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Linear systems theory applied to a horizontally layered crust Jensen, Oliver George 1970

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LINEAR SYSTEMS THEORY APPLIED TO A HORIZONTALLY LAYERED CRUST  by  OLIVER GEORGE JENSEN B.Sc. M.Sc.  University of B r i t i s h University of B r i t i s h  Columbia, 1964 Columbia, 1966  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in t h e Department  of GEOPHYSICS  We  accept t h i s  required  THE  thesis  as c o n f o r m i n g t o the  standard  UNIVERSITY OF BRITISH COLUMBIA December, 1970  In presenting this thesis in partial  fulfilment of the requirements for  an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives.  It  is understood that copying or publication  of this thesis for financial gain shall not be allowed without my written permission.  Depa rtment The University of B r i t i s h Columbia Vancouver 8, Canada  ABSTRACT  E l a s t i c wave p r o p a g a t i o n  i n a multilayered crust with  causally  or a c a u s a l l y a t t e n u a t i n g l a y e r s i s f o r m u l a t e d d i r e c t l y i n terms o f systems t h e o r y .  The  s o l u t i o n of the l i n e a r systems a n a l o g  determines  the wave motions a t the f r e e s u r f a c e , motions o f a l l i n t e r n a l and  the waveforms p r o p a g a t e d  i n t o the mantle i n response  waves g e n e r a t e d w i t h i n the l a y e r i n g o r e n t e r i n g i n t o  boundaries  to p l a n e P o r S  the c r u s t .  p a r t i c u l a r problem s o l v e d i s the d e t e r m i n a t i o n o f the response n-layered crust  to t e l e s e i s m i c P o r S waves i n c i d e n t w i t h  a n g l e at the c r u s t a l base. allow r e f l e c t i o n  domain and  to a n o n - a t t e n u a t i n g  For  of  an  arbitrary  T r i v i a l e x t e n s i o n s of t h i s p r o b l e m would  s o l u t i o n s have been a c c o m p l i s h e d the time domain.  The  i n both  Fourier solution  the restricted  c r u s t i s e q u i v a l e n t t o the s t a n d a r d H a s k e l l m a t r i x  s o l u t i o n o f e l a s t i c wave e q u a t i o n s . the s y n t h e s e s  The  solutions.  Numerical frequency  linear  Direct  time domain s o l u t i o n s  allow  o f seismograms c o n s i d e r i n g i n t e r n a l c r u s t a l a b s o r p t i o n . demonstration  of the u t i l i t y  and  advantages o f the  theory,  the l i n e a r systems f o r m u l a t i o n has been a p p l i e d to s t u d i e s o f the wave response seismic  of the c e n t r a l A l b e r t a c r u s t u s i n g P waves from 6  tele-  events. Frequency domain comparisons between the t h e o r e t i c a l  e x p e r i m e n t a l s p e c t r a l amplitude the  elastic  V/H  r a t i o s have shown t h a t , a l t h o u g h  t h e o r e t i c a l e f f e c t of a t t e n u a t i o n w i t h i n the c r u s t can  considerable, l i t t l e experiment  t h e o r e t i c a l and  be  improvement i n c o r r e l a t i o n s between t h e o r y  has been a c h i e v e d by  a b s o r p t i o n models.  and  Although  considering plausible  significant  e x p e r i m e n t a l V/H  and  crustal  s i m i l a r i t i e s between  r a t i o s were found below 2 Hz,  the little  ii c o r r e l a t i o n was  apparent at h i g h e r  scattering noise partly contributed probable that i n s u f f i c i e n t  frequencies.  Background  to t h i s e f f e c t and  d e t a i l and  a c c u r a c y was  and  i t i s also  available for  the  model c r u s t a l s e c t i o n s . Time domain s y n t h e t i c seismograms have been determined which w e l l c o r r e s p o n d to the e a r l y P coda of s e v e r a l o f the e x p e r i m e n t a l Assumptions on required  the event s o u r c e motions and  the mantle p r o p e r t i e s  to determine i n c i d e n t P waveforms f o r these s o l u t i o n s .  attenuation within  the  c r u s t a l l a y e r i n g was  between the s y n t h e t i c seismograms and  included.  Correlations  the e x p e r i m e n t a l r e c o r d s  and it  t h i s e f f e c t i s p r i m a r i l y due  p o s s i b l e s c a t t e r i n g of i s probable that  the waves w i t h i n  has  to background the c r u s t .  mantle a t t e n u a t i o n  f r e q u e n c y and theory  attenuation  time domains.  mathematics can now  problem and  noise  Furthermore,  assumptions were n o t  A major apparent advantage of the new acausal  It i s  the waveforms used f o r these s o l u t i o n s based  the s o u r c e motion and  c a u s a l and  are Causal  been found to d e c r e a s e r a p i d l y w i t h time f o l l o w i n g the P o n s e t . suggested that  formulation  s o l u t i o n s are p e r m i t t e d  A l s o , the l a r g e body of be  applied d i r e c t l y  to the  inverse  problem.  on  sufficient.  i s that  i n both  the  communications propagation  c o u l d prove u s e f u l i n attempts at the s o l u t i o n of  non-normal i n c i d e n c e  records.  the  iii  TABLE OF  CHAPTER I  CONTENTS  GENERAL INTRODUCTION  1.1  Introduction  1  1.2  Outline of Thesis  4  CHAPTER I I  THE  LINEAR SYSTEM THEORY FOR  A MULTILAYER CRUST  2.1  D e s c r i p t i o n o f the Problem  6  2.2  L i n e a r Systems Approach  6  2.3  The  L i n e a r System Analog f o r Normal I n c i d e n c e  9  2.4  The  Frequency Domain D e s c r i p t i o n  11  2.5  The  L i n e a r System Analog f o r Non-normal I n c i d e n c e  18  2.6  Time Domain F o r m u l a t i o n  27  2.7  Attenuation  34  CHAPTER I I I  and  D i s p e r s i o n i n the System Model  ANALYSIS OF THE  ALBERTA SEISMOGRAMS  3.1  Introduction  38  3.2  The  A n a l y s i s Problem  38  3.3  The  Experiment  3.4  The  C r u s t Model  3.5  Data P r e p a r a t i o n  3.6  The  3.7  Frequency A n a l y s e s - T h e o r e t i c a l S p e c t r a l Amplitudes  59  3.8  Frequency A n a l y s i s - S p e c t r a l R a t i o T e c h n i q u e  63  3.9  T h e o r e t i c a l Spectral Ratios  3.10  Experimental S p e c t r a l Ratios  "  3  9 41 48  Seismograms  50  f  ^ ,  6  8 71  iv  3.11  General Features of S p e c t r a l Ratios  79  3.12  D e t a i l s of the Spectral Ratios  81  3.13  Time Domain Syntheses  85  3.14  Syntheses w i t h L a y e r A t t e n u a t i o n  88  3.15  Impulse S y n t h e t i c C r u s t a l Responses  92  3.16  I n c i d e n t Waveform Models  97  3.17  Seismogram Syntheses  3.18  Comparisons o f the Syntheses 55° E p i c e n t e r  and Records  for  107  3.19  Comparisons o f t h e Syntheses 85° E p i c e n t e r  and Records  for  110  3.20  S y n t h e s i s U s i n g an Input Wavelet Deconvolution  CHAPTER IV  106  O b t a i n e d by  110  CONCLUSIONS  4.1  Summary o f A n a l y s e s  113  4.2  C o n c l u d i n g Remarks and S u g g e s t i o n s  115  REFERENCES  118  APPENDIX I  IMPULSE RESPONSE FUNCTION FOR NORMAL INCIDENCE  121  APPENDIX I I  PHASE ADVANCE TRANSFER FUNCTIONS  128  APPENDIX I I I EXAMPLE BLOCK REDUCTION PROCEDURE  130  APPENDIX IV  136  FORTRAN COMPUTER PROGRAMS FOR TIME DOMAIN SEISMOGRAM SYNTHESES AND FOURIER COMPONENT SOLUTIONS BASED ON THE LINEAR SYSTEMS THEORY  V  L I S T OF TABLES  TABLE I  Crustal Sections  44  TABLE I I  F o r m a t i o n Q's  47  TABLE I I I  E v e n t s Used  57  TABLE IV  A t t e n u a t o r Weights Sampling Increment  i n Analyses f o r the 0.05 Second - •  90  LIST OF FIGURES  FIGURE 1  The h o r i z o n t a l l y m u l t i l a y e r e d medium w i t h n l a y e r s o v e r l y i n g a basement h a l f s p a c e .  FIGURE 2  The i n p u t - o u t p u t system c r u s t .  FIGURE 3  A r e p r e s e n t a t i o n o f the p o s s i b l e i n p u t and o u t p u t waveforms i n l a y e r i under normal incidence.  FIGURE 4  A r a y p a t h r e p r e s e n t a t i o n o f the i n t e r n a l reverberations within layer i giving r i s e t o output t r a n s f o r m s i g n a l s U ( s ) and  diagram f o r a l i n e a r  1  D  i l< >S  +  FIGURE 5  A f u n c t i o n a l b l o c k r e p r e s e n t a t i o n of the l i n e a r system a n a l o g f o r l a y e r i under n o r m a l incidence.  FIGURE 6  A r e p r e s e n t a t i o n o f the "pseudo-coordinate" transformation f o r layer i .  FIGURE 7  A diagram showing t h e n o n - c o i n c i d e n c e o f wave c o n s t a n t phase p o s i t i o n s and a h o r i z o n t a l l a y e r boundary.  FIGURE 8  A functional block representation normal i n c i d e n c e f o r l a y e r i .  FIGURE 9  The f r e e s u r f a c e incidence.  FIGURE 10  A comparison o f a n o r m a l i n c i d e n c e l a y e r impulse response f o r i n f i n i t e Q and c a u s a l l y absorbing c o n d i t i o n s .  FIGURE.11  The time-domain r e p r e s e n t a t i o n o f t h e system a n a l o g o f one l a y e r a t normal i n c i d e n c e .  FIGURE 12  The time-domain r e p r e s e n t a t i o n o f t h e system analog o f one l a y e r f o r non-normal i n c i d e n c e .  FIGURE 13  A map o f t h e s e i s m i c s t a t i o n and o i l w e l l l o c a t i o n s by which the c r u s t a l s e c t i o n was determined.  f o r non-  r e f l e c t i o n f o r non-normal  vii  FIGURE 14  The sedimentary c r u s t a l s e c t i o n o b t a i n e d by i n t e r p o l a t i o n between o i l w e l l g e o p h y s i c a l logs.  42  FIGURE 15  The t h e o r e t i c a l s t a n d a r d seismograph response compared t o a response c a l i b r a t i o n o b t a i n e d by t h e t r a n s i e n t method.  49  FIGURE 16  The v e r t i c a l , r a d i a l h o r i z o n t a l and t r a n s v e r s e h o r i z o n t a l components o f t h e e x p e r i m e n t a l seismograms f o r the events s t u d i e d i n t h i s work.  51  FIGURE 17  The s p e c t r a l amplitude response o f the t h e o r e t i c a l n o n - a t t e n u a t i n g Leduc c r u s t a l s e c t i o n t o a normal incide'nce w h i t e a m p l i t u d e spectrum P-wave i n c i d e n t a t the P r e c a m b r i a n Cambrian boundary.  60  FIGURE 18  The s p e c t r a l a m p l i t u d e response o f t h e t h e o r e t i c a l n o n - a t t e n u a t i n g Leduc c r u s t a l s e c t i o n t o a 30° i n c i d e n t P wave a t t h e Precambrian-Cambrian boundary.  61  FIGURE 19  The s p e c t r a l a m p l i t u d e response o f t h e t h e o r e t i c a l c r u s t a l s e c t i o n t o a 30° i n c i d e n c e P wave a t t h e Precambrian-Cambrian boundary.  62  FIGURE 20  S p e c t r a l V/H r a t i o s f o r v a r i o u s attenuation c o n d i t i o n s compared t o t h e n o n - a t t e n u a t i n g crust r a t i o s .  66  FIGURE 21  V e r t i c a l - t o - v e r t i c a l s p e c t r a l r a t i o s f o r nona t t e n u a t i n g c r u s t models.  70  FIGURE 22  E x p e r i m e n t a l V/H s p e c t r a l r a t i o s o b t a i n e d from the s u i t e o f seismograms.  72  FIGURE 23  S p e c t r a l s i g n a l - t o - n o i s e r a t i o s f o r the v e r t i c a l and h o r i z o n t a l f o r t h r e e e v e n t s .  78  FIGURE 24  T h e o r e t i c a l v e r t i c a l - t o - v e r t i c a l r a t i o s are compared t o those o b t a i n e d from t h e e x p e r i m e n t a l seismograms f o r 30° i n c i d e n c e angle.  84  FIGURE 25  Comparison o f non-normal and normal s y n t h e t i c seismograms.  87  FIGURE 26  The impulse r e s p o n s e o f l a y e r 1 o f t h e c r u s t a l s e c t i o n assuming c a u s a l a t t e n u a t i o n w i t h Q=23.  incidence  89  viii  FIGURE 27  The t h e o r e t i c a l f r e q u e n c y response o f the c r u s t f o r one passage o f an impulse f o r average Q=200 compared to the f r e q u e n c y r e s p o n s e due t o the sum of the time domain a t t e n u a t i o n o p e r a t o r s of T a b l e IV.  91  FIGURE 28  Normal i n c i d e n c e s y n t h e t i c seismograms dependence on Q.  93  FIGURE 29  The v e r t i c a l and r a d i a l h o r i z o n t a l s y n t h e t i c i m p u l s e responses of the t h r e e c r u s t a l s e c t i o n s r e p r e s e n t i n g the t h r e e s t a t i o n s .  94  FIGURE 30  The normal i n c i d e n c e s y n t h e t i c impulse f o r t h e LED c r u s t a l s e c t i o n . - •  95  FIGURE 31  The basement r e f l e c t e d wave s y n t h e t i c impulse r e s p o n s e f o r the LED c r u s t a l s e c t i o n ,  FIGURE 32  The e f f e c t o f v a r i o u s mantle p a t h average Q v a l u e s on an impulse s o u r c e d i s p l a c e m e n t m o t i o n which has t r a v e l l e d 10 minutes at the u n d i s p e r s e d phase v e l o c i t y through the mantle.  100  FIGURE 33  Wavelets c o n s t r u c t e d by c o n v o l u t i o n o f the seismograph response w i t h a b s o r b i n g mantle r e s p o n s e s f o r v a r i o u s T/Q parameters f o r b o t h impulse and s t e p s o u r c e s .  101  FIGURE 34  V e r t i c a l and h o r i z o n t a l component s y n t h e t i c seismograms f o r v a r i o u s T/Q w a v e l e t s o f the form shown i n F i g u r e 32 f o r impulse s o u r c e motions.  102  FIGURE 35  V e r t i c a l and h o r i z o n t a l component s y n t h e t i c seismograms f o r v a r i o u s T/Q w a v e l e t s f o r step source motions.  103  FIGURE 36  Impulse s o u r c e m o t i o n s y n t h e t i c seismograms f o r 55° and 85° e p i c e n t r a l d i s t a n c e s .  105  FIGURE 37  A comparison o f the v e r t i c a l s y n t h e t i c seismograms w i t h the e x p e r i m e n t a l s e i s m o grams of F i g u r e 16 f o r s t a t i o n LED.  108  FIGURE 38  A comparison o f the v e r t i c a l component LED seismogram f o r the V e n e z u e l a event and the r e - s y n t h e s i s u s i n g the U l r y c h w a v e l e t .  111  FIGURE 39  P o l e maps o f the one l a y e r t r a n s f e r f o r normal i n c i d e n c e : Case I .  125  response  function  96  ix  FIGURE 40  P o l e maps o f t h e one l a y e r t r a n s f e r f o r normal i n c i d e n c e : Case I I .  function  FIGURE 41  A r e p r e s e n t a t i o n o f phase advance time c a l c u l a t i o n s n e c e s s a r y f o r non-normal condition analyses.  128  FIGURE 42  The d e s i r e d b l o c k diagram r e p r e s e n t a t i o n i n terms o f t r a n s f e r f u n c t i o n and t r a n s f o r m e d signals.  130  FIGURE 43  B l o c k diagram  130  FIGURE 44  B l o c k diagram o f t h e l i n e a r feedback system.  FIGURE 45  The reduced system b l o c k diagram h a v i n g the l a y e r response t o i n p u t s U ^ ( s ) and V_^(s).  o f p a r a l l e l summing p a t h s . positive  126  131  134  X  ACKNOWLEDGEMENTS  I wish to thank Dr. R. M. E l l i s f o r h i s invaluable a i d and encouragement during the development of this research.  I am also  g r a t e f u l to my advisory committee with Drs. G. K. C. Clarke, R. M. Clowes and R. D. R u s s e l l f o r u s e f u l discussions and c r i t i c i s m s of various phases of  the work. I would also l i k e to acknowledge the contribution of the  Department of Geophysics s t a f f whose preparation of the seismograph f i e l d equipment, recording of the experimental data and preparation of  the d i g i t i z a t i o n equipment was so necessary  to much of this study.  During this experiment, equipment was made a v a i l a b l e by the Seismology D i v i s i o n , Earth Physics Branch and both equipment and f a c i l i t i e s were provided by the University of Alberta. I would also l i k e to thank Judy McLean f o r a f i n e job typing the  manuscript. This research was supported by the National Research Council  (Grant A-2617) and the Defence Research Board of Canada (Grant 9511-76). The author was p a r t l y supported during this research by a University of B r i t i s h Columbia Graduate Fellowship.  CHAPTER I GENERAL INTRODUCTION  .1.1  .•  Introduction Seismologists  have been i n v e s t i g a t i n g the p r o p a g a t i o n of  elastic  waveforms i n h o r i z o n t a l l y m u l t i l a y e r e d media s i n c e the e a r l y development of s e i s m i c and  e x p l o r a t i o n methods i n o r d e r  to determine the  the e f f e c t s o f the s t r u c t u r e on s e i s m i c waves.  obvious p r a c t i c a l a p p l i c a t i o n s i n o i l and m i n e r a l of c r u s t a l p r o p a g a t i o n i s n e c e s s a r y to the seismology. ated by  For  example, much of the  the e l a s t i c p r o p e r t i e s  station.  of the  In a d d i t i o n to  advance of earthquake  of a seismogram i s gener-  crust underlying  a seismograph  Thus i t i s u s e f u l to remove these c o m p l i c a t i n g  to study the d e t a i l s of f o c a l mechanisms and the e a r t h ' s  interior.  the  e x p l o r a t i o n , a knowledge  continuing  character  c r u s t a l structure-  effects in  order  to r e f i n e our knowledge of  Furthermore, s i n c e c r u s t a l i n f o r m a t i o n  is  contained  i n earthquake seismograms, a n a l y s i s p r o c e d u r e s based upon a thorough u n d e r s t a n d i n g of e l a s t i c p r o p a g a t i o n c o u l d be u s e f u l to o b t a i n c r u s t a l s t r u c t u r e from these seismograms. The  first  s a t i s f a c t o r y m a t h e m a t i c a l a n a l y s i s of e l a s t i c wave  p r o p a g a t i o n i n h o r i z o n t a l l y l a y e r e d s t r u c t u r e s was which was  subsequently generalized  s u r f a c e waves by H a s k e l l  (1953).  t h a t of Thomson (1950)  to i n c l u d e both body waves  and  As w e l l as s u r f a c e wave a p p l i c a t i o n s ,  s e v e r a l i n v e s t i g a t o r s (e.g., Phinney, 1964;  Ibrahim, 1969)  have  a p p l i e d the H a s k e l l m a t r i x method to the a n a l y s i s of earthquake body waves.  The  Haskell-Thomson approach o b t a i n s  i n terms of frequency component a m p l i t u d e s ;  the p r o p a g a t i o n s o l u t i o n the more r e c e n t work of  Wuenschel (1960) f o r normal i n c i d e n c e and F r a s i e r (19 70) and  SV motions p e r m i t  For v e r t i c a l  the d e t e r m i n a t i o n  f o r non-normal P  o f time domain c r u s t a l  i n c i d e n c e P waves, e x t e n s i o n s  responses.  o f these methods and o t h e r  a n a l y t i c a l approaches t o the i n v e r s e problem o f d e t e r m i n a t i o n  o f the  l a y e r paramenters from earthquake and e x p l o r a t i o n seismograms have r e c e i v e d considerable Claerbout,  a t t e n t i o n i n the recent  1968;  arbitrary either  the frequency  o r time domain.  L i n e a r systems and analog  propagation  (Lindsey,  procedures  F o r example, t h e more r e s t r i c t i v e one-  p r o b l e m o f v e r t i c a l P-wave p r o p a g a t i o n 1960), a n a l o g  (Rob i n s o n and T r e i t e l , has  f o r the d i r e c t  the v e r t i c a l w i t h i n t h e l a y e r i n g and s o l u t i o n i n  have been used p r e v i o u s l y .  systems  a new f o r m u l a t i o n  l i n e a r systems t h e o r y w h i c h a l l o w s b o t h P and SV motions o f  angle w i t h  dimensional  ( e . g . , G o u p i l l a u d , 1961;  Ware and A k i , 1969).  This t h e s i s presents problem u s i n g  literature  (Sherwood, 1962) and communications  1966) terms.  been d e s c r i b e d by M u e l l e r  has been s t u d i e d i n theory  A l s o , s u r f a c e wave n o r m a l d i s p e r s i o n  and Ewing (1962) by means o f a l i n e a r systems  model. I t i s notable  t h a t when an e l a s t i c wave t r a v e l s through a  s t r u c t u r e d m a t e r i a l , a complex r e c o r d o f t h e e l a s t i c p r o p e r t i e s a l o n g t h e t r a v e l path  i s r e t a i n e d by the wave.  Hence, a n a l y s i s o f t h e wave can  i  provide  s t r u c t u r a l information.  s e i s m i c problem. multilayered first  The r e l a t i v e l y  simple  geometry o f t h e h o r i z o n t a l l y  c r u s t l e a d s one t o e x p e c t t h a t i n v e r s e s o l u t i o n s may be  p o s s i b l e f o r t h i s problem.  a l s o demands t h a t the p r o p a g a t i o n effects  T h i s i s the e s s e n t i a l goal o f the i n v e r s e  However, t h e complete i n v e r s e s o l u t i o n c h a r a c t e r i s t i c s due t o a l l e l a s t i c  i n the c r u s t a r e w e l l u n d e r s t o o d .  extend t h i s  understanding.  This t h e s i s should help to  3 The i n c l u s i o n of a t t e n u a t i o n , both causal and acausal i s new i n t h i s work. feature.  However, extensions of other approaches may a l s o allow the For i n s t a n c e , H a s k e l l (1953) suggested a p o s s i b l e means f o r  i n c l u d i n g attenuation i n h i s formulation;  d e r i v a t i v e s of the H a s k e l l  method (e.g., Wuenschel, 1960; F r a s i e r , 1970) could p o s s i b l y be modified t o permit c r u s t a l absorption. The l i n e a r systems approach has a f u r t h e r advantage that the d i r e c t time and frequency domain problems are cast i n very s i m i l a r forms. Thus, the correspondence of time s e r i e s and s p e c t r a l response characteri s t i c s are p a r t i c u l a r l y apparent. frequency  The equivalence of the time and  s o l u t i o n s by other approaches i s not d i r e c t . One outstanding feature of a systems formulation i s that a great  body of a n a l y t i c a l techniques already e x i s t s , p a r t i c u l a r l y i n automatic c o n t r o l engineering and communications engineering, which now can be a p p l i e d to the problem.  P o s s i b l e f u r t h e r success i n determining  crustal  p r o p e r t i e s from earthquake and r e f l e c t i o n seismograms could be derived by using these mathematical t o o l s .  In the past, communications theory has  been of great i n t e r e s t to.many geophysicists and much recent advance of geophysics has been gained using these methods.  Thus, a d i r e c t formu-  l a t i o n o f the propagation problem i n the systems form may even broaden the i n t e r e s t among these geophysicists already using communications theory methods.  A l a s t c o n s i d e r a t i o n which may be of consequence i s the f a c t  that the systems form of the problem can lead t o d i r e c t mechanical and e l e c t r i c a l analog procedures.  With the recent developments i n micro-  e l e c t r o n i c d i g i t a l l o g i c , f o r example, small hard-wired e f f i c i e n t l y solve propagation problems. domain approaches also e x i s t s .  computers could  The p o s s i b i l i t y of analog time  4 The b a s i c assumptions required by the l i n e a r systems approach are s i m i l a r to those used i n other methods.  D i s t i n c t layers having  i s o t r o p i c v e l o c i t i e s , uniform d e n s i t i e s and welded boundaries Plane r o t a t i o n a l and compressional waves are considered.  are necessary.  Attenuation  with an a r b i t r a r y complex dependence on frequency i s also allowable i n each layer. of  The general s o l u t i o n of the problem requires the  determination  the motions on the free surface and within the medium i n response to  source wave motions generated within the l a y e r i n g or entering i n t o the medium.  The choice of the source l o c a t i o n of the plane wave generation  determines the nature of the p h y s i c a l problem;  e i t h e r the r e f l e c t i o n or  transmission problem i s soluble.  1.2  Outline of Thesis (a)  The systems d e s c r i p t i o n of c r u s t a l transmission of e l a s t i c  waves i s f i r s t obtained i n the frequency  domain for normal incidence  plane waves impinging on a h o r i z o n t a l l y multilayered crust from a basement halfspace.  This r e s t r i c t e d formulation i s then generalized to non-normal  conditions. (b)  The analogous time domain formulation i s derived from the  general frequency d e s c r i p t i o n . (c)  The u t i l i t y  of the theory i s demonstrated i n comparison  analyses with a set of earthquake and nuclear event seismograms recorded in central Alberta.  Both time and frequency analyses as w e l l as seismo-  gram synthesis are obtained. (d)  Extensions of the method are suggested which could allow  analysis of propagation i n crusts possessing complex l i n e a r and nonlinear properties.  The p o s s i b i l i t y of inverse solutions i s also i n d i c a t e d .  5 The properties  of  problem.  The  solutions  is  p r o b l e m has  been  specifically  t e l e s e i s m i c seismograms: obvious noted.  extension  to  that  permit  formulated i s the  normal  to  crustal  incidence  describe transmission reflection  6  CHAPTER I I THE LINEAR SYSTEM THEORY FOR A MULTILAYER CRUST  2.1  D e s c r i p t i o n of t h e Problem In  dimensions  t h i s work e l a s t i c p r o p a g a t i o n i s t o be f o r m u l a t e d i n two  f o r waveforms i n c i d e n t on a h o r i z o n t a l l y l a y e r e d medium from  a h a l f s p a c e beneath.  T h i s model r e p r e s e n t s a t e l e s e i s m i c  arrival  i m p i n g i n g on a l a y e r e d c r u s t from a basement. Each c r u s t a l l a y e r i i s s p e c i f i e d b y i t s - c o m p r e s s i o n a l and s h e a r wave v e l o c i t i e s a., 6.: ray  d e n s i t y p. and t h i c k n e s s n..  The P and SV  a n g l e s i n each l a y e r , d e s i g n a t e d e^ and f ^ r e s p e c t i v e l y , r e p r e s e n t t h e  a n g l e s between t h e r a y p a t h and t h e v e r t i c a l a x i s ( F i g u r e 1).  The f r e e  s u r f a c e d i s p l a c e m e n t , t h e i n t e r m e d i a t e boundary d i s p l a c e m e n t s and t h e d i s p l a c e m e n t waves p r o p a g a t e d  downwards i n t o t h e basement g e n e r a t e d b y a  p l a n e P o r S w a v e f r o n t i n c i d e n t on t h e c r u s t from t h e basement a r e s o u g h t . In  t h i s chapter the normal  i n c i d e n c e wave p r o p a g a t i o n  w i l l be f o r m u l a t e d f i r s t and t h e n g e n e r a l i z e d t o t h e non-normal T h i s approach  s h o u l d most s i m p l y demonstrate  problem problem.  the e s s e n t i a l nature of the  f o r m u l a t i o n w i t h o u t o b s c u r i n g t h e method i n t h e d e t a i l s o f t h e g e n e r a l problem.  Time domain system d e s c r i p t i o n s w i l l  t h e n be d e r i v e d from  these  frequency f o r m u l a t i o n s .  2.2  L i n e a r Systems Approach The  of  l i n e a r system f o r m u l a t i o n d e s c r i b e s wave p r o p a g a t i o n i n terms  f i l t e r o r communications t h e o r y .  I n t h i s manner, t h e c r u s t can be  r e g a r d e d as a f i l t e r system o r communications c h a n n e l w h i c h r e c e i v e s an  7  LAYER  1  LAYER  2  P  v  (^n +  0//  °2  2  2  • • • LAYER  n  fl  LAYER n + 1 ( BASEMENT  7 y  A )y  e  n  n +l  /  F i g . 1. T h e h o r i z o n t a l l y m u l t i l a y e r e d medium w i t h n l a y e r s o v e r l y i n g a basement h a l f s p a c e . E a c h l a y e r i i s s p e c i f i e d by P - and S-wave v e l o c i t i e s ct^ a n d B i ; d e n s i t y p-j_ a n d t h i c k n e s s n ^ . The p l a n e wave f r o m t h e b a s e m e n t i s i n c i d e n t a t a n g l e e , , f o r P wave ( o r f , . f o r S w a v e ) .  n+1  °  input set a  signal  i n the  and o p e r a t e s  case  teleseismic  boundary; the  free  and  all  to  the  output  signal  motions.  set  can be b l o c k the  operator  is  contains  the  are  of  signal  crust  at  cartesian  the  available.  in Figure  the  signal  input  into These  is  basement  coordinates  re-transmitted  also  (or  t r a n s m i s s i o n the  on t h e  that  independent  output  incident  d i a g r a m m e d as  assumption  the  For c r u s t a l  The waveforms  i n t e r n a l boundary motions  Under  create  many o u t p u t s ) .  P o r SV w a v e m o t i o n  surface  relations  response  of  on i t  n+1  the  of  basement  input-output  2.  the  crustal  response  the  input waveform.  is  linear,  Thus,  if  the  two  8  i(t)  OUTPUT S SIGNAL SET  INPUT SIGNAL  CRUST  RESPONSE  Fig. 2. The input-output diagram for a linear system crust. The output signal set O j ( t ) i s obtained from the convolution of the input signal i(t) with the crustal response vector hj(t).  among the input, the output set and the response operator are known, the third i s uniquely determinable.  In particular, i f the input signal and  response operator are known, the output signals can be calculated. A f i l t e r system i s often described in terms of the output signal set generated by specific input signals such as the Dirac delta function <5(t).  This particular response i s called the impulse-response  function set of the system, h.(t).  The output signal set o.(t) i s  J  3  obtained by the convolutions of the input signal i ( t ) with the impulse response function vector h^(t):  iCt'^Ct-t'Jdt' (2-1)  = l(t)*h (t)  F o r a l i n e a r system, components may  an e q u i v a l e n t e q u a t i o n i n terms of complex  frequency  be o b t a i n e d by the L a p l a c e i n t e g r a l t r a n s f o r m a t i o n such  that: Cv.(s) = I ( s ) . H (s)  (2-2)  where s = o+ito i s the complex frequency and  the f u n c t i o n s of the v a r i a b l e  s d e s i g n a t e d w i t h c a p i t a l s a r e the L a p l a c e transformed o  time  functions,  GO  e.g.  I(s) = L{i(t)} = | i ( t ) e ~  S t  dt.  (2-3)  o This•representation relates  the transformed i n p u t s i g n a l to the  output s i g n a l s e t by m u l t i p l i c a t i o n w i t h the system v e c t o r a t each complex frequency s. the system  transfer  function  Both e q u a t i o n s c o m p l e t e l y  characterize  and t h e p a r a l l e l f u n c t i o n s i n the time and frequency  a r e g e n e r a l l y o b t a i n a b l e from each o t h e r through  transformed  domains  the L a p l a c e t r a n s f o r m and  i t s inverse.  2.3  The L i n e a r System Analog The  i s now  l i n e a r system  f o r Normal I n c i d e n c e  analog f o r the simple case o f normal i n c i d e n c e  c o n s t r u c t e d to show the e s s e n t i a l c o n c e p t s .  This  restriction  e l i m i n a t e s any c o n v e r s i o n s between c o m p r e s s i o n a l and r o t a t i o n a l motions at the l a y e r b o u n d a r i e s channels  and l i m i t s the analog to two  equivalent signal  f o r each o f the up and d o w n - t r a v e l l i n g P (or S) waveforms. The m u l t i l a y e r l i n e a r system model may  of i n d i v i d u a l l a y e r s as f o l l o w s .  be developed  On the b o u n d a r i e s  i n terms  o f any one l a y e r i ,  f o u r p o s s i b l e i n p u t - o u t p u t s i g n a l s r e p r e s e n t the up and d o w n - t r a v e l l i n g waveforms;  the s i g n a l s  waveforms to l a y e r i ,  u ^ ( t ) and d ^ ( t ) r e p r e s e n t the i n p u t d i s p l a c e m e n t  while  u  . _ , ( t ) and d, . ( t ) r e p r e s e n t the output  10  Ui_j(t)  LAYER  i-I  dj (t) LAYER i  •  J  l  ,  !  Uj(t) 2  LAYER i+l  =0  dj + |(t)  Fig. 3. A representation of the possible input and output waveforms in layer i under normal incidence.  waveforms from layer i and the equivalent input waveforms to layers i - l and i+l respectively (Figure 3). Between the boundaries, the up-and down-travelling waveforms are not coupled to each other because the propagation characteristics are assumed to be linear.  However, these  waveforms are linked at the boundaries by the standard displacement wave reflection and transmission coefficients.  The four possible input-  output signals are related through four impulse response functions, h. ( t ) , h (t), h. (t) and h (t),representing the layer. These functions l 2 3 itt must be determined from the layer parameters. Convolution of the input x  1  x  11 s i g n a l s and as  impulse  r e s p o n s e f u n c t i o n p a i r s y i e l d s the output  signals  follows:  u _ (t)=h 1  1  l i  (t)*u (t)+h i  (t)Ad (t)  i 2  i  (2-4) d  2.4  The  (t)=h  Frequency It  (t)*u.(t)+h.  (t)*d f t ) .  Domain D e s c r i p t i o n  -'  i s c o n v e n i e n t t o determine  these i n p u t - o u t p u t r e l a t i o n s i n  the f r e q u e n c y r a t h e r than t h e time domain by the L a p l a c e ( o r F o u r i e r ) t r a n s f o r m methods.  The  first  e q u a t i o n above under the L a p l a c e t r a n s -  f o r m a t i o n becomes:  D  i - l  (  s  )  =  H  i  (  s  )  i  U  1  (  s  )  +  H  i  ( 2  s  )  D  i  (  s  )  (2-5) D  In  this  transfer  1+1  13  (s)U.(s)+H  .1  r e p r e s e n t a t i o n each impulse  (s)D.(s). it* 1  response f u n c t i o n c o r r e s p o n d s t o a  function:  H  ( s ) = L{h. k  and each  (s)=H  time s i g n a l  (t)}  corresponds  U ( s ) = L{u.(t)} ±  (2-6)  \ to a transform s i g n a l :  .  (2-7)  ^ T h e e x i s t e n c e o f t h e s e c o n v o l u t i o n i n t e g r a l s demands t h a t the t r a n s f e r f u n c t i o n s H-^_(s) are s e c t i o n a l l y c o n t i n u o u s . Any p h y s i c a l l y r e a s o n a b l e choice of r e f l e c t i o n functions A^(s) w i l l  and t r a n s m i s s i o n c o e f f i c i e n t s f u l f i l l these requirements.  and the a t t e n u a t i o n  12  F i g . 4. A ray path r e p r e s e n t a t i o n of the i n t e r n a l reverberations w i t h i n l a y e r i g i v i n g r i s e to output transform s i g n a l s U ^ ^ C s ) i+i^ ^' a  n  d  D  s  T h e  transmission and r e f l e c t i o n c o e f f i c i e n t s f o r the top and bottom boundaries are t ^ and r ^ and t i ' and r ^ ' r e s p e c t i v e l y . This diagram represents the normal incidence c o n d i t i o n ; the diagram i s expanded h o r i z o n t a l l y f o r clarity.  D i r e c t c a l c u l a t i o n f o r the l a y e r t r a n s f e r f u n c t i o n i s p o s s i b l e on the b a s i s of ray theory by the f o l l o w i n g procedure.  Consider a l l  reverberations due to an input P (or S) wave w i t h i n a l a y e r i (Figure 4). The t r a n s f e r f u n c t i o n H. signal ^^_^( ) s  represents  m a  y  (s) r e l a t i n g input s i g n a l U.(s)  to output  be determined as a s e r i e s i n which the f i r s t term  the d i r e c t path transmission and each advancing term represents  an i n c r e a s i n g order of i n t e r n a l r e v e r b e r a t i o n (1,2,3,....):  13 H  (s) = t { A ( s ) e  ±  1  i  -sT. +A 1  3 J L  -3sT (s)eVr.* (2-8)  + A (s)e  -5sT  r r +...}  5  2  , 2  where the d i s p l a c e m e n t wave t r a n s m i s s i o n and r e f l e c t i o n c o e f f i c i e n t s a t the  upper and lower b o u n d a r i e s a r e r e p r e s e n t e d by t . . r . and t , ' and r . ' 1  respectively.  1  A ^ ( s ) r e p r e s e n t s an a r b i t r a r y a t t e n u a t i o n  X  1  function  complex f r e q u e n c y p e r s i n g l e t r a n s i t t h r o u g h the l a y e r and-T  of the  i s the t r a v e l  time t h r o u g h the l a y e r determined by the wave phase v e l o c i t y and l a y e r thickness.  The e x a c t l i m i t i n g sum o f t h i s s e r i e s i s :  -sT. A,(s)e H  ±  1  (s) =  *  •  1  C  —  „  ~  2 s T  i  i  • (2-9)  rr '  1-A ( s ) e 2  G(s) l-F(s)G(s)  .  (2-10)  t 1  V  -sT where G ( s ) = A ( s ) e  1  i  and F ( s )  = G ^ r ^ r'  S i m i l a r l y , the three remaining t r a n s f e r  V> H  (s) = ± 3  functions  • ' V i  G (s) l-F(s)G(s) 2  f o r layer i are:  <2  . t 'r 1  1  (  U  2  U)  _ '  1  2  )  14  H  (s) = X  *  G(s) l-F(s)G(s)  . t  .  1  (2-13)  1  These i n f i n i t e s e r i e s l i m i t i n g sums c o n s i d e r a l l p o s s i b l e i n t e r n a l b e r a t i o n and The  Laplace  rever-  ray p a t h s w i t h i n a l a y e r f o r the normal i n c i d e n c e  condition.  i n v e r s i o n o f t h e s e t r a n s f e r f u n c t i o n s to o b t a i n an  equivalent  s e t of time domain impulse response f u n c t i o n s i s d i s c u s s e d A p a r a l l e l formulation  f o r S waves r e q u i r e s o n l y the a p p r o p r i a t e  of the shear wave phase v e l o c i t y and  r e f l e c t i o n and  I.  i n Appendix choice  transmission  coefficients. The positive  G(s) analogy o f the form -, \ l-F(s)G(s)  evident  b J  feedback network (Bohn, 1963)  representation  leads  of the l i n e a r system analog  to the s i m p l e  to a f u n c t i o n a l b l o c k  f o r the l a y e r i ( F i g u r e  T h i s system e x a c t l y diagrams a l l f o u r l a y e r t r a n s f e r f u n c t i o n s . example, the i n p u t s i g n a l U\ (s) e n t e r s and  i n p u t s i g n a l and three  the e x a c t  Any  ^ ( s ) becomes the p r o d u c t of  transfer function  (s).  To  The this  S i m i l a r l y , the  number o f such networks, each r e p r e s e n t i n g  combined t o c o n s t r u c t  For  the arrows.  r e m a i n i n g t r a n s f e r f u n c t i o n s o f l a y e r i are d e s c r i b e d  diagram.  5).  the system at the lower boundary  c i r c u l a t e s t h r o u g h o u t the system as i n d i c a t e d by  r e s u l t i n g output response s i g n a l  linear  r  one  i n the  single  l a y e r can  be  a t o t a l m u l t i l a y e r c r u s t l i n e a r system network.  completely  determine the problem, i n p u t s i g n a l s and  s u r f a c e boundary c o n d i t i o n s are r e q u i r e d .  The  free  necessary input s i g n a l  c o n d i t i o n r e l a t e s the i n c i d e n t waveform coming from the basement t o  the  s i g n a l u ( t ) i n the bottom l a y e r n:  n  U  n  ( t )  =  "INPUT ^ ' V l  •  <- > 2  14  15  Uj(s)  Uj_,(s)  -0  Aj(s)e- i sT  Aj(s)e- i sT  t-l Di ( s i y  D ,(s) j +  LAYER i+l  LAYER i  LAYER i -1  F i g . 5. A f u n c t i o n a l b l o c k r e p r e s e n t a t i o n o f the l i n e a r system analog f o r l a y e r i under normal i n c i d e n c e . The heavy v e r t i c a l l i n e s denote l a y e r boundaries.  The c o r r e s p o n d i n g  V  transform  S  )  =  U  INPUT  The f r e e s u r f a c e boundary transmission  s i g n a l s are also s i m i l a r l y r e l a t e d :  coefficient  ( s ) , t :  n+l  '  (2-15)  c o n d i t i o n i s i n c l u d e d by t h e c h o i c e o f the ti=0  and the s i g n a l d i ( t ) = 0 f o r l a y e r 1.  These  16  two  c o n d i t i o n s imply t h a t no wavemotion energy i s l o s t o r g a i n e d a t the  free surface  boundary.  F o r the case o f n l a y e r s above a basement h a l f s p a c e , a s e t o f 2n complex  l i n e a r s i m u l t a n e o u s e q u a t i o n s i n terms o f the i n d i v i d u a l  layer  t r a n s f e r f u n c t i o n s e t s and the s i g n a l s t r a n s f o r m s r e p r e s e n t s the system:  U!(s) = H  2  U (s) = H  3  2  D (s)  1  (s)U (s)+H  2  (s)U (s)+H  3  2  3  1  = Hi  2  2  2  (s)D (s) 2  (s)D (s) 3  (s)U!(s) 3  U (s) = H 3  t t i  (s)U (s)+H I +  l t 2  (s)D (s)  (2-16)  4  D (s) = H , ( s ) U ,(s)+H ,(s)D (s) n n-l n-l n-14 n-l 3  U  n  ( s )  V l  (  W  =  s  )  U  I N P U T  V  =  ( S ) D  n  (  8  ( s ) + H  )  n  ( s ) U 3  n  ( s ) + r  n l- INPUT U  ( s )  +  The s o l u t i o n o f the problem r e q u i r e s the e v a l u a t i o n of each U^(s) D^(s).  and  Then, the d e s i r e d s u r f a c e v e r t i c a l d i s p l a c e m e n t amplitude t r a n s -  form W (s) Q  i s r e l a t e d t o the s i g n a l U j ( s ) by  W (s) Q  = H (s)U!(s) o  (2-17)  17 H (S)  may  q  Figure  5  be o b t a i n e d from the b l o c k diagram noting that  )  t i = 0 ,  s i g n a l , W ( S ) , i s the sum  r j = - l and  of s i g n a l s  Q  D  The  ( s ) = 0 .  1  due  s u r f a c e displacement  to the i n c i d e n t and  waveforms a t the f r e e s u r f a c e boundary. t h e two  f o r layer 1 (refer to  reflected  S i n c e the r e f l e c t i o n  is  -1,  is  twice the i n c i d e n t s i g n a l displacement  coefficient  s i g n a l s have o p p o s i t e s i g n s and d i r e c t i o n s so t h a t amplitude.  the  sum  This incident  -sTi s i g n a l i s formed by U j ( s ) which has been d e l a y e d attenuated  ( m u l t i p l i c a t i o n by A i ( s ) ) and  ( m u l t i p l i c a t i o n by e  reverberated (divided  l  ),  by  —2sT 1+A (s)r e  )  2  1  l  1  so  that: 2Ai(s)e" H  0  <  s )  s T l  •  =  l+A  2 1  (s)r e  ( 2  ~  1 8 )  _ 2 s T  1  In cases i n v o l v i n g o n l y one  o r two  l a y e r s , a s o l u t i o n by  direct  a l g e b r a i c m a n i p u l a t i o n s on the m a t r i x o f the f u n c t i o n a l c o e f f i c i e n t s i s possible.  However, f o r many l a y e r s , the a n a l y t i c a l s o l u t i o n s f o r the  t r a n s f o r m s u r f a c e motion  and  i n v e r s i o n t o g i v e the output feasible.  time domain response  I n l i e u o f the a n a l y t i c a l approach,  simultaneous from  r e f l e c t e d s i g n a l s and  e q u a t i o n s e t a t incremented  T h i s more l i m i t e d computation  f u n c t i o n s i s not  F o u r i e r frequencies (obtained be  accomplished  to g i v e  f u n c t i o n f o r the m u l t i l a y e r e d medium.  a n a l y s i s corresponds  f o r s p e c t r a l responses.  w i l l be p r e s e n t e d i n a l a t e r  Laplace  n u m e r i c a l s o l u t i o n s o f the  the L a p l a c e f r e q u e n c i e s by s e t t i n g s=ico) may  a s t e a d y s t a t e f r e q u e n c y response  the subsequent  chapter.  to the H a s k e l l m a t r i x  numerical  R e s u l t s o f such n u m e r i c a l a n a l y s e s  18 2.5  The L i n e a r System Analog The  f o r Non-normal I n c i d e n c e  f o r m u l a t i o n f o r non-normal i n c i d e n c e p l a n e waveforms  analogously to that  f o r normal i n c i d e n c e .  a r i s e s i n the problem  proceeds  However, immediate c o m p l i c a t i o n  o f h a n d l i n g c o n v e r s i o n s between c o m p r e s s i o n a l  r o t a t i o n a l motions which o c c u r a t each c o n v e r s i o n s o c c u r , the independent  l a y e r boundary.  Because  and  such  P and SV motions must be c o n s i d e r e d  s e p a r a t e l y r e q u i r i n g a f o u r - c h a n n e l r a t h e r than a two-channel a n a l o g approach  (Figure 8). F o r normal i n c i d e n c e , the waveform r a y paths  directly  t o the p o s i t i v e and n e g a t i v e c a r t e s i a n z d i r e c t i o n s . , b u t c o n d i t i o n s , t h e r a y paths Consequently,  determine  f o r non-normal  are n o t c o i n c i d e n t w i t h the c o o r d i n a t e axes.  i t i s c o n v e n i e n t t o c r e a t e a s e t of p s e u d o - c o o r d i n a t e s  which the s e p a r a t e P-and S-wave motions e f f e c t i v e l y vector pair  (Figure 6).  comprise  the r e q u i r e d p s e u d o - c o o r d i n a t e  I t i s evident that a d i f f e r e n t  layer.  the o r t h o g o n a l  s e t and a r e r e l a t e d t o the the P and  S  t r a n s f o r m a t i o n i s r e q u i r e d f o r every  W i t h h o r i z o n t a l l a y e r i n g , the c o m p r e s s i o n a l and s h e a r m o t i o n s  throughout velocity  in  I n each l a y e r , the d i r e c t i o n s a l o n g the r a y paths  c a r t e s i a n c o o r d i n a t e s by the a n g l e s between the n o r m a l and rays.  correspond  the medium p o s s e s s a c o n s t a n t apparent  c determined  by  the i n p u t c o n d i t i o n s .  h o r i z o n t a l phase  F o r example, f o r an  i n p u t P-waveform from the basement:  c = cx  where  n + 1  /sin(e  n + 1  )  (2-19)  i s the P r a y normal a n g l e and ° < ^ i s the P phase v e l o c i t y i n  the basement.  n+  S n e l l ' s law then d e t e r m i n e s  the P and  S ray a n g l e s w i t h  the  n o r m a l i n any o t h e r l a y e r i : c = a / s i n ( e ) = Bj/sinU ) i  i  .  (2-20)  19 Up- and d o w n - t r a v e l l i n g waveforms are thus represented by the p o s i t i v e and negative d i r e c t i o n s of the two axes.  The four p o s s i b l e d i r e c t i o n s on the  two pseudo-coordinate axes determine the four required s i g n a l channels.  F i g . 6 . A r e p r e s e n t a t i o n of the "pseudo-coordinate" transformation f o r layer i . The arrows on the rays of the c a r t e s i a n r e p r e s e n t a t i o n show the p o s i t i v e displacement conventions used throughout t h i s work. The pseudo-coordinate r e p r e s e n t a t i o n i s f o r l a y e r i only. The second diagram shows the P and SV displacements i n l a y e r i transformed i n t o the pseudo-coordinate r e p r e s e n t a t i o n .  20  Z  NODES OF REFRACTED WAVE  NODES OF INCIDENT  WAVE  F i g . 7. A diagram showing t h e n o n - c o i n c i d e n c e o f wave c o n s t a n t phase p o s i t i o n s ( e . g . nodes) and a h o r i z o n t a l l a y e r boundary.  A further complication  a r i s e s f o r non-normal i n c i d e n c e  because  the phase o f a p l a n e wave i s n o t c o n s t a n t a l o n g a boundary o r a l o n g any surface  o f c o n s t a n t depth ( F i g u r e  o f phase advance  (negative  ST .  F o r each l a y e r i , the i n t r o d u c t i o n  time d e l a y ) t r a n s f e r f u n c t i o n s  w i t h the forms  ST .  S  e  7).  P  f o r SV motions and e  f o r P motions (Figure  8) t o t h e l i n e a r  system analog r e - e s t a b l i s h the c o r r e c t phase r e l a t i o n s f o r t h e s i g n a l s a t the b o u n d a r i e s .  F o r a d e s c r i p t i o n and c a l c u l a t i o n o f t h e s e phase  advance t r a n s f e r f u n c t i o n s Just  see Appendix I I .  as each l a y e r i n t h e normal i n c i d e n c e  i z e d by two e q u a t i o n s i n l a y e r t r a n s f e r  functions  problem was  character-  and s i g n a l s , each  layer  /-vST*l-\ .  e  Pj  A .(s)e p  s T  Pj  Fig. 8 . A functional block representation for non-normal incidence for layer i . Note that four channels are shown, two for the up- and down-travelling P-waves and two for the up- and downtravelling SV waves.  22  in  the non-normal problem i s represented by four such equations;  the  transfer function r e l a t i o n s for a l a y e r i i n a matrix form i s :  i-l  u  i - l  v  D  1 + 1  E  i + 1  (  s  (  "  )  s  H  ±  2  •  )  U (s)  (s)H (s)H (s)H (s)" i 13 lu  lj  #  •  •  V (s) ±  • (a)  *  a  •  (s)  *  •  •  where the transforms U\(s) and D^(s)  D (s)  •  H.  (s)  correspond  E (s) i  to the up- and down-  t r a v e l l i n g P waves, the V^(s) and E^(s) correspond t r a v e l l i n g SV waves and the H  (2-21)  i  to the up- and down-  (s) are the t r a n s f e r function s e t f o r the  •Me layer.  A d i r e c t algebraic determination of these transfer functions i s  impractical.  However, i t i s p o s s i b l e to construct d i r e c t l y the l a y e r  l i n e a r systems block diagram representation (Figure 8) by analogy to the normal incidence system and s i m p l i f y i t using "block diagram reduction procedures"  (Kuo, 1962).  For frequency  One example reduction i s performed i n Appendix I I I .  component a n a l y s i s , such reductions are best accomplished  during numerical  analysis;  f o r time domain a n a l y s i s , reductions are  unnecessary. The equivalent l i n e a r system block representation of one l a y e r for  non-normal ray conditions contains four channels interconnected at the  boundaries by the conversion r e f l e c t i o n and transmission c o e f f i c i e n t s c a l c u l a b l e using the Zoeppritz' displacement  r e l a t i o n s (Richter, 1958).  For ray angles less than the c r i t i c a l angle, the s i g n of the r e a l valued Zoeppritz' c o e f f i c i e n t s represents phase changes at r e f r a c t i o n s and  reflections.  S u p e r c r i t i c a l a n g l e s between t h e r a y d i r e c t i o n s and the  boundary normal a n g l e s coefficients  determine complex t r a n s m i s s i o n  and r e f l e c t i o n  r e p r e s e n t i n g phase changes o t h e r than 0 and TT.  system approach d e s c r i b e d such s u p e r c r i t i c a l  The l i n e a r  i n t h i s t h e s i s has n o t been extended t o p e r m i t  conditions.  Thus, t r a p p e d  wave phenomena a r e n o t  described. I n t h e f u n c t i o n a l b l o c k diagram o f F i g u r e 8, t h e s u b s c r i p t i refers  t o the l a y e r i .  separate  The s i n g l e s u b s c r i p t s p and s d e s i g n a t e the  c h a n n e l s f o r the P and SV waveforms;  each f u n c t i o n a l b l o c k so  i n d i c a t e d p e r t a i n s only to the corresponding  wave t y p e .  A  the a t t e n u a t i o n o f a P wave  (s) i s the t r a n s f e r f u n c t i o n representing  d u r i n g one t r a n s i t  t h r o u g h l a y e r i . The double s u b s c r i p t s o f t h e  Zoeppritz' coefficients c a t e d by t h e f i r s t F o r example, t sp conversion  F o r example,  i n d i c a t e conversion  from t h e waveform type  s u b s c r i p t t o the waveform type i s the transmission  i n d i c a t e d by t h e second.  c o e f f i c i e n t which determines t h e  ±  o f S waves from l a y e r i i n t o P waves i n l a y e r i - l ;  the r e f l e c t i o n  r . is PP i  c o e f f i c i e n t which determines t h e P wave t o P wave r e f l e c t i o n  a t t h e lower boundary o f t h e l a y e r i .  The waveform types  c h a n n e l s a r e i n d i c a t e d by P and S symbols. definitions  indi-  of the normal i n c i d e n c e  f o r the s e p a r a t e  A l l o t h e r symbols r e t a i n t h e  case.  An n - l a y e r e d non-normal i n c i d e n c e problem r e q u i r e s the s o l u t i o n of 4n complex l i n e a r s i m u l t a n e o u s e q u a t i o n s discussed  i n the n o r m a l i n c i d e n c e problem:  formed s i m i l a r l y  t o those,  24  Ui(s) = H  V (s)  2  U (s)+H 2  1  = H  2  D (s) = H  :  E (s) = H  x  13  U (s) = H  3  1  V (s) = H  3  1  2  2  2  2  5 9  V (s)+H  2  U (s)+H 2  0  2  D (s)+H  3  2  2  E (s) h 2  V ( s ) + H D (s)+H E ( s ) 6 7 8  2  2  U (s)+H, 1  2  2  2  2  2  Vj(s)  10  U (s)+H 1  U (s)+H 2  3  1  14  V^s)  V (s)+H 2  2  3  3  D (s)+H 2  E (s)  3  2  U (s)+H V ( s ) + H D (s)+H E ( s ) 5 6 7 8 2  3  2  D (s) =H U(s)+H n n-lg n  n - 1  3  V  lo  n  1  2  3  (s)+H n  -  ±  l l  n  2  -  D  (s)+H ~M2  i  n  n  _  E 1  (s)  E (s) = H . U ,(s)+H V ,(s)+H . D (s)+H E (s) n n-1^3 n - l n— 1 ^ n - l n—±15 n - l n-l^g n - l  V  * %p  s )  U n+1  INPUT  (s)  (2-22) V>  =  S  D  'ps^INPUT^  .(s) = H  n+l  +  E  r  U (s)+H  ng n  pp  V (s)+H  n^g n  U n + 1  INPUT  . l l n  D (s)+H n  n  E (s)  l 2  n  ( s )  . (s) = H U (s)+H .V (s)+H D (s)+H E (s) n+1 iij3 n n ^ n 1115 n n 15 n  +  r  ps  U n + 1  INPUT( ) S  25  The the  necessary free surface  free surface  Zoeppritz'  boundary c o n d i t i o n s  r e f l e c t i o n c o e f f i c i e n t s and  t h a t no wavemotions e n t e r from the vacuum h a l f s p a c e The  required  t o the  input  example, the are:  input  conditions  s i g n a l s at the  r e l a t e the  are the  (i.e.  included  requirement (s)=E (s)=0). 1  i n c i d e n t s i g n a l from the basement  lower boundary o f the bottom l a y e r n.  input  condition  V  - Sp  by  t r a n s f o r m s generated by  an i n p u t  For  SV wave  ™  S  )  - INPUT V  n+1  ( S )  ....  (2-23) V  S  )  A solution requires  • Ss  - INPUT V  n+1  evaluation  which a r e  u n s p e c i f i e d by  following  functional operation  on the  ( S )  of a l l s i g n a l s U ^ ( s ) ,  e i t h e r the boundary or i n p u t t h e n r e l a t e s the P and  free surface  boundary to the  s i g n a l s U^(s)  the s o l u t i o n o f the  4n s i m u l t a n e o u s  equations:  U  o  (s) = H  oj  (s)U (s)+H 1  °2  and  D^(s),  V^(s)  and  conditions.  A  SV motions V^(s)  E^(s)  incident  o b t a i n e d from  (s)V (s) 1  (2-24) V  where U of H  o  o  (s) = H  O3  (s) i s the P and  (s) are o b t a i n e d by  V  (s)U (s)+H 1  o  1  (s) the  reduction  rt  Oi+  SV  (s)V (s) 1  1  surface  i n c i d e n t motions.  procedures on  the  The  set  functional block  T2] I f S waves w i t h a r b i t r a r y p o l a r i z a t i o n determine the i n c i d e n t waveform, the"two d i m e n s i o n a l a n a l y s i s i s not s t r i c t l y v a l i d . However, any S motion can be s e p a r a t e d i n t o SV ( v e r t i c a l p o l a r i z a t i o n ) and SH ( h o r i z o n t a l p o l a r i z a t i o n ) components. The response to each component can be c a l c u l a t e d separately.  26  F i g . 9. The f r e e s u r f a c e r e f l e c t i o n f o r non-normal i n c i d e n c e . U and V are t h e i n c i d e n t P- and S-waves and and V R a r e the r e f l e c t e d P- and S-waves. Q  representation  of layer 1 considering  the f r e e s u r f a c e  i n a manner s i m i l a r to t h e normal i n c i d e n c e  case.  boundary  Then, t h e  conditions  surface  m o t i o n t r a n s f o r m s , v e r t i c a l component W ( S ) and h o r i z o n t a l component Q  Y (S), Q  are o b t a i n e d by r e - c o m b i n a t i o n o f the d i s p l a c e m e n t components i n the c a r t e s i a n z and x d i r e c t i o n s due to the P and S V waveforms i n c i d e n t on and  r e f l e c t e d from the f r e e s u r f a c e  ( F i g u r e 9).  The r e f l e c t e d t r a n s f o r m  ' s i g n a l s , U (s) and V ( s ) , a r e r e l a t e d t o the s u r f a c e the Z o e p p r i t z '  free surface  reflection coefficients:  Q  i n c i d e n t s i g n a l s by  27  spi  (2-25)  The  s u r f a c e displacement amplitude c o e f f i c i e n t s  are t h e n :  W (s)=[U (s)+U (s)]cos(e )+[V (s)+V (s)]sin(f ) o  o  R  1  o  R  1  (2-26) Y (s)=[U (s)-U (s)]sin(e )-[V (s)-V (s)]cos(f ) o  o  R  1  Because the a l g e b r a i c forms complex, even the s i m p l e one analytically.  o  R  1  of the t r a n s f e r f u n c t i o n s are so  l a y e r system i s n o t r e a d i l y  A g a i n , n u m e r i c a l methods w i t h incremented  soluble r e a l frequencies  a r e most e a s i l y u s e d . t o d e t e r m i n e the s p e c t r a l r e s p o n s e of the medium.  2.6  Time Domain F o r m u l a t i o n A development  o f the time domain systems  d e r i v e d from the p a r a l l e l f r e q u e n c y d e s c r i p t i o n conditions.  f o r normal  now  incidence  The g e n e r a l i z a t i o n t o p e r m i t non-normal i n c i d e n c e i s n o t  d e s c r i b e d e x p l i c i t l y , but t h e complete t r a n s f o r m approach d e s c r i b e d e a r l i e r system  description i s  response c h a r a c t e r i s t i c s .  methods can be l e s s e f f i c i e n t requirements) i f l i t t l e  system a n a l o g i s p r e s e n t e d .  The  i s o f t e n used i n the a n a l y s i s ^ o f  However, f o r d i g i t a l  ( i n terms  analyses, transform  o f computer time and s t o r a g e  convolution i s necessary f o r a d i r e c t  time domain  solution. Two exist.  First,  the s i g n a l  direct  approaches  the system  t o a time domain l i n e a r systems  solution  o f l i n e a r s i m u l t a n e o u s e q u a t i o n s i n terms  t r a n s f o r m s and l a y e r t r a n s f e r  functions  ( E q u a t i o n s 2-16,  of 2-22)  can be  t r a n s f o r m e d i n t o the e q u i v a l e n t  integrals. standard digital has  system of l i n k e d  T h i s system of e q u a t i o n s i n c o n v o l u t i o n s  convolution  may  be  solved  numerical procedures f o r s o l u t i o n of i n t e g r a l equations computers.  ( e . g . , Carnahan e t a l , 1969).  i n the g e n e r a l  non-normal c a s e .  the s i x t e e n r e q u i r e d  ( F i g u r e 10). reflection  The  convolutions  at the  two  d i g i t a l methods r e q u i r e s  which n e c e s s i t a t e s  the  finite  l o n g as the time r e q u i r e d  at l e a s t as  through the l a y e r i n o r d e r  each l a y e r a r e w e l l c h a r a c t e r i z e d . l a y e r i m p u l s e response f u n c t i o n s  the a t t e n d i n g  that  the  (and  preferably infinite  truncated  5 and  8.  ( F i g u r e 10). the l a y e r  from the  responses  properties  of the  This  of  broadening  Q. more e f f i c i e n t  and  i n terms o f  a n a l o g y to the f r e q u e n c y d e s c r i p t i o n s  Each f r e q u e n c y domain system b l o c k  c o r r e s p o n d s to an e q u i v a l e n t  duration  s i m p l e i m p u l s e t r a i n s but  used i n t h i s work a c c o m p l i s h e s a time domain f o r m u l a t i o n  Figures  short)  the p r e f e r e n t i a l low-pass n a t u r e  dispersion  by  com-  i n c l u s i o n of attenuation  second approach which i s c o m p u t a t i o n a l l y  representations  the  f o r s e v e r a l waveform  longer  i n c r e a s e s w i t h time at a r a t e d e t e r m i n e d by  system b l o c k  time  impulse  reverberation  With the  a r e no  r a t h e r each impulse i s broadened by  The  infinite  Efficient  t r u n c a t i o n of the  must be  and  c o n d i t i o n s , each o f  l a y e r boundaries.  However, f o r a c c u r a c y , the  attenuation  convolutions  impulse amplitudes w i t h i n c r e a s i n g  impulse r e s p o n s e s .  transits  approach  r a t e o f the d e c r e a s e o f a m p l i t u d e depends upon  coefficients  putation using  For non-attenuating  impulse r e s p o n s e s f o r each l a y e r i s an  impulse t r a i n w i t h d e c r e a s i n g  using  However, the  some drawbacks as each l a y e r r e q u i r e s s i x t e e n e q u i v a l e n t  by  transfer  time f u n c t i o n w h i c h can be  f r e q u e n c y f u n c t i o n through the  function  directly  obtained  inverse Laplace transform.  time domain system d e s c r i p t i o n s o f F i g u r e s  11  and  12 have been  of  The  obtained  29  IMPULSE  RESPONSE  Q=00  Q<00  F i g . 10. A comparison o f a n o r m a l i n c i d e n c e l a y e r i m p u l s e response f o r i n f i n i t e Q and c a u s a l l y .absorbing c o n d i t i o n s .  i n t h i s way.  F o r example, f o r l a y e r i a t normal i n c i d e n c e _  domain d e l a y  and a b s o r p t i o n  T  function 5(t-T )*a^(t)  the e q u i v a l e n t  the frequency  i  t r a n s f e r function a^(s)e  i n v e r s e L a p l a c e t r a n s f o r m e d to g i v e absorption  s  ( F i g u r e 5) i s  time domain d e l a y and  ( F i g u r e 11) where a ^ ( t )  d e s c r i b e s the  impulse response o f the l a y e r p e r s i n g l e t r a n s i t d e t e r m i n e d by t h e l a y e r  30  LAYER  i+ l  LAYER i  LAYER  i-l  F i g . 11. The time-domain r e p r e s e n t a t i o n o f the system a n a l o g o f one l a y e r a t normal i n c i d e n c e . Compare to t h e f r e q u e n c y r e p r e s e n t a t i o n o f F i g u r e 5.  absorption properties. later i n this  (The form o f t h i s a b s o r p t i o n  function i s discussed  chapter).  To d e s c r i b e  the e s s e n t i a l n a t u r e o f the time domain d e s c r i p t i o n ,  the normal i n c i d e n c e f o r m u l a t i o n At. the lower boundary T h i s waveform combines  ( F i g u r e 11)  i s now  f u r t h e r demonstrated.  o f l a y e r i , a waveform u ^ ( t ) e n t e r s  the system.  by a d d i t i o n to any waveform u ( t ) which  results  Fig. 12. The time-domain representation of the system analog of one layer for non-normal incidence. Compare to Figure 8.  32  from the r e f l e c t i o n of any u ^ ( t ) i s the output  d o w n - t r a v e l l i n g wave at the lower boundary.  s i g n a l from the system b l o c k r ' which  the l o w e r boundary r e f l e c t i o n c o e f f i c i e n t . delayed  by  the d e l a y o p e r a t o r  6 ( t - T ^ ) and  The  sum  represents  waveform i s then  absorbed a c c o r d i n g  f u n c t i o n a ^ ( t ) d u r i n g i t s t r a n s i t to the top boundary so t h a t  to  the  the  e q u i v a l e n t waveform i n c i d e n t on the top boundary i s :  y i  ( t ) = (u (t)+u (t))*6(t-T )*a (t) 1  r  1  i  (2-27) =  (u (t-T )+u (t-T ))*a (t) i  I f the l a y e r p o s s e s s e s no  The  internal  .  i  attenuation:  1  ,  y (t)  =  u^t-T^+u^t-T^  system b l o c k s  r e f l e c t i o n and  i  =  ±  remaining  r  a (t) t  and  i  (2-28) .  are m e r e l y c o n s t a n t s  (2-29)  d e s c r i b i n g the  other  t r a n s m i s s i o n c o e f f i c i e n t s at the l a y e r b o u n d a r i e s and  i d e n t i c a l to those  o f the f r e q u e n c y  domain system.  F o r example,  are  the  waveform y ( t ) i n c i d e n t at the top boundary o f the l a y e r i s p a r t i a l l y i  t r a n s m i t t e d i n t o the next l a y e r through the t r a n s m i s s i o n t o form ^ _ 2 ^ ^ u  coefficient  t^  :  U  i - l  (  t  )  =  y^(t) i s also p a r t i a l l y  V  Y  i  (  t  )  *  ( 2  ~  3 0 )  r e f l e c t e d back i n t o l a y e r i to form d ^ ( t ) = £ y j_( )  which combines by a d d i t i o n to any  r  ,  t  ;  input down-travelling  waveform d ^ ( t ) .  33  T h i s combined waveform i s d e l a y e d  and  absorbed d u r i n g i t s t r a n s i t  to  the  lower boundary to form:  x (t) =  (d '(t-T )+d (t-T ))*a (t)  ±  x^(t) i s p a r t i a l l y  i  1  reflected  which a g a i n combines by  r  are n e c e s s a r y  non-normal i n c i d e n c e .  two  l a y e r at  convolutions with  f o r each l a y e r ;  (2-31)  the  simplifies  t o the t r a n s i t  the  attenuation  f o u r are r e q u i r e d f o r  I f a t t e n u a t i o n i s not p r e s e n t ,  the form o f  to mere d e l a y s o f the i n p u t waveforms.  the p a s t h i s t o r i e s of the i n p u t f u n c t i o n must be  Two  ~  <  delay operator  equal  .  a d d i t i o n to u ^ ( t ) e n t e r i n g the  F o r normal i n c i d e n c e , o n l y  convolutions  i  at the lower boundary to form u ^ ( t ) = r^'»x^(t)  lower boundary.  and  1  these  Then o n l y  retained f o r a duration  time.  l i n k e d i n t e g r a l equations  thus d e s c r i b e the normal l a y e r  responses:  U  i - l  (  t  )  =  t  i  [  u  i  (  t  _  T  i  )  v  +  t  , i '  d i + 1  (t)]*a.(t) (2-32)  i  r  d.  + 1  ( t ) = t.'[d (t-T.)+ i  a s i m i l a r set of four equations  u._ (t)]*a (t) 1  d e s c r i b e the non-normal  F o r d i g i t a l time s i m u l a t i o n s , each o p e r a t o r the system model must be solutions  i  time i n c r e m e n t e d .  ( s i m u l a t i o n s ) of the n - l a y e r e d  waveform, a s e t of 2n l i n k e d e q u a t i o n s  and  ;  case. signal within  Then, f o r time i n c r e m e n t e d  c r u s t response to an  input  of the above form must be  solved.  34  (4n such e q u a t i o n s  d e s c r i b e non-normal i n c i d e n c e ) .  f o r m u l a t i o n , the necessary  As i n the frequency  i n p u t c o n d i t i o n s a r e d e t e r m i n e d by t h e i n p u t  s i g n a l a t t h e basement boundary;  the c o r r e c t f r e e s u r f a c e  c o e f f i c i e n t s determine the r e q u i r e d boundary c o n d i t i o n s . s i m u l a t i o n , i t i s important  t o choose t h e c o n s t a n t  s m a l l enough as t o p r e v e n t  aliasing  reflection For the d i g i t a l  time i n c r e m e n t t o be  o f the system f r e q u e n c y  response o r  t h e s p e c t r a l c h a r a c t e r o f t h e i n p u t waveform due t o s a m p l i n g .  2.7  Attenuation  and D i s p e r s i o n i n t h e System Model  E l a s t i c wave a t t e n u a t i o n i n an a b s o r b i n g described constant  medium i s u s u a l l y  i n terms o f t h e m a t e r i a l Q f a c t o r (Knopoff, at a l l frequencies, a s p a t i a l absorption  determines the decreasing  of frequency  (2-33)  absorption  to hold while  c i s independent  a c a u s a l a t t e n u a t i o n law r e s u l t s .  (1962) extended t h i s d e s c r i p t i o n o f a t t e n u a t i o n t o a l l o w causality  Futterman  the p r i n c i p l e o f  r e t a i n i n g the e s s e n t i a l l i n e a r nature  (Q remains, c o n s t a n t  distance:  .  f u n c t i o n i s r e a l and t h e phase v e l o c i t y  a non-physical  For a Q  f a c t o r a(w) = £Q^"  a m p l i t u d e o f an harmonic wave w i t h  U(x,co) = U(o,co)exp(-a(to)x)  I f the absorption  1964).  of the  i n f r e q u e n c y ) and wave motion i n o r d e r  t h a t s u p e r p o s i t i o n remains v a l i d .  Subsequent a n a l y s e s  demonstrated t h a t Futterman's t h e o r y w e l l d e s c r i b e d  by Wuenschel  (1965)  attenuation i n r e a l  materials. Under t h e c o n d i t i o n o f c a u s a l a t t e n u a t i o n , a p l a n e wave t r a v e l l e d t h r o u g h d i s t a n c e x can be r e p r e s e n t e d i n t e g r a l i n terms o f t h e o r i g i n F o u r i e r  by a s u p e r p o s i t i o n  coefficients:  having  35  f  u(x,t) =  U(o,w)exp(i[K(u>)x-ut])du>  ,  r 31 1  J  (2-34) where K(oi) = k(a))+ia(u)) . K(oj) a l s o d e s c r i b e s Futterman's model  the d i s p e r s i o n p r o p e r t i e s o f t h e m a t e r i a l . which i s used i n l a t e r  ( A 1 , D 1 )  For  a n a l y s e s K(w) has t h e  form:  o (2-35)  K(w) = •<  to  tii  c  where tu^ i s an a r b i t r a r y considered  c u t o f f angular  t o be i n f i n i t e .  frequency  £  CJ  below which Q can be  F o r f r e q u e n c i e s w e l l above w^,  Futterman  shows t h a t the d i s p e r s e d wave phase V(OJ) and group u(to) v e l o c i t i e s a r e r e l a t e d t o the l i m i t i n g undispersed  v(«o) S u(u) S c ( l -  c, thus d e s c r i b e s condition.  phase v e l o c i t y  lnC^-))"  a lower v e l o c i t y l i m i t  The a b s o r p t i o n  c by:  (2-36)  1  on an i n v e r s e d i s p e r s i o n  t r a n s f e r f u n c t i o n s and impulse  responses  which a r e r e q u i r e d by the l i n e a r systems f o r m u l a t i o n a r e d e s c r i b e d by t h i s a t t e n u a t i o n model.  F o r t h e time domain system:  •[.3] The i n t e g r a l r e p r e s e n t a t i o n f o r F o u r i e r s u p e r p o s i t i o n i n t h i s case i s t h a t n o r m a l l y used i n wave p r o p a g a t i o n problems. Throughout the r e s t o f t h i s t h e s i s , t h e communications theory c o n v e n t i o n i s used.  36  a (t) ±  = | exp(i[K ((o)n -ut])du i  (2-37)  i  —CO  where  i s the thickness of l a y e r i and K^(OJ) the complex  propagation  function determined by the layer Q, phase v e l o c i t y and the cutoff frequency O) . q  This a^(t) i s the Futterman model impulse response ( i . e .  U(o,co)=l) of the l a y e r . frequency  The equivalent frequency  d e s c r i p t i o n ( f o r the  domain l i n e a r system analog) i n Fourier terms i s :  A (a>) = exp(iK (o))n ) i  i  .  .  i  <" > 2  38  I f the causal property f o r attenuation i s not required ( i . e . no phase information i s desired) then the Futterman absorption function can be s i m p l i f i e d to the usual acausal form:  -uT. A (co) - e x p ( - ^ )  (2-39)  i  where  i s the layer Q value and T^ the layer t r a n s i t time. I t should be noted that the Futterman analysis of attenuation  requires cu^ and c.  In p r a c t i c e , however, the dispersed phase v e l o c i t y  v(to) and group v e l o c i t y u(to) are the observable c must be obtained by c a l c u l a t i o n .  quantities while O)  q  and  Because the form of the absorption  impulse response function a^(t) i s only weakly dependent upon the actual cutoff frequency,  can be chosen a r b i t r a r i l y to be very small compared  to the frequencies of i n t e r e s t . ^  Futterman's choice of to =10" o  3  sec  - 1  has  been used i n the computation of the absorption functions used l a t e r i n t h i s thesis.  37  In the p r e c e d i n g d i s c u s s i o n , A_^(a)), and the a b s o r p t i o n  function, either  I t i s o n l y n e c e s s a r y to s p e c i f y the  time and Q c o r r e s p o n d i n g to the wave type t o determine .  the frequency o r time f u n c t i o n The  transfer  time f u n c t i o n , a_^(t), can r e p r e s e n t  P-wave o r S-wave a t t e n u a t i o n . particular transit  the a b s o r p t i o n  desired.  transmission-reflection conditions  a t a f l a t boundary between  two  c a u s a l l y a b s o r b i n g media a r e a f f e c t e d by the c o i n c i d e n t  the  e l a s t i c waves.  not  n e c e s s a r i l y constant at a l l frequencies,  dispersion of  Because the v e l o c i t y r a t i o s between t h e two media a r e  observes t r a n s m i s s i o n  each F o u r i e r  component  and r e f l e c t i o n c o e f f i c i e n t s which a r e d i f f e r e n t from  the n o n - d i s p e r s e d case.  The v e l o c i t y r a t i o s a l s o determine the r a y  d i r e c t i o n s o f the t r a n s m i t t e d  f r e q u e n c y components by S n e l l ' s Law.  I f the  r a t i o s a r e n o t c o n s t a n t , the t r a n s m i t t e d  frequency component r a y d i r e c t i o n s  are n o t a l l c o i n c i d e n t  condition on.dispersion  and a g e o m e t r i c a l  In a f r e q u e n c y domain a n a l y s i s , t h i s d i s p e r s i o n p r o p e r t y problem because t r a n s m i s s i o n  and r e f l e c t i o n  i s n o t a major  c o e f f i c i e n t s and r a y d i r e c t i o n s  can be computed f o r each F o u r i e r component frequency. domain approach, the s e p a r a t i o n  results.  However, i n a time  o f frequency components cannot be accomp- •  l i s h e d and the method becomes an a p p r o x i m a t i o n o f t h e t r u e s o l u t i o n . Wuenschel's  (1965) a n a l y s i s o f a t t e n u a t i o n  Colorado shales  and d i s p e r s i o n i n t h e P i e r r e  suggests t h a t t h e frequency dependence o f t h e d i s p e r s e d  phase v e l o c i t y i s s m a l l over f r e q u e n c y ranges o f s e v e r a l o c t a v e s . :  Con-  s e q u e n t l y , a c o n s t a n t t r a n s m i s s i o n - r e f l e c t i o n c o e f f i c i e n t and r a y d i r e c t i o n a p p r o x i m a t i o n appears t o be v a l i d analyses.'  and i s assumed i n t h e f o l l o w i n g •  38  CHAPTER I I I ANALYSIS OF THE ALBERTA SEISMOGRAMS  3.1  Introduction In  o r d e r t o demonstrate  t h e unique  f e a t u r e s and u t i l i t y  of this  t h e o r y , t h e r e s u l t s o f a n a l y s e s o f seismograms r e c o r d e d i n c e n t r a l  Alberta  w i l l be compared t o l i n e a r systems s o l u t i o n s i n b o t h time and f r e q u e n c y domains.  The e f f e c t s o f c r u s t a l a t t e n u a t i o n on s p e c t r a l r a t i o s w i l l be  shown t o be s i g n i f i c a n t  f o r the low Q v a l u e s which a r e thought t o  c h a r a c t e r i z e c r u s t a l sediments.  I t w i l l be observed t h a t the c o r r e l a t i o n 4  between experiment  and t h e o r y i s b e s t at f r e q u e n c i e s below 1 Hz.  domain s y n t h e t i c seismograms w i l l approach  Time  a l s o be c a l c u l a t e d by the systems  and compared w i t h e x p e r i m e n t a l r e c o r d i n g s .  The e f f e c t s o f  c r u s t a l a t t e n u a t i o n and the assumed form of the i n c i d e n t w a v e l e t w i l l be shown.  A l t h o u g h e x a c t syntheses have n o t been o b t a i n e d , s i g n i f i c a n t  s i m i l a r i t y w i t h experiment w i l l be e v i d e n t . The p r i m a r y aim o f t h e t h e s i s has n o t been to determine the A l b e r t a c r u s t , b u t r a t h e r t o develop the new l i n e a r systems t e c h n i q u e and demonstrate  i t s particular usefulness.  Thus, the n a t u r e of each o f t h e '  f o l l o w i n g e x p e r i m e n t a l a n a l y s e s i s b a s i c a l l y one of example.  3.2  The A n a l y s i s The  Problem  c h a r a c t e r o f the P coda o f a seismogram depends upon:  a) t h e s o u r c e generated waveform, b) the p h y s i c a l p r o p e r t i e s o f the s o u r c e t o r e c e i v e r paths such as Q, mantle  l a y e r i n g , and the n e a r  r e c e i v e r c r u s t a l s t r u c t u r e and c) the response  c h a r a c t e r i s t i c s of the  39  recording instruments.  This l a s t effect  c a l i b r a t i o n o f the seismograph system.  can be w e l l determined by the The combination o f the s o u r c e  waveforms and the mantle path e f f e c t s determines the waveform i n c i d e n t on the base o f the c r u s t which i s n e c e s s a r y f o r . t h e s o l u t i o n t e c h n i q u e s p r e s e n t e d i n t h i s work. The assumed l i n e a r n a t u r e o f e l a s t i c p r o p a g a t i o n i n the c r u s t p e r m i t s i n v e r s e approaches.  F o r example, i f a seismogram  i s known and  the c r u s t a l response i s l i n e a r and known, the c o r r e s p o n d i n g i n p u t wave- . form can be d i r e c t l y  computed.  A l s o , from a known i n p u t s i g n a l and i t s  r e s u l t i n g seismogram, the unique response o f the c r u s t to e l a s t i c waves can be o b t a i n e d .  However, s i n c e t h e r e i s seldom any  comparison  i n f o r m a t i o n on the n a t u r e o f the source or i n c i d e n t t e l e s e i s m i c wave-, forms, t h e s e i n v e r s e approaches utility  are not u s e f u l to demonstrate  o f the p r e s e n t t h e o r y w i t h o u t f u r t h e r assumptions.  the  Rather,  frequency and time domain p r o p a g a t i o n s o l u t i o n s w i l l be compared w i t h a n a l y s e s o f the e x p e r i m e n t a l  3.3  seismograms.  The Experiment ' For a n a l y s e s a chosen s e t of seismograms r e c o r d e d s o u t h of  Edmonton, A l b e r t a d u r i n g a f i e l d program has been used.  of October and November 1968 .  Three p o r t a b l e FM tape r e c o r d i n g s h o r t p e r i o d  seismo-  graph s t a t i o n s a t the U n i v e r s i t y o f A l b e r t a Edmonton S e i s m o l o g i c a l vatory  (our seismograph i s d e s i g n a t e d LED  Chamulka (CHA)  and L a r s e n (LAR)  farms  i n this  Obser-  t h e s i s ) , and a t the nearby  ( F i g u r e 13) r e c o r d e d 16  u s e f u l events from t e l e s e i s m i c d i s t a n c e s .  S i x o f t h e s e were chosen  f o r a n a l y t i c a l comparisons w i t h the t h e o r y (see l a t e r F i g u r e 16, T a b l e III). T h i s r e g i o n o f A l b e r t a i s c h a r a c t e r i z e d by a f l a t l y bedded s e d i m e n t a r y  40  F i g . 13. A map o f t h e s e i s m i c s t a t i o n and o i l w e l l l o c a t i o n s by which t h e c r u s t a l s e c t i o n was d e t e r m i n e d . The w e l l s i t e s a r e d e s i g n a t e d by l e t t e r s and c o r r e s p o n d t o those o f F i g u r e 14. The Dominion Land Survey Range and Township c o o r d i n a t e s a r e shown.  41  s e c t i o n w h i c h can i n g e n e r a l be approximated model.  by the h o r i z o n t a l  layer  Maximum d i p a n g l e s on the n o t a b l y c o n t i n u o u s f o r m a t i o n b o u n d a r i e s  a r e g e n e r a l l y l e s s than 1° except i n anomalous areas such as the Woodbend o i l f i e l d Devonian  (which l i e s beneath  reef structures exist.  s u i t e d t o t h i s problem.  s t a t i o n CHA)  i n which  Leduc-  oil-bearing  T h e r e f o r e , t h e s e r e c o r d s are  ideally  A l s o , the sedimentary s e c t i o n i s w e l l known from  r e f l e c t i o n s e i s m i c s u r v e y s and o i l w e l l b o r e h o l e g e o p h y s i c s f o r s e v e r a l nearby  3.4  drill  The  sites.  C r u s t Model The  on a s i d e .  s t a t i o n l o c a t i o n s formed  a t r i a n g u l a r p a t t e r n 15 k i l o m e t e r s  Over t h e s e r e l a t i v e l y s h o r t d i s t a n c e s , v e l o c i t y and  v i t y l o g s from deep o i l w e l l s  i n the v i c i n i t y p e r m i t t e d easy and  c o r r e l a t i o n s o f the major sedimentary  f o r m a t i o n s t o determine  p o l a t e d p r o f i l e s e c t i o n f o r the upper  crust  r e g i o n a l downward d i p normal N 130°W was well sites  to the s t a t i o n l o c a t i o n s .  s o l u t i o n shows t h a t the fundamental  A  strike i n a direction  and Basham,  to the wavelength  1968).  o f the h i g h e s t  More e x a c t l y , a l i n e a r  systems  r e v e r b e r a t i o n frequency of a l a y e r  the i m a g i n a r y component o f the L a p l a c e f r e q u e n c y s a t  the p r i n c i p a l p o l e o f the t r a n s f e r f u n c t i o n s e t (Appendix  to"one o f two  inter-  the c h a r a c t e r o f a seismogram, a f o r m a t i o n  f r e q u e n c i e s r e c o r d e d by the seismograph.  non-attenuating  an  the i n t e r p o l a t i o n between t h e  (e.g. E l l i s  l a y e r must have a t h i c k n e s s comparable  i s d e t e r m i n e d by  certain  ( F i g u r e 14, T a b l e I ) .  t o the Rocky Mountain  assumed i n o r d e r t o f a c i l i t a t e  In o r d e r t o a f f e c t  resisti-  l a y e r under normal  f r e q u e n c i e s depending  the l a y e r b o u n d a r i e s :  I).  incidence, this condition  For a corresponds  upon the r e f l e c t i o n c o e f f i c i e n t s a t  CRUST SECTION LAYER  A P  SURFACE  -I KM  -2 KM  10 KM  -3 KM  F i g 14. The sedimentary crustal section obtained by i n t e r p o l a t i o n between o i l w e l l geophysical logs. The layer numbers correspond to those i n Table I. The dotted boundaries underlying s t a t i o n CHA show the e f f e c t of the Leduc-Woodbend o i l f i e l d . The control o i l w e l l s i t e s used are designated by l e t t e r s which correspond to those of Figure 13. Extrapolations are indicated by dotted l i n e s .  43  either  f  =  °i  2  T  ^  i (3-1) 1  f  or  °i  4  T  i [4]  where T. i s the layer P-wave (or S-wave) t r a n s i t time.  i determines the minimum frequency berates;  Thus, f o^  component for which the l a y e r rever-  frequencies s i g n i f i c a n t l y lower than this are unseen.  The  sedimentary crust models (Figure 14, Table I) include a l l d i s t i n c t layers for which the t r a n s i t time, T^, i s greater than 0.05 seconds. Consequently, the layer modelling i s s u f f i c i e n t l y detailed to include the fundamental layer reverberation character f o r frequencies markedly l e s s than 10 Hz.  Furthermore, a l l other layer boundaries which showed -  notable v e l o c i t y contrasts were also considered i n the c r u s t a l models, . thus allowing layers for which T^ approximates only 0.025 seconds (f  Q  = 20Hz).  ' -  The frequencies recorded by the seismographs are l i m i t e d  to a band below 5 Hz.  The crust model d e t a i l should be s u f f i c i e n t to  describe reverberations i n t h i s band. The subsedimentary c r u s t a l section included i n the t o t a l c r u s t a l model of Table I follows the gross crust of Cumming and Kanasewich (1966) for  southern Alberta.  The t o t a l crust model i s desirable since the  Mohorovicic d i s c o n t i n u i t y f i r s t causes a part of an incident P wave to be converted into SV motions and p a r t i c u l a r l y a f f e c t s much of the e a r l y character of the h o r i z o n t a l seismograms. [4] The f i r s t condition r e s u l t s i f the product of the two r e f l e c t i o n c o e f f i c i e n t s i s p o s i t i v e ; the second i f the product i s negative. Also for the p o s i t i v e product, there i s a pole on the u) = 0 axis ( i . e . f = 0 ) . This pole determines only the zero frequency character of the i layer response and does not a f f e c t wave propagation (Appendix I ) .  1  TABLE I CRUSTAL SECTIONS LAYER  THICKNESS (km)  FORMATION (TOP)  P-VEL0CITY (km/sec)  cc) DENSITY (gin/  LED  CHA  LAR  LED  CHA  LAR  LED  CHA  LAR  1  Post Alberta  0.62  0.58  0.71  2.7  2.7  2.7  2.2  2.2  2.2  2  Alberta  0.46  0.49  0.49  3.0 .  3.0  3.1  2.3  2.3  2.3  3  Blalrmore  0.31  0.28  0.35  3.7  4.1  4.1  2.4  2.4  2.4  4  Wabumum  0.16  0.23  0.21  5.5  5.6  5.6  2.6  2.6  2.6  5  Ireton  0.17  0.22  0.19  4.0  4.2  4.5  2.4  2.4  2.5  6  Cooking Lake  0.26  0.27  0.30  5.3  5.4  5.5  2.6  2.6  2.6  7  Elk Point  0.19  0.19  0.18  4.6  4.7  4.7  2.5  2.5  2.5  8  Cambrian  0.42  0.40  0.37  4.1  4.2  4.2  2.4  2.4  2.4  9  Precambrian  10.0  10.0  10.0  6.1  6.1  6.1  2.7  2.7  2.7  10  Sub-Layer I  24.0  24.0  24.0  6.5  6.5  6.5  2.8  2.8  2.8  11  Sub-Layer II  11.0  11.0  11.0  7.2  7.2  7.2  2.9  2.9  2.9  12  Mantle Boundary  -  -  -  8.2  8.2  8.2  3.0  3.0  3.0  45  These model c o n s i d e r a t i o n s do n o t  exactly represent  i n c i d e n c e as the fundamental resonance f r e q u e n c i e s as the i n c i d e n t angle  increases.  a r e g e n e r a l l y s m a l l f o r the e f f e c t s a r e to be  But  are s l i g h t l y  t e l e s e i s m i c source  distances, only  However, r e a s o n a b l y  probable  values  small  formation d e n s i t i e s  a v a i l a b l e f o r the A l b e r t a  f o r the S-wave v e l o c i t i e s  were determined assuming P o i s s o n ' s r a t i o i s 1/4  so  that:  B = a //3 i  (3-2)  ±  F o r m a t i o n d e n s i t i e s were o b t a i n e d  from the P-wave v e l o c i t y - d e n s i t y  r e l a t i o n o f Nafe and Drake (Grant  and West, 1965).  Few  angles  expected.  e l a s t i c wave a t t e n u a t i o n p r o p e r t i e s are not  crust.  increased  s i n c e the waveform i n c i d e n t  Independent d a t a on the l a y e r S v e l o c i t i e s , and  non-normal  r e s u l t s have been r e p o r t e d  o f body waves w i t h i n the c r u s t and  f o r the a t t e n u a t i o n p r o p e r t i e s  among these l i t t l e  s t u d i e s o f McDonal e t a l (1958) on the  consistency i s  apparent.  The  thick shale  formations  n e a r P i e r r e , C o l o r a d o a r e among the most w i d e l y  quoted.  How-  ever, o t h e r i n s i t u Q measurements by subsequent workers ( e . g . , T u l l o s and  R e i d , 1969)  agree w i t h 1965)  have n o t  their results.  shown a t t e n u a t i o n c o e f f i c i e n t s which Various  have i n f e r r e d body wave Q's  s t u d i e s do n o t p r o v i d e  authors  ( P r e s s , 1964;  Anderson e t a l ,  from s u r f a c e wave a n a l y s e s .  But  such  the r e s o l u t i o n r e q u i r e d i n the p r e s e n t a n a l y s i s .  Clowes and Kanasewich (1970) e x p e r i m e n t a l l y  obtained  A l b e r t a c r u s t u s i n g deep r e f l e c t i o n seismograms. Q n e a r 300  closely  i n the sediments and  approximately  T h e i r sediment Q appears to be s i g n i f i c a n t l y  1500  Q f o r the c e n t r a l  They determined  average  i n the lower c r u s t .  g r e a t e r t h a n t h a t found  by  46 McDonal et a l f o r s i m i l a r m a t e r i a l s . differences sedimentary  In f a c t , order of magnitude  appear to be common among the s e v e r a l Q's reported f o r crusts.  For t h i s t h e s i s , m a t e r i a l Q values have been obtained from the l i t e r a t u r e (Knopoff, 1964; McDonal e t a l , 1958; Savage, 1965) and correlated with the known formation materials i n the sedimentary p o r t i o n of the A l b e r t a c r u s t .  Within the lower c r u s t , Q's have been estimated  on the basis of the average c r u s t a l attenuation: (Press, 1964).  Com-  prehensive data on the composition of the A l b e r t a sediments has been compiled by the A l b e r t a Society of Petroleum Geologists (1964).  Since  most of the major formations are s u b s t a n t i a l l y homogeneous, the c o r r e l a t i o n s between the formation rock m a t e r i a l and published Q f a c t o r s for the materials should give p l a u s i b l e values. be approximately frequency independent  Q has been reported to  over a wide range of frequencies  between 1 Hz and 10 Hz (Knopoff and MacDonald, 1958 and 1960). 6  Thus,  Q factors which have been published f o r r e l a t i v e l y high frequencies have been assumed to be constant to the Futterman c u t o f f angular frequency U Furthermore,  Q  .  at frequencies below 1 Hz, r e a l i s t i c Q values have almost no  e f f e c t on body wave propagation i n the crust and the assumption constant with frequency to the c u t o f f presents no apparent  that Q i s  difficulties.  Shear and compressional Q's have been assumed to be i d e n t i c a l . The c r u s t a l model Q factors f o r the i n d i v i d u a l layers which have been chosen f o r this work are tabulated and referenced i n Table I I . Of these, only those from the A l b e r t a and post A l b e r t a formations can be expected to be accurate.  These formations are b a s i c a l l y uniform shales  which appear to be contiguous with the P i e r r e , Colorado shales f o r which  47  TABLE I I FORMATION Q's  FORMATION  MATERIAL  REFERENCE  Q' CHOSEN. APPLICABLE  Post A l b e r t a  . sandy s h a l e  23  23  (Pierre  Alberta  shale  30  21-52  Blairmore  shale, sandstone  30  21-52  Wabumum  .dolomite  Ireton  shale  Cooking Elk  100  Lake  Point  V  30  1.  :  .  45-190? 21-52  •  limestone  100 '  45-190?  salt, evaporites  100  70-220  Cambrian  sandstones, . elastics  PreCambrian  granite?  50  200  Sub-Layer I  52  shale)  (limestone)  — .  500  •  1'  ,  :  1  ;  (hard  chalk)  (sandstone)  57-200  ' . 3;"'  • I  I  estimated, 4  500  Sub-Layer I I  1  -  estimated, 4  Q (average f o r whole c r u s t ) = 1 8 6 : Q (average f o r sediments) = 4 0 . The  average  Q 's were computed c o n s i d e r i n g the t r a v e l 1  the i n d i v i d u a l  References:  layers,  1 ) Knopoff  (1964)  2)  McDonal e t a l ( 1 9 5 8 )  3)  Savage  4)  Press  (1965) (1964)  The v a l i d i t y o f the Q's f o r the sub-sedimentary The  times through  r e s u l t s of Press  ( 1 9 6 4 ) apply p a r t i c u l a r l y  layers i s questionable. to Pg waves i n the  t e c t o n i c a l l y a c t i v e r e g i o n o f the Nevada T e s t S i t e .  F o r the remaining  chosen Q's, v a l u e s o b t a i n e d i n the l a b must be assumed t o apply i n s i t u .  48 a t t e n u a t i o n p r o p e r t i e s are w e l l known e t a l (1958) o b t a i n e d beyond 20 Hz  i n this  ( W i l l i a m s and  Q=23 f o r P waves and Q=10 formation.  Burk, 1964).  f o r S waves w i t h  McDonal  frequencies  In o r d e r to conform to the Q assignments  t o o t h e r l a y e r s the Q=23 v a l u e has  been assumed to d e s c r i b e the S wave  attenuation i n this layer also. The  average Q f o r the sedimentary p o r t i o n of the c r u s t above the  P r e c a m b r i a n by  the above model i s a p p r o x i m a t e l y  t h e whole c r u s t i s n e a r  3.5  40 w h i l e  the average Q f o r  200.  Data P r e p a r a t i o n The  FM  tape  recorded  seismograms were d i g i t i z e d w i t h a  r a t e o f 20 p e r second f o r the n u m e r i c a l t h e seismograph s e n s i t i v i t y  rapid f a l l o f f  frequency  so t h a t a l i a s i n g  T r a n s i e n t c a l i b r a t i o n p u l s e s were r e c o r d e d  o f each 5-day tape amplitude  The  from 5 Hz ensures t h a t t h e r e i s l i t t l e  gram c o n t e n t beyond the 10 Hz N y q u i s t are minimal.  analyses.  record.  frequency  sampling  at the  of  seismo-  effects beginning  From t h e s e p u l s e s , t h e a b s o l u t e ground m o t i o n  response of each s t a t i o n component seismograph  was  d e t e r m i n e d u s i n g the t r a n s i e n t c a l i b r a t i o n procedure o u t l i n e d by Deas (1969).  However, the h i g h frequency  p o o r l y determined by  t h i s method s i n c e the d i g i t i z a t i o n  n o i s e swamps the h i g h frequency ( F i g u r e 15).  i n s t r u m e n t a l response c h a r a c t e r i s (quantization)  components i n the c a l i b r a t i o n  pulses  C o n s e q u e n t l y , the t h e o r e t i c a l response f o r f r e q u e n c i e s  beyond 2 Hz was  assumed t o be r e p r e s e n t a t i v e of the systems.  characteristics  of the t h r e e components of any  t o be v i r t u a l l y  identical;  v a r i e d between components. used to e s t a b l i s h a b s o l u t e  one  s t a t i o n were  o n l y the a b s o l u t e s e n s i t i v i t i e s The  The  response designed  significantly  r e s u l t s of the p u l s e c a l i b r a t i o n s were  sensitivities.  49  >-  0.1  05  6.2  [5  £O  FREQUENCY  5'.0 (HZ)  16.  '  F i g . 15. The t h e o r e t i c a l s t a n d a r d seismograph response ( d o t t e d l i n e ) compared t o a response c a l i b r a t i o n o b t a i n e d by the t r a n s i e n t method (solid line).  The n a t u r e of the p r o p a g a t i o n problem i s e s s e n t i a l l y dimensional;  motions o n l y o c c u r v e r t i c a l l y  wave ray p a t h s . it  and h o r i z o n t a l l y  two  along  the  F o r the comparisons of the theory w i t h the seismograms,  i s then n e c e s s a r y  t o perform  components to e s t a b l i s h surface projection  a t r a n s f o r m a t i o n on the N and E  horizontal  the e q u i v a l e n t r a d i a l component motion a l o n g  o f the ray p a t h and  the  the t r a n s v e r s e component normal t o  50 It.  F o r t r a n s f o r m a t i o n , t h e two h o r i z o n t a l seismogram components must  have i d e n t i c a l r e s p o n s e s .  F o r t h i s purpose,  t i v i t i e s were n o r m a l i z e d t o t h e response component o f s t a t i o n LED on October o f t h e i n s t r u m e n t s v a r i e d w i t h time.  a l l s t a t i o n component  sensi-  c a l i b r a t i o n of the v e r t i c a l  4, 1968.  The a b s o l u t e s e n s i t i v i t i e s  E f f e c t s o f temperature  t r a n s i s t o r i z e d a m p l i f i e r s and seismometers  partially  account  on t h e f o r this  effect.  3.6  The Seismograms The s i x events used i n t h e s e comparison  F i g u r e s 16a-f) were chosen p r i m a r i l y s i m p l e P coda.  duration.  f o r t h e i r s h o r t and a p p a r e n t l y  F o r t h e s e e v e n t s , t h e wave motion  o f t h e c r u s t s h o u l d comprise  analyses (Table I I I ,  a basically  i n c i d e n t on t h e b a s e  t r a n s i e n t P wave o f s h o r t  I f the source to r e c e i v e r path i s not s t r u c t u r e d i n i t s  e l a s t i c p r o p e r t i e s , P t o S wave c o n v e r s i o n s do n o t o c c u r a l o n g the p a t h . However, i t s h o u l d be noted t h a t d i s t i n c t and c o n t i n u o u s v e l o c i t y  s t r u c t u r e i n the upper  and d e n s i t y v a r i a t i o n s  mantle  (hence a c o u s t i c impedance  v a r i a t i o n s ) a l o n g the s o u r c e t o r e c e i v e r p a t h must cause some c o n v e r s i o n o f energy  from c o m p r e s s i o n a l t o r o t a t i o n a l motions.  s o u r c e g e n e r a t e d P and.S motions s e v e r a l minutes  F o r these a n a l y s e s ,  a r e assumed t o a r r i v e a t t h e r e c e i v e r  a p a r t w i t h t h e i n c i d e n t waveform e n t i r e l y  compressional.  The P coda i s thus c o n s i d e r e d to be the r e s u l t o f t h e p u r e l y c o m p r e s s i o n a l waveform i m p i n g i n g on t h e base o f t h e c r u s t a t t h e M o h o r o v i c i c discontinuity. Ellis  and Basham (1968) have s u g g e s t e d  t r a n s v e r s e component a m p l i t u d e w i t h time  t h a t an i n c r e a s e i n t h e  ( t h i s i n c r e a s e i s observed f o r  51  EVENT I  VENEZUELA  LED  LAR  R  5 SEC.  F i g . 16a-f. The v e r t i c a l , r a d i a l h o r i z o n t a l and t r a n s v e r s e h o r i z o n t a l components of the e x p e r i m e n t a l seismograms f o r the e v e n t s s t u d i e d i n t h i s work. The P-coda onset f o r each event i s e x a c t l y 5 seconds from the b e g i n n i n g of each r e c o r d . (a) Venezuela  EVENT 2  FIJI ISLANDS  T  5 SEC.  (b)  Fiji  Islands  53 EVENT 3  CHILE  LED  I  CHA  L A R  5  (c)  Chile  SEC.  54  EVENT 4  KURILE IS.  LED  LAR  5 SEC.  (d)  Kurile  Islands  55 EVENT 5  NOVAYA  LED  CHA  R  LAR  5  (e)  SEC.  Novaya Zemlya  ZEMLYA  56  EVENT 6  RYUKYU  . LED  LAR  5  (f)  SEC.  Ryukyu I s l a n d s  IS.  TABLE I I I EVENTS USED IN ANALYSES  DEPTH  MAG  DELTA  10.7°N  62.6°W  97 km  4.8  58.6°  30°  117° E of N  F i j i Is.  20.9°S  178.8°W  607 km  5.7  92.9°  19°  238°  3  Chile  19.6°S  68.9°W  107 km  5.3  82.1°  21°  138°  4  Kurile Is.  49.7°N  155.8°E  35 km  5.5  53.0°  31°  305°  5  Novaya Zemlya  73.4°N  54.9°E  0 km  6.0  53.4°  31°  4°  6  Ryukyu Is.  27.5°N  128.4°E  48 km  5.8  83.4°  21°  308°  LOCATION  LAT  1  Venezuela  2  INCIDENCE  AZIMUTH (S-E)  LONG  EVENT  58  example on the V e n e z u e l a n event may  be  due  recorded  a t s t a t i o n LAR  to s i g n a l g e n e r a t i o n o f n o i s e by  c r u s t a l l a y e r i n g or topographic the r a t e of degradation  features.  i n Figure  scattering within T h i s e f f e c t may  16a)  the  then i n d i c a t e  o f the r e c o r d f o r the purposes of t h i s analysis-.  F o r most o f t h e chosen e v e n t s ,  the r a d i a l and  t r a n s v e r s e motion  ampli-  tudes become comparable w i t h i n 15 t o 30 seconds o f the P o n s e t . f i v e seconds of the r e c o r d has  been used i n the subsequent  domain a n a l y s e s  o f the seismograms.  n o i s e generated  t r a n s v e r s e amplitudes  to exceed the r a d i a l  horizontally layered structures.  CHA  seismograms from the C h i l e a n earthquake ( F i g u r e 16c,  the  amplitudes  However, some r e c o r d s such as  a predominance o f h o r i z o n t a l t r a n s v e r s e m o t i o n . was  frequency  G e n e r a l l y , - one would n o t expect  for  the s t a t i o n CHA  Twenty-  Table  I I I ) show  I t s h o u l d be n o t e d  l o c a t e d on t h e f l a n k o f the Leduc-Woodbend  a r e g i o n i n which t h e r e e x i s t s s i g n i f i c a n t  the  departure  l a y e r i n g i n the s e d i m e n t a r y c r u s t (see F i g u r e 14).  that  oilfield,  from h o r i z o n t a l Indeed, i t appears  t h a t the o i l b e a r i n g Devonian r e e f s t r u c t u r e causes the o v e r l y i n g s e d i m e n t a t i o n l a y e r s to be the s t a t i o n CHA the expected  lifted.  The  does n o t p e r m i t  complicated  subsurface  easy p r e d i c t i o n o f the d e p a r t u r e s  zero t r a n s v e r s e motions.  Also, strongly  f e a t u r e s o f the nearby N o r t h Saskatchewan r i v e r v a l l e y 1 k i l o m e t e r a c r o s s and Furthermore, although anomalous w i t h are n o t .  t h e CHA  (typically records.  record of the.Chilean event,is notably  r e s p e c t to t r a n s v e r s e m o t i o n s , the Novaya Zemlya  T h i s may  l o c a t i o n was  i n d i c a t e t h a t t h e r e i s an a z i m u t h a l  known b e f o r e  from  topographic  100 meters deep) c o u l d a f f e c t - t h e CHA  anomalous t r a n s v e r s e m o t i o n s . CHA  geometry a t  The  dependence of  e s s e n t i a l l y anomalous n a t u r e  analyses  and  the two  records  events  of  the  were i n c l u d e d  59  o n l y f o r the purpose LED  3.7  of comparing  the d i s t i n c t l y h o r i z o n t a l l a y e r i n g a t  and LAR w i t h the more s t r u c t u r e d geometry o f  Frequency  Analyses - T h e o r e t i c a l S p e c t r a l  CHA.  Amplitudes  F o r t h e d e t e r m i n a t i o n of t h e o r e t i c a l f r e q u e n c y r e s p o n s e c h a r a c t e r i s t i c s o f the model c r u s t s , spectrum) s i g n a l s l i n e a r system  impulse  amplitude  ( w h i t e , z e r o phase  r e p r e s e n t i n g P waveforms were used as i n p u t s to the  analogs.  the s u r f a c e m o t i o n s ,  Amplitude  and phase s p e c t r a were c a l c u l a t e d f o r  f o r a l l i n t e r n a l boundary motions  forms r e t r a n s m i t t e d i n t o  the basement h a l f s p a c e .  and f o r the wave-,  F o r non-normal  i n c i d e n c e c o n d i t i o n s , b o t h the v e r t i c a l and h o r i z o n t a l motion components have been o b t a i n e d . between 0 and  5 Hz was  range w h i l e l i m i t i n g The  chosen  A f r e q u e n c y sampling i n t e r v a l o f 0.1  time.  a m p l i t u d e response  p o r t i o n o f the Leduc model c r u s t normal  Hz  t o r e v e a l the major s p e c t r a l peaks i n t h i s  the c o m p u t a t i o n a l  s u r f a c e motion  amplitude  f o r the  sedimentary  ( T a b l e I I I ) above the P r e c a m b r i a n  i n c i d e n c e i s shown i n F i g u r e 17.  The  for  phase s p e c t r a showed m a i n l y  a l a g g i n g t r e n d w i t h f r e q u e n c y due  t o the time d e l a y between the i n p u t  waveform and the s u r f a c e m o t i o n s .  The phase s p e c t r a and  boundary m o t i o n It frequencies  s p e c t r a w i l l n o t be p r e s e n t e d i n t h i s  i s n o t a b l e t h a t r e v e r b e r a t i o n resonances (Figure 17).  The  r e s p o n s e a t a p p r o x i m a t e l y 0.4  f i r s t - resonance  internal  analysis. occur- at- s e v e r a l  peak o f the LED  Hz i s p r i m a r i l y due  t o the r e v e r b e r a t i o n  m u l t i p l e w i t h the l o n g e s t p a t h bounded by the s u r f a c e and the basement. fundamental  (Appendix  I d e s c r i b e s a procedure  crustal  Precambrian  f o r t h e computation  r e v e r b e r a t i o n f r e q u e n c y and harmonics  f o r one  o f the  layer). •  60  F i g . 17. The s p e c t r a l amplitude response o f the t h e o r e t i c a l nona t t e n u a t i n g Leduc c r u s t a l s e c t i o n t o a normal i n c i d e n c e w h i t e amplitude spectrum P-wave i n c i d e n t a t the Precambrian-Cambrian boundary.  S p e c t r a l peaks f o r h i g h e r f r e q u e n c i e s a r e the r e s u l t o f the combined e f f e c t s o f h i g h e r harmonics  o f t h i s r e v e r b e r a t i o n arid p r i m a r y  f o r m u l t i p l e s w i t h s h o r t e r paths  resonances  (which cause h i g h e r frequency r e v e r b e r -  ation) along w i t h t h e i r coincident  harmonics.  The s p e c t r a l amplitude response f o r a 30° ( a t the P r e c a m b r i a n boundary) i n c i d e n t P wave c o n s i s t s o f b o t h v e r t i c a l and h o r i z o n t a l components  (Figure 18).  There i s a marked s i m i l a r i t y i n the  form o f t h e v e r t i c a l amplitude component w i t h t h a t  f o r the normal  i n c i d e n c e c o n d i t i o n a l t h o u g h i t s amplitudes a r e reduced.  Particularly  61  F i g . 18. The s p e c t r a l a m p l i t u d e response o f the t h e o r e t i c a l nona t t e n u a t i n g Leduc c r u s t a l s e c t i o n t o a 30° i n c i d e n t P wave a t the Precambrian-Cambrian boundary. W r e p r e s e n t s the v e r t i c a l component motion, Y the h o r i z o n t a l . Q  Q  the resonance  o f t h e fundamental  r e v e r b e r a t i o n and t h e peaks n e a r 1.6 Hz,  2.3 Hz and 2.9 Hz a r e common t o b o t h r e s p o n s e s .  T h i s i s expected  f o r s m a l l i n c i d e n c e a n g l e s , the P motions w i t h i n t h e l a y e r i n g the v e r t i c a l ponent. vertical;  component w h i l e the SV motions  because  dominate  dominate t h e h o r i z o n t a l com-  I n t h i s example, t h e h o r i z o n t a l a m p l i t u d e s a r e lower than t h e however, f o r S wave i n p u t s o r P wave i n p u t motions  l a r g e r i n c i d e n t a n g l e s the o p p o s i t e c o n d i t i o n would o c c u r . angle a t t h e Precambrian  w i t h much  A 30° i n c i d e n t  boundary c o r r e s p o n d s t o a 42° a n g l e a t the  Mohorovicic d i s c o n t i n u i t y .  T h i s i n c i d e n c e angle c o r r e s p o n d s  to a source  62  at o n l y 25  e p i c e n t r a l d i s t a n c e ( a f t e r Ichikawa,  see Bashaun, 1967). F o r  such nearby s o u r c e s , s i g n a l s from r e f r a c t i v e and r e f l e c t i v e paths • contaminate the e a r l y P. coda. . wavefront  invalidates  A l s o , g e o m e t r i c a l s p r e a d i n g o f the  the n e c e s s a r y  p l a n e wave assumption.  source d i s t a n c e s g r e a t e r than approximately The  Experimentally,  35° s h o u l d be used.  e f f e c t o f c r u s t a l a b s o r p t i o n on the s p e c t r a l  f o r t h e 30° i n c i d e n c e c o n d i t i o n i s shown i n F i g u r e 19.  amplitudes  For the c a l -  c u l a t i o n o f these example s p e c t r a Q=50 was chosen t o r e p r e s e n t l a y e r f o r b o t h P and S motions. f o r t h e sediments  (Table I I ) .  O UJ Q  A c a u s a l , zero phase a t t e n u a t i o n w i t h no S i g n i f i c a n t l y , t h e r e i s a decrease  SECTION  3 0 ° INCIDENT  00  every  T h i s Q i s near t h e computed average  attending d i s p e r s i o n i s considered.  LEDUC  ' . ; .  P WAVE  f-  W  10  0  Yrs —-  Q  CC  o o  O oJ  \  7  \  1.0  '  \ /  /  2.0 FREQUENCY  3.0  4.0  (Hz)  F i g . 19. The s p e c t r a l amplitude response o f t h e t h e o r e t i c a l c r u s t a l s e c t i o n t o a 30° i n c i d e n c e P wave a t the Precambrian-Cambrian boundary. The c r u s t ' i s assumed t o have a Q o f 50 i n every l a y e r . '•'•.  5.0  of  the  and  spectral  horizontal  amplitudes  with  components.  The  exponential  dependence o f  condition.  A  apparent higher  frequencies. per  measure o f physical as  would  cycle  the  of  i n the  the  an  resonance  the  resonant  a slight  effects  of  of  resonances,  the  quality  vertical  I t was  The  on  noted  particularly by  and  the  is a  previously  o f phase  peaks but remain  spectral  r e p r e s e n t s the  this  energy  that effect  velocities  the  form  nearly  amplitudes  crustal  at  direct  demands a d i s p e r s i o n  dispersion  the  describes  a b s o r p t i o n i s the  frequency harmonic  frequency would  of dispersion reasonably  the  i s determined  of the resonance  dependence on  f o r both  frequency which  causal which  model.  shift  with  effect  attenuation.  Futterman  a c a u s a l model  3.8  of  the  unchanged. are  amplitude  minimal, response  attenuating crust.  Frequency To  placement effects of  at  crustal  curve amplitude  the  quality  a t t e n u a t i o n must be  cause  Since  significant  The  frequency  constant Q model g i v e s a n e g a t i v e  amplitudes  r e d u c t i o n i n the  absorbed  such  further  increasing  low  exceed  -  Spectral  Ratio  compare  the  spectral  amplitudes  due  to earthquakes,  of  the  seismograph  frequency signal  n o i s e may  levels  and  amplitude  response.  high  responses  displacements  and  an  a  i t i s necessary  record of  integration  the  of the  made.  than  r e c o n s t r u c t i o n s to  noise  and  Also,  respect  comparisons the  data, the  may  cultural these  velocities  records with  amplitude  the  enhancement  instrument  motion  dis-  t o remove  t e l e s e i s m i c waves.  the ground  necessary before spectral such  ground  T h i s r e q u i r e s .extreme  time would be  apply  with  f r e q u e n c i e s where w i n d , w a t e r  g e n e r a t e d by  are b a s i c a l l y  Rather  Technique  components where m i c r o s e i s m i c and  swamp m o t i o n s  seismograms than  Analysis  rather to  could  be  technique  64  using  s p e c t r a l r a t i o s (Phinney, 1964)  response character ponents can be For  of the  i s more u s e f u l .  S i n c e the  seismographs were n o r m a l i z e d , a l l motion com-  assumed to have been i d e n t i c a l l y a f f e c t e d by a given P a r r i v a l ,  mogram P coda are  the v e r t i c a l and  responses to the  a t the base o f the  crust.  W() o  the  instruments,  radial horizontal seis-  same i n p u t waveform which i s i n c i d e n t  Their Fourier  =  U  frequency  H (c).U v  I N p u T  s p e c t r a may  be  described:  (co)  (3-3) V where H^Cw)  and  U )  H^(OJ) are the  transfer functions i n p u t waveform.  -V ' INPDT  U)  and  1  s  I t i s evident that  =  *  |Y ( )| U  Furthermore, over small h o r i z o n t a l  crustal transfer function  X  distances, on  For  example:  and  1964):  (3-4)  I lyoo | one  may  expect  the base of the  a t e l e s e i s m i c s o u r c e would not v a r y s i g n i f i c a n t l y .  should describe  ratio  H>)  f r e q u e n c y content o f a P a r r i v a l i n c i d e n t  between the v e r t i c a l  arbitrary  the v e r t i c a l - t o - h o r i z o n t a l s p e c t r a l  i n p u t waveform (Phinney,  W (co) | o  radial horizontal  F o u r i e r spectrum of the  t n e  upon the  c o m p l e t e l y independent o f the  V/H  ( U )  t o t a l c r u s t v e r t i c a l and  u'j.NPUT^ ^ i-  a m p l i t u d e r a t i o depends o n l y is  U  that  c r u s t due  T h e r e f o r e , the  component s p e c t r a l a m p l i t u d e s a t two  a n o t h e r r a t i o which i s independent o f the  the  low to  ratios  nearby s i t e s input  waveform.  (3-5) V  u ) 1  V  =  2  U  )  ]  2  '  U  INPUT  (  W  )  and: |W <o )] | o  Vg/Vj  )  |H (o))] |  i  =  v  i  -  |w (co)] | o 2  .  (3-6)  |H (co)] | v 2  These v e r t i c a l - t o - v e r t i c a l component s p e c t r a l amplitude r a t i o s may be u s e f u l i n the comparisons of the c r u s t a l s e c t i o n s at two s t a t i o n s . A l s o , because the v e r t i c a l components u s u a l l y possess high s i g n a l - t o - n o i s e r a t i o s r e l a t i v e to the h o r i z o n t a l components, the accuracy  obtainable  from such analyses should be b e t t e r . Phinney showed that a comparison o f the V/H r a t i o s determined experimentally from long period P-wave seismograms and by d i r e c t c a l c u l a t i o n using the H a s k e l l method could be u s e f u l f o r c r u s t a l i n t e r p r e tations.  Although, h i s work y i e l d e d good comparisons between theory and  experiment, other i n v e s t i g a t i o n s have been l e s s s u c c e s s f u l at short periods (Utsu, 1966;  E l l i s and Basham, 1968).  The work of E l l i s and  Basham.included an area i n c e n t r a l A l b e r t a near that studied i n these analyses.  They ascribed part of t h e i r l i m i t e d success to s c a t t e r i n g  which generates noise from s i g n a l . did not i n c l u d e attenuation.  However, t h e i r H a s k e l l matrix  analyses  The present work w i l l attempt to determine  the s i g n i f i c a n c e of c r u s t a l absorption to s p e c t r a l r a t i o s .  Later, i t  w i l l be shown that f o r short periods and the low Q's of c r u s t a l sediments, the e f f e c t of absorption can be considerable.  F i g . 20a. S p e c t r a l V/H r a t i o s f o r v a r i o u s a t t e n u a t i o n c o n d i t i o n s i n the s e d i m e n t a r y c r u s t compared t o the n o n - a t t e n u a t i n g r a t i o s . The d e s i g n a t e d Q's have been a p p l i e d t o every l a y e r ; the s o l i d l i n e i s f o r i n f i n i t e Q.  F i g . 20b. S p e c t r a l V/H r a t i o s f o r Q v a l u e s i n each l a y e r a c c o r d i n g t o T a b l e I I compared t o n o n - a t t e n u a t i n g r a t i o s . i s f o r i n f i n i t e Q; the dashed l i n e f o r the chosen Q's.  assigned The s o l i d  line  66  SPECTRAL V/H RATIOS  0  1.0  2.0  FREQUENCY  0  1.0  ZO  FREQUENCY  Fig. 20a.  3.0  4.0  5.C  (HZ)  3.0  (HZ)  4.0  5.0  SPECTRAL  1.0  V/H  RATIOS  2.0  FREQUENCY  0  1.0  2.0 FREQUENCY  3.0  4.0  5.0  (HZ)  3.0  4.0  5.0  (HZ)  CHA  o  i.o  2:0  FREQUENCY  Fig. 20b.  3.0  (HZ)  4.0  5.0  68.  The work of E l l i s and Basham i n central Alberta included the analysis of A l events which allowed them to use a s t a t i s t i c a l i n t h e i r experimental comparisons. are  3.9  approach  In this present work only 6 events  used and s t a t i s t i c a l l y v a l i d inferences cannot be deduced.  T h e o r e t i c a l Spectral Ratios T h e o r e t i c a l V/H r a t i o s were computed for the three s t a t i o n crust  models f o r an incident P wave making a 22° angle with the Precambrian upper boundary (which i s equivalent to a 30° incidence angle at the Mohorovicic boundary). and 20b. compared.  The computed ratios are presented i n Figures 20a  Several d i f f e r e n t i n t e r n a l layer attenuation conditions are Since the broad features of the V/H r a t i o s are not greatly  a f f e c t e d by neglecting the large scale layering of the sub-sedimentary A l b e r t a crust ( E l l i s and Basham,  1968),  only the portion of the crust  above the Precambrian has been considered i n these analyses.  The major  e f f e c t of neglecting the three deeper model layers i s to remove closely spaced harmonic peaks due to the long path multiples involving the deep c r u s t a l boundaries.  Smoothing of the computed r a t i o s was not considered  necessary but the frequency sampling increment was  chosen s u f f i c i e n t l y  small to determine a l l of the major s p e c t r a l r a t i o features.  Small  changes i n the P wave incidence angle have l i t t l e e f f e c t on the location of the V/H peaks although the peak amplitudes are modified.  Consequently,  i f the absolute r a t i o amplitudes are not required, the one set of r a t i o s for 22° incidence should be s u f f i c i e n t f o r comparisons with the r a t i o s experimentally determined from the s u i t e of earthquake events with incidence angles ranging from 1 4 ° to 23°.  The t h e o r e t i c a l r a t i o s f o r  22° incidence should underestimate the r a t i o amplitudes f o r the smaller  69  angles  of P-wave i n c i d e n c e . The  curves  designated  describe non-attenuating Q's  represent  the V/H  must be  i n f i n i t e Q ( F i g u r e s 20a  crustal conditions;  ratios  a p p l i e d to each l a y e r . estimate  by  The  f o r the c r u s t model w i t h  Q values  of Table  II  other  the s p e c i f i e d  v a l u e Q=100 i s thought to be  The  20b)  designations with  o f the a b s o r p t i o n i n the A l b e r t a sediments but i m p l a u s i b l y low.  and  a  Q  conservative  the v a l u e Q=25  average 45  f o r the  sediments. I n these frequencies But,  the e f f e c t  above 1 Hz  the LED  and  V/H  of c r u s t a l a b s o r p t i o n i s v i r t u a l l y  absorption effects increase noticeably.  LAR  V/H  lowering Q while the LED  r a t i o comparisons, i t i s apparent t h a t a t  r a t i o peaks near 1.2  Hz  the r a t i o peaks n e a r 1.7  peak n e a r 2.4  Hz  t h e o r e t i c a l Q from i n f i n i t y  insignificant. For  example,  are i n c r e a s e d s h a r p l y  Hz a r e d e c r e a s e d .  i s more than doubled by to 25.  low  by  Furthermore,  decreasing  the  For a l l t h r e e c r u s t a l models, l a r g e  i n c r e a s e s i n the r a t i o a m p l i t u d e s are p a r t i c u l a r l y e v i d e n t beyond 4 for attenuating conditions.  Attenuation  exponentially with  and  frequency  dominant f a c t o r i n V/H The by  useable  o f a harmonic wave i n c r e a s e s  t h e e f f e c t of a b s o r p t i o n must become a  r a t i o s at h i g h  frequencies.  p o s i t i o n s o f s e v e r a l of the major V/H  the a d d i t i o n o f a t t e n u a t i o n .  Therefore,  these  peaks are n o t a f f e c t e d  features should  f o r comparisons of the b a s i c c r u s t a l models w i t h  S i m i l a r l y , the vide separate  Hz  be  experiment.  f e a t u r e s which- s t r o n g l y depend on a t t e n u a t i o n c o u l d i n f o r m a t i o n on the c r u s t a l a b s o r p t i o n  pro-  characteristics.  T h e o r e t i c a l r a t i o s of the v e r t i c a l component s p e c t r a l a m p l i t u d e s from s t a t i o n p a i r s are shown i n F i g u r e 21.  Vertical-to-vertical  have been computed o n l y f o r n o n - a t t e n u a t i n g  c r u s t c o n d i t i o n s because  absorption  c h a r a c t e r i s t i c s at the t h r e e  c l o s e l o c a t i o n s s h o u l d be  ratios the  very  70  SPECTRAL VERTICAL-VERTICAL RATIOS o  UJ O => t  o r6  CHA/LED  s! P  — LAR/LED  <  1.0  2.0  3.0  FREQUENCY  F i g . 21. models.  similar  Vertical-to-vertical spectral  (HZ)  ratios  LAR/LED v e r t i c a l r a t i o s  CHA/LED r a t i o s  are featureless  p o s s e s s more c h a r a c t e r .  LED and LAR c r u s t s h o u l d be s i m i l a r .  models a r e q u i t e s i m i l a r The CHA c r u s t  t h e Leduc-Woodbend o i l f i e l d .  crust  S i g n i f i c a n t l y , the below 3 Hz w h i l e the  This i s not s u r p r i s i n g  r a t i o d e s c r i b e s the r e l a t i v e v e r t i c a l c r u s t a l  on  f o r non-attenuating  and t h e e f f e c t on the r a t i o s must be s m a l l .  theoretical  50  4.0  s i n c e the  r e s p o n s e s and s i n c e the  their  response  characteristics  model d e s c r i b e s an anomalous The d i f f e r e n c e  location  i n the CHA and LED models  i s m a n i f e s t e d by t h e g r e a t e r v e r t i c a l - t o - v e r t i c a l s p e c t r a l  ratio  character.  71  3.10  Experimental  Spectral Ratios  F o r comparisons of e x p e r i m e n t a l  and  theoretical spectral ratios,  d i s c r e t e F o u r i e r s p e c t r a have been computed u s i n g Cooley and  Tukey  the n o r m a l i z e d experimental  (1965) on  records  the f i r s t  spectral ratios  the t r a n s i e n t P-coda has  from the t r u n c a t e d r e c o r d s  computed from these  be  length i s a reasonable  Within  the i n f i n i t e  i n the  admitted.  Cooley-Tukey a l g o r i t h m .  average  ratio calculations.  The  25 second  seconds from the d i g i t i z a t i o n and r e s o l u t i o n o f 0.04  This resolution i s excessive  ratio  computations.  9 point Fejer  record  record the  f o r comparisons  ( t r i a n g u l a r ) running  result  the  The  Hz u s i n g  Smoothing o f the  a p p l i e d to reduce the h i g h l y r e s o l v e d d e t a i l w h i l e s p e c t r a l f e a t u r e s which s h o u l d  (e.g.  duration  compromise f o r the chosen event r e c o r d s .  seconds g i v e s a f r e q u e n c y  s p e c t r a u s i n g 5 and  second time window  r e c o r d would d e c r e a s e the  reduce confidence  s a m p l i n g i n t e r v a l o f 0.05  the t h e o r e t i c a l  s p e c t r a a r e shown f o r  the 25  generally represents  A l s o l a t e r a r r i v i n g phases c o u l d be  with  of  assumed t h a t the spectrum, c a l c u l a t e d  choice of a longer  s i g n a l - t o - n o i s e r a t i o and  l e n g t h o f 25  sample p o i n t s )  g e n e r a l l y decayed t o ground n o i s e l e v e l s  F i g u r e 1 6 a - f ) so t h a t i t may  The  (500  of  f o l l o w i n g the P o n s e t . V e r t i c a l - t o - h o r i z o n t a l  the chosen e v e n t s i n F i g u r e s 2 2 a - f .  transient.  25 seconds  the a l g o r i t h m  experimental  operators  was  r e t a i n i n g the b r o a d  from r e v e r b e r a t i o n s  i n the,upper  sedimentary c r u s t a l l a y e r s . F o r t h r e e chosen event seismogram s e t s , smoothed r a t i o s of F o u r i e r s p e c t r a o f the s i g n a l and  25 seconds o f p r e - e v e n t  v e r t i c a l and h o r i z o n t a l components were compared determine c o n f i d e n c e  l i m i t s on  the computed V/H  the  noise f o r both  ( F i g u r e 23a-c) to ratios.  For these  three  EVENT I VENEZUELA SPECTRAL V/H RATIOS  0  1.0  2.0 FREQUENCY  3.0  4.0  5.C  (HZ)  F i g . 22a-f. Experimental V/H spectral ratios obtained from the suite seismograms. Error bars are included on 3 of the spectral ratios for comparison. These error conditions are thought to be typical. (a)  Venezuela  73  EVENT  2  FIJI ISLANDS  SPECTRAL  8  V/H RATIOS  LED  °-\ o  % SI <  3.0  2.0  1.0  FREQUENCY  5.0  FREQUENCY  Fiji  5.0  (HZ)  2.0  (b)  4.0  Islands  EVENT 3 CHILE SPECTRAL V/H RATIOS  8  UJ  o => Ij  LED  «>|  I -  (c). Chile  74  EVENT 4 KURILE IS. SPECTRAL V/H RATIOS  (d) Kurile. Islands  76  (e)  Novaya Zemlya  77  LAR  1.0  2.0 FREQUENCY  (f)  30 (HZ)  Ryukyu I s l a n d s  4.0  5.0  78  F i g . 23a-c. S p e c t r a l s i g n a l - t o - n o i s e r a t i o s f o r t h e v e r t i c a l ( s o l i d l i n e ) and h o r i z o n t a l ( d o t t e d l i n e ) f o r t h r e e e v e n t s . The n o i s e spectrum was o b t a i n e d from t h e 2 5 seconds p r e c e d i n g the P o n s e t o f each e v e n t .  7  a n a l y s e s , l a r g e s i g n a l - t o - n o i s e r a t i o s were found between 1 and 3 Hz. expected  r o o t mean squared  noise;  spectral ratios  (Figures  The LED V e n e z u e l a r e c o r d i n g showed t y p i c a l r e l a t i v e  only the F i j i  Islands records  a t LED and LAR p o s s e s s e d  pre-event lower  apparent s i g n a l - t o - n o i s e r a t i o s .  Thus the r e l a t i v e e r r o r s found  LED V e n e z u e l a r a t i o i s c o n s i d e r e d  t o be t y p i c a l o f ( o r worse than) the  remaining  ratios  f o r which e r r o r c o n d i t i o n s were n o t o b t a i n e d .  further possible exception is a particularly  General  h i g h frequency  The  Features  theoretical estimate  noise contamination.  f r e q u e n c i e s does n o t appear t o be p o o r ) .  V/H  ratios  a r e much lower than t h e t h e o r y p r e d i c t s . ( t h e  computations f o r t h e l a r g e 22° i n c i d e n c e angle f o r the steeper  angles).  should  under-  The o n l y p o s s i b l e  e x c e p t i o n i s the CHA C h i l e seismograms ( F i g u r e 16c, T a b l e  I I I ) f o r which  a r e e v i d e n t between 0 and 2 Hz b u t r e t u r n t o the anomalous  lower v a l u e s beyond t h i s .  T h i s event  r e c o r d i n g i s a l s o anomalous w i t h  r e s p e c t t o h o r i z o n t a l motions s i n c e t h e t r a n s v e r s e amplitudes radial  there  of Spectral Ratios  r a t i o amplitudes  high ratios  (One  However, the  most n o t a b l e g e n e r a l f e a t u r e o f t h e e x p e r i m e n t a l  i s t h a t t h e amplitudes  f o r the  i s the LAR V e n e z u e l a r e c o r d i n g f o r which  s i g n a l - t o - n o i s e r a t i o at useable  3.11  The  e r r o r s due t o the background n o i s e a r e d e s i g -  nated by t h e e r r o r b a r s on the c o r r e s p o n d i n g 22a,d,e).  exceeded  amplitudes. Reduced r a t i o amplitudes  must r e s u l t  from a r e l a t i v e  enhance-  ment o f t h e h o r i z o n t a l ground motions compared t o the v e r t i c a l m o t i o n s . P o s s i b l e causes i n c l u d e background n o i s e and s i g n a l g e n e r a t e d scattering.  9  It i s significant  n o i s e by  t h a t the h o r i z o n t a l component background  n o i s e was g e n e r a l l y h i g h e r than t h e v e r t i c a l .  T h i s e f f e c t would be  80  expected  t o reduce  the r a t i o a m p l i t u d e s  towards v a l u e s l e s s  f r e q u e n c i e s where the s i g n a l - t o - n o i s e r a t i o i s s m a l l . f o r w h i c h the average  signal-to-noise ratio  appears  than 1 a t  However, r e c o r d s  t o be v e r y h i g h  would then be expected t o show s u b s t a n t i a l l y b e t t e r agreement w i t h theory.  But even those w i t h the l e a s t r e l a t i v e background  e x h i b i t reduced r a t i o amplitudes.  F o r example, the LED  from the C h i l e and Novaya Zemlya events show anomalously amplitudes.  The'computed r a t i o s  ( F i g u r e s 22a,d,e) determine  n o t due  t o background  seismograms low  ratio cal-  t h a t these r e d u c e d a m p l i t u d e s  r a t i o comparisons,  Ellis  and Basham (1968)  t h a t s c a t t e r i n g n o i s e g e n e r a t e d from the s i g n a l c o u l d have  caused p a r t of t h e i r l a c k o f comparison experimental r e s u l t s . irregularities  Scattering  between t h e i r t h e o r e t i c a l  can r e s u l t from b o t h i n t e r n a l  and from s u r f a c e topography which d e t e r m i n e s  f r e e s u r f a c e boundary.  crustal  To e x p l a i n a r e d u c t i o n o f r a t i o a m p l i t u d e s  v e r t i c a l component motions to h o r i z o n t a l .  or p r e f e r e n t i a l l y converts v e r t i c a l  motions  Since a v e r t i c a l to h o r i z o n t a l conversion i s b a s i c a l l y  r e l a t i v e t o c o m p r e s s i o n a l motions experimental r a t i o s .  The  c o u l d e x p l a i n the r e d u c e d  motions  apparent  e f f e c t s o f t o p o g r a p h i c s c a t t e r i n g s h o u l d most  s t a t i o n which was  s t a t i o n w h i c h was  l o c a t e d near a b r o a d v a l l e y  s i t u a t e d w i t h i n 2 km  o f the N o r t h  and  the  CHA  Saskatchewan R i v e r  v a l l e y w h i c h i s a p p r o x i m a t e l y a h a l f km wide and 100 m deep. o n l y the CHA  by  reduces  a P t o S c o n v e r s i o n , a s c a t t e r i n g mechanism which i n c r e a s e d s h e a r  the LED  and  a non-flat  s c a t t e r i n g , i t i s n e c e s s a r y t h a t the mechanism p r e f e r e n t i a l l y  affect  are  n o i s e , a t l e a s t i n the band from 1 t o 3 Hz.  I n t h e i r work on V/H suggested  noise also  f o r which e r r o r c o n d i t i o n s were  culated  the  Surprisingly,  s t a t i o n gave s p e c t r a l r a t i o s which were n o t anomalously  low.  81  An  attenuation  c o n d i t i o n which, p r e f e r e n t i a l l y absorbed P  motions r e l a t i v e t o S motions c o u l d a l s o h e l p However, most e x p e r i m e n t a l e v i d e n c e that S attenuation  i s greater  to e x p l a i n t h i s  condition.  (e.g. McDonal e t a l , 1958) shows  than P w h i c h would determine t h e i n v e r s e  effect.  3.12  D e t a i l s of the S p e c t r a l The  V/H'ratios  common s i g n i f i c a n t  Ratios  from t h e r e c o r d i n g s  s p e c t r a l r a t i o maximum between 0.5 and 1 Hz. (The  r a t i o s computed f o r t h e CHA r e c o r d i n g s peak and i t s v a l i d i t y calculations  i s questionable  show l e s s c h a r a c t e r i n this case).  for this  The t h e o r e t i c a l  f i n d a major r a t i o maximum near 0.7 Hz which c o r r e l a t e s  w i t h the experimental f e a t u r e . LED  o f t h e s i x e v e n t s have a  Most r a t i o s computed f o r t h e LAR and  s t a t i o n e v e n t s show a f u r t h e r common maximum between 1.5 and 2 Hz.  However, f o r t h e LED Novaya Zemlya r a t i o s , t h i s peak i s d i s p l a c e d t o a p p r o x i m a t e l y 1.3 Hz w h i c h i s s u s p i c i o u s  although  the r e c o r d q u a l i t y  and  s i g n a l - t o - n o i s e r a t i o a r e e s p e c i a l l y good.  LED  and LAR show maxima a t 1.6 and 1.7 Hz which a p p a r e n t l y  with these experimental features.  Theoretical ratios f o r  Low f r e q u e n c i e s  correlate  are p r i m a r i l y  a f f e c t e d by t h e s t r u c t u r e s o f the c r u s t which have a s c a l e s i z e o f t h e order should  o f t h e harmonic w a v e l e n g t h .  Since, the g r o s s c r u s t a l f e a t u r e s  be w e l l m o d e l l e d , i t i s n o t s u r p r i s i n g t h a t t h i s  ment between t h e o r y 2 Hz t h e l i t t l e  and experiment e x i s t s .  area  However, b e y o n d . a p p r o x i m a t e l y  c o r r e l a t i o n between t h e t h e o r y  and e x p e r i m e n t  i n d i c a t e t h a t t h e s m a l l s c a l e model s t r u c t u r e i n a d e q u a t e l y the a c t u a l c r u s t s .  of agree-  could  describes  Also, since P - a r r i v a l s i g n a l l e v e l s decrease r a p i d l y  82  w i t h f r e q u e n c y , background n o i s e b e g i n s higher frequencies. low  ( F i g u r e 23)  Beyond -3 Hz,  to p l a y an i n c r e a s i n g r o l e  at  the s i g n a l - t o - n o i s e r a t i o s become so  t h a t good c o r r e l a t i o n s  cannot be e x p e c t e d .  Where the  s i g n a l - t o - n o i s e r a t i o i s n e a r (or below) 1, background n o i s e must l a r g e l y determine the V/H r a t i o s computed s o l e l y  ratios.  But,  even beyond 3 Hz  from the p r e - e v e n t  n o i s e f o r the t h r e e  were found not t o be s i m i l a r to the e q u i v a l e n t event s e q u e n t l y , i f the-background n o i s e can be  the smoothed  ratios.  assumed to be  Con-  a stationary  p r o c e s s , i t does n o t appear t o be the whole cause o f the h i g h d i s c r e p a n c i e s between t h e o r y and e x p e r i m e n t . scattering noise hypothesis. n e a r 3.5  Hz i s n o t e v i d e n t .  s m a l l (as e v i d e n c e d by and  The  events  frequency  T h i s l e n d s w e i g h t to a  cause of the l a r g e LAR  V/H  ratio  peak  However, s i g n a l - t o - n o i s e r a t i o s a r e v e r y  the l a r g e e r r o r b a r s  ( F i g u r e 22d))  beyond 3 Hz  the peak i s p r o b a b l y n o t an i n d i c a t i o n o f c r u s t a l f e a t u r e s . Any  t h e CHA  similarity  C h i l e event  i n the e x p e r i m e n t a l  theoretical ratios for  recordings i s s u r p r i s i n g because t h i s s t a t i o n  the most anomalous i n terms o f s u b s u r f a c e F u r t h e r m o r e , the C h i l e event h o r i z o n t a l motions.  and  l a y e r i n g and  topography.  r e c o r d s showed p a r t i c u l a r l y  Apparent c o r r e l a t i o n s s h o u l d be  was  large transverse  discounted  as  accidental. The not  i n c l u s i o n o f a t t e n u a t i o n i n the t h e o r e t i c a l s o l u t i o n s d i d  improve the agreement between t h e o r y and  the e x p e r i m e n t a l models.  experiment.  Furthermore,  d e t a i l s do n o t d i s c r i m i n a t e the v a r i o u s c r u s t a l  F o r example, a l t h o u g h  the 1.7  Hz peak was  noted  as a p o s s i b l e  i n d i c a t o r of c r u s t a l a t t e n u a t i o n c o n d i t i o n s i t i s apparent i s " no  c o n s i s t e n c y i n i t s r e l a t i v e amplitude  ratio  results.  among the  Q  that there  experimental  83  For the  ideal horizontal  azimuth o f  the P-wave r a y  layering, approach.  the V/H  r a t i o i s independent  However, two  mental r a t i o s show s t r o n g a z i m u t h a l dependence. the V e n e z u e l a  (azimuth 117°  events g i v e n e a r l y  Also,  the LED  r a t i o s are  so n o t i c e a b l e  beneath  (azimuth 138° within  the  For  the  LED  recordings,  layering  but  epicentral distances. the  E of  condition  does n o t  expected  (305° E of  such  error  N)  r a t i o s do  not  azimuthal  Admittedly  explicitly  N)  recordings.  LAR  r e s u l t s seem t o i n d i c a t e  from  A striking  Kurile Is.  a g a i n , the  experi-  ratios  c o r r e s p o n d i n g LAR and  of  are more n o t a b l e than c o r r e l a t i o n s between e v e n t s  events i s s m a l l ,  horizontal  the  s i m i l a r t o about 3 Hz;  s i m i l a r receiver-to-event of  on  (308° E o f N)  e x h i b i t much s i m i l a r i t y . correlations  Chile  LED  s t r o n g s i g n a l band below 3 Hz.  Ryukyu I s .  quite  and  i d e n t i c a l r a t i o character,  l i m i t s , throughout the s i m i l a r i t y i s not  E o f N)  The  pairs  of  that  at  t h i s sample  the  ideal  a p p l y t o the  crust  LED. E x p e r i m e n t a l l y determined s p e c t r a l r a t i o s between s t a t i o n  v e r t i c a l component p a i r s from the  are  shown i n F i g u r e 24a-c.  V e n e z u e l a , C h i l e , Novaya Zemlya and  show t h e . e x p e c t e d r e l a t i v e l y f e a t u r e l e s s beyond 3 Hz The  Fiji  the  and  frequencies. expected. the  Both have more c h a r a c t e r and  Little  c h a r a c t e r from the  experimental r a t i o s ;  than the  appear t o be  LAR/LED r a t i o s  K u r i l e I s . events  r a t i o s below 2.5  r a t i o s become much l a r g e r  Ryukyu r a t i o s a l s o  The  theory  Hz.  generally But,  predicts.  anomalous at the  l a r g e r amplitudes  low than  theoretical ratios correlates  however, the K u r i l e I s . , V e n e z u e l a  with  and  Novaya Zemlya e x p e r i m e n t a l r a t i o s a l l show s i m i l a r c h a r a c t e r below 3 S i n c e the some o f  signal-to-noise  r a t i o s are  t h i s s i m i l a r i t y must be  generally  high at these  Hz.  frequencies,  s i g n i f i c a n t although i t s i n t e r p r e t a t i o n  84 SPECTRAL VERTICAL-VERTICAL RATIOS  FREQUENCY  (HZ)  i s not evident. events  The  show l i t t l e  CHA/LED r a t i o s from  c o r r e l a t i o n with theory.  a r a t i o enhancement r e l a t i v e LAR/LED r a t i o s .  the C h i l e and Novaya Zemlya  The  However, beyond about 3  to t h e o r y i s a l s o observed  enhancement suggests  t h a t the LED  ponent l a c k h i g h f r e q u e n c i e s r e l a t i v e t o the o t h e r two LED  alluvium.  High  and CHA  3 . 1 3  v e r t i c a l comstations.  The  and  i n s t r u m e n t s were l e s s i d e a l l y  l o c a t e d on  c o n t r i b u t e s t o the g r e a t e r n o i s e a t s t a t i o n s  Syntheses  In p r i n c i p l e , i t i s p o s s i b l e t o determine the c r u s t a l t r a n s m i s s i o n response, the output  explicitly.  e x a c t l y one  the i n c i d e n t waveform from  seismogram s e t p r o v i d e d the o t h e r two  the b a s e -  a r e known  I n the f o l l o w i n g  s y n t h e s e s , t h e approach t a k e n i s t o o b t a i n s y n t h e t i c seismograms the response  among  However, the n a t u r e of the p r o p a g a t i o n problem o n l y a l l o w s  d i r e c t knowledge of the r e c e i v e d seismograms.  of t h e system model o f the c r u s t  p a r i s o n s between the s y n t h e t i c and  from  t o i n c i d e n t waveforms  c r e a t e d on t h e b a s i s o f s i m p l e and p l a u s i b l e assumptions.  Then, com-  e x p e r i m e n t a l seismograms are  t o demonstrate the u s e f u l n e s s of the l i n e a r systems approach and validity  on  CHA.  Time Domain  ment and  -  f r e q u e n c y n o i s e i s e a s i l y generated by wind i n u n c o n s o l i -  dated m a t e r i a l s and p r o b a b l y LAR  as f o r the  i n s t r u m e n t s were l o c a t e d on the Canadian s e i s m i c network p i e r  b e d r o c k w h i l e the LAR  Hz  expected the  ( i n p a r t a t l e a s t ) of the A l b e r t a c r u s t models f o r p r o p a g a t i o n  problems a t low f r e q u e n c i e s . I t i s a r g u a b l e t h a t the i n v e r s e approach which s t r i v e s o b t a i n the i n p u t waveform or the c r u s t a l response  f u n c t i o n from  e x p e r i m e n t a l seismograms i s s c i e n t i f i c a l l y b e t t e r . e v i d e n t t h a t assumptions on b o t h  to the  However, i t i s  the n a t u r e of the waveform and  the  86  c r u s t a l r e s p o n s e f u n c t i o n w i l l s t i l l be r e q u i r e d f o r comparison. by  the i n v e r s e approach,  comparisons  can be d i r e c t e d o n l y between com-  p u t e d and assumed s o l u t i o n s .  However, the c r e a t i o n o f s y n t h e t i c  seismograms a l l o w s the d i r e c t  comparison  dependent mental  Thus,  of s o l u t i o n s  (which are  on the a p p l i e d assumptions) w i t h the r e a l i t y o f the e x p e r i -  seismograms. By  s i m u l a t i n g the c o n t i n u o u s l i n e a r systems  c r u s t models  u s i n g the methods d i s c u s s e d i n Chapter I I , time domain responses o f the crus.t to any i n p u t d i s p l a c e m e n t waveform can be c a l c u l a t e d .  A FORTRAN  s i m u l a t i o n program f o r t h e s e s y n t h e s e s i s l i s t e d  IV.  i n Appendix  r e q u i r e m e n t o f t h i s d i g i t a l s y n t h e s i s p r o c e d u r e i s t h a t each transit if  A  layer  time i s an i n t e g e r m u l t i p l e o f the sampling i n t e r v a l .  However,  t h e s a m p l i n g i n t e r v a l i s chosen s m a l l enough to a l l o w a N y q u i s t  f r e q u e n c y beyond the f r e q u e n c i e s o f i n t e r e s t , particularly  restrictive.  work a r e band l i m i t e d program o f Appendix sample i n t e r v a l .  The  the requirement  i s not  e x p e r i m e n t a l seismograms used i n t h i s  t o f r e q u e n c i e s below about 5 Hz.  IV a s s i g n s a minimum l a y e r t r a n s i t  The  FORTRAN  time o f one  time  F o r the f o l l o w i n g s y n t h e s e s the time increment o f  seconds has been used f o r comparisons  0.05  w i t h the d i g i t a l e x p e r i m e n t a l  data. Vertical p o r t i o n o f the LED placement  component s y n t h e t i c responses o f the upper c r u s t model t o an example a r b i t r a r y  waveform a t normal  sedimentary  incident P  dis-  and 30° a n g l es are shown i n F i g u r e 25.  r e v e r b e r a t i o n s of the p r i m a r y motion  l a s t i n g s e v e r a l seconds  due  to long  p a t h m u l t i p l e s w i t h i n the m o d e l l e d c r u s t are p a r t i c u l a r l y n o t a b l e . s m a l l s c a l e h i g h frequency c h a r a c t e r r e s u l t s  The  The  from s h o r t e r p a t h m u l t i p l e s  87  0.5 SEC.  F i g . 25. Comparison of non-normal (a) and normal (b) incidence s y n t h e t i c seismograms. Wavelet (c) i n c i d e n t on the Precambrian-Cambrian boundary at 30° produces s y n t h e t i c seismogram ( a ) , at normal incidence produces s y n t h e t i c seismogram ( b ) .  w i t h i n the thinnest l a y e r s .  For the 30° incidence s y n t h e t i c response,  the l a t e reverberations a r r i v e e a r l i e r than the corresponding reverberations f o r normal incidence.  In frequency s o l u t i o n s t h i s same e f f e c t  appears as a displacement of the r e v e r b e r a t i o n s p e c t r a l peaks toward higher frequencies as the incidence angle increases. also be noted i n Figures 17 and 18).  (This e f f e c t can  For non-normal i n c i d e n c e , the  88  h o r i z o n t a l motion response P and  i s also available.  The  method p e r m i t s  S waveforms r e t r a n s m i t t e d i n t o the basement to be  found b u t  the these  were not g e n e r a l l y used i n t h i s work.  3.14  Syntheses w i t h L a y e r  Attenuation  C a u s a l minimum phase a t t e n u a t i o n w i t h i n each l a y e r o f the c r u s t Is allowed  i n the time domain s y n t h e s e s  a t i o n time o p e r a t o r s a ^ ( t ) (see Chapter impulse  response  by  the i n c l u s i o n o f the  I I ) where each o p e r a t o r i s the  of the a t t e n u a t i n g l a y e r f o r one  work, t h e Futterman  transit.  a t t e n u a t o r impulse  response  c r u s t a l l a y e r f o r m a t i o n w i t h Q=23 i s p r e s e n t e d particularly  significant  n e a r z e r o by 0.1  this  attenuation operators  w h i c h r e p r e s e n t each l a y e r f o r i t s p a r t i c u l a r Q and such  In  (1960) a b s o r p t i o n model (A1,D1) d e s c r i b e d e a r l i e r  has been used t o compute the time incremented  example.of one  attenu-  t h a t i t s response  seconds f o l l o w i n g o n s e t .  transit  time.  o f the p o s t A l b e r t a i n F i g u r e 26.  It is  r a p i d l y decays to v a l u e s The  transit  p o s t A l b e r t a f o r m a t i o n a t i t s P-wave phase v e l o c i t y approximately  0.23  e f f e c t i v e 0.1  second " l e n g t h " o f the o p e r a t o r .  time  through  which have h i g h e r Q's  The  the  (undispersed) i s  seconds which i s c o n s i d e r a b l y l o n g e r than  i n c r e a s e s w i t h d e c r e a s i n g . Q.  An  effective  Therefore, other c r u s t a l l a y e r  this "length" formations  determine a t t e n u a t i o n o p e r a t o r s which a r e even  s h o r t e r r e l a t i v e to t h e i r l a y e r t r a n s i t  times.^  ^ The FORTRAN s i m u l a t i o n program o f Appendix IV demands t h a t the o p e r a t o r i s s h o r t e r than the l a y e r t r a n s i t time w h i c h l i m i t s l a y e r Q's to v a l u e s g r e a t e r than a p p r o x i m a t e l y 5. A l t h o u g h such Q's a r e a l r e a d y u n r e a l i s t i c a l l y low, program m o d i f i c a t i o n s c o u l d be made to a l l o w the v e r y l o n g o p e r a t o r s which c o r r e s p o n d to a r b i t r a r i l y low Q's.  89  0.060.040.02-  0.20  0.25  0 30  SECONDS  F i g . 26. The impulse response o f l a y e r 1 o f the c r u s t a l s e c t i o n ( T a b l e I ) assuming c a u s a l a t t e n u a t i o n w i t h Q=23. Assuming the Futterman a t t e n u a t i o n d e s c r i p t i o n the n o n - a t t e n u a t e d impulse would a r r i v e a t 0.23 s e c o n d s . The amplitude s c a l e i s a r b i t r a r y .  The  time incremented  l a y e r a t t e n u a t o r s must be sampled  p r o p e r l y n o r m a l i z e d t o p r e s e r v e the r e l a t i v e s p e c t r a l r e s p o n s e . the 0.05  second  sampling  f o r the P and S motions  interval,  o f a l l model l a y e r s are g i v e n i n T a b l e IV.  to those g i v e n i n  However, s e v e r a l o f the t h i n n e s t model l a y e r s produce transit  For  these a t t e n u a t o r o p e r a t o r s a r e  a p p r o x i m a t e l y e q u i v a l e n t f o r b o t h P and S motions  s m a l l a t t e n u a t i o n i n one  For  the l a y e r a t t e n u a t o r o p e r a t o r w e i g h t s  each l a y e r , the Q v a l u e s r e p r e s e n t e d by  Table I I .  and  such  t h a t no b r o a d e n i n g o r d i s p e r s i o n o f an  90  TABLE IV ATTENUATOR WEIGHTS FOR THE 0.05 SECOND SAMPLING INCREMENT  FORMATION  Q  P WEIGHTS  .S WEIGHTS  Post A l b e r t a  23  0.839,0.139,0.022  0.621,0.320,0.044,0.015  Alberta  30  0.932,0.06  0.831,0.154,0.015  Blairmore  30  0.953,0.047  0.947,0.053  100  1.0  1.0  30  1.0  1.0  Cooking Lake  100  1.0  1.0  Elk Point  100  1.0  1.0  50  0.979,0.021  0.974,0.026  Precambrian  200  0.946,0.054  0.940,0.060  Sub-Layer I  500  0.915,0.085  0.815,0.168,0.017  Sub-Layer I I  500  0.946,0.054  0.940,0.60  Wabumum Ireton  Cambrian  1  91  F i g . 27. The t h e o r e t i c a l f r e q u e n c y response o f t h e c r u s t f o r one passage o f an impulse f o r average Q=200 compared t o the f r e q u e n c y response due t o the sum o f t h e time domain a t t e n u a t i o n o p e r a t o r s o f T a b l e IV. The v a l u e Q=200 i s n e a r t h e average model Q v a l u e o f 186.  impulse can be seen on t h e 0.05 second  time s c a l e .  F o r these l a y e r s , the  s i n g l e weight v a l u e u n i t y which c o r r e s p o n d s t o i n f i n i t e Q o r z e r o a b s o r p t i o n must be a p p l i e d . ators  f o r one t r a n s i t  The s p e c t r a l r e s p o n s e o f the l a y e r a t t e n u -  o f an impulse through t h e t o t a l  crustal section i s  compared t o the t h e o r e t i c a l s p e c t r a l amplitude r e s p o n s e f o r an average  92  Q equal the  to 200  ( F i g u r e 27).  At  the low  computed l a y e r a t t e n u a t i o n o p e r a t o r s  while  they  s l i g h t l y overestimate  agreement can be  achieved  considered  t o be  that  u n d e r e s t i m a t e the average Q  Q at h i g h  frequencies.  Arbitrarily  by u s i n g s m a l l e r time s a m p l i n g i n t e r v a l s  expense o f i n c r e a s e d c o m p u t a t i o n a l are  frequencies, i t i s evident  time.  However, the  chosen  better  at  the  attenuators  an adequate d e s c r i p t i o n o f a t t e n u a t i o n i n the  following  syntheses. Normal i n c i d e n c e s y n t h e t i c c r u s t r e s p o n s e s f o r the upper mentary p o r t i o n o f t h e c r u s t model and  f o r the waveform used i n  s y n t h e s e s o f F i g u r e 25  of i n c r e a s i n g the c r u s t a l  ( F i g u r e 28). with  The  show the e f f e c t  the a p p r o p r i a t e  attenuators  of Table  d i s p e r s i o n e f f e c t s on the i n i t i a l waveform and  3. 15  frequency  15 i n e v e r y  c h a r a c t e r are e v i d e n t .  The  the  layer.  The  preferential  apparent energy  Impulse S y n t h e t i c C r u s t a l Responses D i r a c d e l t a f u n c t i o n i s o f t e n a convenient  many c o n t i n u o u s  l i n e a r systems problems.  i n f i n i t e amplitude zero  i t is u s e f u l to r e p r e s e n t areas  approximately  e f f e c t i v e l e n g t h o f the r e c o r d a r e a l s o r e d u c e d .  The  with  attenuation  t o a n o t h e r response f o r  significant  removal o f t h e h i g h  chosen to be  IV and  arbitrarily  the  the  i n f i n i t e Q response i s compared to the c r u s t r e s p o n s e  w h i c h Q was  and  sedi-  o f the  the impulse a r e a n u m e r i c a l l y  simulations  and  i n F i g u r e 29.  Rather,  compute  S y n t h e t i c c r u s t responses to a  i n c i d e n c e P i m p u l s e at the M o h o r o v i c i c  vertical  However, n u m e r i c a l  length impulses are not p o s s i b l e .  response i m p u l s e s .  models a r e p r e s e n t e d  input s i g n a l f o r  30°  d i s c o n t i n u i t y f o r the t h r e e  Significant  similarity  i n both  and h o r i z o n t a l component s y n t h e s e s f o r the t h r e e s t a t i o n  models i s e v i d e n t .  the  crustal the crust  0.5  SEC.  F i g . 28. Normal i n c i d e n c e s y n t h e t i c seismograms dependence on Q. I n f i n i t e Q (a) i s compared to the model Q v a l u e s of T a b l e IV (b) and to an average c r u s t a l Q=15 ( c ) . Note the d i s p e r s i o n o f the p r i m a r y arrival. The i n c i d e n t waveform o f F i g u r e 25 has been used.  A l a r g e r s c a l e P impulse normal i n c i d e n c e for  the LED  c r u s t model i s shown i n F i g u r e  30.  synthetic  response  S e v e r a l of the  b e r a t i o n m u l t i p l e s w h i c h cause d e l a y e d impulses are d e s i g n a t e d integer sets.  For t h e s e ,  by  each boundary of the c r u s t model i s numbered  from 0 f o r the s u r f a c e  to 11 f o r the  integer represents  first  the  rever-  c r u s t - m a n t l e boundary.  reflection;  the next r e p r e s e n t s  The the  first  CRUST  IMPULSE  RESPONSE  94  LED  CHA  H —  LAR  H  5  SEG.  Fig. 29. The vertical and radial, horizontal synthetic impulse responses of the three crustal sections representing the three stations.  TRANSMISSION IMPULSE RESPONSE  B  C D E F G 1 SECOND I 1  A: B: C: D: E: F: G: H: I: J: K: L: M: N: 0: P: Q:  LEGEND DIRECT 5-7-5-0 0-4-0 0-7-0 0-3-0-2--0 & 0-3-0-3--0 0-3-0-8--0 & 2-3-2-0 0-2-0 0-3-0 0-5-0 0-8-0 0-3-0-4--0 & 0-7-0-8-•0 & 0-9-0 0-10-0 0-11-0  0-2-0-3-0 0-8-0-3-0  0-4-0-3-0 0-8-0-7-0  F i g . 30. The normal incidence'synthetic impulse response for the LED c r u s t a l section. Various multiples or reverberations are designated by l e t t e r s . The legend describes the i n t e r n a l reverberation paths which cause these l a t e r a r r i v a l s .  BASEMENT REFLECTION IMPULSE RESPONSE LEGEND  A: 11-BASEMENT  D  E F  G  B: 10-BASEMENT C: 9-BASEMENT D: 8-BASEMENT E: 3-BASEMENT F: 1-BASEMENT G: SURFACE BASEMENT H: 0-8-O-BASEMENT  1  SECOND  I 7  1  T  F i g . 31. The basement r e f l e c t e d wave synthetic impulse response f o r the LED c r u s t a l section. Reflections from various i n t e r n a l crust boundaries into the basement are designated by l e t t e r s . One reverberated m u l t i p l e , H, i s also shown. The legend describes the i n t e r n a l reverberation paths causing the l a t e r arrivals.  97  second and so on.  The l a s t  motion i s observed. s u r f a c e and  i n t e g e r r e p r e s e n t s the boundary  P r i m a r y m u l t i p l e s which  then from an i n t e r n a l boundary  surface usually display  first  reflect  on w h i c h from the  before returning  the l a r g e s t a m p l i t u d e s .  t o the  However, some h i g h  a m p l i t u d e impulses from e n t i r e l y i n t e r n a l m u l t i p l e s a r e a l s o F o r example, the m u l t i p l e s d e s i g n a t e d (2-3-2-0) and r e f l e c t e d from the s u r f a c e .  should a f f e c t  e x p e r i m e n t a l seismograms.  The  the e a r l y  t o the  c h a r a c t e r o f the  impulse response d i s p l a c e m e n t wave p r o p a -  g a t e d back i n t o the mantle shows s e v e r a l r e f l e c t i o n s  i n t o t h e mantle  (0-3-0) m u l t i p l e  and s u r f a c e and the (0-8-0) due  P r e c a m b r i a n upper boundary  b o u n d a r i e s o f the c r u s t  evident.  (5-7-5-0) a r e n o t  The p a r t i c u l a r l y s t r o n g  due t o the Wabumum upper boundary  the  ( F i g u r e 31).  from the major  These waveforms r e f l e c t e d  can r e t a i n much c h a r a c t e r c r e a t e d i n c r u s t a l  back  reverber-  ations .  3.16  I n c i d e n t Waveform  Models  The computation o f any r e a s o n a b l e s y n t h e t i c seismogram a r e a l i s t i c i n p u t waveform f o r the l i n e a r systems model which  requires  i s based  upon the known p r o p e r t i e s o f the mantle p a t h and the n a t u r e o f s e i s m i c source motions.  But, because  the form o f P motions a t s o u r c e s are n o t  g e n e r a l l y known, p l a u s i b l e assumptions must be  applied.  F o r the Novaya Zemlya e v e n t , which r e s u l t e d from a S o v i e t underground n u c l e a r t e s t o f a d e v i c e i n the 0.1 e q u i v a l e n c e range, the r e s u l t s  TNT  of Werth and H e r b s t (1967) f o r e x p l o s i o n  waveforms have been c o n s i d e r e d . (Werth and H e r b s t , 1967;  to 1.0 megaton  T h e i r work on s m a l l n u c l e a r e x p l o s i o n s  F i g u r e 1) shows t h a t the r a d i a l d i s p l a c e m e n t  time f u n c t i o n near an e x p l o s i o n s o u r c e has the form o f a p o s i t i v e p u l s e  98  o f s h o r t d u r a t i o n ( l e s s than 0.5 Based  seconds  f o r t h e i r three  on t h e i r e v i d e n c e , an i m p u l s e d i s p l a c e m e n t  time f u n c t i o n  model has been assumed f o r the Novaya Zemlya e v e n t . suggested  examples).  Werth and  t h a t waveforms a t t e l e s e i s m i c d i s t a n c e s are not  a p p r o x i m a t i o n s h o u l d be an adequate  s h o u l d be n o t e d t h a t the f i e l d band o f f r e q u e n c i e s below about  seismograph 5 Hz.  Consequently,  description.  o n l y those  the i n s t r u m e n t bandpass can a f f e c t  an impulse below 5 Hz.  i n f i n i t e s m a l duration i s not  It  Assuming" a l i n e a r n a t u r e f o r the  the " i m p u l s e model" merely  forms l o o k l i k e  p u l s e so  instruments observed o n l y a  wavemotions from t h e s o u r c e through the m a n t l e , frequencies within  Herbst  highly  dependent upon the e x a c t n a t u r e o f the e x p l o s i o n s o u r c e motion t h a t the impulse  source  requires The  source  the seismogram.  t h a t the s o u r c e wave-  exact i n f i n i t e  impulse o f  required.  An impulse model i s a l s o p a r t i c u l a r l y  a t t r a c t i v e because i t  Imposes the minimum p o s s i b l e c h a r a c t e r on the s o u r c e d i s p l a c e m e n t motions.  I t d e t e r m i n e s no s p e c t r a l a m p l i t u d e o r phase c h a r a c t e r and  d e s c r i b e s o n l y an o r i g i n t i m e ;  i t i s the maximum e n t r o p y  (minimum  I n f o r m a t i o n ) waveform. The  chosen  v e r y s h o r t P coda. P coda,  events i n t h i s work were l i m i t e d  to those  S i n c e c r u s t a l r e v e r b e r a t i o n s can o n l y l e n g t h e n the  the i n c i d e n t P wave must have even s h o r t e r d u r a t i o n .  e v e n t s , the LED  r e c o r d i n g o f the V e n e z u e l a event has  This should  the waveforms i n c i d e n t on the c r u s t were a l s o q u i t e  Furthermore,  s i n c e the two  indi-  similar.  events occurred at n e a r l y equal e p i c e n t r a l  d i s t a n c e s from the s t a t i o n s , the e f f e c t s o f the mantle be s i m i l a r .  Among the  c h a r a c t e r which i s  s u r p r i s i n g l y s i m i l a r t o the Novaya Zemlya seismogram. cate that  exhibiting  This implies s i m i l a r i t y  path should also  i n the s o u r c e g e n e r a t e d waveforms,  99  at  least  as seen by  earthquakes  the narrow band i n s t r u m e n t s .  do n o t appear  and t h e s t r i c t  However, the r e m a i n i n g  t o g e n e r a t e such s t r i k i n g l y s i m i l a r  impulse model cannot be used as  seismograms  confidently.  Some s t e p d i s p l a c e m e n t s o u r c e models were a l s o c o n s i d e r e d ands y n t h e s e s u s i n g t h e s e compared t o impulse source s y n t h e s e s .  I t seems  r e a s o n a b l e t h a t a s t e p model c o u l d b e t t e r r e p r e s e n t e l a s t i c motions t h e s o u r c e o f an earthquake  fault.  F o l l o w i n g the l e a d o f Werth and H e r b s t (1967), the e f f e c t o f the mantle t o be due  (1967) and  Carpenter  p a t h on the s o u r c e waveform i s c o n s i d e r e d  o n l y t o a b s o r p t i o n and a t t e n d i n g d i s p e r s i o n .  the Futterman  near  a b s o r p t i o n model obeys a p r i n c i p l e  C a r p e n t e r shows  o f s i m i l a r i t y which he  describes:  F(t,T/Q) =  (Q/T)F (tQ/T)  (3-7)  Q  where F ( t , T / Q ) i s the waveform amplitude at time t f o r an t r a v e l time o f T and a medium w i t h q u a l i t y metrically  Thus, F ( t Q / T ) Q  d e s c r i b e s o n l y the shape o f the waveform and T/Q  characteristic  time which i s the e s s e n t i a l  Futterman model responses p a r e d f o r v a r i o u s mantle station  Q.  event-to-station  travel  characteristic  parameter  para-  is a  of F .  In F i g u r e 32,  to impulse d i s p l a c e m e n t s o u r c e motions Q's  time o f 600  r a n g i n g from 500  seconds.  time parameter  are com-  to 3000 f o r a s o u r c e t o  F o r these r e s p o n s e s , the  ranges from 1.2  seconds  down to 0.2  seconds,  t h e 10 minute t r a v e l time c o r r e s p o n d s r o u g h l y t o the e p i c e n t r a l d i s t a n c e t o the Novaya Zemlya and V e n e z u e l a event s o u r c e s o r a p p r o x i m a t e l y 5 5 ° . The may  be  combined s o u r c e motions  and.mantle a b s o r p t i o n waveforms  f u r t h e r combined w i t h the seismograph  response  to determine  the  100  00  596  598  A  600 SECONDS  602  604  F i g . 32. The e f f e c t o f v a r i o u s mantle p a t h average Q v a l u e s on an impulse s o u r c e d i s p l a c e m e n t motion which has t r a v e l l e d 10 minutes a t the u n d i s p e r s e d phase v e l o c i t y through the mantle. Q=1500 has been found t o be the b e s t r e p r e s e n t a t i o n f o r c o n s t r u c t i o n o f s y n t h e t i c seismograms i n t h i s work.  i n p u t w a v e l e t t o the systems model.  S i n c e the s o u r c e m o t i o n s ,  a b s o r p t i o n and i n s t r u m e n t response c h a r a c t e r i s t i c s o r d e r o f c o m b i n a t i o n does n o t a f f e c t Wavelets  mantle  are a l l l i n e a r  their  the r e s u l t a n t seismogram s y n t h e s e s .  o b t a i n e d by the c o n v o l u t i o n o f the seismograph  impulse  response  and the r e s p o n s e of the mantle  a b s o r p t i o n model t o impulse and s t e p  s o u r c e motions  are p r e s e n t e d i n F i g u r e 33.  f o r v a r i o u s T/Q  w a v e l e t s were o b t a i n e d by C a r p e n t e r u s i n g somewhat d i f f e r e n t response  characteristics.  Similar instrument  10.1  IMPULSE SOURCE  T/Q= 1.2  SEC  0.6  0.2  0.4  STEP SOURCE  0;3  0;6  5 SECONDS F i g . 33. Wavelets c o n s t r u c t e d by c o n v o l u t i o n of the s e i s m o g r a p h response w i t h a b s o r b i n g mantle responses f o r v a r i o u s T/Q parameters f o r b o t h impulse and s t e p s o u r c e s . T/Q=0.4 seconds has been found to be the b e s t r e p r e s e n t a t i o n f o r seismogram s y n t h e s i s at an e p i c e n t r a l d i s t a n c e o f 55°.  IMPULSE  SOURCE AT  55°  T/Q (SEC) 1-2 3  0.6  0.4  o  to  5 SEC. Fig. 34. Vertical and horizontal component synthetic seismograms for various T/Q wavelets of the form shown i n Figure 32 for impulse source motions.  STEPfSOURCE AT 55 e  T/Q (SEC)-12  0.2  o 5  SEC.  Fig. 3 5 . Vertical and horizontal component synthetic seismograms for various T/Q wavelets for step source motions.  104  U s i n g t h e s e computed w a v e l e t s l i n e a r systems model of the LED  f o r the i n p u t s o u r c e to the  c r u s t f o r 30° i n c i d e n c e (which  corres-  ponds t o the 55° t e l e s e i s m i c d i s t a n c e ) a s e t o f s y n t h e t i c seismograms have been c a l c u l a t e d also presented  ( F i g u r e 34).  ( F i g u r e 35).  Comparisons of these syntheses w i t h the  LED Novaya Zemlya and V e n e z u e l a s o u r c e model f o r T/Q of  =0.4  the i n i t i a l motions.  comparison  because  Step source s y n t h e t i c seismograms are  r e c o r d s i n d i c a t e s t h a t the  impulse  seconds b e s t r e p r e s e n t s the f r e q u e n c y c h a r a c t e r (These two p a r t i c u l a r events are used i n t h i s  their epicentral distances result  i n the 30° i n c i d e n c e  a n g l e used i n t h i s s y n t h e s i s ) .  C a r p e n t e r suggested t h a t f o r t e l e s e i s m i c  P waves, T/Q  unity.  T/Q  = 1.2  s h o u l d approximate  seconds  and 0.6  seconds  i n the experimental records.  However, those syntheses f o r  l a c k h i g h f r e q u e n c y c h a r a c t e r observed  Short d u r a t i o n s o u r c e motions which c o u l d  c o n t a i n r e l a t i v e l y more h i g h f r e q u e n c y c h a r a c t e r than an impulse p h y s i c a l l y unreasonable.  I t i s b e t t e r to r e t a i n  r e q u i r e a lower T/Q  seconds  experiment.  =0.4  T h i s T/Q  exceeding t h i s  the impulse model and  v a l u e which appears  v a l u e determines  a mantle  are  to b e s t f i t  Q=1500.  Q's  i n the  mantle  ( t o 6000) f o r P waves w i t h f r e q u e n c i e s n e a r 1 Hz have  been r e p o r t e d by s e v e r a l i n v e s t i g a t o r s t h a t t h i s mantle  (Jackson and Anderson,  a b s o r p t i o n model i s n o t u n r e a l i s t i c .  s y n t h e s e s Q has been h e l d c o n s t a n t to 1500 w h i l e T/Q  1970)  so  For a l l subsequent varies with  travel  time. S y n t h e t i c seismograms f o r i m p u l s i v e d i s p l a c e m e n t s o u r c e s at and  85° e p i c e n t r a l d i s t a n c e s are shown i n F i g u r e 36.  55°  In t h e s e com-  p u t a t i o n s , the i n t e r n a l c r u s t a l a t t e n u a t i o n has been i n c l u d e d u s i n g the a t t e n u a t o r w e i g h t s o f T a b l e IV. determines  T/Q  =0.4  seconds;  F o r the 55° d i s t a n c e , the t r a v e l f o r 85°, T/Q  = 0.6  seconds.  time  Impulse  85° EPICENTER  55° EPICENTER  LEDUC  CHAMULKA  LARSEN  J  U  —\AA_AAA^—.—  5 SEC. Fig. 36. Impulse source motion synthetic seismograms for 55° and 85° epicentral distances. The mantle Q has been chosen to be 1500 for a l l syntheses. Internal crustal absorption (causal) of Table II has been included.  106  s o u r c e w a v e l e t s with, these mantle a b s o r p t i o n model parameters as systems i n p u t s .  A comparison  between a v e r t i c a l  component  a t t e n u a t i n g c r u s t seismogram and t h e c o r r e s p o n d i n g a b s o r b i n g seismogram showed l i t t l e d i f f e r e n c e . apparent h i g h f r e q u e n c y  3. 17  Seismogram  c o n t e n t was  noncrust  Only a s l i g h t r e d u c t i o n i n the evident.  Syntheses  Most o f the r e c o r d e d v e r t i c a l early  were used  component seismograms  possess  c h a r a c t e r which i s q u i t e s i m i l a r t o the e q u i v a l e n t s y n t h e s e s .  particular,  the apparent  s y n t h e t i c responses  f r e q u e n c y o f the f i r s t  few  c y c l e s o f the  compares w e l l t o the f r e q u e n c y n a t u r e o f the  p a r t o f a l l o f the e q u i v a l e n t e x p e r i m e n t a l seismograms. trivial,  s i n c e the mantle Q assumption  i n c i d e n t wave f r e q u e n c y  c o n t e n t was  Venezuela  frequency  records.  The  parable.  c o n t e n t o f the s y n t h e s e s  f o r the the  The h o r i z o n t a l r e c o r d s are n o t so v i s i b l y  This p a r t i a l l y  h o r i z o n t a l motions.  the p o s s i b i l i t y o f good comparisons  Noise  also  w i t h low amplitude  vertical  m o t i o n s i n the l a t e r p o r t i o n s o f the v e r t i c a l component r e c o r d s . is  apparent  t h a t a l l the r e c o r d e d seismograms, a n d - p a r t i c u l a r l y  from the Novaya Zemlya, K u r i l e I s l a n d s and Ryukyu e v e n t s , do not as r a p i d l y as the e q u i v a l e n t s y n t h e s e s . and Ryukyu e v e n t s show d i s t i n c t the f i r s t  25 seconds  com-  r e s u l t s from the g r e a t e r r e l a t i v e n o i s e  c o n t a m i n a t i o n o f the low amplitude reduces  the  o n l y on the Novaya Zemlya and  l a r g e r 85° e p i c e n t r a l d i s t a n c e a l s o f a v o u r a b l y compares w i t h experimental records.  early  T h i s i s not  which a c t u a l l y d e t e r m i n e s  based  In  secondary  which s u g g e s t s  that  Furthermore, arrivals  It those decay  the K u r i l e I s l a n d s  ( p r o b a b l y pP) w i t h i n  the i n c i d e n t waveforms are  l o n g e r and more complex than t h e s e s i m p l e s o u r c e and mantle a b s o r p t i o n  107  models can d e s c r i b e .  I t would be d i f f i c u l t  to c r e a t e the  p o s s i b l e but c o m p l i c a t e d s o u r c e motion models which would d e s c r i b e the e x p e r i m e n t a l r e c o r d s . restrict are  any  dominated  comparisons by  theoretically exactly  Therefore, i t i s reasonable to  t o the i n i t i a l  few seconds  o f r e c o r d which  the b e g i n n i n g of the i n c i d e n t waveform r e s u l t i n g  t h e o n s e t o f s o u r c e motions.  However, even w i t h i n these f i r s t  s e c o n d s , many d i s c r e p a n c i e s between the theory and experiment still  p a r t i c u l a r , the s y n t h e t i c seismograms appear  the amplitude  of the f i r s t p u l s e o f o n s e t .  should r e s u l t  from the i n s t r u m e n t response  g e n e r a t e d by  the v e r y f i r s t passage  the c r u s t to the s u r f a c e . slightly  affect  adequate.  are  exaggerate  T h i s primary c h a r a c t e r to the s u r f a c e motions through  I n t e r n a l c r u s t a l r e v e r b e r a t i o n s can o n l y  o r mantle  Thus , i t becomes obvious  a t t e n u a t i o n models are n o t  completely  Most p r o b a b l y , the s o u r c e d i s p l a c e m e n t s are n o t as abrupt  as the impulse and s t e p d i s p l a c e m e n t models.  But, s i n c e the e x p e r i -  m e n t a l seismograms are contaminated by background correlations  noise, precise  c o u l d not have been expected even from exact  source  models. In  the f o l l o w i n g  and e x p e r i m e n t , used because  the LAR  more d e t a i l e d comparisons  between t h e o r y  r e c o r d i n g s from the V e n e z u e l a event are n o t  of t h e i r h i g h apparent n o i s e  Comparisons o f the Syntheses Neglecting  ponent  to  o f the i n c i d e n t waveform  t h e s e v e r y e a r l y m t o i ons.  t h a t the s o u r c e motion  3.18  few  obvious. In  motion  from  the n o i s y LAR  levels.  and Records Venezuela  seismograms show some motions  f o r 55° E p i c e n t e r  r e c o r d s , a l l v e r t i c a l com-  which are c o r r e l a t a b l e w i t h  the  COMPARISON OF EXPERIMENTAL AND SYNTHETIC SEISMOGRAMS FROM STATION LED  (a)  '  Venezuela  '(e)  F i j i Islands  (b) Kurile Islands  (f)  Chile  (c)  Novaya Zemlya  (g)  Ryukyu Islands  (d)  Synthesis for 55° epicenter  (h)  Synthesis for 85° epicenter  5 SEC. Fig. 37. A comparison of the vertical synthetic seismograms with the experimental seismograms of Figure 16 for station LED.  o  CO  equivalent  s y n t h e t i c motions to beyond 4 seconds f o l l o w i n g o n s e t .  such n o t a b l e  c o r r e l a t a b l e feature i s a strong, short negative  2 seconds w h i c h c o r r e s p o n d s to a c o n s i s t e n t e x p e r i m e n t a l F i g u r e 36 result  and  F i g u r e 16a,d,e;  from the s t r o n g  reflection  a l s o F i g u r e 37).  area  The  s i m i l a r but more compli'cated  synthesized  LED  (compare could  at 1 second experimental  pulses p a r t i c u l a r l y w e l l represent c h a r a c t e r and  those  relative  of  ampli-  Novaya Zemlya seismogram c o n t a i n s more c h a r a c t e r i n t h i s  (near 1 second) than the e q u i v a l e n t  s y n t h e s i s and  This pulse  at  the upper P r e c a m b r i a n boundary.  the K u r i l e I s l a n d s e v e n t s i n b o t h g e n e r a l tude.  feature  Another b r o a d p o s i t i v e p u l s e b e g i n n i n g  appears to c o r r e l a t e w i t h The  pulse  r e v e r b e r a t i o n o f the i n i t i a l motions i n v o l v i n g a  from the f r e e s u r f a c e and  (See F i g u r e 30).  features.  see  One  syntheses.  r e c o r d compare s u r p r i s i n g l y w e l l .  m u l t i p l e s which a r r i v e between 1 and c o n t r i b u t e t o the f o r m a t i o n Apart  2 seconds  The  However, the  several strong  ( F i g u r e 29) must  of t h i s b r o a d c h a r a c t e r i s t i c  from the p r e c e d i n g  CHA  largely  pulse.  comparisons, i t i s a l s o p o s s i b l e t o  c o r r e l a t e the v e r t i c a l component s y n t h e s e s and  the e q u i v a l e n t  experi-  m e n t a l seismograms on a peak to peak b a s i s t o almost 5 seconds f o l l o w i n g onset  (compare F i g u r e  these  l a t e r m o t i o n s w h i c h are lower than those  that these  37).  However, the  theory p r e d i c t s amplitudes f o r observed.  This indicates  l a t e r motions are at l e a s t p a r t i a l l y g e n e r a t e d by  t i n u i n g i n c i d e n t wavemotion at the lower c r u s t a l boundary. could contribute.  Little  significant  of the h o r i z o n t a l seismograms and  some conAlso,  noise  c o r r e l a t i o n i s o b v i o u s between  their equivalent  syntheses.  any  110 3. 19  Comparisons  of the Syntheses  Only the F i j i  and Records  f o r 85° E p i c e n t e r  event v e r t i c a l seismograms d i s p l a y  c h a r a c t e r which d i r e c t l y  initial  c o r r e s p o n d s t o the e q u i v a l e n t s y n t h e s e s  (compare  F i g u r e 37).  However, the C h i l e event r e c o r d s can a l s o be made c o m p a t i b l e  by i n v e r s i o n  ( m u l t i p l i c a t i o n by - 1 ) .  motions  This suggests that  the s o u r c e  f o r t h i s event are b e t t e r r e p r e s e n t e d by a d i l a t a t i o n a l  model r a t h e r than the c o m p r e s s i o n a l impulse model used.  The  seismograms p o s s e s s such v e r y complex e a r l y c h a r a c t e r t h a t are  somewhat ambiguous.  the  narrow p u l s e a t 2 seconds  For a F i j i  The b r o a d p u l s e f e a t u r e a t 1 second f o l l o w e d by i s a l s o e v i d e n t on these s y n t h e t i c  3. 20  the  Chile  The  from t h i s event a l s o appear t o compare q u i t e  comparison  However, the h i g h background  noise level  difficult.  S y n t h e s i s U s i n g an Input Wavelet Ulrych  and  of the e a r l y r e c o r d has been n e g l e c t e d .  f a v o u r a b l y w i t h the t h e o r y . makes a d e t a i l e d  seis-  c o r r e l a t i o n , the s m a l l amplitude p r e c u r s o r t o the  s t r o n g p o s i t i v e motion h o r i z o n t a l motions  Ryukyu  correlations  mograms and a g a i n c o r r e s p o n d s to s i m i l a r f e a t u r e s o f the F i j i records.  impulse  Obtained by D e c o n v o l u t i o n  (1970) has r e c e n t l y developed a method o f s e p a r a t i n g  convolved i n c i d e n t wavelet  from the c r u s t a l impulse  response  f u n c t i o n i n a seismogram u s i n g homomorphic n o n - l i n e a r f i l t e r i n g - ' methods  (Oppenheim e t a l , 1968).  The  t e c h n i q u e r e q u i r e s no  assumptions  on t h e c h a r a c t e r o f the w a v e l e t e x c e p t a c h o i c e o f i t s r e p r e s e n t a t i v e l e n g t h and r e q u i r e s no assumptions linearity.  of c r u s t a l p r o p e r t i e s except  U s i n g h i s homomorphic d e c o n v o l u t i o n p r o c e d u r e on the  V e n e z u e l a seismogram, U l r y c h o b t a i n e d a s h o r t wavelet which was  LED  (30 sample p o i n t s ) i n p u t  used f o r the r e - s y n t h e s i s o f the seismogram by  the  Ill  VENEZUELA-LEDUC  RESYNTHESIS  I 5 SEC.  F i g . 38. A comparison o f the v e r t i c a l component LED seismogram f o r the V e n e z u e l a event (Event 1, F i g u r e 16a) and the r e - s y n t h e s i s u s i n g the Ulrych wavelet. The i n t e r n a l c r u s t a l c a u s a l a b s o r p t i o n o f T a b l e I I has been i n c l u d e d .  l i n e a r systems  approach.  ( F i g u r e 38).  The  seismogram s y n t h e s i z e d w i t h  t h i s w a v e l e t p r o v i d e s a good t e s t o f the c r u s t model and method.  identical  I t i s not s u r p r i s i n g  t h a t the s y n t h e s i z e d seismogram i s n o t  to the e x p e r i m e n t s .  Such a f o r t u i t o u s  an e x a c t c r u s t model and w a v e l e t .  However, the remarkable  s i m i l a r i t y between the t h e o r y and experiment i s extremely encouraging. almost 6 seconds 1.5  seconds  The  r e s u l t would demand degree  f o r the v e r t i c a l  component  easily  c o r r e l a t e d c h a r a c t e r extends  f o l l o w i n g the o n s e t .  S i n c e the i n p u t w a v e l e t was  i n l e n g t h , i t i s obvious t h a t  these l a t e r motions  of  are  to only  112  generated by the c r u s t a l response. the h o r i z o n t a l component. converted  The synthesis s t i l l poorly  represents  Because the h o r i z o n t a l motions are mostly  SV energy which a r i s e s at the i n t e r n a l c r u s t a l boundaries, i t  is apparent that s i g n i f i c a n t model discrepancies must e x i s t .  113  CHAPTER IV CONCLUSIONS  4.1  Summary o f A n a l y s e s Between 0 and 2 Hz most of. t h e e x p e r i m e n t a l  exhibit general  f e a t u r e s which were t h e o r e t i c a l l y p r e d i c t e d by the l i n e a r  systems s o l u t i o n s . were found.  V/H s p e c t r a l r a t i o s  I n p a r t i c u l a r , , expected 0.7 and 1.7 Hz r a t i o peaks  However, except f o r such low f r e q u e n c i e s , the s p e c t r a l r a t i o  comparisons between t h e o r y  and experiment have r e v e a l e d l i t t l e  c o n f i r m a t i o n o f the chosen c r u s t a l models.  further  The o b s e r v e d c o r r e l a t i o n s  below 2 Hz s u g g e s t t h a t on a s c a l e determined by the l o n g wavelengths o f the low f r e q u e n c i e s , the c r u s t models are almost s u f f i c i e n t , b u t on a s m a l l s c a l e , s i g n i f i c a n t model d i s c r e p a n c i e s must e x i s t . a t t e n u a t i o n was v i r t u a l l y peaks w h i c h t h e o r y  indeterminable  i n d i c a t e d should  experimentally  The e f f e c t o f since the s p e c t r a l  reveal c r u s t a l absorption  were n o t c o n s i s t e n t enough t o d e r i v e any c o n c l u s i o n s .  properties  However, the  t h e o r e t i c a l s o l u t i o n s c o n c l u s i v e l y demonstrated t h a t a t t e n u a t i o n a r e c o n s i d e r a b l e beyond 2 Hz f o r a r e a l i s t i c  c r u s t a l absorption  and must be i n c l u d e d i n any comprehensive r a t i o  analysis.  T h i s , however, does n o t imply  a t t e n u a t i o n cannot be important The  at higher  seismo-  that.  frequencies.  v e r t i c a l - t o - v e r t i c a l experimental  r a t i o s showed  c h a r a c t e r as was t h e o r e t i c a l l y p r e d i c t e d below 3 Hz. the s p e c t r a l d e t a i l n e c e s s a r y  model  The- v i s i b l e  e f f e c t s o f c a u s a l c r u s t a l a t t e n u a t i o n o f the time s y n t h e s i z e d grams appeared t o be s l i g h t .  effects  little  Unfortunately,  t o d i s t i n g u i s h and i n t e r p r e t r e l a t i v e  c r u s t a l l a y e r i n g and a b s o r p t i o n p r o p e r t i e s does n o t appear t o be available.  114  Considering and  the e x t r e m e l y s i m p l e  assumptions on s o u r c e motions  mantle p r o p e r t i e s which were used f o r the  time domain s y n t h e s e s ,  s i m i l a r i t y between the e a r l y p o r t i o n s of the t h e o r e t i c a l and experimental  vertical  However, t h e r e was equivalent  component seismograms was  h o r i z o n t a l records  t h e o r e t i c a l s y n t h e s e s and achieved  and  syntheses.  The  using  the s i m p l e s t  some remarkably a c c u r a t e  Ulrych  the  homomorphic  comparison between the successes  of models f o r s o u r c e m o t i o n s , input  the  encouraging.  e x p e r i m e n t . • With t h e s e r e a s o n a b l e  c o u l d expect t h a t b e t t e r c o n c e i v e d  The  most o f  a l s o a d i s t u r b i n g l a c k of c o r r e l a t i o n between  d e c o n v o l v e d i n p u t w a v e l e t determined the b e s t  already  very  the  one  incident wavelets could  provide  syntheses.  p o s s i b l e causes of any  m o d e r a t i o n o f the  success  Background n o i s e , i n e x a c t c r u s t a l models, s c a t t e r i n g and  are v a r i e d .  p o s s i b l e non-  l i n e a r e f f e c t s c o u l d a l l c o n t r i b u t e to d i s c r e p a n c i e s between t h e o r y experiment.  Among these p o s s i b l e f a c t o r s , s c a t t e r i n g e f f e c t s due  i n h o m o g e n i e t i e s i n the l a y e r s , non-plane i n t e r l a y e r b o u n d a r i e s surface  (topography) c o u l d i n v a l i d a t e t h i s s i m p l e  f o r a r e a l s e d i m e n t a r y bedded c r u s t . shown t h a t the e f f e c t r a p i d l y with not  inconsistent with  The  Chernov (Grant  c o r r e l a t i o n s obtained  this s c a t t e r i n g hypothesis.  to  and  l i n e a r systems s o l u t i o n and West, 1965)  of s c a t t e r i n g i n an inhomogeneous medium  frequency.  and  at low  has  increases  frequencies  A l t h o u g h , the  is  chosen  e v e n t s appeared to p o s s e s s l a r g e average s i g n a l - t o - n o i s e r a t i o s , b a c k ground n o i s e to 3 Hz.  dominates the P coda o u t s i d e  T h i s undoubtedly c o n t r i b u t e d  the s i g n a l band from 0.5  to the  lessened  success  at  Hz high  frequencies. It  i s p o s s i b l e that inaccuracy  c r u s t a l models has  been a t t a i n e d u s i n g  and  insufficient  detail in  the i n t e r p o l a t i o n between  the  the  115 available oilwell d r i l l may  be  sites.  C o r r e l a t i o n s between t h e o r y  g r e a t l y improved at h i g h  frequencies  by  and  u s i n g more a c c u r a t e  w h i c h would r e q u i r e more complete knowledge o f the c r u s t . f r e q u e n c y domain computations can f o r any  for d i g i t a l  Although  t h e o r e t i c a l l y exact  time domain s y n t h e s e s .  domain s o l u t i o n s r e q u i r e t h a t motions i s an the 0.05  exact  layer transit  second i n t e r v a l used i n the  Such d i f f i c u l t i e s  SV  transit  the  solutions  any  C o n c l u d i n g Remarks and  p r o b l e m has  sampling i n c r e m e n t .  t h e s i s , the t h i n n e s t  overcome by  reducing  storage  f o r b o t h n o r m a l and  the  interval. sampling  However, p a r a l l e l tolerated.  Suggestions  shown some p a r t i c u l a r l y  non-normal i n c i d e n c e  been s u g g e s t e d f o r the e t i c a l P coda V/H  ratio  standard  useful features.  analyses  demonstrates t h a t  propagation a n a l y s i s using  The  dependence on  has  fact  that  attenuation  the p r e v i o u s  V/H  has  the  theor-  attenuation  models  attenuation  cannot be n e g l e c t e d  s h o r t p e r i o d P-coda.  e f f e c t below 1 Hz, frequencies  include layer  f o r the c r u s t and  c r u s t a l absorption  I t becomes *  although t h i s extension  techniques.  showed the s i g n i f i c a n t  a u t h o r s at low  For  l i n e a r systems s o l u t i o n o f the m u l t i l a y e r p r o p a g a t i o n  already  showed l i t t l e  SV  l a y e r s would  r e q u i r e m e n t s must be  f i r s t mathematical procedure to e x p l i c i t l y  used h e r e i n  are  time  time f o r b o t h P and  d e s i r e d l e v e l of accuracy.  i n computer time and  The  example, the  times p r e c i s e l y e q u a l t o t h i s  c o u l d e a s i l y be  increment to o b t a i n increases  the  For  i n t e g r a l m u l t i p l e of the  e x h i b i t b o t h P and  the  models  model, f u r t h e r assumptions which g e n e r a t e model i n a c c u r a c i e s  necessary  4.2  provide  experiment  beyond 2  i n any  Also, since  complete  attenuation  r a t i o work of  been v a l i d a t e d w i t h r e s p e c t  to  Hz  several  attenuation.  116  The the time and should of  l i n e a r systems approach has frequency  formulations  a i d i n the c o n c e p t i o n  frequency  and  the p a r t i c u l a r advantage t h a t  are c o m p l e t e l y  parallel.  of the r e l a t i o n s between the  time s o l u t i o n s .  Furthermore, two  This  characteristics  independent  viewpoints  s h o u l d p r o v e to be v e r y u s e f u l . The  f o r m u l a t i o n admits a n a l y s i s and s o l u t i o n u s i n g the l a r g e  body o f communications t h e o r y mathematics.  Such methods have, i n the  p a s t , been i n v a l u a b l e f o r the r e d u c t i o n and  a n a l y s i s of  data.  T h e i r f u r t h e r a p p l i c a t i o n to the d i r e c t  propagation producing  problem may  geophysical  formulation of  prove as u s e f u l by p r o v i d i n g new  the  insights  and  g e n e r a l i z a t i o n s of s o l u t i o n s . It  s h o u l d be p o s s i b l e to extend  p r o p e r t i e s which a r e n o n - l i n e a r .  s o l u t i o n s to i n c l u d e  F o r example, a p p r o p r i a t e  crustal  non-linear  o p e r a t o r s which c o u l d d e s c r i b e p o s s i b l e n o n - l i n e a r i t i e s o f  crustal  propagation  formulation  and  c o u l d be  s o l v e d by  the  strictly valid frequency regard.  inserted into  the l i n e a r l y  time s y n t h e s i s approach.  restricted  S i n c e s u p e r p o s i t i o n i s not  f o r n o n - l i n e a r systems, p r e v i o u s methods which r e q u i r e  i n c r e m e n t e d o r modal s o l u t i o n s would not be  sufficient  A l s o , the s o l u t i o n o f s e t s of n o n - l i n e a r d i f f e r e n t i a l  d e s c r i b i n g non-linear propagation  would p r o b a b l y  be much more  in this  equations difficult  t h a n the n o n - l i n e a r e x t e n s i o n o f the time domain systems s o l u t i o n . may  be d i f f i c u l t  to c o n c e i v e n o n - l i n e a r i t i e s  s t r a i n propagation. and  infinitesmal  i n r e a l c r u s t a l m a t e r i a l s where s c a t t e r i n g  a b s o r p t i o n a r e known to o c c u r , n o n - l i n e a r d e s c r i p t i o n s have been  hypothesized  ( e . g . , Walsh, 1966).  Although this  But,  in ideal  It  no  s o l u t i o n s f o r i n c i d e n t S waves were attempted i n  t h e s i s , the l i n e a r systems f o r m u l a t i o n i s d i r e c t l y a p p l i c a b l e .  117  F u r t h e r m o r e , r e f l e c t i o n s e i s m o l o g y c o u l d be extension  o f the normal i n c i d e n c e  i s important  c o n s i d e r e d by  transmission  problem.  a  trivial  However, i t  to n o t e t h a t the systems f o r m u l a t i o n demands p l a n e waves  so t h a t p o i n t s o u r c e s  w i t h i n the c r u s t cannot be  allowed.  Most  other  r e f l e c t i o n s e i s m i c s o l u t i o n s a l s o r e q u i r e p l a n e waves, n o r m a l i n c i d e n c e , and n e g l e c t the e f f e c t s o f g e o m e t r i c a l from e x p l o s i o n The would p e r m i t ideally  spreading  which a c t u a l l y r e s u l t s  sources. systems f o r m u l a t i o n may  a general  provide  a new  s o l u t i o n o f the important  t r a n s i t times and  cannot be  The  I)  r e q u i r e d t o determine l a y e r  I t s h o u l d be p o s s i b l e to  from the e q u i v a l e n t seismogram.  so unambiguous as  thicknesses.  (see Appendix  For  r e l a t i v e a c o u s t i c impedance v a r i a t i o n s f o r b o t h n o r m a l  and non-normal i n c i d e n c e c o n d i t i o n s . this information  which  i n v e r s e problem.  l a y e r e d c r u s t s , the l a y e r t r a n s f e r f u n c t i o n s  appear to c o n t a i n a l l the i n f o r m a t i o n  viewpoint  Inverse  obtain  T h i s approach, however,  to determine l a y e r v e l o c i t i e s , d e n s i t i e s and  s o l u t i o n s are now  being  attempted.  a u t h o r f e e l s t h a t s t u d i e s u s i n g the l i n e a r systems  f o r m u l a t i o n c o u l d prove to be generalized propagation  and  extremely f r u i t f u l  inverse solutions.  f o r b o t h more  118 REFERENCES  A l b e r t a S o c i e t y of P e t r o l e u m G e o l o g i s t s . 1964. western Canada.  G e o l o g i c a l h i s t o r y of  Anderson, D.L., Ben-Menahem, A. and Archambeau, C.B. 1965. Attenuation of s e i s m i c energy i n the upper mantle. J . Geophys. Res. 70, pp. 1441-1448. Basham, P.W. 1967. Time domain s t u d i e s ' o f s h o r t p e r i o d t e l e s e i s m i c P phases, M.A.Sc. T h e s i s , U n i v e r s i t y o f B r i t i s h Columbia. Bohn, E.V. 1963. The Wesley, Reading,  transform p. 100.'  a n a l y s i s of l i n e a r systems. ' • .  Carnahan, B., L u t h e r , H.A., and W i l k e s , J.O. methods. W i l e y , New York. p. 80.  1969.  Applied  Addison-  numerical,  C a r p e n t e r , E.W. 1967. T e l e s e i s m i c s i g n a l s c a l c u l a t e d f o r underground, underwater and atmospheric e x p l o s i o n s . G e o p h y s i c s , _32, pp. 17-32. C l a e r b o u t , J.F. 1968. S y n t h e s i s of a l a y e r e d medium from i t s a c o u s t i c t r a n s m i s s i o n response. G e o p h y s i c s , 3_3, pp. 264-269. Clowes, R.M. and Kanasewich, E.R. 1970. 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A t t e n u a t i o n of shear and c o m p r e s s i o n a l waves i n P i e r r e Shale. Geophysics, 23, pp. 421-439. M u e l l e r , S. and Ewing, M. 1962. S y n t h e s i s of n o r m a l l y d i s p e r s e d wave t r a i n s by means of l i n e a r systems t h e o r y . Lamont G e o l o g i c a l O b s e r v a t o r y , Columbia U n i v e r s i t y , C o n t r i b u t i o n No. 559. Oppenheim, A.V., S c h a f e r , R.W. and Stockham, T.G., J r . 1968. Non-linear f i l t e r i n g of m u l t i p l i e d and convolved s i g n a l s . P r o c . IEEE 56, pp. 1264-1291. Phinney, R.A. "1964.. S t r u c t u r e of the E a r t h ' s c r u s t from s p e c t r a l b e h a v i o u r of l o n g p e r i o d waves. J . Geophys. Res. 69, pp. 2997-3017. P r e s s , F. 1964. Res. 69_, pp. Richter, C F . p. 670.  S e i s m i c wave a t t e n u a t i o n i n . t h e c r u s t . 4417-4418.  1958.  Elementary s e i s m o l o g y .  Freeman, San  J . Geophys.  Francisco,  Robinson, E.A. and T r e i t e l , S. 1966. S e i s m i c wave p r o p a g a t i o n i n l a y e r e d media i n terms o f communication t h e o r y . Geophysics, 31, pp. 17-32. Savage, J . C. 1 9 6 5 . A t t e n u a t i o n of e l a s t i c waves i n g r a n u l a r mediums. J . Geophys. Res. 7_0, pp. 3 9 3 5 - 3 9 4 2 .  120  Sherwood, J.W.C. 1962. The s e i s m o l i n e , an analog seismograms. G e o p h y s i c s , 27, pp. 19-34.  computer o f  Thomson, W.T. 1950. T r a n s m i s s i o n o f e l a s t i c waves through s o l i d medium. J . A p p l . Phys. _21, pp. 89-93.' T u l l o s , F.N. and sediments.  R e i d , A.C. Geophysics  1969. 34, pp.  theoretical  a stratified  Seismic attenuation i n Gulf coast 516-528.  Ulrych, T.J. 1971. A p p l i c a t i o n of homomorphic d e c o n v o l u t i o n to seismology. Submitted f o r p u b l i c a t i o n . Walsh, J.B. 1966. S e i s m i c wave a t t e n u a t i o n i n rock due J . Geophys. Res. 71, pp. 2591-2599.  to f r i c t i o n .  Ware, J.A. and A k i , K. 1969. Continuous and d i s c r e t e i n v e r s e - s c a t t e r i n g problems i n s t r a t i f i e d e l a s t i c medium. J . A c o u s t . Soc. Amer. 45, pp. 911-921. Werth, G.C. and H e r b s t , R.F. 1963. Comparisons of amplitudes of seismic/.' waves from n u c l e a r e x p l o s i o n s i n f o u r mediums. J . Geophys. Res. 68, pp. 1463-1475. W i l l i a m s , G.D. and Burk, C.F., J r . 1964. Canada. A l b e r t a S o c i e t y of Petroleum Chapter 12. Wuenschel, P.C. Geophysics  G e o l o g i c a l H i s t o r y of Western G e o l o g i s t s , Calgary.  1965. D i s p e r s i v e body waves - an e x p e r i m e n t a l 30, pp. 539-551.  study.  121  APPENDIX I IMPULSE RESPONSE FUNCTION FOR NORMAL INCIDENCE Al.1  Laplace The  Inversion of Transfer  Functions  normal i n c i d e n c e t r a n s f e r f u n c t i o n s e t :  H. ( ) = , g<?> S  V  U  i ^  S ;  l-F(s)G(s)  2  H a  ( )  °  S  ±  V  N  • t  ±  l-F(s)G(s)  3  2  (  s  )  l-F(s)G(s)  KS)  " l-F(s)G(s)  (  V i • t 'r t  ( A 1 _ 1 )  i i r  i  fc  -sT where G ( s ) = A ,(s)e  and  j  must be i n v e r s e L a p l a c e  F(s) = G C s ) ' ^ '  transformed t o obtain the c h a r a c t e r i s t i c l a y e r  impulse response s e t :  h Rather than d i r e c t l y  k  (t) = L  _ 1  {H  ^  (s)} .  attempt t o i n v e r t t h e c l o s e d forms above, i t i s  e a s i e r to return to the s e r i e s representation term by term. theory  and a c c o m p l i s h i n v e r s i o n  T h i s approach i s u s e d , f o r example, i n t r a n s m i s s i o n  line  ( e . g . , Bohn, 1963, C h a p t e r 11) because the c l o s e d forms a r e  generally unsuitable of a g e n e r a l  f o r Laplace  absorption  inversions.  Also, with  the i n c l u s i o n  f u n c t i o n A ^ ( s ) , the p o l e s e t r e q u i r e d f o r d i r e c t  122  inversion  cannot be  calculated.  Thus f o r :  -3sT  -sT H  ±  (s)= t {A (s)e j L  Kk^Cs)*  i  1  -5sT  r r i  +A  , i  5 i  (s)e  \  2  *  ±  '  2  +  ••*> (Al-2)  the i m p u l s e r e s p o n s e s e r i e s  i s e a s i l y determined by term by term  inversion:  h. ( t ) = t . { a . ( t ) * 5 ( t - T . ) + a , ( t ) * a , ( t ) * a , ( t ) * S ( t - 3 T )r r. ,1^ i i i l I I i l l  1  (Al-3) + a (t)*...*a (t)*5(t-5T )r ±  where  Noting  i  a (t) =  i  2 i  r  2 i  +  ...}  L~ {A (s)} 1  ±  that:  -sT H  i  (  s  )  =  2  ( i' i r  A  t  H  and  we may f u r t h e r  i  (  s  )  =  3  H  (s) =  ( s ) e  'r  > i/ > H  —sT  ("VSi 6 0 0  1 ) H  -r-  s  H  ix  ( s )  < -*> M  (s)  determine:  h. l  2  (t) = r ' a.(t)*6(t-T.)*h. (t> 1 1 l ±i  (Al-5)  123  t *r. h, (t) = - ft - i -a.(t)*5(t-T.)*h. (t) 13 i l l  'i  1  (Al-5)  t,  1  and  h, (t) = iu t  For non-attenuating may  be represented  *h. (t)  .  11  ±  conditions, a ^ ( t ) = l and the four response functions i n s i m p l i f i e d i n f i n i t e s e r i e s form:  >i (t) = t  I  ±  n=l  1  h.±2 (t) =  t r ±  ±  (r.^'T  I  n=l  X  5(t-[2n-l]T ) i  (r.r.'V^Vt^nT^ (Al-6)  h  i  h.  and  1(  Al.2  ( T )  V i  =  r  3  ^ (r r *) " 6(t-2nT ) n=l n  i  i  (t) = t . ' I ( r . r » ) * n=l 1  1  1  i  n  1  6(t-[2n-l]T )  1  Poles and Zeros of the Normal Incidence  1  Transfer  Functions  The four t r a n s f e r functions (equation Al-1) each have a common denominator which determines the pole set at complex frequencies for  which: -2sT l-(r r )A. (s)e ,  i  i  2  1  = 0  (Al-7)  124  S i n c e the A ^ ( s )  can r e p r e s e n t any  s p e c i f i c p o l e s e t can be the non-attenuating  g e n e r a l a t t e n u a t i o n f u n c t i o n , no  determined w i t h o u t  case, A^(s) = 1 .  assumptions on A ^ ( s ) .  For  A p h y s i c a l constraint (conservation  of energy) demands t h a t the r e f l e c t i o n c o e f f i c i e n t s  a t the two  elastic  b o u n d a r i e s o f l a y e r i obey:  < 1  s i s a complex number and  (Al-8)  the e x p o n e n t i a l must obey the  periodic  relationship:  -2sT  e  where n i s any  i  -2sT  = e  i  (Al-2) f o l l o w s :  1  e  the n a t u r a l l o g a r i t h m o f t h i s  2 ( 1 ^ - 3 ^ )  and  (Al-9)  arbitrary integer.  Thus a s o l u t i o n of e q u a t i o n  Taking  i2nir  •e  =  (Al-10)  equation:  -ln(r  r^)  (Al-11)  t h e r e a r e an i n f i n i t e number o f s o l u t i o n s f o r s:  S  r  = ^-{inir+ -|ln(r r ^ ) }  (Al-12)  Two c o n d i t i o n s on the c o e f f i c i e n t s y i e l d d i f f e r e n t general pole patterns:  Case I  -  0 < r.r.'< 1 i l  Thus l n ( r ^ r ^ ' ) i s negative r e a l and the pole p o s i t i o n s are shown i n Figure 39.  The poles are p e r i o d i c w i t h the rate to = TT/T. ( f = — — ) i 2T 1 the f i r s t non-zero frequency pole i s at f j = . 1  LU  + s  2T  £  -/  ;lniVi'l  Fig.. 39. Pole maps of the one l a y e r t r a n s f e r f u n c t i o n f o r normal i n c i d e n c e : Case I .  126  Case I I  1 < r.r.' < 0 l l  Set r ^ r . ' =  [r.r.' le* . 7 1  Thus l n ( r . r . ' ) i s complex w i t h  principal  value: (Al-13)  l n ( r . r . ' ) = l n l r . r . ' l+iir . i i l l 1  The p o l e s  = ~  1  {i(n+l/2)+l/21n| r  | } are shown i n F i g u r e 40.  the p o l e s are p e r i o d i c w i t h r a t e f =' p o l e has  the f r e q u e n c y component f  +s +s  =  2T  ±  4T,  '  However, the  fundamental.  These fundamental  poles  LU  + S;  +s7  2T  £  F i g . 40. P o l e maps o f the one i n c i d e n c e : .Case I I .  Again,  l a y e r t r a n s f e r f u n c t i o n f o r normal  1  a t fi  -  o r  \  f o r Case I and f i  = °  — f o r Case I I r e p r e s e n t i  r e v e r b e r a t i o n resonance f r e q u e n c i e s due t o the l a y e r ; r e p r e s e n t the harmonics. For non-normal  7  t h e fundamental  the p e r i o d i c  poles  Each t r a n s f e r f u n c t i o n has z e r o s a t i n f i n i t e s. i n c i d e n c e , the computation o f p o l e s and zeros  i s much more d i f f i c u l t because the n a t u r e t r a n s f e r f u n c t i o n i s h i g h l y complicated  o f the denominator o f t h e  (Appendix I I I ) .  However,  s i m i l a r g e n e r a l f e a t u r e s s h o u l d p r e v a i l f o r i n c i d e n c e a n g l e s which a r e not l a r g e .  2  128  APPENDIX I I PHASE ADVANCE TRANSFER FUNCTIONS  Consider  any l a y e r i w i t h o r i g i n as shown i n F i g u r e  4 1 . The  d i r e c t P wave r a y w h i c h e x i t s t h e upper boundary at, x=0, z=n^ e n t e r s the l o w e r boundary a t x=a, z=0. The phase a t a g i v e n i n s t a n t a t ( , 0 ) a  is t i m e advanced r e l a t i v e t o t h e phase a t p o i n t P and hence a t t h e origin  (0,0) a c c o r d i n g  t o the trigonometric  relation:  x = (n.tan(e.)sin(e.))/a. p. i l i i  .  (A2-1)  LAYER i  (a,0)  (b,0)  F i g . 41. A r e p r e s e n t a t i o n o f phase advance time c a l c u l a t i o n s n e c e s s a r y f o r non-normal c o n d i t i o n a n a l y s e s .  129  Similarly for the direct S wave entering at (b,0), the time advance between (b,0) and point Q i s :  T  = (n tan(f )sin(f.))/g.  s^  1  1  1  1  .  (A2-2)  These advance times represent any one transit from one boundary to the other for the P and S waves respectively. Higher orders of reflections are included through further multiplication by identical advance functions.  The included advance transfer functions of layer in Figure 8  exactly establish the correct phase relations for any possible combinations of internal reverberations. The phase advance transfer functions and the layer delay functions may be combined into single delay functions: -sT'  -sT* and  e  e  i = T -x =—cos(eJ P P a i n  where  T' P  ±  ±  ±  (A2-3)  ±  i -x =-r cos(fJ i i i n  and  T' S  i  = T S  i  S  B  1  .  (A2-4)  130  APPENDIX I I I EXAMPLE  It represent  BLOCK REDUCTION  PROCEDURE  i s d e s i r a b l e to obtain a s i n g l e t r a n s f e r f u n c t i o n H(s) to  a system component  ( F i g u r e 42) such t h a t V ( s ) = H ( s ) V . ( s ) .  o  1  V(s)  Vj (s)  0  F i g . 42. The d e s i r e d b l o c k diagram r e p r e s e n t a t i o n i n terms o f t r a n s f e r f u n c t i o n and t r a n s f o r m e d s i g n a l s .  Two common system b l o c k diagrams can be e a s i l y using Case I  the standard -  conformed t o t h i s  b l o c k diagram r e d u c t i o n methods (Kuo, 1963).  Summed p a r a l l e l systems  ( F i g u r e 43):  H(s) 2  F i g . 43.  Block  diagram o f p a r a l l e l  summing p a t h s .  form  131  V  (s) =  (H (s)+H (s))V.(s) 1  (A3-1)  2  O  X  H(s) = HiCsJ+^Cs)  Set:  Case I I  -  Continuous  (A3-2)  (feedback systems) l o o p s ( F i g u r e  44)  Hj(s) V >  = l-H (s)H (s) *  S  1  2  (A3-3)  V > S  H!(s) Set:  H(s) =  1  r  '  (A3-4.)  l-H (s)H (s) 2  H[(s)  V(s) 0  H(s) 2  Fig.  44.  B l o c k diagram  o f the l i n e a r p o s i t i v e  feedback  system.  132  Example  Reduction: Refer to Figure 8 during the following reductions.  Given the equation:  U,  (s) = H. (s)U.(s)  (A3-5)  X  1—JL  which relates an up travelling P wave entering layer i to the equivalent up travelling P wave entering layer i-1, determine  H  It  i !  (  s  )  =  U  i_l  (  s  )  /  i  U  (  s  )  '  ( A 3 _ 6 )  Is f i r s t necessary to determine the P and SV motions incident on the  upper boundary due to the P and SV motions at the lower boundary. For simplicity, l e t the forward absorption and delay transfer functions be represented by (see Appendix I I for definition of T  Gi(s) = A  p  G (s) = A 2  (s)e  .  (s)e  1  1  and T ' ) :  (A3-7)  (A3-8)  i  S  and the return transfer functions:  B!(s) = r  r  .G^s)  B (s) = r r ,G (s) p s sp ' 2  z 2  ±  B (s) = r  r  3  S  i  p  i  ,Gi(s) p  p  i  (A3-9)  133  Bi*(s) = r  ss  r ±  B (s) = r 6  PP  B (s) 7  = r  r ±  r  ss^  s  ,G (s) 2  P ±  Ps  .G^s)  (A3-9)  ±  ss^  ,G (s) 2  Combining the p a r a l l e l t r a n s f e r f u n c t i o n s (Case I) the s i m p l i f i e d  block  r e p r e s e n t a t i o n of F i g u r e  with  its  r e t u r n path  function  45 i s p o s s i b l e .  F u r t h e r combining G j ^ s )  t r a n s f e r f u n c t i o n and G ( s )  with i t s r e t u r n path t r a n s f e r  2  form:  G (s) X  F  l  (  s  )  =  l-G (s)(B (s)+B (s)) 1  1  2  (A3-10) G (s) 2  ?  2  (  S  )  =  i-G (s)(B (s)+B (s)) 2  Since i t i s intended  7  8  o n l y to o b t a i n the l a y e r r e s p o n s e t o the i n c i d e n t  P wave, t h e o n l y non-zero i n p u t s i g n a l i s U (s) . t r a n s f e r f u n c t i o n ( o b t a i n e d by b r e a k i n g is F (s)F (s)(B (s)+B^(s))(B (s)+B (s)). 1  2  3  5  6  The  the c o n t i n u o u s The  t r a n s f e r f u n c t i o n s combine to form (Case I I ) :  t o t a l open l o o p l o o p of F i g u r e  open l o o p and  forward  44)  134  U'(s)  B (s)  +  3  t  B (s) 4  :  V,(s)&  V'(s)  F i g . 4 5 . The reduced system block diagram having the l a y e r response to inputs U^(s) and V ^ ( s ) .  U'(s)  Fi(s)  l-F (s)F (s)(B (s)+B (s))(B (s)+B (s)) 1  2  5  6  3  4  (A3-11) Fi(s)F (s)(B (s)+B (s))  V'(s)  uTTsj"  2  5  6  l - F ( s ) F ( s ) ( B ( s ) + B ( s ) ) (B (s)+B (s)) 1  2  5  6  3  tt  The. f i n a l transmission of these P and SV motions i n c i d e n t on the upper boundary through the upper boundary determines U\ ^ ( s ) :  135  U, , ( s ) = t U*(s)+t V'(s) i-1 pp sp ±  so  (A3-12)  1  that:  U  H,  (s) =  i - l  (  s  )  F  l  (  s  )  '  t  D D  + F  l(s)F2(s)(B (s)+B (s))-t 5  6  1  U (s) ±  1  l-F (s)F (s)(B5(s)+B (s))(B (s)+B (s)) 1  2  s  3  l t  (A3-13) Each t r a n s f e r f u n c t i o n H  ( s ) can b e ' o b t a i n e d i n a s i m i l a r  •k A l l 16 t r a n s f e r f u n c t i o n p o s s e s s the same denominator.  manner,  136  APPENDIX IV FORTRAN COMPUTER PROGRAMS FOR TIME DOMAIN SEISMOGRAM SYNTHESES AND FOURIER COMPONENT SOLUTIONS BASED ON THE LINEAR SYSTEMS THEORY  The f o l l o w i n g FORTRAN  IV-G programs were used f o r the  n u m e r i c a l s o l u t i o n s o f the t r a n s m i s s i o n problems i n t h i s I:  thesis.  NON-NORMAL INCIDENCE CRUSTAL FOURIER TRANSFER  FUNCTION PROGRAM T h i s program was used f o r the frequency domain function  transfer  solutions. I I : NON-NORMAL INCIDENCE SYNTHETIC SEISMOGRAM FOR ANGLES  LESS THAN  CRITICAL  T h i s program was used f o r the time domain s y n t h e s e s f o r non-normal i n c i d e n c e c o n d i t i o n s .  seismogram  On an IBM  System 360 Model 67 computer o p e r a t i n g under t h e M i c h i g a n T e r m i n a l System, the c o m p u t a t i o n a l time was a p p r o x i m a t e l y 0.005 second p e r l a y e r p e r time i n c r e m e n t f o r n o n - a t t e n u a t i n g c r u s t models. The c o m p u t a t i o n a l time was s l i g h t l y i n c r e a s e d f o r a t t e n u a t i n g  layer  models. •III:  SUBROUTINES  A l l subprograms r e q u i r e d by the above two programs a r e following.  INCIDENCE  •»Uri"iMOKMAl * : ; ;  *  *  *  *  *  *  *  *  *  *  *  *  *  ;|::;:  *  *  CRUSTAL *  :|:  *  *  *  :1:*  *  FOURIER *  *  *  *  *  *  *  *  TRANSFS* *i\t  *  :|c  *  *  *  *  *  *  .-UNCTION  PROGRAM  *  *  *  *  *  ;s  5  *  *  *i\:  *  *  *  *  •  :\:  PROGRAM D E S C R I P T I O N TOE PROGRAM USES THE FPJHUJENCY DOMAIN L I N E A R SYSTEMS C R U S T A L MODEL FOR COMPUTATION Oh SAMPLED F RE OU EM CY C O M P O N E N T FOUR t E R T R A N S F E R FONCTIONS PROGRAM L I M I T A T I O N S •i"«X 11 H !i i MO.  OF  LAYERS-10  MAXIMUM MO. O F FREQUENCY POINTS C A L C U L A B L F=3 0 '\ C A U S A L 7. E RO'-'P HAS t A T T E N U A T I O N I S ASSUMED «•«••> THE PROGRAM MAY J -. G E N E R A L I Z E D TO C A U S A L A T T E N U A T I O N BY ALLOWING COMPLEX A B S O R P T I O N F N C S . 'INPUT CARDS . CARD 1: T I T L E CARD  2 : NO.  OF  LAYERS,. NO.  OF  FREQUENCY  POINTS* I N I T I A L  FREQUENCY',  F I N A L FRF.OUF.NCY 3 : OUTPUT L I S T AND ATTENUATOR F L A G S OUTPUT L I S T NFLAG 1 = 0 ATTENUATION NFL AG 2 = 0 4: FLAGS TO CHOOSE PROGRAM O P T I O N S "••« . RAT 10 PROGRAM MR/\TIO = 0 D I S P L A C E M E N T PROGRAM NOISPL=0 5 F F : L A Y E R T H I C K N E S S , P V E L O C I T Y , S V E L O C I T Y , D E N S I T Y , 0 FO< 0 FOR S V . I F THE ATTENUATOR F L A G = 0 S K I P THE it SPEC IF ICAT IONS CARD 6: P OR S-'WAVE I N C I D E N C E ANGLE VII TH NORMAL «•» THE INPUT NAVE TYPE IS D E T E R M I N E D BY WHICH ANGLE I S N U W . F . R O . FOR NORMAL IMC IDE N C F VERY SMALL I N C I D E N C E ANGLES MAY USED W I T H O U T D E T E C T A B L E i=RR-') I 01 MEMS I Or! AL P ( 11 ) , B ET ( 1.1 ) ,RHO( 11 ) , II ( 10 ) , : ( 11 ) , I- ( 11 ) , T P P ( 2 1 ) ,TPS( 21 1) ,TSP( 21 ) ,TSS ( 21 ) , R S P ( 2 1 ) ,RSS ( 21 ) ,RPS( 21 ) , R P P ( 21 ) , FR F0{ 5 0 ) , VP( 11 ) 2 , V S ( 1 1 ) , T P ( 1 0 ) , T S ( 1 0 ) , A P ( 1 0 ) , A S ( 1 0 ) , E X P P( 1 0 ) ,E X P S( I D ) , S T A T ( 1 5 ) , 0 X ( 3 1 1 ) ,W7. ( 11 ) , PANGLE ( 11 ) , S ANGL E ( 11 ) , A ( 4 0, 41 ) , ii { 4 0 , 4 1 ) , X( 4 0 , 1 ) , S INE( 11 4 ) , S TMF( 11 ) , COSE ( 11 ) ,COSF { 11 ) ,TAUP( 10 ) , TAILS ( 10 ) , R A T ( SO ) ,OAMP( 1 1 , 50 ) 5 , 0 P H S ( 1 1 , 5 0 ) ,WAMP(1L,50) , W P H S ( 1 1 , 5 0 ) , 0 P ( 1 0 ) ,0S( 10) COMPLEX E X P P , E X P S ,G1 ,G2 ,A ,H1 ,H2,H3 ,H4 ,X ,DET, B, 0 1 , 02 , 0 3 , 0 4 , 0 5 , 0 6 , 0 7 1,D8,P1 ,S1 ,0X ,W/ DATA P 1/3.141 5 9 2 7 / READ TIT'. Ir CARD: BO C H A R A C T E R S TO BE P R I N T E D ON OUTPUT RE A I'M 5,2) STAT •I :•! I T E ( 6 ,2 ) S TA T CARD SKIP S <IP CARD SUP SKIP CARD  C C  READ I N P U T : NUMBER 1.1 F L A Y E R S NUMBER OF FREQUENCY P f l l NTS » I NI T I A L A NO F I N A L RE DO EN I Y R E A I") ( 5 ,1 () ) N »I" M A X , E R E Q ( 1 ) , F R E Q ( (•'! M AX)  FREOIJENCY  i.|P=N+l l-INC = ( F R B P h i M A X ) - F R E O d ) ) / F L O A T (MHAX-1) OO I I=2,Mi-1/\X 1 . F R E 0 ( I ) =FRE'> ( I«l ) +F INC C READ I-.'PUT: FLAGS TO DETERMINE. OPTIONS K E A O ( 5 , 3 ) N FL A G l , r-l FLA G2 REAI'X ii ,3 ) MR AT 111,WO IS PL I F ( M R AT 10 .E0..0 .AND .NO IS PL .EO.O) STOP R IT E ( 6 ,2 9 ) R E A D ( 5 , 1 1 ) ANGINP , ANG I MS I F ( M F L A G 5 . E O .0) GO TO 13 C READ I N P U T : L A Y E R T H I C K N E S S , P V E L O C I T Y , S V E L O C I T Y , D E N S I T Y , A T T E N U A T I O N C CONSTANTS FOR P AMD SV MOTIONS REA 0 ( 5 , 1 2 ) ( H ( 1 ) , A L P ( I ) » B E T ( I ) ,RHO(I ) , O P ( I ),DS( I ) , I = 1,N ) GO TO 111 C READ I N P U T : L A Y E R PARAMETERS WITHOUT A T T E N U A T I O N C O N S T A N T S "=> T H I S O P T I O N C I S USED WHEN ATTENUATOR F L A G IS SET TO 0 13 R E A D ( 5 , 1 2 1 ) ( H ( I ) , A L P ( I ) , B E T ( I ) , RHO(I ) , I = 1,N) 111 R E A D ! 5 , 1 2 1 ) DUMMY,ALP ( M P . ) , B E T ( N P ) , R H O ( M ? ) W R I T E ( 6 , 1 4 ) ( I , A L P ( I ) , B E T ( I ) ,RHO(I ) , H ( I ) , I = 1 , M ) WRITE(6,14) NP,ALP(MP),BET(MP),RHO(NP) EOO I V A L E N C E ( V P , A L P ) , ( V S , 8 E T ) C R E F L E C T I O N AND T R A N S M I S S I O N Z O E P P R I T Z ' C O E F F I C I E N T S ARE CALCULATED  Ai-!GIi-]P=A!v'GINP=:--PI/180. ;  C  '  '  A N G I N S = ANGTMS-: P 1/180 . CALL COEGEN(N,VP,VS,RHO,AMGINP,ANGINS,E,F,TPP,TPS,RPP,RPS,TSP,TSS, 1RSP,RSS) L A Y E R T R A N S I T T I M E S AND PHASE ADVANCE TIMES ARE CALCULATED 0 0 5 1=1,N COS E( I ) = C O S ( E ( I ) ) S INE( I ) = S IN( E( t ).) ' COS F( I ) = C O S ( F ( I ) ) S INF( I ) = S IN( F'( t ) ) T P t I )=H( I ) / ( V P ( I ) * C O S E ( I ) ) T S ( I )'=H( I ) / ( V S ( I ) * 0 O S r ( I )) TAOP( I) =H( I ) * S I N E ( I ) * S I N E ( I ) / (VP { I )*COSE ( I ) )  T/MJS( I ) = h!( I )*S I N F ( I ) *S I N F { I ) / ( V S ( I ) * C O S F ( I ) ) 5 CONTINUE C OS E ( N P) = C OS (E (N P ) ) S INE( NP ) =S INI E (NP ) ) 0 OSE( N P ) = COS (i- ( N P ) ) S I N F ( i M P ) =SIN( F (IMP ) ) C C A L C U L A T I O N OF TRANSFER MATRIX V S . F RF OUENC/ IF ( N F L A G 2 .NE .0 ) GO TO 2 2 1 |J() 2 2 2 I = l , i M . C A B S O R P T I O N FUNCTIONS ARE SET TO ONE FOR NO A B S O R P T I O N A P ( I ) = 1. 222 A S( I ) = 1 . 221 CONTINUE 00 99 H=1,MMAX W = PI*i-RE(.)(M)*2. C C A L C U L A T I O N OF A B S O R P T I O N F U N C T I O N S OF FREQUENCY 1 F( MFLAG2 .EC) .0 ) GO TO 24 00 22 1=1,N A S( I ) = E X P ( » 0 , 5 * W * T S ( I ) / Q S ( I ) ) 22 . AP( I)=EXP(«0.5*W*TP(I ) / P P ( I ) ) 24 CONTINUE C C A L C U L A T I O N OF OF D E L A Y TRANSFER F U N C T I O N S AT FREQUENCY 0 0 4 1 = 1,1-1 A1.= W * ( TP ( I )™TAUP ( I ) ) A 2=U* ( TS ( I ) T A U S ( I ) ) E X P P( I ) = C M PL X ( C O S ( A 1 ) , " S I N ( A 1 ) ) E XPS ( I ) = C M P L X ( C O S ( A 2 ) , " S I N ( A 2 ) ) 4 CONTINUE <HAX = 4*N LMA XP =K MAX+ 1 0 0 2 0 K = l ,KMA'X DO 20 L = 1,LMAXP 20 A ( K , L ) =0.  '  COMPONENT  M  . M Ui  C  F G R i-i  HA I N  DO 21 C  21  DIAGONAL  OF  MATRIX  ALL  VALUES="1.  K=l , K M A X  A ( K , K ) = « 1 . COMPOTE  REMAINING  IF(N.EO.l) 00  A  MATRIX  COMPONENTS  GO T O 2 3  I=2,N  G1=AP(I )*EXPP  ( I )  G2 = AS ( I ) * E X P S  ( t)  J=I+NP C  S O i i R O O T I N E .TRAMS.I.  COMPOTES  C  FOR OP"TRA VELL ING  OUTPUT  T R A N S l ( R P  CALL  1 ) , G l , G 2  ,H1,H2  01 =T P P ( I D2 = T P P (  J  T H E COMMON  PARTS  UF  THE LAYE '  TRANSFER  ( I ) , R P S ( I  ) ,RSP(I ) ,RSS(I  ) , * P P ( J )  , R P S ( J ) , R S P ( J ) , R S S ( J  ,H3,H4)  )*HL+TSP(  I )*H3  I )*H2+TSP( I )*H4  D3=TPS(I)*H1+TSS(I)*H3 D4=TPS(  I)*H2+TSS(  I )*H4  D 5= R P P ( J ) * G 1 D6=RPS(J)*G1 0Y=RSP(J)*G2 D8=RSS(J)*G2 <=4*I-7 L=K+4 .<P=K + 1 LP=L+1 C C  INPUT-OUTPUT A(  RELATIONS  l)( I'-l ) / U (  I ) )  A(K,L)=D1 C  A ( 0( I " 1 ) / V ( A(K,  C  I ) )  LP)=02  A ( V ( I»l)/0( I ) ) A(KP,L)=D3  C  A( V( I " l ) / V ( I ) ) A(KP,LP)=D4 L = K+2 LP=L +1  C  A(U.( I » l ) / D ( I ) ) A ( K , L ) = 0 5 * 0 1 + 0 6*02  C  A ( 0 ( T "1 ) / E ( I ) )  FUN;i"IONS  SIGNALS  ARE INDICATED  BY  THE FOLLOWING  COMMENT  CARDS  C C C C  C C C C C  C C C C  A ( K , L P )=D7*D1+08*02 A ( V ( I ~ l ) / D ( I )) A ( K P , L ) = D5*D3+06*D4 A(V( I"1 ) / E ( I ) ) A(KP,LP)=D7*D3+D8*04 • ., SUBROUTINE TRANS 1 COMPUTES THE COMMON P A R T S OF THE L A Y E R T R A N S F E R F U N C T I O N S FOR DOWN-TRAVELLING OUTPUT SIGNALS C A L L TRANS 1 ( R P P ( J ) , R P S ( J ) , R S P ( J ) t R S S ( J ) , R P P ( I ) , R P S ( I ) , R S P ( I ) , R S S ( I 1) ,G1,G2,H1,H2,H3 ,H4) 01=TPP(J)*H1+TSP(J)*H3 D2=TPP ( J ) * H 2 + T S P ( J )*H4 D3=TPS( J) *H1+TSS ( J )*H3 D4=TPS(J)*H2+TSS(J)*H4 D5=RPP(I)*G1 Q6=RPS(I)*G1 •D7 = RSP( I ) *G2 08=RSS(I)*G2 K=4*I-1 L = K=4 KP=K+1 LP=L+1 A ( D ( I + l ) / 0 ( I) ) A(K,L)=D1 INPUT«OUTPUT R E L A T I O N S ARE I N D I C A T E D * BY THE FOLLOW ING COMMENT CARDS A ( D{ I + l )/E ( I ) ) A(K,LP)=D2 A ( E ( I +1 ) / 0 ( I ) ) A(KP,L)=D3 A ( E ( I +1) / E ( I ) ) A(KP,LP)=D4 L = K™2 LP=L+1 A ( D ( I + l )/U(I )) A( K ,L)=D5*D1.+ D 6 * 0 2 ' A(0(I+1)/V(I)) A( !< ,LP) =07*01 +08*02 A( E( I+1 ) /U( I ) ) • • A ( K P , L ) = 0 5 *D3+06*04 A( E ( I+ 1 )/V ( I ) )  Al I\K,LPJ =U/*D3+D8*D4 6  CONTINUE  2 3  CONTINUE  C  CALCULATE  TRANSFER  MATRIX  COMPONENTS  FOR  LAYER  1  1=1 J=1+NP G1 = A P ( 1 ) * E X P P ( 1 ) G 2 = A S ( 1 ) * E X P S ( I ) CALL  TRANS 1(R P P ( J ) , R P S ( J ) , R S P ( J ) R S S ( J )  T  R P P ( I ) , R P S ( I ) , R S P (  1 ) , G 1 , G 2 , H 1 , H 2 , H 3 , H 4 ) 0 1 = T P P ( J ) * H 1 + T S P ( J ) * H 3 D 2 = T P P ( J ) * H 2 + T S P ( J ) * H 4  '  .  D3=TPS<J)*H1+TSS(J)*H3 D 4 = T P S (J ) * H 2 + T S S ( J  )*H4  D5=RPP(I ) *G1 D6=RPS(I)*G1 D7=RSP(I)*G2 0 C  8=RSS(I)*G2  A ( D ( 2 ) / 0 ( 1 ) ) A(3,l)=D5*DL+D6*D2  C  A ( 0 ( 2 ) / V ( l ) ) A ( 3 , 2 ) = 0 7 * 0 1 + 0 8 * 0 2  C  A( E ( 2 ) / 0 ( l  ))  A ( 4 , I ) = 0 5 * 0 3 + D 6 * D 4 C  .  A ( E ( 2 ) / V ( l ) ) A ( 4 , 2)  =07*03+08*04  K=4*N"3 KP=K+1 KPP=KP+1 <PPP=KPP+1 L=K + 4 C  THE  CONSTANT  VECTOR  I F ( A M G I N S . E O . O . )  DETERMINING GO  TO  7  INPUT  CONDITIONS  IS  COMPUTED  I),RSS(  I  I F( ANG INP.1:0.0. ) A( U ( N ) / U ( N + 1 ) ) A ( K »L )=™T P P ( N P ) A(V(N)/U(N+1)) A ( K P t L )=" T P S ( N P )  GO T O  8  ^  A(D(N+1)/U(N+1)) A(KPP,L)=«RPP(NP) A ( E ( M + 1 ) / U ( N + l )) A(KPPP,L)=«RPS(NP) GO T O 9 A ( U ( N ) / V ( N + 1 )) A( K ,1, ) = « T S P ( N P ) A ( V( N) / V ( N + 1 ) )  ' '  A(KP,L)=-TSS(NP)  •  A ( D( N + 1 ) / V ( N + l ) )  .  •  •  A(KPP,L)="RSP(NP) A ( E ( N+ 1 ) / V ( N +1 ) ) A(KPPP,L)=«RSS(NP) CONTINUE  DO  30 K=l KMAX f  DO 30 1=1 , L M A X P B(K,L)=A(K,L) CALL GAUC0M(KM4X,8,X,DET,40,41) 1=1 J=1+NP C A L L T R A N S 1 ( R P P ( I ) ,RPS ( I ) ,RSP U 1) , G I , G 2 , H 1 , H 2 , H 3 , H 4 )  ) , R S S ( I ) , R P P ( J ) ,RPS'.( J ) ,R.SP( J ) , R S S ( J  COMPOTE TRANSFER FUNCTION THROUGH UPPER P1 = H 1 * X ( 1 t l ) + H 2 * X ( 2 , l S1=H3*X(1,1)+H4*X(2,1) C A L C U L A T I O N OF S U R F A C E D I S P L A C E M E N T S — D1=P1*RPP(1) D2=S1*RSP(1) D3 = S 1 * R S S ( I ) D4=P1*RPS(1)  )  05=(P1+D1+D2)*COSE(1) D6=(P1-D1-D2)*SIN£(1 ) D7=(S1+D3+D4)#SINF(1) 0 8 = ( «S 1 + 0 3 + 0 4 ) * C O S F ( 1 )  '  ,  '  •  tAY-R  TO  COMPLEX  FREE  SURFACE  BOUNDARY  UX(1)=D6+D8 WZ{  1 )=D5 + D7  I F ( H D I S P L . E O . O ) C  COHPUTATIUN DO  32  I ••!=  OF  1=2  GO  TO  INTERIOR  321 BOUNDARY  DISPLACEMENTS  —  COMPLEX  »NP'  I"l  L4=4*IM L3=L4"1 L2 = L3' 'l :  L=L2"1 D 1 = X ( L , 1 ) * C O S E ( I M ) + X ( L 3 , l ) * C O S E ( I ) D2 = X ( L , 1 ) * S I N E ( I M ) - X ( L 3 , 1 )  * S I N E ( I )  D 3 = X ( L 2 , 1 ) * S I N F ( I M ) + X ( 1 4 , 1 ) * S I N F ( I ) D4==X(L2,1 )*COSF( OX'( I 32  WZ(  )  INTERNAL  DISPLACEMNTS  I ) =01+03  C  COMPUTE  C  AMPLITUDE DO  I M ) + X ( L 4 , l ) * C O S F ( I  )=D2+D4 SURFACE  33  ANO 1=  AND  PHASE  BOUNDARY  V S .  FREQUENCY  IN  TERMS  OF  COMPONENTS  1,NP  U X I = A I MA G ( U X ( I ) ) UXR=REAL (UX(  I ) )  W Z I = A I MA G ( W Z ( I WZR = R E A L ( W Z ( I  '  ) )  U A M P ( I , M ) = S Q R T ( U X R #U X R + U X I # U X I  )  WAMPtI,M)=SORT(WZR*WZR+WZI*WZI)  '  UPHStI,M)=ATAN2(UXI,UXR) 33  WPHS( 00  I,M)=ATA N2(WZI  2 5  1=1  PANGLE(  I)  SANGLE(I 2 5  •  ) )  *  ,WZR)  ,NP = E U )=F(  ) * 1 8 0 . / P I I  ) * 1 8 0 . / P I  CONTINUE  )  I F ( N F L A G 1 . E O . O ) W R I T E ( 6 , 1 8 )  GO  TO  26  FREO(M)  W R I T E ( 6 , 1 5 ) ( ( A ( K , L ) , L = 1 , L M A X P ) , K = 1 , K M A X ) W R I T E ( 6 , 1 5 ) 26  ( X ( L , 1 ) , L = 1 , K M A X )  CONTINUE  .  ,  "  WRITE! 6,16)  ' .  WRITE(6,17) ( I ,UX(  I ) ,WZ ( I ) ,UAMP{  1 , P A ' N G L E t I ) , S A N G L E ( I ) , 1=1  ,NP )  I ,M ) , U P H S <  I , M ) , W A M P t ( ,M>  M ,WPHS(  I,  M)  WRITE(6,19) 2 7 321  OET  CONTINUE I F ( N R A T I O . 6 O . 0 ) OF  GO  TO  SPECTRAL  322  C  COMPUTATION  C  BOUNDARY  SPECTRAL  RATIOS  COULD  ALSO  8E  OBTINED  C  BOUNDARY  SPECTRAL  RATIOS  COULD  ALSO  BE  OBTAINED  UXI = AIMAG(UX  AMPLITUDE  RATIOS  FOR  SURFACE  MOTIONS--  INTERNAL  (1 ) )  U X R = R E A L (UX ( 1 ) ) WZI=AIMAG(WZ(1)) WZR=REAL(WZ(1)) RAT(M)=SORT(WZR*WZR+WZI*WZI)/SORT(UXR*UXR+UXI*UXI) 3 22  CONTINUE  99  CONTINUE IF(NRATIO.NE.O) IF(NRATIO.NE.0)  WRITE(6,110) WRITE(6,11)  (F  REQ(M),RAT(M),M=1,MMAX)  STOP 2  F O R M A T ( 1 5 A 4)  3  FORMAT( 511  10  )  FORMAT(212,2F6.2)  11  FORMAT!2F6.2)  110  FORMAT('  12 121 14 15 16 17  '  FREQ  •  -  RATIO')  FORMAT!6F8.4) FORMAT(4F8.4) F O R M A T ( 14 , 4 F 9  .3)  F 0 R M A T ( T 8 ( 1 X , F 5 . 2 ) / ) F 0 R M A T ( 3 X , I 2 , 1 X , 1 0 F 1 1 . 4 ) FORMAT ( / / 2 X , 5 H L A Y E R 15HU-PHS 2 4 H I MAG  6X,5HW-AMP 7X,4HREAL  8X , 9 H U - C 0 M P L E X 6X,5HW»PHS  9X,5HU=AMP  6X., 5 H S - A N G / 12X ,41-IREAL  6X, 7X,  7X,4HIMAG/)  18  FORMAT(//35X,10HMATRIX  19  FORMAT(/l5H  29  FORMAT!IX,5HLAYER END  1 3 X , 9 H W - C 0 M P l. E X  6X,5HP=ANG  FOR  DETERMIMANT=  F 6 . 2 , 1 7 H C Y C L E S  PER  SECOND  //)  2 E 1 7 . 7 / / / )  2X,5HP-»VEL  4 X , 5 H S « V 6 L  3X,7HDENSITY 1  3X,5H0EPTH  /)  NON-NORMAL  INCIDENCE SYNTHETIC  SEISMOGRAM FOR  ANGLES LESS THAN C R I T I C A L  * * * * * * * * * ***** * * * * * * * * * * * ******** * * * * * * * ****** * * * * * * * * * * * * * * * ******* * * PROGRAM .  DESCRIPTION  r HE  PROGRAM  SYNTHESIS PROGRAM  •  OSES  OF  THE  TIME  DOMAIN  LINEAR  SAMPLED  TIME  SERIES  SYNTHETIC  MA X I M U M  NO.  OF  NO.  OF  HELD  MAXIMUM  NO.  OF  ATTENUATOR  MODEL  FOR  LAYERS=20  ATTENUATORS  MAXIMUM IN POT  CRUST  SEISMOGRAMS  LIMITATIONS  MAXIMUM r HE  SYSTEMS  NO.  POINTS  MOST  OF.  lit  OUTPUT  PER  LAYER=200  OPERATOR  WEIGHTS  PER  L A Y E R = 20  CAUSAL TIME  P0INTS=100()  CARDS:  CARD  1:  CARD  2:  FLAG  TITLE  CARD  3:  NO. P  FOR OF  ATTENUATION  LAYERS,  INCIDENCE  DETERMINES CARD  4 F F :  LAYER  FOLLOWING  CARD5FF: CARD THE  P  6 F F :  AND  THICKNESS,  P  INPUT  CARDS S  ATTEN  TIME  FOR  INCIDENCE OF P  NO  SAMPLE  THE  ATTENUATION POINTS,  ANGLE.  INPUT  VELOCITY,  WAVE S  ENTER  TIME  THE.  00  '  INCREMENT  N O N Z E R O  ANGLE  FORM.  VELOCITY,  DENSITY  BASEMENT  ARE  INCLUDED  ATTENUATOR WEIGHTS  WAVEFORM DEVICE  S  FORM  TO  MODEL:  OF  ANGLE, THE  CONTINUE THE  NO.  TIME  ONLY  LENGTHS  AND  S A T T E N  SERIES  IF  FOR  SAMPLE  NATTEN LAYERS  WEIGHTS POINTS  1 FOR  OF  IS  NOT  TO  N  EQUAL  LAYERS  ANY  1  LENGTH  TO  TO  00  N  ARE  READ  FROM  * * * * *  * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *  0 I M E N S ION  4  E POLA Y ( 2 0 ) , E S D I A Y ( 2 0 ) , F  4ES< 2 0 , 2 0 0 ) , F P ( 2 0 , 2 0 0 ) , F S ( 2 0 , 2 00) $TPP(41 ),TPS(41 ) , T S P ( 4 l  ,BET(21 ) ,RHO(21 ) , H ( 2 1 ) , A I N (  SO)  ,WSORF ( 1000)  ,E(21  ) ,F(21  DATA  P I / 3 . 1 4 1 5 9 2 7 /  INITIALIZE 0 0  10  ALL  1=1,20  E P 0 L A Y(  I)=0.  F S D L A Y ( I ) = 0 . FSOLAYt  I )  =0.  OUTPUTS  ,  , R P P ( 4 1 ) , R P S ( 4 1 ) , R S P ( 4 1 ) , R S S ( 4 1 ) ,  ) , T S S ( 4 1 ) , N T A O P ( 2 0 ) , N T A 0 S ( 2 0 ) , S TA  $ 1) SO)  P D L A Y ( 2 0 ) , F S 0 L A Y ( 2 0 ) , E P ( 2 0 , 200)  1000)  ,POUT(  ) , A T T E N P ( 2 0 , 2 0 )  10U0),SOOT(  T ( 2 0 ) , A L P ( 2  1000),USURF(  , A T TENS ( 2 0 , 2 0 ) ,J'P(  20)  100  ,JS(  Z  * *  F P O L A Y ( I ) =0.' NTAUP(I)=0 . NTAUS(I)=0. OU 10 J = 1 , ? 0 0 E P ( I , J ) = (). ES(I,J)=0. F P ( I , J ) = (). FS(I,J)=0. ;-lr'1IN = 0  READ!5,1) STAT W R I T E ( 6 ,1 ) S TA T R E A D ( 5 , 9 ) MATTEN R E A Q ( 5 , 2 ) i M , r W A X ' r i i M C , ANG INP , AN"G INS t  i\jP = N + l  i\J|-i= |\]L-I1 REA0(5,3) ( H ( I ) , A L P ( I ) , B E T ( I ) , R H O ( I ) ,1 = 1 ,N ) READ(5,3) DUMMY,ALP(NP)»8ET(NP),RHQ(NP) WRITE(6,4) W R I T E ( 6 , '5 ) ( I , A L P ( I ) , 3 ET ( I ) , RHO ( I ) , H ( I ) , [ = 1, N ) W R I T E ( 6 , 5 ) MP , A L P ( M P ) , B E T ( N P ) ,RHO(NP) ANGINP=AM GIM P * P I / 1 8 0 . A N G IM S = A M G INS * P I / 1 80 . C A L C U L A T E T R A N S M I S S I O N AND R E F L E C T I O N C O E F F I C I E N T S . C A L L CUEGEM I N , A L P,B E T , R H O , A N G I N P , A N G I N S , E , F , T P P , T P S , R P P , R P S , T S P , STSS , R S P , R S S ) C A L C U L A T E T R A N S I T T I M E S AND DELAYS 00 50 1 = 1 ,N ... TAOP=H(I)*COS(Ell))/ALP(I) TAOS = H (T ) *COS ( F ( I ) ) / 3 ET ( I ) NTAOPI I ) = I F I X ( T A U P / T I N C + 0 . 5 ) NTAUS( I ) = I F I X ( T A U S / T I N C + 0 . 5 ) I F ( M T A U P ( I ) .EO.O ) N T A U P ( I ) = 1 I F ( N T A U S ( I ) .EQ . 0 ) N T A U S ( I ) = 1 CONTINUE I F ( N A T T E N . E Q . O ) GU TO 21 RE A 0 IIM MODEL A T T E N U A T I O N P A R A M E T E R S READ IN A T T E N U A T I O N OPERATOR L E N G T H S REA0(5,9) ( J P ( I ) , J S ( I ),I=1,N) R E « 0 A T I E NO A TOR '..'EIGHTS  21 C 9 9  98  22 23 C C C C  C  WRITE(6,101) 0 0 21 1=1 ,N JPLIM=JP( I ) .) SL I ;•'= J S ( I ) R E A U ( 5 , 1 0 2 ) ( A T T E N P U ,K) ,K=1 , J P L I M ) R EA 0 ( 5 , 1 0 2 ) (ATT E N S ( I , < ) , < = 1 , J S L I M ) W R I T E R , 1 0 3 ) ( A T T E N P I I ,K) , K = 1 , J P L I M ) W R I T E ( 6 , 1 0 3 ) ( A T T E M S ( I ,<) ,< = 1 , J S L I 1 ) CONTINUE WRI TE( 6 , a) REAO INPUT WAVEFORM FROM O E V I C E 4 R EA 0 ( 4 , 7 , ENO =99 ) ( A IN ( I ) , I = 1, 1 0 0 0 ) M5N0=I 0 0 98 I = MENO ,MMAX A I N ( I ) = 0 .0 CONTINUE 00 100 H=1,MMAX 1 F ( ANG IMP .NE . 0 . 0 ) GO TO 22 S I N PUT= A I N ( M ) P I N PUT = 0 .0 GO TO 2 3 PINPUT = AIN(M ) SIMPUT=().() CUNT INOF UPDATE A L L DELAY OUTPUTS FOR A L L L A Y E R S I F ( W A T T E N . E O . O ) GO TO 24 A T T E N U A T I N G MODEL S Y N T H E S I S DELAY O P E R A T I O N S DELAY 0 0 28 1 = 1 ,N NT P = N T A U P ( I ) NTS = N T A U S ( I ) MTPM = MTP«1 NTSM=NTS«1 EP I N T P = E P ( I ,NTP) F P I M T P = F P ( I , NT P ) ES I N T S = E S ( I ,NTS ) F S I NTS = F S ( I , NTS ) A T T E N U A T I O N OPERATOR C O N V O L U T I O N S  FO* ALL L A Y E R S  F. P O L A Y ( I ) = E P I N T P * A T T E N P (  I »1)  ESULAY(I)=ESINTS*ATTENS(1,1) F P O L A Y U  )=FPINTP*ATTENP(I  ,1)  F S O L A Y ( I ) = F S I N T S * A T T E N S ( I , 1 ) i)D  2 5  K=l  ,NTPM  IF(K.GE . J P ( I )  )  GO  TO  26  E P ( I ,MTP"K + 1 ) = E P ( I , N T P ™ < ) + E P I N T P * A T T E N P ( FP( I , N T P ™ K + 1 ) =FP ( I , N T P « K ) +FP I N T P * A f  GO 2 6 2 5  TO  [ , K + 1)  T E N P ( I , K+1 )  2 5  EP( I , N T P « K + 1 ) = E P ( I  ,NTP«K)  F P ( I ,I\ITP-K + 1 ) = F P ( I  ,NTP"K)  CONTINUE OO  2 8  K=1  ,NTSN  IF(K.GE . J S ( I )  GO  )  TO  27  ES ( I , N T S - K + l )=ES ( I , N T S » ; < )+ES I N T S * A T T E N S ( I ,K+1 FS( I , N T S - K + 1 ) = F S ( I , N T S " K ) + F S I N T S * A f " T E N S ( I  GO . 2 7  TO  ES(  )  ,K+1)  8  2  I ,NT S«K+1)= ES(I  ,NTS-K)  F S ( I » N T S " K +1 ) = F S ( I , N T S K ) r j  2 8  CUNT I NOE  GO  TO  30  C  MON  24  CONTINUE  C  C 3  ATT EMUA T I N G  MODEL  SYNTHESIS  DELAY 0 0  30  1= 1 ,M  NT P= N T A U P ( I  )  NTPM=NTP«1 NTS=NTAOS(I) NTSM=NTS«1EPDLAY(  I ) = E P ( I ,IMTP )  E S O L A Y (I FSOLAY(  )=  FPOLAYII  301  ES(I,NTS)  I)=FS( )=FP  I  ,NTS)  ( I  ,NTP)  00  301  EP(  I , M T P » K + 1 )=EP(  K=1,NTPM  F P ( I , NT P 0 0  302  , j  I  ,MTP«'<)  K + 1) = F P ( I , N T P  K=l  , ! ,  K )  , NTSM  E S ( I , N T S-K+1)= ES( I ,NTS «K) 30 2  F S ( I , M T S - K +1  )= F S ( I , N T S - K )  DELAY  OPERATIONS  •  CONTINUE 3 0I-1PUTE WAVE FORMS R E T R A N S M I T T E D INTO BASEMENT AT T I M E S A M P L E <A P O U T P T = T S P ( M + N P ) * F S D L A Y ( N ) + T P P ( N + N P ) * F P D L A Y ( N)+RS° ( NP ) * S I NPUT + RPP S( NP ) *P INPUT SOUTPT=TSS(N+NP)*FSDLAY(N)+TPS(N+NP)*FPDLAY(N)+RSS(NP)*SINPUT+RPS $(NP)*PINPUT J U N C T I O N UPDATE TO TIME S A M P L E M BOTTOM L A Y E R E P ( M , 1 ) = TS P ( N P ) *S IN PUT +T PP ( NP ) *P IN PU T + RP P ( N+N P ) *FP DL AY ( N ) + RS P ( N + N P $)*FSULAY(N) E S ( N , 1 ) = T S S ( N P ) * S I N P U T + T P S ( N P ) * P INPUT + R S S ( N + N P ) * F S D L A Y ( N ) + R P S ( N + N P $)*FPDLAY(N) F P ( N , 1 ) = T S P(N+NP»1)*FS D L A Y ( N " 1 ) + T P P ( N +NP"1)*F P D L A Y ( N - l ) + R S P ( N ) * E S D $ L A Y ( N ) + R P P ( N ) * E P D L A Y(N ) FS ( M , 1 ) = TS S ( N +N P» 1 ) * FS DL A Y ( N™ 1) +T P S ( N +N P •• 1 ) *F P DL A Y ( N-1 ) +R S S ( N ) * E S D $LAY(N)+RPS(N)*EPDL4Y(M) INTERMEDIATE LAYERS I F ( N . E O . 2 ) GO TO 41 DO 40 I=2,NM E P ( 1,1 ) = T S P ( I + L ) * E S D L A Y ( I + 1 ) + T P P ( I + 1 ) * E P D L A Y ( [ + 1 ) + R S P ( I + N P ) * F S D L * Y $ ( I ) +RPP ( I +:•! P) *F PD.LA Y ( I ) . ES( I ,1 ) = T S S ( I + L )*ES OLA Y ( I + l ) + T P S ( I + l ) * E P D u A Y ( I + 1 ) + R S S ( I + N P ) * F S D L A Y S( I ) +RPS( I + N P ) * F P 0 L A Y ( I ) r P ( I , 1 ) = TSP( I+NP-1)*FSOLAY(I»l)+TPP(I+NP»1)#FPDLAY(I-l)+RSP( I ) * E S O $ LA Y ( I ) + R P P ( I ) * E P 0 L A Y ( I ) F S ( I,1 ) = T S S ( I + M P - 1 ) * F S D L A Y ( I - l ) + T P S ( I + N P - l ) * F P D L A Y ( I » 1 ) + R S S ( I ) * E S D $LAY(IJ+RPS(I)*EPDLAY(I) CONTINUE TOP L A Y E R EP(1,1)=TSP(2)*ESDLAY(2)+TPP(2)#EPDLAY(2)+RSP(1+NP)*FSDLAY(1)+RPP( S l + NP) *F P D L A Y ( 1 ) ESC 1 * 1 ) = T S S ( 2 ) * E S D L A Y ( 2 ) + T P S ( 2 ) * E P D L A Y ( 2 ) + R S S ( 1 + N P ) * F S D L A Y ( 1 ) + R P S ( $ 1 + NP) * F P D L A Y ( 1 ) FP(1,1}=RSP{1)*ESDLAY(1)+RPP(1)#EPDLAY(1) FS(1,1)=RSS(1)*ESDLAY(1)+RPS( 1)*EPDLAY(1) END OF J U N C T I O N UPDATE TO TIME SAMPLE M COMPOTE S U R F A C E MOTIONS FROM I M P I N G I N G S AND P WAVEFORMS PZ=( E P U L A Y U ) * ( 1 ,+RPP ( 1 ) ) +ESOL AY ( 1 ) * R S P ( 1) ) *COS( E ( 1 ) ) PX = ( E POLAY ( 1) *C1..«RPP ( 1 ) )»ESOLAY ( 1 )*RSP ( 1 ) ) # S I N ( E ( 1 ) )  S Z = ! ES O L A Y ( 1 ) * ( 1 , + R S S ( 1 ) ) + E P O L A Y ( 1 ) * R P S ( SX = { " E S O L A Y ( 1 ) * ( ! . » R S S ( 1 ) ) + E  1 ) )*St M ( F ( 1 ) . )  POL A Y { 1 ) * R P S ( 1 ) ) * C O S ( F ( 1 ) )  00=PX+SX WO=PZ+SZ  XM=FLOAT(M»I)*rIN: C  WRITE OUTFIRST 2 0 0 TIME SAMPLES FOR CHECK IF(M .LT.200) WR I T E ( 6 , 6 ) X M , P I N P U T , S I N P U T , P U U T T,SOUT PT,WO,UO SAVE OUTPUT S U R F A C E MOTIONS ANO B A S E M E N T WAVEFORM T I M E S E R I E S F O R L A T E R P L O T T I N G OR WRITEOOT P O U T ( M) = P O I J T P T S O U T ( M ) = S OUT PT USURF(M)=U() WSURF(M)=WO 3  C C  100 1  CONTINUE STOP FORMAT!20A4)  2 3 4 5  FORMAT(12,14,3F8.4) FORMAT!4FB.4) F O R M A T {» L A Y E R " P«VEL FORMAT!2X,12,IX,4F10.4)  6 7 8  F0RMAT(F8.4,6E17.7) FORMAT! 10F8 .5 ) F O R M A T (' TIME INPUT S BASEMENT S REFL VERT  9  FORMAT(2012)  101  FORMAT)'  1)2  DENSITY  FORMAT(16F5.3) FORMAT! 1 6 F 6 . 3 ) .  1)4  FORMAT ( 1 0 E 1 0  .3)  WEIGHTS  DEPTH')  P PULSE INPUT S PULSE 8 A S E -1E N f SURF MOTION HORZ S U R F MOTION') .  FILTER  103  END  S-VEL  F O R P AND  . S  MOTIONS  P  REEL  '  IN LAYERS  1 TO N  1  )  ZOEPPRITZ *  * * * * * *  C06 FF ICI  S08ROOT INE  GENERATION  PROGRAM  * * * * * * * * * * * * * * * * * * sis * * *  .  0 I S C R.I P T I O N  VARIABLES M :  EN f  * * * * * * * i'f  * *  TRANSMITTED  NUMBER  OF  FROM  MAIN  LAYERS P  ANO  S  A NG IN p , A N G IN S :  ALP ,BET ,RHO:  THE  INCIDENT  E , F :  THE  INTERNAL  TPP,  RPP  E T C :  SUBROUTINE  VELOCITIES  LAYER  P  P  AND  ANO AND S  S  LAYER  ANGLES  AT  WITH  ZOEPPRITZ'  COEFFICIENTS  FOR  FOR  BOUNDARY  TO  BOTTOM  D E N S I T I E S  ANGLES  M+2  THE  THE  BASEMENT  BOONO A R /  NORMAL  T O P - B O U N D A <Y  1  TO  N+1  2N  C 0 E GE N ( N , A L P , 8 ET , R H 0 , A N G I N P A N G I N S , E , F , T t  P P , T PS , R P P ,  1 R P S , T S P , T S S , R S P , R S S ) MCCAMY  NOTATION  0 I M EMS ION £1)  , T S P ( 1)  FOR  ZOEPPRITZ'  ALP I 1 ) ,8ET(1  I F ( A N G I M P . E O . O . )  GO  TO  2  E(NP)=ANGIMP C = A L P ( N P ) / S I N ( E ( N P ) TO  )  3  F(NP)=ANGINS C = 8 E T ( N P ) / S I N ( F ( N P ) ) CONTINUE 00  4  I  =1,NP  X = A L P U ) / C Y = B E T ( I ) / C I F I X . G T . l . )  GO  TO  10  E ( I ) =A"fAN2 ( X , S Q R T ( 1 . » X * X ) ) IF{Y.GT F( GO  I)=  . 1 .  GO  )  I  I ) = 1.5707964 TO  14  W R I T E ( 6 ,11 F(  GO  4  W R I T E ( 6 ,11 E(  )  TO  ATAN 2 ( Y , SORT  TO  I ) = 1  1 ) , F ( 1 ) , T P P ( 1 ) , T P S (  ,TSS ( 1 ) , R S P ( 1 ) , R S S ( 1 ) , W ( 2 , 2 ) , Z ( 2 , 3 )  NP=M+1  GO  EQUATIONS  ) ,RHO( 1 ),E(  )  I  .5707964  15 (l-.-Y*Y))  1 ) , R P P ( 1  , C O F ( 2 , 1 )  ) ,RPS(  GO 11 4 C  TO  4  FORMAT(47H •  CRITICAL  REFLECTION™  NO  TRANSMISSION  TO  LAYER  12)  CONTINOE FREE  SURFACE  REFLECTION  COEFFICIENTS  T PP( 1 ) =0 . TPS(  1)  TSP(  1) = 0 .  =0.  T S S ( 1 ) = 0 . W( 1 , 1 ) = « S I N ( 2 . * E ( 1  ) )  W ( 2 , 1) =+C O S ( 2 . * F ( L  ) )  W ( 1 , 2 ) = + A L P ( 1 ) * C 0 S ( . 2 . * F ( 1 ) ) / B E T ( 1 ) W( 2 , 2 )  = + B E T ( 1 ) * S I N ( 2 - * F (1 ) ) / A I . P ( 1 )  00  20  L=  0 0  20  M=l  20  ,2  Z(L,H)=W(L,M) Z(1  , 3)  Z(2  ,3)=+W  CALL RPP( RPS(  21  l,2  = « W ( 1 , 1 ) . (2,1)  G A O E L I ( 2 , Z , C O F , O E T , 2 , 3 ) 1 )=COF(1,1 1) = C 0 F ( 2  00  21  L = l ,2  0 0  21  M=l  Z ( L ,M)=W  )  ,1 ) '  ,2 (L  •  , ,M)  Z ( 1 , 3 ) = - W ( 1 , 2 ) Z(2,3)=W CALL RSP(  (2 , 2 )  G A O E L I ( 2 , Z , C O F , D E T , 2 , 3 ) 1)=COF(1,1  )  R S S ( 1) = C O F (2 ,1 ) 00  12  J = I" K= 12  I=2,NP  1  I  CALL 00 J=I  C O E F F ( A L P , 6 E T , R H 0 , E , F , T P P , T P S , R P P , R P S ,  13  T S P , T S S , R S P , R S S , I , J , K )  1=1,N  +l  KNP=I+NP 13  CALL  C O E F F ( A L P , B E T , R H O , E , F , T P P , T P S , R P P , R P S , T S P , T S S , R S P , R S S , I , J , K N P  1 ) RETURN •EMU  SECOND  COEFFICIENT  SUBROUTINE  ****** *******************  THIS  SUBROUTINE  IS  REQUIRED  BY  -COEGEN  * * : ; : * * * * * * * * * * =1: * * * * * * * * * * * * * * * * * * * * * * * * * * * * *  SUB R O U T I N E  *  C O E F F ( A L P • B E T , R H O , E , F , T P P , T P S , R P P , R P S , T $ P , T S S , R S P , R S S , I  1, J , K ) 0 I  i-i E N S I O N  $1)  , T S P ( 1)  A L P ( I ) , B E T ( 1 ) , RHO( 1 ) , E ( ,TSS(1 ) ,RSP(1 ) ,RSS(  C ( l , l ) = ™ S I N ( E ( C( 1 , 2 ) =  +COS(F  [  , F ( 1 ) , T P P ( 1 ) , T P S ( 1) ,  R PP (  1 ) , RP S (  ))'  ( I ) )  C ( 1 , 3 ) = - S I N ( E ( J C( 1 , 4 ) = + C O S ( F  h)  1 ) , C ( 4 , 4 ) , Z ( 4 , 5 ) , C O F ( 4 , 1 )  ))  (J ) )  C ( 2 , l ) = + C O S ( E ( C{2,2)= + SIN(F  I ) ) ( I ) )  C ( 2 , 3 ) = - C O S ( E ( J  ) )  C ( 2 , 4 ) = « S I N ( F ( J  ) )  C ( 3 , l ) = + C O S ( 2 . * F ( [ ) ) C ( 3 , 2 ) = + BET(I C ( 3,3) C(3  ) / A L P ( I ) * S I N ( 2 . * F ( I ) )  = + R H O ( J ) / R H O ( I ) * A L P ( J ) / A L P ( I ) *CO  ,4)= + R H O ( J ) / R H O { I ) * 3 E T ( J ) / A L P U  S{ 2 . * F ( J ) )  )*SINC  2 . * F ( J ) )  C ( 4 , 1 ) = " S I N ( 2 . * E ( I ) ) C ( 4 , 2 ) = + A L P ( I ) / B E T ( I ) * C O S ( 2 . * F ( I ) ) C(4,3)=RHO(J  )  / R H O ( I ) * ( ( 8 E T ( J ) / B E T ( I ) ) * * 2 ) * A L P ( I ) / A L P l J ) * S I N ( 2 . * E ( J  1) ) C ( 4 , 4 ) = - R H 0 ( J J / R H J ( I ) *(  ( B ET ( J ) / B E T ( I ) ) * * 2 ) * A L P ( I ) / B E T ( J ) * C O S (  U) ) DO  2 0  L = l , 4  DO  20  0 = 1 ,4  Z ( L , Z(  1  i'l)  =C ( L ,  (A )  , 5 ) = " S IN ( E ( I ) )  Z ( 2 , 5 ) = « C O S ( E ( I ) ) Z ( 3 , 5 ) = + C O S ( 2 . * F ( I ) ) Z ( 4 , 5 ) =+SIN(2.*E ( I ) ) CALL  G A O E L 1 ( 4 , 7 . , C O F ,OET  R P P ( K ) = C O F ( l , l ) RPS(K)=C0F(2,1) TPP(  :<) = C 0 F ( 3  ,1 )  •  ,4 , 5 )  2 . * F (  TPS( K)=C'OF (4,1 00 00 21  Z .  (  21  L=l ,4  21  .14 = 1  )  ,4  L t\A) =C ( L  ,M )  Z( 1 , 5 ) = + C 0 S ( F ( I  ) )  Z( 2 , 5 ) = " S I N ( F ( (  ) )  Z ( 3 , 5 ) = +BFT(I  ) / A L P ( I ) * S I N ( 2 . * F ( I  ) )  Z ( 4 , 5 ) = « A I. P ( I ) / B E T ( I ) * C O S ( 2 . * F ( I ) ) CALL  G A O E L I ( 4 , Z , C O F , O E T , 4 , 5 )  R S P ( K ) = C O F { I ,1 ) K S S ( K ) = C 0 F ( 2 , 1  )  T S P ( <) = C O F ( 3 , 1 ) T S S ( K ) = C U F ( 4 , 1 KETOKM EN 0  )  C  BASIC  C  THIS  C  # * * * * * * * * *:;= * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *  C  LAYER  TRANSFER  FUNCTION  GENERATION  PROGRAM  * * * * * * ** * ** * * * * * * * * * * * * * * * * * * * * * ** * # * * * * * * * * * * * SUBROUTINE  S UB R O U T I N E  IS  CALLED  BY  THE  TRANSI(RPPI,RPS  FOURIER  COEFFICIENT  MAIN  PROGRAM  I , R S P I , R S S I , R P P J , R P S J , R S P J , * S S J , G 1 , G 2 , H I  1,H2,H3,H4) COMPLEX  RPPI ,RPSI ,RSPI ,RSSI  , R P P J , R P S J , R S P J , R S S J , G l , G 2 , H I , H 2 ,  1H3.,H4,81 ,B2 ,B3 ,B4 ,B5 , B 6 , B 7 ,B8  ,DEN  B1=G1*RPPJ B3 = I U * R S P I 31=B1*RPPI 66=G1*RPSJ B8=B6*RSPI B6=B6*RPPI B5=G2*RSSJ B7 =  B5*RSSI  B5=B5*RPSI B 4 = G2 * R S P J •3 2 = 8 4 * R P S I B4=B4*RSSI B 1 = B 1 + .B2 B3=B3+B4 B5=B5+B6 B7=B7+B8 B2=G1/( 1 . - G l * S l B4=G2/(  • )  1 .»G2*.B7 )  O E N = l . / ( l . ~ 8 2 * B 3 * B 4 * B 5 ) H1 = B 2 * U E N H2=B4*H1 H3 = B 5 * H 2 H2=B3*H2 H4 = 8 4 * U E N RETURN  .  •  '  C  GAUSS  r  *  C C C C C C C C C  PROGRAM 01 S C R I P T ION TRANSMITTED VARIABLES . N: N U M B E R O F EOOATIONS A: I N P U T C O E F F I C I E N T S ( C O N S T A N T VECTOR [ N ROW N+1) X: O U T P U T S O L U T I O N VECTOR 0 ET : C HEC f< 0 ET E R -I I N A N T 1 0 : M A I N P R O G R A M D I M E N S I O N OF F I R S T D I M E N S I O N OF A J O : M A I N P R O G R A M D I M E N S I O N O F S E C O N D D I M E N S I O N OF A * * * * * * * * * -f * * * * * * * * * * * -:- * * * * * * * * * * * * * * * * * * * * * * * * * * * *  C  11  12 C  21 22  *  * *  *  SUBROUTINE  ELIMINATION * *  *  *  * *  *  *  *  *  * *  *  *  *  *  *  *  *  *  *  *  FOR *  *  *  *  REAL *  «  *  *  SIMULTANEOUS *  S U B R O U T I N E G A U E L I ( N , A , X ,OET , 10 , J O ) DIMENSION A ( I D , J O ) ,X(ID,1 ) NP=N+L NM=N«1 OET=1.0 TRIANGULAR IZATION DO L 2 K = 1 , N M KP=K+1 R=1.0/AK,<) U E T = O E T * A ( K ,K ) DO 1 1 J = < P , N P A( K , J ) = R * A ( K , J ) 0 0 12 I = K P , N S=A(I,K) . DO 12 J=KP,NP A( I , J ) =A ( I , J ) ~ S * A (K , J ) •DET = O E T * A ( N , N ) BACK S U B S T I T U T I O N X ( N , I ) =A ( N ,N P ) /A ( N ,N ) DO 2 2 I = 1 , N M K = N-I KP=K+1 S=A(K,NP) DO 2 1 J = K P , N S= S » A ( K , J )#X(J , 1 ) X( K , 1 ) =S RETURN 'END ,  *  *  *  *  *  *  *  ;|: *  *  *  EOOATIONS *  *  *  s|: *  *  :1c * * * * *  * *  *  *  GAUSS  ELIMINATION  SUBROUTINE  FOR  COMPLEX  S[MULTANEOUS  EOUATIONS  * * * ** * ** * *** * * ** * ** * ** * * * ** * * ** * #* * * * *** * * * * ** * * ** ** * * * * * * ** * **  PROGRAM  INSCRIPTION  TRANSMITTED N: A:  INPUT  X:  DOT POT  DET:  *  * *  VARIABLES  NUMBER  OF  £00AT  IONS  COEFFICIENTS SOLUTION  CHECK  (  CONSTANT  VECTOR  MAIN  PROGRAM  DIMENSION  OF  FIRST  JO:  MAIN  PROGRAM  DIMENSION  OF  SECOND  * :|c * * * * * *  *  SOBROUTINE GAOSS 0 H E N S  GAOCOM(N ,  A,X  ION  N+1)  DIMENSION DIMENSION  OF OF  A A  * * * ;\: * * * * * # « * * * * * * * * * * * * * * * * * * * * * * * s|s * * :;: *  ELIMINATION  COMPLEX  ROW  DETERMINANT  ID: *  IN  VECTOR  *  A , X , D E T , I D , J O )  SUBROUTINE  FOR  COMPLEX  SIMULTANEOUS  EQUATIONS  O E T , S  A ( ID , J D )  , X ( 1 0 ,1  )  NP=M+1  NM=N"1. 0ET=1  .0  TRIANGULAR DO  12  IZATION  K=1,NM  KP=K+1 R = 1 . 0 / A ( K , K ) DET = DET*A 00  11  (:<,K)  J=KP,NP  A ( K , J ) = R * A ( K , J ) DO  12  S = A( 00  I=KP,N  I ,K ) 12  J = KP  ,NP  A( I , J ) = A ( I , J ) - S * A ( K , J ) ' D E T = 0 ET BACK  A (N , N )  SUBSTITUTION  X(N,1) = A ( N , N P ) / A ( N , N ) DO  22  1=1,MH  K=N"I KP=K + 1 S=A(K,IMP) 0 0  21  J = KP,N  S=S"A(K,J X(K,1)=S RETURN F MO  ) * X l J , l )  "  (  

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