  stacked  stack  should  w i t h  (2=100,  sweeps  t r i a d  a  i s  have the  markedly  ( F i g u r e s  A.6).  C o n c l u s i o n s .  s e i s m i c octaves  present  e f f e c t s  Q of  the  c h o i c e may  p o s s i b l e to  a  o f f e r  u s e f u l  e f f e c t s  be  and  band  the  the  the  the  of  s o u r c e s ,  method.  survey  c o u l d  the  by  the  such  a d d i t i o n a l  the  d a t a ,  avoided  l o c a t i o n  proposed  to  input  as  Systems  long  number of  of the  d e l e t e r i o u s  j u d i c i a l  c h o i c e  s i g n a l .  Such  V i b r o s e i s ® ,  and  r e c o r d i n g  be  of  a p p l i c a b i l i t y  r e f l e c t i o n  p a r t i a l l y  tuned  d e g r a d a t i o n  r e l a t e d  Given  may  worth  i n  and  p r o c e s s i n g  a l r e a d y  r e a d i l y  designed  a d j u s t e d  to  the  method.  Regardless  Q  r o l e  s e i s m i c  w i t h  Combisweep  o c t a v e - s t a c k  c o n s t a n t ,  to  be i n  a  s i g n a l .  width  i n v o l v e d  run  a  d i s p e r s i o n  band i s  i n  p l a y s  produces  model  prove  c o s t s  which  r e c o r d s ,  constant  to  p a i r  examples,  o v e r a l l  D i s p e r s i o n ,  of  T h i s  i m p l i c a t i o n s  the  from to  s i m i l a r  two  lower  i n f e r i o r  a  the  these  r e s u l t s ,  f.  i n  sweeps.  of  model,  the  a b s o l u t e  a n a l y s e s  i n s i g h t s  i n t o  v a l i d i t y  such the  as  the  processes  of one  a  c o n s t a n t , presented  a f f e c t i n g  and  or  n e a r l y  here  can  a f f l i c t i n g  the  p r o p a g a t i n g  s e i s m i c  w a v e l e t .  237  FIGURE A . l C o r r e l o g r a m s  f o r  s y n t h e t i c  (a)  produced  by  a  (b)  produced  by  s t a c k i n g  three  s i n g l e  (c)  produced  by  s t a c k i n g  three  o v e r l a p p i n g  a l l  data  of  the  f i n a l  bandwidth.  symmetric  a r e  waveform  s i n g l e  seismograms:  same with  m u l t i - o c t a v e  w e l l  d e f i n e d  sweep, octave  sweeps,  band  sweeps,  A l l methods  p o l a r i t y  r e s u l t  i n  a  238  (00  LOO  FRtOUtNCT (HZ]  |80  FRtoi.ifNcr  (HZ)  100  FIGURE A.2 The (a) (b) (c) The were not The a t t thr  f i r s t t h produced produced produced lobes i n tapered  r e e by by by (a) and e the p y r a m i d second t r i e n u a t e d v e r s i ee s p e c t r a  s p e c t r a correspond to a s i n g l e m u l t i - o c t a v e s t a c k i n g three s i n g l e s t a c k i n g three o v e r l a p a r e due to Gibb's e f f overlapped s l i g h t l y t a l e f f e c t of s t a c k i n g a d of s p e c t r a ((d) o n s of the seismograms  the data of F i g u r e A . I : sweep, octave sweeps, p i n g band sweeps, e c t . In (b) the s p e c t r a o a v o i d g a p s , and in (c) we Combisweep data ( e ) ) , correspond to the which produced the f i r s t  239  -to.o  0.0  4  * J  C o r r e l o g r a m s (a)  p r o d u c e d  (b) (c) a l l l o s t s i n g stac  p r o d u c e d produced data a r e a f t e r ab l e octave k ( b ) .  for by by by of out sw  FIGURE A.3 Q=150) d i s p e r s e d s y n t h e t i c seismograms (with a s i n g l e m u l t i - o c t a v e sweep, s t a c k i n g three s i n g l e octave sweeps, s t a c k i n g three o v e r l a p p i n g band sweeps, t h e same f i n a l b a n d w i d t h . P o l a r i t y begins t o be 1.5 s , h o w e v e r , t h e e f f e c t i s much worse f o r the eep. The best r e s u l t i s f o r t h e s i n g l e octane  240  FIGURE C o r r e l o g r a (2=125) : (a) p r o d u c (b) p r o d u c (c) produc a l l data l o s t a f t e r s i n g l e oc stack ( b ) w i t h time,  m s  f o r attenuated  ed by ed by ed by a r e o about tave .Due t and t  A.4  d i s p e r s e d  s y n t h e t i c  seismograms  ( w i t h  a s i n g l e m u l t i - o c t a v e sweep, s t a c k i n g three s i n g l e octave sweeps, s t a c k i n g three o v e r l a p p i n g band sweeps, f t h e same f i n a l bandwidth. P o l a r i t y begins t o be 1.5 s , h o w e v e r , t h e e f f e c t i s much worse f o r t h e sweep. The best r e s u l t i s f o r t h e s i n g l e octave o t h e a t t e n u a t i o n , t h e energy d i e s down r a p i d l y h e high f r e q u e n c i e s a r e l o s t .  -20.0  0.0  20.0  -8.0  FIGURE (a) The c o r r e l o g r a m f o r an a t t seismogram produced u s i n g the s i n g F i g u r e A.4b) (b) The same a f t e r recovery of the in the t e x t ) . T h i s e f f e c t may be r e p e a t i n g t h e higher frequency c o r r e l o g r a m s . (c) (d)  The The  spectrum spectrum  of ( a ) . of ( b ) .  2.0  12.0  A.5 e n u a t e d d i s p e r s e d s y n t h e t i c l e octave stack method (as per high f r e q u e n c i e s (as d e s c r i b e d mimicked i n %he f i e l d by sweeps and then s t a c k i n g the  242  -M.O  o.o  -eo.o  ao.o  FIGURE C o r r e l o g r a m s  f o r  (a)  produced  by  s t a c k i n g  t h r e e  (b)  produced  by  s t a c k i n g  two  1.4  The  s i n g l e  (a)  produces  even  though  d i s p e r s e d  o c t a v e both  method data  are  s i n g l e  the  ao.o  A.6  s y n t h e t i c  of  o.o  seismograms octave  o c t a v e same  ( w i t h  (?=100):  sweeps,  sweeps, a  b e t t e r  f i n a l  f i n a l  bandwidth.  p r o d u c t ,  REFEREED  PUBLICATIONS:  1. J o n e s , I . F . , Mansinha, L , a n d S h e n , P . Y . , 1 9 8 2 , On t h e double e x p o n e n t i a l frequency-magnitude r e l a t i o n o f e a r t h q u a k e s : B u l l e t i n o f t h e S e i s m o l o g i c a l S o c i e t y o f A m e r i c a , v . 7 2 , p. 23732375. 2. E l l i s , R.M., J o n e s , I.F., G r e e n , M.J., Mereu, R.F., Ka Hyndman, R.D., McMe Vancouver I s l a n d s e i s study o f a c o n v e r g e n t v.20, p. 719-741. 3.  Clowes,  1983, S e i s m i c N a t u r e , v.303,  S p e n c e , G.D., C l o w e s , R.M., W a l d r o n , D.A., A.G., F o r s y t h , D.A., M a i r , J . A . , B e r r y , nasewich, E.R., Cumming, G.L., H a j n a l , Z., c h a n , G.A., a n d L o n c a r e v i c , B.D., 1 9 8 3 , The m i c p r o j e c t : a CO-CRUST o n s h o r e - o f f s h o r e margin: Canadian J o u n a l o f E a r t h S c i e n c e s ,  R.M.,  E l l i s ,  r e f l e c t i o n s p . 668-670.  4. Chapman, O l d e n b u r g , D.W., marine s e i m i c d a t for a b y s a a l p l a A c o u s t i c a l S o c i e t  R.M.,  from  H a j n a l ,  t h e  Z.,  a n d J o n e s ,  s u b d u c t i n g  I.F.,  l i t h o s p h e r e ? :  N.R., Levy, S., J o n e s , I.F., S t i n s o n , K., a n d P r a g e r , B.T., 1 9 8 4 , I n v e r s i o n o f deep ocean a t o r e c o v e r v e l o c i t y and d e n s i t y i n f o r m a t i o n i n s e d i m e n t s . S u b m i t t e d t o t h e J o u r n a l o f t h e y o f A m e r i c a .  5. J o n e s , I . F . , a n d Levy, S., 1 9 8 4 , The a p p l i c a t i o n o f Karhunen-Loeve t r a n s f o r m a t i o n i n i n m u l t i c h a n n e l s e i s m i c p o c e s s i n g : ( i n v i t e d paper f o r G e o p h y s i c a l P r o s p e c t i n g ) . 6. D.W.,  Levy,  S.,  J o n e s ,  1984, Complex 7.  U l r y c h ,  I . F . ,  Common  T . J . ,L e v y ,  U l r y c h ,  S i g n a l  T . J . ,  A n a l y s i s .  S., Oldenburg  I n  D.W.,  a n d  O l d e n b u r g ,  p r e p . . a n d J o n e s ,  1984, A p p l i c a t i o n s o f t h e complex Karhunen-Loeve in e x p l o r a t i o n s e i s m o l o g y : i n p r e p . .  OTHER  t h e d a t a  I . F . ,  t r a n s f o r m a t i o n  PUBLICATIONS: 1.  M a s t e r ' s  J o n e s , t h e s i s ,  2. U l 1983, A p p e x p l o r a t i o annual mee  I.F.,. U n i v .  r y c h , T . l i c a t i o n n g e o p h i ting o f  1980, Western  A s p e c t s  o f  g l o b a l  s e i s m i c i t y :  O n t a r i o .  J . , Levy, S., Oldenburg D.W., a n d J o n e s , I.F., s o f t h e Karhunen-Loeve t r a n s f o r m a t i o n i n s i c s : paper S 6 . 5 , p r o c e e d i n g s o f t h e 53 r d . t h e S o c i e t y o f E x p l o r a t i o n G e o p h y s i c i s t s .  3. Levy, S., U l D.W., 1 9 8 3 , A p p l i c a t i o e x p l o r a t i o n s e i s m o l o g annual meeting o f t h e  r y c h , T . J . , J o n e s , I.F., and Oldenburg, n s o f c o m p l e x common s i g n a l a n a l y s i s i n y : p a p e r S 6 . 6 , p r o c e e d i n g s of-t h e 53 r d . S o c i e t y o f E x p l o r a t i o n G e o p h y s i c i s t s .  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Country Views Downloads
United States 66 9
India 55 0
China 19 7
Russia 5 0
Germany 5 7
Japan 4 0
Malaysia 3 0
Poland 3 0
Unknown 3 0
Saudi Arabia 2 1
Austria 2 0
Canada 2 4
Republic of Korea 2 1
City Views Downloads
Unknown 53 15
Mountain View 28 0
Ashburn 25 0
Shenzhen 9 4
Bangalore 9 0
Warangal 7 0
Chennai 4 0
Qingdao 4 2
Tokyo 4 0
Mumbai 4 0
Beijing 3 0
Bremerhaven 3 0
Jinan 3 1

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}
Download Stats

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0052986/manifest

Comment

Related Items