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Reflected wave propagation in a wedge Ishill, Hiroshi 1969

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REFLECTED WAVE IN  PROPAGATION  A WEDGE  by "  HIROSHI B.Sc, M.Sc,  I SHII  Tohoku U n i v e r s i t y , 1963 T o h o k u U n i v e r s i t y , 1965  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENTS  FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY in  t h e Department of GEOPHYSICS  We a c c e p t  THE  this  thesis  as c o n f o r m i n g  UNIVERSITY OF BRITISH S e p t e m b e r , 1969  to the  COLUMBIA  In p r e s e n t i n g an  this  thesis  in partial  advanced degree a t the U n i v e r s i t y  the  Library  I further for  shall  make i t f r e e l y  agree that  permission  f u l f i l m e n t of the requirements f o r of B r i t i s h  Columbia,  I agree  that  a v a i l a b l e f o r r e f e r e n c e and S t u d y . f o rextensive  copying of this  thesis  s c h o l a r l y p u r p o s e s may be g r a n t e d b y t h e Head o f my D e p a r t m e n t o r  by  h i s representatives.  of  this  written  thesis  It i s understood  for financial  gain  shall  permission.  Department The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  Date  2-UU  Sept  Columbia  .  \°.t> °l  that  copying or p u b l i c a t i o n  n o t be a l l o w e d w i t h o u t  my  ii ABSTRACT The b e h a v i o r o f e l a s t i c body waves i n a d i p p i n g l a y e r o v e r l y i n g an e l a s t i c medium has been t h e o r e t i c a l l y i n v e s t i g a t e d by a m u l t i p l e r e f l e c t i o n f o r m u l a t i o n . Although the d i f f r a c t e d wave i s not i n c l u d e d i n t h i s f o r m u l a t i o n , i t s importance  i s s t u d i e d by i n v e s t i g a t i o n o f  the amplitude d i s c o n t i n u i t i e s w i t h i n "the wedge. For a plane SH wave i n c i d e n t a t the base o f the d i p p i n g l a y e r p e r p e n d i c u l a r t o s t r i k e , a s e r i e s s o l u t i o n has been o b t a i n e d .  Numerical v a l u e s o f the amplitude, phase  and phase v e l o c i t y a r e c a l c u l a t e d on the s u r f a c e . p r o p a g a t i n g i n the up-dip d i r e c t i o n the amplitude  For waves versus  frequency curves f o r a c o n s t a n t depth t o the i n t e r f a c e change s l o w l y w i t h i n c r e a s i n g d i p f o r d i p angles l e s s than 20°. However f o r waves p r o p a g a t i n g i n the down-dip d i r e c t i o n the c h a r a c t e r o f the amplitude curves change r a p i d l y .  In these  c a s e s , i t i s found t h a t the d i f f r a c t e d wave p l a y s an important r o l e .  In a d d i t i o n t o s a t i s f y i n g the boundary c o n d i t i o n s  at the s u r f a c e and the lower boundary of. the wedge, the d i f f r a c t e d wave must a l s o s a t i s f y a d d i t i o n a l c o n d i t i o n s along a d i p p i n g i n t e r f a c e between the wedge boundaries due t o the g e o m e t r i c a l nature o f the r e f l e c t e d wave s o l u t i o n .  It i s  found t h a t the phase v e l o c i t i e s vary r a p i d l y w i t h both p e r i o d o f the wave and depth t o the i n t e r f a c e . For i n c i d e n t plane P and SV waves, the complexity o f the problem  due t o the converted waves does not a l l o w the  s o l u t i o n t o be expressed i n s e r i e s form.  However, a com-  p u t a t i o n a l scheme has been developed which a l l o w s the  iii  calculation waves.  of the  For both  incident  of displacements the  surface.  for  incident  are  there  and  SV  curves peaks  As  in  the  Rayleigh  an the  elastic  final  of  line  study  medium.  by  the  role  the  a  ratio  o f the  SV propa-  versus  interface  energy  do  not  15°.  have  The  wave  are  diffracted  relations  media are  between  determined  in a cylindrical  direction  system  is interpreted  i s to determine  displacements  sources  contributions  determined  with  than  using  wave.  wave f o r m u l a t i o n t h e impulsive  P and  the o u t g o i n g  direction  complex p r o p a g a t i o n  The  by  on  ratios  to d i p  g r e a t e r than  between e l a s t i c  terms o f p r o p a g a t i o n the  to  values  calculated  incident  amplitude  depth  caused  problems  interface  and  are  displacement  For  the  to determine  subsidiary  waves a t an  the  f o r dip angles  maximum d i s c o n t i n u i t i e s calculated  ratios  i n t h e down-dip d i r e c t i o n  f o r constant  reflected  SV waves, n u m e r i c a l  P waves.  d i r e c t i o n oL, (3 = 1 2 0 ° ,  significant  wave.  that  to the m u l t i p l y  waves a r e much more s e n s i t i v e  for incident  frequency  also  P and  displacement  I t i s found  waves p r o p a g a t i n g gation  d i s t u r b a n c e due  A due  o f SH  waves  formal t o head  and  Using  a  reflected  to p e r i o d i c  i n t h e wedge  solution  e v a l u a t i o n o f the  steepest descent.  due  by  i s found  reflected  integrals  ray paths,  the  overlying by  waves  by  and  the  which are method  contributions  iv  of  t h e i n t e g r a l s have b e e n  interpreted.  The r a n g e o f  existence  o f head waves has b e e n e x a m i n e d and t h e d i s c o n -  tinuities  associated  the  case  of a free  the  dispersion  with  diffracted  or r i g i d  relation  waves  studied.  In  l o w e r b o u n d a r y o f t h e wedge,  has b e e n  determined.  V  TABLE OF CONTENTS Page ABSTRACT  i i  L I S T OF FIGURES  v  L I S T OF TABLES ACKNOWLEDGEMENTS CHAPTER  CHAPTER  1  GENERAL  INTRODUCTION  Preliminary  1.2  Summary o f P r e v i o u s  1.3  Scope o f T h i s  '  Remarks  i  x  i  i 1  Studies  2  Thesis  4  MULTIPLE REFLECTION OF PLANE SH WAVES BY A DIPPING LAYER  CHAPTER  i  x i i  1.1  2  i  2.1  Introduction  2.2  Wave E q u a t i o n  2.3  Reflection  2.4  Multiple  2.5  Numerical  3  7 7  and F u n d a m e n t a l  and R e f r a c t i o n  Reflection  Solution  8  Coefficients  9  Solution  Computations  f o r a Wedge  and D i s c u s s i o n  2.5.1  Amplitude  Disctoninuity  2.5.2  Surface Amplitude  2.5.3  Phase V e l o c i t y  20  a t Q-<^>^~J\Z 20  Characteristics  at the Free  MULTIPLE REFLECTION OF PLANE P AND  Surface  21cL 30  SV WAVES  BY A DIPPING LAYER  32  3.1  Introduction  3.2  Equations  3.3  Reflection  3.4  Computation o f Displacement of a Dipping Layer  of Motion  14  32 and B o u n d a r y C o n d i t i o n s  and R e f r a c t i o n  Coefficients  32 36  i n t h e Case 46  vi  3.5  Displacement  Discontinuities  3.6  Surface Displacements  50  and D i s p l a c e m e n t s  Ratios  CHAPTER 4  53  3.6.1  Incident  P  53  3.6.2  Incident  SV  56  HEAD AND REFLECTED WAVES FROM AN SH LINE SOURCE IN A DIPPING LAYER OVERLYING AN E L A S T I C MEDIUM 4.1  Introduction  4.2  Equation of Motion  4.3  Steady  State  4.4  Formal Source  Steady  4.5  61 and B o u n d a r y C o n d i t i o n s  P l a n e Wave S o l u t i o n State  4.5.2  -  62 65  Solution f o r a Line 72  E v a l u a t i o n o f the F i r s t Series the I n t e g r a l 4.5.1  61  Term o f  C o n t r i b u t i o n from t h e S a d d l e P o i n t ( R e f l e c t e d Waves) Contribution Point  (Head  from  73  76  the Branch  Waves)  4.6  Aperiodic  4.7  I n t e r p r e t a t i o n o f the T r a v e l  4.8  Range o f E x i s t e n c e  4.9  Discontinuities  4.10  Dispersion  78  Solution  80 Time,  o f Head Waves  82 84 88  E q u a t i o n f o r t h e Lower  B o u n d a r y F r e e and R i g i d  94  4.11  The H o r i z o n t a l  96  4.12  Computation  Layer S o l u t i o n  o f Displacement  Seismograms  99  vii  CHAPTER  5  SUMMARY, CONCLUSIONS  5.1  Summary  5.2  Suggestions  AND FURTHER  STUDIES  and C o n c l u s i o n s f o r Further  105 105  Studies  BIBLIOGRAPHY  109 111  APPENDIX  I  ENERGY RELATIONS  113  APPENDIX  II  EXPRESSION OF A FREE.RAYLEIGH WAVE USING COMPLEX ANGLES  116  EVALUATION OF THE SECOND SERIES TERMS OF THE INTEGRAL  120  APPENDIX  III  viii  L I S T OF FIGURES FIGURE 2-1  2-2  PAGE C y l i n d r i c a l c o o r d i n a t e system used i n t h i s problem. Reflection inclined  2-3  2-4  2-5a  2-5b  2-6a  2-6b  and r e f r a c t i o n  a t an a r b i t r a r y  (T  0 '  2/) 10  at a boundary angle  6tk -  H  M u l t i p l e r e f l e c t i o n and r e f r a c t i o n f o r a w e d g e - s h a p e d medium w i t h a wave i n c i d e n t with propagation d i r e c t i o n oL •  15  D i s p l a c e m e n t d i s c o n t i n u i t y a l o n g t h e edge o f o u t g o i n g r e f l e c t e d wave f o r u n i t a m p l i t u d e i n c i d e n t waves w i t h p r o p a g a t i o n d i r e c tion cL . "  22  Amplitude s u r f a c e f o r the parameters d i p a n g l e and (r=£/3jLH/CbiT f o r an i n c i d e n t wave w i t h p r o p a g a t i o n d i r e c t i o n d = 60°.  24  Amplitude s u r f a c e f o r the parameters dip a n g l e and < T ~ 2 s f 3 7 Z H / C b t T f o r an i n c i d e n t wave w i t h p r o p a g a t i o n d i r e c t i o n Oi = 120° .  25  Amplitude s u r f a c e f o r the parameters propagation direction cL and fr=j2/?JJGH/CbiT f o r a h o r i z o n t a l boundary.  27  Amplitude s u r f a c e gation direction for a d i p angle  28  f o r the parameters propaoL and <S*-Zt/JJZ H / C b i T 10°.  2-7  Amplitude s u r f a c e f o r the parameters d i p a n g l e and T—JZ,JLT/CMT f o r an i n c i d e n t wave w i t h p r o p a g a t i o n d i r e c t i o n oL = 6 0 ° .  2- 8  Phase v e l o c i t y ( C u / C b i ) c u r v e s v e r s u s CT = ^ ^ 3 7 0 H / 6 b i T f o r a d i p angle o f 10° and p r o p a g a t i o n d i r e c t i o n cL . The t h i n h o r i z o n t a l l i n e s a r e the phase v e l o c i t i e s for the h o r i z o n t a l l y layered case.  31  R e f l e c t i o n and r e f r a c t i o n o f waves a t a b o u n d a r y i n c l i n e d a t an a r b i t r a r y a n g l e 8dL w i t h the nomenclature f o r angles b e t w e e n r a y s and t h e h o r i z o n t a l and boundary s u r f a c e s i n d i c a t e d .  37  3- 1  .  ix  3-2  3-3  3-4  3-5  3-6  R e f l e c t i o n o f waves a t a f r e e s u r f a c e w i t h n o m e n c l a t u r e f o r a n g l e s between r a y s and t h e f r e e s u r f a c e i n d i c a t e d .  44  Flow d i a g r a m s h o w i n g t h e c o m p u t a t i o n a l scheme u s e d t o c a l c u l a t e t h e a m p l i t u d e s and p r o p a g a t i o n d i r e c t i o n s o f the r e f l e c t e d waves i n t h e wedge and t h u s the d i s p l a c e m e n t and d i s p l a c e m e n t r a t i o a t any p o i n t .  49  Maximum d i s p l a c e m e n t d i s c o n t i n u i t y o f t h e r a d i a l component from the e x i t i n g P waves and t a n g e n t i a l component from the e x i t i n g SV waves f o r an i n c i d e n t P wave w i t h p r o p a g a t i o n d i r e c t i o n s oi - 60° and oi = 1 2 0 ° .  52  Maximum d i s p l a c e m e n t d i s c o n t i n u i t y o f t h e r a d i a l component from t h e e x i t i n g P waves and t a n g e n t i a l component from t h e e x i t i n g SV waves f o r an i n c i d e n t SV wave w i t h p r o p a gation directions = 60° and ^ = 120°.  54  H o r i z o n t a l and v e r t i c a l d i s p l a c e m e n t s v e r s u s the parameter (T = Z/TjZH/CaiT for incid e n t P waves w i t l i p r o p a g a t i o n d i r e c t i o n s cL = 60° and — 120 f o r the r a n g e o f d i p a n g l e s 3°^6 ^30°.  55  D i s p l a c e m e n t r a t i o s V/H v e r s u s t h e p a r a m e t e r §- = zJKlLH/CatT f° i n c i d e n t P waves w i t h propagation directions o C = 60° and oC « 120° f o r the range o f d i p a n g l e s B°— 6^.= 3 0 °  57  H o r i z o n t a l and v e r t i c a l d i s p l a c e m e n t s v e r s u s the parameter c? ~Zf3 7Z\A/C(x\T for incid e n t SV waves w i t h p r o p a g a t i o n d i r e c t i o n s (S = 60° and ($,= 120° f o r t h e r a n g e o f d i p angles $°£9<i&30 .  58  D i s p l a c e m e n t r a t i o s H/V v e r s u s the p a r a m e t e r 6~" -2,J~3K,H/Ca.iT f o r i n c i d e n t SV waves w i t h p r o p a g a t i o n d i r e c t i o n s (3 = 60° and .|S = 120 f o r t h e r a n g e o f d i p a n g l e s S%d^k30°.  60  Geometry o f t h e p r o b l e m : the l i n e source ( S ) i s l o c a t e d a t (d, 0 ) and t h e r e c e i v e r ( R. ) a t ( T , 0 ) i n t h e wedge bounded by t h e f r e e s u r f a c e ( Q=—S\ ) and t h e b o u n d a r y (0 = 0ji, ) b e t w e e n t h e two m e d i a .  64  <K  3-7  r  3-8  0  3- 9  y  4- 1  X  4-2  4-3  4-4  4-5  4-6  4-7  4-8  4-9  4-10  The o i i - p l a n e (0Cr = X + l ^ ) on which Re(A-s)7>0 and the r e g i o n s of p o s i t i v e and n e g a t i v e I m ( A s ) , separated by the curves bg and L^g , indicated. Notat i o n : 5 - s a d d l e - p o i n t ; B , B' - branch p o i n t s ; L - o r i g i n a l path of i n t e g r a t i o n ; Lis - path of s t e e p e s t descent through the saddle p o i n t ; L M , Viz. - paths of branch l i n e i n t e g r a l ; U8 - branch cut Re(As)=0; and bg - curve along which lm(/ls) = 0 •  ^5  B a s i c ray paths used i n p h y s i c a l i n t e r p r e t a t i o n of c o n t r i b u t i o n s from branch and saddle p o i n t s .  83  Ray paths of the head and r e f l e c t e d waves expressed by the f i r s t s e r i e s term of the integrals.  85  Maximum value of Qx f o r which the head waves shown i n F i g u r e 4-4 e x i s t versus the r a t i o of source to o b s e r v a t i o n d i s t a n c e s . The o b s e r v a t i o n and source p o i n t s at 5° from the f r e e s u r f a c e .  87  Maximum value of the wedge angle ( 6 i + Qz ) f o r which the head waves of the types shown i n F i g . 4-4 e x i s t f o r an o b s e r v a t i o n p o i n t at 5° from the f r e e s u r f a c e and d / T = l O . O .  89  D i s c o n t i n u i t i e s i n medium (1) due to i n t e r a c t i o n of the wave with the v e r t e x . The l i n e d areas i n d i c a t e the r e g i o n s f o r which the geom e t r i c wave from t h e l a s t r e f l e c t i o n e x i s t s with the term from which i t a r i s e s i n d i c a t e d i n brackets.  90  R e l a t i v e amplitudes of the displacement d i s c o n t i n u i t i e s due to a plane i n i t i a l wave c l o s e to the x - a x i s f o r p r o p a g a t i o n upward ( 771 = -) and downward ( 771 = +) .  92  Coordinate system f o r the h o r i z o n t a l l a y e r case w i t h the source ( S ) at ( d l , 0 ) and , the r e c e i v e r ( R. ) at . Three were Hi ck C  cases f o r which t h e o r e t i c a l seismograms calculated. The parameters used were: = 9.59 km, H^= 3.00 km, 0 - 99.6 km, - 10.0 km, and the displacement parameter = 0.05 sec.  97  100  xi  4-11  4-12  4-13  A-l  A-2  A-3  Ray p a t h s w h i c h c o n t r i b u t e r e t i c a l seismograms.  t o . t h e theo-  D i s p l a c e m e n t s o f t h e component waves f o r t h e g e o m e t r i e s g i v e n i n F i g u r e s 4-10a, 4-10b and 4-10c.  102  S y n t h e s i z e d seismograms r e s u l t i n g t h e d i s p l a c e m e n t s o f F i g u r e 4-12.  104  from  C o o r d i n a t e system used to c a l c u l a t e the complex a n g l e o f a f r e e R a y l e i g h wave.  117  The c ^ - i - p l a n e showing b r a n c h c u t s and i n t e g r a l p a t h s f o r e v a l u a t i o n o f the second s e r i e s term o f the i n t e g r a l s . Notation: B , C - branch p o i n t s ; £ saddle point; L - o r i g i n a l path of integration; L s - path of steepest d e s c e n t t h r o u g h s a d d l e p o i n t ; and L L (I— 1 , Z • - -) - paths of branch line integral.  123  Ray p a t h s o f t h e head waves e x p r e s s e d by t h e s e c o n d s e r i e s term o f t h e i n t e g r a l s w i t h the f o u r c o m b i n a t i o n s o f 77L(+, — ) and £ ( \ , Z ) c o r r e s p o n d i n g to the f o u r s e c o n d s e r i e s terms o f t h e i n t e g r a l in ( A - 3 . 1 ) .  130  LIST  TABLE 1  101  OF  TABLES PAGE  Notation  used  i n Figure  3-3.  50  xii  ACKNOWLEDGEMENTS  I wish Ellis of  f o r h i s guidance  discussion  Smylie  h i s comments  t o D r . R. M.  and f o r many  o f the e n t i r e  on C h a p t e r  and e n c o u r a g e m e n t  Helpful  4.  hours  investigation.  a r e due t o D r s . R. D. R u s s e l l  the U n i v e r s i t y  and D. E.  and D r . G. K. C. C l a r k e  I appreciate the constant  o f Dr. R u s s e l l  d u r i n g my  studies  of British  Columbia.  discussions  a r e a c k n o w l e d g e d w i t h my  c o l l e a g u e Mr. 0. G. J e n s e n , program  thanks  and e n c o u r a g e m e n t  f o r reading the manuscript  interest at  my s i n c e r e  during the course  Thanks  for  to express  f o r p l a n e waves  who a l s o  incident  p r o v i d e d me w i t h h i s  on a h o r i z o n t a l l y  layered  system.  I would Akio  like  t o e x p r e s s my a p p r e c i a t i o n  Takagi, Chief, Akita  granting  studies  at the U n i v e r s i t y  This manuscript  This  Research and  Tohoku U n i v e r s i t y f o r  e d u c a t i o n a l l e a v e and t o P r o f e s s o r Z i r o  suggested  Council  Observatory,  of British  was t y p e d by M i s s  s t u d y was s u p p o r t e d  o f Canada  R. D. R u s s e l l .  Fellowship  (Grant  acknowledged.  year  Judi  Kalmakoff.  Research  and t h e D e f e n c e  of British of this  who  Columbia.  9511-76) t o D r s . R. M.  A University  d u r i n g the second  Suzuki  by t h e N a t i o n a l  ( G r a n t A-2617) t o D r . R. M. E l l i s Board  to Professor  Columbia  study  Ellis  Graduate  i s gratefully  CHAPTER 1 GENERAL  1.1  Preliminary  Remarks  Elastic determination constitution lysis have the  are  of'the  zontally  and  dipping models.  However,  t h e ana-  number o f m o d e l s  (1968),  successful region  o f Clowes  has p r o v e d  an  average hori-  t o be a d e q u a t e i n  Ibrahim  (e.g.,  only  depend  point  as i n d i c a t e d by t h e Hence  Phinney  ( 1 9 6 9 ) ) have  beneath the o b s e r v a t i o n  which  reflection  i t i s necessary  to i n v e s t i g a t e the b e h a v i o r  l a y e r to o b t a i n  parti-  as body wave a m p l i t u d e s  e t a l (1968) .  which  structures.  and hence t h e  body wave a p p l i c a t i o n s  be g e o l o g i c a l l y complex  important  concerning  waves have b e e n  path  ana-  instrumentation  for horizontally layered  and Basham  been m o d e r a t e l y  studies  internal  o f o u r knowledge  surface  i n the  d e c a d e new  advances i n  by t h e l i m i t e d  formulation  However,  on a l o c a l i z e d  and t h e  In t h e p a s t  the p r o p a g a t i o n  layered  (1964), E l l i s  roles  f o r i n t e r p r e t a t i o n as t h e y y i e l d  over  cases.  structure  with  seismology,  useful  important  o f the e a r t h .  a v a i l a b l e - mainly  structure  may  coupled  properties  In e a r t h q u a k e  most  crustal  o f the e a r t h .  are r e s t r i c t e d  cularly  very  to r a p i d expansion  seismic  lyses  waves p l a y  techniques lead  INTRODUCTION  an u n d e r s t a n d i n g  o f waves  in a  o f t h e more  complex  2  1.2  Summary o f The  tures  has  Haskell at  the  ing  Previous  i n t e r p r e t a t i o n of h o r i z o n t a l l y layered  been dominated  (1953, 1960, base of  the  by  the  1962).  He  boundary c o n d i t i o n s the  to  eventually  input For  next  wave a t  of  considered  wave  obtains  displacements  the  and  obtaining  SV  stresses  be  vertical  h o r i z o n t a l displacements.  the  experimental  ratio  dent  SV,  with  theoretical Haskell  ture. wave  the  V/H  experimental  Haskell's  formulation  the  by  the  matrices  ratio  the  motion. input  of  the  incident and  for  t h e n be  to d e t e r m i n e  boundary  between  surface  For  can  apply-  from one  frequency,  ratio,  ratios  and  inci-  compared  crustal  i s also applicable  P,  to  struc-  surface  studies.  Several  studies  for n o n - p a r a l l e l boundaries  b e e n done, m a i n l y r e l a t i n g Nagumo  (1961) and  Sato  wedge-shaped medium. source He  taking  versus H/V  input  f r e q u e n c y domain  f u n c t i o n may and  by  system  a relation  waves, the  eliminated  an  propagator  l o w e r b o u n d a r y and  i n c i d e n t P and  struc-  theoretical studies  a h o r i z o n t a l l y layered  which c a r r y the  Studies  to  surface  (1963) d e a l t w i t h Hudson s t u d i e d  i n a wedge-shaped medium w i t h  obtained  waves.  a s o l u t i o n composed  SH  SH  Hudson waves  of m u l t i p l y  lower  (1963),  in a  waves f r o m a  a rigid  have  line  surface.  reflected  3  and  diffracted  gated  waves.  the e f f e c t  Using  of d i f f r a c t i o n  by means o f an a p p r o x i m a t e and  found  (  w  n  e  a liquid  and  wave a m p l i t u d e  T,  two d i m e n s i o n a l  layer  mode s o l u t i o n s dispersion fraction  exist.  problem  boundaries  w h i c h he f o u n d Lapwood  overlying  relations  to incident  free  T  e  a  rigid  wave p r o p a g a t i o n i n He f o u n d  the s o l u t i o n  o f t h e wave.  Nagumo  Sato  he  s t u d i e d the d i f -  SH p u l s e p a r a l l e l  and c a l c u l a t e d diminished  and A l s o p  corner  t o one o f t h e  diffracted  rapidly  that  investigated  o f SIT waves a t an o b t u s e - a n g l e d plane  pulse  decreases  the v e r t e x ) .  bottom.  (1961) , Kane and Spence  (1964) , M c G a r r  investi-  are the d i s t a n c e s o f the  elastic  From  he  o f the d i f f r a c t e d  and o b s e r v a t i o n p o i n t , from  considered  due  r  solution  a t t h e apex o f t h e wedge  form  that the d i f f r a c t e d  as source  this  wave  away from  forms  the v e r t e x .  (1963) , Hudson and  (1967) and o t h e r s  Knopoff  have s t u d i e d  R a y l e i g h wave t r a n s m i s s i o n i n a wedge-shaped medium. Lapwood on  investigated  one o f t h e f r e e  integral  boundaries  from  a line  of a right  t r a n s f o r m a t i o n and a p p r o x i m a t i o n  Kane and Spence considered  pulse  angle,  free  procedures.  (1963) and Hudson and K n o p o f f  boundaries.  tion  procedure  The f i r s t  and t h e l a t t e r  i n order  authors  employed  to calculate  source  using  (1964)  R a y l e i g h wave t r a n s m i s s i o n on e l a s t i c  with  technique  wave forms  employed  wedges an  itera-  a Green's f u n c t i o n  transmission  coefficients  4  o f the R a y l e i g h wave.  Using  method, McGarr and A l s o p transmission  an approximate v a r i a t i o n a l  (1967) computed the 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 R a y l e i g h waves normally  dent on v e r t i c a l d i s c o n t i n u i t i e s . o n l y been a few  studies  (Fuchs  Conversely,  (1966)) and  and  inci-  there have  Kane (1966))  on the e f f e c t of n o n - p a r a l l e l boundaries f o r body waves; nevertheless  body waves c o n s t i t u t e an i n i t i a l  s e c t i o n of a  seismogram which i s very o f t e n used i n a n a l y s e s . synthesized g a t i n g along  seismograms due  Fuchs  to a primary P s i g n a l propa-  the median plane i n a s o l i d wedge w i t h  free  b o u n d a r i e s , by t a k i n g a summation of r e f l e c t e d waves. determined the d i s p e r s i o n of the body waves and motion.  He  particle  Kane employed a t r e e diagram which i s o b t a i n e d  r e f l e c t i n g the wedge r a t h e r than the rays and which c a r r i e s nine p i e c e s of d a t a . t h e o r e t i c a l seismograms due  a  by  vector  Thus, he c a l c u l a t e d  to an input plane P p u l s e  f o r the t e l e s e i s m i c response o f an a r r a y o f s t a t i o n s l o c a t e d on a u n i f o r m l y  dipping c r u s t .  In t h i s way  demonstrated the s i g n a l d i s t o r t i o n e f f e c t s o f the  he geometry.  However the amplitude c h a r a c t e r i s t i c s which are used f o r i n t e r p r e t a t i o n o f c r u s t a l s t r u c t u r e were not nor was 1.2  investigated  the d i f f r a c t e d wave.  Scope o f T h i s The  Thesis  o b j e c t i v e o f t h i s t h e s i s i s to extend the  o f body wave p r o p a g a t i o n f l e c t e d wave f o r m u l a t i o n . f r a c t e d waves are not  theory  i n a dipping structure using a reAlthough the forms o f the  i n v e s t i g a t e d , determination  amplitude d i s c o n t i n u i t i e s due  of  difthe  to the r e f l e c t e d wave w i t h i n  the wedge i n d i c a t e s i t s importance.  5  First, at  the base  case  reflection  teristics terms  o f depth  d i p angle.  the  last  i s obtained  as a g u i d e  of Chapters  tion  and  solution  scheme  are given  In C h a p t e r  an e l a s t i c  reflected  wave  the  integrals  i s studied  does n o t  the c a l c u l a -  and p h a s e v e l o c i t i e s . relations  As  direction  medium  formulation.  source  direction  interpreted.  o f S H waves i n a dipping  i s investigated  from  a  layer  using a  The c o n t r i b u t i o n s  waves a r e d e t e r m i n e d  sub-  b e t w e e n waves a t  i n terms o f the p r o p a g a t i o n  and i m p u l s i v e l i n e  and r e f l e c t e d  which allows  4, p r o p a g a t i o n  overlying  head  The c o m p l e x i t y  i s developed  t h e complex p r o p a g a t i o n  periodic  problems  t o be o b t a i n e d ; however, a com-  problems , the energy  a boundary  i n Chapter. 2  3, t h e c o r r e s p o n d i n g p r o b l e m  o f the d i s p l a c e m e n t  sidiary  from  to d i f f r a c t e d  t h e more d i f f i c u l t  P and S V waves.  a series  putational  which r e s u l t  3 and 4.  incident  allow  charac-  p e r i o d o f t h e wave,  T h i s development  for solving  by  on t h e s u r f a c e  and w h i c h a r e r e l a t e d  In Chapter for  A solution  calculated  The d i s c o n t i n u i t i e s  waves a r e d e t e r m i n e d .  incident  and t h e a m p l i t u d e  to the i n t e r f a c e ,  reflection  S H wave  i s c o n s i d e r e d as i n t h i s  are present.  and p h a s e v e l o c i t y  and  serves  2, a p l a n e  of a dipping layer  no c o n v e r t e d waves  multiple  in  i n Chapter  due t o  by e v a l u a t i n g  by t h e method o f s t e e p e s t d e s c e n t  and a  6  comparison made w i t h a h o r i z o n t a l l y l a y e r e d case the case o f n u m e r i c a l examples. o f head waves i s determined  through  The range o f e x i s t e n c e  and the d i s c o n t i n u i t i e s  a s s o c i a t e d w i t h d i f f r a c t e d waves s t u d i e d . The  study i s summarized and s u g g e s t i o n s made  f o r f u r t h e r i n v e s t i g a t i o n s i n the f i n a l The  chapter.  t h e o r y f o r m u l t i p l y r e f l e c t e d waves as developed  i n t h i s t h e s i s c o u l d serve as a u s e f u l s t a r t i n g p o i n t f o r the study o f d i f f r a c t i o n .  Techniques  t h e o r y o f d i f f r a c t i o n as developed  such as the g e o m e t r i c a l by K e l l e r  (1962) appear t o  be a p p l i c a b l e ; however, they may not be p r a c t i c a l due t o the complexity  introduced.  In t h i s theory f o r s m a l l wavelengths,  K e l l e r uses d i f f r a c t i o n laws s i m i l a r t o laws o f r e f l e c t i o n and r e f r a c t i o n which are d e r i v e d from Fermat's p r i n c i p l e . Away from the d i f f r a c t i n g s u r f a c e s , he i s able t o use d i f f r a c t e d rays j u s t l i k e o r d i n a r y r a y s .  By the use o f the  r e f l e c t e d wave s o l u t i o n and such a d i f f r a c t e d wave i t may be p o s s i b l e to o b t a i n a more s a t i s f a c t o r y o f e l a s t i c waves i n a wedge.  procedure,  description  7  CHAPTER 2 MULTIPLE REFLECTION OF PLANE SH BY A DIPPING LAYER  2.1  Introduction The  c a l c u l a t i o n of  o f waves p r o p a g a t i n g been g r e a t l y Haskell  mining  the  waves.  the  1962).  proved  crust  regional  and  structure  to  allow may  be  useful  g e o l o g i c a l l y complex. the  teristics  a localized  structure  required  observed  to  explain  As  an  at  the  initial  a plane  SH  an free  strike  the  boundaries  the  layered  body wave they  s t a t i o n which  on  the  important charac-  F e r n a n d e z and this  the  Careaga  t y p e may at  La  be  Paz.  o f body waves i n t e r a c t i n g  surface  will  been  waves  It i s , therefore,  elastic  wave i n c i d e n t  d i r e c t i o n of  beneath  a model o f  study  surface  have o n l y  body wave o b s e r v a t i o n s  b e t w e e n the  using  deter-  i n t e r p r e t a t i o n as  surface.  that  a wedge o v e r l y i n g  reflected  area  of  for  i n t e r p r e t a t i o n , the for  has  this  c l o s e l y enough t o  e f f e c t of dipping  (1968) have s u g g e s t e d  the  a p o w e r f u l method  conform  may  study  on  may  only  to  formulation  Even though f o r s u r f a c e  a successful not  media  a p p l i c a t i o n of  upper mantle  depend  with  matrix  The  t o be  successful.  amplitudes  be  amplitude c h a r a c t e r i s t i c s  However, body wave a p p l i c a t i o n s  moderately  theory  s i m p l i f i e d by  has  the  in h o r i z o n t a l l y layered  (1953, 1960,  formulation  to  WAVES  on be  the  medium, a l l waves and  the  dipping  internally layer  wedge p e r p e n d i c u l a r  considered.  The  due  to  objective  8  is  t o c a l c u l a t e the amplitude  distance period vious  from  the v e r t e x ,  o f t h e wave. workers,  waves w i l l observation explicitly  points  from  On t h e b a s i s  t h e most  that  important  d i s t a n t from  investigated  wave, t h e b o u n d a r y tions  depth  i t i s expected  play  c h a r a c t e r i s t i c s i n terms o f  i n this  conditions  the s u r f a c e ,  and t h e  o f the r e s u l t s o f prethe m u l t i p l y  reflected  r o l e i n a seismogram a t  the v e r t e x study.  and w i l l  For the d i f f r a c t e d  are expressed  made t o i n d i c a t e i t s i m p o r t a n c e  be  and c a l c u l a -  i n particular situa-  tions.  This b e t w e e n wave  simple types  more d i f f i c u l t as  being  case  i n which  serves  problems  as a g u i d e  right.  wave, r e f r a c t e d wave and r e f l e c t e d  points  by e v a l u a t i n g  and s a d d l e  reflection, tant  step  coupling  f o r s o l v i n g the  Further,  as  surface  wave components a r e  the c o n t r i b u t i o n o f p o l e s ,  points  branch  r e s p e c t i v e l y i n terms o f m u l t i p l e  the s o l u t i o n o f the p r e s e n t  leading  i s no  o f i n c i d e n t P and SV waves as w e l l  o f i n t e r e s t i n i t s own  obtained  there  problem  i s an impor-  t o t h e s o l u t i o n o f t h e s e more  complex  problems.  2.2  Wave E q u a t i o n In  found ( T ,0  this  convenient  and F u n d a m e n t a l  problem with t o choose  Solution  a dipping  a cylindrical  boundary  i t is  coordinate  H, ) r e l a t e d t o a c a r t e s i a n s y s t e m  (%  }  system  -"fr ^ 2L ) as  9  2-1.  shown i n F i g u r e t h e x-y  plane,  displacement variation  For  a plane  the m o t i o n  has  o f the  SH  wave p r o p a g a t i n g  i s independent  o n l y a z-component. form  the  >  of  Z  Assuming  equation  and a  in  the  time  of motion  ~ct~9±F becomes  fr*  +  where We  ( 2 , 1 )  in cylindrical  ^ f r £  +  =  b  choose  ^ f e  o;/C  as  the  coordinates  r  fundamental  2.3  A propagating  We w i t h waves  i n the  only non-zero  Reflection now from  (Figure  2-2).  written  as  and  cL  (2.2)  0  s  of this  i s a plane  equation  wave o f  ampli-  direction.  component  of stress  is  Refraction Coefficients  c o n s i d e r two medium The  "  solution  which  The  u =  b  A • (3 tude  K)  +  (2)  elastic  incident  solutions  media d i v i d e d on  i n media  the (1)  by  0=  interface and  (2)  can  be  0^  12  L-fe ,rcos(0-|s;  .  b  itbzrcos(e-oL).  .IXz~ AL o  L%irco$(e-r)  -4- Afji'S  -  The in  boundary  conditions  displacement  fore  a t the i n t e r f a c e  and s t r e s s .  require  O — Bd.  At  wo  continuity  can there-  write  (2.5)  ffao)|~( Pae) The  condition  z  of equality  o f phase a t  G — OeL  leads to  £ cos(e -p) = ^ c o s ( e ^ - ^ ) = ^ c o S ( e a - T ) b l  d  Using  A ji. r  (2.3) and s u b s t i t u t i n g  (2.4) i n t o  ( 2 . 5 ) , we  ( 2  . ) 6  obtain  S S i n(6di- op - A S (n(9^- g>  =  A S i n ( 0 d - {$)- S s i n ( O d - T )  AL  (2.7)  Arf--,  5sin(Gd-^)-gsin(edi-r)  Ai  Asin('0A-p)-SS/n(0d-T)  where Using angles (Figure  & =  C^Z/CM  and  (2.6) and t h e g e o m e t r i c cL, $  t  T  a  2 - 2 ) , we have  n  d  t  h  e  % —  / (z/ M-\ /  /  relationships  angles  ;  [,p  between the ;  l  T  13  (2.8) 7 0  is = e + -^- - s i n ' ^ c o s C e a - o O ) d  and  (2.9)  S i n ( e d - p ) = .-7 i - ( i / ^ ) c o s ( ^ - o c ) a  e .<oc< 6a -v- .^c  with  d  Finally,  An  substituting  sIn (0d-  (2.9) i n t o  oC) +  (2.7) we  CI / S)  cos  a  obtain  ("^"oQ"  Sin(0 -o/o-(|/g)yA--cos-(0d-5y  AL  d  (2.10)  • asin(eA-oO  Arf • _ Ac  Sin(0d-oC)- (1/ S)^A -cos^Oci-.oc) A  T h u s , we and  refraction  gation  boundary lowing  been  coefficients  direction,  In  have  Aff ' Ai ~ =  to denote  the r e f l e c t i o n  i n terms o f the i n i t i a l  the d i p a n g l e ,  and  the e l a s t i c  t h e c a s e where t h e waves a r e i n c i d e n t  from medium  ( 1 ) , t h e same p r o c e s s  yields  propa-  constants.  on t h e the  fol-  equations:  A$m(0d-oO-r  AL  able  §/7^A*C0S*(  0d.-op (2.11)  2&S\n(6<k~oc)  A S i n ( 0 d - o l ) i - \SJ \-  A^cos^-oO  14  and  '  P  (2.12)  -ir^+Sirr^ACosce^-oc))  =  with  0^H- T t <  If  cL < . 0 * + £ 7 G  A COS (0d-oC)> A  a  must be r e p l a c e d solution  by  -  I  then  J A^COS^CO^  to remain f i n i t e For  I  at  waves i n c i d e n t  J|-A  a C  OS*(e ~oO~ A  — Oi) ~ T  f  o  r  t  h  e  infinity.  on t h e f r e e  surface,  we have  (2.13)  T = Z7L with  2.4  0  -  cL  <  OL <  Multiple  Reflection  Consider the  boundary  Jb  Q=  a wave 0^  Solution  A ^ u " ^ ^ T C O S ( e - oi) A^'O incident b  from medium  a r e s u l t i n g r e f l e c t e d wave of  f o r a Wedge  ^''  (2) ( F i g u r e  V  ,  » '/  A  n  «A -A- -e * '  o  lt ^TCOS(0-T,) b  i  l  k  b  r c o 8 ( e  2-3) and assume  and r e f r a c t e d  t h e forms  -^  on  wave  I M * 2-3. M u l t i p l e r e f l e c t i o n and r e f r a c t i o n f o r a wedge--. Eig. 2 Siped a v e i n c i d e n t w i t h propagat i o n d i r e c t i o n cC m  e  d  i  u  m  w  i  t  k  a  W  16  Using are  A  equations  (2.8) and  (2.10),  the b o u n d a r y  conditions  satisfied for  "_  A,  -  sin(e^-oO-r  (i/S)v A "~cos -(6d-oC) /  a  2  ssn(e -^)~(!/s)/A"-cos-(aa"oC) d  Tj  =  (2.14)  ZdcK-i-Z'JZ-oL  and  /\  =  2$\n(6rj-cL)  Si n(e -oC) - ( i / §) / ^ c o s ^ c e ^ - o c ) d  (2.15)  Pi = O d - T To  satisfy  a reflected  - Sin'(•S'COS(edL-oC))  the boundary c o n d i t i o n at  9 = 0  , we  assume  wave  i> ,rcos(e- r , o b  By e q u a t i o n provided  (2.13),  the boundary  condition is satisfied  that  To s a t i s f y  the boundary  assume t h e r e f l e c t e d  c o n d i t i o n s at  0 = 0^  and r e f r a c t e d waves  l-fe |TCOS(0-Ti) b  A*/A',- A L - 6 ^  / / =  A l - A • AL' S ^ ^ L  (  B  T G O S  ^ ~ 0  ^  , we  must  17  which  •*  satisfy  the boundary  conditions f o r  A s i n ( 0 d - V H 8/1 - A cos*(aL- r / ) a  (2.16)  and  (2.17)  using  equations These  not ^  an i n f i n i t e <  cases  V*,  < X  (2.11) and steps  are then  process + 6JL  with  repeated.  f o r i t terminates o  IL  r  t h e wave p r o p a g a t e s  collision  (2.12).  <  T ^ <  where  L -  Tl m a x  JO + 0dL  The l a s t  i n medium 1 i s o f t h e form  r c  wheneveri n these  down t h e wedge w i t h o u t  the b o u n d a r i e s .  A,(7cA.)e^  However, i t i s  °  S ( e  -^  term  further  o f the s e r i e s  18  This  at  gives  rise  <£> — JZ '71  0 =  fracted  As ®~  the we  a discontinuity  .  Therefore,  waves must be  displacement ing  to  and  stress  boundary  shall  see  $ri~  a  i n the ""  S  n  at  reflection  o  s  t  c  at  and  Q — 0  s  e  solution  s  S  (1963) has  the  diffracted  wave d e c r e a s e s  of  the  ray  discontinuity  distant  from  a  H  usually  of  points  m  0  0 =  dominates the  .  that the  TC0S(9_T  w  -  be  then  ^^  at the  seismo-  amplitude  T  We  of  satisfy-  0^  a\\ ay from  hence w i l l  5  as  indicating  vertex.  M^A^fTtAje^  dif-  discontinuity  rapidly and  the  continuity  well  shown t h a t  the  K=A,f:(JcA )e^  the  for  give  as  Further,  theory  will  0 = (h — JG  section, a  displacement,  solution  that  gram.  surface  Sato  next m  the  form  conditions  ~"  multiple  of  i n the  the small  region at  write  (2.18)  )  and  N =A,i(^Aje^  i r c o s ( e  ' T  }  (2.19)  19  where A  ~  \  ,  :  _  (2.20)  (2.21)  sin ( e * - o l ) - ( c o s ^ e I = ^  (2.22)  ft-  e  A  + f - - s i n ^ i - c o s c e A - o c ) )  (2.23)  TTt max  L  (2.24)  |_j — TTt max  The s o l u t i o n ing  ]  c a n t h e n be w r i t t e n  on w h e t h e r t h e l a s t  Q _=  0^  .  A  with  +  i s from  c a s e we  for  0=0  or  have  oie4<-}c ,  /  '^i Ni M,. =  reflection  In t h e f i r s t  U i = -N,+ Ni'  i n one o f two forms depend-  '  ^or r ~ ^ ^ e ^ e L  (2.25) A  20  and  i n t h e s e c o n d c a s e we  have  (2.26)  TO < TJ, <  W I T H  The velocity  C  v  (H)  a m p l i t u d e A, t h e p h a s e  , and t h e p h a s e  0 = c o n s t a n t ) may be  ( i n the d i r e c t i o n  written as:  A -  y H r C e C U O * *  I  m  ( U  (  (2.27)  f  (2.28)  / Re(u,)-rx;—  lm(^i)—  ~  v-  x  2.5  Numerical Computation For  for  the n u m e r i c a l computations, the values JJL = ( 8 $Z X  correspond to the c r u s t  Haskell  t  and  chosen  Ct>z/C \-\ b  - u p p e r m a n t l e model u s e d by  (1960) .  2.5.1 As reflection, rise  and D i s c u s s i o n  t h e p a r a m e t e r s were  which  /  Amplitude  Discontinuity at  discussed  i n the previous  which  does  not c o l l i d e  section,  the l a s t  w i t h a boundary,  to a d i s p l a c e m e n t d i s c o n t i n u i t y  gives  and c o r r e s p o n d i n g t o  ( 2  2  9  )  21  "tKis~a  ^''-function  i n the s t r e s s at  @~  3^  •  The  magnitude o f the displacement d i s c o n t i n u i t y versus d i p angle i s shown i n F i g u r e 2-4 f o r v a r i o u s angles o f i n c i d e n c e applicable  to t e l e s e i s m i c  magnitude i s small describes tinuity  In the case where the  the r e f l e c t e d wave s o l u t i o n  the p h y s i c a l  problem.  adequately  However, i f the d i s c o n -  i s l a r g e , then a d i f f r a c t e d wave with a l a r g e  tude i n the r e g i o n continuity  of  d — '^—TC  i s required  ampli-  to p r o v i d e  i n displacement and s t r e s s .  We see that the  waves.  as  up-dip d i r e c t i o n  f o r the i n c i d e n t wave p r o p a g a t i n g i n (oL \ ]0 <  £  ) , the d i s c o n t i n u i t y  0  i s small  f o r d i p angles l e s s than 15°. However, f o r i n c i d e n t waves propagating  i n the down-dip d i r e c t i o n ( O C > ^ 0 ) the d i s -  placement d i s c o n t i n u i t y dip  angles.  i s large  f o r some ranges o f small  In these cases the d i f f r a c t e d wave i s impor-  t a n t because the i n t e r n a l l y r e f l e c t e d wave propagates out o f the wedge a f t e r a small ever, f o r surface expected that  points  number o f r e f l e c t i o n s .  distant  from the v e r t e x , i t i s  the r e f l e c t e d wave amplitude w i l l  good approximation i n most r e g i o n s to the true as  the d i s c o n t i n u i t y s u r f a c e  How-  becomes d i s t a n t  give a amplitude  from the f r e e  surface. I t should a l s o be p o i n t e d out that to the d i s c o n t i n u i t y w i t h i n  i n addition  the wedge, d i s c o n t i n u i t i e s are  21a  generated by the v e r t e x on r e f l e c t i o n o f the incoming wave and each r e f r a c t i o n  i n t o medium ( 2 ) .  In the r e f l e c t e d  wave t h e o r y these appear as displacement and s t r e s s continuities  radiating  from the v e r t e x .  these w i l l not normally be l a r g e wedge except c l o s e  dis-  The e f f e c t o f  on the s u r f a c e o f the  t o the v e r t e x as the amplitude  decreases  r a t h e r r a p i d l y w i t h d i s t a n c e and the wave w i l l be p a r t i a l l y reflected  at the lower boundary o f the wedge.  i n any study o f d i f f r a c t e d waves t h e i r r e l a t i v e tance should be  2.5.2  However impor-  investigated.  Surface Amplitude  One e f f e c t o f i n t e r e s t  Characteristics i s the e f f e c t o f a v a r i a -  t i o n o f d i p angle on the amplitude c h a r a c t e r i s t i c s  at the  s u r f a c e f o r a c o n s t a n t depth to the boundary and f i x e d  22  O  o  CO Q LLJ Q  DIP ANGLE Fig» 2-4  0  (DEGREES)  Displacement d i s c o n t i n u i t y along the edge of outgoing r e f l e c t e d wave f o r u n i t amplitude incident waves with propagation d i r e c t i o n cLo  23  propagation tude  direction.  s u r f a c e s are p l o t t e d  and  JO H / C  for  oL = 60°  then  and  amplitude  For  slowly with 25°.  b t  T  120°.  see  that  the  and  are  dent  waves p r o p a g a t i n g  respectively.  amplitude  but  the d i s c o n t i n u i t y  of  rapid  multiple fully  physical  Cbl  except  these  period.  cL  = 120°  is  This  curves  for  inci-  indicated  cL  = 60°,  wave i s l i k e l y  oL = 1 2 0 ° ,  situation  larger  range  i n these  characteristics,  as p r e s e n t e d  t o p l a y an  this  that  but  here  does  that  the  important  diffracted  i n t h e down-dip d i r e c t i o n  the for  o v e r most o f t h e  amplitude  reflection solution  direction  0 = ^ — 7 0  indicates  fre-  The  for a propagation d i r e c t i o n  considered.  the  down-dip  for this  near  with  considered.  and  change  range  characteristics  reasons  are  direction  change r a t h e r r a p i d l y  is significant  viewpoint,  and  i n the  wave w i t h  o f the  angle  characteristics  i s small at  changes o f the  propagating  and  propagation  amplitude  d e s c r i b e the p h y s i c a l  fracted  to the  a propagation d i r e c t i o n  discontinuity  angles  H  i n the up-dip  o f the  For  Q^<^Z\°  dip  typical  One  F i g u r e 2-4.  fixed  of dip angles  2-5a  ampli-  boundary)  incident  range  the  ( H i s depth  d i p angle  characteristics  2-5b  2-5b dip  wave w i t h  the  increasing  quency over  and  f o r the parameters  For  incident  However, f o r an  amplitude  2-Sa  s u r f a c e s f o r v a r y i n g d i p angle  an  o(, = 6 0 ° , we  in  In F i g u r e s  role.  of  regions  the not difFrom  a  wave f o r waves  arises  as  t h e wave  24  PROPAGATEON DIRECTION » SO°  2  JlwH C,T b  2-5a  a  Amplitude s u r f a c e f o r t h e parameters d i p a n g l e and =JZ/3"7GH/CbiT f ° i n c i d e n t wave w i t h propagation d i r e c t i o n oC=60°<> r  a  n  25  o  ro  BamndiAiv P i g . 2-5b. Amplitude s u r f a c e f o r t h e parameters d i p a n g l e and 6^=£/3 70H/CbiT f o r a n i n c i d e n t wave w i t h p r o p a g a t i o n d i r e c t i o n oi=[ZQ° •  26  only  collides  propagates leads  o u t o f t h e wedge  Hence we  waves w i t h  cribes  Also the  parameters  dip  angle.  angles range  curve  These  change r a p i d l y  f o r a constant  2-6a and 2-6b f o r d i p are s i m i l a r  amplitude  i n the  oscillation  g r e a t e r than with  Figure  directions.  with  between  amplitude  angle.  for this  = 0  no d i p . F o r  2-4 t h a t we e x p e c t role  t h a t the  direction  110° t h e  increasing  the amplitude  and d i p a n g l e  versus  I t s h o u l d be n o t e d  to the curve  tion  ^JSL£-  surfaces for  o f 10° and p r o p a g a t i o n  to p l a y a s i g n i f i c a n t  =  des-  this  wave  Finally,  adequately  direction  two g r a p h s  more r a p i d l y .  from  i n t h e down-  direction  directions  s h o u l d be n o t e d  and  45° t o 90° i n t h a t f o r i n -  cT" = 49 as compared  curves  ampli-  direction  f o r a d i p angle  propagation  solution  are the amplitude  = 45° has one a d d i t i o n a l  and  -r;  wave  a r e shown i n F i g u r e s  propagation  oscillate  of significant  propagating  cT~ and p r o p a g a t i o n  of propagation  curves  wave, w h i c h  problem.  of interest  These  before i t  f o r large d i p angles  angles  the r e f l e c t e d  o f 0° and 1 0 ° .  creasing  and h e n c e t h i s  see t h a t except  the p h y s i c a l  few t i m e s  wave, i s s t i l l  large incident  direction  oL  the boundary a very  to the d i f f r a c t e d  tude.  dip  with  range  Again i t the d i f f r a c t e d of propaga-  s u r f a c e s f o r the parameters  ( F i g u r e 2-7) a r e c o n s i d e r e d .  The  DIP  Pig  0  ANGLE  = 0°  2-6& Amplitude surface f o r the parameters propagation d i r e c t i o n oL and 6"=Zj3 7tH/C\,\T horizontal 0  f  "boundary.  o  r  a  DIP  ANGLE • 10°  135  4-i UJ Q  •  2  0.  /  <  0  7  14  21  28  2 73 C  Pig.  2-6b.  b,  35  42  49  / /  60  / 45  TTH  ...  T  Amplitude s u r f a c e f o r the f o r a d i p a n g l e 0^=10°.  parameters  p r o p a g a t i o n d i r e c t i o n cL and  _  (T = £ / 3 ~ X . H / G b i T  PROPAGATION  14  21  28  35  42  49  56  DIRECTION - 6 0 °  63  70  77  2 IT r  P i g * 2-7. Amplitude s u r f a c e f o r t h e parameters d i p a n g l e and T=2JDr/CbiT an i n c i d e n t wave w i t h p r o p a g a t i o n d i r e c t i o n ©1=60° *  30  major of dip  f e a t u r e o f these  oscillation angle  vertex  the s p e c t r a l rapidly  2.5.3  i s the i n c r e a s e i n the r a t e  f o r the amplitude  increases.  to change  curves  Hence  versus . T  c h a r a c t e r o f a seismogram  with, a c h a n g i n g  Phase V e l o c i t y  a d i p angle .  the h o r i z o n t a l l y present depth  dip angle.  at the Free  until  layered  to boundary.  incidence  i s reached  Surface  the o s c i l l a t i o n s  to v a r i a t i o n s interface.  curves  from  (thin  lines)  directions  the p e r i o d  as d i s p e r s i o n i s  o f t h e wave and  o f the phase  layered  then decrease.  velocity  propagation  which corresponds  the h o r i z o n t a l l y  angle  to v e r t i c a l  case.  Beyond  Clearly,  layered  this  measure-  on a wedge-shaped medium  with both p e r i o d  are p l o t t e d  the phase v e l o c i t y f o r  increasing  f o r the h o r i z o n t a l l y  d e v i a t e markedly  from  The a m p l i t u d e  ments o f p h a s e v e l o c i t y  the  case  increases with  an a n g l e  angle  markedly  w h i c h depends oh b o t h  oscillations  i s expected  o f 10° f o r v a r i o u s p r o p a g a t i o n  These d i f f e r  as t h e  f o r a c o n s t a n t d i s t a n c e from t h e  In F i g u r e 2-8, t h e p h a s e v e l o c i t y for  curves  will  c a s e due  o f t h e wave and d e p t h t o  30a  An  of  the  large  phase v e l o c i t y which o c c u r  for  incident  dip  explanation  direction  gate  toward  reversed dent  and  angles  i s as  the  in  the  layered  vertex  and  p r o p a g a t e out ( oC  direction will contribution  follows.  to  transfer  of  the  be  their  wedge.  amplitudes resulting  phase v e l o c i t y w i t h  function  situation.  large  the  compared  waves  in  to  the  the  i n the  t h i s c a s e waves  t h e n have  = 75°) , t h e  still  In  variations  up-  propa-  direction For  small  inci-  upon r e v e r s a l  of  in a significant little  change  horizontally  31  A1I0013A Pigo  2-8o  3SVHd  Phase v e l o c i t y GV/C-bi c u r v e s v e r s u s 6"=£>/3 70H/CbiT f o r a d i p a n g l e of 10° and p r o p a g a t i o n d i r e c t i o n cC o The t h i n h o r i z o n t a l l i n e s a r e the phase v e l o c i t i e s f o r the h o r i z o n t a l l y l a y e r e d c a s e *  32  CHAPTER 3 MULTIPLE REFLECTION OF PLANE P AND BY A DIPPING LAYER  3.1  Introduction , In t h e p r e v i o u s c h a p t e r  SH waves  incident  A solution  incident  a t any a n g l e  obtained  f o r the amplitude,  In t h i s  chapter  considered.  expand  3•2  o f Motion  Equations  In t h i s a cylindrical  motion T cal  problem,  system  of  components.  has b e e n  velocity.  use d i s p l a c e -  Consequently,  (T  &  (e.g.,  c a l c u l a t i o n s to the present  available for  Conditions  i t i s again convenient  ( % ,^ , %  coordinates are:  f o r waves  structure  in their  Q  )  %  to choose  ) related to  ) as shown i n F i g u r e 2-1.  P and SV waves p r o p a g a t i n g  0  who  crustal  and B o u n d a r y  c o - o r d i n a t e system  i s independent  and  investigators  limited  con-  P and SV waves i s  t h e number o f m o d e l s  purposes.  For plane  of incident  layered structures.  interpretation  a cartesian  p h a s e and p h a s e  to i n t e r p r e t  1964) have b e e n  will  of a plane  p e r p e n d i c u l a r to s t r i k e  Experimental  horizontally analysis  by m u l t i p l e r e f l e c t i o n  the case  characteristics  Phinney,  the problem  a t t h e b a s e o f a d i p p i n g l a y e r was  sidered.  ment  SV WAVES  i n t h e x-y p l a n e , the  and t h e d i s p l a c e m e n t  The e q u a t i o n s  o f motion  has o n l y in cylindri-  33  ^ at* ~ U + W a r  ae .  T  -  C3  "  1}  where:  2CO = a  -™(T.lAe)  and  7p " 9 0 ^  i s density; and  , displacements  The  stress  components  r  y/.  and  \X  r  (3.4)  , Lame's T  i n the  are expressed  constants;  9  and  directions.  by  = » ( M i _ Jig. , J _ 9 U ^ Using we  r  equations  (3.3) and (3.4) i n t h e e q u a t i o n s  of motion,  obtain  9  * ®  at*  • .-a i  r/.V * 3  ar  1  T  0  r  i  '  ^  I  f  1  •  a  - ^ \  ae* '  „  1  7  J  1  34  and  i n the s t r e s s  r e l a t i o n s , we  obtain  (3.9)  (3.10)  9T J - ^ L  where"  and  , t h e P and  Ci  S wave v e l o c i t i e s r e s p e c t i v e l y .  Assuming  a time  variation  ~ c *() of  the form  (3.8)  u  ^  , equations  (3.1),  (3.2),  (3.7) and  become  1 a® (3.11)  T  (3.12)  (3.13) .2. b where  ^  (3.14)  ^&  CO  a.  Substituting  Q  Gb  CT. (3.11) and (3.12) i n t o  and  V  9  ar  2 1  I 2 r a r ' r^sd*  (3.9) and ( 3 . 1 0 ) ,  +  we  have  (3.16)  35  We  choose  and  as t h e f u n d a m e n t a l  solution  (3.14) ,  a  p l a n e waves p r o p a g a t i n g i n t h e respectively. (3.11),  Substituting  ( 3 . 1 2 ) , (3.15)  expressions  and  cyC  (3.16)  |3  and  directions solutions  we have t h e stresses:  i'kbTcosfe-6)  Fsiri(e-0)e  L  P  (3.i7)  s'\n(_e-oL)Q  e-  l  .  (3.18)  Kb  -^c:Fcos(e-p)sm(e-p)e * i  fe =  into  following  •,  2i  (3.13)  F'6  d  the fundamental  f o r d i s p l a c e m e n t s and  Z  n  of equations  -z  f c%{ E  cos(e-OL)S  in( e ^  T  C  e  + (l~zcos^e-p))Fe  0  b r c o s ( 0  S  '  p )  ( 0  l l b r c o S ( e  ^H  (3.20)  36  3.3  Reflection In  this  coefficients and  and R e f r a c t i o n section  t h e r e f l e c t i o n and r e f r a c t i o n  i n terms o f the i n i t i a l  the e l a s t i c constants  sider  two e l a s t i c m e d i a  waves  from medium  3-1). (3.14),  Coefficients  be c o n s i d e r e d .  separated  (2) i n c i d e n t  The s o l u t i o n s i n media  will  propagation  0=  by  0^  First  con-  with  on t h e i n t e r f a c e  o f the equations  direction  (Figure  o f m o t i o n , (3.13) and  (1) and (2) c a n be w r i t t e n  as  lto^TCOS(e~oLrJi) (3.21)  o3 ,= a  D  i n  e^  T C O S ( 9  ^ [) +  r t  e  where 60  The  boundary c o n d i t i o n s  at  are  (3.22)  re, = r e  37  CO  CD  b  cT  CD ti CD  R e f l e c t i o n and r e f r a c t i o n of waves at a boundary i n c l i n e d at an a r b i t r a r y angle 0* with the nomenc l a t u r e f o r angles between rays and the h o r i z o n t a l and boundary s u r f a c e s . i n d i c a t e d .  38  For  a s o l u t i o n o f t h e form  ments and s t r e s s e s expressions  n  -  i n medium  (3.21),  the d i s p l a c e -  (1) and medium  (2) a r e f r o m  (3.17) t o (3.21)  [-Larncffl  ^  . L % a \ r c o s ( e - d  ^-o^CinCQSCe-oOe^  - ^ p  i  n  -t  e -  ) e ^  0  3  0  e  T  C  0  S  (  D s i n ( e - p )&  1 W  c  o  s  S  i  r A  n  (  P  rJt  r<-CM  -t-l^— C  Khz.  0  r x  sin(e~c^rx)6  6  -  ^  - ^ C  e  -  r  £  39  -^P.CblcosCe-p^sinCe-MB^e^ ™^"^ 1  (3.24)  + {I-*cos*Ce  -M  6 ^  (  e  cosCe-^sinO-ooCine^  0  0  l  C  °  S  + {i-^e-piD „e *  r  w r C 0 S ( e  I  + Application diately  o f the boundary  to the e q u a l i t y  SV waves  yields  ia.cos(e*-oi) ft  b 4  which  i s Snell's  j ^ =  Law  boundary  equations  t  ,  '  conditions  t  t  0  5  (3.22)  o f phase which  (  5  -  ^"^  p )  ' ' . '  leads  for incident  r  t  )  immeP or  respectively  cos (e*- p) J  The lowing  ( i ^ - M } ^  _  f l 2 C 0 S ( e A  _^  ) =  ^cosceu-Pm) .cs-25)  = t ,cos(0Aa  expressed  i n a cosine  conditions  i  b  l  c os (©A -  p ) r5  form.  (3.22) t h e n y i e l d  the  fol-  40  CD.  ^ Vccx.  CO.  crT I  S  5 '  CD  c  00  o _ o ,  —  •  o ll  f  cS  X  i CO.  I  i I  CD  8  cn. I  CQ.  CD  CO  o  O  c\>  CA5  o  I  o  CD  o  CO  o  3  OQ.  CD  ,5 ^  CO O  o  l O  CCu I  CN(  t  CO O O  8 ^  'co CD  9  Q4  CO  5  CAi  CO  o o  X  co  cA>  -6  o  o  CD  II  8  CD  cO  I  <  CD  8 .E Q_| CO  •Q_i  y  CD  Sr CD  y  41  To an  solve  incident oCr-J-  a  incident medium dent  n  equations  oi SL  P o r SV wave, t h e a n g l e s §r£  d  "lust  and b o u n d a r y  D  R  determined  e  angles.  ( 2 ) , the f o l l o w i n g  from  (3.26) i n t h e c a s e o f e i t h e r $TJL  »  i n terms  o f the  F o r a P wave i n c i d e n t  geometric  '  relationships  from  are evi-  F i g u r e 3-1.  I  06=  0^ + "^7 70 -  boL  oC c = 0^ + -Jr JO — LoC ^. r  T:  for  0^<^oC^ 0^4"  '  (3.25) we t h e n  n  [  $rSL=  %+X " x  t 5 f n  ^r** x 5  7 0  1  case.  (X,<^  27l~oC  2  (3.28)  (  l(u /C )CoS(0d-cC)} {  a2  value o f the inverse  It i s easily  A.-^.-25 <c  +  c  ™sin  where t h e p r i n c i p a l  obtain  ~ {t ^/Ca )COS(0^-o6)} _  for  ^ S  e^^jr-7ltS\n {cOS(3A-ct)}^Zed  ztr^  each  l J s  verified •  that  For incident  sine  i s taken i n  (3.28) a l s o  holds  SY waves, we  find  (3.29)  42  (3.29)  Hence, (3.28),  i n (3.26) and s u b s t i t u t i o n  t h e r e f l e c t i o n and r e f r a c t i o n  incident  P wave c a n be d e t e r m i n e d .  Q-ft = 0 the  Din—0  by s e t t i n g  coefficients  Similarly,  and t h e u s e o f (3.29) a l l o w s  r e f l e c t i o n and r e f r a c t i o n  by  of  f o r an setting  us t o d e t e r m i n e  coefficients  f o r an  incident  SV wave. For  P waves i n c i d e n t  on t h e b o u n d a r y f r o m medium  (1) we d e t e r m i n e t h e f o l l o w i n g  OLTJL= dd  -f \  expressions  j c - s i n'{COS(0A-  f o r the a n g l e s .  oi)}^ze^-i-zjc-oi  PrX^^+ir^-Sin' {(G ,/Ca,)COS(0^~o6)} l  b  ^  J  cL $ = 6 + - j r ^ * r  Pr± and  A  = 6* + j r ^  Sin  (3.30)  ^{{Uz/c^COSidcK-oL)}  +5m {(C ^/Ca|)COS(0 A-o(.)} H  <  b  f o r S waves  ^ r x ^ ^ ^ i : ^ - - S I n {(G /c )C0S(e^-(S)} ,  a(  bl  o  ^  = 6* H-  (3  JG + sy n ' { ( G W C , ) c o s f ^ - b  jS;}  Equations tion  may  t h e n be u s e d  to determine  and r e f r a c t i o n c o e f f i c i e n t s f o r waves  (1) by  medium ing  (3.26)  (3.30)  the use o f  incident  (3.31)  and  the r e f l e c -  and t h e  from follow-  substitutions:  and  Cb2~^  Obi  are  i n t h i s case  DTJ)_  '  s  n  Finally, 0—0  (3.14)  i n medium  ©,= A  i  n  u  ^  we  3-2).  (1) c a n  tions  a  n  C-rJL  (2) and  incident  The s o l u t i o n s  be w r i t t e n  of  between the a m p l i t u d e  ^  ^  •  and  as  (3.32) r,tbircos(e-p jL)  as b e f o r e we  r  find  t h e equa-  c o e f f i c i e n t s t o be  cos (3 si n i V TJL  ~~COSoL jzS\noLrJL  •\-ZC0S*p JL  £ C O S I  •-4-v *Cospsfng  r  n  (3.13)  r  t h e same p r o c e d u r e  a  on t h e f r e e  -f- A x6 ' •  3r$-  d  (1) .  c o n s i d e r waves  (Figure  Ar^-  that  i n medium  i t b \ r c o s ( e - § )  Following  noted  e  i n medium  e  -  l d  amplitudes  amplitudes  surface  o  T  (3.33)  b  co§oLS\r\cL For seen  to hold  P wave  incidence  the f o l l o w i n g  r e l a t i o n s are  and  the  free  surface  indicated.  45  COScL sL— r  £0S  COSdL  ^ r J L — ( C b i / C a \ ) COS  Sinod=  oi  ^i-cosvT  od = -§-J"c +  5\r\ {cosoi}-=Z7io-oL  and  SV wave  ]  rJL  f o r an i n c i d e n t  C0So^=(CAi/Cb»)C0S(3 COSp ji= r  COS £  Sin p- /"Pcos ^" 2  cCrx = x X . + 51 n" { ( G A > / C b i ) c O S £ > 1  Using P  (3.34) and (3.35) i n ( 3 . 3 3 ) , we o b t a i n  f o r incident  waves  ^ •__ 4^bfcos^sino(,/i-i/^cos cxl :  ~ (\-z  v£ cos* oiT .  V  ;<lcoSct S i n o t ( I  p r  ^  ^  -  zVbi  COS*oQ  c o s - ^ s inotyi -• v £ c o s ^ -t (1 ~ z v T c o s V J * 2  b  (3.36)  ^ i n  46  and  for incident  S waves  -zv^cosps\r\$(\-2cos p)  A  z  cos p s i n (3 /  rjL  - c o s ^ + (i - z cos p  a  " ^  A  1 n  (3.37) P^O  — ~  In  L  expressions  (3.36) and ( 3 . 3 7 ) ,  K  i t s h o u l d be  noted  t h a t when t h e argument A i n a s q u a r e  root  then  \f~/\  f o r the s o l u -  -~L\J~/\  must be r e p l a c e d by  tion  to remain  finite  tude  coefficients  at i n f i n i t y .  i n this  form,  .  i s negative,  As a c h e c k on t h e a m p l i -  the energy  flux  equations  have b e e n d e r i v e d i n A p p e n d i x I . <>  3.4  Computation To  we must From  deteirnine  (3.23) we  see t h a t  coefficient  the process  i n order  incident  angles  matically tion  amplitudes this  that  o f a l l waves w h i c h  and p r o p a g a t i o n d i r e c t i o n complex  the cases  g r e a t e r than  angles  for total  the c r i t i c a l  involved i n results.  Layer  a t any p o i n t i n a wedge  r e q u i r e s the c a l c u l a t i o n  o f computation,  employed  i n t h e Case o f a D i p p i n g  the amplitude  sum t h e complex  amplitude In  of Displacement  o f the  f o r each  have  wave.  been  r e f l e c t i o n and angle  In A p p e n d i x  o f a R a y l e i g h wave w r i t t e n i n terms  arrive.  are auto-  I I , investiga-  o f complex  angles  47  shows t h a t gation of  the r e a l  direction  part  o f the angle  indicates  and t h e i m a g i n a r y p a r t  the propa-  g i v e s the decrease  amplitude.  As tion  a P and S wave  and from  each  arise  reflection  from  from  the i n i t i a l  the f r e e  t h e b o u n d a r y between m e d i a , t h e r a y s  refrac-  s u r f a c e and  i n c r e a s e i n number  71+ I Z  as  where  reflection direction  fl  process  i s between  a further  neglecting  70  o u t o f t h e wedge.  artificial  (the  displacement  ment  amplitudes  free  s u r f a c e i n the absence  note  that  tudes for  (K  Further, M  initial for  o f the  For computation  was  as t h e Kf  and  t h a n o—{CT^  less  (  i t i s important to  and p r o p a g a t i o n d i r e c t i o n  at a l l p o i n t s .  Hence,  ffl  r  may  i f the a m p l i -  then  easily  of reflected  be  the t o t a l calculated.  waves depend  TTL o f t h e i n p u t wave, i t i s i m p o r t a n t  refraction  waves w h i c h may  and e a c h generate  reflection further  for a  ( TTl ) a r e d e t e r m i n e d  i n t h e wedge,  a t any p o i n t and  i n t r o d u c e d by  boundary).  ) and p r o p a g a t i o n d i r e c t i o n s  o f motion  pur-  wave w o u l d have on t h e  purposes,  a l l waves r e v e r b e r a t i n g  case  a r e n o r m a l i z e d by t h e d i s p l a c e -  the i n c i d e n t  computational  i s t h e same  amplitude  on  which  f o r i n that  t e r m i n a t i o n was  amplitudes  the amplitudes  wavefront  and 70+'9^  a l l waves whose a m p l i t u d e  For  The  i s t e r m i n a t e d whenever the p r o p a g a t i o n  t h e wave p r o p a g a t e s poses  i s the order o f r e f l e c t i o n .  to store  waves.  The  directly  at the  Kl  and  rather  48  complex c o m p u t a t i o n examination for in  o f the flow c h a r t  the r e f r a c t e d  the amplitude  t h e P wave  amplitude and  OTls  later f^l  and  a  r  also  e  T)1  are then c a l c u l a t e d  order  propagates £  the  .  refracted  parameters Cbz./Ga\= respond  again f i r s t  by t h e n e x t  S and i t s r e s u l t i n g calculation  will  these  0.73  time  P  by t h e S wave generate  of this  are then  f o c u s s e d on t h e P ) ,  lowest  order u n t i l  finally  waves a r e e x a m i n e d .  Upon  and h o r i z o n t a l  displacement  ratio  for different  computation b  II  vector) u n t i l  the v e r t i c a l  the v a l u e s chosen  C i / C a i = 0. £ 7 8 < t  were  f o r the r a y  or the amplitude i s  and t h e v e r t i c a l - h o r i z o n t a l  t h e n be c a l c u l a t e d  In  H$  (  as P ( a t t h e same  The waves g e n e r a t e d  of this  displacement may  the system  out o f the system  generated  completion  t h e wedge, t h e  i n a v e c t o r which  i n the s u b s i d i a r y  (with a t t e n t i o n  those  i n t h e wedge.  and s t o r e d  and h i g h e r o r d e r waves t h e y may  examined then  through  f^g. and TTLs  than  to c a l -  o f t h e S wave  temporarily stored  TVb  and s t o r e d  can be u s e d  and p h a s e a t any p o i n t  and  by  t o i n v e s t i g a t e waves due t o S \</ave c o n v e r s i o n ,  which propagates  less  later  i s t o be f o l l o w e d t h r o u g h  be u s e d  either  they  M  3-3).  calculated  and p r o p a g a t i o n d i r e c t i o n )  storing  (Figure  waves a r e f i r s t  vectors i n order that  culate As  scheme used, c a n b e s t be u n d e r s t o o d  and  t o c r u s t - u p p e r mantle  ,  Caz/C&\~  | . I'7 £  •  f o r the I .^£>7 which c o r -  model e m p l o y e d by H a s k e l l  (1962).  llnout  parameters LN=i  Determine M and m f o r : r e f r a c t e d P .and S and s t o r e i n STP , STS. Store M i n RECS(N). s  Determine whether r e f l e c t i o n f r o m f r e e s u r f a c e o r boundary. Determine M and m f o r r e f l e c t ed P and 8 f o r i n c i d e n t P and s t o r e i n STP,STS. S t o r e M ,m i n RECS(N). s  s  Determine wether r e f l e c t i o n from f r e e s u r f a c e o r boundary. Determine M and m f o r r e f l e c ed P and S f o r i n c i d e n t S ( u s i n g Ms and m from RECS(N)) and s t o r e i n STP and STS. S t o r e Ms,m i n RECS(N) . 5  s  No Mo Yes  No  Yes  i) < b  Yes  Yes  N = N-  No  M= 0  C a l u c u l a t i o n of t o t a l displacement and v e r t i c a l - h o r i z o n t a l displacement r a t i o  F i g . 3-3. Flow diagram showing t h e c o m p u t a t i o n a l scheme used t o c a l u c u l a t e t h e a m p l i t u d e s and propag a t i o n d i r e c t i o n s of- t h e r e f l e c t e d waves i n the wedge and t h u s the d i s p l a c e m e n t and d i s placement r a t i o a t any p o i n t . N o t a t i o n i s g i v e n i n T a b l e 1.  IE  Print [End]  VO  50  3-3.  T a b l e 1.' N o t a t i o n used i n F i g u r e  STP(K,2) - complex s t o r a g e m a t r i x f o r p a m p l i t u d e s and p r o p a g a t i o n d i r e c t i o n s STS(K,2) - complex s t o r a g e m a t r i x f o r S a m p l i t u d e s and p r o p a g a t i o n d i r e c t i o n s RECS(L,2) - complex m a t r i x t o t e m p o r a r i l y , r e t a i n S a m p l i t u d e s and p r o p a g a t i o n d i r e c t i o n s of S r a y s which may generate f u r t h e r s i g n i f i c a n t amplitudes N-1 - no. o f r e f l e c t i o n s a wave has undergone M - amplitude m - complex p r o p a g a t i o n d i r e c t i o n S u b s c r i p t s p and s i n d i c a t e P and S wave t y p e s  3.5  Displacement As  Discontinuities  discussed  i n the previous  chapter,  the l a s t  r e f l e c t i o n w h i c h does n o t c o l l i d e w i t h b o u n d a r i e s rise  t o a d i f f r a c t e d wave w h i c h  solution the  a p p e a r s as a d i s p l a c e m e n t  displacement  diffracted placement  at  large  expected with  wave  discontinuity i s required  and s t r e s s  adequately tinuities  i n the r e f l e c t e d  describes will  from  t o be a d e q u a t e  distance.  large  only  continuity  problem.  When  a small  t h e r e f l e c t e d wave  the p h y s i c a l  require  distances  to p r o v i d e  and h e n c e  wave  discontinuity.  i s small,  gives  Large  i n dissolution discon-  d i f f r a c t e d waves; however,  the vertex  the s o l u t i o n  is still  as d i f f r a c t e d waves d e c r e a s e  rapidly  51  For  incident  discontinuity  o f both  P waves t h e m a g n i t u d e o f t h e maximum the r a d i a l  P wave and t h e t a n g e n t i a l  component  component  from  wave i s shown i n F i g u r e 3-4 f o r i n c i d e n t gation  directions  should  be n o t e d .  dip  cL -  60° and 1 2 0 ° .  The d i s c o n t i n u i t y  p r o p a g a t i o n i s much  tion.  This  before  t h e wave p r o p a g a t e s  tinuity that as  from  since  rapid  an e x i t i n g  reflected  (calculation  i n t h e c a s e o f down-  fewer  reverberations  occur  The d i s c o n -  i n comparison  i n the amplitude  dip angles.  result  interval  large  i s associated  with  The  to  result  different  particularly  when an SV wave g e n e r a t i n g  P wave r e a c h e s  wave w h i c h may be o f much  Several points  direc-  maximum P wave d i s c o n t i n u i t y  point  propa-  f o r the up-dip  changes  waves f o r d i f f e r e n t  decreases observed  waves w i t h  S  than  P waves i s r e l a t i v e l y  t h e maximum d i s c o n t i n u i t y  the l a s t  the e x i t i n g  o u t o f t h e wedge.  f o r S waves and r a p i d  exiting  The  i s expected  larger  from  the c r i t i c a l  at the next  angle.  calculated  = 0.25°) i s t h e n due t o a n o t h e r  lower  amplitude.  52  30nindlAIV 3AI1V~I__ Pig., 5-4o Maximum displacement d i s c o n t i n u i t y of the r a d i a l component from the e x i t i n g P waves and t a n g e n t i a l component from the e x i t i n g S.Y waves f o r an incident P wave with propagation d i r e c t i o n s <x>60° and cC=12.0° »  53  a  placement  n  i n c I d e n t SV wave (Figure 3-5), the, d i s -  d i s c o n t i n u i t i e s are n e g l i g i b l e f o r d i p angles (9=  l e s s than 21° f o r an i n c i d e n t wave w i t h  Q0°  indi-  c a t i n g that t h i s s o l u t i o n v e r y c l o s e l y approximates the complete  solution.  A g a i n the d i s c o n t i n u i t y  i s larger f o r  the i n c i d e n t wave p r o p a g a t i n g i n the down-dip d i r e c t i o n . However i n the case f o r discontinuity  j3  = 120° the l a r g e  f o r the outgoing P wave i s f o r l a r g e d i p  angles r a t h e r than the s m a l l e r d i p angles as found f o r the i n c i d e n t P wave c a s e . reason f o r t h i s  ( O^^JZO" )  The p h y s i c a l  i s not c l e a r .  As d i s c u s s e d i n Chapter 2, d i s c o n t i n u i t i e s  exist  i n medium (2) due to the r e f l e c t i o n o f the i n c i d e n t wave and r e f r a c t i o n o f waves back i n t o the lower medium.  Except  c l o s e t o the v e r t e x , the amplitudes o f the r e s u l t i n g  dif-  f r a c t e d waves are expected to be s m a l l .  3.6  Surface Displacements and Displacement R a t i o s 3.6.1  Incident P  H o r i z o n t a l and v e r t i c a l displacements are p l o t t e d versus the parameters  (T=^/3~JGH/CaiT *  to the i n t e r f a c e ) and i l l u s t r a t e d  CH  i n F i g u r e 3-6.  i s depth For an  53a  initial  propagation  changes  very  zontal  slowly with  displacement  the major changes from  Figure  r a y becomes in  direction  3-4,  o f 60° t h e v e r t i c a l  increasing  changes  dip angle.  we  see t h a t t h e r o l e  important.  c h a r a c t e r o f both  For  The  r a t h e r more r a p i d l y ;  i n character occur  QC  for  0^  of the  component  however  JZ>0°  where  diffracted  = 1 2 0 ° , a more r a p i d  the v e r t i c a l  hori-  and h o r i z o n t a l  change  surfaces  54  3Cin±ndlAIV Pig,* 3-5c  3AI1V13H  Maximum d i s p l a c e m e n t d i s c o n t i n u i t y o f t h e r a d i a l component from the e x i t i n g P waves and t a n g e n t i a l component from t h e e x i t i n g S T waves, f o r a n i n c i d e n t S¥" wave w i t h p r o p a g a t i o n d i r e c t i o n s @=60°and p = 1 2 0 °  0  55  00  in u  o \~ O <_ U> ce o <  O  <  CC  1o  IU  ce a z o to  o  cc CL  X  o Iz UJ  s UJ  o  < _l  CL </>  o z o to  a  Y<  O  < o I-  CL  o H o  LU CC Q  Z o  < a. o cc CL  co cc Ui Io <  CC  <. X  (J UJ  £  UJ o < _J 0.  z o N CC o  UJ  CL co  a  § 3 H Q:  >  oo o  . & O 5 CM  z o Io  £  < cc < X  o  UJ  1-  z g  UJ o < _l CL  z  cc U J a s  CO o  X  o  5 g B ^  Ui >  o <0  UJ Io < ce <  Z  a  to O  cc  CO  s o  cc CL  X  oo Q  z o N ce o X  «> *  M  O  3anindwv  P i g . 3 - 6 . H o r i z o n t a l and v e r t i c a l displacements versus the parameter <y=;i/3JtH/C,T f o r i n c i d e n t P waves with propagation d i r e c t i o n s oC=60° and oC~120 f o r the range o f d i p angles 5 ° ^ © ^ 3 0 ° • a  d  is  evident with  true is  increasing  i n the h o r i z o n t a l  discontinuity  diffracted of dip  be  in this  important  greater  than  that  the  range  V/H  interest  direction less  than  10°.  p e a k s move t o  i n c r e a s e markedly  i n amplitude.  ratios  change much more r a p i d l y  the peaks moving  this  t h e V/H  case  the  ratio  3.6.2  to  i s almost  Incident  feature  i s that  For  6^  (5  of  = 120°,  again.  until  very  angles  values  oL  are  s  the  angles However  9<K~=£Q°  for  6^  for variable  i t i s seen  that  character.  the One  displacement particular  f o r dip angles  g r e a t e r than  18°,  of horizontal  displacement  becomes s h o r t  corresponding this  larger  the  SV  significant  variation  that  ratios  For d i p  decreases  constant  From F i g u r e 3-8, surfaces- e x h i b i t  For  even at s m a l l d i p  increasing  amplitude  which are o f  oL - 60° , t h e  10°, the  noted  that  o v e r most o f t h e  shown i n F i g u r e 3-7.  are  f o r dip angles  the  indicated  analysis  similar  the  case  surface  e  noted  ratios  propagation  of  i t s h o u l d be  n  displacement  initial  with  t  angles.  in practical  and  This i s particularly-  0dL/M7°  for  However  curves  wave may  The  in  where  rather featureless.  the  dip angle.  amplitude  small for  |3  = 60°.  change o c c u r s b e f o r e d i f f r a c t e d  the p e r i o d  It should  and be  waves become  INCIDENT PROPAGATION  C  Pig.  3-7«  DIRECTION • 60 °  P  WAV Lb PROPAGATION  OIT  DIRECTION =120° -  c  a i T  Displacement r a t i o s ? / B v e r s u s the parameter 6"=A^3JcH/CaiT f o r incident P waves with propagation d i r e c t i o n s oC=60° and o(=120 f o r the range of d i p angles 5% 0 ^ 3 0 ° . 0  58  59  significant. oscillations 10° w h i l e decrease  For  (B  = 120° t h e h o r i z o n t a l  become v e r y  the v e r t i c a l i n amplitude  same t i m e  displacement  small f o r d i p angles  displacement with  oscillations  increasing  the p e r i o d o f the v a r i a t i o n  d i p angle  dip  curves  H/V  (Figure 3-9).  a n g l e s marked d i f f e r e n c e s  curves tion to  ratio  are e v i d e n t .  direction  |3  the h o r i z o n t a l  becomes  At the l a r g e r  = 60° we component  featureless.  from  than  more s l o w l y and a t t h e  lengthens.  The above f e a t u r e s a r e most e v i d e n t placement  greater  i n the  F o r even  the h o r i z o n t a l l y d i p angles  see the r a p i d  dissmall layered  f o r propaga-  oscillations  and f o r J Q — 1 2 0  0  the  ratio  due  INCIDENT SV PROPAGATION  DIRECTION  WAVES  =60 °  60  PROPAGATION  DIP =30°  DIP =30°  25°  25°  20"  20«  10'  10'  5  DIRECTION  = 120°  5  0  'ai T  'at T  Pig... 3-9. Displacement r a t i o s H / V v e r s u s t h e parameter fr--£/3JoH/C*fT f o r i n c i d e n t ST waves w i t h p r o p a g a t i o n d i r e c t i o n s (3 =60° and ^=120° f o r t h e range o f d i p a n g l e s 5%Q ^30° » o  A  61  CHAPTER 4 HEAD AND REFLECTED WAVES FROM AN DIPPING LAYER OVERLYING AN  4.1  A  Introduction A number o f w o r k e r s  (1963), the  and  Hudson and  propagation  of  Hudson  in  a rigid  the  two  case  parts  of  - the  solutions. sisting often  used  a dipping chapter and  and  layer  author w i l l  i n an  In  Chapter  input,  the an  due  to  an  establish  waves by  a line  of  observation and  into  been w e l l  one  method  the  for  source  refracted  the  of  an  of  However, f o r from  the  are  the  for present  solution for  an  SH  e l a s t i c medium.  layer  i s sought which  distant  In  problem  this  con-  studied  problem  a dipping  a solution  point  seismogram  e l a s t i c medium.  d i f f r a c t e d wave t e r m .  reflected  of  not  SH  integration  clude  has  a solution  found. turbance  plane  divided  r e f l e c t e d waves w h i c h  2,  of  be  solution  the  part  e l a s t i c wedge o v e r l y i n g  reflection By  could  the  early  theoretically investigate  source  that  i n wedge-  d i f f r a c t e d wave  multiply  overlying  investigated  d i f f r a c t e d waves out  Hudson  r e f l e c t e d and  in interpretation  the  will  line  head  and  lower boundary  multiply  (1961),  ( 1 9 6 4 ) ) have  (1963) p o i n t e d  However, t h e  of  ( e . g . , Lapwood  Knopoff  surface  shaped media.  the  SH LINE SOURCE IN E L A S T I C MEDIUM  of  has  type,  multiple been  the  does n o t a  disin-  transient  vertex  waves e a r l i e r t h a n t h e  receives diffracted  62  waves w h i c h fore of  the present the i n i t i a l  solution as  result  of branch  line  others.  existence angles For  collisions  solution section  should  with  apply  the v e r t e x . to the  o f the seismogram.  by Honda and Nakamura integrals  In t h i s  The  f o r comparison with  responding  to those  wave v e l o c i t y velocity  (1954) f o r e v a l u a t i o n  are determined  (1960)  the case  of Haskell  of a horizontal  ^-Cbz/c^—  Cbl  1.2-7  >  a n c  *  3.64  =  layer.  parameters  (1960) a r e u s e d  layer  of  for various dip  the f o l l o w i n g e l a s t i c  i n the upper  ratio  technique  t h e wave forms and t h e r a n g e s  o f t h e head waves  a l l computations  formal  and as a p p l i e d by Emura  way  There-  composition  i s e v a l u a t e d by t h e s t e e p e s t d e s c e n t  recommended  and  from  cor-  i n the S  km/sec, t h e  the r i g i d i t y  ratio  S =/<*//<i= 1.88 As tions ing  Further,  lem of  4.2  o f the f i n i t e  series  the d i s c o n t i n u i t i e s  the d i f f r a c t e d  problem tion  are obtained  from  o f p o l e s , t h e s u r f a c e wave p r o b l e m  poles  with  s u r f a c e waves  i s separated  and d i f f r a c t e d  t e  of Motion propagation  found  The s o l u t i o n  i s t h e r e f o r e an i m p o r t a n t  Equation  i n displacement  to f i n d -  step  associated  and h e n c e  the d e t e r m i n a t i o n  due t o o t h e r waves.  surface  reduces  e x p r e s s i o n o f our s o l u t i o n .  waves have b e e n from  the c o n t r i b u -  this  o f the s o l u -  o f the p r e s e n t  prob-  f o r the c o n s i d e r a t i o n  waves.  and B o u n d a r y o f SH waves  Conditions through  a system con-  63  sisting f  of  ,  an  and  elastic  a dip  6, •+• d&  angle JJL  be  free surface  .The  boundary between the tesian  system  ( %  and  Z  %  }  '0  , E  ( T  %~TCO$Q  » ^=TSln0 '  ( ck.  7  0  by  a line  ) i n the For  of  Ei  , and  Then, assuming equation  of  )  source  ( S  becomes  where  ) °f  the has  a time v a r i a t i o n  A  car-  cylindrical  SH  of  The  waves l o c a t e d , a t system.  motion only the  motion i s  a  is  independent  z-component.  form  Q,  , the  motion  l/=l,£  <?"D  in cylindrical  C^i=  .  coordinate  displacement  the  . the  will  standard r e l a t i o n s h i p s =  above p r o b l e m ,  the  to  2, 2  a n d  V U,= 7 ^ - r - T Obi  the  and  Q —  is  elastic  ( F i g u r e 4-1)  Q — —Q\  is  media  by  cylindrical  the  j \  ) is related  coordinates  generated  density  elastic  > density  > o v e r l y i n g an  medium o f r i g i d i t y considered.  jX\  medium o f r i g i d i t y  sjj^lffl  (4.1)  coordinates  i s  t  n  e  velocity  o f the  S waves  and  Free  surface  S(d o) 5  (2)  Pigo 4-lo Geometry of the problems.the l i n e source ( s ) i s located at (d,0) the r e c e i v e r (R) at (r,0) i n the wedge bounded by the f r e e surface (0=~6,) and the boundary (0=Sb) between the two media 0  65  The  only non-zero  The  boundary c o n d i t i o n s then  (Pae),=  0  component  o.t  0=~e  of stress i s  become  (4.4)  l  and  d = Q  (kt  To solution  s o l v e the l i n e  satisfying  be  obtained.  by  integration  cal  4.3  (4.5)  z  source  problem,  a plane  the boundary c o n d i t i o n s w i l l  The l i n e of this  source  solution  wave  first  c a n t h e n be o b t a i n e d  s o l u t i o n with r e s p e c t to the c y l i n d r i  angle.  Steady  State Plane  The expressed  initial  Wave  Solution  displacement  due t o a p l a n e wave i s  i n t h e form  - ^ , { ( d - ^ ) C o S o i . i , + |^-|5Jno6Ljb  =  A.: • e  66  c(.i  where of  i s the angle  t h e SH waves  luation  b e t w e e n t h e x - a x i s and. wave  and may  o f the e f f e c t  take  due t o a l i n e  r e p r e s e n t s waves d o w n g o i n g r  In the  ii  those  cases  former  been  the f r e e  expressions on w h e t h e r  in detail.  with  first  direction  and w h e t h e r  and t h e  2, t h e  i n a dipping  layer  by m u l t i p l e  manner.  any o b s e r v a t i o n p o i n t i n t h e wedge,  the i n i t i a l  the motion  four  depending  o f t h e wave  is positive  the f i n a l  reflection  i s from t h e  surface  ( 2 ) . Using  (1) o r t h e f r e e  as t h e d e r i v a t i o n  7  the boundaries,  The s o l u t i o n  i n t h e same  IXQ  the x - a x i s .  In Chapter  o f SH waves  are r e q u i r e d to express  negative  cedure  surface f i r s t .  i s obtained  from  interact  0  f o r 0 "x. 0  the x - a x i s , while  the d i p p i n g boundary  and r e f r a c t i o n  For  boundary  with  investigated  reflection  or  where t h e waves  collides  with  reflection has  from  i n the eva-  F o r 07  source.  r e p r e s e n t s waves u p g o i n g  Q  latter  on complex v a l u e s  normal  o f (2.18) and  t h e same  (2.19),  these  proexpres-  sions are  ^  ( N ) = A  ^lSN  e  4(N)=A,C(70A^)e  S(H')=A;E:(£A)e 2  tt  (4.7)  67  where t h e maximum numbers o f r e f l e c t i o n s |\j  and  (4.7)  ^ ]  /  v  a r e s e e n by e x a m i n a t i o n  from the boundary  o f the phase i n  t o (4.10) t o be d e t e r m i n e d by  K, + e * Z  zfNHM +^ N G ^  Jt-Q\  (  4.ii)  or  (4.12)  and  [\j by /  (4.13)  or  7 0 + ^  The  ^(M'+Oe.+^N'e*^  expressions  reflected  C _ A  f o r the d i r e c t  from the f r e e  surface  (4.14)  wave and t h e wave  once  a r e f r o m (4.6)  J>~^b\^CO$(cii+\Q\)~it \(kCO$cll b  7) -r\'v,& 0  (4.15)  and  A)  — 0  r\i&  (4.16)  68  The  reflection  a downgoing expression  coefficients  wave w i t h (2.20);  f o r a wave w h i c h s t a r t e d as  respect  to the x-axis  are s i m i l a r to  namely  tf ASin(&+-*(1HX (4.17) and to  *  f o r a wave w h i c h s t a r t e d as an u p g o i n g wave w i t h  respect  the x - a x i s i s  As In{ e ^ l l e ^ ^ (4.18)  z\  and  §  The due  for  solution  to the i n i t i a l  conditions  A=Cb^/Cbl  a r e d e f i n e d by  on and c l o s e  disturbance  a n d  &~Mz/Ml  to the s u r f a c e  and s a t i s f y i n g  the boundary  (4.4) and (4.5) i s , ( u s i n g t h e same method as  t h e d e r i v a t i o n o f 2.25 and 2.26)  14 = So + So + S* ( N -1) + 5 ^ (N - 0 + -ST(N- 0 + S l ( M - o (4.19) for  conditions  (4.11) and  < = 5 o + ^ + S!(N-O +  (4.13)  S1(N-O+S~(N')+SI(N ; /  (4.20)  69  for  c o n d i t i o n s (4.11) and  u ; = 5 + s> s>)+  (4.14)  SI(N)+s;v-o+ si  0  (4.21) for  c o n d i t i o n s (4.12) and  (4.13)  < = S + S~ + S*(N) + Sl(N)+ S~(N')+ Sl(lM') 0  (4.22) for  c o n d i t i o n s (4.12) and ( 4 . 1 4 ) .  out  that  tion  these  formal  f o r an i n c i d e n t  oCl  solutions  are not a p h y s i c a l  p l a n e wave ©.£ a p a r t i c u l a r  b u t a r e t h e p l a n e wave  conditions  I t s h o u l d be p o i n t e d  from which the l i n e  forms s a t i s f y i n g source  solu-  real  angle.  the boundary  solution will  be  obtained. The  last  (4.10) g i v e r i s e stress  which  terms o f t h e s e r i e s to d i s c o n t i n u i t i e s  serve  fracted  waves.  a later  section.  expressions  (4.7)-  i n displacement  and  as b o u n d a r y c o n d i t i o n s f o r t h e d i f -  These w i l l  be i n v e s t i g a t e d  N e g l e c t i n g these  last  in detail in  terms o f t h e s e r i e s ,  u;=5 +So+s>-i)+jSi(N-o + sr(N'-o+sioi'-o 0  is  valid  everywhere For  source, and  i n medium  1.  e v a l u a t i o n o f the displacement  i t i s convenient  (4.15) and (4.16) as  to express  formulae  due t o a  line  (4.7)-(4.10)  70  5>)=A±(7tA;)e^ 71=1  H=i  ^=1 \fc=i  k  , R ; | C O S K _ e ; i )  /  7  (4.26)  b> -  A l 6  b^A-G  (4.27)  (4.28)  where p = +  (4.29)  71  n2  .  •  t&n6  (4.30)  =  u i  •  tan  8^.  2  (4.31)  =  (4.32)  (4.33)  tane = rsin|e|/{dL0  rco$\e\]  tan6j= TSinf^i-e)/{A-rcos(^a,+e?} .  72  4,4  Formal  Steady  State  In o r d e r of  a line  Solution f o r a Line  to g e n e r a l i z e the r e s u l t s  source,  Source to the case  the operator  TC + I oo  c^cxl ^  ~kb\j  (4.35)  -loo is  a p p l i e d to the p l a n e  wave  solution.  displacement  IXo  due t o t h e i n i t i a l  written  (4.6)  as  using  In p a r t i c u l a r , t h e disturbance  c a n be  7t+loo  -Loo  ~t0O  Equation  (4.36) c a n e a s i l y  Nakamura  (1960).  When  mated by t h e a s y m p t o t i c  U  o  = /\  / ^ l y _ '  l  which  e  be d e r i v e d from {^, i-s l a r g e , 0  the r e s u l t s o f  (4.36) c a n be a p p r o x i -  formula,  ~ ^ b | P ^ - t f  1  (4.37)  (to  are the outgoing  waves  from  the l i n e  source.  73  Using solution can  be  (4.19) t o (4.22) and  i n t h e wedge c o r r e s p o n d i n g  obtained  (4.23) t o ( 4 . 2 8 ) , to a l i n e  source  the (4.36)  as  f  (4.38)  4.5.  E v a l u a t i o n of the F i r s t In  this section  Series  Term  the i n t e g r a t i o n  o f the  Integral  o f t h e terms w h i c h  a r e p r o d u c e d by waves w h i c h a r e r e f l e c t e d once by t h e dary  between  evaluation tions  wreen  due  the media  o f the h i g h e r o r d e r t o waves t w i c e  the media  series  reflected  are c a l c u l a t e d  ( 4 . 2 3 ) - ( 4 . 2 6 ) and lowing  are e v a l u a t e d .  ( 4 . 3 8 ) , we  (As a g u i d e terms, the  boun-  to the contribu-  from the boundary b e t -  i n the Appendix see t h a t  they  I I I . ) From  have  the  fol-  forms:  (4.39) - loo where  (4.40)  74  Jc. 1 Z  and  =  (4.17)  a  n  and ( 4 . 1 8 ) ,  and  (  i 7 Y l ~ H" ., —  0 ^ 0 ^  are given For  taken  i n the plane  (4.39)  of and  this  contains  the  o i l -plane  (Figure To  very £,  but  i sonly  are given  absorptive small  the  =  6 ~ 0  t  n  e  i t i s assumed  0,^  .  is  The i n t e g r a n d A^COS^^^-rol*)  C0S(^  1TU  -+-c/_r,)  ), B'(©B' ^ ~ 6 0 ~ ^ ' ° ' ) "  that  1  =  index.  t h e medium i s  -yi— T l o ~ i 0  where  <  (This  assumption  r e s u l t s which correspond  to £ — ^ Q  to f a c i l i t a t e B  "  axis of  refractive  positive quantity.  The b r a n c h p o i n t  [ 3 / ^ 1 — p a r a l l e l  B  ~ TL ,  =  L  path  l o c a t e d on t h e r e a l g ( 6  R,^  to (4.32).  by t h e r e l a t i o n  by s e t t i n g  thef i n a l  a technique  integrals.)  (4.29)  and  f u n c t i o n \$~\J\—  the two-valued  evaluation,  does n o t a f f e c t  201+0.*  >  are therefore  i s a very  =  R.6(A-s)/ 0  f o r which  4 - 2 ) where C O S 6 o  slightly  and $  by the e q u a t i o n s  at the points  facilitate  From t h e e u q a t i o n s  i n t e g r a t i o n , theo r i g i n a l  i t s branch points and  •  evaluation o f the  i s t h e n d i s p l a c e d by  t o the p o s i t i v e imaginary  a x i s on  ot^-plane. We c h o o s e  which  the branch  cut given  by  R*Q[/\-S)—0  i s d e f i n e d by:  cos ( x + ^ )  s  (n (x+$  m  ) cos h u- s i n h y = n 1 0  c o s t * + 0*) co s h " ^ - s i n (#+<f>™) s! n rf ^ > A  n  C 0  4  •  4  1  }  y  76  L  In F i g u r e 4-2, and  the  signs of  B  Im(As)in  by p l u s and m i n u s .  £^ b  origin, L on  b{  0  is the  R-IACOS(O(.I- eJJ)  OT  P, i s shown by  along which hold,  are  can t h e r e f o r e  r e g i o n where  hatching.  can be  replaced  passes  through  around  the branch  by  The  L$  and  B  that  t h e y a r e on t h e  Each  o f them  second  the  a n c i  and  •  path  shifted  I m ( C O S (OCL — 8|j2.)J<^ 0 d i s t a n c e from path  L  ,  Im(s)^ 0  ( L i , L>z. ) >' where  the s a d d l e p o i n t point  be  original  Im(Sl)ldl)<0  the r e l a t i o n s  indicated  d i s t a n c e s from  v a n i s h e s at a l a r g e  the o r i g i n  ln\(Xs)=0  is  B  v a n i s h e s a l o n g the  , t h e path, o f i n t e g r a t i o n The  ^  O^t-plane  When f o r l a r g e  t h e Riemann s u r f a c e s .  ,  The  ( L i , L.2.)  dotted lines  Riemann s h e e t where  i s drawn a l o n g t h e p a t h  goes  denote  RLs^A-s")^ 0 .  of steepest descent,  cosOfc-Ocosh^.l  <' ) 4 42  and  c o 5 ( x - 0 ^ ) c o s h ^ = c o s ( 0 - e^)  (4.43)  B  respectively.  4.5.1  Contribution From  axis.  (4.42),  from  L,  s  In t h e n e i g h b o u r h o o d  the S a d d l e  makes an a n g l e  Point ^jp-  o f the s a d d l e p o i n t  (Reflected with the  the  x-  Waves)  77  following  approximations  <*l-e£=  are v a l i d :  ftf*  (4.44)  and  C0StdLi-da)=\-l?7z The  contour  (4.45)  integral  Ls  along  f o r (4.39) i s t h e n  (4.46)  Expanding Watson's  t h e term lemma  ^S-Ai  where,  i f  A,  i^l)  (Jeffreys,  /  A, ( 0  0  ( J L  >  6  near 1956)  ( J  the saddle p o i n t we  and u s i n g  obtain  J6  (4.47)  B  Sin(0,  w  + Q - S^/l/A^CoS (0, + 2  w  9£)  and, if Q,^ < 6g  (4.49)  78  where  4.5.2  C o n t r i b u t i o n from Next  L|  thec o n t r i b u t i o n  o f steepest descent .  ( 4  -  L|  i sconsidered.  path  ,  3 9 )  u  the Branch  around  B  a r e taken a  n  ^ tend  Sin(^VoU)'  should  Waves)  along along the  t o ^jp- + 0 ^ 4~ L<2° (  w e  h a v e  f r o m  = t ,Ai x b  /  It  (Head  the i n t e g r a l  a n d L,^  Setting  L l L a  from  Point  be n o t e d  0  that along  near  B  m  ,  (4.5 0)  Tn.  that  Im(A- )>0.  and  ( 4  S  From  (4.43)  along  Li^,  we c a n w r i t e  cosC^r~0^)-cos(e -0a)-iT: B  x>o  C 4  . 2) 5  therefore  C  U  E  ,  =  L^/Sin(oC -e^) c  (4.53)  '  5 1 )  79  along for  {JZ  very  B  near  small  .  X{  and  ot I = 0 B + W-+  Putting V"  , we have  i,V  (4.54)  approximately  (u+lu)sin(e -eu)=L'"c 8  On t h e o t h e r  hand,  S m ( c x U + (f^)-  along  L^.  ( 4 # 5 5 )  we  have  S \ n ( 0 + VL+ IV") o  (4.56)  Zl - I/A* and  from  Hence,  along  -A j r i U + L U )  (4.55)  i n the l i m i t  , we have  when  using  (4.51)  M •V5in(0 -6 ^) B  As CO  1  '  (  4.58)  80  we  have  |  -IK  (i-i/A r{5i (e -e-)p' i  n  L  4.6  Aperiodic For  convenient  ^  (4.60)  Solution  computation of s y n t h e t i c  t o choose a d i s p l a c e m e n t  seismograms,  i t is  o f t h e form  A>0 , O O  the o p e r a t i o n  0 - 0 0 on  (4.37),  (4.47) and  (4.60) we o b t a i n t h e f o l l o w i n g  tions (1)  y --/y  Direct  ^  Waves  . A  i-  e  B  0(t)=- ^ Performing  R £ COS (66-6,7)+  \  solu-  81  where  d o / C  t = D  (2)  Waves r e f l e c t e d once  JO  Uc=-A'i  A  from t h e  interface  AT(C)  X  7TV  Cos {|- tan l l ^ i ^ L - t -  A.  c  9|i  for  >  Uc=-A  (4.62)  7 0  $B  A  t  x (4.63)  C O S ) 4 - t a n ' - ^ ^ - i - ^ - -r-^jpj* tit  \  £  (3)  U  l"2  i & ~  A  /Cbi (R-)^  |  {sin(e -e-)} ^ x 3/  6  m  {iTp^yr  ^  y^[SL/C'b\ .  Head Waves  ZJZJC-S /A(I~I/A^  = A-  L  cosl-Ltan <^&L  + f jcj (4.64)  where  f  L  „  u -« |i «A. — H  R * CO S(6 -  8-  •  . Cbi "  6^)  82  4.7  I n t e r p r e t a t i o n of In a s t u d y  in  a horizontal  contributions  and  associated  we  will  Travel  from branch  the  time  the  and  find  that  they  points  factors  of  the  time  f a c t o r s of  given  the  a point  (1954) a saddle  source  evaluated point  contributions  Similarly, in this  point  a  and  n  H^lil  d  with  section  h e a d wave t r a v e l ^p\jL  by  saddle  and  these  r e f l e c t e d and  are  from  Nakamura  r e f l e c t e d waves.  determine  Time  o f wave p r o p a g a t i o n  l a y e r , Honda and  the  head waves and  the  branch p o i n t  times ' con-  tributions .  Consider  '  SA+  first  path  SAR.  i-  n  Figure  4-3.  Then  S'R.  AR.  C ,  S A f l  the  Cbi  b  __ JJTcosdz-rcos(ea^ejj^-f  {AsV/^Ws^iel^e)}"  c b| +  (4.65) Also,  / S B C R  the  t r a v e l time  C  bt  along  '  eisin e  . - ,  the  path  S8CR,  i s given  by  Cb, A  r  + ^kcose*-rcos(&-e;  ^sine^rsinCe^e)] tan(e +e*) J B  rsin(0^-e)  +  c iSm(0 +02) b  B  6  4-3o  Basic ray paths used i n p h y s i c a l i n t e r p r e t a t i o n of contributions from branch and saddle p o i n t s o  84  ( c i c o s 0 ^ - r c o s ( e ^ e ) ) c o s ( e + e*)  G  B  (4.66)  +(dsiK\0 +r5in(e -e))5in^B-f & 0 } a  A  -bl  since  cos(e +6;0== C M / C M . B  Hence we have v e r i f i e d tn tj,.  as t h e r e f l e c t e d  0  f o r )TL= 4~ /  and  a  X—  ^d  H"^JJI  0TL the  and  c  Jl  a  n  I ^  earlier  i  n  In a s i m i l a r t  e  r  P  than  should  ,  a  n  d  r  e  t  e  d  manner,  D  for different 4-4.  "fc,  0  values  of  Obviously, f o r  d i s t a n t from  the v e r t e x ,  these  the d i f f r a c t e d  waves w h i c h  are pro-  o f waves w i t h  adequately  the v e r t e x .  describe  waves  Hence  this  the e a r l y s e c t i o n o f  seismogram.  4.8  Range o f E x i s t e n c e The  evaluated.  includes  branch point  the process  In F i g u r e  a c l o s e d contour which  o f Head  range o f e x i s t e n c e  mined by c o n s i d e r i n g is  e  point  d u c e d by c o l l i s i o n s solution  .  , ra $y \SL  of  and head wave r e s p e c t i v e l y  as shown i n F i g u r e  observation  arrive  the  the i n t e r p r e t a t i o n  Waves o f head waves c a n be d e t e r by w h i c h  4-2, when  which connects  with  the i n t e g r a l  ©B/'^IJZ.  »  w  e  the o r i g i n a l  c o n t r i b u t i o n s o f the saddle  on a p p l i c a t i o n o f C a u c h y ' s  c  a  f°  n  path  hand when  0g<^ Q\j>  >  a  and t h e  theorem.  On t h e  c l o s e d contour  cannot  be  r m  and  point  17V  other  (4.39)  (b)  (a)  CO  (d)  (c)  T?i£  4-4• Ray paths of the head and r e f l e c t e d waves expressed by the f i r s t s e r i e s term of the i n t e g r a l s .  86  made w i t h o u t point.  Therefore  from b o t h  the i n t e g r a l  i n the f i r s t  the branch p o i n t  (reflected a saddle clear  excluding  waves) w h i l e  point  that  the c r i t i c a l  h e a d waves  is  (head waves) and s a d d l e  i n the second  by  the use o f (4.17),  •  rsinJ^gj+gggH^_  ~  d-rcosUdi+zo^-T-e)  Figure  waves whose p a t h s 0 =^°  ~U„  ft  A  ^  ~K- z-ze  4-5, t h e r a n g e o f e x i s t e n c e are i l l u s t r a t e d  a b s c i s s a i s the r a t i o  for  C4.es)  d  i n Figure  and t h e o b s e r v a t i o n  (  distances  (4.31),  ( 4  {  Q  In  The  Djj2  c o n d i t i o n we  _^  for  A N C L  equations:  r o s  c  i s only It i s  (4.30),  the c r i t i c a l  point  the existence  UB  Obtaining  the following transcendental  (k-rcostze^^^J"  there  waves).  ( 4 . I S ) and ( 4 . 2 9 ) ,  (4.32) r e s p e c t i v e l y and u s i n g  tan1-  case  c o n d i t i o n to decide  0|JL  DB -  the branch  we have c o n t r i b u t i o n s  contribution (reflected  of  obtain  case  around  from  o f source  t h e apex w h i l e  point  w h i c h head waves e x i s t .  6 9 )  T4.70)  o f head  4-4 i s shown  on t h e l i n e  to observation  the o r d i n a t e  .  8~Q.  point  i s t h e maximum  As e x p e c t e d  from  the small  Pig.  4-5.  Maximum v a l u e o f 6 f o r w h i c h - t h e head waves shown i n P i g . 4-4. e x i s t v e r s u s t h e r a t i o o f s o u r c e t o o b s e r v a t i o n d i s t a n c e s . The o b s e r v a t i o n and s o u r c e p o i n t s a r e 5° f r o m t h e f r e e s u r f a c e . Z  88  path are  d i f f e r e n c e , t h e range o f e x i s t e n c e close  as a r e ( c ) and  decreases  with  decreasing  surface ratio  o f 10.0, F i g u r e  0|  (b) e x i s t  4-6  S\  source.  , the d i p angle linearly  4.9  Discont i n u i t i e s Discontinuities  with  a boundary  because changes  sion with not  the vertex  considered  goihg  f o r which  from  t h e h e a d waves (a) f o r head  collision  o f t h e wave  through  bination  to the boundary  zero.  in a diffracted  solution.  we  When b o t h  This  areas  initially  have f o u r c a s e s  i n d i c a t e the regions  colli-  wave w h i c h i s up-  o f the l a s t o f t h e com-  o f t h e d i s c o n t i n u i t y as shown i n F i g u r e  cross-hatched  waves'of  and s t r e s s i n  the free surface  passes  are considered,  t o a change  decreases.  the f i r s t  results  in this  distance  that with i n -  and down-going waves and t h e i n t e r f a c e  reflection  The  I t i s noted  i n displacement  oi\,  b e t w e e n t h e m e d i a as  the f r e e  , which corresponds  increases while  ( c ) and (d) i t l i n e a r l y  (1) a r i s e  a t 5° from  to o b s e r v a t i o n - v e r t e x  type  medium  and  shows t h e r a n g e o f e x i s t e n c e o f  changing  o f the l i n e  creasing and  point  and a s o u r c e - v e r t e x  depth  of reflections  also  cL/T  an o b s e r v a t i o n  head waves w i t h of  ( d ) . The r a n g e o f e x i s t e n c e  i n c r e a s i n g number  ratio  For  o f (a) and (b)  4-7.  f o r which the  0  5  SO  15  9. ( D E G R E E S ) F I g o  4-6.  Maximum value of the wedge angle (6|+6 ) f o r which the head waves of the types shown i n Pig* 4-4» e x i s t f o r an observation point at 5° from the f r e e surface and d / r = 1 0 0 » A  o  (a)  (b)  o  (c) Pig,  4_7.  (d)  D i s c o n t i n u i t i e s i n medium ( l ) d u e t o i n t e r a c t i o n o f t h e wave w i t h t h e v e r t e x . The l i n e d a r e a s i n d i c a t e t h e r e g i o n s f o r w h i c h t h e g e o m e t r i c wave f r o m t h e l a s t r e f l e c e x i s t s w i t h the term from which i t a r i s e s i n d i c a t e d i n b r a n k e t s .  91  geometric tions  wave from  the  last  (4 .11) - (4 .14)  the  equations  continuity  S i  .6=  si  e = j^-ziu'+oe^-zti'e^  (ft)  lem.  discontinuities  from  vertex  (m = -) ing  the  In o r d e r  displacement the  and  to  complete  tant  and  downward  this  (m=+) are  role. the  early  the  a discrepancy  of the  plane  physical  waves small  shown i n F i g u r e the  to the the  vertex w i l l  toward  result-  is  02,<C!3°.  for  wave p l a y s  later  the  wave s o l u t i o n  seismogram  arrive  my  4-8.  formulation should  p a r t o f the  7 4  upward  The  solution  diffracted  .  prob-  incident angles  reflected  complete  of  estimation of  have b e e n e x a m i n e d .  However, t h i s  waves f r o m  indicate  at v e r y  geometry  0^,^2,1°  for  (4.72)  solution  propagating  approximation  describe  dis-  ( 4  discontinuities,  For  However  of  (4.7i)  obtain a quantitative  discontinuities  a good  lines  equa-  7C-zN0i-2Ne^  The solution  From  are  e = 2(N-i)6i-i-xNe^-7c  s;  o)  of these  •+  (OS,  ®  reflection exists.  an  impor-  adequately  as  the  diffracted  than  the  initial  phases.  It in  Figures  lation  of  i s seen t h a t c o i n c i d e n c e 4-7b  the  and  4-7c leads  discontinuities.  o f the  to at  least  Two  special  discontinuities p a r t i a l cancelcases  are  of  ,  Fig* 4 - 8 „  R e l a t i v e amplitudes of the displacement d i s c o n t i n u i t i e s due to a plane i n i t i a l wave close t o t h e x-axis f o r propagation upward ( m = - ) and downward (m=+)o  93  interest  as t o t a l c a n c e l l a t i o n (1)  results.  Lower boundary f r e e or r i g i d  When the lower boundary o f the wedge i s e i t h e r free  or r i g i d  The  condition  >  6  +  0  *  =  then  A-fo and A-fe, are +1 or -1  respectively,  N+ N'  leads to (4.71) = (4.74) or (4.72) = (4.73) so that the two l i n e s o f the d i s c o n t i n u i t i e s discontinuities  exist.  are c o i n c i d e n t and no  Hence the s o l u t i o n  i s complete and  no d i f f r a c t e d waves e x i s t . (2)  Surface  Source  I f the l i n e source i s p l a c e d i n the s u r f a c e ( 6,= For  0  ) then from  the p a r t i c u l a r  (4.17) and (4.18)  A t = A-fe. +  situation  N + N'  •  we have (4.71) = (4.74) or (4.72) = (4.73) and the two discontinuities  coincide,  t i n u i t i e s exist  i n medium ( 1 ) .  In t h i s d i s c u s s i o n  hence i n t h i s case no d i s c o n -  the d i s c o n t i n u i t i e s  ( 2 ) are a g a i n expected to be l e s s medium ( 1 ) .  i n medium  important than those i n  94  4.10  Dispersion In  for is  this section  a dipping either  A^>  Equation  structure  a i r or r i g i d .  and-A-^ For  f o r t h e Lower B o u n d a r y the d i s p e r s i o n  i n the simple  equation  Free  and R i g i d  i s derived  c a s e where medium  When medium  (2) i s a i r o r  (2)  rigid,  become +1 o r -1 r e s p e c t i v e l y .  2Tn.(9\-Td&)<^  |  w  e  c  a  n  write  cosC^i+^m(0,+e4> = cosoCL-^7n(e,+e )sinoCu A  In  this  case  S*(N)=A -e L  •  K  ±  0  6  (4.75)  <VKIWA P  ' " ^  6  (4.76)  95  Similarly  the expressions  obtained.  Operating  for  S,(N)  a  n  c  S^CH)  !  c  a  n  °e  with  -loo  poles  I  appear  from  the r e l a t i o n  —  \y  which  \J  yields  Sin(-fe ,rce,+ejsmoCL)=o b  c o s ( f e i T ( 0 , + 6^)sincx:L) For  oLi  real  , we c a n t h e n  =o write  (niz (4  where  C^i  i s the phase v e l o c i t y  ( 0<Coci<CJC  This  ).  as  expression  COSoil  ~  Cb\/C>n  i s t h e same as t h a t  obtained  b y Nagumo  Further,  i f we p u t T ( G i - t © ^ ) ~ H  the  o f a h o r i z o n t a l l a y e r ) , the d i s p e r s i o n r e l a t i o n  case  (1961) f o r a s l o p i n g r i g i d  (4.77) c o i n c i d e s w i t h Nagumo  (1961) has c a l l e d  formal  p h a s e and g r o u p v e l o c i t y velocities.  H  i s the depth i n  t h a t o f the h o r i z o n t a l l y  case.  observed  (  Cyi  bottom.  layered  and %L = - ^ - s —  to d i f f e r e n t i a t e  the  from t h e  96  4.11  The H o r i z o n t a l It  horizontal solution  i s interesting layer  Solution to derive the s o l u t i o n  fora  u s i n g my m e t h o d as t h e t r a n s i t i o n o f t h e  to the horizontal  for obtaining layer.  Layer  layer  c a s e may s u g g e s t a m e t h o d  t h e s u r f a c e wave s o l u t i o n s  f o rthe dipping  F o r t h e d i f f r a c t e d wave p r o b l e m , i t i s u s e f u l t o  s t u d y t h i s t r a n s i t i o n as t h e q u a n t i t a t i v e behaviour o f the d i s c o n t i n u i t i e s z e r o d i p a n g l e may i n d i c a t e  and q u a l i t a t i v e  as they a p p r o a c h z e r o f o r  the nature of the d i f f r a c t e d  solution. The same ( X-, ^ ) c o o r d i n a t e s y s t e m i s u s e d the  x - a x i s now b e i n g h o r i z o n t a l a t ( cL , 0  is placed  (Figure  ) i n the layer  4-9).  of thickness  E m p l o y i n g t h e same p r o c e d u r e as f o r t h e d i p p i n g obtain the displacement  f o r t h e time  with  The s o u r c e H-H1  + H2.  l a y e r , we  variations  70 + loo  e , <~7  A  -f-e  u f ~i1l |{(4-;<)cos b  Wat  "T  -L^bi{(^^cosoCu+(^(nH,+ n H z ) + ^ ) s i n o C L } C  •T  O  .  - it ! b  + e.  {(<L-PQ COSoLl  0 H, + U H _ ) + U ) 5 ' Hoci} I J J  -i& RoCOS(cLi~6 ) bl  bl I n i -loo  + (Z( (71.+  6  0  -HL \RoCOS b  {d-i-6~)  +6 (4.78)  Free  surface  S (d o) X9  x  <  x  R ( x y) 8  (!)  7TTT  (2)  y Pigo 4-9» Coordinate system f o r the h o r i z o n t a l layer case with the source (S) at ( d , 0 ) and the r e c e i v e r (R) at ( x y ) B  9  98  (4.78)  \ O  "I-  j- &0C  g  where  ASinoCo-Wi-^cosV.t  (4.79)  A,=  t a n d~^=(z H,+^) / (ck- PC) R X L I  t(\ndnr{z(w~HO~^/(ck-x)  / W ^ M ^ H r HO - l j  =  t a n dtz^  (znH +  VrVfa-pc)  (4.80) As e q u a t i o n  (4.78) i s o£ t h e same  f o r m as  (4.38), formulae  (4.62) o r (4.63) and (4.64) c a n be a p p l i e d f o r t h e r e f l e c t e d and  h e a d waves r e s p e c t i v e l y .  Therefore  the v a r i a t i o n s o f 7Tl <m. D  t h e w a v e f o r m s depend QQ the  .  Equation  formal  only  on t h e v a l u e s  (4.78) c a n a l s o  dipping layer  solution,  rtfljj. , 6 ^  be d e r i v e d f r o m i f we  take  and (4.38),  the l i m i t  as  rsin6i=Hi  <xr\A  rs\n9^H  (4.81)  99  In t h e c a s e appear forms  from  contributions  to the h o r i z o n t a l  surface series  waves  form w h i c h  For reflected culated  could  to a normal  investigate  i f the  finite  into  a  mode e x p r e s s i o n .  Seismograms  wave, h e a d waves, and waves  the boundary, d i s p l a c e m e n t s  are again those  trans-  have b e e n  shown i n F i g u r e 4-10.  employed  once  by H a s k e l l  cal-  Elastic  (1960) .  C o n t r i b u t e to the seismogram  Ray  a r e shown  F i g u r e 4-11.  plot  component waves  o f F i g u r e 4-12 ray paths  feature to  As o u r s o l u t i o n  c a n be c h a n g e d  of Displacement  o f waves w h i c h  s u r f a c e waves  of a dipping layer  f o r the three cases  The  the  layer,  s o l u t i o n , we  corresponds  the d i r e c t  from  constants  in  layer  i n the case  Computation  paths  of poles.  e x p r e s s i o n o f our s o l u t i o n  compact  4.12  of a horizontal  inspection  indicated  by l e t t e r e d  and r e f l e c t e d  of equations  head waves d e c r e a s e as  and t h e a r r i v a l  i s the s m a l l amplitude  the d i r e c t  waves  a r e shown on t h e t i m e - d i s p l a c e m e n t  as  significantly,  waves do n o t u n d e r g o illustrated.  waves.  large  arrows.  A  detailed compared  This i s expected  (66) w h i c h  [ / (  show  from  that the  and t h e r e f l e c t e d  Although  t h e wave forms  c o r r e s p o n d i n g to  o f t h e h e a d waves  (65) and  I/("R^A)'  times  the t r a v e l  o f the r e f r a c t e d  changes  times and  f o r the three  changed  reflected  cases  100  F i g . 4-10.  Three c a s e s f o r which t h e o r e t i c a l seismograms were c a l u c u l a t e d . The parameters used were: H,= 9.59 km, -H = 3.00 km, D= 99-6 km, d= 10.0 km, and the d i s p l a c e m e n t parameter c= 0.05 sec. A  101  102  Pig„ 4-12. D i s p l a c e m e n t s o f t h e component waves f o r t h e g e o m e t r i e s g i v e n i n P i g s . 4-10a„. 4-10b and 4-10c 9  o  103  Figure  4-13 r e p r e s e n t s  from  t h e components  very  different..  recognizable the  wave  wave  of Figure  f o r t h e head wave  o f the d i r e c t  wave  feature  i s the l a t e  (e) i n t h e c a s e  reflected tance  waves w i l l  and  4.62 km/sec r e s p e c t i v e l y ,  contribute  to the s e c t i o n  diffracted  arrival  layer.  arrival  between t h e o b s e r v a t i o n p o i n t o f medium  layer. phases.  More  (1) and medium  multiply  As t h e d i s -  and t h e v e r t e x  i s 10.0 km  (2) a r e 3.64 km/sec  the d i f f r a c t e d  waves  hardly  shown h e r e as  4 to 5 sec a f t e r  i s due t o a h e a d wave o f s m a l l  the v e r t e x .  A very  o f the r e f l e c t e d  o f the seismogram  wave a r r i v i n g  are a l l  ( c ) and t h e r e f l e c t e d  a p p e a r as l a t e r  the v e l o c i t i e s  arrivals  look  (b) i s embedded i n  o f the h o r i z o n t a l  and  with  wave  synthesized  The s e i s m o g r a m s  (d) i n t h e c a s e o f t h e h o r i z o n t a l  noticeable  the  4-12.  However, t h e d i f f e r e n t  except  forms  the seismograms  the f i r s t  amplitude  interacting  104  (a)  23  * a Travel time  25  29-  (sec)  d  (b)  23  Fig*  4-13»  t a  25  S y n t h e s i z e d seismograms r e s u l t i n g placements o f F l g o 4-12o  29  from the d i s -  105  CHAPTER 5 SUMMARY, CONCLUSIONS AND  5.1  Summary In  in  and C o n c l u s i o n s t h i s paper,  a dipping layer  investigated t h e models  FURTHER STUDIES  the behavior  overlying  of elastic  an e l a s t i c  medium has been  i n terms o f body waves i n o r d e r  available  waves  f o r the i n t e r p r e t a t i o n  t o expand of crustal  structure.  In plane  Chapter  SH i n c i d e n t  pendicular examples  a t the base o f a d i p p i n g l a y e r  t o s t r i k e has b e e n d e v e l o p e d  presented.  direction  2, t h e r e f l e c t e d wave s o l u t i o n  with  angle  r e f l e c t e d wave s o l u t i o n  collide  case  along  with  approximates  t h e edge  size  hence  as a g u i d e  con-  However, f o r waves  o f the f i n a l  wave w h i c h  becomes l a r g e .  of s i g n i f i c a n t  of this discontinuity  serves  t h e com-  as t h e b o u n d a r y  t h e wave has r e v e r b e r a t e d o n l y a v e r y is still  that  d i r e c t i o n , the displacement  the i n t e r f a c e s  t h e wedge and h e n c e The  closely  satisfied.  i n t h e down-dip  discontinuity  7 5 ° ) , i t i s found  f o r small d i p angles  are approximately  propagating  not  S waves  (45°<^ oi <C  the  ditions  i n the up~dip  o f i n c i d e n c e i n the range o f that  teleseismic  solution  and p e r -  and n u m e r i c a l  F o r waves p r o p a g a t i n g  for  plete  fora  lias  In t h i s  few t i m e s  within  amplitude.  been d e t e r m i n e d  as t o w h e t h e r  does  and  the r a y s o l u t i o n  106  is  applicable.  reflected waves of  For a t r a n s i e n t  waves w i l l  and h e n c e e v e n  solution  section  should  arrive  i n displacement  final  as w e l l  to  a t the base  investigated  developed  using  Due  solution  i s not presented  to the c o m p l e x i t y  however, a c o m p u t a t i o n a l amplitudes  ment  waves  provide  t h e edge o f t h e  ratios  directions. frequency  i n t h e wedge  of this  up-dip  I t i s found  reflected  coordinate  problem, a  series  done f o r t h e SH p r o b l e m ;  directions In t h i s  of a l l the c o n t r i way  the d i s p l a c e m e n t  due t o r e f l e c t e d  examples  i n both  of a  a cylindrical  of displacements  at the s u r f a c e are p r e s e n t e d  curves  incident  and p e r p e n d i c u l a r  scheme i s g i v e n by w h i c h t h e  are determined.  Numerical  propagating  for  wave, must  o f P and SV waves i n -  by means  as was  and p r o p a g a t i o n  any p o i n t  found.  initial  at. t h e s u r f a c e and t h e  3, t h e b e h a v i o u r  system.  at  type  o f the  along  of a dipping layer  has b e e n  solution  buting  imposed  this  between t h e m e d i a .  strike  wave  discontinuities,  and s t r e s s  as t h o s e  In C h a p t e r cident  the d i f f r a c t e d  The d i f f r a c t e d  continuity  boundary  than  to the composition  o f a seismogram.  wave  earlier  f o r large  apply  i n p u t t o t h e wedge, t h e  (oC, ^  that  f o r constant  waves may and  displace-  for incident  = 6 0 ° ) and down-dip  the displacement depth  ratios  to i n t e r f a c e  P and SV waves p r o p a g a t i n g  be  waves (oL, ^ versus  become  i n t h e down-dip  flat  = 120°)  107  direction  f o r dip angles  different  from  the case  wave p r o p a g a t i n g peaks  are  greater  large  than  note  incident  for large  10°  at  curves  tion  than  is  that  the  transient  again apply  used  in this  real  the  seismogram. chapter  example  p a r t o f the the  to the  than  the  imaginary  of a free angle  diffracted reflected of the  4,  informa-  i t is large  even  wave  reflected  waves f o r a  wave  solution  initial  sec-  direction  i n Appendix  II  R a y l e i g h wave t o show t h a t  indicates  the p r o p a g a t i o n  p a r t g i v e s the d e c r e a s e  In C h a p t e r  be  complex p r o p a g a t i o n interpreted  of dip  ratio  the d i f f r a c t e d  composition  been  for  P waves.  However, s i n c e t h e  The  has  than would  that  i n p u t t o t h e wedge, t h e  o f the  using  earlier  to y i e l d  d i s c o n t i n u i t y may  indicating  parti-  therefore that  down-dip d i r e c t i o n ,  amplitude.  arrive  tion  and  displacement  of s i g n i f i c a n t  waves w i l l  i n the  and  curves  displacement  the  angles  frequency  ratio  I t appears  P  = 60°),  to s m a l l changes  t h e V/H  dipping interfaces  small dip angles  should  are  the  feature of  displacement  P waves.  F o r waves- p r o p a g a t i n g  for  H/V  A  For  for dip  o f SV waves w o u l d be more l i k e l y  concerning  found  and  to lower  a r e much more s e n s i t i v e  for incident  a study  shift  This i s very  (  direction  decreasing dip.  the  small dip angles  15°.  direction.  dip angles  the peaks  i s that  SV  of up-dip  i n the u p - d i p  become n a r r o w e r w i t h cular  g r e a t e r than  the p r o p a g a t i o n  the  direction  of  amplitude.  of, SH  waves from  a  108  line has  source been  investigated  A formal has  solution  of  time  points,  The  two  and  series  f o r both  i n t e r m s o f head  head waves do  Using  changes  the  not  due  o f the  which  cases,  section  are  also  been  through do  the  found  solution  seismogram.  can  be  range  i n the  examples.  greatly; markedly  Discontinuities the  cases  i s the  the  compari-  seismogram  times.  special  and  numerical  waves.  be  solution  not. d i f f e r  synthetic  wave s o l u t i o n  o f the  the  associated with For  this  s m a l l e r than Hence  an  inter-  reflected  o f head waves may  in arrival  studied.  reflected  other  has  arrivals  to changes  wave have been the  layer  and  have been  and  appear.  same t e c h n i q u e ,  c h a r a c t e r o f the  displacement  initial  waves  a harmonic  p o i n t s are  the d i p p i n g l a y e r  forms  however, the  the  formulation.  terms o f t h e i n -  contributions  the b r a n c h  of a h o r i z o n t a l  wave  that  medium  include diffracted  e x i s t e n c e o f the v a r i o u s types  s o n made w i t h  In  first  variation  integral  determined.  in  The  using ray paths  i n the  case  an e l a s t i c  using multiple r e f l e c t i o n  to o b t a i n displacements  aperiodic  saddle  overlying  have b e e n e v a l u a t e d u s i n g t h e method o f s t e e p e s t  descent  preted  layer  w h i c h does n o t  been o b t a i n e d .  tegral  If  i n a clipping  diffracted  i t is  complete applied  found solution. to  the  109  5,2  Suggestions As  of  a r e s u l t of t h i s study,  investigation (1)  at  w i t h an e l a s t i c  seismogram  base  f o r an  incident  pulse. The c a l c u l a t i o n o f t h e a m p l i t u d e  of a multiple horizontal  developed  lines  a r e suggested.:  i n a wedge  (2) tics  the f o l l o w i n g  The c a l c u l a t i o n o f a s y n t h e t i c  a station  p l a n e wave  and  f o r Further Studies  r e f l e c t i o n i n the case  layers  by a c o m b i n a t i o n  i n this thesis  and H a s k e l l ' s  characteris-  o f both  dipping  o f the technique method  (Haskell,  1953). (3) wedge  The p r o b l e m  overlying  reflected  waves n e g l e c t i n g (4)  flected of for  an e l a s t i c  The e x a c t  waves  a line  medium  in a  t h e d i f f r a c t e d waves.  solution  i n a dipping  time  sources  i n terms o f h e a d and  i n terms o f m u l t i p l y r e -  and m u l t i - r e f l e c t e d  source  transient  o f P and SV l i n e  variations  head waves  layer with using  i n the case  an e l a s t i c  base  t h e method o f C a g n i a r d  (1962). (5) in  An i n v e s t i g a t i o n o f s u r f a c e  the presence  medium. appear  of a dipping  In t h e c a s e from  approaches  overlying  of a horizontal  a contribution zero,  layer  of poles.  the s o l u t i o n  found  wave  layer,  propagation  an  elastic  surface  waves  When t h e d i p a n g l e i n Chapter  4  reduces  110  to  the c a s e  of a h o r i z o n t a l  in  the presence  if  the  finite  of a dipping  series  solution  f o r m which, c o r r e s p o n d s (6) using  the  An.attack  the m u l t i p l e  continuities  layer.  found  layer  could  surface be  mode  the problem  d i f f r a c t e d , waves.  solution  in a  compact  expression. o f d i f f r a c t e d waves  r e f l e c t i o n wave s o l u t i o n in this  waves  investigated  can be w r i t t e n  to a normal on  Hence  which  and  are  the  dis-  related  to  Ill  BIBLIOGRAPHY  C a g n i a r d , L . , 1962. 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T o h o k u U n i v . , S e r . 5, G e o p h y s i c s , 6, 70-84. Hudson, J . A.,  196 3.  Geophys.  J.  SH waves R.A.S.,  7,  i n a wedge - s h a p e d 517-546.  medium,  112  Hudson, "J. A. and Knopoff, L. , 1964. T r a n s m i s s i o n and r e f l e c t i o n o f s u r f a c e waves at a corner 2, R a y l e i g h waves, J. Geophys. Res., 69, 281-289. Ibrahim, A. B. , 1969. D e t e r m i n a t i o n o f c r u s t a l t h i c k n e s s from s p e c t r a l behavior o f SH waves, Bull. Seism. Soo. Amer., S9_, 1247-1258. J e f f r e y s , H. and J e f f r e y s , B. S., 1956. Methods o f mathem a t i c a l p h y s i c s , Cambridge Uriiv. P r e s s , Cambridge, England. Kane, J . and Spence, J . , 1963. R a y l e i g h waves t r a n s m i s s i o n on e l a s t i c wedges, Geophysics, 28_, 715-723 . Kane, J . , 1966. T e l e s e i s m i c response o f a uniform d i p p i n g c r u s t (Part I o f a s e r i e s on c r u s t a l e q u a l i z a t i o n of s e i s m i c a r r a y s ) , Bull. Seism. Soo. Amer., 56, 841-859.  K e l l e r , J . B., 1962. G e o m e t r i c a l theory o f d i f f r a c t i o n , J. Aooust. Soo. Am., 5_2, 116-130. Lapwood, E. R., 1961. The t r a n s m i s s i o n o f a R a y l e i g h p u l s e round a c o r n e r , Geophys. J. R. Astr. Soo., 4_, 174-196.  McGarr, A. and A l s o p , L. E., 1967. T r a n s m i s s i o n and r e f l e c t i o n o f R a y l e i g h waves at v e r t i c a l , boundaries, J. Geophys. Res., 72_, 2169-2180. Nagumo, S. , 1961. E l a s t i c wave p r o p a g a t i o n i n a l i q u i d l a y e r o v e r l y i n g a s l o p i n g r i g i d bottom, J. Seism. Soo. Japan, 14_, 189-197. Nakamura, K., 1960. Normal mode waves i n an e l a s t i c p l a t e ( 1 ) , S c i . Rep. Tohoku Univ., Ser. 5, Geophysics, 12,  Phinney,  44-62.  R. A., 1964. S t r u c t u r e o f the e a r t h ' s c r u s t from s p e c t r a l behavior o f long p e r i o d body waves, J. Geophys. Res., 69, 2997-3017.  Sato, R., 1963. D i f f r a c t i o n o f SH waves a t an obtuse-angled c o r n e r , J. Phys. Earth, 11_, 1-17.  ,  113  APPENDIX I ENERGY RELATIONS  As the  text,  used the  a c h e c k on t h e a m p l i t u d e  t h e method u s e d  to derive expressions incident,  To velocities IL—  reflected  calculate 1X  waves.  e n e r g i e s , we n o t e  that the  to the displacements  The e n e r g y  by m u l t i p l y i n g  the area  flux  energy  ^y the v e l o c i t y  o f wavefront  involved.  equations  per unit  of propagation  F o r a P wave  on  t h e s u r f a c e , i t s energy  flux  to  t h e sum o f t h e e n e r g i e s  i n the r e f l e c t e d  waves.  IA. by  f o r t h e waves c a n t h e n  the k i n e t i c  ~~^f(^r~^~ l^e')  volume,  X  between  LCOU. and h e n c e may be o b t a i n e d d i r e c t l y f r o m  obtained  and  partition  and r e f r a c t e d  kinetic  derived i n  e t a l (1957) h a s b e e n  f o r energy  are related  (3.17) and ( 3 . 1 8 ) . be  by Ewing  relations  per unit  incident  a r e a must be e q u a l and r e f r a c t e d  We have  c*2 ctn  \s\n(oLi~d )\ d  ~ ~Z?2. Caz Crjt ^2. (SI n (oirsi- 0<0| +  2 ? cti Drx c x z  A  b  Is fn  QO|  f o l l o w i n g c o m p u t a t i o n a l l y more u s e f u l  (A-1.1)  form  i s obtained  114  using  (3.28)  ( C m /  1  HvJlCiJ  |Sin(6k-o6)|  •+ >Y 1 A/Wf Yi-Ci/v^)^cos^Ce^-oO 1  The  corresponding  equation  •'"  W ^ W V P W  (A-1.2)  f o r S waves i s  |sin(6k-p)|  + 4§\M I Din/  I Sln(6d-P)|  For  P and SV waves  incident  on t h e b o u n d a r y  the  r e l a t i o n s are r e s p e c t i v e l y  from  medium ( 1 ) ,  115  s i n ( e ^ - O L ) |  , c ^ / C r * A  v 17  B  + Si  Vbi/  1 Bin,  ^£COS%d(k-d5  (A-1.4)  (A-1.5)  116  APPENDIX I I EXPRESSION OF A FREE RAYLEIGH WAVE USING  In t h e c a l c u l a t i o n angles  have  reflection  been  o f the displacements,  used i n order  and i n c i d e n t  i n v o l v e d i n the r e s u l t s .  not  produced i n t h i s  an e l a s t i c The  g r e a t e r than Although  ANGLES  complex  of total  the c r i t i c a l  R a y l e i g h waves a r e  problem, the e x p r e s s i o n o f Rayleigh  o f complex  angles  half-space with  solution  that the cases  angles  are  waves i n t e r m s  COMPLEX  i n t h e medium  free  i s of interest. 0~O  surface  Consider (Figure A - l )  c a n be w r i t t e n as  (A-2.1)  0> K  The  i V  .c^  boundary  T C O S (  °-^  c o n d i t i o n s at  S — 0  are  ee = o (A-2.2)  T0=O Substituting (3.15)  (A-2.1)  and ( 3 . 1 6 ) , we  into  the boundary  conditions using  have  (l-^VbtcosVOA^+  BJ?SI n  ^ c o s ^ =  0 (A-2.3)  118  and  V  b  C O S p g .  C O S d ^  [  From  ( A - 2 . 3 ) , we  (A-2.4)  have  ~ ^v^s\noi cosd,c s\n^cos$^o ?<  Substituting  ( A  i  ( A - 2 . 4 ) , and w r i t i n g  % — COSbL^  ' 2  5 )  V — t ^ i  and  gives  | G ( i - v ) ? c Assuming  The  ±  1.0  we  cos  + ( i 6 - ^ ) ^ + - ^ ^ - i 7 3 = relation,  Z>~J^-  of t h i s equation  t o COS olfi= ±  88  a.rccos(r arc  Poisson's  r e a l root  responds  3  •  1,88^  Recall ing  0 ( A  . . 2  6 )  , yields  i s %=3.  which  and u s i n g  (A-2.4),  corCOS  the r e l a t i o n s  B)=JC-arccosn p = C&rccoshp  (p=rea\  >i)  obtain  oCp.= 1.-2^-7 L Pa=o.^o68L  or or  jo-\.2.<t7 I  jo™o.4-oG8o  (A-2.8)  119  I f i n equations  cos(p±  (A-2.1), we use  1%) =  cospcosh% f" :  ls\n?s\nh%  S i n ( p ± L ^ ) = S i n p c o s h S T icosps\r\]r\% we have  O  ^±l^rCi.S8^cose±i i . F ^ 7 s m a )  A  (A-2.9)  -  o  +Lt  b l  r(|.o88CoS0±Lo.^73sfn6)  60  0.^2.78.,  _  %  .  We see t h a t the d i l a t a t i o n and r o t a t i o n propagate the v e l o c i t y  (A-2.10)  aA  with  0. J{ |'j-Cfc>| which c o i n c i d e s w i t h the v e l o c i t y c  c  of the f r e e R a y l e i g h wave.  As a r e s u l t we see t h a t f o r  a R a y l e i g h wave w r i t t e n i n terms o f complex a n g l e s , the r e a l p a r t o f the angle i n d i c a t e s the p r o p a g a t i o n  direction  and the imaginary p a r t g i v e s the decrease o f amplitude w i t h the two s o l u t i o n s o f (A-2.8) r e p r e s e n t i n g waves propagating i n opposite d i r e c t i o n s  ( 0  O^Hoi JC  ).  120  APPENDIX I I I EVALUATION  As in  a guide  the series,  results  OF THE SECOND SERIES  a summary  of higher  order  second  terms  o f t h e e v a l u a t i o n p r o c e d u r e s and  f o r t h e c o n t r i b u t i o n s b y waves t w i c e  the boundary between The  t o computation  TERMS OF THE INTEGRAL  the e l a s t i c  terms o f t h e s e r i e s  media  have  reflected  are evaluated  from  here.  t h e form  (A-3.1) -LOO  where  ASin(^+o6L)-§yi-A"cos^^o6 )  ^ _ 1  z  L  A  s i n( 0r+ou)+S/1  As in (  0 ^ oa)+S / 1 and  and  1=  From  equations  -tfcosXtf*-*  771 = +  ou)  (A-3.2)  -tfcosXA+oil)  , —  (4.17) a n d (4.18)  (A-3.3)  and  0 ^  have b e e n  g i v e n by e q u a t i o n s  (4.29) t o (4.32)*.  121  As  the integrands  o f (A-3.1) c o n t a i n  the  expres-  sions  and  which is  (A- 3.4)  are both  required  two-valued, a four-sheeted  for their  which the four  (P^sz)  =  and is  sheets  coalesce  0  •  assumed t o be v e r y  sheets  representation.  F°  r  Riemann s u r f a c e  The b r a n c h c u t s ,  are defined  R.s(A's0 O =  by  e v a l u a t i o n purposes  slightly  absorptive  I , I I , I I I and IV a r e d e f i n e d  along  t h e medium  as b e f o r e .  corresponding  The  to the  combinations  „ „, (A-3.5,  (Re(Asi)<o,Re(A )>o)  , ( R e M > o , Re(A .)>o)  (Re(A i)>o,Re(A )<o)  , (Re(A )<o,Re(A«a)<o).  S2  S  sa  respectively. on 0  any s h e e t  The o r i g i n a l  path o f i n t e g r a t i o n  ^^vanishing  distance  from t h e o r i g i n .  on  I I where t h e r e l a t i o n s  and  hold As  sl  o f t h e Riemann s u r f a c e  i&b\fi-2Si.COS(oLi,  sheet  S2  along  an e x a m p l e , when  the path a t a l a r g e path  [_J  Im(S\)T\oLl)<iO L QQ ^>  shifted  f o r the f a c t o r  The o r i g i n a l  along  c a n be  . Qc,  i s taken  , Im(Asi)*C0  ;  122  where  (A- 3.6)  Q = ATCCOSO/A) 0  t h e o r i g i n a l p a t h c a n be s h i f t e d t o  (bi ,  Ltjj, )  a  s  shown  the saddle p o i n t ( L1  ;  ) go a r o u n d  respectively, of  [j  i n Figure A-2. £>  each  steepest descent  Lis  ,  the branch  points  }  passes  S  , and t h e c o n t o u r s  ( L 3 Ly- ) and  ( L»3 £  ;  through  LyO  and  and 3  one o f them b e i n g drawn a l o n g t h e p a t h g i v e n by  I  cos(x-d?i)cosh*&=  (A-3.7)  cosoc-e^coshty = cos(eo- eS)  .  (A-3.8)  and COS(X-0j£)COSh where  ^ =  C O S ( 0  B  -  0 ^ )  (A-3.9)  123  Re X , < o s  Re X  n  m  Re X ReX  S 2  S I  S 2  >o  >o >o  Re X  S)  >o  Re X  S 2  <o  Re X , <o s  Re X < o S2  E i g . A-2.  The oci-plane showing b r a n c h c u t s and i n t e g r a l p a t h s f o r e v a l u a t i o n of the second s e r i e s term o f the i n t e g r a l s . N o t a t i o n : B,C - branch p o i n t s ; S - s a d d l e point; L - o r i g i n a l path of i n t e g r a t i o n ; L - path o f s t e e p e s t d e s c e n t t h r o u g h s a d d l e p o i n t ; and L{, ("i= 1,.2 ••••) - p a t h s o f b r a n c h l i n e i n t e g r a l * s  124  Integral  Around The  evaluated  C  contour  integrals  by t h e same p r o c e d u r e  along  ( L3,L»f)  as b e f o r e .  c a n be  By n o t i n g t h e  relations  CA-3.10)  for  the path  variation  Vu\.  near  C  the c o n t r i b u t i o n  on s h e e t  I, f o r a harmonic  to the displacement  time  i s found t o  be  where  Integral  Around The  evaluated  B  contour  i n t e g r a l s along  by t h e same p r o c e d u r e  ( \j  K  }  as b e f o r e .  L^,)  c  a  n  also  be  By n o t i n g t h e  125  relations  He(XsO>0 , I m ( A s O > 0 , (A-3.13)  f o r the path  L.z near  B  l y i n g on sheet I I , f o r a harmonic  time v a r i a t i o n the c o n t r i b u t i o n t o the displacement i s found to  be  .  -  =  :  A  J / I J G S  -  :  :  ^  \  '  :  f  i  |  •  (A-3.14)  I n t e g r a l through  S  The contour i n t e g r a l along  L 5  l u a t e d by the same procedure as b e f o r e .  can a l s o be evaFor a harmonic  time v a r i a t i o n the c o n t r i b u t i o n to the displacement i s found t o be  u -KJ^& s  AT(C) A : ( C ) o  t f e M ! C i  ^  l  (A-3.15)  126  V* ^ B V ' ^  where f o r  A Sin A  S  O  (0r+  c  -  h J \ - t f c o s ^ +  in ($zT+ 65) +  e  %)  iJ\-£?cos*(0r+eZ) (A-3.16)  Al(C> A  if  (ej).+ g/i-^  e >e^>0c  C  o s ^ ^ eS.) +  :  B  At@£) tanf,=  if  s in  (A-3.17) =  same as  (A-3.16)  Asinc^+e^)  6& > 6c >  (A-3.18)  dza  (A-3.19)  tan^  Asin(^a^)  (A-3.20)  127  Aperiodic  Solution When t h e m o t i o n s a r e a p e r i o d i c  </>(t)=-t  the  A>0 , O O  c  operation  0  applied lowing Head  4 *  and v a r y as  -<?o t o (A-3.11),  (A-3.14) and (A-3.15) y i e l d s  the f o l -  solutions:  waves,  jE{\-\ ?r* /£  (  R  -)3A  |  S  j  n  (  e  c  _  •  * (A-3.21)  l/f X A  J pz___J"jl+  A  Sin (f^+ ds)-Sj 1 - ^ 0 0 5 " ^ + 6s) _sinC0r+aB)+§/i-A^ost$r-feB) (A-3.23)  128  where  u  T,„ =  —  n  ——  & Reflected  fj  G | b  waves,  ;  .  A  7"' ' f  A;(O-AM)  JT:  for  :  V  ,..  e >0 7>0c B  2  i l = -A-  3  COS  0  1  A  +^  +  Y  3ft  129  for  e  B  70c>e_jL .m  where  R  L^=  n  w-  l  ............  /Cbl  (A-3.  I f the branch p o i n t s are s m a l l e r than the saddle p o i n t s head waves do not appear.  The ray paths f o r a r r i v a l s  which t r a v e l along p a r t o f the path as head waves are shown i n Figure A - 3 . Hence, except  f o r d i f f r a c t e d waves, we can f o r m a l l y  o b t a i n a complete s y n t h e t i c seismogram i n the case o f a d i p ping order  l a y e r by a p p l y i n g  t h i s procedure to the t h i r d and higher  s e r i e s terms o f the formal  integral solution.  130  From  branch point  B  From branch point m =  i n (A-3.-1).  +  C  PUBLICATIONS  Nakamura, K. and I s h i i , H., 1965. R e f r a c t i o n of e x p l o s i v e sound waves from a l i n e source i n a i r i n t o water, S c i . Rep. Tohoku Univ., Ser. 5, Geophysics, 16, 90-107. Tohoku Univ. A f t e r s h o c k s O b s e r v a t i o n Group, 1966. Observat i o n of a f t e r s h o c k s o f an earthquake happened o f f Oga-Peninsula on 7th, May, 1964, Tohoku D i s a s t e r P r e v e n t i o n Research Group Report, 85-101. I s h i i , H. and T a k a g i , A., 1967. T h e o r e t i c a l study on the c r u s t a l movements, Part I. The i n f l u e n c e of s u r f a c e topography (Two-dimensional SH torque s o u r c e ) , S c i . Rep. Tohoku Univ., Ser. 5, Geophysics, 19, 77-94. I s h i i , H. and T a k a g i , A., 1967. T h e o r e t i c a l study on the c r u s t a l movements, Part I I . The i n f l u e n c e of h o r i z o n t a l d i s c o n t i n u i t y , S c i . Rep. Tohoku Univ., Ser. 5, Geophysics, 19, 95-106. I s h i i , H. and E l l i s , R. M., M u l t i p l e r e f l e c t i o n of plane SH waves by a d i p p i n g l a y e r , B u l l . Seism. Soc. Amer. (accepted f o r p u b l i c a t i o n ) .  

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